Mechanics of non-holonomic systems a new class of control systems 3540323511 [PDF]

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Foundations of Engineering Mechanics Series Editors: V.I. Babitsky, J. Wittenburg

Foundations of Engineering Mechanics Series Editors:

Vladimir I. Babitsky, Loughborough University, UK Jens Wittenburg, Karlsruhe University, Germany

Further volumes of this series can be found on our homepage: springer.com Soltakhanov, Sh. Kh., Yushkov, M.P., Zegzhda, S.A. Mechanics of non-holonomic systems, 2009 ISBN 978-3-540-85846-1 Skubov, D., Khodzhaev, K.S. Non-Linear Electromechanics, 2008 ISBN 978-3-540-25139-2 Slivker, V.I. Mechanics of Structural Elements, 2007 ISBN 978-3-540-44718-4 Elsoufiev, S.A. Strength Analysis in Geomechanics, 2007 ISBN 978-3-540-37052-9 Awrejcewicz, J., Krysko, V.A., Krysko, A.V. Thermo-Dynamics of Plates and Shells, 2007 ISBN 978-3-540-37261-8 Wittbrodt, E., Adamiec-Wojcik, I., Wojciech, S. Dynamics of Flexible Multibody Systems, 2006 ISBN 3-540-32351-1 Aleynikov, S.M. Spatial Contact Problems in Geotechnics, 2007 ISBN 3-540-25138-3 Skubov, D.Y., Khodzhaev, K.S. Non-Linear Electromechanics, 2007 ISBN 3-540-25139-1 Feodosiev, V.I., Advanced Stress and Stability Analysis Worked Examples, 2005 ISBN 3-540-23935-9

(Continued after index)

Lurie, A.I. Theory of Elasticity, 2005 ISBN 3-540-24556-1 Sosnovskiy, L.A., TRIBO-FATIGUE · Wear-Fatigue Damage and its Prediction, 2005 ISBN 3-540-23153-6 Andrianov, I.V., Awrejcewicz, J., Manevitch, L.I. Asymptotical Mechanics of Thin-Walled Structures, 2004 ISBN 3-540-40876-2 Ginevsky, A.S., Vlasov, Y.V., Karavosov, R.K. Acoustic Control of Turbulent Jets, 2004 ISBN 3-540-20143-2, Kolpakov, A.G. Stressed Composite Structures, 2004 ISBN 3-540-40790-1 Shorr, B.F. The Wave Finite Element Method, 2004 ISBN 3-540-41638-2 Svetlitsky, V.A. Engineering Vibration Analysis - Worked Problems 1, 2004 ISBN 3-540-20658-2 Babitsky, V.I., Shipilov, A. Resonant Robotic Systems, 2003 ISBN 3-540-00334-7 Le xuan Anh, Dynamics of Mechanical Systems with Coulomb Friction, 2003 ISBN 3-540-00654-0

Sh. Kh. Soltakhanov · M. P. Yushkov · S. A. Zegzhda

Mechanics of non-holonomic systems A New Class of control systems With 37 Figures

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Series Editors: V.I. Babitsky University Loughborough Department of Mechanical Engineering Loughborough LE11 3TU, Leicestershire United Kingdom

J. Wittenburg Universit¨at Karlsruhe Fakult¨at Maschinenbau Institut f¨ur Technische Mechanik Kaiserstrasse 12 76128 Karlsruhe Germany

Authors: Prof. Dr. Shervani Kh. Soltakhanov Academy of Sciences of the Chechen Republic Grozny Chechen Republic Russia 364906

Prof. Dr. Mikhail P. Yushkov St. Petersburg State University Dept. Mathematics & Mechanics Universitetsky Pr. 28 St. Petersburg Russia 198504

Prof. Dr. Sergei A. Zegzhda St. Petersburg State University Dept. Mathematics & Mechanics Universitetsky Pr. 28 St. Petersburg Russia 198504

ISBN: 978-3-540-85846-1

e-ISBN: 978-3-540-85847-8

Foundations of Engineering Mechanics ISSN: 1612-1384

e-ISSN: 1860-6237

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Reviewers: Dr phys.-math. sci., prof. A. V. Karapetyan (Moscow State University), Dr phys.-math. sci., prof. V. S. Novoselov (St.Petersburg State University)

Annotation

A general approach to the generation of equations of motion of holonomic and nonholonomic systems with the constraints of any order is proposed. The system of equations of motion in generalized coordinates is regarded as one vector relation, represented in a space tangential to a manifold of all possible positions of system at given instant. The tangential space is partitioned by the equations of constraints into two orthogonal subspaces. In one of them for the constraints up to the second order, the law of motion is given by the equations of constraints and in another, for ideal constraints, it is described by the vector equation not involving reactions of constraints. In the whole space the law of motion involves the Lagrange multipliers. It is shown that for the holonomic and nonholonomic constraints up to the second order, these multipliers can be found as the function of time, positions of system, and its velocities. The application of the Lagrange multipliers to holonomic systems permits us to construct two new methods for determining the normal frequencies and normal forms of oscillations of elastic systems and also to propose a special form of equations of motion for system of rigid bodies. The nonholonomic constraints, the order of which is greater than two, are regarded as program constraints such that their validity is provided by the existence of generalized control forces, which are determined as the functions of time. The closed system of differential equations is obtained which makes it possible to find both these control forces and generalized Lagrangian coordinates. The theory suggested is illustrated by examples of spacecraft motion. It is shown that instead of using the Pontryagin maximum principle it is expedient to apply the generalized Gauss principle for solving the problems of vibration suppression (damping). The book is primarily addressed to specialists in analytical mechanics.

v

Preface to the English edition The first equations of motion of nonholonomic mechanics not including the Lagrange multipliers have been reported at a scientific seminar in 1895 and published in 1897 by the world famous specialist in hydromechanics, academician of the Soviet Union Academy of Sciences Sergei Alekseevich Chaplygin (1869–1942). One of his favourite pupils, who worked under the direct supervision of S. A. Chaplygin since 1929 till 1941, was Professor Nikolai Nikolaevich Polyakhov (1906–1987). In 1952–1987 N. N. Polyakhov was the head of the mechanics department of the Faculty of Mathematics and Mechanics of Leningrad University and in charge of the chair of theoretical and applied mechanics, then since 1977 he headed the chair of hydromechanics. As well as his teacher, Nikolai Nikolaevich successfully studied not only problems of hydromechanics (he has created, in particular, the mathematical theory of a water propeller), but being at the head of the chair of theoretical and applied mechanics, he also turned to studying nonholonomic mechanics. N. N. Polyakhov published his first works in this direction in 1970–1974. Since 1975 the investigations under the supervision of N. N. Polyakhov and with his personal participation had been regularly conducted. They were summed up in Chapters "Motion with constraints"and "Variational principles in mechanics"of the treatise for universities "Theoretical Mechanics"by N. N. Polyakhov, S. A. Zegzhda, and M. P. Yushkov, which was published in 1985 by the Leningrad University Press and reprinted in 2000 by the "Vysshaya Shkola"("Higher School") Publishing House. After the decease of N. N. Polyakhov (January 27th 1987), the direction in nonholonomic mechanics, which he had established, began to be developed by his pupils: Professors of Saint Petersburg University S. A. Zegzhda and M. P. Yushkov, and Professor of Chechen State University, the head of the department of mathematics and theoretical physics of the Academy of Sciences of the Chechen Republic Sh. Kh. Soltakhanov, a graduate from Polyakhov’s chair. Their collaborative work, to which they had devoted so many years, was completed in 2002 when the monograph "Equations of motion of nonholonomic systems and variational principles in mechanics"was published at Saint Petersburg University. In 2005 the Moscow Publishing House "Nauka"("Science") published the second revised and improved edition of this book "Equations of motion of nonholonomic systems and variational principles in mechanics. A new class of control problems". This very book, extended for the English edition, is offered to readers. This book is dedicated to the 100th anniversary of the birth of our teacher Professor N. N. Polyakhov. vii

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Preface to the English edition

The prominent specialist in nonholonomic mechanics, Professor J. G. Papastavridis (Professor of Georgia Institute of Technology, USA) has undertaken the work of Editor-in-Chief while editing the book in English. It was translated by Dr in phys.-math. sciences E. A. Gurmuzova. Gratefull acknowledgement is made by the authors to all those who helped in preparation of this book — E. L. Belkind, D. N. Gavrilov, D. V. Lutsiv, A. E. Mel’nikov, A. A. Nezderov, E. M. Nosova, G. A. Sinilshchikova, K. K. Tverev, S. V. Zaykov. The authors are deeply indebted to the "Springer"Publishing House, and the publisher of the series "Foundations of Engineering Mechanics"Professor V. Babitsky and the editor Dr Ch. Baumann, for their valuable advice that contributed greatly to improving this book. It is due to their recommendations that the contents of the book has been considerably extended. The authors are also grateful to Ms C. Wolf and Ms V. Jessie who contributed much to preparing the book for the press. While writing the book, the authors gave much attention to the role of the Lagrange multipliers in analytical mechanics. Holonomic and classical nonholonomic mechanics are presented in the framework of one approach, in this case the properties of constrained motion which are typical of one particle (mass point) can be observed in any mechanical systems with the finite number of degrees of freedom. Such an approach makes it possible to construct also mechanics for the motion of systems with any-order constraints, which are considered as programming ones. Reaction forces of these constraints are interpreted as control forces that provide the motion of system under realization of the program given as an additional system of differential equations, the order of which is higher than two. Thus, a new class of control problems is introduced. The offered theory is illustrated through solving two examples of motion of real mechanical systems with three-order constraints imposed on their motion. In the monograph much attention is given to studying practical problems. Along with solving a number of classical problems (for instance, problems dealing with investigation of a car motion with possible slipping of driving wheels and sideslip), a number of new methods for solving important practical problems is proposed in the book. The reader is asked to pay special attention to the two of them (see Chapter VI). The first method makes it possible to find natural frequencies and natural modes of vibration of an elastic body system in terms of the known natural frequencies and natural modes of the system’s separate elastic bodies (its parts). Due to the dynamical consideration of the finite number of natural modes and quasistatical consideration of the higher ones of the system’s elements, the lower frequencies can be determined with a high accuracy from an algebraic equation. The second method is concerned with the problems of damping the vibration of mechanical systems. Instead of the commonly used method that is based on the minimization of the functional of control force squared, it is offered to apply the generalized Gauss principle stated in the monograph for

Preface to the English edition

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solving similar problems. As a result, the control force can be constructed as a polynomial in time. During the given time this force transfers smoothly a system from one state to another, in particular, to the state of rest. Especially important for the authours is the fact that their work is published in the English language, which plays today in scientific communication the same role as Latin in the Middle Ages. In this regard this monograph can be useful for English scientists. It will help them to get acquainted with a rather great number of works by Russian scientists. Saint Petersburg — Grozny — Berlin, 2008 Sh. Kh. Soltakhanov, M.P. Yushkov, S.A. Zegzhda

Table of Contents Chapter I Holonomic Systems 1. Equations of motion for the representation point of holonomic mechanical system . . . . . . . . . . . . . . . . . . . . . . . . 2. Lagrange’s equations of the first and second kinds . . . . . . . . 3. The D’Alembert–Lagrange principle . . . . . . . . . . . . . . . . 4. Longitudinal accelerated motion of a car as an example of motion of a holonomic system with a nonretaining constraint . . . . . Chapter II Nonholonomic Systems 1. Nonholonomic constraint reaction . . . . . . . . . . . . . . . . . 2. Equations of motion of nonholonomic systems. Maggi’s equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. The generation of the most usual forms of equations of motion of nonholonomic systems from Maggi’s equations . . . . . . . . . 4. The examples of applications of different kinds equations of nonholonomic mechanics . . . . . . . . . . . . . . . . . . . 5. The Suslov–Jourdain principle . . . . . . . . . . . . . . . . . . . 6. The definitions of virtual displacements by Chetaev . . . . . . .

1 1 4 12 15 25 25 28 38 45 66 74

Chapter III Linear Transformation of Forces 77 1. Some general remarks . . . . . . . . . . . . . . . . . . . . . . . . 77 2. Theorem on the forces providing the satisfaction of holonomic constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 3. An example of the application of theorem on the forces providing the satisfaction of holonomic constraints . . . . . . . . . . . . 88 4. Chetaev’s postulates and the theorem on the forces providing the satisfaction of nonholonomic constraints . . . . . . . . . . . . 92 5. An example of the application of theorem on forces providing the satisfaction of nonholonomic constraints . . . . . . . . . . . . 97 6. Linear transformation of forces and Gaussian principle . . . . . 100 Chapter IV Application of a Tangent Space to the Study of Constrained Motion 105 1. The partition of tangent space into two subspaces by equations of constraints. Ideality of constraints . . . . . . . . . . . . . . 105 2. The connection of differential variational principles of mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 xi

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Table of Contents 3. Geometric interpretation of linear and nonlinear nonholonomic constraints. Generalized Gaussian principle . . . . . . . . . . 113 4. The representation of equations of motion following from generalized Gaussian principle in Maggi’s form . . . . . . . . . . . 119 5. The representation of equations of motion following from generalized Gaussian principle in Appell’s form . . . . . . . . . . . 121

Chapter V The Mixed Problem of Dynamics. New Class of Control Problems 1. The generalized problem of P. L. Chebyshev. A new class of control problems . . . . . . . . . . . . . . . . . . . . . . . . 2. A generation of a closed system of differential equations in generalized coordinates and the generalized control forces . . . . 3. The mixed problem of dynamics and Gaussian principle . . . . . 4. The motion of spacecraft with modulo constant acceleration in Earth’s gravitational field . . . . . . . . . . . . . . . . . . . 5. The satellite maneuver alternative to the Homann elliptic motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter VI Application of the Lagrange Multipliers to the Construction of Three New Methods for the Study of Mechanical Systems 1. Some remarks on the Lagrange multipliers . . . . . . . . . . . . 2. Generalized Lagrangian coordinates of elastic body . . . . . . . 3. The application of Lagrange’s equations of the first kind to the study of normal oscillations of mechanical systems with distributed parameters . . . . . . . . . . . . . . . . . . . . . . . 4. Lateral vibration of a beam with immovable supports . . . . . . 5. The application of Lagrange’s equations of the first kind to the determination of normal frequencies and oscillation modes of system of bars . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Transformation of the frequency equation to a dimensionless form and determination of minimal number of parameters governing a natural frequency spectrum of the system . . . . . . . . . . 7. A special form of equations of the dynamics of system of rigid bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. The application of special form of equations of dynamics to the study of certain problems of robotics . . . . . . . . . . . . . . 9. Application of generalized Gaussian principle to the problem of suppression of mechanical systems oscillations . . . . . . . . .

125 125 128 131 137 144

149 150 152

154 160

165

173 178 181 183

Chapter VII Equations of Motion in Quasicoordinates 193 1. The equivalence of different forms of equations of motion of nonholonomic systems . . . . . . . . . . . . . . . . . . . . . 193

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2. The Poincar´e–Chetaev–Rumyantsev approach to the generation of equations of motion of nonholonomic systems . . . . . . . . 201 3. The approach of J. Papastavridis to the generation of equations of motion of nonholonomic systems . . . . . . . . . . . . . . . 207 Appendix A The Method of Curvilinear Coordinates 1. The curvilinear coordinates of point. Reciprocal bases . . . . . . 2. The relation between a reciprocal basis and gradients of scalar functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Covariant and contravariant components of vector . . . . . . . . 4. Covariant and contravariant components of velocity vector . . . 5. Christoffel symbols . . . . . . . . . . . . . . . . . . . . . . . . . 6. Covariant and contravariant components of acceleration vector. The Lagrange operator . . . . . . . . . . . . . . . . . . . . . . 7. The case of cylindrical system of coordinates . . . . . . . . . . . 8. Covariant components of acceleration vector for nonstationary basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9. Covariant components of a derivative of vector . . . . . . . . . .

213 213 215 216 217 218 220 222 225 227

Appendix B Stability and Bifurcation of Steady Motions of Nonholonomic Systems

229

Appendix C The Construction of Approximate Solutions for Equations of Nonlinear Oscillations with the Usage of the Gauss Principle

235

Appendix D The Motion of Nonholonomic System without Reactions of Nonholonomic Constraints 1. Existence conditions for “free (unconstrained) motion” nonholonomic system . . . . . . . . . . . . . . . . . 2. Free motion of the Chaplygin sledge . . . . . . . . . . 3. The possibility of free motion of nonholonomic system active forces . . . . . . . . . . . . . . . . . . . . . .

239 of . . . . . . 239 . . . . . . 240 under . . . . . . 243

Appendix E The Turning Movement of a Car as a Nonholonomic Problem with Nonretaining Constraints 245 1. General remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 2. The turning movement of a car with retaining (bilateral) constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246 3. The turning movement of a rear-drive car with nonretaining constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 4. Equations of motion of a turning front-drive car with non-retaining constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 5. Calculation of motion of a certain car . . . . . . . . . . . . . . . 258 6. Reasonable choice of quasivelocities . . . . . . . . . . . . . . . . 260

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Appendix F Consideration of Reaction Forces of Holonomic Constraints as Generalized Coordinates in Approximate Determination of Lower Frequencies of Elastic Systems 263 Appendix G The Duffing Equation and Strange Attractor

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References

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Index

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Dedicated to the 100th birthday of our teacher Prof. Nikolai Nikolaevich Polyakhov

Introduction This treatise is the English translation of the second edition of the book "Equations of motion of nonholonomic systems and the variational principles of mechanics" , published by St. Petersburg University Press in 2002. The second edition is considerably improved. First of all, particular attention has been given to the mixed problem of dynamics (for detail, see below), which formulates a new class of control problems in the case when the program of motion is given by an additional system of high-order differential equations. The theory, suggested for the solutions of such problems, is illustrated by a number of new examples. Both editions are devoted to the development of ideas of nonholonomic mechanics, considered before in the treatise for universities: "Theoretical mechanics" (Chapters "Motion with constraints" and "Variational principles of mechanics") of N. N. Polyakhov, S. A. Zegzhda, and M. P. Yushkov, which has been published by Leningrad University Press in 1985 and reprinted by Publishing house "Higher school" in 2000 [189]. The first works [185] in this direction are published by prof. N. N. Polyakhov in 1970–1974. Beginning from 1975 at the chair of theoretical and applied mechanics of mathematical and mechanical faculty of Leningrad State University these problems have been studied under the conduction and direct participation of N. N. Polyakhov down to the decease of Nikolai Nikolaevich in year 1987. In the present treatise a new definition of ideal constraint, extended to a high-order nonholonomic constraint, is given. Finally, the theory of generation of equations of motion for a certain new class of problems is constructed. Following academician S. S. Grigoryan we shall call such problems the mixed problems of dynamics since they have the features of both the direct and inverse problems of dynamics. Really, on the one hand the motion of mechanical system, described by the generalized coordinates q = (q 1 , . . . , q s ), is determined by the given generalized active forces Q = (Q1 , . . . , Qs ) on the other hand it is necessary that these generalized coordinates are also the solution of the following additional system of differential equations (n)

fnκ (t, q, q, ˙ ..., q ) = 0,

κ = 1, k ,

k  s,

(1)

where n is any integer number. The characteristics, of motion of mechanical system, for the validity of which it is necessary to find the additional forces Λ = (Λ1 , . . . , Λk ), are given by equations (1). xv

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For completing the work the substantial role played the discussion in Moscow State University in the science seminar conducted by academician V. V. Rumyantsev and prof. A. V. Karapetyan. If conditions (1) are satisfied, then, in fact, a certain control problem is solved in which the program of motion is given by system of differential equations (1). Following the terminology, accepted in the nonholonomic mechanics, these equations can be called n-order nonholonomic constraints. However, in essence, as is pointed out above, they are a program of motion and therefore it is more rational to call them program constraints. The possibility of application of the generalized Gauss principle to investigation of problems on vibration suppression is shown, this principle is stated in the monograph and is valid for non-holonomic high-order constraints. It turned out that solving similar problems with the help of the Pontryagin maximum principle, which is commonly used in such cases, can be interpreted as solving a mixed problem of dynamics. The monograph gives a comparision of these two principles through solving the problem of vibration suppression of mathematical pendulums that are attached to a moving trolley. The discussion of the suggested technique for solving problems of vibration suppression at the Institute of problems of mechanics of RAS (Russian Academy of Scienes), supervised by academician F. L. Chernous’ko, has played a decisive role in the final version of this technique. The book consists of the introduction, the survey of main stages of development of nonholonomic mechanics, seven chapters, seven appendices, and references. The survey of main stages of development of nonholonomic mechanics gives a short description of main directions of study in nonholonomic mechanics. In the first chapter a notion of the point that represents a motion of a holonomic mechanical system is introduced. To generate Lagrange’s equations of the first and second kinds, the approach demonstrating their unity and generality is applied. This approach permits us to write Lagrange’s equations in the form, which can be used in the case of both one material point and arbitrary mechanical system with finite or infinite numbers of degrees of freedom. The notion of ideal holonomic constraints is considered from the different points of view. The connection of the obtained equations of motion with the D’Alembert–Lagrange principle is analyzed. The longitudinal motion of a car with acceleration and possible sideslip of driving wheels is considered as an example of motion of a holonomic system with a nonretaining constraint. In the second chapter from the analog of Newton’s law the Maggi’s equations, which are the most convenient equations of the nonholonomic mechanics, are deduced. From Maggi’s equations the most useful forms of equations of motion of nonholonomic systems are obtained. The connection of Maggi’s equations with the Suslov–Jourdain principle is considered. The notion of ideal nonholonomic constraints is discussed. In studying nonholonomic systems the approach, applied in Chapter I to analysis of motion of holonomic

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systems, is employed. The role of Chetaev’s type constraints for the development of nonholonomic mechanics is considered. For the solution of a number of nonholonomic problems, the different methods are applied. In the third chapter the linear transformation of forces is introduced. In this case for holonomic systems the notion of ideal constraints and the relation for virtual elementary work are used. By the transformation of forces, Lagrange’s equations of the first and second kinds are obtained. The theorem of holonomic mechanics is formulated according to which the given motion over the given curvilinear coordinate can be provided by means of an additional generalized force corresponding to this coordinate. For nonholonomic systems the linear transformation of forces is introduced by using Chetaev’s postulates. In this case with the help of generalized forces, corresponding to the equations of constraints, the family of basic equations of nonholonomic mechanics is obtained in compact form. The theorem of holonomic mechanics is formulated according to which the given change of quasivelocity can be provided by one additional force corresponding to this quasivelocity. The application of the formulated theorems of the holonomic and nonholonomic mechanics is demonstrated on the example of the solution of two problems on the controllable motion connected with the flight dynamics. At the end of this chapter the linear transformation of forces is used to obtain Gauss’ principle. In the fourth chapter by means of the introduction of a tangent space, a system of Lagrange’s equations of the second kind is represented in the vector form. It is shown that the tangential space is partitioned by the equations of constraint into the direct sum of two subspaces. In one of them the component of a vector of acceleration of system is uniquely determined by equations of constraints. The notion of ideality of holonomic constraints and nonholonomic constraints of the first and second orders is analyzed. This notion is extended to high-order constraints. The relationship and equivalence of the differential variational principles of mechanics are considered. It is given a geometric interpretation of the ideality of constraints. The generalized Gauss principle is formulated. By means of this principle, for nonholonomic systems with third-order constraints equations in Maggi’s and Appell’s forms are obtained. In the fifth chapter the law of motion of mechanical system, represented in the vector form, is applied to the solution of the mixed problem of dynamics. The essence of the problem is to find additional generalized forces such that the program constraints, given in the form of a system of differential equations of order n  3, are satisfied. The notion of a generalized control force is introduced. It is proved that if the number of program constraints is equal to the number of generalized control forces, then the latter can be found as the time functions from the system of differential equations with respect to generalized coordinates and these forces. The conditions, under which this system of equations has a unique solution, are determined. The conditions are also obtained under which for the constraints of any order the motion control is realized according to Gauss’ principle. Thus, the theory is created with the

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help of which a new class of control problems can be solved. This theory is used to consider two problems, connected with the dynamics of spacecraft motion. In the first problem a radial control force, providing the motion of spacecraft with a modulo constant acceleration, is determined as the time function. In the second problem we seek the law, of varying in time the radial and tangential control forces, by which a smooth passage of spacecraft from one circular orbit to another occurs. In the sixth chapter the Lagrange multipliers are used to construct three new methods for the study of mechanical systems. The first of them corresponds to the problem of determining the normal frequencies and normal forms of oscillations of elastic system, consisting of the elements with known normal frequencies and normal forms. In this method the conditions of connection of elastic bodies to one another are regarded as holonomic constraints. Their reactions equal to the Lagrange multipliers are the forces of interaction between the bodies of system. Using the equations of constraints, the system of linear uniform equations with respect to the amplitudes of the Lagrange multipliers for normal oscillations is obtained. By the solution of this system the normal frequencies and normal forms of complete system are expressed in terms of the normal frequencies and normal forms of its elements. An approximate algorithm for determining the normal frequencies and normal forms, based on a quasistatic account of higher forms of its elements, is proposed. A development of this method is given in Appendix F. The second method suggested is connected with the study of the dynamics of system of rigid bodies. In this case the Lagrange multipliers are introduced for the abstract constraints taking into account that the number of introduced coordinates, by which the kinetic energy of rigid body has a simple form, is excessive. In this case the elimination of the Lagrange multipliers leads to a new special form of equations of motion of rigid body. This form is utilized to describe a motion of a dynamic stand, which lets us to imitate the state of pilot in cabin in extremal situations. The third method is applied to a problem of the vibration suppression (damping) of mechanical systems. It is based on reducing the problem given to a mixed problem of dynamics, which is solved with the help of the generalized Gauss principle. In the seventh chapter it is shown that all known types of equations of motion of nonholonomic systems are equivalent since they can be obtained from the invariant vector form of the law of motion of mechanical system with ideal constraints. The nonholonomicity of constraints, which does not allow the equations of motion to be represented in the form of Lagrange’s equations of the second kind, appears most clearly if the equations of motion of nonholonomic system are written in independent quasicoordinates. In the case of linear constraints these equations are generated here by three different methods. This allows us to consider the problem of nonholonomicity from three different points of view. In this chapter the vector form of equations of dynamics, found in Chapter IV, is used to generate the equations

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in quasicoordinates and the equations of Poincar´e–Chetaev–Rumyantsev. A geometric interpretation of the equations of Poincar´e–Chetaev–Rumyantsev is given. The approach of Poincar´e–Chetaev–Rumyantsev to the generation of equations of motion of nonholonomic systems is compared with other approaches. In Appendix A the kinematics of point in curvilinear coordinates is considered. The formulas obtained are extended to the motion of any mechanical system. The main material of treatise is considered in the frame of the theory developed in the Appendix. Therefore, generally speaking, it should be recommended to read this book beginning from this Appendix. Appendix B contains a short review of the works, devoted to the questions of the existence, stability, and branching of steady motion of conservative nonholonomic systems. This appendix is the plenary report of A. V. Karapetyan with the same title, which was read at the International science conference on mechanics "The third Polyakhov readings" (St.Petersburg, February 4-6, 2003). In Appendix C Gauss’ principle in integral form is applied to the construction of approximate solutions of equations of nonlinear oscillations, in particular, of the solutions, which are obtained by the Bubnov–Galerkin method. In Appendix D the motion of nonholonomic systems in the case when the reactions of constraints are absent is investigated. By the Mei Fengxiang terminology such a motion is called a free (unconstrained) motion of nonholonomic system. The free motion of the Chaplygin sledge is considered. The realizing of free motion of nonholonomic systems under external forces is discussed. In Appendix E the possibillity of sideslip of both front and rear wheels of a car while turning is considered. Solving this problem with nonretaining constraints leads to necessary consideration of four possible types of the car motion. The computational results of motion of a certain car are presented. In Appendix F the problem of determining the lower frequencies of a mechanical system that consists of elastic bodies connected to each other by holonomic constraints is considered. It is shown that reaction forces of these constraints can be regarded as the generalized Lagrange coordinates. In the method suggested the equation for determining the lower frequencies is not transcendental but an algebraic one. A number of examples illustrate that this approximate method makes it possible to define the first natural frequency to a high accuracy. Appendix G is devoted to studying the possible arising of strange attractors for lateral vibration of a beam with fixed supports. The list of names in the text is the same as in the treatise published in the "Nauka" Publishing House. The order of references is also the same. The references are complemented at the end with some new items, a part of new works is added to old items. The theory considered is illustrated by many examples. The computations, connected with the solution of problems, are due to S. V. Almazova,

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O. V. Almazov, Yu. A. Belousov, E. S. Bolgar, A. B. Byachkov, D. N. Gavrilov, I. N. Drozd, E. S. Drozd, T. N. Dudareva, E. Yu. Leont’eva, Lee Yang, A. E. Mel’nikov, A. A. Nezderov, Yu. L. Nikiforova, E. M. Nosova, T. N. Pogrebskaya, G. A. Sinilshchikova, A. V. Smal, N. S. Smirnova, V. P. Sysik, K. K. Tverev, A. A. Fedorov, L. G. Fedorchenko, N. G. Filippov, N. A. Khor’kova, Chen Yu, Yu. S. Sheverdin, A. E. Shevtsov, A. V. Shkondin, M. A. Yushkevich. The authors are grateful to all of them for their help. As it was shown in the preface to this edition, following the advice of the publisher of the series "Foundations of Engineering Mehanics" of the "Springer" Publishing House Professor V. Babitsky and the editor of the Publishing House Dr Ch. Baumann the volume of the treatise has been increased. The work at the new text was carried out with the assistance of S. V. Almazova (§§ 2, 6 of Chapter VI and Appendix F), A. B. Byachkov (§ 4 of Chapter I and Appendix E), D. N. Gavrilov (Appendix F), Lee Yang (Appendix D), A. E. Mel’nikov (§ 4 of Chapter VI and Appendix G), A. A. Nezderov (§ 4 of Chapter I and Appendix E), E. M. Nosova (§ 4 of Chapter I and Appendix E), G. A. Sinilshchikova (Appendix F), Yu. A. Belousov and Chen Yu (§ 2 of Chapter II), M. A. Yushkevich (§ 4 of Chapter VI). This new text (excluding §§ 2, 4 of Chapter VI and Appendices D and G) was translated by T. V. Zhil’tsova. Besides, the Preface to the English edition, § 4 of Chapter I, § 9 of Chapter VI, Appendices E, F, and G were translated by Dr G. A. Sinilshchikova. All new ideas of the book have undergone the careful check and kindly critique of the scientific editor of the book, head of Chair of theoretical and applied mechanics of the Mathematics and Mechanics Faculty of St.Petersburg State University, laureate of State Prize RF, honored worker of science RF, Prof. P. E. Tovstik. The authors wish to acknowledge their great indebtedness to Petr Evgen’evich Tovstik for his able assistance during the whole period of constructing the suggested theory and writing this book. The authors will be very grateful to readers for sending their comments on this book. E-mail: [email protected]; [email protected]

Survey of the main stages of development of nonholonomic mechanics The theory of motion of nonholonomic systems is of obvious scientific interest. Already in the investigations of I. Newton, L. Euler, I. Bernoulli, J. Bernoulli, J. D’Alembert, and J. Lagrange in studying the problems on the rolling of rigid bodies without slide we find elements with distinctive features of motion of systems with nonholonomic constraints. For the solution of similar problems, S. Poisson [374. 1833] makes use of the general theorems of dynamics. In the book [379. 1884] E. Routh considers the problem on the rolling of rigid body without sliding on a fixed surface and reduces it to quadratures in many sophisticated cases, for example, in the case of the rolling of rigid uniform ball on a cylindrical surface with cycloidal section. The motion of rolling bodies is discussed by P. Appell [266. 1899]. The interesting problem on the rolling without sliding of a ball with a gyroscope inside is considered by D. K. Bobylev [16. 1892]. In the case when the center of mass of complete system is at the center of ball he solves this problem, having expressed all sought unknowns via elliptical functions. N. E. Zhukovsky [67. 1893] shows that if in a spherical shell there is an additional ring and the moments of inertia are chosen in a suitable way, then the study of this problem is simplified. He also gives a geometrically descriptive consideration. All these problems were solved correctly by different methods and many authors. However at the end of the 19th century and early in the 20th the attempts to solve the typical nonholonomic problems, applying the methods of holonomic mechanics, lead to many well-known errors, which played a substantial role in the making of nonholonomic mechanics. So, in 1885 and 1886 to generate equations of motion of heavy body rolling without sliding on a fixed plane, C. Neumann [366] applies usual Lagrange’s equations of the second kind. However he understands presently that for the solution of similar problems it is necessary to use more complicated Lagrange’s equations with multipliers [366. 1887–1888]. In 1899 C. Neumann has solved totally this problem [366]. A more specific problem is solved by E. Lindel¨of [352. 1895]. In this problem he considered a body that is bounded by a surface of revolution and has a center of inertia on the axis of revolution being a dynamic axis of symmetry of body. The forces are assumed to be conservative, in which case the force function depends on the coordinates of a contact point of body only. Following the treatise of S. Poisson [374], E. Lindel¨of suggests to consider in place of the general theorems of dynamics the Hamilton principle or Lagrange’s equations of the second kind, obtained from this principle. Having written two equations of nonholonomic constraints, he applies them to the construction of kinetic energy and assumes erroneously that the nonholonomicity of this problem is completely accounted and therefore Lagrange’s equations of the second kind can be generated. Naturally, the system of differential

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equations obtained turns out a more simple than the truth-value system and was solved therefore in quadratures. The works of E. Crescini [296. 1889] and G. Schouten [382. 1899] contain also similar errors. For a nonholonomic system E. Crescini uses erroneously the Hamilton–Jacobi equations and G. Schouten the Lagrange’s equations of the second kind. P. Molenbrock [364. 1890] and a number of other scholars also ignore a differential nature of nonholonomic constraints. Even one of the future creator of nonholonomic mechanics, L. Boltzmann, admits a similar error [276. 1885]. He applies Lagrange’s equations to the study of a revolution of frictional and toothed gear wheels with a nonholonomic constraint, which gives a proportional dependence of angular rates of wheels. This error was corrected by L. Boltzmann in 1902 [277] only. P. Appell is interested in the elegant outwardly but untrue in essence solution of E. Lindel¨of to an extent that he uses it as an example of applying Lagrange’s equations of the second kind in § 452 of the first edition of his treatise on the theoretical mechanics [265. 1896]. But in the second edition in 1898, based on the investigations of J. Hadamard [311. 1894] and A. Vierkandt [398. 1892], he writes: ". . . the results of E. Lindel¨of are inaccurate. I have pointed out this error to E. Lindel¨of in 1898 and have made a correction in the following edition of my "Trait´e" ". The first to remark the substantial error, committed by E. Lindel¨ of, and to inform the author was S. A. Chaplygin. October 25, 1895 he reported the results on this topic on a meeting of the physics division of the Association of lovers of natural sciences, anthropology, and ethnography. S. A. Chaplygin notes that ". . . on the first pages of his work . . . E. Lindel¨of committed an important error, by which the equations obtained turned out more simple than the intrinsic ones and provided a seeming achievement of author". In this report S. A. Chaplygin presented first his equations of motion for nonholonomic systems. Two years hence he found the correct solution of the E. Lindel¨of problem and published the results in the paper [239]. It is of interest to note that, apparently, for the visualization of solution, S. A. Chaplygin generates the equations of motion for the E. Lindel¨of problem, no using his own equations but applying the principles of the motion of the center of mass and of conservation of angular momentum of system, in which case he preliminarily introduces a friction force and eliminates it then from the equations found. For the sake of generality, S. A. Chaplygin considers a body connected with a gyroscope and reduces the solution to quadratures, which are simplified in the case considered earlier by D. K. Bobylev [16]. After S. A. Chaplygin, the E. Lindel¨of problem was solved by D. Korteweg [336. 1899], using Routh’s equations with the multipliers of constraints, and by P. Appell [268. 1900] and P. V. Voronets [41. 1903] in the framework of the equations they suggest, and by many other scholars. Thus, the work of E. Lindel¨of [352] gave great impetus to the making and development of nonholonomic mechanics. We remark that in these first problems it was rather difficult to take in to account correctly the fact that the constraints considered

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are differential. For example, in the above-mentioned work [336], D. Korteweg describes in detail the errors of G. Schouten, E. Lindel¨of, P. Molenbrock, and P. Appell but at the same time he himself commits a similar error when he attempts to create the theory of small oscillations with nonholonomic constraints. As an autonomous division of Newtonian mechanics the nonholonomic mechanics was formulated in the work of H. Hertz "Die Prinzipien der Mechanik in neuem Zusammenhange dargestellt" [317. 1894]. The terms the holonomic and nonholonomic systems were introduced by Hertz. N. Ferrers [306. 1872], and E. Routh [379. 1877] were among the first to suggest the correct equations of motion with nonholonomic constraints. N. Ferrers introduced the expressions for Cartesian velocities in terms of generalized velocities, and E. Routh wrote the equations with the Lagrange multipliers. In 1877 in the third edition of his "Dynamics of a system of rigid bodies" for linear non-holonomic constraints E. Routh introduces such form, which is usually called now in literature the Lagrange equations of the second kind with multipliers [59]. S. A. Chaplygin [239. 1895, 1897] was the first to suggest the equations of motion without Lagrange multipliers. He introduces the certain conditions, which are to be satisfied by the linear equations of constraints, the forces, and the relation of kinetic energy (such systems were called then the Chaplygin systems), and transforms the form of kinetic energy, using the equations of constraints. As a result, he discriminates in the left-hand side of equations of motion the group of addends of the type of the Lagrange operator. The rest addends characterize the nonholonomicity of system and go to zeros in the case of integrability of differential equations of constraints. It is to be noted that, practically, all the considered problems of nonholonomic mechanics were of the type of the Chaplygin systems and therefore these equations had a widespread application. In 1901 P. V. Voronets [41] generalizes Chaplygin’s equations to the cases of noncyclic holonomic coordinates and of nonstationary constraints. The work of S. A. Chaplygin acted into notice many outstanding scholars of the day. The different forms of the equations of motion for nonholonomic systems without Lagrange multipliers were proposed. These are the equations of V. Volterra [399. 1898], L. Boltzmann [277. 1902], G. Hamel [313. 1904] and others. The obtained different types of the equations of motion for nonholonomic systems are generated in the quasicoordinates and have a usual structure of Lagrange’s equations of the second kind with corrective additive terms of nonholonomicity. We remark that in parallel with extending Chaplygin’s equations P. V. Voronets [41] generated also the equations of motion in quasicoordinates. These investigations are represented completely in his Master dissertation [41. 1903]. The equations, generated by P. V. Voronets, L. Boltzmann, and G. Hamel, are highly similar in the form to each other and are obtained almost at the same time. This explains why in the modern scientific literature different authors give them different names.

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For the study of the dynamics of nonholonomic systems the other forms of equations, which did not also involve the Lagrange multipliers, were suggested. First of all, it is Appell’s equations, which are represented shortly in the works [267. 1899] and completely in 1900 [269]. In these equations a notion of the energy of accelerations (the term is suggested by A. SaintGermain [381. 1900]) is used. It is of interest for us that in the work [268. 1900] Appell applies this method to the solution of the E. Lindel¨of problem. In 1924 J. Tz´enoff [393] generates the equations of the mixed type, which involve both the energy of accelerations, and the kinetic energy. J. Schouten [383. 1928] finds later the equations, possessing a contravariant structure. Just as in the case of the equations of Boltzmann–Voronets–Hamel, the equations of the Appell type were also obtained by certain other scientists. For example, J. W. Gibbs [309] presented similar equations already in 1879, the analogous equations were also obtained apparently independently of Appell by P. Jourdain [325. 1904], the same ideas were presented in the works by H. Hertz [ 317, p. 224, 371]. We remark that the equations of G. Maggi [355. 1896] were obtained, in fact, at the same time as by S. A. Chaplygin but the contemporaries did almost not notice them. These equations, not involving the Lagrange multipliers, are represented in quasicoordinates and are the linear combinations of Lagrange’s equations of the second kind. Maggi’s equations are highly convenient to solve the problems of nonlinear nonholonomic mechanics [189, 286, 327] and to generate the equations of motion for systems of rigid bodies [221]. However, apparently, at present they are insufficiently known. For example, utilizing the original definition of ideal constraints, V. N. Suchkov [222. 1999] generates generalized Lagrange’s equations, which are coincident with Maggi’s equations with an accuracy to multipliers. In 1901 G. Maggi himself published the note [356], in which he has shown that Volterra’s equations just as Appell’s equations can be obtained from the equations, suggested by him already in 1896 in his book on mechanics [355]. In the treatise for universities [189] from Maggi’s equations the main forms of Lagrange’s equations of motion of nonholonomic systems are obtained. Maggi’s equations and the relations for reactions of nonholonomic constraints are discussed in the work of J. Papastavridis [370]. A new direction in obtaining equations of motion is due to H. Poincar´e [373. 1901]. V. V. Rumyantsev [203. 1994, p. 3] writes that "the remarkable idea of Poincar´e [373] that the equations of motion for holonomic mechanical systems can be generated by using a certain Lie transitive group of infinitely small transformations was extended then by Chetaev [247, 248, 292] to the case of nonstationary constraints and dependent variables when the transformation group is intransitive. Chetaev transforms Poincar´e’s equations to the form of canonical equations and develops the theory of integration of these equations". The Poincar´e–Chetaev theory is extended to the nonholonomic linear systems in the works of L. M. Markhashov, V. V. Rumyantsev, and Fam Guen [149, 203, 229]. In 1998 V. V. Rumyantsev [203] extends the

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Poincar´e–Chetaev equations to nonlinear nonholonomic constraints. Therefore these equations might be called the equations of Poincar´e–Chetaev– Rumyantsev. As V. V. Rumyantsev [203] remarks, these equations are general equations of nonholonomic mechanics and the rest of types of equations of motion can be deduced from them. In the work [81. 2001] a geometric interpretation of the Poincar´e–Chetaev–Rumyantsev equations is given. In parallel with the obtaining of different forms of equations of motion the variational principles, used in the nonholonomic mechanics, were developed (a detail survey, with a comprehensive bibliography, of variational principles of mechanics can be found in the works of V. N. Shchelkachev and J. Papastavridis [254, 370]). In 1894 H. Hertz in his outstanding "Die Prinzipien der Mechanik in neuem Zusammenhange dargestellt" [317] showed that the Hamilton principle in its classical formulation cannot be applied to nonholonomic systems. In the introduction to the mentioned book he explains this on the example of a ball, rolling mechanically without sliding. The elegant proof of the same fact was also given by H. Poincar´e [372. 1897]. The Hamilton–Ostrogradsky principle is extended first to the stationary nonholonomic systems by O. H¨older [318. 1896]. A. Voss [400. 1900] extends this result, using curvilinear coordinates, to the case of nonstationary constraints. Almost at the same time as by Voss the similar investigations were also made by P. V. Voronets [41. 1901] and G. K. Suslov [219. 1901] and, what is of interest, all these works are published in the same number of journal. We remark that the Hamilton–Ostrogradsky principle was also extended to the case of nonholonomic system with two free parameters by S. A. Chaplygin. The possibility of applying the integral variational principles of mechanics to the study of motion of nonholonomic systems is discussed in the works of V. S. Novoselov, V. V. Rumyantsev, A. S. Sumbatov and others [171, 200, 217]. When obtaining the equations of motion for nonholonomic systems, many scholars made use of the D’Alembert–Lagrange principle but in this case it was necessary to complete a definition of the notion of virtual displacements with nonholonomic constraints. For this purpose P. Appell [265] and J. W. Gibbs [309] introduce the virtual displacements by the rules, according to which the virtual displacements coincide really with virtual velocities, what is quite natural. However P. Jourdain [326. 1908–1909] connects the corresponding principle of nonholonomic mechanics exactly with the notion of virtual velocities. Note that G. K. Suslov [218. 1900] has formulated practically the same principle but with somewhat modified terminology. In this connection the variational differential principle for nonholonomic systems might be called the Suslov–Jourdain principle [187]. E. Delassus [298. 1913] suggests to call it the analytic form of the generalized D’Alembert principle. The investigations, devoted to applying the Suslov–Jourdain principle, are still in progress (see, for example, the work [285. 1993]). In the case of applying in nonholonomic mechanics the principles of D’Alembert–Lagrange, Jourdain, and Gauss together we need to investigate

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the connection between the differential variational principles of mechanics. Already early in the 20th century this question was considered (for example, in the work of R. Leitinger [343. 1913]) but a widespread study of this problem was begun in the work of N. G. Chetaev [245. 1932–1933] and was completed by the research of V. V. Rumyantsev [199. 1975–1976]. Much attention is being given to this direction up to day [124. 2004, 288. 1989, 387. 1995]. N. G. Chetaev [245] introduces the important notion of nonholonomic mechanics, namely the virtual displacements of system with the nonlinear nonholonomic constraints (the constraints of Chetaev’s type). A. Przeborski [375. 1931–1932] introduces a similar axiom of ideality of nonholonomic constraints while extending Maggi’s equations to the case of nonlinear nonholonomic constraints. Much attention to the discussion of these conditions is also given by L. Johnsen [324]. In justice to P. Appell, V. S. Novoselov calls such conditions the Appell–Chetaev conditions and introduces "A-displacements" term [173] for the corresponding virtual displacements. J. Papastavridis [370. 1998, 1999, 2002] calls these conditions the definition of Maurer–Appell–Chetaev–Hamel for virtual displacements with nonlinear nonholonomic constraints. These conditions are the main apparatus of investigation in the nonholonomic mechanics (see, for example, the works of V. S. Novoselov [169, 170] and the recent works [348, 365]; in the works [349. 1994] the connection of the model of Chetaev with the model of Vacco is established). Particular attention of scholars has been given to the problems, posed by the classics of nonholonomic mechanics. For example, the motion of heavy body of revolution in the statement of S. A. Chaplygin [239] was studied by A. S. Sumbatov [217] and A. P. Kharlamov [235] (for other similar investigations, see below), the development of the theorem on reducing multiplier [242] can be found in many papers and in the treatise of Yu. I. Neimark and N. A. Fufaev [166], the modification of Gauss’ principle, suggested by N. G. Chetaev [246] (Chetaev’s principle), was extended by V. V. Rumyantsev [198, 199], and so on. Much attention was also given to the construction of the new forms of equations of motion for nonholonomic systems and to the extension of the known types of equations to a more wide class of constraints: A. Przeborski [375] extended Maggi’s equations to the case of nonlinear nonholonomic constraints, V. S. Novoselov [169] suggested the equations of Chaplygin’s type and the equations of the type of Voronets–Hamel, which allow the Hamel equations with nonlinear constraints [313, 314] to be applied to nonstationary and nonconservative systems, J. Papastavridis [370. 1995] extended the domain of applying the Boltzmann–Hamel equations, the different forms of equations were suggested by J. Nielsen [367], D. Mangeron and S. Deleanu [360], Bl. Dolaptschiew [301, 302], G. S. Pogosov [182], N. N. Polyakhov [185], M. F. Shul’gin [255.1950], I. M. Shul’gina [256], and others. In many works there are discussed the different forms of equations of motion (for example, [301, 302, 310, 341]). The convenient matrix form of equations of nonholonomic mechanics is given by Yu. G. Martynenko [146. 2000]. Ya. V. Tatarinov

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suggested the new form of equations of nonholonomic (and holonomic) mechanics [225]. This form incorporates the known representations of equations of motion, in which case many of addends is obtained by means of the formal Poisson bracket. F. Udwadia and R. Kalaba [394. 1992] obtained the equations taking into account the presence of constraints that are linear with respect to generalized accelrations. The Kane equations [330] made themselves conspicuous especially in the western literature. Their geometric interpretation can be found in the work by M. Lesser [345]. By means of these equations a number of problems of nonholonomic mechanics was solved. Many studies [280, 300, 345, 363, 384, 408] showed a direct relationship between Kane’s equations, Maggi’s equations, and the Gibbs–Appell ones. In 1906 J. Quanjel [377] obtained a canonical form of equations of motion for nonholonomic systems, using Lagrange’s equations with multipliers. These results were developed by S. Dautheville, L. Johnsen and T. P¨oschl [324, 377]. A. J. Van der Schaft and B. M. Maschke [396] obtained the equations of motion for nonholonomic systems close in form to Hamilton’s equations. The Jacobi method for two curvilinear coordinates was generalized to nonholonomic systems by S. A. Chaplygin [242. 1911]. A canonical form of the equations of nonholonomic systems was also obtained by N. N. Polyakhov [185]. The mathematical questions, related to the above problem, are discussed in the foundational work of V. V. Kozlov [112]. One of theories of the integration of differential equations of nonholonomic mechanics was suggested by I. S. Arzhanykh [5]. In 1939 V. V. Dobronravov [58] extended the Hamilton–Jacobi theorem to the case of canonical system of nonholonomic equations. However in the work [162. 1953] Yu. I. Neimark and N. A. Fufaev, having a bearing on their own investigations [161–163], challenge the work of V. V. Dobronravov [58], regarding that the results obtained are valid for holonomic systems only. At the same time they also challenge the accuracy of generation of the V. Volterra equations [399]. We remark that V. V. Dobronravov did not agree with the said critique [58. 1952]. This discussion emphasizes the fact that the theory of motion of nonholonomic systems is rather complicated. The works of E. A. Bolotov [18. 1904], G. K. Pozharitskii [183. 1961], V. V. Rumyantsev [196. 1961-2006], V. A. Samsonov [205. 1981-2005], A. S. Sumbatov [217. 1982], and others are devoted to the motion with nonideal constraints. Special problems of analytic mechanics are the problems with nonretaining constraints. The notion of such holonomic constraints was introduced first by M. V. Ostrogradskii in generalizing the principle of virtual displacements and the D’Alembert principle to the similar systems [176]. The motion with releasing constraints was also considered by G. K. Suslov [220]. The general case of the motion with nonretaining constraints is studied especially completely by means of the modern mathematical tools in the works by A. P. Ivanov and A. P. Markeev [86, 87], and is stated in the treatise of V. F. Zhuravlev and N. A. Fufaev [72]. This book summarizes the numerous results of different works devoted to this topic (see the bibliography at the

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end of the book) and permits us to make use of the powerful theory of analytic mechanics for the study of a wide class of different substantial in practice problems: the motion of vibroimpulsive and vibratory displaced systems, the rolling of different systems with possible slide and so on. Among similar problems the questions of repeated collisions are completely considered, for example, in the treatise of R. F. Nagaev [160] and the possibility of lateral sliding of car in the book of M. A. Levin and N. A. Fufaev [130], releasing from nonholonomic constraints during tne turning movement of wheeled robot vehicles and cars in works [253]. A successful simulation of the motion of nonholonomic systems with one-sided constraints is given in the work of I. I. Kossenko [122]. The classical problems of nonholonomic mechanics are the problems on a rolling of bodies on rigid surface. After the works of Kh. M. Mushtari [158. 1932] and Yu. P. Bychkov [29. 1965–1966] such problems were actively studied by A. V. Karapetyan [96], A. S. Kuleshov [127], A. P. Markeev [141– 144], V. K. Poida [184], V. V. Rumyantsev [201], V. A. Samsonov [205], Ya. V. Tatarinov [224], V. N. Tkhai [228], N. A. Fufaev [234], A. P. Kharlamov [235], E. I. Kharlamova [237], V. Ya. Yaroshchuk [263], Dong Zhiming, Yang Haixing [303, 410], T. Yamamoto [409] and others (for example, by L. D. Akulenko and D. D. Leshchenko [3]). The recent results and the current status of this question are given in the fundamental treatise of A. P. Markeev [143. 1992]. In this book there is a comprehensive bibliography on this topic. A new approach, concerning the interaction of body with a surface, is represented in the work of V. F. Zhuravlev [70. 1998–1999]. In many problems on a motion of bodies without slide on a fixed surfaces, particular attention has been given to the integration of a system of differential equations. However the particularly many works, devoted to the mathematical questions of the integrability of equations of motion for nonholonomic systems, was published beginning from the end 70s of 20th century. These are the works of A. A. Afonin, A. V. Borisov, A. A. Burov A. P. Veselov, L. E. Veselova, A. V. Karapetyan, A. A. Kilin, V. V. Kozlov, S. N. Kolesnikov, A. S. Kuleshov, I. S. Mamaev, A. P. Markeev, N. K. Moshchuk, Yu. N. Fedorov, V. A. Yaroshchuk and others [11, 20, 22, 26, 27, 37, 38, 112. 1985, 127, 142, 155. 1986, 230. 1988, 263, 278, 279, 331]. Among these investigations we can remark, in turn, the works of V. V. Kozlov [112. 1985] and A. P. Markeev [141. 1983]. Note that in many said papers together with the study of motion of the above-mentioned classical ball of Bobylev–Chaplygin the problems of G. K. Suslov [220] and L. E. Veselova [39] are also considered. In the works of A. V. Borisov, A. A. Kilin, I. S. Mamaev [278, 279] a possible hierarchy of the dynamics of the rolling bodies considered is suggested. A sui generis encyclopedia of this scientific direction is the treatise [19], which involves both the already published and specially written works devoted to the study of the dynamics of rolling bodies. The great difficulties over a long period moment were connected with the study of stability of nonholonomic systems. For example, even E. Whittaker [405], repeating the errors of F. Klein and D. Korteweg [336], regarded that the

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differential equations of small oscillations with holonomic and nonholonomic constraints have the same form. O. Bottema was one of the first to explain correctly in 1949 the influence of nonholonomicity of system on its stability (see also the recent work [281]). A detail consideration of stability of nonholonomic systems is given in the works of M. A. Aiserman and F. R. Gantmacher [264], I. S. Astapov [10], R. M. Bulatovich [24], D. V. Zenkov [85], A. A. Zobova [101], A. V. Karapetyan [94–97, 99, 100, 204], T. R. Kane and D. A. Levinson [331], G. N. Knyazev [108], V. V. Kozlov [113], A. S. Kuleshov [127, 333], A. P. Markeev [143], Yu. I. Neimark and N. A. Fufaev [165, 166], A. N. Obmorshev [175], M. Pascal [180], C. Risito [378], V. V. Rumyantsev [197, 201, 204], L. N. Semenova [207], Lilong Cai [350], A. Nordmark and H. Ess´en [369], Zhu Haiping and Mei Fengxiang [414], J. Walker [403], P. Hagedorn [312] and others. The works of V. I. Kalenova, A. V. Karapetyan, V. M. Morozov, M. A. Salmina, E. N. Sheveleva [91–93, 423] are devoted to the questions of stability and stabilization of a steady motion of nonholonomic systems. It is of highly interest for us the investigations of stability of a revolution of Celtic rattlebacks. The unusual peculiarity of their revolution was remarked first by G. T. Walker still in 1895 [402]. The detailed survey (with the main literature on the topic considered) of modern investigations of a steady motion of nonholonomic systems can be found in the work of A. V. Karapetyan and A. S. Kuleshov, published in the above-mentioned book [19] (see also Appendix B in the present treatise). We remark that, basing on the above works of I. S. Astapov, A. V. Karapetyan, A. P. Markeev, M. Pascal, on the numerical approach in the work [351] and using computer calculations, A. V. Borisov, I. S. Mamaev, and A. A. Kilin obtained that in the motion of the Celtic rattlebacks the chaos and attractors [19] may occur. A good simulation of Celtic rattleback is given in the work of I. I. Kossenko and M. S. Stavrovskaia [123, 337]. The motion of a Celtic rattleback with friction and sliding has been studied in the works by T. P. Tovstik [427]. The works of V. S. Novoselov [172], V. A. Sapa [206], M. F. Shul’gin and I. M. Shul’gina [257] and many foreign scholars ([308], Luo Shaokai, Mei Fengxiang, Qiao Yongfang, Zhang Jiefang and others) are devoted to the study of a motion of nonholonomic systems with variable mass. A new direction in the study of stochastic nonholonomic systems is given in the works of N. K. Moshchuk and I. N. Sinitsyn [155, 156]. The theory of motion of nonholonomic systems is successfully applied to the solution of different technical problems, which occur in the theory of motion of bicycle and motor cycle (M. Bourlet, M. Boussinesq, E. D. Dikarev, S. B. Dikareva, E. Carvallo, A. M. Letov, I. I. Metelitsyn, V. K. Poida, N. A. Fufaev [184, 282, 283, 289]), in the theory of motion of car (N. E. Zhukovsky, P. S. Lineikin, L. G. Lobas, Yu. I. Neimark, V. K. Poida, N. A. Fufaev, E. A. Chudakov [68, 72, 132, 133, 166, 184, 251, 253]), in the theory of interaction of wheel with road (V. G. Vil’ke, V. Gozdek, M. I. Esipov, A. Yu. Ishlinskii, M. V. Keldysh, I. V. Novozhilov, P. Rokar, N. A. Fufaev [40, 103, 167]), in different machines with variators of speed (I. I. Artobolevskii, I. I. Vul’fson, Ya.

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Survey of the main stages of development

L. Geronimus, V. A. Zinov’ev, A. I. Kukhtenko, A. V. Mal’tsev, V. S. Novoselov, B. A. Pronin, I. I. Tartakovskii [8, 47, 128, 170, 261]), in the theory of motion of electromechanical systems (A. V. Gaponov, V. A. Dievskii, O. Enge, G. Kielau, A. Yu. L’vovich, P. Maißer, Yu. G. Martynenko, F. F. Rodyukov, J. Steigenberger [45, 57, 136, 137, 145, 304, 359, 391]), and in many other fields of engineering (for example, the rotor breaking-in on rigid bearing [53]). In recent years the investigations are performed which are devoted to a motion of sportsman on a skateboard and a snakeboard (Yu. G. Ispolov, B. A. Smol’nikov [320. 1996], A. S. Kuleshov [126. 2004-2007], F. Pfeiffer, M. Foery, H. Ulbrich [432. 2006]). M. A. Levin and N. A. Fufaev have described a complicated nonholonomic interaction of tire with road, using a phenomenological model of rolling a deformable wheel [72, 130]. This model permits us to find the force and moment, which are a result of interacting wheel with road when a car moves. According to the suggested approach the motion of system is described by usual Lagrange’s equations of the second kind. Such an approach was used by E. V. Abrarova, A. A. Burov, S. Ya. Stepanov, D. P. Chevallier for the generation of equations of motion, which were applied then to the investigations of stability of a steady motion of complicated motor system, consisting of an articulated vehicle with hitching [2]. In 1981 in the work [186] it was shown that the acceleration of system can be resolved into two orthogonal components, one of which is completely determined by the equations of nonlinear ideal nonholonomic constraints. This result was presented in the treatise for universities [189] in 1985. In 1999 Yu. F. Golubev also obtains the resolutions for nonlinear nonholonomic constraints [50]. Similar resolutions for linear nonholonomic constraints, were obtained in 1989 by J. Storch and S. Gates, in 1991 by H. Brauchli and V. V. Velichenko, in 1992 by W. Blajer, M. Borri, C. Bottasso, P. Mantegazza, H. Ess´en [31, 275, 280, 284, 305, 392]. They applied the matrix calculus and obtained the equations, which permit for a motion and reactions of holonomic and nonholonomic constraints to be found for the system of bodies connected to each other. The projective method, proposed by them, is really a sui generis form, of Maggi’s equations, convenient for computer calculations. For applying the methods of computer algebra to the problems of mechanics, the treatise of D. M. Klimov and V. M. Rudenko [107] is especially useful. In the work by F. E. Udwadia and R. E. Kalaba [394. 1992] for defining the constraint reactions represented as linear second-order nonholonomic constraints the matrix calculus is used. In so doing, the partition of the whole space by the constraint equations into two orthogonal subspaces is automatically done by using the Moore and Penrose generalized inverse, which was already offered in 1920 [422]. Basing on their form of the equations of motion, they give a new version of the Gauss principle. Similar problems are highly actual in studying the problems of robotics. In this case it is rationally to use the treatises of G. V. Korenev [121], G. F. Moroshkin [154], D. E. Okhotsimskii and Yu. F. Golubev [178], E. P. Popov,

Survey of the main stages of development

xxxi

A. F. Vereshchagin and S. L. Zenkevich [191], J. Wittenburg [406], the works of V. A. Malyshev [139], P. Maißer [357, 358], and others. A new effective approach to the generation of compact equations of motion for system of rigid bodies is proposed by V. A. Konoplev in the works [119, 120] and is generalized in the treatise [120. 1996]. The questions of the dynamics and control by mobile wheeled robots are discussed in the works of V. I. Babitsky, A. Shipilov [435], V. N. Belotelov, V. I. Kalyonova, A. V. Karapetyan, A. I. Kobrin, A. V. Lenskii, Yu. G. Martynenko, V. M. Morozov, D. E. Okhotsimskii, M. A. Salmina [146-148, 423]. The individual question of nonholonomic mechanics is a question on a possible realization of nonholonomic constraints (the investigations of A. V. Karapetyan, C. Carath´eodori, V. V. Kozlov, I. V. Novozhilov, V. V. Kalinin, N. A. Fufaev and others [95, 115, 168, 233, 287]). Already in the early days of nonholonomic mechanics this question is actively discussed in the works of P. Appell, E. Delassus and others [270, 271, 298, 299]. The example of Appell– Hamel [270, 271, 315], considered from the point of view of the possibility of mechanical constructing the nonlinear nonholonomic constraints, was of special interest. Now the scholars also discuss often this example [171, 274, 290, 321, 376, 408]. The incorrectness of limit passage, used by P. Appell and G. Hamel, was remarked by Yu. I. Neimark and N. A. Fufaev [164]. In the work [321] it is shown that the limit passage, applying by Appell and Hamel, reduces, in fact, the problem of rolling the disk to the study of a motion of ball. Thus, in the nonholonomic mechanics it is assumed that in the case of motion of rigid bodies without slide and when there exist knife-edges the linear nonholonomic constraints can occur only. The limits of application of the theory of motion of nonholonomic systems are considerably extended by the introduction of servoconstraints due to A. Beghin and P. Appell [4, 13]. The theory of servoconstraints was actively developed by V. I. Kirgetov [105]. The tools of nonholonomic mechanics turn out yet more useful for the solution of a number of control problems (see, for example, the works of S. Deneva, V. Diamandiev, V. V. Dobronravov, Yu. G. Ispolov, B. A. Smol’nikov, K. Jankowski, E. Jarz¸ebowska, L. Steigenberger, Mei Fengxiang, J. Parczewski and W. Blajer, [52, 60, 89, 322, 323, 362, 371]). In this case the role of nonholonomic constraints is played by a program of motion and the reaction of such constraints is a control force. The theory of motion of systems with program constraints and the investigations of stability of computational process with provision for that the equations of constraints are satisfied approximately can be found in the works of A. S. Galiullin, I. A. Mukhametzyanov, R. G. Mukharlyamov, and V. D. Furasov [43, 157]. We remark that the program of motion can be given in the form of differential equations of a higher order than the first. Therefore the theory of nonholonomic systems with high-order constraints becomes actual. The works of Bl. Dolaptschiew, D. Mangeron, S. Deleanu, G. Hamel, J. Nielsen, L. Nordheim, J. Tz´enoff [61, 140, 238, 301, 314, 315, 360, 367, 368, 393] are devoted to the motion with high-order constraints. At present, this

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theory is actively developed, for example, in the works of Yu. A. Gartung, V. V. Dobronravov, Do Sanh, Yu. G. Ispolov, V. I. Kirgetov, B. G. Kuznetsov, M. A. Matsura, Mei Fengxiang, B. N. Fradlin, L. D. Roshchupkin, M. A. Chuev, I. M. Shul’gina, K. Jankowski, F. Kitzka, J. Stawianowski, R. Huston and others [46, 62, 88, 105, 125, 150, 159, 177, 193, 232, 252, 256, 316, 322, 335, 353, 362, 385, 389, 415]. However the numerical realization of similar theories in the cases when they are applied to some concrete problems is missing and therefore the validity of these theories is not tested. In this regard, examples of ideal linear third-order nonholonomic constraints describing the motion of a spacecraft with a constant modulo acceleration and its smooth transfer from one circular orbit to another one are of particular interest [78. 2005, 79, 214]. Early in the 20th century the applying of the tensor methods to the mechanics of nonholonomic systems results in the occurrence of a new field of geometry, namely the nonholonomic geometry. This field of geometry is developed in the works of V. V. Vagner, G. Vranceanu, A. Wundheiler, Z. Gorak, A. M. Lopshits, P. K. Rashevskii, J. Synge, J. Schouten, and W. Chow [30, 134, 194, 208, 223, 295, 383, 401]. The mathematical aspects of nonholonomic mechanics are considered in the works of V. I. Arnol’d, A. M. Vershik, A. P. Veselov, L. E. Veselova, V. Ya. Gershkovich, C. Godbillon, V. V. Kozlov, M. Le´on, L. M. Markhashov, A. I. Neishtadt, N. N. Petrov, P. Rodrigues, D. M. Sintsov, S. Smale, L. D. Faddeev, D. P. Chevallier and others [6, 7, 33, 35, 38, 48, 63, 149, 181, 209, 293, 344, 349, 354, 386, 397, 413]. Particular significance for their understanding has the treatises of V. I. Arnol’d [6], A. D. Bruno [23], and B. A. Dubrovin, S. P. Novikov, A. T. Fomenko [63], C. Truesdell [418]. Note that the present survey and references do not contain unfortunately many highly important works. A more detailed survey of variational principles of mechanics and equations of motion for nonholonomic systems and also the extensive bibliography can be found in the works of Yu. I. Neimark and N. A. Fufaev [166], J. Papastavridis [370. 1998, 2002], B. N. Fradlin [231], and V. N. Shchelkachev [254]. The interesting survey of the methods and problems of nonholonomic mechanics is given in the work of Mei Fengxiang [362].

Chapter I HOLONOMIC SYSTEMS

In this chapter we introduce a notion of the point that represents a motion of mechanical system. To generate Lagrange’s equations of the first and second kinds we make use of the approach demonstrating their unity and generality. This approach permits us to write Lagrange’s equations in the form, which can be used both in the case of one material (mass) point and of arbitrary mechanical system with finite or infinite numbers of degrees of freedom. The notion of ideal holonomic constraints is considered from the different points of view. The connection of the obtained equations of motion with the D’Alembert–Lagrange principle is analyzed. The longitudinal motion of a car with acceleration is considered as an example of motion of a holonomic system with a nonretaining constraint.

§ 1. Equations of motion for the representation point of holonomic mechanical system A simple and geometrically descriptive generation of equations of motion for holonomic mechanical systems is based on applying the notion of representation point, introduced by H. Hertz. The notion of representation point, in particular, is considered in the works [25, 135, 185]. We give here their results. Consider the motion of N material points with the masses mν , ν = 1, N . Their position in three-dimensional space in the Cartesian coordinates Ox1 x2 x3 can be characterized by the radius-vectors rν = xν1 i1 +xν2 i2 +xν3 i3 , ν = 1, N . If on the motion of system it is imposed the holonomic constraints f κ (t, x11 , x12 , x13 , . . . , xN 1 , xN 2 , xN 3 ) = 0 ,

κ = 1, k ,

(1.1)

then the vector equations of motion have the form mν ¨rν = Fν + R′ν ,

ν = 1, N .

(1.2)

Here Fν = Xν1 i1 + Xν2 i2 + Xν3 i3 is a resultant of forces, acting on the ′ ′ ′ i1 + Rν2 i2 + Rν3 i3 is a constraint reaction imposed ν-th point, R′ν = Rν1 on the ν-th point. To vector equations (1.2) correspond the following scalar differential equations ′ mν x ¨νj = Xνj + Rνj ,

ν = 1, N ,

1

j = 1, 2, 3 .

(1.3)

2

I. Holonomic Systems

We use a continuous numbering for the projections of radius-vectors, forces, and reaction of constraints: xµ = xνj , µ = 3(ν − 1) + j ,

′ Rµ′ = Rνj ,

Xµ = Xνj , ν = 1, N ,

j = 1, 2, 3 ,

µ = 1, 3N .

(1.4)

We also assume that mµ = mν

for

µ = 3ν − 2, 3ν − 1, 3ν ,

ν = 1, N .

(1.5)

Then equations (1.3) can be rewritten as mµ x ¨µ = Xµ + Rµ′ ,

µ = 1, 3N .

(1.6)

Introduce the following notations M=

3N 1 mµ , m  µ = mµ /M , 3 µ=1 ν=1   Yµ = Xµ / m µ , Rµ = Rµ′ / m µ , N 

mν ≡

yµ =



m  µ xµ ,

(1.7)

µ = 1, 3N .

In this case equations (1.6) become

M y¨µ = Yµ + Rµ ,

µ = 1, 3N .

(1.8)

We introduce in the 3N -dimensional Euclidean space the unit vectors j1 , . . . , j3N of Cartesian coordinates. Then scalar equations (1.8) correspond to the following vector equation MW = Y + R ,

(1.9)

where one uses the 3N -dimensional vectors: ˙ =y ¨, W=V

y = yµ j µ ,

Y = Yµ jµ ,

R = R µ jµ ,

µ = 1, 3N .

Further, in the products a summation in the corresponding limits of repeating indices is implied. The point of the mass M , the position of which in the 3N -dimensional space is given by the radius-vector y, is called an representation point. For this point equation (1.9) have the form of foundational law of mechanics for constrained motion of one point and therefore we shall call, for short, vector equation (1.9) Newton’s second law of motion In the 3N -dimensional Euclidean space (l = 3N −k) a set of the equations of holonomic constraints f κ (t, y) = 0 ,

y = (y1 , . . . , y3N ) ,

κ = 1, k ,

(1.10)

corresponding to original equations (1.1), define an l-dimensional surface on which the representation point is at the moment t. Transition formulas (1.4),

1. Equations of motion for the holonomic mechanical system

3

(1.5), (1.7) permit us to define the motion of the representation point in terms of the known motion of system in three-dimensional space, and vice versa, if the motion of the representation point in a 3N -dimensional space is known, then the same formulas take this motion to the motion of N material points in usual three-dimensional space. In the case of one point and one constraint, which is given by the equation f 1 (t, y) = 0 ,

y = (y1 , y2 , y3 ) ,

yµ = xµ ,

µ = 1, 3 ,

the constraint reaction can be represented as R = Λ1 ∇ f 1 + T 0 = N + T 0 . Here T0 is orthogonal to the normal component N. It is important for us that the mathematical equation of holonomic constraint gives the direction of the vector N only. At the same time the value and direction of the vector T0 must be given by the additional characteristics of constraint, which depend on its physical realization. The demonstrative example of the constrained motion of material point is a spherical pendulum. Obviously, the change of the length of pendulum l by the given law, i. e. the fact that the constraint f 1 (t, y) = y12 + y22 + y32 − l2 (t) = 0

(1.11)

holds can be provided by the force N, directed along the normal to the sphere, given at the moment of time by equation (1.11). In particular, if the constraint is physically realized by means of the retraction of a thread, then the constraint reaction is the tension N. Thus, for the study of the motion of spherical pendulum it is necessary to assume T0 = 0. The holonomic constraint, imposed on the point, is said to be ideal in the case T0 = 0. The example of motion with nonideal constraint is the motion of the point over a rough surface. In this case for the characteristic of T0 is often used Coulomb’s law v T0 = −k1 |N| , (1.12) |v| where k1 is a coefficient of friction. In the case of the one material point and two constraints, given by the following equations f κ (t, y) = 0 ,

κ = 1, 2 ,

yµ = xµ ,

µ = 1, 3 ,

the reaction R of these two constraints takes the form R = Λκ ∇ f κ + T0 ,

κ = 1, 2 .

Here T0 is orthogonal to the vectors ∇ f κ , κ = 1, 2. Then the point moves in a line, for example, in the case of stationary constraints in a circle (mathematical pendulum). The force T0 is lacking if the reaction R has no a component,

4

I. Holonomic Systems

directed at a tangent to the line on which the point is situated at this moment of time. Such constraints are called ideal. An additional consideration of holonomic constraints will be continue in § 1 of the next chapter. Using the analogy between a one material point and the point that represents the motion of mechanical system, in the general case we assume that R = Λκ ∇ f κ + T 0 ,

κ = 1, k ,

(1.13)

and we shall say that constraints (1.10) are ideal if T0 = 0. For ideal constraints (1.10) Newton’s second law (1.9) for the representation point can be written as (1.14) M W = Y + Λκ ∇ f κ . Projecting this vector equation on the axes of Cartesian coordinates, we obtain Lagrange’s equations of the first kind. If the vector equation is projected on the axes of curvilinear coordinates chosen specifically, then we arrive at Lagrange’s equations of the second kind. Thus, equation (1.14) shows the unity of two kinds of Lagrange’s equations. These equations are considered in more detail in the following division

§ 2. Lagrange’s equations of the first and second kinds Projecting vector equation (1.14) on the axes of Cartesian coordinates and recurring from the variables yµ to the variables xµ and from the quantities Yµ and Rµ to the quantities Xµ and Rµ′ , respectively, we obtain Lagrange’s equations of the first kind ¨ µ = X µ + Λκ mµ x

∂f κ , ∂xµ

µ = 1, 3N ,

κ = 1, k .

(2.1)

Equations (2.1) involve 3N + k unknowns x1 , . . . , x3N , Λ1 , . . . , Λk and therefore they can be considered together with equations of constraints (1.1). Now we eliminate the unknowns Λ1 , . . . , Λk from the above system. For this purpose we differentiate twice with respect to time the equations of constraints: ∂2f κ ∂2f κ ∂2f κ ∂f κ d2 f κ ≡ +2 x˙ µ + x ¨µ = 0 , x˙ µ x˙ µ∗ + 2 2 dt ∂t ∂t∂xµ ∂xµ ∂xµ∗ ∂xµ κ = 1, k ,

µ, µ∗ = 1, 3N ,

and substitute the relations for x ¨µ from equations (2.1). Then we have a system of linear nonuniform algebraic equations in unknowns Λ1 , . . . , Λk . Using the notion of the representation point, we represent the elements of the determinant of this system as g∗κ

∗

κ

∗

= ∇f κ · ∇f κ ,

κ, κ ∗ = 1, k .

2. Lagrange’s equations of the first and second kinds

5

Below we shall assume that holonomic constraints (1.1) are such that ∗ |g∗κ κ | =  0. The validity of this condition permits us to find the quantities Λκ , κ = 1, k, as the functions of the variables t, xµ , x˙ µ , µ = 1, 3N . Note that this condition and the analytic relations for the functions Λκ (t, xµ , x˙ µ ), κ = 1, k, were obtained and studied first by G. K. Suslov [220] and A. M. Lyapunov [138]. Substituting the functions Λκ (t, xµ , x˙ µ ), κ = 1, k, into formulas (2.1), we obtain 3N differential equations of the functions x1 , . . . , x3N . This system is convenient for numerical integration since it is solvable for the higher derivatives x ¨µ . We introduce now for the representation point the curvilinear coordinates q 1 , . . . , q 3N . Multiplying scalarly equation (1.14) by the vectors of fundamental basis eσ = ∂y/∂q σ , σ = 1, 3N , we obtain (M y¨µ − Yµ )

∂yµ ∂f κ ∂yµ = Λκ , σ ∂q ∂yµ ∂q σ

σ, µ = 1, 3N .

(2.2)

We shall regard the quantities q λ , λ = 1, l, l = 3N − k, as free independent curvilinear coordinates. The rest of coordinates, by assumption, are the following q l+κ = f κ (t, y), κ = 1, k. Then for q l+κ = 0, κ = 1, k, equations of constraints (1.10) are satisfied. In this case we have  0, σ = l + κ , ∂f κ ∂yµ ∂q l+κ ∂yµ κ l+κ ∇ f · eσ = = = δσ = ∂yµ ∂q σ ∂yµ ∂q σ 1, σ = l + κ . Thus, the vectors el+κ = ∇ f κ , κ = 1, k, are the vectors of a reciprocal basis, which is introduced as eσ · eτ = δτσ ,

σ, τ = 1, s .

Relations (2.2) can be represented as two systems of equations: (M y¨µ − Yµ ) (M y¨µ − Yµ )

∂yµ = 0, ∂q λ

∂yµ = Λκ , ∂q l+κ

λ = 1, l , κ = 1, k .

(2.3) (2.4)

Passing in equations (2.3), (2.4) from the variables yµ to the variables xµ and from the quantities Yµ to the quantities Xµ , we obtain (mµ x ¨ µ − Xµ ) (mµ x ¨ µ − Xµ )

∂xµ = 0, ∂q λ

∂xµ = Λκ , ∂q l+κ

λ = 1, l , κ = 1, k .

(2.5) (2.6)

6

I. Holonomic Systems

Taking into account the Lagrange relations mµ x ¨µ

d ∂T ∂T ∂xµ = − σ, ∂q σ dt ∂ q˙σ ∂q

T =

3N  mµ x˙ 2µ , 2 µ=1

σ = 1, 3N ,

and the generalized forces Qσ = Xµ

∂xµ ∂rν = Fν · σ = Y · eσ , σ ∂q ∂q

σ = 1, 3N ,

equations (2.5) and (2.6) can be written as usual Lagrange’s equations of the second kind: ∂T d ∂T − λ = Qλ , λ = 1, l , (2.7) dt ∂ q˙λ ∂q d ∂T ∂T − l+κ − Ql+κ = Λκ , dt ∂ q˙l+κ ∂q

κ = 1, k .

(2.8)

Equations (2.7) are, in essence, the equations of motion. Then for the given initial data we can obtain the law of motion of the system q λ = q λ (t) ,

λ = 1, l .

(2.9)

Representing equations (2.8) in explicit form, assuming that q l+κ = q˙l+κ = q¨l+κ = 0, κ = 1, k, and substituting functions (2.9), we get the generalized reactions Λκ , κ = 1, k, as time functions. If we find q¨λ = q¨λ (t, q, q) ˙ from equations (2.7) and substitute these relations in (2.8), then we obtain the ˙ functions Λκ = Λκ (t, q, q). We remark that if equation (1.14) is scalarly multiplied by the vectors eσ , σ = 1, s, we obtain M Wσ = Qσ + Λκ δσl+κ ,

σ = 1, s ,

κ = 1, k .

Then by (2.7) and (2.8) we have M Wσ = M W · eσ =

d ∂T ∂T − σ. σ dt ∂ q˙ ∂q

Further we shall often use this representation of covariant components of the vector M W in terms of the Lagrange operator. This implies that   d ∂T ∂T σ M W = M Wσ e = − σ eσ . dt ∂ q˙σ ∂q The Lagrange operator involves the following quantities pσ =

∂T , ∂ q˙σ

σ = 1, s ,

7

2. Lagrange’s equations of the first and second kinds

which are called generalized impulses. We shall show that they can be regarded as covariant components of the vector p = M V = pσ eσ , where V = dy/dt is a velocity of representation point. Really, we have T =

3N 

mµ

µ=1

then

3N  x˙ 2µ M y˙ µ2 = = 2 2 µ=1

M V2 M = = 2 2



∂y ∂y + σ q˙σ ∂t ∂q

∂T = M V · eσ , ∂ q˙σ

eσ =

2

M = 2



∂y + q˙σ eσ ∂t

2

,

∂V . ∂ q˙σ

Taking the derivative of the vector p = M V with time, we obtain     d ∂T d ∂T d(M V) ∂T τ ∂T τ MW ≡ e − τe = eτ + τ e˙ τ . , dt dt ∂ q˙τ ∂q dt ∂ q˙τ ∂ q˙ In this case from the identity e˙ τ · eσ = −eτ · e˙ σ it follows that

∂T ∂V = M V · σ = M V · e˙ σ , σ ∂q ∂q

and we have e˙ σ =

∂V . ∂q σ

Thus, the vector of velocity V of mechanical system, its acceleration W, the fundamental basis eσ , and the derivatives of it with respect to time e˙ σ , σ = 1, s, can be introduced by the relation for the kinetic energy of system. This will be used in Chapter IV. So, for ideal holonomic constraints equations of motion (2.5) (or (2.7)) can be separated from equations of reactions (2.6) (or (2.8)). Lagrange’s equations of the second kind (2.7) are obtained for the independent coordinates q λ , the number of which is equal to the number of degrees of freedom l. The left-hand sides of these equations is determined by the relation for kinetic energy and the right-hand sides by the relation for virtual elementary work. Taking into account that these scalar quantities (the kinetic energy and virtual elementary work) can be introduced for any mechanical system, it is natural to assume that equations (2.7) describe the law of motion of any mechanical system, the position of which is uniquely given by a set of independent Lagrangian coordinates q λ , λ = 1, l. This

8

I. Holonomic Systems

generalization of Lagrange’s equations of the second kind to the case of any mechanical system, consisting of as rigid bodies as solids, can be regarded as a postulate similar to the other postulates of physics. Notice that for the above choice of generalized coordinates the vectors el+κ , κ = 1, k, of reciprocal basis are equal to the vectors ∇ f κ such that as equations (1.14) show, the vector of reaction for ideal constraints is decomposed into these vectors. Therefore it is convenient to consider two subspaces: the L-space with the basis e1 , . . . , el and the K-space with the basis el+1 , . . . , e3N . These two subspaces are orthogonal to each other, in which case in the first of them it is considered a motion of system (a subspace of motions) and in the second subspace the generalized reactions (a subspace of reactions). Consider now the notion of ideal holonomic constraints from another point of view. Using the mentioned above subspaces, the acceleration of representation point can be written as the sum W = WL + WK , WL = W λ eλ , WK = Wl+κ el+κ , WL · WK = 0 . In this case by (1.13), the equation (1.9) can be replaced by two equations M WL = YL + T0 ,

(2.10)

M W K = Y K + Λκ ∇ f κ .

(2.11)

Here YK = Ql+κ el+κ ,

YL = Y − YK .

Vector equation (2.11) is equivalent to the following k scalar equations Λκ = M Wl+κ − Ql+κ ,

κ = 1, k .

(2.12)

We shall show that the vector WK , given by the components Wl+κ , is completely determined by the equations of constraints, i. e. the quantities Wl+κ can be obtained as the functions of time t, the generalized coordinates q λ , and the generalized velocities q˙λ (λ = 1, l), using the equations of constraints only. In this case relations (2.12) imply that the generalized reaction Λκ as the functions of the same variables for the given generalized forces Ql+κ can also be obtained using the equations of constraints. Really, differentiating twice equations of constraints (1.10) with time, we obtain ∂2f κ ∂2f κ ∂2f κ y˙ µ y˙ µ∗ , y˙ µ − ∇f κ · W = − 2 − 2 ∂t ∂t∂yµ ∂yµ ∂yµ∗ (2.13) µ, µ∗ = 1, 3N . κ = 1, k , Using the transition formula (from the Cartesian coordinates to the generalized ones) q λ = f∗λ (t, y) ,

λ = 1, l ,

q l+κ = f κ (t, y) ,

κ = 1, k ,

9

2. Lagrange’s equations of the first and second kinds and taking into account that q l+κ = q˙l+κ = 0 ,

κ = 1, k ,

we have yµ = yµ (t, q) ,

q = (q 1 , . . . , q l ) ,

∂yµ ∂yµ + λ q˙λ . ∂t ∂q

y˙ µ =

Substituting these relations into the right-hand side of relations (2.13), we get q = (q 1 , . . . , q l ) ,

∇ f κ · W = χκ (t, q, q) ˙ ,

κ = 1, k.

(2.14)

Taking into account that W = Wl+κ el+κ + W λ eλ ,

∇ f κ = el+κ ,

λ = 1, l ,

κ = 1, k ,

we obtain g∗κ

∗

κ

κ, κ ∗ = 1, k ,

Wl+κ ∗ = χκ (t, q, q) ˙ ,

(2.15)

where g∗κ

∗

κ

∗

= ∇f κ · ∇f κ ,

κ, κ ∗ = 1, k .

∗

Assuming, as above, that |g∗κ κ | =  0, and solving the system of linear algebraic equations (2.15), we obtain ∗

∗ κ (t, q, q) ˙ , Wl+κ = gκκ ∗χ

κ, κ ∗ = 1, k .

(2.16) ∗

∗ κ κ Here gκκ ). ∗ are elements of a matrix inverse to the matrix (g∗ K From relations (2.16) it follows that the vector W is completely determined by equations of constraints. The effect of the equations of constraints on the vector W is given by formulas (2.14). Since ∇ f κ · WL = 0, κ = 1, k, these formulas can be rewritten as ∇ f κ · WK = χκ (t, q, q) ˙ , κ = 1, k .

This implies that the constraints are satisfied for any vector WL . Therefore the form of equations of constraints does not give any information on the vector WL . From equation (2.10) it follows that the constraints can effect only indirectly on the vector WL via the vector T0 , which, in no way, is directly connection with the equations of constraints. The generalized forces Ql+κ and the equations of constraints completely determine the normal component N = Λκ ∇ f κ of the vector of reaction R only. Therefore the ideal constraints, for which T0 = 0, can be called the constraints, which completely defined by their analytic representations. Thus, for ideal constraints, Newton’s second law, written in the L-space, has the same form as for free system: M WL = YL .

10

I. Holonomic Systems

Lagrange’s equations of the first and second kinds can be applied also to the study of the dynamics of elastic systems with distributed parameters. Taking into account this generality of Lagrange’s equations, it is useful consider the case when the initial coordinates of the mechanical system of general type are its Lagrangian coordinates q σ , the number of which is equal to s if the constraints are missing. The equations of ideal holonomic constraints are assumed to be given in the form f κ (t, q) = 0 ,

κ = 1, k .

(2.17)

For simplicity, consider the case when the system consists of N material points. The position of system is described by the curvilinear coordinates q σ , σ = 1, s, s = 3N . We introduce now the new coordinates q∗ρ = q∗ρ (t, q) ,

q σ = q σ (t, q∗ ) ,

ρ, σ = 1, s ,

and the new fundamental and reciprocal bases e∗τ =

∂q σ eσ , ∂q∗τ

eρ∗ =

∂q∗ρ τ e , ∂q τ

ρ, σ, τ = 1, s .

(2.18)

The coordinates q∗1 , . . . , q∗l (l = s − k) are chosen arbitrary, the rest of new coordinates is defined by the functions q∗l+κ = f κ (t, q), κ = 1, k. Then the imposition of constraints (2.17) means that q∗l+κ = 0, κ = 1, k. From formulas (2.18) it follows that el+κ = ∗

∂f κ τ e = ∇f κ , ∂q τ

and therefore the reaction of ideal holonomic constraints (2.17) can be represented as . R = Λκ el+κ ∗ Multiplying Newton’s equation M W = Y + Λκ el+κ ∗

(2.19)

scalarly by the vectors e∗1 , . . . , e∗s , we obtain (M Wσ − Qσ ) (M Wσ − Qσ )

∂q σ = 0, ∂q∗λ

∂q σ = Λκ , ∂q∗l+κ

λ = 1, l , κ = 1, k .

Taking into account M Wσ

∂q σ = M Wτ∗ ; ∂q∗τ

Qσ

∂q σ = Q∗τ , ∂q∗τ

σ, τ = 1, s ,

(2.20) (2.21)

11

2. Lagrange’s equations of the first and second kinds we have

d ∂T ∂T − λ = Q∗λ , λ dt ∂ q˙∗ ∂q∗

λ = 1, l ,

d ∂T ∂T − l+κ − Q∗l+κ = Λκ , dt ∂ q˙∗l+κ ∂q∗

κ = 1, k .

From the first system of these equations we can obtain a motion in new independent coordinates and from the second one a generalized reaction of constraints (2.17). The method to find the motion and reactions is described above for systems (2.7), (2.8). Multiplying scalarly equation (2.19) by the vectors of fundamental basis e1 , . . . , es of the initial system of coordinates q 1 , . . . , q s , we obtain d ∂T ∂T ∂f κ − = Q + Λ , σ κ dt ∂ q˙σ ∂q σ ∂q σ

σ = 1, s .

(2.22)

These equations involve s + k unknowns q 1 , . . . , q s , Λ1 , . . . , Λk , which have to be solved together with equations of constraints (2.17). This was distinctive for classical Lagrange’s equations of the first kind (2.1) when the Cartesian coordinates were used. The equations (2.22) are written in the curvilinear frame, in which case constraints (2.17) are imposed on the generalized coordinates, and therefore these equations, following N. V. Butenin and N. A. Fufaev [28], we shall call Lagrange’s equations of the first kind in generalized coordinates. In the literature equations (2.22) are also called Lagrange’s equations of the second kind with multipliers (see, for example, [59]). V. V. Rumyantsev [202, p. 23] writes: "Lagrange notes that such cases can occur when for the sake of simplicity of calculations it is useful to preserve a greater number of variables and to make use of indefinite multipliers". Equations (2.22) permit us, as is cited above, to describe the motion of mechanical systems in redundant coordinates. If for the coordinates q σ (σ = 1, s) constraints (2.17) are lacking, then they become usual Lagrange’s equations of the second kind. Equations (2.22) will be repeatedly used further, in particular, with their help in Chapter VI some new methods to study the motion of mechanical systems will be constructed. It is often convenient to represent equations (2.22) in explicit form but not by the Lagrange operator. Assuming that in the coordinates q σ , σ = 1, s, the kinetic energy of system has the form T = we have

M gαβ q˙α q˙β , 2

α, β = 0, s ,

q0 = t ,

q˙0 = 1 ,

∂f κ , M (gσρ q¨ρ + Γσ,αβ q˙α q˙β ) = Qσ + Λκ ∂q σ   1 ∂gσβ ∂gσα ∂gαβ Γσ,αβ = , + − 2 ∂q α ∂q β ∂q σ ρ, σ = 1, s ,

α, β = 0, s .

(2.23)

12

I. Holonomic Systems

Multiplying equations (2.23) by the coefficients g στ , which are the elements of the matrix inverse to the matrix with the elements gστ and summing over σ from 1 to s, we obtain ∂f κ , ∂q σ σ, τ = 1, s , α, β = 0, s .

M (¨ q τ + Γταβ q˙α q˙β ) = Qτ + Λκ g στ Γταβ = g στ Γσ,αβ ,

Qτ = g στ Qσ ,

§ 3. The D’Alembert–Lagrange principle Equations (2.22) involve the equations of constraints. To express the system of these equations in the form of one scalar relation only, without the equations of constraints, we consider the admissible (virtual) displacements of mechanical system. The mathematical definition of these displacements can be given in the following way. We introduce the two systems of generalized Lagrangian coordinates uniquely determining the position of mechanical system before the holonomic constraints are imposed on it. Suppose, these coordinates are related as q σ = q σ (t, q∗ ) ,

q∗ρ = q∗ρ (t, q) ,

ρ, σ = 1, s .

In differential form these constraints between coordinates are the following δq σ =

∂q σ ρ δq , ∂q∗ρ ∗

δq∗ρ =

∂q∗ρ σ δq , ∂q σ

ρ, σ = 1, s .

This implies that the quantities δq σ and δq∗ρ are the partial differentials of the functions q σ (t, q∗ ) and q∗ρ (t, q), respectively, computed at the moment t. They are called admissible (virtual) displacements or the variations of coordinates. The quantities δq σ can be regarded as the contravariant components of the vector δy of virtual displacement of system. In the frame q∗ρ the same vector δy is given by a family of the quantities δq∗ρ . Consider now the mechanical system with ideal holonomic constraints, which are given by equations f κ (t, q) = 0 ,

κ = 1, k .

Then, assuming q∗l+κ = f κ (t, q) ,

κ = 1, k ,

l = s−k,

and making use the validity of the equations of constraints, we have δq∗l+κ =

∂q∗l+κ σ ∂f κ σ δq = δq = ∇ f κ · δy = 0 , σ ∂q ∂q σ

κ = 1, k .

(3.1)

13

3. The D’Alembert–Lagrange principle

Whence it follows that if we multiply each of equations (2.22) by δq σ and summarize then over all σ, then we arrive at the relation   ∂T d ∂T σ − − Q (3.2) σ δq = 0 . dt ∂ q˙σ ∂q σ This relation, generated as a consequence of equations (2.22), give the D’Alembert–Lagrange principle. We shall show how equations (2.22) can be obtained from this principle. Multiplying each of relations (3.1) by Λκ , summarizing over all κ from 1 to k, and subtracting this sum from relations (3.2), we have   d ∂T ∂f κ ∂T − σ − Qσ − Λκ σ δq σ = 0 . (3.3) dt ∂ q˙σ ∂q ∂q Equation (3.3) is satisfied for any Λκ and any values δq σ (σ = 1, s), satisfying equations (3.1). From system of equations (3.1) it follows that only l values of δq σ are independent and the rest of the variations of coordinates can be expressed via them. Choose the quantities Λκ so that in relation (3.3) the coefficients of dependent variations of coordinates are equal to zero. The coefficients of the rest of variations of coordinates are also equal to zero since these variations are arbitrary and independent. Thus, in relation (3.3) all the coefficients of δq σ are equal to zero and this means that equations (2.22) follows from the D’Alembert–Lagrange principle (3.2). In the case when the mechanical system consists of N material points and the initial Lagrangian coordinates q σ are Cartesian coordinates xµ (µ = 1, 3N ) of the points of system the D’Alembert–Lagrange principle (3.2) can be represented in the form ¨µ − Xµ ) δxµ = 0 , (mµ x

(3.4)

since in this case we have

mµ x˙ 2µ . 2 Relation (3.4) can also be represented as T =

(mν ¨rν − Fν ) · δrν = 0 , where the vector δrν = δxν1 i1 + δxν2 i2 + δxν3 i3 =

∂rν λ δq ∂q λ

is the vector of virtual displacement of the ν-th point. By formulas (3.1) the coordinates δxµ , µ = 1, 3N , of the vectors δrν , ν = 1, N , satisfy the relations ∂f κ δxµ = 0 , ∂xµ

κ = 1, k .

14

I. Holonomic Systems

We consider now the notion of ideal holonomic constraints from the point of view of the D’Alembert–Lagrange principle. This principle (3.2) can be rewritten in the vector form  M W − Y · δy = 0 ,

or

R · δy = 0 .

(3.5)

Then from equations (3.1) it follows that the reaction of ideal holonomic constraints can be represented as R = Λκ ∇ f κ , i. e. the vector of reaction can be decomposed into the vectors el+κ = ∇f κ , ∗ κ = 1, k, of the basis of K-space (the subspace of reactions). The equations of holonomic constraints give the l-dimensional surface V (t, q) on which at moment t the representation point is to be situated. To the curvilinear coordinates q∗1 , . . . , q∗l corresponds the basis e∗1 , . . . , e∗l , situated in the plane T (q, V ) tangent to the surface V (t, q). The vectors δy of the virtual displacements of representation point (the subspace of virtual displacements) lie in this plane. Thus, the D’Alembert–Lagrange principle in the form (3.5) states that for ideal holonomic constraints the subspace of reactions (K-space) is orthogonal to the subspace of virtual displacements (L-space). We show that the representation of the D’Alembertian–Lagrange principle in the form (3.5) is the generalization of usual notion of ideality of constraint for one material point to the case of representation point. Consider this condition for the system of N material points in the case when we use Cartesian coordinates and the equations of constraints are represented in the form (1.10). By formulas (1.4), (1.7) condition (3.5) takes the form R · δy = Λκ ∇ f κ · δy = Λκ

∂f κ ∂f κ δyµ = Λκ δxµ = 0 . ∂yµ ∂xµ

(3.6)

If we introduce the reaction  κ  ∂f ∂f κ ∂f κ i1 + i2 + i3 , R′ν = Λκ ∂xν1 ∂xν2 ∂xν3 applied to the ν-th point in usual three-dimensional space by all constraints, then in place of formula (3.6) we have R′ν · δrν = 0 .

(3.7)

This relation is usually regarded as a definition of ideality of holonomic constraints, imposed on the motion of N material points. It is of hardly explainable axiomatic nature. However in fact, as is shown above, condition (3.7) is a generalization of usual notion of constraints ideality for one material point to the case of representation point.

4. Longitudinal accelerated motion of a car

15

§ 4. Longitudinal accelerated motion of a car as an example of motion of a holonomic system with a nonretaining constraint Problem definition. Longitudinal acceleration of a car with possible slipping of its driving (front or rear) wheels is considered. The wheels are assumed to be perfectly rigid, undeformable. The calculation model of a frontdrive car is presented in Fig. I. 1. The car is supposed to consist of the body on the springs with shock absorbers, in which viscous drag forces of both front (1) and rear (2) double wheels are taken into account. C1 and C2 are the centers of the front and rear wheels, correspondingly. B1 and B2 are the attachment points of the springs and shock absorbers to the car body.

Fig. I. 1

Let us introduce the notation: M , M1 , M2 are masses of the body, the front and rear wheels, correspondingly; J, J1 , J2 are their moments of inertia about the centers of mass ; χ, χ1 , χ2 are the coefficients of viscous drag forces acting on the body and in the shock absorbers, correspondingly; c1 , c2 are the stiffness coefficients of the springs of front and rear wheels; k1st , k1dyn (or k2st , k2dyn ) are the static and dynamic coefficients of Coulomb’s friction force for the driving front (or rear) wheels; r1 , r2 are the coefficients of rolling friction for the front and rear wheels; R1 , R2 are the radii of the corresponding wheels. Note that we consider a reduced moment of inertia, which includes rotating masses of the cardan shaft, details of the transmission gear-box, the flywheel, the crankshaft, etc., connected to the driving wheels, as the moment of inertia of driving wheels (according to Fig. I. 1 it is J1 ). In this case the quantity J1 depends, in general, on the relation between angular velocities of the driving wheels and the engine. The engine acts on the driving wheels with some drive moment Θdr , specified as a time function. In Fig. I. 1 the drive moment is applied to the driving wheels.

16

I. Holonomic Systems

Equations of the car motion are to be obtained to determine, when slipping begins at the stage of acceleration and when it stops. Rolling motion is the main mode of vehicles’ operation. When rolling without slipping, the instantaneous center of velocity of the wheel is located at the point of contact between the wheel and the road. Let us introduce a fixed frame of reference Oxy, the y-axis being directed vertically upwards. The abscissa x = x(t) and the ordinate y = y(t) of the centre of mass C of the car body, as well as the angle of the body rotation ϕ = ϕ(t) about the centre of mass C, will be considered as generalized coordinates. Suppose, that in the state of static equilibrium of the body we have y = 0 and ϕ = 0. The condition of rolling without slipping for a front-drive car can be written in the form of the first order differential linear constraint x˙ = ϕ˙ 1 R1 ,

(4.1)

where ϕ1 is the angle of rotation of front driving wheels. In a similar manner, for a rear-drive car the equation of constraint shoud be written as x˙ = ϕ˙ 2 R2 , where ϕ2 = ϕ2 (t) is the angle of rotation of rear driving wheels. The constraint (4.1) is holonomic: integrating expression (4.1) yields x = ϕ1 R1

(x = ϕ2 R2 ) .

The constraint (4.1) is nonretaining, the car is released from it when the driving wheels begin to slip. When slipping occurs, instead of the condition (4.1) the following inequality x˙ ϕ˙ 1 > R1 begins to be fulfilled. Note, that when the car is accelerating, this inequality can not be opposite in sign. So, the constraint under consideration is unilateral. The slipping occurs, when the horizontal reaction force of the road towards the driving wheels of the car reaches some "limit"value, which is related with the static Coulomb friction force. Let us start analysis with the case when the constraint (4.1) is satisfied. Motion without slipping. In the absence of slipping of the driving wheels the kinetic energy of the car T takes the form    J1 J2 1 2 2 2 . T = M + M1 + M2 + 2 + 2 x˙ + M y˙ + J ϕ˙ 2 R1 R2 The system potential energy Π is equal to the potential energy of the springs deformation. We shall measure it from the position of static equilibrium. Let

17

4. Longitudinal accelerated motion of a car

us introduce auxiliary coordinates for each of the springs: y1 = y1 (t) and y2 = y2 (t), which are the vertical upward displacements of the points B1 and B2 from the position of static equilibrium. By expressing y1 and y2 in terms of y and ϕ, we obtain Π=

1 (c1 y12 + c2 y22 ) = 2

(4.2) 1 (c1 (y + L1 ϕ)2 + c2 (y − L2 ϕ)2 ) . 2 Here Lk (k = 1, 2) are the horizontal distances from the point C to the points Bk (see Fig. I. 1). The Rayleigh scattering function corresponding to the resistance forces takes the form 1 R = (χx˙ 2 + χ1 y˙ 12 + χ2 y˙ 22 ) = 2 (4.3) 1 ˙ 2 + χ2 (y˙ − L2 ϕ) ˙ 2) . = (χx˙ 2 + χ1 (y˙ + L1 ϕ) 2 Let us take into consideration the drive moment Θdr and the moment of rolling friction of the wheels. The elementary work of generalized forces applied to the driving front wheels is the product of the drive moment Θdr by the angular displacement δϕ1 and the moment of rolling friction by the angular displacement δϕ1 . The elementary work of generalized forces applied to the driven rear wheels is the product of the moment of rolling friction by the angular displacement δϕ2 . As a result, we obtain =

δAx = Θdr δϕ1 − N1 r1 δϕ1 − N2 r2 δϕ2 . Here N1 and N2 are the vertical components of the road reaction towards the first and the second pair of wheels, correspondingly. Expressing the angular displacements δϕ1 and δϕ2 in terms of the elementary displacement δx implies   Θdr r1 r2 δAx = δx . − N1 − N2 R1 R1 R2 Now determine the reactions N1 and N2 . They depend on gravity forces of the wheels, the statical reactions of the body towards the wheels N1st and N2st , and the dynamical corrections to them, which are due to the vertical movement of the body and its rotation. Besides these, the damping forces, which are generated by shock absorbers when the body is oscillating, should be also taken into account. These forces are calculated by the Rayleigh scattering function (4.3) and applied to the body at the points B1 and B2 . At the same time, according to the third Newton law they act from the shock absorbers to the axles of front and rear wheels. Thus, we have M gL2 , L1 + L2 M gL1 = . L1 + L2

N1 = M1 g + N1st − c1 (y + L1 ϕ) − χ1 (y˙ + L1 ϕ) ˙ ,

N1st =

˙ , N2 = M2 g + N2st − c2 (y − L2 ϕ) − χ2 (y˙ − L2 ϕ)

N2st

18

I. Holonomic Systems

Finally, the influence of the drive moment on the body should be taken into consideration. This moment is opposite in direction to the drive moment Θdr and is the same as the drive moment in value. The elementary work of this moment is (4.4) δAϕ = Θdr δϕ . Note that taking into consideration the influence of this moment on the body rotation is of principal importance. Unfortunately, it is not taken into account in a number of studies. Using the written above expressions for the kinetic and potential energies, Rayleigh’s function, and the elementary work, we write the Lagrange equations of the second kind (M + M1∗ + M2∗ ) x ¨=

    Θdr M L2 M L1 r1 g r2 g − M1 + − M2 + + R1 L1 + L2 R1 L1 + L2 R2     c1 r1 c1 r1 L1 c2 r2 c2 r2 L2 + y+ ϕ − χx+ ˙ + − R1 R2 R1 R2     χ1 r1 χ2 r2 χ2 r2 L2 χ1 r1 L1 + − + y˙ + ϕ˙ , (4.5) R1 R2 R1 R2

M y¨ = −(c1 + c2 )y − (c1 L1 − c2 L2 )ϕ− − (χ1 + χ2 )y˙ − (χ1 L1 − χ2 L2 )ϕ˙ , J ϕ ¨ = Θdr − (c1 L1 − c2 L2 )y − (c1 L21 + c2 L22 )ϕ− − (χ1 L1 − χ2 L2 )y˙ − (χ1 L21 + χ2 L22 )ϕ˙ ,

J1 J2 , M2∗ = M2 + 2 . R12 R2 The system of differential equations (4.5) describes the motion of a frontdrive car in the absence of driving wheels slipping. The analogous equations of motion for a rear-drive car differ from system (4.5) in that the expression Θdr /R1 should be replaced with Θdr /R2 in them. Note that the second and the third equations of system (4.5) do not contain the variable x and can be integrated apart from the first equation.

where M1∗ = M1 +

Motion with slipping. From kinematics standpoint, slipping of the driving front wheels occurs when the point K1 of contact with the road ceases to be an instantaneous velocity center of the driving wheels, that is when ϕ˙ 1 = x(t)/R ˙ 1 . It is obvious that in this case one should introduce a new "independent"generalized coordinate ϕ1 , which defines a rotation angle of the driving front wheels relative to the initial position. As a result, the expression for the system kinetic energy takes the form    J2 1 T = (4.6) M + M1 + M2 + 2 x˙ 2 + M y˙ 2 + J ϕ˙ 2 + J1 ϕ˙ 21 . 2 R2

19

4. Longitudinal accelerated motion of a car

Expressions (4.2) and (4.3) for the potential energy Π and Rayleigh’s function R, as well as the elementary work (4.4) are the same as in the case of motion without slipping. The elementary work δA′x , that is done in moving through displacement of the car body δx = δϕ2 R2 by Coulomb’s friction force k1dyn N1 and by the rolling friction moment of the driven wheels, is δA′x = k1dyn N1 δx − r2 N2 δϕ2 =



k1dyn N1 −

r2 N2 R2



(4.7)

δx .

When slipping occurs, the driving front wheels are affected by the slipping friction force k1dyn N1 , the rolling friction moment, and the drive moment Θdr . Therefore δAϕ1 = (Θdr − k1dyn N1 R1 − r1 N1 )δϕ1 .

(4.8)

From expressions (4.6), (4.2), (4.3), (4.4), (4.7), (4.8) it follows that the system of the Lagrange equations for a front-drive car moving with slipping takes the form     r2 g M L2 M L1 dyn ∗ k1 g − M2 + ¨ = M1 + + (M + M1 + M2 ) x L1 + L2 L1 + L2 R2 +



   c2 r2 c2 r2 L2 dyn dyn y − c1 k1 L1 + ϕ − χx+ ˙ − c1 k1 R2 R2

+



   χ2 r2 χ2 r2 L2 dyn dyn − χ1 k1 y˙ − χ1 k1 L1 + ϕ˙ , R2 R2

M y¨ = −(c1 + c2 )y − (c1 L1 − c2 L2 )ϕ − (χ1 + χ2 )y˙ − (χ1 L1 − χ2 L2 )ϕ˙ , J ϕ¨ = Θdr − (c1 L1 − c2 L2 )y − (c1 L21 + c2 L22 )ϕ −

(4.9)

− (χ1 L1 − χ2 L2 )y˙ − (χ1 L21 + χ2 L22 )ϕ˙ , 

M L2 J1 ϕ¨1 = Θdr − M1 + L1 + L2



(r1 + k1dyn R1 )g+

+ c1 (r1 + k1dyn R1 )y + c1 (r1 + k1dyn R1 )L1 ϕ+ + χ1 (r1 + k1dyn R1 )y˙ + χ1 (r1 + k1dyn R1 )L1 ϕ˙ .

20

I. Holonomic Systems

Analogous equations for a rear-drive car appear as     r1 g M L1 M L2 (M + M1∗ + M2 ) x k2dyn g − M1 + ¨ = M2 + + L1 + L2 L1 + L2 R1 +



   c1 r1 c1 r1 L1 ϕ − χx+ ˙ − c2 k2dyn y + c2 k2dyn L2 + R1 R1

+



   χ1 r1 χ1 r1 L1 ϕ˙ , − χ2 k2dyn y˙ + χ2 k2dyn L2 + R1 R1

M y¨ = −(c1 + c2 )y − (c1 L1 − c2 L2 )ϕ − (χ1 + χ2 )y˙ − (χ1 L1 − χ2 L2 )ϕ˙ , J ϕ¨ = Θdr − (c1 L1 − c2 L2 )y − (c1 L21 + c2 L22 )ϕ− − (χ1 L1 − χ2 L2 )y˙ − (χ1 L21 + χ2 L22 )ϕ˙ , 

M L1 J2 ϕ¨2 = Θdr − M2 + L1 + L2



(4.10)

(r2 + k2dyn R2 )g+

+ c2 (r2 + k2dyn R2 )y − c2 (r2 + k2dyn R2 )L2 ϕ+ + χ2 (r2 + k2dyn R2 )y˙ − χ2 (r2 + k2dyn R2 )L2 ϕ˙ . Note that the second and third equations of systems (4.9) and (4.10) are unchanged in comparison with the case of motion without slipping. They can be also integrated apart from the first equations. The conditions of occurrence and termination of slipping. The last equation of system (4.9) for a front-drive car can be rewritten as J1 ϕ¨1 = Θdr − r1 N1 − k1dyn N1 R1 . In the absence of slipping we should set ϕ1 = x/R1 in this equation, and the quantity k1dyn N1 should be replaced by the friction force Fdr . Note that this friction force Fdr is a driving force in the problem concerned, that is why it has a subscript "dr". The force Fdr is less than the static Coulomb friction force k1st N1 . Therefore, the dynamic condition of motion of the driving font wheels without slipping takes the form r1 N1 J1 x ¨ Θdr − − 2 = Fdr < k1st N1 . R1 R1 R1

(4.11)

4. Longitudinal accelerated motion of a car

21

Hence, the system of equations (4.5) can be used as long as the values of N1 and x ¨, calculated from it, satisfy inequality (4.11). Recall that for calculating the reaction N1 the quantities y, ϕ, and their derivatives should be known. If at some instant t1 inequality (4.11) is violated, and the slipping of the front wheels begins, then system (4.9) should be integrated. Now the driving wheels are affected by the dynamic Coulomb friction force k1dyn N1 . As stated at the beginning of the section, the non-holonomic constraint imposed on the driving wheels is nonretaining and unilateral. Therefore, if when integrating system (4.9) at some instant t2 the equality ϕ˙ 1 = x/R ˙ 1 is fulfilled, then it means that the slipping of driving wheels has terminated and constraint (4.1) is restored. Starting with the time instant t2 , one should come to integrating the system of differential equations (4.5). So, the condition of slipping termination is the fulfillment of the equality ϕ˙ 1 (t) =

x(t) ˙ . R1

(4.12)

The analogous condition of motion without slipping for a rear-drive car and the condition of slipping termination are r1 N1 J2 x ¨ Θdr − − 2 = Fdr < k2st N2 , R2 R2 R2 ϕ˙ 2 =

x˙ . R2

The example of problem solving. As an example consider the acceleration of a hypothetical front-wheel drive car. Let us assume that the period of time under consideration is equal to 50 seconds. At the initial instant of time (t0 = 0) the car is immovable and starts to accelerate under the action of the drive moment (the moment is measured in N · m, time t is measured in seconds):  πt   750 sin , t  37 , 40 (4.13) Θdr =   750 sin 37π , t > 37 . 40

The values of parameters used in calculations are: M = 955 kg (including masses of the front and rear axles: 515 kg and 440 kg, correspondingly); J = 1394.2 kg·m2 ; χ = 10 N·s·m−1 ; L1 = 1.17 m; L2 = 1.307 m; M1 = M2 = 14 kg; J1 = 21.52 kg·m2 ; J2 = 1.076 kg·m2 ; R1 = R2 = 0.392 m; c1 = c2 = 20000 N/m; χ1 = χ2 = 1000 N·s·m−1 ; r1 = r2 = 0.0024 m; 2 k1st = 0.3; k1dyn = 0.25; the gravitational acceleration is g = 9.8 m/s . The passage from system (4.5) to system (4.9) and then back again to system (4.5) is performed on the basis of realization of conditions (4.11)

22

I. Holonomic Systems

Fig. I. 2

and (4.12). It is explained by Fig. I. 2, corresponding to the numerical data specified above. In this figure the static Coulomb friction force Ffrst = k1st N1 and the dynamic Coulomb friction force Ffrdyn = k1dyn N1 are shown by solid lines, and the driving friction force Fdr =

Θdr r1 N1 J1 x ¨ − − 2 . R1 R1 R1

(4.14)

is shown by a dashed line As follows from the figure, the driving friction force increases from the instant of time t0 = 0 till the instant t1 = 14.004 s, but inequality (4.11) is fulfilled. Therefore, system (4.5) is to be integrated. Starting with the instant of time t1 = 14.004 s, there comes the second stage of the car acceleration — motion with slipping. It is described by system (4.9), the initial data for which are found from the values of functions that are the solution of system (4.5) for t1 = 14.004 s. When integrating system (4.9), we determine the instant of time t2 , when equality (4.12) is fulfilled. In the example given t2 = 38.747 s. At this moment of time t2 constraint (4.12) is imposed instantly, and the third stage of the car acceleration begins — the resumption (recovering) of motion without slipping. It is described by system (4.5), the initial data for which are found from the values of functions that are the solution of system (4.9) for t2 = 38.747 s. The passage from system (4.9) to system (4.5) when t2 = 38.747 s, caused by instantaneous imposition of constraint (4.12), is followed by the jump of acceleration x ¨ (see Fig. I. 3): for t1 < t < t2 the acceleration x ¨ is found from system (4.9), and for t > t2 it is found from system (4.5), in which case ¨/R1 . Therefore, for t = t2 we have ϕ¨1 = x x ¨(t2 − 0) = x ¨(t2 + 0) .

4. Longitudinal accelerated motion of a car

23

Fig. I. 3

This acceleration jump is connected with decrease of the value of traction (force of cohesion) of the front wheels with the road from the value k1dyn N1 down to the value Fdr , defined by formula (4.14), in which x ¨ is calculated from system (4.5) with the initial data taken from the end of motion described by system (4.9). This jump of traction of the front wheels with the road is characterized by the segment A3 A4 in Fig. I. 2. It is an interesting feature of the car acceleration in the presence of the driving wheels slipping.

Fig. I. 4

24

I. Holonomic Systems

Fig. I. 5

The further check of condition (4.11) allows us to make a conclusion that under the given law of variation of the drive moment (4.13), the new slipping of the driving wheels does not occur till the 50th second. Therefore, in Fig. I. 2 the change of the driving force acting on the driving wheels is characterized by the arc OA1 when time changes from t0 till t1 ; at the instant t1 we have the force jump A1 A2 caused by the difference between the static and dynamic coefficients of Coulomb friction (traction coefficients); when time changes from t1 till t2 the driving force is described by the arc A2 A3 , which is almost horizontal. At the instant of time t2 the jump A3 A4 of the driving force occurs; and the force Fdr changes according to the segment A4 A5 when t > t2 . The change of the generalized coordinates y, ϕ is shown in Fig. I. 4, and that of the coordinates x, ϕ1 is shown in Fig. I. 5. We see that at the stage of the driving wheels slipping, the coordinate ϕ1 rapidly increases. Note that, if tyres are compliant and the traction coefficient depends continuously on the speed, the curves in figures I. 2 and I. 3 will be replaced with continuous ones. However, a sharp change of the driving force and acceleration at the instants t1 and t2 remains.

C h a p t e r II NONHOLONOMIC SYSTEMS

From the analog of Newton’s law, Maggi’s equations are deduced which are the most convenient equations of the nonholonomic mechanics. From Maggi’s equations the most useful forms of equations of motion of nonholonomic systems are obtained. The connection between Maggi’s equations and the Suslov–Jourdain principle is considered. The notion of ideal nonholonomic constraints is discussed. In studying nonholonomic systems the approach, applied in Chapter I to analysis of the motion of holonomic systems, is employed. The role of of Chetaev’s type constraints for the development of nonholonomic mechanics is considered. For the solution of a number of nonholonomic problems, the different methods are applied.

§ 1. Nonholonomic constraint reaction Consider the Cartesian coordinates Ox1 x2 x3 with the unit vectors i1 , i2 , i3 . If on the motion of material point of the mass m it is imposed the nonholonomic constraint ϕ(t, x, x) ˙ = 0,

x = (x1 , x2 , x3 ) ,

(1.1)

then the second Newton’s law can be represented as mw = F + R′ ,

(1.2)

where F = (X1 , X2 , X3 ) is an active force, acting on the point, and R′ = (R1′ , R2′ , R3′ ) is constraint reaction (1.1). Consider the vector R′ . We differentiate equation of constraint (1.1) with respect to time: ϕ˙ ≡

∂ϕ ∂ϕ ∂ϕ x˙ k + x ¨k = 0 , + ∂t ∂xk ∂ x˙ k

Together with the usual vector ∇ ϕ =

k = 1, 2, 3 .

(1.3)

∂ϕ ik we introduce the new vector ∂xk

∇ ′ ϕ proposed by N. N. Polyakhov [185]: ∇′ϕ =

∂ϕ ik . ∂ x˙ k

Then equation (1.3) can be rewritten as ∂ϕ + ∇ϕ · v + ∇′ϕ · w = 0 . ∂t 25

(1.4)

26

II. Nonholonomic Systems

Multiplying scalarly equation (1.2) by ∇ ′ ϕ and equation (1.4) by m, we obtain R′ · ∇ ′ ϕ = −m ′

 ∂ϕ ∂t

 + ∇ϕ · v − F · ∇′ϕ .

This implies that the vector R can be represented in the form ∇ ′ ϕ + T0 = N + T 0 , R′ = Λ∇ ∂ϕ ∇ϕ · v + F · ∇ ′ ϕ + m∇ m , T0 · N = 0 . Λ = − ∂t ∇′ ϕ|2 |∇

(1.5)

Note that the only component N of constraint reaction depends on (1.1), in which case by formulas (1.5) it is defined as a certain function of t, x, x. ˙ In particular, equations (1.1) and (1.2) are also valid for T0 = 0. The nonholonomic constraints of such type we shall called ideal. If T0 = 0, then the construction of the vector T0 should be described separately, based on the additional characteristics of the physical realization of constraint (1.1). As a rule, T0 essentially depends on the quantities |N| and, in lesser degree, on t, x, x. ˙ Consider the partial case of holonomic constraint, namely f (t, x) = 0 .

(1.6)

Represent it in the form of (1.1): ∂f ∂f ϕ ≡ f˙ = + x˙ k = 0 . ∂t ∂xk Then we have

∂ϕ ∂f = , ∂ x˙ k ∂xk

and therefore for holonomic constraint (1.6) the vector ∇ ′ ϕ, introduced above, coincides with the usual vector ∇ f . Here, as is shown in Chapter I, the vector N is directed along a normal to the surface, given by equation (1.6), and the vector T0 lies in the plane tangential to this surface. In particular, if equation (1.6) gives a certain material surface, on which the point must move, then for the ideally burnished surface (for ideal holonomic constraint) we have T0 = 0. Otherwise we need to point out a rule for construction of the vector T0 , for example, to give Coulomb’s law (1.12) from Chapter I. Assume now that on the motion of material point it is imposed two nonholonomic constraints ˙ = 0, ϕκ (t, x, x)

x = (x1 , x2 , x3 ) ,

κ = 1, 2 .

Arguing as above, we obtain ∂ϕκ + ∇ ϕκ · v + ∇ ′ ϕκ · w = 0 , ∂t

κ = 1, 2 .

27

1. Nonholonomic constraint reaction

The differential equation of motion has, as before, the form of (1.2). This law permits us to eliminate the vector w from the previous relations and to write them as   ∂ϕκ ∇ ϕ κ · v + F · ∇ ′ ϕκ , κ = 1, 2 . R′ · ∇ ′ ϕκ ≡ R′κ = − m + m∇ ∂t This implies that if we represent the vector R′ as the sum R′ = Λκ ∇ ′ ϕκ + T0 ,

(1.7)

where T0 is a certain unknown vector orthogonal to the vectors ∇ ′ ϕκ , then the coefficients Λκ can be found from the following system of equations ∇′ ϕ1 |2 + Λ2∇ ′ ϕ1 · ∇ ′ ϕ2 = R′1 , Λ1 |∇ ∇′ ϕ2 |2 = R′2 . Λ1∇ ′ ϕ1 · ∇ ′ ϕ2 + Λ2 |∇ Thus, the components Λκ ∇ ′ ϕκ of the vector R′ are uniquely defined by equations of constraints and the force F. We remark that a similar reasoning can also be used for two holonomic constraints since in this case we have ∇′ ϕκ = ∇f κ . Therefore if there are the two holonomic constraints, then the reaction can be represented as R′ = Λκ ∇ f κ + T0 ,

κ = 1, 2 .

We consider now the motion of representation point under the condition that there exist k nonholonomic constraints: ˙ = 0, ϕκ (t, y, y)

κ = 1, k .

(1.8)

Then like the motion of one material point we can write MW = Y + R ,

(1.9)

which has the form of the second Newton’s law in the vector form. In the sequel relation (1.9) is called the second Newton’s law just as in Chapter I. Using formula (1.7) in the case of representation point, we have R = N + T0 ,

N = Λκ ∇ ′ ϕκ ,

∇ ′ ϕκ =

∂ϕκ jµ , ∂ y˙ µ

T0 · N = 0 .

(1.10)

In Chapter IV the notion of a tangent space to the manifold of all possible configurations of a mechanical system will be introduced. The set of the Lagrange equations of the second kind describing a motion of the unconstrained mechanical system is written in this space as a single vector-valued equality which has a form of the second Newton law. This makes it possible to generalize formulas (1.5), (1.7) not only to mechanical systems, consisting of the finite number of particles, but also to arbitrary mechanical systems with the finite number of degrees of freedom.

28

II. Nonholonomic Systems

Thus, the conclusion on the structure of the constraint reaction obtained for a single particle is of general nature. It is fundamental. Expressions (1.5), (1.7), (1.10) for the reaction force were obtained by N. N. Polyakhov in 1974 [185]. Later these results were included into the treatise (manual for universities) [189]. In 2001 O. M. O’Reilly and A. R. Srinivasa [416] devoted their work to deriving and discussion of expressions (1.5).

§ 2. Equations of motion of nonholonomic systems. Maggi’s equations Assume that the nonlinear nonholonomic constraints, imposed on a motion of system, in the curvilinear coordinates q = (q 1 , . . . , q s ) have the form ϕκ (t, q, q) ˙ = 0,

κ = 1, k .

(2.1)

In the case of the motion of system of N material points s = 3N . Now we pass from the variables q˙ = (q˙1 , . . . , q˙s ) to the new nonholonomic variables v∗ = (v∗1 , . . . , v∗s ) by formulas v∗ρ = v∗ρ (t, q, q) ˙ ,

ρ = 1, s .

(2.2)

If the solvability conditions are satisfied, then we can write the inverse transformation q˙σ = q˙σ (t, q, v∗ ) , σ = 1, s . (2.3) Assuming that the derivatives of functions (2.2), (2.3) are continuous, we can introduce the two systems of linearly independent vectors: ετ =

∂ q˙σ eσ , ∂v∗τ

ερ =

∂v∗ρ τ e , ∂ q˙τ

ρ, τ = 1, s .

(2.4)

Since the product is as follows ∂v∗ρ ∂ q˙σ ε · ε τ = σ τ = δτρ = ∂ q˙ ∂v∗ ρ

 0, 1,

ρ = τ , ρ=τ,

vectors (2.4) can be regarded as the vectors of the fundamental and reciprocal bases. We shall say that bases (2.4) are nonholonomic. By assumption, the equations of constrains (2.1) are such that ∗

∇ ′ ϕκ · ∇ ′ ϕκ | = |∇  0,

κ, κ ∗ = 1, k .

In this case in transition formulas (2.2) the last functions can be given in the following way ˙ , v∗l+κ = ϕκ (t, q, q)

l = s−k,

κ = 1, k .

(2.5)

2. Equations of motion of nonholonomic systems Maggi’s equations

29

Therefore if constraint (2.1) is satisfied, then we have v∗l+κ = 0. Then by formulas (2.4) we have ε l+κ =

∂ϕκ τ e ≡ ∇ ′ ϕκ , ∂ q˙τ

κ = 1, k .

We introduce two subspaces orthogonal to each other with the nonholonomic bases {εε1 , . . . , ε l } and {εεl+1 , . . . , ε s } and call them L-space and Kspace, respectively. Decompose the acceleration vector into the following two components W = WL + WK ,

 λε λ , WL = W

l+κ ε l+κ , WK = W

WL · WK = 0 .

Here the wavy sign denotes that the components of acceleration vector are taken for nonholonomic bases (2.4) but not for the usual fundamental and reciprocal bases. The second Newton’s law (1.9) is replaced then by the following two equations: (2.6) M WL = YL + RL , M WK = YK + RK .

(2.7)

Differentiating the equations of constraints (2.1) with respect to time and taking into account that the vector W can be represented as W = (¨ q σ + Γσαβ q˙α q˙β )eσ , we obtain

σ = 1, s ,

α, β = 0, s ,

ε l+κ · W = χκ ˙ , 1 (t, q, q) κ κ ∂ϕ ∂ϕ σ ∂ϕκ σ α β ˙ =− q˙ + Γ q˙ q˙ , χκ − 1 (t, q, q) ∂t ∂q σ ∂ q˙σ αβ κ = 1, k , α, β = 0, s .

These equations are similar to equations (2.14) of Chapter I. This implies that the vector WK as the function of t, q σ , q˙σ , σ = 1, s is uniquely determined by constraints equations. By equation (2.7) for the given force YK the vector WK can be obtained by means of the constraint reaction RK = N = Λκ ∇ ′ ϕκ . Unlike the above the component WL is independent of the mathematical definition of the equations of constraints. It can be determined from equation (2.6) for any vector RL , in particular, for RL ≡ T0 = 0 if in L-space the equation of properly motion has the form M WL = YL . It is naturally to call nonholonomic constraints (2.1), which do not influence the vector WL , ideal. For these constraints the vector of reaction is as follows R = RK = N = Λκ ∇ ′ ϕκ .

(2.8)

30

II. Nonholonomic Systems

By formulas (1.9) and (2.8) the second Newton’s law for ideal nonholonomic constraints has the form M W = Y + Λ κ ∇ ′ ϕκ .

(2.9)

Multiplying this equation by the vectors ε λ , λ = 1, l, we obtain Maggi’s equations ∂ q˙σ  = 0, λ = 1, l , (2.10) M Wσ − Qσ ∂v∗λ where d ∂T ∂T M Wσ − Qσ = − σ − Qσ , σ = 1, s . dt ∂ q˙σ ∂q For linear nonholonomic constraints these equations have been obtained by Maggi in 1896 [355]. Later for nonlinear nonholonomic constraints, by means of the generalized principle of D’Alembert–Lagrange they have been generated by A. Przeborski [375]. Integrating equations (2.1), (2.10) with the given initial data, we can find the law of motion of the system q σ = q σ (t) ,

σ = 1, s .

(2.11)

Multiplying equation (2.9) by the vectors ε l+κ , κ = 1, k, we obtain the second group of Maggi’s equations  ∂ q˙σ M Wσ − Qσ = Λκ , ∂v∗l+κ

κ = 1, k .

(2.12)

For the given law of motion of system (2.11), the generalized reactions of nonholonomic constraints (2.1) can be determined as the time functions from the above equations. Formulas (2.12) do not give directly the quantities Λκ as the functions of t, q, q. ˙ They can be found from the following equations ε l+κ · W = χκ ˙ 1 (t, q, q),

WK =

1 (YK + Λκ ∇ ′ ϕκ ) . M

Thus, for nonholonomic constraints the introduction of nonholonomic bases (2.4) permits us to obtain the two subspaces K and L. These subspaces turn out orthogonal to each other and in studying the problems in these subspaces it is convenient to make use of Maggi’s equations (2.10) and (2.12). Maggi’s equations are highly convenient to consider the motion of nonholonomic systems. It is to be noted that they are valid for any nonholonomic constraints, including the nonlinear ones. Most of the well-known forms of equations, describing the motion of nonholonomic systems, can be obtained from these equations (for detail, see the next section), for example, Chaplygin’s equations   l+κ ∗ ∂bλ∗ ∂bl+κ ∂T d ∂(T ) ∂(T ) λ q˙λ = Qλ , − + − ∗ λ λ l+κ λ λ dt ∂ q˙ ∂q ∂ q˙ ∂q ∂q (2.13) ∗ κ = 1, k , λ, λ = 1, l ,

2. Equations of motion of nonholonomic systems Maggi’s equations

31

if the equations of constraints (2.1) are represented as 1 l λ q˙l+κ = bl+κ λ (q , . . . , q )q˙ ,

λ = 1, l ,

κ = 1, k ,

(2.14)

or the Hamel–Boltzmann equations ∗ d ∂T ∗ ∂T ∗ ∂T ∗ λ , − + γλ(l+κ)λ∗ π˙ λ = Q λ λ l+κ dt ∂ π˙ ∂π ∂ π˙ κ = 1, k , l = s−k, λ, λ∗ = 1, l, ,

(2.15)

for the nonholonomic constraints of the form σ ϕκ (t, q, q) ˙ ≡ al+κ σ (q)q˙ = 0,

κ = 1, k ,

σ = 1, s ,

(2.16)

if in place of formulas (2.2), (2.3) there are introduced the following relations between the time derivatives of the generalized coordinates q 1 , . . . , q s and of the quasicoordinates π 1 , . . . , π s : π˙ ρ = aρσ (q)q˙σ ,

q˙σ = bσρ (q)π˙ ρ ,

ρ, σ = 1, s .

(2.17)

In Chaplygin’s equations the symbol (T ) denotes, as usual [59], the kinetic energy in which the generalized velocities q˙l+κ , κ = 1, k, are replaced by relations (2.14). Similarly, in the Hamel–Boltzmann equations T ∗ denotes the kinetic energy if in it the quantities q˙σ , σ = 1, s, are replaced by their relations in unknowns π˙ ρ , ρ = 1, s. Recall that the analytic representations of nonholonomic constraints (2.16) give k last quasivelocities π˙ l+1 , . . . , π˙ s in λ , formulas (2.17). Besides, equations (2.15) involves the generalized forces Q which correspond to the quasivelocities π˙ λ (λ = 1, l) : σ

 λ = Qσ ∂ q˙ , Q ∂ π˙ λ

λ = 1, l ,

σ = 1, s ,

(2.18)

and the objects of nonholonomicity γλ(l+κ)λ∗   l+κ ∂aσ ∂al+κ τ σ τ , − γλ(l+κ)λ∗ = bλ bλ∗ ∂q τ ∂q σ λ, λ∗ = 1, l ,

κ = 1, k ,

(2.19)

σ, τ = 1, s .

The derivatives ∂T ∗ /∂π λ are computed by formulas ∂T ∗ ∂ q˙σ ∂T ∗ = , ∂π λ ∂q σ ∂ π˙ λ

λ = 1, l ,

σ = 1, s .

The following equations [169, 314]   d ∂T ∗ ∂T ∗ ∂T d ∂ q˙σ ∂ q˙σ λ , =Q − − σ − dt ∂v∗λ ∂π λ ∂ q˙ dt ∂v∗λ ∂π λ

(2.20)

λ = 1, l

are more general than Chaplygin’s equations. Whence it follows that in the case of linear stationary transformations (2.17) with respect to quasivelocities

32

II. Nonholonomic Systems

and generalized velocities we can obtain Chaplygin’s equations. Therefore V. S. Novoselov calls the above equations the equations of Chaplygin’s type. Similarly, the equations more general than the Hamel–Boltzmann ones are the equations of Hamel–Novoselov [169, 314]   ∂T ∗ ∂T ∗ ∂ q˙σ d ∂v∗ρ ∂v∗ρ d ∂T ∗ λ , =Q − + − λ = 1, l , dt ∂v∗λ ∂π λ ∂v∗ρ ∂v∗λ dt ∂ q˙σ ∂q σ

which are obtained also for nonlinear constraints (2.1). V. S. Novoselov calls these equations the equations of the Voronets–Hamel type (for detail, see § 1 Chapter VII). In the equations of Chaplygin, Hamel–Boltzmann, and those similar to them, the authors have made an attempt to discriminate the Lagrange operator. Then the rest of addends in the left-hand sides of equations characterize the nonholonomicity of system. Therefore in the case of the integrability of constraints the differential equations become usual Lagrange’s equations of the second kind of holonomic mechanics. Equations (2.13), (2.15), and the similar ones are generated for the concrete forms of usually linear nonholonomic constraints of the type (2.14), (2.16) and therefore they are useful for solving the corresponding problems. As a rule, such equations make it possible to obtain the minimal number of equations of motion. For example, the left-hand sides of Chaplygin’s equations (2.13) involve only the unknown q 1 , . . . , q l and after integration of these equations the rest of the coordinates q l+1 , . . . , q s can be found from equations of constraints (2.14). As distinct from this, Maggi’s equations are valid, as is mentioned above, for any of nonholonomic constraints, including the constraints nonlinear in generalized velocities. An important point is that for generating differential equations of motion (2.10) we need to apply a single equitype technique to all problems: after the choice of the generalized coordinates q 1 , . . . , q s the expressions for the left-hand sides of usual Lagrange’s equations of the second kind are generated; the transition formulas (2.2) to nonholonomic variables are introduced, in which case the last of them take account of the relations of nonholonomic constraints by means of formulas (2.5); the inverse transformation (2.3) is found and after differentiating it with respect to new nonholonomic variables, the equations of motion (2.10) are generated. Here two remarks can be given which are useful from the computational point of view. Firstly, for numerical integrating system (2.10) together with constraints (2.1) it is necessary previously to differentiate the latter with respect to time and to obtain the equations linear in generalized accelerations. These equations and Maggi’s ones are the system of linear nonuniform algebraic equations in unknown q¨1 , . . . , q¨s . Solving them, we obtain the system of differential equations convenient for numerical integration. Secondly, in the case of nonlinear nonholonomic constraints (2.1) the obtaining of inverse transformations (2.3) from formulas (2.2) may turn out difficult. To avoid this we need to to compile the matrix of derivatives (∂v∗ρ /∂ q˙σ ),

2. Equations of motion of nonholonomic systems Maggi’s equations

33

ρ, σ = 1, s, using formulas (2.2), and to find then the inverse matrix (∂ q˙σ /∂v∗ρ ), ρ, σ = 1, s, the elements of which are used for generating Maggi’s equations. Consider one more type of equations of nonholonomic mechanics. In the case of ideal constraints (2.1) equation (2.9) can be represented as M W = Y + Λκ

∂ϕκ τ e . ∂ q˙τ

(2.21)

Multiplying scalarly equation (2.21) by the vectors of fundamental basis eσ , σ = 1, s, of the original system of curvilinear coordinates, we obtain the following relation M Wσ = Qσ + Λκ

∂ϕκ , ∂ q˙σ

σ = 1, s ,

which can be rearranged in the form d ∂T ∂T ∂ϕκ − σ = Qσ + Λκ σ , σ dt ∂ q˙ ∂q ∂ q˙

σ = 1, s .

(2.22)

These equations are usually called Lagrange’s equations of the second kind with undetermined multipliers [59]. Together with the equations of nonholonomic constraints (2.1) they give a system of s + k differential equations in s + k unknowns q σ , σ = 1, s, Λκ , κ = 1, k. This is the reason why, like equations (2.22) of Chapter I, they can be called Lagrange’s equations of the first kind in generalized coordinates for nonholonomic systems [28]. If the original system of coordinates are Cartesian, then we have q σ = yσ ,

eσ = eσ = jσ ,

ϕ (t, y, y) ˙ = 0,

σ = 1, s , κ = 1, k ,

κ

and equations (2.22) take the form M y¨σ = Yσ + Λκ

∂ϕκ , ∂ y˙ σ

σ = 1, s .

(2.23)

Equations (2.23) are usual Lagrange’s equations of the first kind for nonholonomic constraints rearranged for representation point. F. Udwadia and R. Kalaba [394. 1992] derived the equations of dynamics in the matrix form taking into consideration the presence of nonholonomic constraints up to the second order with the help of the Moore and Penrose generalized inverse. In their opinion "the equations of motion obtained in this paper appear to be the simplest and most comprehensive so far discovered". Note that the partition of the whole s-dimensional space into the direct sum of the K-space and L-space by means of constraint equations (2.1) actually corresponds to the application of Moore and Penrose generalized inverse. (A more general case for the constraints that are linear in generalized accelerations is considered in Chapter IV). This partition led to expressions

34

II. Nonholonomic Systems

(2.12) for generalized reactions. Substituting them into equations (2.22), we obtain ˙ q¨τ = Bσ (t, q, q) ˙ , Aστ (t, q, q)   ∗ ∂ q˙σ ∂ϕκ , Aστ = M gστ − gσ∗ τ ∂v∗l+κ ∂ q˙σ ∗ σ∗ ∂ϕκ ∂ q˙σ ∂ϕκ α β ∂ q˙ ∗ Bσ = Qσ − Qσ∗ l+κ + M Γ q ˙ q ˙ − σ ,αβ l+κ σ ∂ q˙ ∂ q˙σ ∂v∗ ∂v∗ σ, σ ∗ , τ = 1, s , α, β = 0, s , κ = 1, k . −M Γσ,αβ q˙α q˙β , These formulae do imply the Udwadia–Kalaba equations q¨τ = Aτ σ (t, q, q) ˙ Bσ (t, q, q) ˙ ,

σ, τ = 1, s ,

where Aτ σ are elements of the matrix inverse to the matrix (Aστ ). We note that these equations can be also derived with the help of the linear force transformation, which will be introduced in the next chapter, and elimination of generalized reaction forces Λκ , κ = 1, k, from equations (2.22) in a simillar way as it was described for holonomic systems in § 2 of Chapter I. V. V. Rumyantsev [203] thinks that the most general equations of nonholonomic mechanics are the Poincar´e–Chetaev equations. They were introduced by H. Poincar´e [373] and N. G. Chetaev [247, 248, 292] for holonomic systems. The mathematical problems, concerning their structure, and their place in the new theory of Hamiltonian systems were considered by V. I. Arnol’d, V. V. Kozlov, A. I. Neishtadt [7], L. M. Markhashov [149], and others. In the sequel they were generalized to nonholonomic systems due to the works of L. M. Markhashov [149], V. V. Rumyantsev [203], and Fam Guen [229]. As is shown by V. V. Rumyantsev [203], all the rest of types of the equations of motion for nonholonomic mechanics with the linear nonholonomic constraints of the first kind can be obtained from the Poincar´e–Chetaev equations. We assume that these constraints have the form l+κ σ (t, q) = 0 , v∗l+κ = al+κ σ (t, q)q˙ + a0

κ = 1, k ,

σ = 1, s .

(2.24)

Arguing as in the work [203], we supplement equations (2.24) with the following relations v∗λ = aλσ (t, q)q˙σ + aλ0 (t, q) ,

λ = 1, l ,

l = s−k,

σ = 1, s ,

which implies that the generalized velocities can uniquely be represented as q˙σ = bστ (t, q)v∗τ + bσ0 (t, q) ,

σ, τ = 1, s .

Introducing, for short, the notions [203] q0 = t ,

q˙0 = 1 ,

v∗0 = 1 ,

a0α = b0α = δα0 ,

α = 0, s ,

2. Equations of motion of nonholonomic systems Maggi’s equations

35

we have β v∗α = aα β q˙ ,

q˙β = bβα v∗α ,

α, β = 0, s .

Denote the Lagrange function L = T − Π, which was found as the function of variables t, q σ , v∗σ , σ = 1, s, by L∗ (t, q, v∗ ). In these notions the Poincar´e– Chetaev equations for nonholonomic systems with constraints (2.24) are the following [203]: d ∂L∗ ∂L∗ ∂L∗ ∂L∗ λ , = cρµλ v∗µ ρ + cρ0λ ρ + bσλ σ + Q λ dt ∂v∗ ∂v∗ ∂v∗ ∂q λ, µ = 1, l, ,

(2.25)

ρ, σ = 1, s .

 λ = bσ Qσ are generalized nonpotentional forces, corresponding to the Here Q λ quasivelocities v∗λ , λ = 1, l, and cρµα and cρ0λ are the coefficients, given in the form  γ   ρ  ∂bβ δ ∂aγ ∂aρδ γ δ ∂bγα δ ρ ρ b − b − γ b α bβ , = cαβ = aγ ∂q δ α ∂q δ β ∂q δ ∂q (2.26) α, β, γ, δ = 0, s , ρ = 1, s . As V. V. Rumyantsev emphasizes [203], the function L∗ , entering into equations (2.25), depends, generally speaking, on all quasivelocities v∗ρ , ρ = 1, s, and therefore the equations of constraints (2.24) v∗l+κ = 0, κ = 1, k, should be used only after the generation of equations (2.25). By (2.26) for holonomic constraints we have cl+κ αβ = 0, κ = 1, k, α, β = 0, s, and in this case the above remark does not refer to holonomic systems. Equations (2.25), supplemented with the equations q˙σ = bσλ (t, q)v∗λ + bσ0 (t, q) ,

λ = 1, l ,

σ = 1, s ,

are the closed system of equations in unknowns q σ , σ = 1, s, and v∗λ , λ = 1, l. The number of sought independent variables is minimal and the differential equations in unknowns as q σ as v∗λ are of the first kind. This is the advantage of equations (2.25) in contrast with Maggi’s equations. The Hamel–Novoselov and Poincar´e–Chetaev equations are also considered in Chapter VII, where they are obtained by three different approaches. Finally, we give some important remarks. In studying the motion of rigid body linear nonholonomic constraint (2.24) occurs, in particular, in the case when the projection of the velocity v of the point M of rigid body on the direction of the unit vector j of body is equal to zero by virtue of its interaction with another body. This example of nonholonomic constraint is the most routine one. Therefore we consider it in more detail. We shall show that the assumption that the considered constraint is ideal means that the force, applied to the point M of body in the result of its interaction with another body, is equal to Λj if the equation of constraint is as follows ϕ = v · j = aσ (t, q)q˙σ + a0 (t, q) = 0 ,

σ = 1, s ,

s  6.

36

II. Nonholonomic Systems

Here v is the velocity of the point M and q σ ,σ = 1, s, are generalized coordinates of rigid body. By assumption, the constraint is linear and therefore the unit vector j can only depend on the time t and on the generalized coordinates q σ , σ = 1, s, but not on the generalized velocities q˙σ , σ = 1, s. Equations (2.22) implies that for the proof of such assertion it is sufficient to show that the generalized forces Rσ , corresponding to the force Λj, can be represented in the form ∂ϕ Rσ = Λ σ . ∂ q˙ Really, by definition, we have Rσ = Λj ·

∂r , ∂q σ

where r = r(t, q) is the radius vector of the point M . The velocity v of the point M is as follows ∂r ∂r v= + σ q˙σ . ∂t ∂q Hence

∂v ∂r = σ, σ ∂q ∂ q˙

and we have Rσ = Λj ·

∂v , ∂ q˙σ

σ = 1, s ,

σ = 1, s .

The vector j is independent of the variables q˙σ , σ = 1, s. In this case the quantities Rσ , σ = 1, s, can be represented as Rσ = Λ

∂(v · j) ∂ϕ =Λ σ , ∂ q˙σ ∂ q˙

σ = 1, s ,

which was to be proved. Consider another example from the dynamics of rigid body, related to the problem on the controllable motion of rigid body but not the problem on its rolling or sliding motion. We assume that it is necessary to realize the free motion, of rigid body, such that the projection of the vector of instantaneous angular velocity ω on the fixed axis j is a given function of time t and the generalized coordinates q σ , σ = 1, 6. Thus equation of linear nonholonomic constraint (2.24) is assumed to be given in the form ϕ = ω · j + a0 (t, q) = 0 .

(2.27)

We shall show that from equations (2.22) it follows that the "ideal"realization of such program of motion is possible by means of an additional system of

2. Equations of motion of nonholonomic systems Maggi’s equations

37

forces such that its force resultant is equal to zero and the resultant moment about the center of mass is equal to Λj. Suppose, ρ ν are the radius-vectors of the points of application of the additional forces Fν , the number of which is equal to n. By definition, we have Rσ =

n 

ν=1

n

Fν ·

n

  ∂ρρν ∂vν ∂ = Fν · σ = Fν · σ (vC + ω × rν ) . σ ∂q ∂ q ˙ ∂ q˙ ν=1 ν=1

(2.28)

Here vC is a velocity of the center of mass and rν is a radius vector of the point of application of the additional force Fν relative to the center of mass. By virtue of the problem setting we have n 

Fν = 0 ,

ν=1

n 

rν × Fν = Λj .

(2.29)

ν=1

Therefore relations (2.27) and (2.28) yield Rσ =

n 

ν=1

=

n 

n

Fν ·

ω × rν )  ω ∂(ω ∂ω = Fν · σ × rν = ∂ q˙σ ∂ q ˙ ν=1

(rν × Fν ) ·

ν=1

ω ω ∂ω ∂ω ∂ϕ = Λj · σ = Λ σ . ∂ q˙σ ∂ q˙ ∂ q˙

In the above proof the fact that the unit vector j is that of fixed frame is not used. We need only in the fact that this vector is independent of the generalized velocities q˙σ , σ = 1, 6. It can be a vector, which is of a given dependence of time and generalized coordinates, and therefore it can, in particular, be the unit vector, rigidly connected with body. The essential distinction between the considered problem and the problem on the rolling or sliding motion of rigid body is that the validity of constraint (2.27) can be provided by different families of the additional forces Fν ,ν = 1, n, satisfying condition (2.29) while in the problem on the rolling or sliding motion the validity of constraint is provided by the one additional force Λj, applied to the contact point M . It is also important that the generation of this force as the function of the variables t, q, q˙σ , σ = 1, s, is given by the contact interaction of two bodies. This force can be eliminated and the motion can be found from the equations of constraints and, for example, from Maggi’s equations, which do not involve the Lagrange multipliers. For controllable motion, the generation of the moment Λj is realized by the control system and only after applying the found control moment Λj the motion can satisfy equation (2.27).

38

II. Nonholonomic Systems § 3. The generation of the most usual forms of equations of motion of nonholonomic systems from Maggi’s equations

We obtain now the mentioned above forms of equations of motion of nonholonomic systems from Maggi’s equations. Chaplygin’s and Voronets’ equations. Suppose that on the system are imposed the stationary linear nonholonomic constraints, the equations of which take the form q˙l+κ = βλl+κ (q)q˙λ ,

λ = 1, l,

κ = 1, k .

(3.1)

Then, assuming v∗λ = q˙λ , v∗l+κ

= q˙

l+κ

we have ∂ q˙µ = δλµ = ∂v∗λ

−

λ = 1, l,

βλl+κ (q)q˙λ ,

 1, 0,

∂ q˙l+κ = βλl+κ , ∂v∗λ

κ = 1, k ,

µ = λ, µ = λ , λ = 1, l,

λ, µ = 1, l , κ = 1, k .

From these relations it follows that for nonholonomic constraints, given by (3.1), Maggi’s equations (2.10) can be represented as M wλ + M wl+κ βλl+κ = Qλ + Ql+κ βλl+κ , λ = 1, l,

κ = 1, k .

(3.2)

Suppose that the kinetic energy T is independent of the generalized coordinates q l+κ and Ql+κ = 0 (κ = 1, k). Then equations (3.2) have the form ∂T d ∂T d ∂T − λ + βλl+κ = Qλ , λ = 1, l . (3.3) λ dt ∂ q˙ ∂q dt ∂ q˙l+κ Transform equation (3.3). By means of equation of constraints (3.1), we eliminate all velocities q˙l+κ from the relation for the kinetic energy T , and denote by T∗ the obtained expression for kinetic energy. In this case the relations hold ∂T∗ ∂T ∂T ∂ q˙l+κ ∂T ∂T = + = λ + l+κ βλl+κ , (3.4) ∂ q˙λ ∂ q˙λ ∂ q˙l+κ ∂ q˙λ ∂ q˙ ∂ q˙ ∂T ∂T ∂ q˙l+κ ∂T ∂T ∂βµl+κ µ ∂T∗ = + = + q˙ , ∂q λ ∂q λ ∂ q˙l+κ ∂q λ ∂q λ ∂ q˙l+κ ∂q λ λ, µ = 1, l .

(3.5)

3. The generation of the most usual forms of equations

39

We assume that the coefficients βλl+κ are independent of q l+κ , κ = 1, k. Then, differentiating in time relation (3.4), we obtain d ∂T∗ d ∂T ∂T dβλl+κ l+κ d ∂T = = + β + λ dt ∂ q˙λ dt ∂ q˙λ dt ∂ q˙l+κ ∂ q˙l+κ dt =

∂T ∂βλl+κ µ d ∂T l+κ d ∂T + β + q˙ , λ dt ∂ q˙λ dt ∂ q˙l+κ ∂ q˙l+κ ∂q µ λ, µ = 1, l .

(3.6)

Computing the quantities d(∂T /∂ q˙λ )/dt and ∂T /∂q λ by formulas (3.6) and (3.5) and substituting them into equations (3.3), we get d ∂T∗ ∂T∗ ∂T − λ − l+κ dt ∂ q˙λ ∂q ∂ q˙ κ = 1, k ,



 ∂βµl+κ µ ∂βλl+κ q˙ = Qλ , − ∂q µ ∂q λ

(3.7)

λ, µ = 1, l .

These equations were obtained by S. A. Chaplygin [239]. Now we eliminate the dependent velocities q˙l+1 , q˙l+2 , ..., q˙l+k from the expressions ∂T /∂ q˙l+κ in equations (3.7), using equations of constraints (3.1). Then we get the system of l equations in unknown functions q 1 , q 2 , ..., q l . Thus, Chaplygin’s equations permit us to determine q 1 (t), q 2 (t), ..., q l (t) independently of constraints (3.1) and to find then the rest of q l+1 (t), q l+2 (t), ..., q l+k (t) from equations (3.1). Suppose, the coefficients βλl+κ satisfy the following conditions ∂βµl+κ ∂βλl+κ − = 0, ∂q λ ∂q µ

κ = 1, k ,

λ, µ = 1, l .

(3.8)

According to the assumption that the coefficients βλl+κ are independent of q l+κ (κ = 1, k) this implies that they take the form βλl+κ =

∂U l+κ , ∂q λ

λ = 1, l ,

κ = 1, k .

(3.9)

Here U l+κ are the functions of coordinates q 1 , q 2 , ..., q l . Substituting relations (3.9) into equations (3.1), we obtain q l+κ = U l+κ (q 1 , q 2 , ..., q l ) ,

κ = 1, k .

Thus, the coordinates q l+κ result from the rest. Therefore if conditions (3.8) are satisfied the motion is described by usual Lagrange’s equations. Now we generate the equations of motion, obtained by P. V. Voronets [41. 1901]. Consider a mechanical system with constraints given in the form (3.1) without the additional assumptions, which arrive to Chaplygin’s equations.

40

II. Nonholonomic Systems

In the case when the kinetic energy T depends on all coordinates Maggi’s equations (3.2) are the following   d ∂T d ∂T ∂T ∂T βλl+κ = Qλ + Ql+κ βλl+κ , − + − dt ∂ q˙λ ∂q λ dt ∂ q˙l+κ ∂q l+κ (3.10) λ = 1, l . κ = 1, k , Arguing as above, we reduce these equations to the Voronets ones. Relations (3.5) preserve their form. In accordance with that the coefficients βλl+κ depend now on all q σ , relations (3.6) take the form d ∂T∗ d ∂T ∂T ∂βλl+κ µ l+κ d ∂T = + β + q˙ + λ dt ∂ q˙λ dt ∂ q˙λ dt ∂ q˙l+κ ∂ q˙l+κ ∂q µ ∂T ∂β l+κ + l+κ λl+ν βµl+ν q˙µ , ∂ q˙ ∂q

κ, ν = 1, k ,

(3.11)

λ, µ = 1, l .

In the considered case, together with (3.5) and (3.11) we need to account for the following relations   ∂T ∂T ∂βµl+ν µ l+κ ∂T∗ l+κ βλ = βλ + l+ν l+κ q˙ . ∂q l+κ ∂q l+κ ∂ q˙ ∂q This relation, together with relations (3.5) and (3.11), permits us to represent equations (3.10) as ∂T∗ ∂T ∂T∗ d ∂T∗ l+κ µ − λ − βλl+κ l+κ − l+κ βλµ q˙ = dt ∂ q˙λ ∂q ∂q ∂ q˙ = Qλ + where l+κ βλµ =

Ql+κ βλl+κ

,

λ, µ = 1, l ,

(3.12)

κ = 1, k ,

∂βµl+κ ∂βλl+κ ∂βλl+κ l+ν ∂βµl+κ l+ν − + β − β . ∂q µ ∂q λ ∂q l+ν µ ∂q l+ν λ

Equations (3.12) are called Voronets’ equations. The equations of motion (3.12) together with equations of constraints (3.1) are the system of differential equations for obtaining the functions q σ (t), σ = 1, s. In the case of constrained motion of system acted by forces, which have a potential, equations (3.12) take the form ∂(T∗ + U ) ∂T ∂(T∗ + U ) d ∂T∗ l+κ µ − − βλl+κ − l+κ βλµ q˙ = 0 , dt ∂ q˙λ ∂q λ ∂q l+κ ∂ q˙ λ, µ = 1, l , κ = 1, k . In the partial case when the coordinates q l+1 , q l+2 , ..., q l+k , corresponding to the eliminated velocities, do not enter into the relations for kinetic and potential energies in explicit form and also into the equations of constraints, Voronets’ equations (3.12) coincide with Chaplygin’s equations (3.7).

3. The generation of the most usual forms of equations

41

The equations in quasicoordinates (the Hamel–Novoselov, Voronets–Hamel, and Poincar´ e–Chetaev equations). As is known, the projections of the vector of instantaneous angular velocity ω on the fixed axes cannot be regarded as the derivatives with respect to certain new angles, which uniquely determine the position of rigid body. Similarly, it may turn out that the quantities v∗ρ , which are a one-to-one function of the generalized velocities q˙σ , cannot be regarded as derivatives with respect to the certain new coordinates q∗ρ . Therefore the quantities v∗ρ are called quasivelocities and the variables π ρ , given by formulas  t0 v∗ρ dt , πρ = t

are called quasicoordinates. In the relation for the kinetic energy T the generalized velocities q˙σ are changed by the quasivelocities v∗ρ . The function thus obtained is denoted by T ∗ . Consider, which form can have Maggi’s equations, represented as  σ  d ∂T ∂ q˙ ∂T − − Q = 0, σ = 1, s , λ = 1, l , (3.13) σ σ σ dt ∂ q˙ ∂q ∂v∗λ when used the function T ∗ . Taking into account the relations ∂T ∗ ∂T ∂ q˙σ = , ∂v∗λ ∂ q˙σ ∂v∗λ

∂T ∗ ∂T ∂T ∂ q˙ρ = + , ∂q σ ∂q σ ∂ q˙ρ ∂q σ

ρ, σ = 1, s , we have

λ = 1, l ,

   d ∂T ∂ q˙σ d ∂T ∂ q˙σ = − dt ∂ q˙σ ∂v∗λ dt ∂ q˙σ ∂v∗λ d ∂T ∗ ∂T d ∂ q˙σ ∂T d ∂ q˙σ − σ = − σ , λ λ ∂ q˙ dt ∂v∗ dt ∂v∗ ∂ q˙ dt ∂v∗λ   ∂ q˙σ ∂T ∗ ∂T ∂ q˙ρ ∂T ∂ q˙σ = − ρ σ = ∂q σ ∂v∗λ ∂v∗λ ∂q σ ∂ q˙ ∂q σ ∗ ∂T ∂ q˙ρ ∂ q˙σ ∂ q˙ ∂T − ρ σ λ. = λ σ ∂v∗ ∂q ∂ q˙ ∂q ∂v∗ 

(3.14)

(3.15)

In the double sum in the right-hand side of relation (3.15) we exchange the indices of summing ρ and σ. As a result we have ∂ q˙σ ∂T ∗ ∂T ∂ q˙σ ∂ q˙ρ ∂T ∂ q˙σ = − . ∂q σ ∂v∗λ ∂v∗λ ∂q σ ∂ q˙σ ∂q ρ ∂v∗λ

(3.16)

Consider the operator ∂ ∂ q˙σ ∂ = , ρ ∂π ∂v∗ρ ∂q σ

ρ, σ = 1, s .

(3.17)

42

II. Nonholonomic Systems

Under the assumption v∗ρ = π˙ ρ = q˙∗ρ it passes into the operator of partial derivative with respect to the new coordinate q∗ρ since we have ∂ q˙σ ∂ ∂ q˙σ ∂ ∂q σ ∂ ∂ = = = ρ. ρ ρ σ σ ∂v∗ ∂q ∂ q˙∗ ∂q ∂q∗ρ ∂q σ ∂q∗ By (3.17) relation (3.16) takes the form ∂T ∗ ∂T ∂ q˙σ ∂T ∂ q˙σ = − . ∂q σ ∂v∗λ ∂π λ ∂ q˙σ ∂π λ According to relation (3.14) this implies that Maggi’s equations (3.13) take the form   ∂T ∗ ∂T d ∂ q˙σ ∂ q˙σ d ∂T ∗ − − − = Q∗λ , dt ∂v∗λ ∂π λ ∂ q˙σ dt ∂v∗λ ∂π λ (3.18) σ = 1, s , λ = 1, l . Here Q∗λ = Qσ

∂ q˙σ . ∂v∗λ

(3.19)

Equations (3.18) are called, sometimes, Chaplygin’s type equations [169]. Consider the partial case when the generalized velocities q˙σ and the quasivelocities v∗ρ are related by the following linear uniform stationary relations v∗ρ = ασρ (q)q˙σ ,

q˙σ = βρσ (q)v∗ρ ,

ρ, σ = 1, s ,

(3.20)

and the equations of constraints are the following v∗l+κ ≡ ασl+κ (q)q˙σ = 0 ,

κ = 1, k .

(3.21)

In this case, using relations (3.20) and operator (3.17) and taking into account that after performing the operations of differentiation it can be assumed that v∗l+κ = 0 (κ = 1, k), we have d ∂βλσ ρ ∂βλσ ρ µ d ∂ q˙σ = βλσ (q) = q˙ = β v = λ ρ dt ∂v∗ dt ∂q ∂q ρ µ ∗ ∂ q˙ρ ∂β σ ∂β σ ρ, σ = 1, s , λ, µ = 1, l ; = v∗µ µ λρ = v∗µ λµ , ∂v∗ ∂q ∂π ∂ q˙σ ∂ q˙ρ ∂ q˙σ ∂ q˙ρ ∂βµσ µ = = v = ∂π λ ∂v∗λ ∂q ρ ∂v∗λ ∂q ρ ∗ ∂βµσ = v∗µ λ , ρ, σ = 1, s , λ, µ = 1, l . ∂π Then equations (3.18) take the form d ∂T ∗ ∂T ∗ ∂T − − σ λ λ dt ∂v∗ ∂π ∂ q˙ σ = 1, s ,



 ∂βµσ µ ∂βλσ v = Q∗λ , − ∂π µ ∂π λ ∗ λ, µ = 1, l .

(3.22)

3. The generation of the most usual forms of equations

43

These equations are usually called Chaplygin’s equations in quasicoordinates [166, 169]. Note that equations (3.18) and (3.22) should be considered together with the equations of nonholonomic constraints. Equations (3.18) and (3.22) involve as the function T ∗ as the function T . We reduce now Maggi’s equations (3.13) to the form that involves the function T ∗ only. The following relations ∂T ∗ ∂v∗ρ ∂T = , ∂ q˙σ ∂v∗ρ ∂ q˙σ

ρ, σ = 1, s ,

yield the relation    d ∂T ∂ q˙σ ∂ q˙σ d ∂T ∗ ∂v∗ρ = = dt ∂ q˙σ ∂v∗λ ∂v∗λ dt ∂v∗ρ ∂ q˙σ   ∂T ∗ ∂ q˙σ d ∂v∗ρ d ∂T ∗ ∂v∗ρ ∂ q˙σ + . = ρ dt ∂v∗ ∂ q˙σ ∂v∗λ ∂v∗ρ ∂v∗λ dt ∂ q˙σ 

Since

we have

∂v∗ρ ∂ q˙σ = δλρ = ∂ q˙σ ∂v∗λ 

d ∂T dt ∂ q˙σ



 1, 0,

ρ = λ, ρ = λ ,

∂ q˙σ d ∂T ∗ ∂T ∗ ∂ q˙σ d ∂v∗ρ = + . ∂v∗λ dt ∂v∗λ ∂v∗ρ ∂v∗λ dt ∂ q˙σ

(3.23)

Taking into account the relations ∂T ∗ ∂T ∗ ∂v∗ρ ∂T = + ∂q σ ∂q σ ∂v∗ρ ∂q σ and operator (3.17), we obtain ∂T ∂ q˙σ ∂T ∗ ∂T ∗ ∂ q˙σ ∂v∗ρ = + . σ λ λ ∂q ∂v∗ ∂π ∂v∗ρ ∂v∗λ ∂q σ Then from the above and formulas (3.19) and (3.23) it follows that Maggi’s equations (3.13) can be represented in the form   ∂T ∗ ∂T ∗ ∂ q˙σ d ∂v∗ρ ∂v∗ρ d ∂T ∗ − + − σ = Q∗λ , dt ∂v∗λ ∂π λ ∂v∗ρ ∂v∗λ dt ∂ q˙σ ∂q (3.24) λ = 1, l . ρ, σ = 1, s , Equations (3.18) and (3.24) can be applied to both holonomic and nonholonomic systems with as the linear with respect to velocities ideal constraints as the nonlinear ones. In the case when the time does not enter into the kinetic energy and the equations of constraints in explicit form, equations (3.18) and (3.24) were obtained by G. Hamel [314] and in the general case by V. S. Novoselov [169]. Therefore we shall call these equations the Hamel– Novoselov ones.

44

II. Nonholonomic Systems

In the case when the quasivelocities are defined by formulas (3.20) and the constraints are given by equations (3.21) we have ρ ρ ρ ∂ q˙σ d ∂v∗ρ σ dασ σ ∂ασ τ σ τ ∂ασ µ = β = β q ˙ = β β v , λ λ λ µ ∂v∗λ dt ∂ q˙σ dt ∂q τ ∂q τ ∗ ρ ρ ∂ q˙σ ∂v∗ρ σ ∂ατ τ σ τ ∂ατ µ = β q ˙ = β β v , λ λ µ ∂v∗λ ∂q σ ∂q σ ∂q σ ∗

ρ, σ, τ = 1, s ,

λ, µ = 1, l .

Then (3.24) takes the form ∗ d ∂T ∗ ∂T ∗ ρ µ ∂T − + c v = Q∗λ , ρ ∗ λµ dt ∂v∗λ ∂π λ ∂v∗  ρ  ∂ασ ∂ατρ ρ − σ βλσ βµτ , cλµ = ∂q τ ∂q

ρ, σ, τ = 1, s ,

(3.25)

λ, µ = 1, l .

In the case l = s these equations and the relations for the coefficients cρστ were obtained first by P. V. Voronets in 1901 [41]. In 1904 for l < s these results were obtained once more by G. Hamel [313]. Therefore these equations are usually called the Voronets–Hamel equations but G. Hamel itself called them the Euler–Lagrange equations. We remark that in the literature they are also called the Hamel–Boltzmann equations. Together with the works of P. V. Voronets, H. Poincar´e obtains [373] the equations, which are highly close to equations (3.25). Poincar´e’s equations correspond to the case when in equations (3.25) for l = s the coefficients cρστ are constant and the forces are expressed via the forcing function U : Q∗τ = βτσ

∂U , ∂q σ

σ, τ = 1, s .

In this case equations (3.25) can be represented in the form, suggested by H. Poincar´e: ∗ ∗ d ∂L∗ ρ σ ∂L σ ∂L = c v + β , L∗ (q, v∗ ) = T ∗ + U , στ ∗ τ dt ∂v∗τ ∂v∗ρ ∂q σ ρ, σ, τ = 1, s .

(3.26)

When generated equations of motion (3.26) H. Poincar´e made use of the group theory. The approach of Poincar´e was developed then in the works of N. G. Chetaev, L. M. Markhashov, V. V. Rumyantsev, and Fam Guen. They generalized Poincar´e’s equations to the case when the coefficients cρστ are not constant and the motion is acted by as potential as nonpotential forces. Besides, V. V. Rumyantsev considered the case of nonlinear first-order nonholonomic constraints. These equations, describing the motion of nonholonomic systems, are called the Poincar´e–Chetaev–Rumyantsev equations. For detail, see Chapter VII.

45

4. The examples of applications § 4. The examples of applications of different kinds equations of nonholonomic mechanics

E x a m p l e II .1 . The motion of a double-mass system with holonomic and nonholonomic constraints (the application of Maggi’s equations). Consider in the horizontal plane Oxy the motion of two points M1 (x1 , y1 ) and M2 (x2 , y2 ) with masses m. They are connected by a rigid rod whose mass is ignored and length is 2l (Fig. II. 1,a). The other similar examples of the systems of finite numbers of material particles with nonholonomic constraints are considered in the work [84]. The short runner with clinches (a skate) is horizontally fixed at the middle point C of the rod at right angles. The runner has a knife-edge and therefore it allows the displacement without friction along the knife-edge but it does not allow the motion in perpendicular direction. We assume that since the runner is of sufficiently small length and has clinches, the system can freely rotate about its center. The following holonomic constraint (x2 − x1 )2 + (y2 − y1 )2 = (2l)2 is imposed on the motion of points. Then the position of system is uniquely defined by three parameters. As the generalized coordinates we shall regard the Cartesian coordinates x, y of the middle point of rod and the angle θ between the direction of the rod and the axis Oz: q1 = x , Then we have

q2 = θ ,

˙ sin θ , x˙ 1 = x˙ + θl ˙ sin θ , x˙ 2 = x˙ − θl

q3 = y .

˙ cos θ , y˙ 1 = y˙ − θl ˙ cos θ . y˙ 2 = y˙ + θl

(4.1)

(4.2)

Now we obtain the equation of nonholonomic constraint. Since the runner is at the point C of the middle of rod, this point may have only the velocity perpendicular to the axis of rod. The projections of velocity of any of two points of rigid body on the straight line, passing through these points, are equal. Since there exists a skate, the velocity of middle point of rod v has no a projection on the axis of rod and therefore the velocities v1 and v2 of the points M1 and M2 also have no this projection. This condition can be represented as y˙ 1 x˙ 1 = . x˙ 2 y˙ 2 With (4.2) this implies that ˙ x˙ cos θ + y˙ sin θ) = 0 . θ( The above equation is satisfied for θ˙ = 0 or under the condition x˙ cos θ + y˙ sin θ = 0 .

(4.3)

46

II. Nonholonomic Systems

In the case θ˙ = 0 the angle θ is constant and therefore we have a translational motion for the linear displacement of the point C. Such a motion is realized for a long runner, which opposes the rotation of system round the point C. Since we consider the case of short runner, the nonholonomic constraint is given by (4.3). The represented in Fig. II. 1,a system with nonholonomic constraint (4.3) may explain, in particular, the motion, on one skate, of the vertically standing figure-skater and in the case θ˙ = 0 the motion of the skater on racing skates. Note that constraint (4.3) is satisfied for θ˙ = 0 and for θ˙ = 0. In this case ˙ x˙ cos θ + y˙ sin θ) = 0 is more general we cannot assume that the equation θ( than equation (4.3) and therefore we cannot treat its as the example of nonlinear nonholonomic constraint. Similarly, the functional relations, obtained in the other examples of the work [84], cannot also be regarded as nonlinear nonholonomic constraints. For generation of equations of motion we obtain first the relation for the kinetic energy T . By (4.2) we have T =

 m 2 x˙ 1 + y˙ 12 + x˙ 22 + y˙ 22 = m x˙ 2 + y˙ 2 + θ˙2 l2 . 2

This implies the relation

M W1 = 2m¨ x,

M W2 = 2ml2 θ¨ ,

M W3 = 2m¨ y,

(4.4)

where M = 2m is a mass of representation point. In accordance with the general theory the new velocities v∗1 , v∗2 , v∗3 are ˙ v 3 = x˙ cos θ + y˙ sin θ. ˙ v∗2 = q˙2 ≡ θ, introduced by formulas v∗1 = q˙1 ≡ x, ∗ Then we obtain x˙ ≡ q˙1 = v∗1 ,

θ˙ ≡ q˙2 = v∗2 ,

y˙ ≡ q˙3 =

v∗3 − v∗1 cos θ . sin θ

(4.5)

By (4.4), (4.5) Maggi’s equations (2.10) take the form 2m¨ x − Q1 + (2m¨ y − Q3 )(− ctg θ) = 0 , 2¨ 2ml θ − Q2 = 0 .

(4.6)

We remark that the second equation coincides with usual Lagrange’s equation of the second kind, which corresponds to the generalized coordinate θ since ˙ in the equation of nonholonomic constraint (4.3) is lacking the velocity θ. System of equations (4.6) must be supplemented by the equation of constraint (4.3). Differentiating it in time, we obtain x ¨ cos θ − x˙ θ˙ sin θ + y¨ sin θ + y˙ θ˙ cos θ = 0 .

(4.7)

Solving the system of equations (4.6) and (4.7) as the system of algebraic linear nonhomogeneous equations in unknowns x ¨, y¨, θ¨ and representing the

47

4. The examples of applications

obtained results as the system of six first-order differential equations, we have x˙ = vx , y˙ = vy , θ˙ = ωz , v˙ x = ωz (vx sin θ − vy cos θ) cos θ + (Q1 sin θ − Q3 cos θ) sin θ/(2m) , v˙ y = ωz (vx sin θ − vy cos θ) sin θ − (Q1 sin θ − Q3 cos θ) cos θ/(2m) , ω˙ z = Q2 /(2ml2 ) . This normal form of the system of differential equations is convenient to use the numerical integration methods. For the computation of generalized reaction of nonholonomic constraint by formula (2.12) we have now the following relation Λ = (2m¨ y − Q3 )/ sin θ . Consider the motion of system acted by the force F = Fx i + Fy j, imposed at the point C, and when there exists the moment N = Nz k. Besides, we = −µv1 , Fresist = −µv2 also take into account the resistance forces Fresist 1 2 (µ = const), applied at the points M1 , M2 (Fig. II. 1,a), in which case to generalized coordinates (4.1) correspond the following generalized forces: Q1 ≡ Qx = Fx − 2µx˙ ,

Q2 ≡ Qθ = Nz − 2µl2 θ˙ ,

Q3 ≡ Qy = Fy − 2µy˙ .

For concrete computation we assumed that m = 7 kg, l = 1 m, µ = 0.6 N·s/m, Fx = Fy = 2 N. In Fig. II. 1,b are given three trajectories, which the point C traces in the time 15 s for Nz1 = 1 N·m, Nz2 = 0.65 N·m, Nz3 = 0.3 N·m. The initial data are zero. E x a m p l e II .2 . The motion of figure-skater (the application of Chaplygin’s equations). Now we apply Chaplygin’s equations to the solution of the

Fig. II. 1

48

II. Nonholonomic Systems

Fig. II. 2

following problem: find the motion of lop-sided figure-skater on the short skate A (Fig. II. 2). We introduce the moving and stationary coordinates; Axyz and Oξηζ, respectively,. The motion is acted by the resistance force Fresist = −κ1 vC and the drag torque Nresist = −κ2ω , C is a center of mass of figure-skater. Since the figure-skater can move only along a skate going round at a time, the constraint, imposed on the system considered, consists in that the velocity of the point A is always directed along the moving axis Ax, i. e. its projection vAy on the axis Ay is equal to zero at each moment of time. Denote the unit vectors of stationary coordinates Oξηζ by i1 , j1 , k1 and the coordinates of a center of gravity in the stationary coordinates by ξC , ηC . The coordinates of a center of gravity in the moving coordinates Axyz are assumed to be the following: xC = α, yC = β. As the generalized coordinates of system we regard the coordinates of the point A and the angle between the axes Ax and Oξ, namely q1 = ξ ,

q3 = η ,

q2 = θ .

We obtain now the equation of constraint. Represent the constraint in terms of projections of the vector vA on the fixed axis Oξη, taking into account that ˙ 1 + ηj vA = vAξ i1 + vAη j1 = ξi ˙ 1. The projection of the vector vA on the axis Ay has the form vAy = −ξ˙ sin θ + η˙ cos θ . Then the equation of constraint vAy = 0 can be written as ϕ(t, q 1 , q 2 , q 3 , q˙1 , q˙2 , q˙3 ) ≡ −ξ˙ sin θ + η˙ cos θ = 0 .

(4.8)

The kinetic energy is determined by the K¨onig theorem: T =

 1  ˙ ˙ ˙ cos θ − β sin θ))2 + k 2 θ˙2 , (4.9) M (ξ − θ(α sin θ + β cos θ))2 + (η˙ + θ(α C 2

49

4. The examples of applications

where kC is a radius of inertia of body relative to the axis, passing through the center of gravity and perpendicularly to the plane of motion, M is a mass of system. After the transformation in accordance with the equation of constraint, the relation for kinetic energy takes the form  1  ˙ sin θ + β cos θ))2 +(ξ˙ tg θ+ θ(α ˙ cos θ−β sin θ))2 +k 2 θ˙2 . (T ) = M (ξ˙ − θ(α C 2

Now we write Chaplygin’s equation in unknown coordinate ξ:  3     3 ∂b1 d ∂(T ) ∂(T ) ∂T ∂b3 ∂b2 ∂b3 + − 1 ξ˙ + − 1 θ˙ = Qξ . − dt ∂ ξ˙ ∂ξ ∂ η˙ ∂ξ ∂ξ ∂ξ ∂θ

(4.10)

By the above notions Chaplygin’s equation for constraint takes the form q˙3 = b31 q˙1 + b32 q˙2 ,

b31 = tg θ ,

b32 = 0 .

In this case equation (4.10) can be represented as   d ∂(T ) ∂T ∂b31 ˙ + − θ = Qξ . dt ∂ ξ˙ ∂ η˙ ∂θ Using the relation for kinetic energy, we obtain 2 ¨ cos θ − θ˙2 α cos θ = Qξ cos θ . ξ¨ + ξ˙θ˙ tg θ − θβ M

We generate now the equation of motion over the coordinate θ. Performing similar numerical computation, we obtain Qθ cos2 θ γ 2 cos2 θθ¨ − β cos θξ¨ + (α cos θ − β sin θ)ξ˙θ˙ = , M 2 where γ 2 = α2 + β 2 + kC . The generalized forces, acting on the system, are the following

Qξ = −κ1 ξ˙ ,

Qη = −κ1 η˙ ,

Qθ = −κ2 θ˙ .

(4.11)

Finally, we obtain a system of differential equations in Chaplygin’s form, describing the motion of a figure-skater in the case when its center of mass lies not above the skate: ¨ cos θ − θ˙2 α cos θ = −κ1 ξ˙ cos2 θ/M , ξ¨ + ξ˙θ˙ tg θ − θβ γ 2 cos2 θθ¨ − β cos θξ¨ + (α cos θ − β sin θ)ξ˙θ˙ = −κ2 θ˙ cos2 θ/M ,

(4.12)

η˙ = ξ˙ tg θ . Note that the considered motion of a figure-skater is one of possible interpretations of the motion of Chaplygin’s sledge. One more problem, related to Chaplygin’s sledge, is considered in Appendix D.

50

II. Nonholonomic Systems

E x a m p l e II .3 . The motion of figure-skater (application of Maggi’s equations). We generate Maggi’s equations for the problem considered in Example II. 2. The frames of reference and the generalized coordinates are introduced as above. Then the relations for the kinetic energy T and the covariant components of generalized forces Qξ , Qθ , Qη are given by formulas (4.9) and (4.11). The equation of constraint (4.8) has the form ξ˙ tg θ − η˙ = 0 .

(4.13)

We introduce the new nonholonomic variables in the following way: v∗1 = ξ˙ ,

v∗2 = θ˙ ,

v∗3 = ξ˙ tg θ − η˙ .

Having performed the change of the old variables to the new ones, we obtain the following inverse transformation ξ˙ = v∗1 ,

θ˙ = v∗2 ,

η˙ = v∗1 tg θ − v∗3 .

Using these formulas, we can compute the derivatives: ∂ q˙1 = 1, ∂v∗1 ∂ q˙1 = 0, ∂v∗2 ∂ q˙1 = 0, ∂v∗3

∂ q˙2 = 0, ∂v∗1 ∂ q˙2 = 1, ∂v∗2 ∂ q˙2 = 0, ∂v∗3

∂ q˙3 = tg θ , ∂v∗1 ∂ q˙3 = 0, ∂v∗2 ∂ q˙3 = 1. ∂v∗3

Using the computed coefficients in Maggi’s equations (2.10) and performing some simplifications, we obtain the differential equations of motion for the system κ1 β α − θ˙2 = − (ξ˙ + η˙ tg θ) , ξ¨ + η¨ tg θ − θ¨ cos θ cos θ M ¨ sin θ + β cos θ) = − κ2 θ˙ . γ 2 θ¨ + η¨(α tg θ − β sin θ) − ξ(α M

(4.14)

These equations should be integrated together with equation of constraint (4.13). Compare the obtained results with those, get in Example II. 2. Using the method of Chaplygin, we change in system (4.14) the quantities η˙ and η¨ to their expressions from the equation of nonholonomic constraint (4.13). Then we have   κ1 α 1 β ¨ ¨ ˙ ˙ − θ˙2 = − (ξ˙ + ξ˙ tg2 θ) , ξ + tg θ ξ tg θ + ξ θ 2 − θ¨ cos θ cos θ cos θ M   1 2¨ ¨ ˙ ˙ γ θ + ξ tg θ + ξ θ 2 (α cos θ − β sin θ)− cos θ ¨ sin θ + β cos θ) = − κ2 θ˙ . −ξ(α M

51

4. The examples of applications

After the transformations we arrive at the system 1 α tg θ β κ1 ξ¨ 2 + ξ˙θ˙ 2 − θ¨ − θ˙2 = − ξ˙ , cos θ cos θ cos θ cos θ M (α cos θ − β sin θ) κ2 β 2¨ ˙ ˙ ¨ + ξθ = − θ˙ . γ θ−ξ 2 cos θ cos θ M It is easily remarked that multiplying these equations by cos2 θ, we obtain Chaplygin’s equations (4.12), generated in Example II. 2. Thus, Maggi’s equations give a more simple method to find the equations of motion than Chaplygin’s equations, in which case it is not required that the mechanical system satisfies additional conditions. It is sufficient only to generate the relations for the kinetic energy and generalized forces, to choice rationally new nonholonomic variables, to find the derivatives of inverse transformation, and to construct the linear combinations of the Lagrange operators. Besides, by equation (2.12) we can easily write the relations for generalized reactions of nonholonomic constraints. For the considered problem we obtain 1 Λ ¨ cos θ − β sin θ) − θ˙2 (α sin θ + β cos θ) + κ1 ξ˙ tg θ . = ξ¨ tg θ + ξ˙θ˙ 2 + θ(α M cos θ M In Fig. II. 3 are given the results of numerical integrating the system of differential equations for 10 s. Here we assumed that γ 2 = 0.07 m 2 ,

κ1 /M = 1 s −1 ,

κ2 /M = 0.02 m 2 · s −1 ,

˙ ξ(0) = 5 m · s −1 , η(0) = 0 , ˙ θ(0) = 0 , θ(0) = 12.5 s −1 , α = 0 ,

ξ(0) = 0 ,

Fig. II. 3

η(0) ˙ = 0, β = 0.

52

II. Nonholonomic Systems

Fig. II. 4

E x a m p l e II .4 . The motion of car in a sweep (the application of the Hamel–Boltzmann equations). Consider a motion of car (Fig. II. 4), consisting of a body of the mass M1 and a front axis with the mass M2 . Suppose, they have the moments of inertia J1 and J2 about the vertical axes through their centers of mass, respectively. The front axis can rotate about their vertical axis through its center of mass. The masses of wheels and backward axis, regarded as separate parts, are assumed to be negligible. The motion of car is subject to the force F1 (t), acting along its longitudinal axis Cx, and to the moment L1 (t), rotating the front axis. In this case F1 (t), L1 (t) are the given functions of time. In addition, we take into account the resistance force F2 (vC ), acting in the direction opposite to the direction of the velocity vC ˙ which is applied to of the center of mass C of body, the drag torque L2 (θ), the front axis and is opposite to the angular velocity of its rotating, and the righting moment L3 (θ). A similar scheme was considered in the work [132] as the simplified mathematical model for the car motion on a sweep. At present it can be of interest when studying wheeled robot vehicles [146-148, 423]. We generate the Hamel–Boltzmann equations for the study of the motion of this system. The motion of car in the horizontal plane is considered in the fixed coordinates Oξηζ. The car position is given by the following generalized coordinates: q 1 = ϕ is an angle between the longitudinal axis of car Cx and the axis Oξ, q 2 = θ is an angle between the front axis and the perpendicular to the axis Cx, and q 3 = ξC , q 4 = ηC are coordinates of the point C. On the motion of car the two nonholonomic constraints are imposed which express that the sideways sliding motion of the backward and front axes of car is missed. Their equations can be written similarly to formula (4.8) from Example II. 2: −ξ˙B sin ϕ + η˙ B cos ϕ = 0 , (4.15) −ξ˙A sin(ϕ + θ) + η˙ A cos(ϕ + θ) = 0 .

53

4. The examples of applications

Here ξA ,ηA ,ξB ,ηB are the coordinates of the centers of mass for the front and backward axes of car. We assume that the distances between the centers of mass of these axes and the center of gravity of the body of car are equal to l1 and l2 , respectively. Then the equations of nonholonomic constraints (4.15) take the form ϕ1 ≡ −ξ˙C sin ϕ + η˙ C cos ϕ − l2 ϕ˙ = 0 , ϕ2 ≡ −ξ˙C sin(ϕ + θ) + η˙ C cos(ϕ + θ) + l1 ϕ˙ cos θ = 0 .

(4.16)

We introduce the quasivelocities by formulas π˙ 1 = ϕ˙ , π˙ 2 = θ˙ , π˙ 3 = −ξ˙C sin ϕ + η˙ C cos ϕ − l2 ϕ˙ , π˙ 4 = −ξ˙C sin(ϕ + θ) + η˙ C cos(ϕ + θ) + l1 ϕ˙ cos θ ,

(4.17)

i. e. in formulas (2.17) the coefficients aρσ (q), ρ, σ = 1, 4, have the form a11 = 1 ,

a22 = 1 ,

a42 = l1 cos θ ,

a31 = −l2 ,

a33 = − sin ϕ ,

a43 = − sin(ϕ + θ) ,

a34 = cos ϕ ,

a44 = cos(ϕ + θ) .

To formulas (4.17) corresponds the inverse transformation q˙2 ≡ θ˙ = π˙ 2 , q˙1 ≡ ϕ˙ = π˙ 1 , q˙3 ≡ ξ˙C = b3 π˙ 1 + b3 π˙ 3 + b3 π˙ 4 , 4

q˙ ≡ η˙ C = where

1 b41 π˙ 1

+

3 b43 π˙ 3

+

4 b44 π˙ 4

(4.18)

,

b31 = (l1 cos ϕ cos θ + l2 cos(ϕ + θ)/ sin θ , b33 = cos(ϕ + θ)/ sin θ ,

b34 = − cos ϕ/ sin θ ,

b41 = (l1 sin ϕ cos θ + l2 sin(ϕ + θ))/ sin θ , b43 = sin(ϕ + θ)/ sin θ ,

b44 = − sin ϕ/ sin θ .

The rest of coefficients aρσ and bσρ , are equal to zero. Thus, in transformation (2.17) the matrices (aρσ ) and (bσρ ) are obtained. Now we can compute the coefficients of nonholonomicity by formulas (2.19). We have γ133 = −γ331 = b33 cos ϕ + b43 sin ϕ , γ134 = −γ431 = b34 cos ϕ + b44 sin ϕ , γ241 = −γ142 = l1 sin θ + b31 cos(ϕ + θ) + b41 sin(ϕ + θ) , γ143 = −γ341 = γ243 = −γ342 = γ144 = −γ441 = γ244 = −γ442 =

b33 b34

cos(ϕ + θ) + cos(ϕ + θ) +

The rest of quantities γλ(l+κ)λ∗ , are equal to zero.

b43 b44

sin(ϕ + θ) , sin(ϕ + θ) .

(4.19)

54

II. Nonholonomic Systems

The kinetic energy of system consists of the kinetic energies of the body and the front axis and is computed by formula 2 2 + η˙ C ) + J ∗ ϕ˙ 2 + J2 θ˙2 + 2J2 ϕ˙ θ˙ + 2M2 l1 ϕ(− ˙ ξ˙C sin ϕ + η˙ C cos ϕ) , 2T = M ∗ (ξ˙C

M ∗ = M1 + M2 ,

J ∗ = J1 + J2 + M2 l12 . (4.20)

The generalized forces, acting on the car, can be represented as Q1 ≡ Qϕ = 0 , ˙ − L3 (θ) , Q2 ≡ Qθ = L1 (t) − L2 (θ) Q3 ≡ Qξ = F1 (t) cos ϕ − F2 (vC )ξ˙C /vC ,

(4.21)

C

Q4 ≡ QηC = F1 (t) sin ϕ − F2 (vC )η˙ C /vC , Then by (2.18) we have

vC =

 2 + η˙ 2 . ξ˙C C

 1 = (F1 (t) cos ϕ − F2 (vC )ξ˙C /vC )b31 + (F1 (t) sin ϕ − F2 (vC )η˙ C /vC )b41 , Q

 2 = L1 − L2 − L3 , Q  3 = (F1 (t) cos ϕ − F2 (vC )ξ˙C /vC )b33 + (F1 (t) sin ϕ − F2 (vC )η˙ C /vC )b43 , Q  4 = (F1 (t) cos ϕ − F2 (vC )ξ˙C /vC )b3 + (F1 (t) sin ϕ − F2 (vC )η˙ C /vC )b4 . Q 4 4 (4.22) Using formulas (4.18) and (4.20) we can construct the relation for T ∗ :   2T ∗ = π˙ 12 M ∗ (β13 )2 + (β14 )2 +

  + J ∗ + M2 l12 + 2M2 l1 β14 cos ϕ − β13 sin ϕ +       + π˙ 22 J2 + π˙ 32 M ∗ (β33 )2 + (β34 )2 + π˙ 42 M ∗ (β43 )2 + (β44 )2 +     + π˙ 1 π˙ 2 2J2 + π˙ 1 π˙ 3 2M ∗ β13 β33 + β14 β34 + 2M2 l1 β34 cos ϕ − β33 sin ϕ +     + π˙ 1 π˙ 4 2M ∗ β13 β43 + β14 β44 + 2M2 l1 β44 cos ϕ − β43 sin ϕ +    + π˙ 3 π˙ 4 2M ∗ β33 β43 − β34 β44 .

Omitting some tedious calculations and using formulas (2.20), (4.19), and (4.22), we obtain the Hamel–Boltzmann equations (2.15) for the problem considered:

55

4. The examples of applications

 2   2 2 2 ∗ 2 ∗ l2 + l1 cos θ + 2l1 l2 cos θ ¨ ϕ¨ + J2 θ− J + M2 l1 + 2M2 l1 l2 + M sin2 θ (l1 + l2 )2 M ∗ cos θ ˙ ϕ˙ θ = − sin3 θ     F2 (vC )ξ˙C 3 F2 (vC )η˙ C 4 b1 + F1 (t) sin ϕ − b1 , = F1 (t) cos ϕ − vC vC ¨ = L1 (t) − L2 (θ) ˙ − L3 (θ) . J2 (ϕ¨ + θ) (4.23) Note that we need to solve this system together with equations of constraints (4.16). As an example, we consider the motion of the hypothetical compact motor car with M1 = 1000 kg ,

M2 = 110 kg ,

J2 = 30 kg · m 2 ,

J1 = 1500 kg · m 2 ,

l1 = 0.75 m ,

l2 = 1.65 m

and with the following power characteristics: F2 (vC ) = κ2 vC , κ2 = 100 N · s · m −1 , ˙ = κ1 θ˙ , κ1 = 0.5 N · m · s , L1 (t) = 15 N · m , L2 (θ) L3 (θ) = κ3 θ , κ3 = 100 N · m .

F1 (t) = 2500 N ,

The results of the numerical solution of nonlinear system of differential equations (4.16), (4.23) are given in Fig. II. 5. In computing we make use of the following initial data ϕ(0) = 0 ,

ϕ(0) ˙ = 0,

θ(0) = π/180 rad ,

ξ˙C (0) = 0.00176856 m · s −1 ,

ηC (0) = 0 ,

Fig. II. 5

˙ θ(0) = 0,

ξC (0) = 0 ,

η˙ C (0) = 0.000018008 m · s −1 .

56

II. Nonholonomic Systems

E x a m p l e II .5 . The turning movement of a car (the application of Maggi’s equations and the Lagrange equations of the first kind). Consider now the motion of car from Example II. 4 by means of Maggi’s equations. We make use of the same curvilinear coordinates. Then the equations of constraints take the form (4.16) and the kinetic energy and the generalized forces are given by formulas (4.20) and (4.21), respectively. We introduce the following new nonholonomic variables v∗1 = ϕ˙ , v∗2 = θ˙ , v∗3 = −l2 ϕ˙ − ξ˙C sin ϕ + η˙ C cos ϕ , v∗4 = l1 ϕ˙ cos θ − ξ˙C sin(ϕ + θ) + η˙ C cos(ϕ + θ) and write the inverse transformation q˙2 ≡ θ˙ = v∗2 , q˙1 ≡ ϕ˙ = v∗1 , q˙3 ≡ ξ˙C = β 3 v 1 + β 3 v 3 + β 3 v 4 , 4

q˙ ≡ η˙ C = where

1 ∗ β14 v∗1

+

3 ∗ β34 v∗3

+

4 ∗ β44 v∗4

(4.24)

,

β13 = (l1 cos ϕ cos θ + l2 cos(ϕ + θ)/ sin θ , β33 = cos(ϕ + θ)/ sin θ ,

β43 = − cos ϕ/ sin θ ,

β14 = (l1 sin ϕ cos θ + l2 sin(ϕ + θ))/ sin θ , β34 = sin(ϕ + θ)/ sin θ ,

(4.25)

β44 = − sin ϕ/ sin θ .

In this case the first Maggi’s equation has the form (M W1 − Q1 )

∂ q˙1 ∂ q˙3 ∂ q˙4 + (M W − Q ) + (M W − Q ) = 0. 3 3 4 4 ∂v∗1 ∂v∗1 ∂v∗1

(4.26)

˙ the second Since the equations of constraints do not involve the velocity θ, Maggi’s equation is Lagrange’s equation of the second kind: M W2 − Q2 = 0 .

(4.27)

The relations M Wσ can be computed in terms of the kinetic energy by formulas d ∂T ∂T M Wσ = − σ, σ = 1, 4 . dt ∂ q˙σ ∂q Then by (4.20), (4.21), (4.24), (4.25) equations of motion of car (4.26), (4.27) can be represented as   ∗ ¨ 2 θ¨ + (M ∗ β13 −M2 l1 sin ϕ)ξ¨C+ J +M2 l1 (l1 −β13 sin ϕ+β14 cos ϕ) ϕ+J +(M ∗ β14 + M2 l1 cos ϕ)¨ ηC = M2 l1 ϕ˙ 2 (β13 cos ϕ + β14 sin ϕ)+     + F1 (t) cos ϕ − F2 (vC )ξ˙C /vC β13 + F1 (t) sin ϕ − F2 (vC )η˙ C /vC β14 , ˙ − L3 (θ) . J2 (θ¨ + ϕ) ¨ = L1 (t) − L2 (θ) (4.28)

57

4. The examples of applications

If the initial data and the analytic representations of the functions F1 (t), ˙ L3 (θ) are given, then after numerical integrating nonF2 (vC ), L1 (t), L2 (θ), linear system of differential equations (4.16), (4.28), we find the law of motion of car ϕ = ϕ(t) ,

θ = θ(t) ,

ξC = ξC (t),

ηC = ηC (t) .

(4.29)

Now we can determine generalized reactions. For the second group of Maggi’s equations we have ∂ q˙3 ∂ q˙4 + (M W − Q ) , 4 4 ∂v∗3 ∂v∗3 ∂ q˙3 ∂ q˙4 Λ2 = (M W3 − Q3 ) 4 + (M W4 − Q4 ) 4 , ∂v∗ ∂v∗ Λ1 = (M W3 − Q3 )

or the same in the expanded form: Λ1 =[M ∗ ξ¨C −M2 l1 (ϕ¨ sin ϕ+ ϕ˙ 2 cos ϕ)−F1 (t) cos ϕ+F2 (vC )ξ˙C /vC ]β33+ +[M ∗ η¨C +M2 l1 (ϕ¨ cos ϕ− ϕ˙ 2 sin ϕ)−F1 (t) sin ϕ+F2 (vC )η˙ C /vC ]β34 , Λ2 =[M ∗ ξ¨C −M2 l1 (ϕ¨ sin ϕ+ ϕ˙ 2 cos ϕ)−F1 (t) cos ϕ+F2 (vC )ξ˙C /vC ]β 3+ ∗

+[M η¨C +M2 l1 (ϕ¨ cos ϕ− ϕ˙

2

4 sin ϕ)−F1 (t) sin ϕ+F2 (vC )η˙ C /vC ]β44

.

Taking into account (4.29), we obtain the law of variation of the generalized reactions Λi = Λi (t), i = 1, 2. Using these functions we can study the conditions, under which nonholonomic constraints (4.16) are satisfied. If the reaction forces are equal to the Coulomb friction forces, then these constraints need not be satisfied and the car can begin to slide along their axes. Note that in Appendix D the motion of nonholonomic systems in the absence of reaction forces of nonholonomic constraints is considered, and in Appendix E the turning movement of a car with possible side slipping of its wheels is studied. Thus, Maggi’s equations can be generated in the same easy manner as Lagrange’s equations of the second kind. For ideal nonholonomic constraints, Maggi’s equations decompose into two groups. The first group, together with the equations of constraints, permits one to find the law of motion of a nonholonomic system and then the generalized reactions can be determined from the second group. We notice that for generating the Hamel–Boltzmann equations it is required much greater calculations than for Maggi’s equations. It is of interest to compare the obtained Maggi’s equations (4.28) with the Hamel–Boltzmann equations (4.23). The second equations of these systems are coincide. If we obtain the relations ξ¨C and η¨C from the equations of constraints and substitute them into the first equation of system (4.28), then we obtain the first equation of system (4.23). We could also write the Lagrange equations of the first kind in curvilinear coordinates (see equations (2.22) of the present Chapter). In the problem considered they have a form:

58

II. Nonholonomic Systems J ∗ ϕ¨ + J2 θ¨ − M2 l1 ξ¨C sin ϕ + M2 l1 η¨C cos ϕ = −Λ1 l2 + Λ2 l1 cos θ , ˙ − L3 (θ) , ¨ = L1 (t) − L2 (θ) J2 (θ¨ + ϕ) M ∗ ξ¨C − M2 l1 ϕ¨ sin ϕ − M2 l1 ϕ˙ 2 cos ϕ = ˙ − Λ1 sin ϕ − Λ2 sin(ϕ + θ) , = F1 cos ϕ − L2 (θ) M ∗ η¨C + M2 l1 ϕ¨ cos ϕ − M2 l1 ϕ˙ 2 sin ϕ = = F1 sin ϕ − k2 η˙ C + Λ1 cos ϕ + Λ2 cos(ϕ + θ) .

The four equations given include four unknown generalized coordinates and two unknown Lagrange multipliers, so they have to be solved together with the equations of constraints (4.16). This is characteristic (peculiar) for the Lagrange equations of the first kind. If we differentiate in time the equations of constraints and with the help of them eliminate generalized reaction forces, then we get Maggi’s equations of motion (4.28), as well as formulas for defining Λ1 и Λ2 . E x a m p l e II .6 . The rolling of ellipsoid on a rough plane (the generation of Maggi’s equations). It is to be noted that the specific form of Maggi’s equations depends essentially on the choice of the variables v∗ρ . By successful choice we can considerably simplify calculations, concerning the reduction of the problem to the system of differential equations, represented in normal form. As an example, consider the rolling of homogeneous rigid body of ellipsoidal form on the fixed plane. The center of ellipsoid coincident with centroid is taken as the origin of the moving coordinates Cxyz, the axes of which are rigidly fixed with the axes of body (Fig. II. 6). We assume that the plane π, on which the ellipsoid rolls, coincides with the plane Oξη of the fixed system of coordinates Oξηζ. Denote by ξ, η, ζ the coordinates of the center of

Fig. II. 6

59

4. The examples of applications

ellipsoid relative in the fixed frame. The velocity of the contact point p can be computed by formula −−→ vp = vC + ω × CP . For the rolling without sliding the velocity of the point p is equal to zero and therefore the equation of constraint can be represented as    iξ iη iζ    −−→ ˙ ξ + ηi ˙ ζ + ωξ ω η ω ζ  = 0 . (4.30) vC + ω × CP = ξi ˙ η + ζi    ξ0 η0 ζ0 

Here ξ0 , η0 , ζ0 are the coordinates of the points P in the frame with ξ1 η1 ζ1 , the axes of which ξ1 , η1 , ζ1 are parallel to the axes ξ, η, ζ of the fixed coordinates, respectively. It can be shown that the quantities ξ0 , η0 , ζ0 are computed by formulas − ξ0 ζ = (a2 − b2 ) sin θ cos ψ sin ϕ cos ϕ+ + (c2 − a2 sin2 ϕ − b2 cos2 ϕ) sin ψ cos θ sin θ , − η0 ζ = (a2 − b2 ) sin ψ sin θ sin ϕ cos ϕ+ + (a2 sin2 ϕ + b2 cos2 ϕ − c2 ) cos ψ cos θ sin θ ,

 ζ0 = −ζ = − a2 sin2 θ sin2 ϕ + b2 sin2 θ cos2 ϕ + c2 cos2 θ . Here a, b, c are the semiaxis of ellipsoid; ψ, θ, ϕ are the Euler angles, giving the orientation of the system of coordinates Cxyz with respect to the frame with ξ1 η1 ζ1 . Vector equation (4.30) is equivalent to three scalar equations for nonholonomic constraints: ϕ1 ≡ ξ˙ + ωη ζ0 − ωζ η0 = 0 , ϕ2 ≡ η˙ + ωζ ξ0 − ωξ ζ0 = 0 , ϕ3 ≡ ζ˙ + ωξ η0 − ωη ξ0 = 0 .

(4.31)

As generalized Lagrangian coordinates we regard the coordinates ξ, η, ζ of the center of mass and the Euler angles ψ, θ, ϕ. The numerical computation of the kinetic energy of ellipsoid in these coordinates is based on applying the K¨onig theorem. Then we have Jω ω 2 M ˙2 (ξ + η˙ 2 + ζ˙ 2 ) + . 2 2 The quantity Jω ω 2 can be represented as T =

Jω ω 2 = Aωx2 + Bωy2 + Cωz2 , where A, B, C are the moments of inertia of the ellipsoid about the axes x, y, z, respectively. Since the ellipsoid, by assumption, is a homogeneous rigid body, we get A=

M (b2 + c2 ) , 5

B=

M (c2 + a2 ) , 5

C=

M (a2 + b2 ) . 5

60

II. Nonholonomic Systems

The projections ωx , ωy , ωz of the vector ω on the axes of moving coordinates Cxyz are the following: ωx = ψ˙ sin θ sin ϕ + θ˙ cos ϕ , ωy = ψ˙ sin θ cos ϕ − θ˙ sin ϕ , ωz = ψ˙ cos θ + ϕ˙ . These formulas allow one to compute the covariant components of the vector M W M Wη = M η¨ , M Wζ = M ζ¨ , M Wξ = M ξ¨ , d ∂T d ∂T d ∂T ∂T ∂T ∂T M Wϕ = − , M Wψ = − , M Wθ = − . ˙ ˙ dt ∂ ϕ˙ ∂ϕ dt ∂ ψ ∂ψ dt ∂ θ ∂θ Since the explicit relations for Wϕ , Wψ and Wθ are lengthy, they are omitted. The quantities ωξ , ωη , ωζ , entering into the equations of constraints (4.31), take the form ωξ = ϕ˙ sin ψ sin θ + θ˙ cos ψ , ωη = −ϕ˙ cos ψ sin θ + θ˙ sin ψ , ωζ = ϕ˙ cos θ + ψ˙ . ˙ v 2 = η, ˙ v 3+κ = ϕκ , κ = 1, 3, ˙ v∗3 = ζ, In this case if we assume that v∗1 = ξ, ∗ ∗ then by reason of the complicated dependence of the functions ϕκ on the velocities q˙σ the relations ∂ q˙σ /∂v∗λ turn out highly awkward and therefore final Maggi’s equations are also complicated. The task turn out more simple if as the free variables v∗λ we choose the angular velocities ωξ , ωη , ωζ . It can be shown that if the quasivelocities v∗ρ are given by formulas v∗1 = ωξ , v∗2 = ωη , v∗3 = ωζ , v∗4 = ξ˙ + ωη ζ0 − ωζ η0 , v∗5 = η˙ + ωζ ξ0 − ωξ ζ0 , v∗6 = ζ˙ + ωξ η0 − ωη ξ0 , then we get

∂ ξ˙ = 0, ∂ωξ ∂ ϕ˙ sin ψ , = ∂ωξ sin θ

∂ η˙ = ζ0 , ∂ωξ

∂ ζ˙ = −η0 , ∂ωξ

∂ ψ˙ sin ψ cos θ , =− ∂ωξ sin θ

∂ ξ˙ = −ζ0 , ∂ωη cos ψ ∂ ϕ˙ , =− ∂ωη sin θ ∂ ξ˙ = η0 , ∂ωζ

∂ η˙ = 0, ∂ωη

∂ ζ˙ = ξ0 , ∂ωη

∂ ψ˙ cos ψ cos θ , = ∂ωη sin θ ∂ η˙ = −ξ0 , ∂ωζ

∂ θ˙ = cos ψ , ∂ωξ

∂ θ˙ = sin ψ , ∂ωη

∂ ζ˙ = 0, ∂ωζ

61

4. The examples of applications ∂ ϕ˙ = 0, ∂ωζ

∂ ψ˙ = 1, ∂ωζ

∂ θ˙ = 0. ∂ωζ

Substituting these relations into Maggi’s equations, we obtain them in explicit form. This example demonstrates that the problems on the rolling of one body on the surface of the other are complicated even under the assumption that the constraint, given by equation (4.30), is ideal. The dynamics of the rigid body, contacting with a rigid surface, is considered in the treatise of A. P. Markeev [143]. The new theory of the interaction of a rolling rigid body with a deformable surface is supposed by V. F. Zhuravlev [70]. E x a m p l e II .7 . Equations of motion and the technical realization of the Appell–Hamel problem (the generation of Maggi’s equations and Lagrange’s equations of the first kind in the case of nonlinear nonholonomic constraints). The example of P. Appell [270, 271] on the motion of a special nonholonomic system (Fig. II. 7, a) was of fundamental importance for the development of Analytic mechanics. This problem has intensively been considered, especially in journal "Rendiconti del circolo matematico di palermo"(1911– 1912). Some works were due to E. Delassus. He considered the example of Appell in more detail in the work [298] and in his book [299]. This problem was also studied by G. Hamel [315, p. 502–505]. Until the present time the problem of Appell–Hamel is of interest for scientists (see, for example, the works [274, 376, 408]). In the example of Appell–Hamel the motion of a disk with knife-edge on the horizontal plane Oξη is considered. The horizontal axis of disk through its center C is fixed in a weightless frame, the stubs of which can slide on the plane without friction (Fig. II. 7, a). The frame prevents the overturn of disk. The disk is rigidly fixed with a coaxial cylinder. The nonstretchable thread, which is turned over the two blocks fixed with the frame, is winded on the cylinder. To the end of thread the mass m is hung, the descent of which results in the rolling of disk. The axis of the mass descend is spaced at ρ from the contact point B of the disk with horizontal plane. We also assume that the frame is fixed with the parallel to BC smooth rail, preventing the swinging of mass. The disk and cylinder have the radii a and b, respectively. Denote the angle between the plane of the rolling of disk and the axis Oξ by θ, the angle of rotation of the wheel about its axis by ϕ, the coordinates of the mass m by x, y, z, the coordinates of the point B by ξ, η. Obviously, the coordinates are related as x = ξ + ρ cos θ ,

y = η + ρ sin θ .

(4.32)

On the motion of system it is imposed the linear nonholonomic constraints ξ˙ = aϕ˙ cos θ ,

η˙ = aϕ˙ sin θ ,

z˙ = −bϕ˙ .

(4.33) (4.34)

62

II. Nonholonomic Systems

Fig. II. 7

Taking into account constraints (4.33) and (4.34), G. Hamel generates the equations of motion for the considered system [315]. Further, he analyzes the limiting case as ρ → 0. In this case it is necessary to consider only the change of the coordinates x, y, z of the mass m, in which case the following nonlinear nonholonomic constraint occurs: ϕ1 ≡ x˙ 2 + y˙ 2 −

a2 2 z˙ = 0 . b2

(4.35)

P. Appell also considered a similar passage to the limit, introducing the parameter α, which is the quotient of the moment of inertia of the disk about its diameter to the quantity ρ. The problem of Appell–Hamel was considered most completely and in more detail by Yu. I. Neimark and N. A. Fufaev in the paper [164] entering into the book [166], which became the classical monograph on the nonholonomic mechanics. They notice [166, p. 227, 228] that ". . . the system, considered by P. Appell and G. Hamel, with nonlinear nonholonomic constraints can be obtained from the nonholonomic system with linear constraints by means of the passage to the limit as ρ → 0. However in this case we have the depression of the system of differential equations, i. e. their degeneracy and therefore it is not known in advance wether the motions of the limit system (ρ = 0) coincide with the limit motions of nondegenerate system as ρ → 0. Therefore the question remains open: to what extent the equations of motion for degenerated system describe properly the motion of the original system with the vanishingly small ρ?". The authors performed the "investigation, which was based on the study of the motions of nondegenerate system for

63

4. The examples of applications

Fig. II. 8

ρ > 0 and ρ < 0, the limit motions of nondegenerate system as |ρ| → 0, and the motions of degenerated system. This investigation implies that the motions of degenerated system differs essentially from the limit motions and therefore the example of nonholonomic system with nonlinear nonholonomic constraints is incorrect". Thus, when used the mentioned above passage to the limit, P. Appell and G. Hamel in place of the study of the original system investigated the degenerated system. We regard the motion of the obtained degenerated system as the separate problem of mechanics: there is the mass m with the coordinates x, y, z, on the motion of which is imposed nonlinear nonholonomic constraint (4.35). Note that in the model of P. Appell and in the corresponding model of V. S. Novoselov [171] when the mass is connected with the disk by the set of inertialess pinions (Fig. II. 7, b) the case ρ = 0 can easily be carried out technically (Fig. II. 8, a, b). However for ρ = 0 in the mentioned above models the satisfaction of constraints (4.33) remains essential. In this case the constraints yield the following relation, imposed on the projections of velocities of the mass m: y˙ = x˙ tg θ . (4.36) Here we take into account that for ρ = 0 by virtue of formulas (4.32) we have ˙ y˙ = η. x˙ = ξ, ˙ In studying the degenerated system constraint (4.36) is not accounted and constraint (4.35) is only introduced such that, by assumption, the velocity of the center of disk can have any direction. This means that, by using constraints (4.35) only, we replace the motion of disk by the motion of ball.

64

II. Nonholonomic Systems

Thus, in studying the degenerated system it should be required that constraint (4.36) is satisfied, i. e., together with the coordinates x, y, z, it is necessary to look to the change of the variable θ. The neglect of the masses of disk, frame, and blocks results in the degeneracy of system and therefore the variable θ turns out to be the “massless” coordinate. If this coordinate is ignored, then it is impossible to describe the motion of massless ball by means of the motion of massless disk. The technical realization is obviously hard in the case when the connection between the velocity of descending the load and the velocity of the center of ball is provided by means of the nonstretchable thread or the system of pinions. However it is possible to study the motion of the mass m with the coordinates x, y, z under the condition that constraint (4.35) is satisfied only. For this purpose we consider the following control problem: the motion of material point with the mass m is realized in such a way that by virtue of (4.35) its vertical velocity is varied proportionally to the velocity of the motion of its trace in horizontal plane. The realization of such problem can be provided by means of the new technical means. We generate Maggi’s equations and Lagrange’s equations of the first kind, using exactly such problem setting. So, we consider the problem on the space motion of material point with nonlinear nonholonomic constraint (4.35). In this case the generalized coordinates are the following q1 = x ,

q2 = y ,

q3 = z .

(4.37)

We introduce the following new nonholonomic variables: v∗1 = x˙ ,

v∗2 = y˙ ,

v∗3 = x˙ 2 + y˙ 2 −

a2 2 z˙ . b2

(4.38)

In the Appell–Hamel problem Maggi’s equations take the form ∂ q˙1 ∂ q˙2 ∂ q˙3 + (M W2 − Q2 ) 1 + (M W3 − Q3 ) 1 = 0 , 1 ∂v∗ ∂v∗ ∂v∗ 1 2 ∂ q˙ ∂ q˙ ∂ q˙3 (M W1 − Q1 ) 2 + (M W2 − Q2 ) 2 + (M W3 − Q3 ) 2 = 0 , ∂v∗ ∂v∗ ∂v∗ 1 2 ∂ q˙ ∂ q˙ ∂ q˙3 (M W1 − Q1 ) 3 + (M W2 − Q2 ) 3 + (M W3 − Q3 ) 3 = Λ . ∂v∗ ∂v∗ ∂v∗ (M W1 − Q1 )

(4.39)

∂ q˙σ , σ, ρ = 1, 3, for numerical com∂v∗ρ putation of which it is required the transformation inverse to transformation (4.38). However, the obtaining of such transformation is a difficult task since nonholonomic constraint (4.35) is nonlinear. Therefore we obtain these derivatives in the following way. Compute the matrix  ρ ∂v∗ , σ, ρ = 1, 3 . (ασρ ) = ∂ q˙σ

These relations involve the derivatives

65

4. The examples of applications By (4.38) we have α11 = 1 ,

α21 = 0 ,

α31 = 0 ,

α12 = 0 ,

α22 = 1 ,

α32 = 0 ,

α13 = 2x˙ , α23 = 2y˙ , α33 = −2a2 b−2 z˙ . Find the matrix (βρσ ) inverse to the matrix (ασρ ). Then we obtain β11 = 1 ,

β21 = 0 ,

β31 = 0 ,

β12 = 0 ,

β22 = 1 ,

β32 = 0 ,

β13

β23

β33

2

= h x/ ˙ z˙ ,

2

= h y/ ˙ z˙ ,

It is important for us that (βρσ ) =

(4.40) 2

= −h /(2z) ˙ ,

 ∂ q˙σ 

∂v∗ρ For the considered problem we have

2

2

2

h = b /a .

.

T = m(x˙ 2 + y˙ 2 + z˙ 2 )/2 ,

Π = mgz .

Then, using formulas (4.40), we can generate Maggi’s equations (4.39): x ¨ + (m¨ z + mg)(h2 x/ ˙ z) ˙ = 0,

(4.41)

m¨ y + (m¨ z + mg)(h2 y/ ˙ z) ˙ = 0,

(4.42)

2

˙ = Λ. (m¨ z + mg)(−h /(2z))

(4.43)

If we now consider Lagrange’s equations of the first kind m¨ x = Λ2x˙ ,

m¨ y = Λ2y˙ ,

m¨ z = −mg + Λ(−2z/h ˙ 2) ,

(4.44)

then it is easily seen that they are the linear combination of equations (4.41)– (4.43). We need to solve Lagrange’s equations of the first kind (4.44) together with the equation of constraint (4.35), in which case the fact that the unknowns involve the reaction Λ makes to be rather sophisticate the solution. At the same time the determining of the motion itself from equations (4.35), (4.41), (4.42) turns out more simple since the reaction can be found from equation (4.43). In addition, using Maggi’s equation, the form of reaction can be obtained also in original curvilinear coordinates (4.37). In fact, in the case of ideal nonholonomic constraint (4.35) we have ∇ ′ ϕ1 = Λ R = N = Λ∇

∂ϕ1 τ e = (m¨ z + mg)(−h2 /z)( ˙ xi ˙ + yj ˙ − h−2 zk) ˙ . (4.45) ∂ q˙τ

By (4.45) the direction of horizontal constraint (4.35) is opposed to the horizontal component of velocity of the mass m, what is distinctive feature of

66

II. Nonholonomic Systems

the motion of ball. Consider also nonholonomic constraint (4.36), which is convenient to rewrite as ϕ2 ≡ y˙ − x˙ tg θ = 0 .

(4.46)

Then, by reason of the existence of constraint (4.46), together with the reaction R given by (4.45), we also need to consider the reaction R∗ = Λ∗∇ ′ ϕ2 = Λ∗ (− tg θi + j) . The latter provides the satisfaction of constraint (4.46) and is distinctive feature of the motion of disk. In the case of the massless coordinate θ the value of this angle for ρ = 0 does not enter into the system of equations of motion and therefore it is natural to consider the rolling of the ball in place of the disk. For ρ = 0 we might consider the rolling of the massless disk but in this case there must exists the mechanism for the disk to be oriented in the corresponding way. The indefiniteness, which occur in determining the angle θ, can be put out if either ρ = 0 is satisfied for the massless disk, frame, and blocks, either for ρ = 0 with due regard any of these masses. In place of the account of their masses we can evidently consider not the material point of the mass m but the body of the same mass, which, descending in the rail mentioned above, rotates together with the frame about the axis BC. The examples of employing the Hamel–Novoselov equations are given, in particular, in the work [65] and the examples of the application of the Poincar´e–Chetaev equations in the works [149, 203, 229]. § 5. The Suslov–Jourdain principle Consider the vector  = vσ eσ , V

vσ ≡ q˙σ ,

σ = 1, s .

(5.1)

 differs from the velocity of representation In the general case this vector V point since ∂y ∂y ∂y V= + σ q˙σ = + vσ eσ . ∂t ∂q ∂t

Above the new variables v∗ρ were introduced by formulas (2.2) in place of the variables vσ ≡ q˙σ . By assumption, there is also inverse transformation (2.3). We emphasize that in the mentioned above transformations the time t and the coordinates q σ are regarded as parameters. Introduce the variations δ ′ vσ and δ ′ v∗ρ of the variables vσ and v∗ρ , defining them as the partial differentials of these functions, related as δ ′ vσ =

∂ vσ ′ ρ δv , ∂v∗ρ ∗

δ ′ v∗ρ =

∂v∗ρ ′ σ δ v , ∂ vσ

ρ, σ = 1, s .

(5.2)

67

5. The Suslov–Jourdain principle Recall that in formulas (2.2) we make use of relations (2.5) and therefore ∂ϕκ ′ σ δ v = 0 , ∂ vσ

δ ′ v∗l+κ =

σ = 1, s ,

κ = 1, k ,

(5.3)

and formulas (5.2) take the form δ ′ vσ =

∂ vσ ′ λ δv , ∂v∗λ ∗

δ ′ v∗λ =

Consider the vector

∂v∗λ ′ σ δ v , ∂ vσ

 = δ ′ vσ eσ = δ′ V

σ = 1, s ,

∂ vσ ′ λ δ v eσ = δ ′ v∗λε λ , ∂v∗λ ∗

λ = 1, l .

(5.4)

(5.5)

 given by relation (5.1), the new and construct together with the vector V, vector  =V  + δ′ V  = ( v σ + δ ′ vσ )eσ = (q˙σ + δ ′ vσ )eσ . V

 into the We substitute the coordinates q˙σ + δ ′ vσ of the generalized velocity V κ equations of constraints (2.1) and expand the functions ϕ (as the function of the variables q˙σ only) in the Taylor series in the neighborhood of the point with the coordinates (q 1 , . . . , q s ), corresponding to time t:  + o(|δ ′ V|)  , ˙ + ∇ ′ ϕκ · δ ′ V ϕκ (t, q, q˙ + δ ′ v) = ϕκ (t, q, q)

κ = 1, k . (5.6)

These relations imply that if for the point with the coordinates (q 1 , . . . , q s )  is kinematically possible, then with an at time t the generalized velocity V  =V  + δ′ V  is also kinematically accuracy to the first order the velocity V possible under the condition  = 0, ∇ ′ ϕκ · δ ′ V

κ = 1, k .

(5.7)

 satisfying equation (5.7), describes the Thus, a set of the vectors δ ′ V,  allowed by kinematically possible changes, of the generalized velocity V, 1 constraints at time t when the system is in the position (q , . . . , q s ). The  satisfying relations (5.7), is called a variation of the arbitrary vector δ ′ V,  generalized velocity V. Since the variations δ ′ v∗λ are linear independent, the family of Maggi’s equations (2.10) is equivalent to one equation  ∂ q˙σ ′ λ M Wσ − Qσ δ v = 0, ∂v∗λ ∗

which by virtue of formulas (5.4) can be represented as  M Wσ − Qσ δ ′ vσ = 0

(5.8)

68

II. Nonholonomic Systems

or by (5.5) in the vector form: 

 = 0. M W − Y · δ′ V

(5.9)

It is important for us that these equations are independent of the choice of free variables v∗λ . They are obtained as a sequence of equations of motion (2.10) and therefore as a sequence of Newton’s equation (2.9), written for ideal nonholonomic constraints with reaction (2.8). We remark that by formulas (5.9), (2.8), (2.9) we have  = 0, (5.10) R · δ′V

i. e. the reaction of ideal nonholonomic constraints is orthogonal to the vector of variation of generalized velocity. We obtain now Maggi’s equations from relation (5.9) being regarded as  has the form (5.5), then scalar product the initial one. Since the vector δ ′ V (5.9) is as follows ∂ q˙σ ′ λ  δ v = 0, M Wσ − Qσ ∂v∗λ ∗

which implies in accordance with linear independence of variations that δ ′ v∗λ , λ = 1, l. So, we arrive at Maggi’s equations (2.10). Thus, relation (5.9) can be regarded as a differential variational principle of mechanics, according to which for systems with ideal retaining nonholonomic constraints the scalar product of the vector of constraint reaction by the variation of generalized velocity is equal to zero. This principle has been formulated in 1908–1909 by P. Jourdain [326] and in 1900 by G. K. Suslov [218], who named it a universal equation of mechanics. That is why it is reasonable to refer this principle as the Suslov–Jourdain principle.

E x a m p l e II .8 . The equations of motion for Novoselov’s reducer (the generation of equations of motion by means of the Suslov–Jourdain principle). We generate the equations of motion for friction reducer, which was considered first by V. S. Novoselov [170]. The reducer (Fig. II. 9) transmits the rotation from shaft 1 to shaft 2 and consists of disk A, rigidly fixed with shaft 1, wheel B, freely rotating on shaft 3, shaft 2 with cylinder C, and a centrifugal regulator by the masses K and N and a spring with the deflection rate c1 . The motion of muff D of regulator with the help of a cable, turned over fixed blocks O1 and O2 , and a spring with the deflection rate c2 results in the displacement of shaft 3 with wheel B and changes the distance ρ between the average circle of wheel B and the axis of shaft 1. Wheel B has the radius a. The sizes given are the following: P N = N L = LK = KP = l. The position of the friction reducer is given by the following generalized coordinates: the rotation angles of shafts q 1 = ϕ1 and q 2 = ϕ2 and the distance q 3 = x of muff D from joint L. As is shown in Fig. II. 9, the distance ρ is related to x as x − ρ = c ≡ const .

69

5. The Suslov–Jourdain principle

Fig. II. 9

We consider a system with the nonholonomic constraint ϕ(t, q 1 , q 2 , q 3 , q˙1 , q˙2 , q˙3 ) ≡ (x − c)ϕ˙ 1 − Rϕ˙ 2 = 0.

(5.11)

If the sliding is missed, then constraint (5.11) gives the condition that the rotational velocities of the points of contact of wheel B with disk A and cylinder C are equal. The kinetic and potential energies are defined as  R2 1 JA ϕ˙ 22 +JC ϕ˙ 22 +mD x˙ 2 +mB ρ˙ 2 +JB 2 ϕ˙ 22+ T = 2 a   2 2 2 x l x ˙ ϕ˙ 22 + 2 +2mN l2 − , 4 4l −x2 1 1 Π = c1 (δ1 + x − x0 )2 + c2 (δ2 + x0 − x)2 . 2 2 Here δ1 , δ2 are the static deformations of springs with the deflection rates c1 and c2 , respectively, x0 is the static declination of muff D from joint L. For this system the Suslov–Jourdain principle is as follows (M W1 − Q1 )δ ′ ϕ˙ 1 + (M W2 − Q2 )δ ′ ϕ˙ 2 + (M W3 − Q3 )δ ′ x˙ = 0 .

(5.12)

The relation between the variations of velocities have the form ∂ϕ ′ ∂ϕ ′ δ ϕ˙ 1 + δ ϕ˙ 2 = 0 . ∂ ϕ˙ 1 ∂ ϕ˙ 2

(5.13)

Therefore in equation (5.12) the variations δ ′ ϕ˙ 2 and δ ′ x˙ are independent. Express from relation (5.13) the variation δ ′ ϕ˙ 1 in terms of δ ′ ϕ˙ 2 and then

70

II. Nonholonomic Systems

from equation (5.12) we obtain (M W1 − Q1 )

R + (M W2 − Q2 ) = 0 , x−c

M W3 − Q3 = 0 .

(5.14) (5.15)

Here Q1 = M1 , Q2 = −M2 are the moments of forces, impressed upon shafts 1 and 2, respectively, and Q3 = −∂Π/∂x. From the general theory it follows that the equations obtained coincide with Maggi’s equations. We remark that the second of them is usual Lagrange’s equation of the second kind since the coordinate x is holonomic. Since d ∂T ∂T M Wσ = − σ, σ = 1, 3 , σ dt ∂ q˙ ∂q then equations (5.14) and (5.15) can be rewritten as R R ϕ¨1 + J(x)ϕ¨2 − mN xx˙ ϕ˙ 2 = M1 − M2 , x−c x−c 2l2 x 1 mN x˙ 2 = c1 (−δ1 −x+x0 )+c2 (−x+x0 +δ2 ) . m(x)¨ x + mN xϕ˙ 22++ 2 2 (4l +x2 )2 (5.16) Here 1 R2 J(x) = JC + JB 2 + mN (4l2 − x2 ) , a 2 2mN l2 m(x) = mB + mD + 2 . 4l − x2 The equations of motion (5.16) together with equation of constraint (5.11) give a closed system for determining the functions ϕ1 (t), ϕ2 (t), x(t). It is to be noted that if we substitute equation of constraint (5.11), differentiated with respect to time, into the first equation of system (5.16), then the equations can be written in the Appell form. Such equations were also generated by A. I. Lur’e [135]. This example was also considered by Ya. L. Geronimus [47]. His results coincide with equations (5.16). JA

E x a m p l e II .9 . The motion of mechanical system with a fluid flywheel (the generation of the equations of motion for nonholonomic systems with the help of the Suslov–Jourdain principle and Lagrange’s equation of the first kind in generalized coordinates). The fluid flywheel consists of two centrifugal wheels filled by oil: a driving torus and a turbine. The driving torus is fixed with the motor shaft and while rotating, by means of blades and centrifugal force, it speeds up the oil, which falls with a great velocity on the blades of turbine, setting the latter in motion. The turbine is placed on the shaft of consumer and therefore by means of the fluid flywheel the rotation is imparted from the leading shaft to the driven one, in which case the connection between them turns out nonrigid. At present the fluid flywheels gain a wide application in different powerful transmissions, in the starters of

71

5. The Suslov–Jourdain principle

Fig. II. 10

gas turbines, the drives of pumps, the conveyors of hoisting machines, and so on. The study of transient processes in similar arrangements is of importance since the transient regime averages about sixty percents of their operation time. Consider one of possible approaches to the study of transient processes in systems with fluid flywheel or with fluid converter, which differs from the fluid flywheel by an additional wheel (of reactor). We shall regard mechanical systems with hydrotransmissions as first-order nonholonomic systems. This permits us to eliminate the reaction and to obtain the equation of motion, which is to be integrated together with the equation of constraint. Denote by ω1 and L1 an angle velocity and a moment generated by a motor, respectively, by J1 a moment of inertia of impeller and driving parts of motor, by ω2 and L2 an angle velocity and a drag torque produced by customer, respectively, by J2 a moment of inertia of a turbine and driven parts of device. Suppose that for the racing of system the following characteristics of the motor L1 = L1 (ω1 ) and the customer L2 = L2 (ω2 ) hold which taken off for the regimes being steady-state (see. Fig. II. 10; these and all subsequent numerical data are taken from the work [106]). The quantities L1 , L2 are given in Newton-meters, (N·m), ω1 , and ω2 in the seconds in the minus first degree (s−1 ), t in seconds (s). The process of the racing of system can be partitioned into three stage. At the first stage after a starting of motor its moment L1 , applied to the driving parts of device, is used for their racing and for the racing of fluid in the fluid flywheel. When the flow occurs in the work chamber of fluid flywheel the moment L occurs on its fixed driving torus. At the end of the first stage the value of the moment L becomes sufficient for the initiation of driven parts (L2 for ω2 = 0) and we have the second stage when the turbine begins to rotate with the ascending angle velocity ω2 . The racing of system at the third stage is characterized by increasing the angle velocity ω2 with deacceleration of flow, in which case the moment L, generated by the turbine, is greater than the moment, which the driving torus

72

II. Nonholonomic Systems

Fig. II. 11

imparts to a flow. At this stage the moment L generally decreases. When its value for the certain angle velocity ω2 becomes equal to the moment of customer L2 , the racing is ended and the system operates in the steady-state regime for L1 = L = L2 . The analysis of experimental and computed investigations of racing processes for the systems with different relative moments of inertia J = J1 /J2 and with different characteristics of a motor and a customer shows that in view of a great power of motor the angle velocity of driving shaft varies slightly and the angle velocity of driven shaft varies essentially at the initial period and tends asymptotically to a certain constant value as the mode of operation tends to the steady-state regime when we have ω1 = ω2 + const. For the racing of system with hydrotransmission the graphs of the functions ω1 = ω1 (t) and ω2 = ω2 (t) have a specific form shown in Fig. II. 11. These graphs demonstrate that for as nonstationary as stationary regimes we have ω1 = ω2 . Therefore between the angle velocities of the driving and driven shafts there exists a certain functional relation, which can be regarded as a nonholonomic constraint. Since from the graphs of the functions ω1 = ω1 (t) and ω2 = ω2 (t) we can obtain the relation for the angle velocities as a function of time, then the equation of nonholonomic constraint can be represented as ϕ(t, ω1 , ω2 ) ≡ ω2 − i(t)ω1 = 0 .

(5.17)

The kinetic energy of system is defined by the following relation J1 ω12 J2 ω22 + . 2 2 Let us generate the equations of motion. The Suslov–Jourdain principle for this system takes the form T =

(M W1 − Q1 )δ ′ ω1 + (M W2 − Q2 )δ ′ ω2 = 0 .

(5.18)

The relation between the variations of angle velocities is given by the following formula ∂ϕ ′ ∂ϕ ′ δ ω1 + δ ω2 = 0 . (5.19) ∂ω1 ∂ω2

73

5. The Suslov–Jourdain principle Hence from equation (5.18) with relations (5.17) and (5.19) we have J1

dω1 dω2 + i(t)J2 = L1 − i(t)L2 . dt dt

(5.20)

Equation (5.20) together with equation of constraint (5.17) give the closed system for determining the functions ω1 (t) and ω2 (t). Equation of motion (5.20) can also be obtained in a different way. We write Lagrange’s equations of the first kind in generalized coordinates for nonholonomic system (2.22): d ∂T ∂T − = L1 + R1 , dt ∂ω1 ∂ϕ1 d ∂T ∂T − = −L2 + R2 . dt ∂ω2 ∂ϕ2

(5.21)

Here the generalized reactions are defined as R1 = Λ

∂ϕ = −i(t)Λ , ∂ω1

R2 = Λ

∂ϕ = Λ. ∂ω2

Eliminating Λ from system (5.21), we obtain equation of motion (5.20). When passing to the stationary regime, the relation ω1 = ω2 + const is satisfied, in which case we have R1 |ω1 =ω2 +const = −Λ

∂ϕ  = −Λ ,  ∂ω1 ω1 =ω2 +const

i. e. −R1 = R2 . The model suggested coincides with the model considered in the work [106] since in the equations of this work L1 = (J1 + J ∗ )

dω1 + L, dt

L = J2

dω2 + L2 , dt

the correction J ∗ ε1 , which accounts for the moment, initiating the racing of flow in rotor blades, is equal to zero. In the above equations R1 = −R2 = −L. In the work [106] the quantity J ∗ and the moment L, transmitted by a fluid flywheel, are accounted empirically. Having solved the system of equations (5.17), (5.20), we can find the reaction and, by that, the moment developed by the fluid flywheel. The system of equations (5.17), (5.20) was numerically integrated by computer. The computing permits us to obtain the following relations: the behavior of the angle velocities of the driving and driven shafts in time, the change of the moments on the driving and driven shafts, the behavior of the moment, transmitted by fluid flywheel, in time. The certain results of computing are represented in Fig. II. 12. Thus, the method suggested permits us, using the experimental data ω1 = ω1 (t), ω2 = ω2 (t), to describe nonstationary processes in systems with

74

II. Nonholonomic Systems

Fig. II. 12

hydrotransmission. In this case we take into account an error of experiment only while in the other methods, moreover, the additional inaccuracy occurs by reason of using the approximate theory for the account of hydrodynamic processes. § 6. The definitions of virtual displacements by Chetaev As is noted in the review of the main stages of the development of nonholonomic mechanics, the definition of nonholonomic system was introduced first by Hertz [317] in 1894. He was the first to pay attention to the possible existence of such kinematical constraints, which do not impose some restrictions on the possibility of the passing of system from one position to another. The development of nonholonomic mechanics was essentially due to the work of E. Lindel¨of [352], in which with the help of usual methods of holonomic mechanics there were obtained the incorrect equations of motion of nonholonomic system. In particular, S. A. Chaplygin points to this error and suggests his own method [239] to obtain the equations of motion. The error similar to Lindel¨ of’s error was also made by C. Neumann [366], what was repeatedly remarked in the literature (see, for example, [41]). Further, in 1899, C. Neumann gives already the certainly valid equations of motion [366]. For the description of motions of nonholonomic systems, P. Appell, L. Boltzmann, V. Volterra, P. V. Voronets, G. Hamel, J. Gibbs, Bl. Dolaptschiew, G. A. Maggi, L. M. Markhashov, J. Nielsen, V. S. Novoselov, G. S. Pogosov, A. Przeborski, V. V. Rumyantsev, J. Schouten, Fam Guen, N. Ferrers, J. Tz´enoff, S. A. Chaplygin, M. F. Shul’gin, and another authors suggest a number of different methods to generate the differential equations of motion. Some of them are considered, for example, in the treatises [59, 166]. At the time of intensive development of nonholonomic mechanics many scholars often obtained similar results and therefore the same forms of equations of motion have different names. The investigations, connected with the

75

6. The definitions of virtual

possible applications of these equations to more wide classes of nonholonomic constraints, are being continued to the present (see, for example, [370]). In generating the equations of motion of nonholonomic systems, most authors made use of the D’Alembert–Lagrange principle that they extended to the case of the system under consideration. In this case they need to clarify what should be regarded as a virtual displacement for the given type of constraint. V. V. Kozlov [114, p. 60, 61] notes that ". . . for such method for constructing the dynamics, the hypotheses must involve the definition of virtual displacements"and "even in the simplest case of stationary integralable constraint the definition of virtual displacements is the independent hypothesis of dynamics". Exactly such hypothesis (see below (6.3)) was brightly formulated by N. G. Chetaev. He aims [245, p. 68] ". . . at the introduction of the notion of virtual displacement for nonlinear constraints in such a way that to save as D’Alembert’s, as Gaussian principles . . . ". For the sake of generality, we consider an arbitrary mechanical system, the position of which is given by the generalized coordinates q σ , σ = 1, s. Suppose that on this system is imposed the following nonlinear nonholonomic constraints ϕκ (t, q, q) ˙ = 0, κ = 1, k , k < s. (6.1) We remark that the nonholonomicity of these constraints makes itself evident in the fact that in spite of their existence the passage of system from any its position with the coordinates q0σ , σ = 1, s, to another with the coordinates q1σ , σ = 1, s, is kinematically possible. According to N. G. Chetaev for the real motion of the system considered the D’Alembert–Lagrange principle   d ∂T ∂T σ − − Q (6.2) σ δq = 0 dt ∂ q˙σ ∂q σ must be satisfied. We assume that the kinetic energy T has the form T =

M gαβ (t, q)q˙α q˙β , 2

α, β = 0, s ,

q0 = t ,

q˙0 = 1

and the generalized forces Qσ are given in terms of the functions of the time t, the coordinates q σ , and the generalized velocities q˙σ (σ = 1, s). N. G. Chetaev also assume that the quantities δq σ , entering into the D’Alembert–Lagrange principle (6.2), satisfy the following conditions ∂ϕκ σ δq = 0 , ∂ q˙σ

κ = 1, k .

(6.3)

Nonlinear nonholonomic constraints (6.1) such that conditions (6.3) are assumed to be valid, were named the constraints of the Chetaev type. As is shown in the previous section, the general principle of nonholonomic mechanics is the Suslov–Jourdain principle (5.9) or (5.8) in which the variations of velocity must satisfy conditions (5.3). The virtual displacements,

76

II. Nonholonomic Systems

allowed by the constraints of the Chetaev type, are satisfied exactly such conditions (6.3). Therefore the generalized principle of D’Alembert–Lagrange, which in the case of Chetaev’s postulate permits us to apply the usual D’Alembert–Lagrange principle (6.2) to the study of nonholonomic systems, coincides with the Suslov–Jourdain principle. The comparison of formulas (6.3) and (5.3) implies, in turn, that the virtual displacements (δq 1 , . . . , δq s ), introduced by Chetaev for nonlinear nonholonomic constraints, coincide with the variations of the generalized velocity (δ ′ v1 , . . . , δ ′ vs ). Like the holonomic problems from formula (5.10) it follows that the reaction of constraints of the Chetaev type is orthogonal to the virtual displacements satisfying conditions (6.3). Differential forms (6.3) show that the scalar products of the vectors δy = δq σ eσ by the vectors ∇ ′ ϕκ ≡ ε l+κ , κ = 1, k, are equal to zero. Formulas (5.7) stress even this orthogonality. Thus, conditions (6.3) suggested by N. G. Chetaev, which give the axiomatic definition of virtual displacements with nonlinear nonholonomic constraints, show the possibility of the transition, in the nonholonomic mechanics, from the vectors given on the manifold of possible positions of mechanical system to the vectors given on the manifold of possible velocities of system. Relations (6.3) have played an important role in the development of nonholonomic mechanics. The different forms of conditions of type (6.3) were also introduced by the other famous scholars, for example, by J. W. Gibbs [309], P. Appell [265], and A. Przeborski [375]. J. Papastavridis [370. 1997, 2002] refers to conditions (6.3) as the Maurer–Appell–Chetaev–Hamel definition. Note, that much attention to this definition of virtual displacements with nonlinear nonholonomic constraints is paid in the works of Norwegian scientist L. Johnsen [324], which are not too famous. We shall repeatedly remark further the role of Chetaev’s type constraints for obtaining the cited results.

C h a p t e r III LINEAR TRANSFORMATION OF FORCES

In this Chapter the linear transformation of forces is introduced. In this case for holonomic systems the notion of ideal constraints and the relation for virtual elementary work are applied. By the transformation of forces, Lagrange’s equations of the first and second kinds are obtained. The theorem of holonomic mechanics is formulated by which the given motion over the given curvilinear coordinate can be provided by an additional generalized force corresponding to this coordinate. For nonholonomic systems the linear transformation of forces is introduced applying Chetaev’s postulates. In this case with the help of generalized forces, corresponding to the equations of constraints, the family of fundamental equations of the nonholonomic mechanics is obtained in compact form. The theorem of holonomic mechanics is formulated according to which the given change of quasivelocity can be provided by one additional force corresponding to this quasivelocity. The application of the formulated theorems of the holonomic and nonholonomic mechanics is demonstrated on the example of the solution of two problems on the controllable motion connected with the flight dynamics. At the end of this chapter the linear transformation of forces is used to obtain the Gauss principle.

§ 1. Some general remarks In § 1 of Chapter I for studying the motion of representation point under constraints (1.10) f κ (t, y) = 0 ,

y = (y1 , . . . , y3N ) ,

κ = 1, k ,

(1.1)

the following differential equations (1.8) are used: M y¨µ = Yµ + Rµ ,

µ = 1, 3N .

(1.2)

In equations (1.1) and (1.2), the general number of which is equal to 3N + k, the unknowns are the functions yµ and the reactions Rµ , µ = 1, 3N . Thus, the number of unknowns exceeds the number of equations by 3N − k. There arises the natural question wether the present problem can be solved and we can attain that the solution of problem is unique and the number of equations is equal to the number of unknown. It turns out that for the answer to that important question of analytic mechanics the type of constraints: holonomic or nonholonomic, is nonessential. Therefore for the sake of generality, we assume that the constraints can be given by either the equations ˙ = 0, ϕκ (t, y, y) 77

κ = 1, k ,

(1.3)

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III. Linear Transformation of Forces

or the equations ˙ y¨) = al+κ ˙ y¨µ + al+κ (t, y, y) ˙ = 0, ψ κ (t, y, y, µ (t, y, y) 0 κ = 1, k ,

µ = 1, 3N .

(1.4)

˙ can also depend nonlinearly on velocities. Note that the functions ϕκ (t, y, y) Differentiating of equations of constraints (1.1) twice and equations (1.3) once with respect to time, we can represent all types of equations of constraints in uniform differential form (1.4). Then for holonomic constraints (1.1) we obtain al+κ = µ

∂f κ , ∂yµ

al+κ = 0

∂2f κ ∂2f κ ∂2f κ y˙ µ y˙ µ∗ , + 2 y ˙ + µ ∂t2 ∂t ∂yµ ∂yµ ∂yµ∗

κ = 1, k ,

µ, µ∗ = 1, 3N

and for nonholonomic constraints (1.3) we have = al+κ µ

∂ϕκ , ∂ y˙ µ

al+κ = 0

κ = 1, k ,

∂ϕκ ∂ϕκ y˙ µ , + ∂t ∂yµ

µ = 1, 3N .

As is known [194, 295], given nonholonomic constraints, it is impossible, in principle, to introduce the less number of new variables via which the coordinates yµ , µ = 1, 3N , can uniquely be expressed. Therefore we can use the equations of constraints only for that to change from the unknowns Rµ to the new unknowns Λκ , the number of which must be equal to the number of constraints and via which the reactions Rµ can uniquely be expressed. This problem of analytic mechanics can be solved in the following way. At first we suppose that the constraints are lacking. Then the vector equation of motion is as follows MW = Y . For concrete initial data it permits us to find uniquely a subsequent motion of mechanical system when the force Y is a given function of time, a position of system, and its velocities. Suppose now that the force Y is lacking and the system moves mechanically. At time t = t0 when the system is in a position with the coordinates yµ0 and has the projections of velocities y˙ µ0 (µ = 1, 3N ), the constraints are imposed on the motion of system. According to the releasability principle this leads to the occurrence of the force of reaction R. Therefore, beginning from time t = t0 , Newton’s second low can be written as MW = R . This equation allows us to find uniquely the subsequent motion in the case when by the equations of constraints the vector R can be found as the function of time, position, and velocities of system.

79

1. Some general remarks

In the case of simultaneous operation of active forces and the forces, generated by constraints, Newton’s secondlaw takes the form MW = Y + R .

(1.5)

Obviously, in this case for the given initial data the subsequent motion can also be found only when the force R is represented as a function of time, coordinates, and velocities of system. Clear up how this can be made. Consider the vectors εl+κ = al+κ jµ , µ

l = 3N − k ,

κ = 1, k .

(1.6)

For holonomic and nonholonomic constraints (1.1), (1.3), (1.4) vectors (1.6) can be represented, respectively, in the form

ε l+κ =

ε l+κ =

∂f κ jµ = ∇ f κ , ∂yµ

ε l+κ =

∂ϕκ j µ = ∇ ′ ϕκ , ∂ y˙ µ

∂ψ κ jµ = ∇ ′′ ψ κ , ∂ y¨µ

l = 3N − k ,

(1.7)

κ = 1, k ,

µ = 1, 3N .

The introduction of the vectors ε l+κ permits us to represent equations of constraints (1.4) in the following way: ˙ , ε l+κ · W = χκ (t, y, y)

χκ = −al+κ , 0

κ = 1, k .

(1.8)

These relations imply that in the 3N -dimensional Euclidean space we use it is rational to consider a subspace, the basis of which are the vectors ε l+κ , κ = 1, k. In this case the 3N -dimensional Euclidean space can be represented as a direct sum of this subspace and the orthogonal to it complement, the basis of which are the vectors ε λ , λ = 1, l, satisfying the relations εl+κ · ελ = 0 ,

κ = 1, k ,

λ = 1, l .

Denote the introduced subspaces by K-space and L-space. These subspaces permit us to represent the acceleration of representation point as the following sum W = WL + WK ,

λ ελ , WL = W

WL · WK = 0 .

l+κ ε l+κ , WK = W

(1.9)

It is to be noted that this representation of the vector W corresponds to the fixed values of the variables t, yµ , y˙ µ (µ = 1, 3N ). In fact, the idea of introduction of K-space is based on the Chetaev postulate on the ideality of nonlinear nonholonomic constraints [245]. This postulate can be interpreted as the requirement of the orthogonality of the vectors of virtual displacements of system to the vectors ε l+κ , κ = 1, k.

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III. Linear Transformation of Forces

Using relations (1.9), we can replace equation (1.5) by two following equations M WL = YL + RL , (1.10) M WK = YK + RK . Here

λ ε λ , YL = Q

(1.11)

 l+κ ε l+κ , YK = Q

 λ ελ , RL = R RK = Λκ εl+κ .  l+κ of the vector RK are denoted by Λκ since The covariant components R in the sequel they turns out equal to the Lagrange multipliers. Vector equation (1.11) is equivalent to the following k scalar equations l+κ − Q  l+κ , Λκ = M W

κ = 1, k .

(1.12)

l+κ , is comWe show that the vector WK , given by the components W pletely determined by the equations of constraints. Really, relations (1.8) and (1.9) give ∗ l+κ ∗ = χκ (t, y, y) ˙ , κ, κ ∗ = 1, k , (1.13) hκκ W where

∗

∗

hκκ = ε l+κ · ε l+κ ,

κ, κ ∗ = 1, k .

If ∗

|hκκ | =  0,

(1.14)

then the solution of system of linear algebraic equations (1.13) can be represented as l+κ ∗ = hκ ∗ κ χκ (t, y, y) W ˙ , κ, κ ∗ = 1, k . (1.15) ∗

Here hκ ∗ κ are the elements of a matrix inverse to the matrix of elements hκκ . Thus, the vector WK , entering into equation (1.11), is a vector, which as the function of time, a position of system, and its velocities is uniquely determined by equations of constraints. The second vector YK in equation (1.11) is assumed to be the function of the same arguments. Hence by (1.11) the vector RK also can be found as a function of time, position, and velocities of system. The representation of this vector in the form RK = Λκ ε l+κ

implies that its determination is reduced to the numerical computation of the quantities Λκ , κ = 1, k, by formulas (1.12) and (1.15). Thus, for the equations of constraints to be satisfied it is necessary to add the force RK to the active force YK . We shall show that this condition is also sufficient. Really, the influence of equations of constraints on the vector W is given by formulas (1.8). Since ε l+κ · WL = 0, κ = 1, k, formulas (1.8) can be represented as ε l+κ · WK = χκ (t, y, y) ˙ ,

κ = 1, k .

(1.16)

1. Some general remarks

81

It follows that the constraints are satisfied for any vector WL and the form of equations of constraints does not allow us to say anything about the vector WL . Therefore without violating the condition that the equations of constraints are satisfied we can assume in equation (1.10) that RL = 0. Thus, we can solve the problem, formulated above: under which conditions and in which form the vector of reaction R can be expressed via the quantities Λκ , the number of which is equal to the number of constraints, and how this vector can be represented in terms of a function of time, a position of system, and its velocities. Firstly, this problem can be solved in the case when the equations of constraints are independent, i. e. condition (1.14) is satisfied and, secondly, when the vector RL , which does not directly connected with the equations of constraints, is equal to zero. The procedure of numerical computation of the vector R = RK is shown as part of the study. The holonomic and nonholonomic constraints under the assumption RL = 0, are called ideal. This implies that these constraints are completely defined by their analytic representations. We pay attention to the following fact, resulting from the above. In the case of free mechanical system the vector of acceleration W is defined as a function of time, coordinates, and velocities by Newton’slaw of the form W = W(t, y, y) ˙ = Y(t, y, y)/M ˙ . In the case of the existence of constraints in K-space, the vector WK as a function of time, coordinates, and velocities is uniquely defined by equations of constraints (1.8). In other words, in this subspace the motionlaw is given by equations of constraints (1.16). In K-space Newton’s secondlaw itself is written, if necessary, only for that the force of reaction RK can be found by means of thislaw. The equations of constraints cannot affect the vector WL belonging to L-space and therefore in this subspace the vector WL under ideal constraints can be found by Newton’s secondlaw, i. e. from the equation WL = WL (t, y, y) ˙ = YL (t, y, y)/M ˙ . Multiplying the last vector relation by the vectors ε λ , generating the basis of L-space, we obtain  M WL − YL · ε λ = 0 , λ = 1, l .

Since WK · ε λ = 0 and YK · ε λ = 0, the system of these scalar equations can be represented as  M W − Y · ελ = 0 , λ = 1, l . (1.17) Supplementing system of equations (1.17) with equations (1.8), we obtain the closed system of equations since in this case the vector W as a function of time, position, and velocities of system is given in the whole space. Note

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III. Linear Transformation of Forces

that system of equations (1.17) does not contain the reactions of constraints. The concrete form of them depends on the form of a given system of the vectors ε λ , λ = 1, l, orthogonal to the vectors ε l+κ , κ = 1, k. For holonomic systems such equations were obtained by Lagrange (see Chapter I), for firstorder nonholonomic systems they were suggested by P. Appell, L. Boltzmann, P. V. Voronets, G. Hamel, G. A. Maggi, L. M. Markhashov, V. S. Novoselov, ˙ Rumyantsev, Fam Guen, S. A. Chaplygin and others (see Chapter II). V.V. The equations with high-order nonholonomic constraints will be discussed in Chapter V. So, under ideal constraints Newton’s law can be written as M W = Y + Λκ ε l+κ ,

κ = 1, k .

(1.18)

Therefore the motion of representation point under ideal holonomic or nonholonomic constraints is described by Lagrange’s equations of the first kind, which give with (1.7) for constraints (1.1), (1.3), and (1.4), respectively, the following relations M y¨µ = Yµ + Λκ

∂f κ , ∂yµ

µ = 1, 3N ,

κ = 1, k ,

(1.19)

M y¨µ = Yµ + Λκ

∂ϕκ , ∂ y˙ µ

µ = 1, 3N ,

κ = 1, k ,

(1.20)

M y¨µ = Yµ + Λκ

∂ψ κ , ∂ y¨µ

µ = 1, 3N ,

κ = 1, k .

(1.21)

From relations (1.12), (1.15) it follows that the Lagrange multipliers (generalized reactions) Λκ , entering into these equations, are uniquely determined by the equations of constraints and the active forces in the form of the functions of t, y, y. ˙ Hence, by equations (1.19)–(1.21) we can always find a motion, satisfying the equations of constraints. Thus, we obtain the answer to the question how the number of unknowns can be diminished and the reactions Rµ can be expressed via the quantities Λκ , the number of which is equal to the number of constraints. To depend upon the form of ideal constraints, we obtain the following representations for reactions: R µ = Λκ

∂f κ , ∂yµ

µ = 1, 3N ,

κ = 1, k ,

(1.22)

R µ = Λκ

∂ϕκ , ∂ y˙ µ

µ = 1, 3N ,

κ = 1, k ,

(1.23)

R µ = Λκ

∂ψ κ , ∂ y¨µ

µ = 1, 3N ,

κ = 1, k .

(1.24)

Further we shall show that the form itself of these linear relations between the quantities Rµ and Λκ permits us to answer many questions and, in particular, to establish why the quantities Λκ is called a generalized reaction.

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2. Theorem on the forces of holonomic constraints

The structure of linear transformations (1.22)–(1.24), as will be shown further, allows us to separate the equations for determining the motion, from the equations for obtaining the generalized reactions. We remark that by representations (1.22)–(1.24) we can construct a single solution only in the case when condition (1.14) is satisfied.

§ 2. Theorem on the forces providing the satisfaction of holonomic constraints If the representation point is free, then any of its positions can be given by the Cartesian coordinates as well as by the curvilinear ones. In the general case these coordinates are related as yµ = yµ (t, q) ,

q σ = q σ (t, y) ,

µ, σ = 1, s ,

s = 3N ,

(2.1)

and depend on time. The formulas, introduced in the present section, will be written, where possible, in the form, which admits the development in the case of any mechanical system. Therefore in the sequel the curvilinear coordinates of representation point will be called generalized Lagrangian coordinates or, simply, generalized coordinates and their number will be assumed equal to s. Newton’s secondlaw, describing the motion of representation point, in the case of free system takes the form MW = Y .

(2.2)

Multiplying this vector relation by the vectors of the fundamental basis of the introduced curvilinear system of coordinates eσ , σ = 1, 3N , we obtain Lagrange’s equations of the second kind ∂T d ∂T − σ = Qσ , dt ∂ q˙σ ∂q

σ = 1, s ,

s = 3N ,

(2.3)

which can be represented in explicit form as M (gστ q¨τ + Γσ,αβ q˙α q˙β ) = Qσ , σ, τ = 1, s , Here Γσ,αβ = eα =

α, β = 0, s ,

s = 3N .

  1 ∂gσα ∂gσβ ∂gαβ , + − 2 ∂q β ∂q α ∂q σ

∂y , gαβ = eα · eβ , q0 = t , q˙0 = 1 , ∂q α σ = 1, s , α, β = 0, s , s = 3N .

(2.4)

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III. Linear Transformation of Forces

Given constraints (1.1) it is rational to involve the functions, entering into the equations of constraints, in the system of functions, realizing the transition to generalized coordinates, i. e. to put q λ = f∗λ (t, y) , q

l+κ

=

f∗l+κ (t, y)

λ = 1, l , = f (t, y) , κ

l = 3N − k , κ = 1, k .

(2.5)

The independent coordinates q λ , λ = 1, l, are chosen arbitrary but in such a way that the coordinates yµ , µ = 1, 3N , can uniquely be expressed from system (2.5) via the coordinates q σ , σ = 1, 3N . Together with the introduction of generalized coordinates we consider generalized forces, using the relation for virtual elementary work. The invariant differential form, by which a virtual elementary work is given, can be represented as the chain of relations δA = Yµ δyµ = Yµ

∂yµ σ ∂q σ σ δq = Q δq = Q δyµ . σ σ ∂q σ ∂yµ

Whence it follows that the forces Yµ and the generalized forces Qσ are related linearly, namely Yµ = Qσ

∂q σ , ∂yµ

Qσ = Yµ

∂yµ , ∂q σ

µ, σ = 1, 3N .

Then, the reactions Rµ and the generalized reactions Rσ are also related linearly as Rµ = Rσ

∂q σ , ∂yµ

Rσ = Rµ

∂yµ , ∂q σ

µ, σ = 1, 3N .

(2.6)

When generated Lagrange’s equations of the first kind (1.19) we show that the imposition of ideal holonomic constraints (1.1) leads to the occurrence of reactions, given by formulas (see formulas (1.22)) R µ = Λκ

∂f κ , ∂yµ

κ = 1, k ,

µ = 1, 3N .

Taking into account relations (2.5) and (2.6), we obtain Rλ = 0 ,

λ = 1, l ,

Rl+κ = Λκ ,

κ = 1, k .

(2.7)

Thus, the generalized reactions, corresponding to the arbitrary generalized coordinates q λ , λ = 1, l, are equal to zero and the quantities Λκ , κ = 1, k, are equal to the generalized reactions, corresponding to the equations of constraints. In other words, from Lagrange’s equations of the first kind it follows that the imposition of each ideal constraint results in the occurrence of generalized reaction, providing the satisfaction of this constraint.

85

2. Theorem on the forces of holonomic constraints This implies that under the constraints q l+κ = f κ (t, y) = 0 ,

κ = 1, k ,

in the left-hand sides of equations (2.4) it is to be assumed q l+κ = q˙l+κ = q¨l+κ = 0, κ = 1, k. In the right-hand sides nothing is added to the generalized forces Qλ , λ = 1, l, but the generalized reactions Λκ , κ = 1, k, are added to the generalized forces Ql+κ . In this case it is rational to divide system of equations (2.3) by two groups d ∂T ∂T − λ = Qλ , λ dt ∂ q˙ ∂q

λ = 1, l ,

∂T d ∂T − l+κ − Ql+κ = Λκ , dt ∂ q˙l+κ ∂q

κ = 1, k .

(2.8) (2.9)

Thus, Lagrange’s equations of the second kind (2.8) results, in fact, from Lagrange’s equations of the first kind (1.19). As was already noted in Chapter I, for the given initial conditions from equations (2.8) we can find the law of motion of system q λ = q λ (t) ,

λ = 1, l ,

and determine then by formulas (2.9) the generalized reactions Λκ , κ = 1, k, in terms of the functions of time. For determining the generalized reactions, we make use of equations (2.9), in particular, in the case when in studying the dynamics of different mechanisms it is necessary to account for the dry friction forces, which are distinctive for nonlinear nonideal holonomic constraints. In this case by means of a rational choice of the generalized coordinates q l+κ , κ = 1, k, the generalized reactions Λκ , κ = 1, k, can be related directly to the forces of safe pressure of a system of elements on roughened surfaces. Then the dry friction forces, entering into the right-hand sides of equations (2.8), are connected directly with generalized reactions. In this case it is necessary to consider systems of equations (2.8) and (2.9) together. Recall that to compute the left-hand sides of equations (2.9) it is necessary also to know the relation for kinetic energy in terms of the coordinates q l+κ , κ = 1, k, and to assume that q l+κ = q˙l+κ = q¨l+κ = 0, κ = 1, k, but only after representing the left-hand sides in explicit form (2.4). In studying the motion under the releasing holonomic constraints we also need to obtain generalized reactions from equations (2.9). Lagrange’s equations of the second kind (2.8) are convenient for us since they can be constructed by means of the relations for kinetic energy and virtual elementary work. Therefore, as is already remarked, these equations can be applied to any mechanical system. Marvelling at the perfection of Lagrange’s equations, L. Pars [179, p. 90] writes: "The work of Lagrange [340] is a capital source of ideas of analytic mechanics and is legally considered as one of superlative spiritual achievements of humankind".

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III. Linear Transformation of Forces

We pay attention to that the established above connection between Lagrange’s equations of the first kind (1.19) and those of the second kind (2.8), (2.9) implies the following theorem on the forces providing the satisfaction of holonomic constraints: The motion, under which one of generalized coordinates is a given function of time, can be obtained by the introduction of one additional generalized forces, corresponding to this coordinate. A direct corollary of this theorem is a more general assertion that the motion, given at the same time by several coordinates, can be provided by the same number of the corresponding additional generalized forces. In other words, if in system of equations (2.8), (2.9) we assume q l+κ = F κ (t), κ = 1, k, then from this system we can obtain the functions q λ (t), λ = 1, l, as well as the functions Λκ (t) , κ = 1, k. Really, assuming that the constraints are given by the following equations f κ (t, y) = f∗l+κ (t, y) − F κ (t) = 0 ,

κ = 1, k ,

we can conclude that the satisfaction of each of them can be provided by the corresponding generalized reaction Λκ , κ = 1, k. The quantities Λκ , as is shown in § 1, are uniquely defined by the equations of constraints and by the active forces. If the expression for reactions is obtained, then from equations (1.19) we can find a motion, i. e. the functions yµ (t), µ = 1, 3N , and then we obtain the functions q σ (t), σ = 1, 3N . Since system of equations (2.8), (2.9) is equivalent to equations (1.19) from this system we can determine the functions q λ (t), λ = 1, l, as well as the functions Λκ (t) , κ = 1, k. The above theorem can easily be proved directly, without a reference to the results, obtained earlier. Below we give this proof. For definiteness, we assume that the only given function of time is the coordinate q s . Adding in equations (2.4) the quantities Λs only to the generalized force Qs , we obtain M (gρτ q¨τ + Γρ,αβ q˙α q˙β+ + 2Γρ,sτ q˙s q˙τ ) = Qρ − M (gρs q¨s + Γρ,ss (q˙s )2 + 2Γρ,0s q˙s ), ρ, τ = 1, s − 1 , σ

α β

M (gsσ q¨ + Γs,αβ q˙ q˙ ) = Qs + Λs ,

α, β = 0, s − 1 ,

(2.10)

σ = 1, s ,

(2.11)

α, β = 0, s .

The sum gρτ q˙ρ q˙τ (ρ, τ = 1, s − 1) is a positively defined quadratic form. Therefore the matrix with the elements gρτ (ρ, τ = 1, s − 1) is nonsingular. In this case system (2.10) is solvable for q¨τ (τ = 1, s − 1) and can be integrated for the given initial data. Then we find the functions q τ (t) (τ = 1, s − 1). Substituting them and the function q s (t) into the left-hand side of equation (2.11), we obtain the additional generalized force Λs (t) , providing the given motion over the coordinate q s . Note that the assertion proved does not mean that the motion over any coordinate does not effect the motion over the rest of coordinates. It means

87

2. Theorem on the forces of holonomic constraints

only that it is unnecessary to apply another additional forces except for the force, corresponding to the chosen coordinate. Really, the function q s (t) enters into the right-hand side of system (2.10) and therefore effects the functions q τ (t) (τ = 1, s − 1). Lagrange’s equations of the first and second kinds result from the above theorem. Really, from this theorem, generalized to the case of few coordinates, it follows that under the holonomic constraints q l+κ = f κ (t, y) = 0 ,

κ = 1, k ,

in the linear transformations of forces (2.6) the generalized reactions must be given in the form (2.7). In this case linear transformations (2.6) can be represented in the following way: Rµ Rµ

∂yµ = Rλ = 0 , ∂q λ

∂yµ = Rl+κ = Λκ , ∂q l+κ R µ = Λκ

∂f κ , ∂yµ

λ = 1, l ,

µ = 1, 3N ,

(2.12)

κ = 1, k ,

µ = 1, 3N ,

(2.13)

κ = 1, k ,

µ = 1, 3N .

(2.14)

Taking into account the relations M y¨µ

d ∂T ∂T ∂yµ = − σ, σ σ ∂q dt ∂ q˙ ∂q

µ, σ = 1, 3N ,

which have been proved by Lagrange, and also the fact that Rµ = M y¨µ − Yµ ,

Qσ = Yµ

∂yµ , ∂q σ

we can conclude that from relations (2.12) it follows Lagrange’s equations of the second kind (2.8), from relations (2.13) the equations for determining of generalized reactions (2.9), and from relations (2.14) Lagrange’s equations of the first kind (1.19). Thus, relations (2.12)–(2.14), which are, in essence, a short analytic form of the theorem, involve all the types of equations of holonomic mechanics. They also permit us to show the covariance property of Lagrange’s equations of the second kind, i. e. the fact that these equations in terms of the ∗ independent coordinates q λ and q∗λ , λ, λ∗ = 1, l, are related as Rλ

∂q λ = R∗λ∗ = 0 , ∂q∗λ∗

λ, λ∗ = 1, l ,

where Rλ =

d ∂T ∂T − λ − Qλ , dt ∂ q˙λ ∂q

R∗λ∗ =

d ∂T ∂T − λ∗ − Q∗λ∗ . dt ∂ q˙∗λ∗ ∂q∗

88

III. Linear Transformation of Forces ∗

Really, relations (2.12), written in terms of the variables q∗λ , λ∗ = 1, l, yield that Rµ

∂yµ ∂yµ ∂q λ ∂q λ = R = R = R∗λ∗ = 0 , µ λ ∗ ∗ ∂q∗λ ∂q λ ∂q∗λ ∂q∗λ∗

λ, λ∗ = 1, l ,

which was to be proved. We now turn to Lagrange’s equations of the second kind (2.8) and remark that this system of equations is equivalent to one vector relation M WL = YL ,

(2.15)

represented in the vector space, the basis of which is the vectors eλ , λ = 1, l. The index L of the vectors in equation (2.15) points out a dimension of space, in which this equation is represented. Deleting the index L, we arrive at equation (2.2). Note that in this case case in equations (1.16) we have ε l+κ = el+κ = ∇ f κ , κ = 1, k, and in equations (1.17) ε λ = eλ , λ = 1, l. The vector form of a system of Lagrange’s equations of the second kind was used here for the simplest holonomic mechanical system with a finite number of mass points. In the case of arbitrary mechanical system, consisting of as rigid bodies as solids, the system of Lagrange’s equations of the second kind is also equivalent to one vector relation, represented in the space tangential to the manifold of all possible positions of mechanical system (in more detail, see Chapter IV).

§ 3. An example of the application of theorem on the forces providing the satisfaction of holonomic constraints In the present section and then in § 5 we study some problems on a guidance of mass point on target by the methods of analytic mechanics. The pursuitlaw to depend upon its form can be regarded as the ideal holonomic or nonholonomic constraint, respectively. In this case the constraint is a sought control force. Among the works, devoted the study of controllable motion with applying the theory of constrained motion it should be noted the work of V. I. Kirgetov [105]. We remark that not only the problem on the approach to target is highly actual but the opposite problem when one studies the optimal deviations of target from the object, which points to the target by various ways. (see, for example, [244]). Example III. 1 . The target guidance by the method of parallel approach. In the case when the aircraft of the mass m moves in horizontal plane Oxy we account for the tractive force P, directed along the velocity v, and the force of air drag Ra , operating in opposite direction. It is required to find the

89

3. An example of the application of theorem

control force R, under which the described below method provides a guidance on target moving by the knownlaw ξ = ξ(t) ,

η = η(t) .

(3.1)

We consider a target guidance by the scheme of parallel approach [129]. As is known, in this case in moving the aircraft the line of sighting a target must move in parallel itself, what provides the continuous guidance of aircraft at the point of instantaneous collision. If for t = 0 the aircraft (rocket) is at the origin of coordinates M0 (0, 0) and the target have the following coordinates ξ(0) = ξ0 ,

η(0) = η0 ,

then the line of sight is directed along the straight line y=

η0 (x − ξ) + η . ξ0

(3.2)

It means that if the target has the position (ξ(t), η(t)), then the coordinates x, y of rocket satisfy equation (3.2), which can be rewritten as (x − ξ) sin ϕ0 + (η − y) cos ϕ0 = 0 ,

tg ϕ0 = η0 /ξ0 .

In other words, on the coordinates of aircraft it is imposed the nonstationary holonomic constraint f (t, x, y) ≡ (x − ξ) sin ϕ0 + (η − y) cos ϕ0 = 0 .

(3.3)

Consider a new system of coordinates Oq 1 q 2 , which is rotated clockwise relative to the original system by the angle π/2 − ϕ0 . In these two systems the coordinates of aircraft are related by the transition formulas: q 1 = x sin ϕ0 − y cos ϕ0 , x = q 1 sin ϕ0 + q 2 cos ϕ0 ,

q 2 = x cos ϕ0 + y sin ϕ0 ,

(3.4)

y = −q 1 cos ϕ0 + q 2 sin ϕ0 .

(3.5)

We make use of the following linear transformations of forces: ∂q 1 ∂q 2 + R2 , ∂x ∂x ∂x ∂y R1 = Rx 1 + Ry 1 , ∂q ∂q

Rx = R1

∂q 1 ∂q 2 + R2 , ∂y ∂y ∂x ∂y R2 = Rx 2 + Ry 2 . ∂q ∂q Ry = R1

(3.6)

Substitute the expressions of x and y from (3.5) into equation of constraints (3.3). Then we obtain q 1 = ξ sin ϕ0 − η cos ϕ0 .

(3.7)

We pay attention to the fact that according to the first formulas of (3.4) 1 in the right-hand side of this relation there is the projection qtarg of the

90

III. Linear Transformation of Forces

radius-vector of target on the axis Oq 1 and therefore relation (3.7) can be rewritten as 1 (t) , q 1 = qtarg

1 qtarg (t) = ξ(t) sin ϕ0 − η(t) cos ϕ0 .

(3.8)

Thus, we can formulate the following problem: it is required to find the additional force, under which for the given active forces a motion of the mechanical system in the system of coordinates Oq 1 q 2 is provided such that the coordinate q 1 is varied by given low (3.8). By the theorem, proved in the previous section, such a motion can be obtained by the introduction of one additional force Λ, corresponding to the coordinate q 1 . Thus, forlaw (3.8) to be satisfied it is sufficient to put R1 = Λ ,

R2 = 0 .

In this case linear transformations of forces (3.6) take the form Rx = Λ sin ϕ0 ,

Ry = −Λ cos ϕ0 ,

Λ = Rx sin ϕ0 − Ry cos ϕ0 ,

Rx cos ϕ0 + Ry sin ϕ0 = 0 .

(3.9) (3.10)

Construct the projections of given active forces, acting on the aircraft: ˙ , Y = (P − Ra ) y/v ˙ , X = (P − Ra ) x/v ∂x ∂y = (P − Ra ) q˙1 /v , Q1 = X 1 + Y ∂q ∂q 1 ∂x ∂y = (P − Ra ) q˙2 /v , Q2 = X 2 + Y ∂q ∂q 2 v 2 = x˙ 2 + y˙ 2 = (q˙1 )2 + (q˙2 )2 .

(3.11)

The projection Rx , Ry for aircraft is as follows Rx = m¨ x−X,

Ry = m¨ y−Y .

(3.12)

Therefore to linear transformation (3.9) correspond Lagrange’s equations of the first kind m¨ x = X + Λ sin ϕ0 ,

m¨ y = Y − Λ cos ϕ0 ,

(3.13)

and to linear transformation of forces (3.10) Lagrange’s equations of the second kind m¨ q 1 − Q1 = Λ , (3.14) m¨ q 2 = Q2 .

(3.15)

It is easily seen that, taking into account the simplest algebraic transformations, we can obtain equations (3.14), (3.15) from equations (3.13) and vice versa. This example demonstrates the reciprocity of two kinds of Lagrange’s equations.

91

3. An example of the application of theorem

Given the initial data and function (3.8) and using relations of forces (3.11) we can numerically integrate equation (3.15) and obtain the law of variation of the coordinate q 2 : q 2 = q 2 (t) .

(3.16)

Using functions (3.8) and (3.16), we can obtain then from formula (3.14) the control force Λ = Λ(t) , (3.17) which provides thelaw of a guidance of aircraft on a target moving bylaw (3.1) (or (3.8)). After determining function (3.17) by formulas (3.9) we can find the components of control force in the system Oxy. As an example of concrete numerical computation we consider a motion of target by the followinglaws (t is given in seconds, ξ, η in meters): (I) ξ(t) = v0 t + ξ0 , (II) ξ(t) = v0 t cos ϕ0 + ξ0 ,

η(t) = η0 ,

(3.18)

2

η(t) = −9.812 t /2 + v0 t sin ϕ0 + η0 . (3.19)

In Fig. III. 1 curves 11, 12 show the trajectories of the target, moving bylaws (3.18) and (3.19), and curves 21, 22 the corresponding motions of aircraft, respectively,. In this case for a hypothetical aircraft (rocket) we put m = 200 kg ,

P = 2500 N ,

Ra = 0.01v 2 N ,

ξ0 = η0 = 5000 m , ϕ0 = π/4 , v0 = 194.44 m/s , x(0) = y(0) = 0 , x(0) ˙ = v0 cos ϕ0 , y(0) ˙ = v0 sin ϕ0 .

(3.20)

Fig. III. 1

We consider now a technical realization of control forces. For the aircraft pursues a target by laws (3.18) or (3.19), it is necessary together with a motor, which generates the tractive force P, to use an additional motor, which generates the required by value and direction control force R. One can utilize only one motor if it can vary a tractive force by value and direction, i. e. can generate the tractive force P∗ such that P∗ = P + R .

(3.21)

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III. Linear Transformation of Forces

Fig. III. 2

In Fig. III. 2 are represented the hodographs of the vectors PI∗ and PII ∗, providing the motion of aircraft, which pursues by the method of parallel approach the target moving bylaw (I) (3.18) or bylaw (II) (3.19). The arrows in the figure point out the increasing of time. The corresponding curves are denoted by digits 41 and 42. In Chapter VI according to Lagrange’s multipliers a special form of equations of the dynamics of rigid bodies system is obtained. This form of the equations of dynamics is used for a motion control of the platform of robotic stand by means of the bars of variable length. The orientation of platform is given by the positions of the vector of a center of mass and those of the unit vectors of principal central axes of inertia. The vector equations with respect to these four vectors are constructed. The stresses in bars, regarded as control parameters, enter into the equations linearly. If the position of platform is given by six generalized coordinates, which are the lengths of bars, then the theorem of holonomic mechanics we use in the present section becomes descriptive. Recall that according to this theorem the motion, under which one of generalized coordinates is given by a function of time, can be obtained by means of the introduction of one additional force, corresponding to this coordinate. § 4. Chetaev’s postulates and the theorem on the forces providing the satisfaction of nonholonomic constraints Introduce the generalized forces, corresponding to the equations of constraints. Then from the D’Alembert–Lagrange principle, extended to the case of Chetaev’s type constraints, the base complex of equations of nonholonomic mechanics in compact form can be obtained. Let us formulate the theorem, involving this complex of equations. For the sake of generality, we consider an arbitrary mechanical system, the position of which is defined by the generalized coordinates q σ , σ = 1, s. Suppose that this system is under the nonlinear nonholonomic constraints ˙ = 0, ϕκ (t, q, q)

κ = 1, k ,

k < s.

(4.1)

93

4. Chetaev’s postulates and the theorem

Recall that the nonholonomicity of these constraints consists in that in spite of their occurrence the transition of the system from any its position with the coordinates q0σ , σ = 1, s, to other position with the coordinates q1σ , σ = 1, s, is kinematically possible. N. G. Chetaev showed that for the real motion of the considered system, the D’Alembert–Lagrange principle   ∂T d ∂T − σ − Qσ δq σ = 0 (4.2) dt ∂ q˙σ ∂q is satisfied. The kinetic energy T is given in the form T =

M gαβ (t, q)q˙α q˙β , 2

α, β = 0, s ,

q0 = t ,

q˙0 = 1 ,

where M is a mass of complete system and the generalized forces Qσ are assumed to be the function of time t, the coordinates q σ , and the generalized velocities q˙σ (σ = 1, s). N. G. Chetaev assume that the quantities δq σ , entering into the D’Alembert–Lagrange principle, satisfy the following conditions (see formulas (6.3) of Chapter II) ∂ϕκ σ δq = 0 , κ = 1, k . (4.3) ∂ q˙σ Equations (4.2) and (4.3) we shall call further the Chetaev postulates. Now we consider the constraint of the Chetaev postulates with the generation of Maggi’s equations, which are based on the introduction of generalized forces, corresponding to quasivelocities. Using the principle of releasability of constraints (4.1), we consider their generalized reactions Rσ and represent Lagrange’s equations in the form d ∂T ∂T − σ − Qσ = Rσ , dt ∂ q˙σ ∂q

σ = 1, s .

(4.4)

From Chetaev’s postulates (4.2), (4.3) it follows that the quantities Rσ can be represented as ∂ϕκ , σ = 1, s , (4.5) Rσ = Λκ ∂ q˙σ where Λκ are Lagrange’s multipliers. Thus, the Chetaev postulates (4.2), (4.3) are equivalent to one postulate, which is given by formulas (4.5). Constraints (4.1) are nonholonomic and therefore their left-hand sides can be regarded only as the certain quasivelocity v∗l+κ , l = s − k, i. e. ˙ , v∗l+κ = ϕκ (t, q, q)

κ = 1, k .

(4.6)

We supplement this system of quasivelocities by the following quasivelocities ˙ , v∗λ = v∗λ (t, q, q)

λ = 1, l ,

94

III. Linear Transformation of Forces

in such a way that the transition from the generalized velocities q˙σ to the quasivelocities v∗ρ (ρ, σ = 1, s) is one-to-one, i. e. there exists the following inverse transformation q˙σ = q˙σ (t, q, v∗ ) ,

σ = 1, s .

(4.7)

v∗λ ,

λ = 1, l, not necessarily must be quasivelocities, Note that the quantities they can be generalized velocities. A distinctive example of the usage of quasivelocities is dynamic Euler’s equations dωx A + (C − B) ωy ωz = Lx , dt dωy (4.8) + (A − C) ωz ωx = Ly , B dt dωz + (B − A) ωx ωy = Lz . C dt Here A, B, C are the moments of inertia of rigid body about the principal axes of inertia x, y, z, rigidly bound with body, and ωx , ωy , ωz and Lx , Ly , Lz are the projections of the vector of the instantaneous angle velocity ω and of the principal moment of exterior forces L about the center of mass on these exes, respectively. The quantities ωx , ωy , ωz are, as is well known, the quasivelocities since they cannot be regarded as time derivatives with respect to the certain three new angles, uniquely related to the Euler angles. We regard the moment Lx as the generalized force, corresponding to the quasivelocity ωx . We can similarly regard the moments Ly and Lz . Note that for the forces, having potential, the generalized forces, corresponding to quasivelocities, were introduced by Poincar´e [373]. N. G. Chetaev [248] extends then this notion on the forces of any nature. In this case the relation between the generalized velocities and the quasivelocities is assumed to be linear. Consider now the case when this relation are nonlinear. Above we assumed that to the generalized coordinates q σ correspond the generalized forces Qσ and the transition from the quasivelocity v∗ρ to the generalized velocities q˙σ (ρ, σ = 1, s) is given by formulas (4.7). Then the gen ρ , corresponding to quasivelocities v∗ρ , are given by formulas eralized force Q σ

 ρ = Qσ ∂ q˙ ρ , Q ρ = 1, s . (4.9) ∂v∗ Consider also the inverse linear transformation of forces, which is given by the relation ρ  ρ ∂v∗ , Qσ = Q σ = 1, s . (4.10) ∂ q˙σ  ρ by the reacReplacing in formulas (4.9) and (4.10) the forces Qσ and Q ∗  , respectively, we obtain tions Rσ and R ρ

σ  ∗ = Rσ ∂ q˙ ρ , R ρ ∂v∗

∗ Rσ = R ρ

∂v∗ρ , ∂ q˙σ

ρ, σ = 1, s .

(4.11)

95

4. Chetaev’s postulates and the theorem

 ∗ρ is a generalized reaction, corresponding to the quasivelocity v∗ρ (ρ = Here R 1, s). Consider now postulate (4.5) from the point of view of general formulas (4.11). Relations (4.5) and (4.6) implies that postulate (4.5) means that in formulas (4.11) it should be assumed the following  ∗λ = 0 , R

λ = 1, l ,

 ∗l+κ = Λκ , R

κ = 1, k .

(4.12)

The postulate (4.5), as is remarked above, is equivalent to Chetaev’s postulates (4.2) and (4.3). Therefore relations (4.11) and (4.12) can be regarded as one of forms of notation of Chetaev’s postulates. Relations (4.12) imply that it is rational to divide the transition formulas from the quantities Rσ to the  ∗ into two groups: quantities R ρ Rσ

∂ q˙σ = 0, ∂v∗λ

λ = 1, l ,

Rσ

∂ q˙σ = Λκ , ∂v∗l+κ

κ = 1, k ,

which can be represented in accordance with relations (4.4) as  σ  ∂ q˙ ∂T d ∂T − σ − Qσ = 0, λ = 1, l , dt ∂ q˙σ ∂q ∂v∗λ   d ∂T ∂ q˙σ ∂T − − Q = Λκ , κ = 1, k . σ σ σ dt ∂ q˙ ∂q ∂v∗l+κ

(4.13) (4.14)

Equations (4.13), obtained from the Chetaev postulates, are Maggi’s equations. From these equations the many famous forms of equations of motion for nonholonomic systems can be obtained (see § 3 Chapter II). We now consider one of theorems of nonholonomic mechanics. As is known, the introduction of generalized coordinates leads to the introduction of generalized forces, corresponding to these coordinates. In § 2 of the present Chapters we prove the theorem expressing the following property of generalized forces: the motion such that one of generalized coordinates is a given time function, can be obtained by the introduction of one additional generalized force corresponding to this coordinate. Above, the introduction of generalized forces, corresponding to quasivelocities, was demonstrated on an example of dynamic Euler’s equations (4.8). Considering these equations once more, we can conclude that if, for example, the quasivelocity ωx is the given function of time, then we can obtain  ∗x , corsuch motion by means of introduction of one additional moment R responding to the quasivelocity ωx . Extending this example to an arbitrary mechanical system, we arrive at the following theorem on the forces, under which the nonholonomic constraints are satisfied: Let the quasivelocities be the given time functions v∗l+κ = χκ ∗ (t), and be related with generalized velocities as v∗l+κ = ϕκ ˙ , ∗ (t, q, q)

κ = 1, k .

(4.15)

96

III. Linear Transformation of Forces

On the rest of quasivelocities v∗λ , λ = 1, l, no restrictions are imposed. Then for the mentioned above motion to be realized, it is sufficiently to apply the  ∗ = Λκ , corresponding to the quasivelocities additional generalized forces R l+κ l+κ v∗ , κ = 1, k. Relations (4.15) can be represented as ˙ = ϕκ ˙ − χκ ϕκ (t, q, q) ∗ (t, q, q) ∗ (t) = 0 ,

κ = 1, k .

(4.16)

We observe that they coincide, by form, with the equations of constraints (4.1) and therefore we can use the theory of constrained motion. In § 1 of this Chapter it was shown that for first-order nonholonomic constraints to be satisfied, it is sufficient to add the forces Rµ , given in the form of (1.23), to the active forces Yµ : R µ = Λκ

∂ϕκ , ∂ y˙ µ

µ = 1, 3N ,

κ = 1, k .

When passed from the Cartesian coordinates to the generalized ones, the above relations take the form Rσ = Λκ

∂ϕκ , ∂ q˙σ

σ = 1, 3N ,

κ = 1, k .

Relations (4.11), (4.12) imply that the quantities Λκ , κ = 1, k, are additional generalized forces, which provide the satisfaction of constraints (4.16). This means that in order that the quasivelocity v∗l+κ , κ = 1, k, are the given functions of time, it is sufficient to add the corresponding to them additional  ∗ = Λκ , κ = 1, k. generalized forces R l+κ Thus, to each equation of nonholonomic constraint can be assigned a generalized force, controlling this constraint. Equations (4.15) can also be regarded as the equations of noncomplete program, of motion, given in quasivelocities. Thus, the present theorem can also be used to study a controllable motion. System of differential Maggi’s equations (4.13), supplemented by equations (4.16) differentiated in time, can be reduced to the following system ˙ , q¨σ = F σ (t, q, q)

σ = 1, s .

(4.17)

Substituting these relations into the left-hand sides of equations (4.14), we obtain the additional generalized forces Λκ in the form ˙ , Λκ = Λκ (t, q, q)

κ = 1, k .

Note that the quantities Rσ as the functions of the same variables can be obtained if relations (4.17) are substituted into formulas (4.4). By the theorem we have the following three base groups of relations: Rσ

∂ q˙σ = 0, ∂v∗λ

λ = 1, l ,

97

5. An example of the application of theorem Rσ

∂ q˙σ = Λκ , ∂v∗l+κ

Rσ = Λκ

∂ϕκ , ∂ q˙σ

κ = 1, k ,

(4.18)

σ = 1, s .

Using relations (4.4), they make transition to Maggi’s equations (4.13), (4.14), and Lagrange’s equations of the first kind in curvilinear coordinates for nonholonomic systems, respectively: ∂ϕκ d ∂T ∂T − σ = Qσ + Λκ , σ dt ∂ q˙ ∂q ∂ q˙σ

σ = 1, s .

So, we can say that the present theorem involves the main complex of equations of nonholonomic mechanics. The first two groups of relations (4.18) follows from the third relation, which, as already was remarked, is equivalent to Chetaev’s postulates (4.2) and (4.3). Thus, using Chetaev’s postulates, we show that the fundamental equations of nonholonomic mechanics can be constructed by using the theorem, according to which the generalized forces corresponding to quasivelocities permit us to control the varying of these quasivelocities.

§ 5. An example of the application of theorem on forces providing the satisfaction of nonholonomic constraints Example III .2 . A target guidance by pursuit method. Consider the problem, formulated in Example III.1 in the case when the aircraft pursues a target by the pursuit method. As is known [129], in the method of target guidance, the vector of velocity of aircraft is continuously directed to the moving target. For such motion the following relation y˙ x˙ = x−ξ y−η

(5.1)

must be satisfied. Here the coordinates of target ξ, η are assumed to be given functions of time (3.1). We shall regard formula (5.1) as a nonstationary nonholonomic constraint, imposed on the motion of the point M (x, y) being an aircraft (rocket): ϕ(t, x, y, x, ˙ y) ˙ ≡ (y − η)x˙ − (x − ξ)y˙ = 0 .

(5.2)

The equations of motion are generated by the theorem of nonholonomic mechanics, proved in § 4. We introduce the new quasivelocity v∗1 , v∗2 , related with the initial velocities x, ˙ y˙ as v∗2 = (y − η)x˙ − (x − ξ)y˙ . (5.3) v∗1 = x˙ ,

98

III. Linear Transformation of Forces

Then it is easy to find the inverse transformation x˙ = v∗1 ,

y˙ = (y − η)v∗1 /(x − ξ) − v∗2 /(x − ξ) .

(5.4)

Introduce a linear transformation of forces (4.11) using transition formulas (5.3) and (5.4). We pay attention that in the example under consideration the usual Cartesian system x, y is regarded as the original system of the coordinates q 1 , q 2 . Therefore in formulas (4.11) in place of R1 , R2 we take Rx , Ry . Thus, we have 1 2  ∗ ∂v∗ ,  ∗ ∂v∗ + R Rx = R 1 2 ∂ x˙ ∂ x˙

1 2  ∗ ∂v∗ ,  ∗ ∂v∗ + R Ry = R 1 2 ∂ y˙ ∂ y˙

 ∗ = Rx ∂ x˙ + Ry ∂ y˙ , R 1 ∂v∗1 ∂v∗1

 ∗ = Rx ∂ x˙ + Ry ∂ y˙ . R 2 ∂v∗2 ∂v∗2

(5.5) (5.6)

For the condition of guidance (5.2) to be satisfied it is necessary for the introduced quasivelocity v∗2 to be equal to zero: v∗2 = 0 .

(5.7)

By the theorem of nonholonomic mechanics, proved in § 4, the given change of the quasivelocity (5.7) can be provided by the one additional force Λ, corresponding to this quasivelocity v∗2 , i. e. in order to satisfylaw (5.7) it is sufficient to assume that  ∗2 = Λ . R

 ∗1 = 0 , R

In accordance with formulas (5.3), (5.4) the linear transformations of forces (5.5), (5.6) are the following: Rx = Λ(y − η) , 0 = Rx + Ry

Ry = Λ(ξ − x) ,

(5.8)

1 . ξ−x

(5.9)

y−η , x−ξ

Λ = Ry

In these transformations the reactions Rx , Ry take the form (3.12). Then transformations (5.8) lead to Lagrange’s equations of the first kind m¨ x − X = Λ(y − η) ,

m¨ y − Y = Λ(ξ − x) ,

(5.10)

and transformations (5.9) to Maggi’s equations m¨ x − X + (m¨ y−Y) Λ=

y−η = 0, x+ξ

m¨ y−Y . ξ−x

(5.11) (5.12)

It is easily seen that eliminating Λ from equations (5.10) according to (5.12), we obtain the equation of motion coincident with the Maggi’s equations (5.11).

5. An example of the application of theorem

99

Equation of motion (5.11) and equation of constraint (5.2) give a closed system. For representing it in normal form we differentiate equation of constraint (5.2) with respect to time. Finally, we obtain (η˙ x˙ − ξ˙y)(y ˙ − η) (P − Ra )x˙  + , (x − ξ)2 + (y − η)2 m x˙ 2 + y˙ 2 (P − Ra )x˙ (ξ˙y˙ − η˙ x)(x ˙ − ξ) +  . y¨ = 2 2 (x − ξ) + (y − η) m x˙ 2 + y˙ 2

x ¨=

(5.13)

In Fig. III. 3 curves 31, 32 show the trajectories of aircraft, obtained as a result of the integration of system (5.13) in the cases when the target moves by laws (I) (3.18) and (II) (3.19), respectively. The characteristics of aircraft are given by formulas (3.20). Let R = Rx i + Ry j be the sought control force. Taking into account that the motion of aircraft must satisfy equations (5.13) and the equations m¨ x = X + Rx ,

m¨ y = Y + Ry ,

we obtain that Rx =

m(η˙ x˙ − ξ˙y)(y ˙ − η) m(ξ˙y˙ − η˙ x)(x ˙ − ξ) , Ry = , 2 2 2 (x − ξ) + (y − η) (x − ξ) + (y − η)2  m|ξ˙y˙ − η˙ x| ˙ R = Rx2 + Ry2 =  . (x − ξ)2 + (y − η)2

The force Q = P + Ra is directed along the tangent line to the trajectory of aircraft. The force R, as follows from formulas (5.1) and (5.8), is perpendicular the vector of aircraft velocity. Thus, the force Q is proportional to the tangential acceleration and the force R to the normal one. In the case when the control is realized by one motor, which can generate the required traction P∗ of the given value and direction, arguing similar to that at the end of Example III.1, we can determine the vector P∗ by formula (3.21). In Fig. III. 4 are represented the hodographs of the vectors

Fig. III. 3

100

III. Linear Transformation of Forces

Fig. III. 4

PI∗ and PII ∗ , providing the motion of aircraft, which pursues by the pursuit method a target, moving by law (I) (3.18) or by law (II) (3.19), respectively. In the figure arrows corresponds to increasing the time. The corresponding curves are denoted by the symbols 51 and 52.

§ 6. Linear transformation of forces and Gaussian principle The motion of free mechanical system of the general type in generalized coordinates is described by Lagrange’s equations of the second kind (2.3). We change from the variables q σ to the new coordinates q∗ρ by formulas q∗ρ = q∗ρ (t, q) ,

q σ = q σ (t, q∗ ) , |∂q∗ρ /∂q σ | =  0.

ρ, σ = 1, s ,

(6.1)

Let the given functions of time be the variables q∗l+κ = q∗l+κ (t, q), l = s − k, κ = 1, k (noncomplete program of motion) only. The generalized forces Rσ , which must be added to the forces Qσ in equations (2.3) for the mentioned above program to be satisfied, we seek in the form Rσ = R∗ρ

∂q∗ρ , ∂q σ

R∗ρ = Rσ

∂q σ , ∂q∗ρ

ρ, σ = 1, s .

(6.2)

The variables q∗λ , λ = 1, l, are free coordinates. Therefore the corresponding to them additional forces R∗λ can be assumed equal to zero. In this case relations (6.2) take the form Rσ = R∗l+κ

∂q∗l+κ , ∂q σ

Rσ

∂q σ = 0, ∂q∗λ

κ = 1, k ,

λ = 1, l .

Rσ

∂q σ = R∗l+κ , ∂q∗l+κ

101

6. Linear transformation of forces and Gaussian principle Discriminate from these relations the following equations Rσ

∂q σ = 0, ∂q∗λ

λ = 1, l .

The latter is, in fact, a short form of notation for Lagrange’s equations of the second kind in the new free variables q∗λ . This results from the proved above covariance property of Lagrange’s equations of the second kind. In a similar way, if we have the following transformation of velocities v∗ρ = v∗ρ (t, q, q) ˙ ,

q˙σ = q˙σ (t, q, v∗ ) , |∂v∗ρ /∂ q˙σ |

ρ, σ = 1, s ,

=  0,

(6.3)

then the corresponding to it transformation of forces is as follows ρ  ∗ ∂v∗ , Rσ = R ρ ∂ q˙σ

σ  ∗ = Rσ ∂ q˙ ρ , R ρ ∂v∗

ρ, σ = 1, s .

(6.4)

For noncomplete program in the case when the given functions of time are the variables v∗l+κ , l = s − k, κ = 1, k, only, in relations (6.4) we can put  ∗ = 0, λ = 1, l. R λ Analogously, for the linear transformation of accelerations ˙ q¨σ + aρ0 (t, q, q) ˙ , w∗ρ = aρσ (t, q, q) q¨σ = bσρ (t, q, q) ˙ w∗ρ + bσ0 (t, q, q) ˙ , |aρσ | =  0,

(6.5)

ρ, σ = 1, s ,

it is rational to introduce a transformation of forces by formulas ρ  ∗ ∂w∗  ∗ ρ =R Rσ = R ρ ρ aσ , σ ∂ q¨

∂ q¨σ  ∗ R = Rσ bσρ , ρ = Rσ ∂w∗ρ

ρ, σ = 1, s , (6.6)

and for noncomplete program in the case when there are given only the  ∗ variables w∗l+κ , l = s − k, κ = 1, k, we can put R λ = 0, λ = 1, l. ∗  ∗ ρ = R  λ = 0, λ = 1, l, relations (6.4), (6.6) yield, in particuIn the case R lar, the relations Rσ

∂ q˙σ = 0, ∂v∗λ

Rσ

∂ q¨σ = 0, ∂w∗λ

λ = 1, l ,

which are Maggi’s equations and Appell’s equations, respectively. Differentiating twice the transformation of coordinates (6.1) with respect to time and assuming q¨∗ρ = w∗ρ , we arrive at linear relations (6.5). Similarly, differentiating the transformation of velocities (6.3) with respect to time and denoting v˙ ∗ρ = w∗ρ , we obtain the relations, which can be represented in the form (6.5). However linear transformations (6.5) can also be introduced in the case when for certain ρ there do not exist the coordinates q∗ρ and

102

III. Linear Transformation of Forces

quasivelocities v∗ρ such that we can put q¨∗ρ = w∗ρ and v˙ ∗ρ = w∗ρ . In this sense formulas (6.6) are of a more general form of the transformation of forces. We shall show that the equations Rσ

∂ q¨σ = 0, ∂w∗λ

λ = 1, l ,

(6.7)

give Gaussian principle in the free variables w∗λ . A concrete form of equations (6.7) depends on a choice of variables w∗λ . We reduce a family of equations (6.7) to the form invariant under a choice of the free variables w∗λ . For this purpose we write equations (6.5) in differential form: δ ′′ w∗ρ = aρσ δ ′′ q¨σ , δ ′′ q¨σ = bσρ δ ′′ w∗ρ , ρ, σ = 1, s . The primes of the derivative δ underline that we compute the partial derivatives for the fixed t, q σ , q˙σ . The assumptions that δ ′′ w∗l+κ = 0, κ = 1, k, and the derivatives δ ′′ w∗λ , λ = 1, l, are regarded as arbitrary and independent variables means that the variables w∗l+κ are to be given and the variables w∗λ under the condition |aρσ | =  0 are chosen arbitrary. The linear independence of the quantities δ ′′ w∗λ permits us to represent system of equations (6.7) as a unique equation ∂ q¨σ ′′ λ δ w∗ = Rσ δ ′′ q¨σ = 0 . (6.8) Rσ ∂w∗λ In the next Chapter we shall show that the family of Lagrange’s equations of the second kind  M Wσ ≡ M gστ q¨τ + Γσ,αβ q˙α q˙β = Qσ + Rσ ,   1 ∂gσβ ∂gσα ∂gαβ Γσ,αβ = , + − 2 ∂q α ∂q β ∂q σ τ, σ = 1, s ,

q0 = t ,

α, β = 0, s ,

q˙0 = 1 ,

describing the motion of mechanical system of any structure, can be represented as a one vector relation written in tangential space MW = Y + R ,

W = Wσ eσ ,

Y = Qσ eσ ,

R = Rσ eσ .

Let the vector W be given in the contravariant form: W = W σ eσ ,

W σ = q¨σ + Γσαβ q˙α q˙β .

Since the Christoffel symbols Γσαβ is independent of accelerations, we have δ ′′ W σ = δ ′′ q¨σ , and therefore the sum Rσ δ ′′ q¨σ can be represented as the scalar product R · δ ′′ W. Then equation (6.8) has the form R · δ ′′ W = 0 .

(6.9)

6. Linear transformation of forces and Gaussian principle

103

Taking into account that the force Y is independent of the accelerations q¨σ , we obtain δ ′′ R = δ ′′ (M W − Y) = M δ ′′ W . Therefore in place of equation (6.9) we can write δ ′′ R2 = 2R · δ ′′ R = 0 .

(6.10)

It means that the vector R is chosen from the condition of a minimality of its modulus. Relations (6.10), expressing Gaussian principle, can be represented in usual form: Z = M (W − Y/M )2 . (6.11) δ ′′ Z = 0 , Formulas (6.10) and (6.11) give Gaussian principle in invariant form and equation (6.7) gives it in terms of the variables w∗λ . We remark that this principle is here the principle of optimal choice of control forces, providing the given conditions of motion. The additional discussion of Gaussian principle will be given in the next Chapter.

C h a p t e r IV APPLICATION OF A TANGENT SPACE TO THE STUDY OF CONSTRAINED MOTION

By means of a tangent space we introduce, a system of Lagrange’s equations of the second kind is represented in the vector form. It is shown that the tangential space is partitioned by equations of constraints into the direct sum of two subspaces. In one of them the component of an acceleration vector of system is uniquely determined by the equations of constraints. The notion of ideality of holonomic constraints and nonholonomic constraints of the first and second orders is analyzed. This notion is extended to high-order constraints. The relationship and equivalence of differential variational principles of mechanics are considered. A geometric interpretation of the ideality of constraints is given. Generalized Gaussian principle is formulated. By means of this principle for nonholonomic systems with third-order constraints the equations in Maggi’s form and in Appell’s form are obtained. § 1. The partition of tangent space into two subspaces by equations of constraints. Ideality of constraints We assume that the motion of free mechanical system in the generalized coordinates q σ , σ = 1, s, is described by Lagrange’s equations of the second kind [189] ∂T d ∂T − σ = Qσ , dt ∂ q˙σ ∂q α, β = 0, s , σ = 1, s ,

M gαβ q˙α q˙β , 2 q0 = t , q˙0 = 1 ,

T =

(1.1)

where Qσ is a generalized force corresponding to the coordinate q σ and M is a mass of the whole system. Consider a manifold of all positions of the considered mechanical system at time t. Fix a certain point of this manifold, given by the coordinates q σ , σ = 1, s. Suppose, the old and new coordinates of this point are related by formulas q σ = q σ (t, q∗ ) , q∗ρ = q∗ρ (t, q) , ρ, σ = 1, s , or in differential form δq σ =

∂q σ ρ δq , ∂q∗ρ ∗

δq∗ρ =

∂q∗ρ σ δq , ∂q σ

ρ, σ = 1, s .

The quantities δq σ and δq∗ρ , related by the these expressions, are called contravariant components of the tangent vector δy and the whole set of the 105

106

IV. Application of a Tangent Space

vectors δy is called a tangent space to the above-introduced manifold at the given point [63]. It is useful to represent the vector δy as δy = δq σ eσ ,

σ = 1, s ,

and to regard a family of the vectors eσ as a fundamental basis of tangent space in the system of coordinates q σ . We introduce an Euclidean structure in a tangent space, making use of the invariance of the positively defined quadratic form ∗

∗

(δy) 2 = gστ δq σ δq τ = gσ∗ ∗ τ ∗ δq∗σ δq∗τ ,

σ, τ, σ ∗ , τ ∗ = 1, s .

Here gστ and gσ∗ ∗ τ ∗ are coefficients, entering into the relation of kinetic energy in terms of the coordinates q σ and q∗ρ (ρ, σ = 1, s), respectively. They, thus, prescribe the metric tensor, by which the scalar product of the vectors a = aσ eσ and b = bτ eτ can be represented as a · b = gστ aσ bτ ,

gστ = eσ · eτ ,

σ, τ = 1, s .

The components δq σ , σ = 1, s, of the tangent vector δy are also called the variations of the coordinates q σ or the admissible (virtual) displacements. By definition, the generalized forces Qσ , entering into system of equations (1.1), are the coefficients of the variations of coordinates δq σ in the expression for the virtual elementary work δA. Using through numbering µ = 1, 2, 3, . . . for denoting as the Cartesian coordinates of the points of forces application as the projections of these forces, we can write δA = Xµ δxµ . Taking into account that δxµ =

∂xµ σ ∂xµ ρ δq = δq , σ ∂q ∂q∗ρ ∗

we obtain δA = Qσ δq σ = Q∗ρ δq∗ρ , where Qσ = Xµ

∂xµ , ∂q σ

Q∗ρ = Xµ

(1.2)

∂xµ ∂q σ . ρ = Qσ ∂q∗ ∂q∗ρ

Relation (1.2) is a linear invariant differential form of the vector δy. The coefficients Qσ and Q∗ρ of this form, in terms of the coordinates q σ and q∗ρ , respectively, become the components of the covariant vector Y [63]. Taking into account the Euclidean structure of tangent space, we can represent the quantity δA as the scalar product δA = Y · δy ,

Y = Qσ eσ ,

σ = 1, s .

107

1. The partition of tangent space

Here eσ , σ = 1, s, are the vectors of reciprocal basis, which are given by the following relations  0, σ = τ , eσ · eτ = δτσ = 1, σ = τ . Then from the relations gστ = eσ · eτ we have eτ = gστ eσ ,

eσ = g στ eτ .

The coefficients g στ are the elements of the matrix inverse to the matrix with the elements gστ . The introduction of the covariant vector Y, using the relation for the virtual elementary work δA, permits us to regard system of equations (1.1) as a one vector relation MW = Y . (1.3) Here

  1 d ∂T ∂T W = Wσ e = − σ eσ = M dt ∂ q˙σ ∂q  σ    τ α β σ = gστ q¨ + Γσ,αβ q˙ q˙ e = W eσ = q¨σ + Γσαβ q˙α q˙β eσ ,   1 στ ∂gτ β ∂gτ α ∂gαβ σ στ Γαβ = g Γτ,αβ = g , + − 2 ∂q α ∂q β ∂q τ σ

τ, σ = 1, s ,

(1.4)

α, β = 0, s .

Thus, by formulas (1.3) and (1.4) we can introduce the acceleration vector W for arbitrary mechanical system with s degrees of freedom. Consider now a constrained motion. By the releasability principle the imposition of constraints leads to occurrence the reaction force R and therefore Newton’s second law can be written in the following way: MW = Y + R . The reaction force arises out of the acceleration, generated by constraints. Therefore it is necessary to clear up the influence of constraints on generating the vector W. Consider first nonlinear first-order nonholonomic constraints, given by the relations f1κ (t, q, q) ˙ = 0, κ = 1, k . Differentiating these constraints in time, we obtain f2κ (t, q, q, ˙ q¨) ≡ al+κ ˙ q¨σ + al+κ ˙ = 0, 2σ (t, q, q) 20 (t, q, q) κ = 1, k ,

l = s−k.

(1.5)

Note that the linear second-order nonholonomic constraints can be given in the same form. The holonomic constraints give relations (1.5) after a double differentiation in time.

108

IV. Application of a Tangent Space

The introduction of a tangent space and the vector W, given by formulas (1.4), allows us to represent system of equations (1.5) in the vector form: ε l+κ · W = χκ ˙ , 2 (t, q, q) σ ε l+κ = al+κ 2σ e ,

l+κ l+κ σ α β χκ 2 = −a20 + a2σ Γαβ q˙ q˙ ,

κ = 1, k ,

(1.6)

α, β = 0, s .

The vectors ε l+κ , κ = 1, k, corresponding to constraints (1.5), are assumed to be linearly independent. Therefore in an s-dimensional tangential space we can consider a subspace with the basis consisting of these vectors (K-space). Then the whole space can be represented as the direct sum of this subspace and its orthogonal complement with the basis ε λ , λ = 1, l (L-space), in which case we have ε λ · ε l+κ = 0 , λ = 1, l , κ = 1, k . We remark that this partition of a tangent space by the equations of constraints corresponds to the fixed values of the variables t, q σ , q˙σ (σ = 1, s). Substituting the acceleration W given by W = WL + WK , λ ελ , WL = W

l+κ ε l+κ WK = W

WL · WK = 0 ,

(1.7)

in equations (1.6), we obtain

l+κ ∗ = hκ ∗ κ χκ (t, q, q) W ˙ , 2

κ, κ ∗ = 1, k .

(1.8)

Here hκ ∗ κ are elements of the matrix inverse to the matrix with the ele∗ ments hκκ , given by the following relations ∗

∗

hκκ = ε l+κ · ε l+κ ,

κ, κ ∗ = 1, k .

(1.9)

The vectors ε l+κ , κ = 1, k, are linearly independent and therefore we have ∗

|hκκ | =  0.

(1.10)

Taking into account relations (1.7), we represent Newton’s second law as two equations M WL = YL + RL , λ

RL = Rλε λ ,

YL = Q ε λ , MW

 l+κ ε l+κ , YK = Q

K

=Y

K

λ = 1, l ,

K

+R ,

RK = Λκ ε l+κ ,

(1.11)

κ = 1, k .

Here R = RL + RK is a constraint reaction, in which case the components Rl+κ of the vector RK are denoted by Λκ since they are just the Lagrange multipliers. If condition (1.10) is satisfied, then relations (1.7)–(1.10) imply

2. The connection of differential variational principles of mechanics

109

that the vector WK is uniquely defined by the equations of constraints in terms of a function of the variables t, q σ , q˙σ . Thus, in K-space the law of motion is given by equations of constraints and takes the form (1.6). The component of constraint RK , which occurs in this case, is computed by means of the second equation of system (1.11). The equations of constraints cannot influence the vector WL since it can be eliminated from equations (1.6). Therefore we have only indirect influence of constraints on the component of acceleration WL via the vector RL . In particular, the equations of constraints can also be satisfied for RL = 0. Such constraints are called ideal. Thus, the influence of ideal constraints on the acceleration W is completely defined by their analytic representations. Pay attention, that it was necessary to represent all kinds of constraints in a single differential form (1.5) to find out how the constraints influence the generation of the constraint reaction. It is this form of writing the constraint equations that makes it possible to show that the whole space is partitioned by the constraint equations into two orthogonal subspaces. As this takes place, the analytical expressions for constraint reactions were found. We note, that for the first time these results were given in the monograph [189. 1985]. As was shown in "Survey of the main stages of development of nonholonomic mechanics"the same results were obtained with the help of matrix calculus by F. E. Udwadia and R. E. Kalaba [394] in 1992. § 2. The connection of differential variational principles of mechanics Taking the partial derivative δ ′′ of both sides of relations (1.6) for fixed t, q , q˙σ , we obtain σ

ε l+κ · δ ′′ W = ε l+κ · δ ′′ WL = 0 ,

κ = 1, k .

(2.1)

Then from relation RK = Λκ ε l+κ it follows that RK · δ ′′ W = 0. For ideal constraints we find RK = R = M W − Y , and therefore (M W − Y) · δ ′′ W = 0 .

(2.2)

δ ′′ (W − Y/M )2 = 0 .

(2.3)

Hence The relation represents Gaussian principle. This principle is obtained here from the fact that constraints are ideal but it can also be regarded as a definition of ideal constraints. Consider now how the condition of ideality of constraints can be described with the usage of definition of the vector of reaction R in terms of a covariant vector, which is given by the relation for the invariant differential form δAR = R · δy .

110

IV. Application of a Tangent Space

Relations (2.1) and (2.2) yield that the quantity δAR , which is given on a set of virtual displacements δy satisfying the system of equations ε l+κ · δy = 0 ,

κ = 1, k ,

(2.4)

is equal to zero for R = RK , i. e. for RL = 0. Thus, the condition of the ideality of constraints (1.5) is as follows (M W − Y) · δy = 0 ,

(2.5)

which is the generalization of the notation of the D’Alembert–Lagrange principle. Taking into account that ε l+κ =

∂f1κ σ e = ∇ ′ f1κ , ∂ q˙σ

relations (2.4) and (2.5) become ∂f1κ σ δq = 0 , ∂ q˙σ 

κ = 1, k ,

 d ∂T ∂T − σ − Qσ δq σ = 0 . dt ∂ q˙σ ∂q

(2.6) (2.7)

The connection of Gaussian principle (2.2), (2.3) with the generalized D’Alembert–Lagrange principle (2.5) — (2.7) was considered by N. G. Chetaev [245, p. 68], using another approach. As is already mentioned in § 6 of Chapter II, he imposed conditions (2.6) on virtual displacements and attempted to ". . . introduce a notion of virtual displacement for nonlinear constraints so that the D’Alembert principle and Gaussian principle . . . to be saved together". Note that in the same way this question is considered in the paper of G. Hamel, published in 1938 [314]. As is already mentioned in Chapter II, the Chetaev–Hamel conditions (2.4), (2.6) have played the great important role in the development of the nonholonomic mechanics. From these conditions, in particular, it follows that for the fixed t, q σ , q˙σ the tangential space is partitioned by the equations of constraints into the subspaces K and L. In the works [31, 280] the partition into two orthogonal subspaces is given in the matrix form. The authors make use of this partition in order to eliminate the Lagrange multipliers from equations of motion. In these works for the study of the dynamics of system of rigid bodies the computer algorithms are constructed. The partition into two orthogonal subspaces is also used in the works [50, 275, 305, 392]. We now turn to equation (2.5). This equation together with equations (2.4) shows that for ideal constraints the component of reaction vector is lacking in the subspace such that for fixed values of the variables t, q σ , q˙σ , σ = 1, s, any acceleration WL is kinematically admissible. Therefore the D’Alembert–Lagrange principle (2.5) is, in fact, a principle of virtual accelerations. It was called first a principle of virtual velocities [4] and then a general

2. The connection of differential variational principles of mechanics

111

(fundamental) equation of dynamics. It was applied to linear nonholonomic constraints, in particular, in the works [379, 398]. A detail survey of the works on the nonholonomic mechanics can be found in the work [370. 1998, 2002]. In the case of holonomic constraints, given by the equations f0κ (t, q) = 0 ,

κ = 1, k ,

(2.8)

the restrictions on the vector W can also be represented in the form of (1.6). In this case the vectors ε l+κ , entering into equations (1.6) and (2.4), are the following ∂f κ εl+κ = 0σ eσ = ∇f0κ , κ = 1, k , ∂q i. e. they coincide with the usual gradients to surfaces, given by the equations of constraints. In the particular case of the equilibrium of mechanical system the D’Alembert–Lagrange principle passes to the principle of virtual displacements, namely Qσ δq σ = Xµ δxµ = 0 ,

(2.9)

in which case by virtue of the constraints given by equations (2.8), missing with time t, the quantities δq σ satisfy the relations ∂f0κ σ δq = 0 , ∂q σ

κ = 1, k .

The conditions, under which the D’Alembert–Lagrange principle and, in particular, the principle of virtual displacements (2.9) are satisfied, are considered in detail in the treatises of G. K. Suslov [220] and A. M. Lyapunov [138]. In this connection G. K. Suslov writes [220, p. 380]: "Many attempts were made to find a strong proof of the principle of virtual displacements . . . ". Considering in detail two such attempts realized by Lagrange and Amp´ere, he arrived at the following general conclusion "any proof of the considered principles . . . , strictly speaking, cannot be called a proof, i. e. a reduction to the acknowledged verities only". A similar conclusion was made by A. M. Lyapunov. In the present section we show how principles (2.7), (2.9) are connected with the restrictions, imposed on the acceleration vector of system W by means of the equations of constraints. Consider equation (2.2). It implies that by ideal constraints the invariant differential form δAR = R·δy goes identically to zero on a set of the tangential vectors δy such that, as is shown by V. V. Rumyantsev [199], they are the following τ 2 ′′ τ 2 ′′ δy = δ W= δ WL . (2.10) 2 2 Here τ is an infinitely small time interval, introduced by Gauss.

112

IV. Application of a Tangent Space

By formulas (1.5) and (1.6) and taking the partial differential δ ′ of the equations of nonholonomic constraints for the fixed values of the variables t and q σ , σ = 1, s, we obtain δ ′ f1κ =

∂f1κ ′ σ δ q˙ = ∇ ′ f1κ · δ ′ V = ε l+κ · δ ′ V = 0 , ∂ q˙σ

κ = 1, k ,

where δ ′ V = δ ′ q˙σ eσ . Then equations (2.4) yields that, following the work [199], we can represent the tangential vector δy, written in the form (2.10), as as δy = τ δ ′ V .

(2.11)

This relation permits us to identify the notion of virtual velocity and virtual displacement (see the notion: "virtual velocities (displacements) in the work [202, p. 5]). By the change (2.11) the generalized D’Alembert–Lagrange (2.5) principle pass to the Suslov–Jourdain principle   d ∂T ∂T − σ − Qσ δ ′ q˙σ = 0 . dt ∂ q˙σ ∂q A detailed analysis of the differential and integral variational principles of mechanics and a detailed survey of the works, devoted to them, can be found in the treatise of V. N. Shchelkachev [254]. The representation of the vector δy, entering into the generalized D’Alembert–Lagrange (2.5) principle in the form of (2.10), shows clearly the substance of these principles. It established that in the case of ideal constraints in L-space, which the vectors δ ′′ WL belong to, the reaction is lacking since the equations of constraints cannot effect the acceleration WL and generate it. They generate the acceleration WK only, what is provided by the reaction RK together with the active force YK . In other words, in L-space the reaction is lacking since in it the equations of constraints "do not constrain"the mechanical system to have the acceleration WL . The expression: "do not constrain is taken from the formulation of Newton’s first law in the translation of A. N. Krylov. Formulas (2.10) and (2.11), as was emphasized by V. V. Rumyantsev [199], confirm the conclusion of N. G. Chetaev on the compatibility of differential variational principles of Gauss, D’Alembert–Lagrange, and Suslov–Jourdain. Generalizing the approach to the notion of ideality of constraints, we arrive at the following definition: the nonholonomic constraints of any order, which are linear in high derivatives, are ideal if for the fixed values of the variables t, q σ , q˙σ , σ = 1, s, the equations of constraints discriminate in tangential space the L-space such that the mechanical system is not constrained by equations of constraints to have the acceleration WL different from the acceleration given by Newton’s law M WL = YL .

113

3. Generalized Gaussian principle

In § 3 of Chapter V the problem is considered: what constraints are ideal for n  3 ?

§ 3. Geometric interpretation of linear and nonlinear nonholonomic constraints. Generalized Gaussian principle Consider the velocity vector V of mechanical system, using the vector of generalized impulse ∂T p = σ eσ , ∂ q˙ and putting, by definition, that V = p/M = Vσ eσ = (gστ q˙τ + gσ0 )eσ = = (gστ q˙τ + gσ0 )g σρ eρ = V ρ eρ = (q˙ρ + gσ0 g σρ )eρ ,

(3.1)

ρ, σ, τ = 1, s . Denoting e0 = gσ0 eσ ,

σ = 1, s ,

we have gαβ = eα · eβ ,

V = q˙α eα ,

α, β = 0, s .

The introduction of the velocity vector V makes it possible to represent a kinetic energy of an arbitrary mechanical system and a vector of its acceleration in just the same form as in studying the motion of one mass point. This permits us to use in the theory of constrained motion of arbitrary mechanical system the apparatus of analytic and differential geometry, extended to the case of s-dimensional Euclidean space. Then we have M Vσ V σ M V2 = , 2 2 ˙ = V˙ σ eσ + Vσ e˙ σ = V˙ σ eσ + V σ e˙ σ , W=V T =

(3.2) σ = 1, s .

Taking into account that e˙ σ · eτ = −eσ · e˙ τ , we obtain Wτ = W · eτ = V˙ τ − V · e˙ τ ,

τ = 1, s .

In this case relations (1.1), (1.4), (3.1), (3.2) yields the relations (n)

∂V ∂W ∂V eσ = σ = = , (n+1) ∂ q˙ ∂ q¨σ σ ∂ q

σ = 1, s ,

(3.3)

114

IV. Application of a Tangent Space (n)

e˙ σ =

Γτσα q˙α

∂V ∂V 1 ∂W 1 eτ = σ = = , ∂q 2 ∂ q˙σ (n + 1) (n) ∂ qσ σ = 1, s . α = 0, s

(3.4)

(n)

(n)

The notation q σ and V , correspond to the n-th derivatives in time of the function q σ and the vector V, respectively. By formulas (3.3) and (3.4) for computing the n-th derivative in time of the vector M V we can use the generalized operator of Appell (n) (n)

MV =

∂Tn (n+1) ∂ qσ

M V2 , Tn = 2

σ

e ,

n  0,

T0 = T ,

and the generalized operator of Lagrange

(n) 1 ∂Tn−1 d ∂Tn−1 MV = eσ , − (n−1) dt (n) σ σ n ∂q ∂ q (for detail, see [252]). From formulas (3.2) and (3.4) we obtain that the contravariant compo(n)

(n−1)

nents of the vectors V and V are related as  (n−1)   (n) d (n−1) σ V · eσ = V ·e + V · eτ Γστα q˙α , dt σ, τ = 1, s ,

(3.5)

α = 0, s .

(n)

This relation between the derivatives q σ and the covariant and contravari(n−1)

ant components of the vector

V

for the fixed values of the variables

(n−1) σ

t, q σ , q˙σ , . . . , q , σ = 1, s, gives geometric interpretation as linear, as nonlinear nonholonomic constraints of any order. Consider first the constraints of the first order. Relations (3.1) implies that the equations of linear nonholonomic constraints l+κ σ ˙ ≡ al+κ f1κ (t, q, q) 1σ (t, q) q˙ + a10 (t, q) = 0 ,

κ = 1, k ,

l = s−k,

impose the following restrictions on the velocity vector V: ε l+κ · V = χκ 1 (t, q) , κ = 1, k ,

l+κ l+κ στ χκ 1 = −a10 + a1σ g gτ 0 ,

σ, τ = 1, s .

In the present case the vectors ε l+κ = ∇ ′ f1κ =

∂f1κ σ σ e = al+κ 1σ (t, q) e , ∂ q˙σ

κ = 1, k ,

(3.6)

115

3. Generalized Gaussian principle

which partition, under condition (1.10), the tangential space into the subspaces K and L, are independent of the generalized velocities q˙σ , σ = 1, s. System of equations (3.6) is similar to system (1.6). Therefore we can conclude that the component VK of the vector V as the function of the variables t, q σ , σ = 1, s, is uniquely defined by equations of constraints. The space, which the vectors V belong to, has Euclidean structure. In this case from system of equations (3.6) it follows that the set of velocities V, admitted by linear first-order nonholonomic constraints, is an l-dimension plane. For s = 2, l = k = 1, we have a straight line (Fig. IV. 1). Fixed the values of the variables t and q σ , σ = 1, s, by means of formulas (3.1) we can relate the generalized velocities q˙σ , σ = 1, s, and the components of the vector V. This implies that for the fixed values of the quantities t, q σ , σ = 1, s, the system of equations of nonlinear first-order nonholonomic constraints can be regarded as a system of equations, which gives an l-dimensional surface in the space of the velocities V. For s = 2, l = κ = 1, this is a curve on a plane. In particular, it is the closed curve shown in Fig. IV. 2.

Fig. IV. 1

Fig. IV. 2

In the case of linear nonholonomic constraints for the fixed values of t and q σ , σ = 1, s, and for any velocity V, admitted by constraints, the mechanical system has the same component VK of the vector V (Fig. IV. 1). In particular, for the uniform equations of constraints this component is equal to zero. The technical realization of such constraint can be easily made: for example, it is the component of velocity perpendicular to a knife-edge and equal to zero. In the case of nonlinear constraints the directions ε λ , λ = 1, l, along which the equations of constraints admit the infinitely small changes of the vector V for the fixed values of the variables t and q σ , σ = 1, s, and, respectively, the directions ε l+κ , κ = 1, k, along which such changes are impossible, depend on the vector V (Fig. IV. 2). This substantially complicates the technical realization of nonlinear nonholonomic constraints as the constraints between the mass bodies. Another point of view, connected with the existence of nonlinear nonholonomic constraints, is discussed in the treatises [72, 166].

116

IV. Application of a Tangent Space Consider the nonlinear second-order nonholonomic constraints f2κ (t, q, q, ˙ q¨) = 0 ,

κ = 1, k .

(3.7)

At present, we cannot cite an example such that even a linear secondorder nonholonomic constraint is a result of a certain mechanical interaction of mass bodies. (An exception is the work [335]). Therefore we shall regard constraints (3.7) as equations of a certain program of motion (as program constraints). We can obtain a descriptive representation on the restriction, which one constraint (3.7) imposes on the vector W for s = 2, if in Fig. IV. 2 the vector V is changed to the vector W. Note that if in Figs. IV. 1 and IV. 2 we change the vector V and its com(n−1)

(n−1)

(n−1)

ponents VK and VL to the vectors V , VK , VL , respectively, then Fig. IV. 1 corresponds to linear n-order constraint and Fig. IV. 2 to nonlinear that. Consider now second-order constraints. For the linear in accelerations constraints the assumption that they are ideal, as shown above, results in the minimality of Gaussian function Z = (W − Y/M )2 ,

(3.8)

given on a set of the accelerations W, admitted by constraints, for the fixed t, q σ , q˙σ , σ = 1, s. We extend this property of ideality on nonlinear constraints (3.7). Necessary conditions of minimality of the function Z for constraints (3.7) are the following [210] δ ′′ f2κ = ∇ ′′ f2κ · δ ′′ W = 0 , ′′

κ = 1, k , ′′

2M δ Z = (M W − Y) · δ W = 0 .

(3.9)

We can represent the derivatives δ ′′ f2κ , κ = 1, k, in above form since by relations (1.4) the vector δ ′′ W can be represented as δ ′′ W = δ ′′ q¨σ eσ . Introduce the Lagrange multipliers, using the classical approach [210]. Then from system of equations (3.9) we obtain the relation M W = Y + Λκ ∇ ′′ f2κ ,

(3.10)

which in passing to the scalar form becomes the system of Lagrange’s equations of the first kind. Note that for the nonlinear in accelerations constraints this system is used in the work [60]. Consider a geometric interpretation of equation (3.10). We consider first the case of linear constraints, given by equations (1.5). For the first-order nonholonomic constraints and the holonomic constraints, we obtain these

117

3. Generalized Gaussian principle

equations by differentiating in time one and two times, respectively, the equation of these constraints. Then for n = 2, n = 1, and n = 0 the vectors ∇ ′′ f2κ , κ = 1, k, entering into equation (3.10), can be represented as σ l+κ , ∇ ′′ f2κ = al+κ 2σ e = ε

κ = 1, k .

For n = 1 and n = 0 we have ε l+κ = ∇ ′ f1κ ,

ε l+κ = ∇ f0κ ,

κ = 1, k ,

respectively. The l-dimensional plane in a space of the accelerations W is given by system of equations (1.5), represented in vector form (1.6). This geometric approach is illustrated in Fig. IV. 3, which corresponds to the motion of one mass point with one constraint. From all acceleration vectors, admitted by constraints, we discriminate the vector WK that is minimal in absolute magnitude. To construct this vector, we drop a k-dimensional perpendicular from the origin of coordinates on the considered plane. The crosspoint of this perpendicular and the plane corresponds to the end of the vector WK , beginning from the origin of coordinates. This implies that the vector WK is uniquely defined by equations of l+κ ε l+κ . Adding to the vecconstraints and can be represented as WK = W K tor W the arbitrary vector WL , which is in the plane, we obtain the vector W, satisfying the equations of constraints. Using vector Lagrange’s equation of the first kind (3.10), we choose then from all these vectors a unique acceleration W such that a distance between the point, given by the vector Y/M , and the l-dimensional plane considered is minimal. This method to choice the vector W is geometric interpretation of the notion of ideality of constraints. To this simple and natural choice of the vector W from all accelerations, admitted by constraints, corresponds Gaussian principle of virtual accelerations, which, as shown in § 2, can, in turn, be represented in the form of the D’Alembert–Lagrange principle. We proceed now to the case when equations of constraints (3.7) depend nonlinearly on the generalized accelerations q¨σ , σ = 1, s. Using formulas

Fig. IV. 3

118

IV. Application of a Tangent Space

(1.4), we obtain the relation between the quantities q¨σ and the components of the vector W. This allows us to regard system of equations (3.7) for the fixed values of the quantities t, q σ , q˙σ , σ = 1, s, as a system of equations, which gives an l-dimensional surface in the space of the accelerations W. The acceleration W, satisfying equation (3.10) and equations of constraints (3.7), corresponds to the point, of this surface, which is at the minimal distance from the end of the vector Y/M . The difficulties occur in the case when this point is not single. The difficulties also occur in the case when the vectors ∇ ′′ f2κ , κ = 1, k, depend nonlinearly on the generalized accelerations q¨σ , σ = 1, s. In the case when this dependence is linear the solution of problem, as shown in a concrete example in Chapter V, can be sufficiently simple. Now we turn to equations (3.9). Following the work of V. V. Rumyantsev [199], we consider the vector δy =

τ 2 ′′ δ W. 2

Then system of equations (3.9), represented in scalar form, is as follows ∂f2κ σ δq = 0 , ∂ q¨σ 

κ = 1, k ,

 d ∂T ∂T − σ − Qσ δq σ = 0 . dt ∂ q˙σ ∂q

(3.11) (3.12)

It follows that for the nonlinear in velocities nonholonomic constraints, the approach of N. G. Chetaev [245] to the virtual displacements δq σ , σ = 1, s, which enter into the D’Alembert–Lagrange principle (3.12), can also be extended to the nonlinear accelerations of constraints (3.7) by means of formulas (3.11). For the first time this was made by G. Hamel in 1938 [314]. For the nonlinear in accelerations constraints, to each point of l-dimensional surface, introduced above, corresponds its own tangential plane, while for the constraints, depending linearly on the generalized accelerations q¨σ , σ = 1, s, we have a unique plane, given by equations (1.6). The point, of this surface, for which the tangential plane, containing the vectors δy = (τ 2 /2) δ ′′ W, is introduced, is at the shortest distance from the end of the vector Y/M . Then for the nonlinear in accelerations constraints, in equation (3.12), represented in vector form (M W − Y) · δy = 0 , (3.13) the directions of the vector of reactions R = M W − Y and the vectors δy depend on the vector Y. Unlike this case for the linear in accelerations constraints (1.5), the directions of these vectors are independent of the vector Y. Generalized Gaussian principle. Consider now linear third-order con˙ is given by equastraints. In this case an l-dimensional plane in the space W ... tions of these constraints. Replacing in Fig. IV. 3 the quantities x ¨k by x k , K K ˙ W ˙ ,W ˙ L, k = 1, 2, 3, and the vectors W, W , WL , Y by the vectors W,

119

4. Gaussian principle in Maggi’s form

˙ respectively, we obtain the picture, illustrating a geometric interpretation Y, of generalized Gaussian principle. According to this principle the function ˙ − Y/M ˙ )2 Z1 = (W ˙ admitted by the constraints for the is minimal on a set of the vectors W, σ σ σ fixed values of the variables t, q , q˙ , q¨ , σ = 1, s [59. 1976, 177. 1997, 188. 1983, 252. 1974]. This assertion is equivalent to that the following variation ˙ − Y/M ˙ δ ′′′ (W )2 = 0

(3.14)

is equal to zero. Here three accents after the differentiation symbol δ mean that the third derivatives in time of generalized coordinates is varied only. The above reasonings can be extended to the case of constraints of any order. Thus, if the linear nonholonomic constraints of any order (program constraints) are given in the form of linear differential equations of order (n + 2), then it is rational to construct a system of differential equations, which completes the given system, on the principle that (n)

(n)

δ (n+2) (W − Y /M )2 = 0 ,

n  1.

(3.15)

Here the index (n) points out an order of derivative in time with resoect to the vector and the index (n + 2) means that a partial derivative is computed (n+1)

for the fixed t, q σ , q˙σ , . . . , q σ . Formula (3.15) is a generalization of Gaussian principle to the case of high-order nonholonomic constraints. Note that, in using principle (3.15), at the initial moment of time we assume that all of the coordinates q σ and all their derivatives up to be order (n + 1) and therefore the vector R and its derivatives up to be order (n − 1) are given. We make use of generalized Gaussian principle (3.14) for the certain types of equations of motion of nonholonomic systems for the linear third-order constraints to be found. Besides, generalized Gaussian principle will be used in § 4 of Chapter V when studying the motion of a spacecraft with constant in modulo acceleration and in § 9 of Chapter VI when solving the problem of damping the vibration of elastic systems. § 4. The representation of equations of motion following from generalized Gaussian principle in Maggi’s form Generalized Gaussian principle for constraints of any of orders was considered first by M. A. Chuev in 1974 [252]. He also suggests the different forms, of notation for equations of motion, following from this principle [59, 252]. In the present and next sections we consider Maggi’s and Appell’s forms for linear third-order constraints only [76].

120

IV. Application of a Tangent Space So, we assume that the linear third-order differential constraints ... α∗l+κ = cl+κ ˙ q¨) q σ + cl+κ (t, q, q, ˙ q¨) = 0 , σ (t, q, q, 0 κ = 1, k ,

l = s−k,

(4.1)

are imposed on the motion of a system, the position of which is given by the generalized coordinates q σ , σ = 1, s. As shown in § 1, the system of Lagrange’s equations of the second kind can be reduced to one vector relation equivalent to Newton’s second law: MW = Y + R , where the vectors belonging to the space tangential to the manifold of virtual positions of system can be represented as   d ∂T ∂T MW = − σ eσ = M Wσ eσ , Y = Qσ eσ , R = Rσ eσ . dt ∂ q˙σ ∂q Given constraints (4.1), for a generalized Gaussian principle to be used, this expression for Newton’s second law is differentiated in time. Then we obtain MU = P + G ,

˙ , U=W

˙ , P=Y

˙ . G=R

(4.2)

According to formulas (4.2), generalized Gaussian principle (3.14) takes the form (4.3) δ ′′′ (M U − P)2 = 0 . In this formula three accents after the differentiation symbol δ mean that the partial derivative is computed for the fixed t, q σ , q˙σ , q¨σ , σ = 1, s. By (4.2) principle (4.3) is as follows δ ′′′ G2 = 0 . Thus, given constraints (4.1), in accordance with generalized Gaussian principle the vector G is chosen to be minimal in absolute magnitude. We rewrite principle (4.3) in the form ... (M Uσ − Pσ ) δ ′′′ q σ = 0 . (4.4) Using formula (A.52) from Appendix A, we have ˙ σ − Q˙ σ − Γρ (M Wρ − Qρ ) q˙τ . M Uσ − Pσ = M W στ Then, we supplement system (4.1) with the equations ... ˙ q¨) q σ + cλ0 (t, q, q, ˙ q¨) , λ = 1, l , l = s − k . α∗λ = cλσ (t, q, q,

(4.5)

(4.6)

The family of ... equations (4.1) and (4.6) can be regarded as the transition formulas from q σ to α∗ρ , σ, ρ = 1, s. If det [cρσ ] is not equal to zero, then we can write the inverse transformation ...σ q = hσρ (t, q, q, ˙ q¨) α∗ρ + hσ0 (t, q, q, ˙ q¨) , ρ, σ = 1, s . (4.7)

121

5. Gaussian principle in Appell’s form It follows that

... δ ′′′ q σ = hσρ δ ′′′ α∗ρ ,

ρ, σ = 1, s .

However, since the constraints (4.1) are valid, we obtain δ ′′′ α∗l+κ = 0, κ = 1, k, and therefore ... δ ′′′ q σ = hσλ δ ′′′ α∗λ , λ = 1, l , ρ, σ = 1, s . (4.8) Substituting formulas (4.8) into principle (4.4), we have (M Uσ − Pσ ) hσλ δ ′′′ α∗λ = 0 ,

λ = 1, l .

Since the variations δ ′′′ α∗λ , λ = 1, l, are independent, we obtain the following equations of motion of system (M Uσ − Pσ ) hσλ = 0 ,

λ = 1, l ,

which by formulas (4.5) can be represented finally in the form   ˙ σ − Q˙ σ − Γρ (M Wρ − Qρ ) q˙τ hσ = 0 , MW λ = 1, l . στ λ

(4.9)

(4.10)

Equations (4.9) have the same structure as Maggi’s equations and therefore they can be called the equations, represented in Maggi’s form, for thirdorder constraints (4.1). The law of motion can be found after the solution of system of equations (4.10) and (4.1), in which case for their integration the initial values of generalized coordinates, velocities, and accelerations have to be given.

§ 5. The representation of equations of motion following from generalized Gaussian principle in Appell’s form Since the vectors eσ , σ = 1, s, can be represented by formulas eσ =

˙ ∂W ∂W = ...σ , σ ∂ q¨ ∂q

˙ = U, W

the quantities M Uσ , σ = 1, s, introduced in the previous section, take the form ... ˙ q¨, q ) ∂U ∂S1 (t, q, q, ...σ M Uσ = M U · eσ = M U · ...σ = , (5.1) ∂q ∂q where S1 = M U2 /2. According to relations (4.7) and (5.1) equations (4.9) are the following ... ... ∂S1 ∂ q σ ∂qσ ...σ = P , λ = 1, l , σ = 1, s . (5.2) σ ∂ q ∂α∗λ ∂α∗λ

122

IV. Application of a Tangent Space

Now we represent equations (5.2) as ∂S1 = Pλ∗ , ∂α∗λ where Pλ∗

λ = 1, l ,

(5.3)

... ∂qσ = Pσ . ∂α∗λ

Taking into account that Pλ∗

... ∂U ∂(P · U) ∂qσ = Pσ =P· = , ∂α∗λ ∂α∗λ ∂α∗λ

equations (5.3) have the form ∂(S1 − P · U) = 0, ∂α∗λ

λ = 1, l .

(5.4)

We introduce in place of the function S1 − P · U the function  2 1 2 M P P = > 0, Z1 = S1 − P · U + U− 2M 2 M for which the following relation ∂(S1 − P · U) ∂Z1 = , λ ∂α∗ ∂α∗λ is satisfied since ∂P2 /∂α∗λ = 0. In this notation equations of motion (5.4) are the following ... ∂Z1 ∂ q σ ∂Z1 = ...σ = 0, λ = 1, l . ∂α∗λ ∂ q ∂α∗λ Represent these equations in the form of the scalar products ∇ ′′′ Z1 · ε λ = 0 , where ∇ ′′′ =

∂ ... eσ , ∂qσ

ελ =

λ = 1, l ,

(5.5)

... ∂qσ eσ = hσλ eσ . ∂α∗λ

Comparing equations (5.5) and equations (4.9), we obtain M U − P = ∇ ′′′ Z1 .

(5.6)

From equations (5.5) it follows that for the value U, corresponding to a real motion, the function Z1 (U) has the value, which is minimal in comparison with the value Z1 (U1 ) for any other U1 , which is kinematically admissible for the same t, q σ , q˙σ , q¨σ . This shall be shown below.

123

5. Gaussian principle in Appell’s form The vector U can be represented as U = UL + UK ,

UL · UK = 0 .

Here UL = U λε λ ,

UK = Ul+κ ∇ ′′′ f3κ ,

λ = 1, l ,

κ = 1, k ,

in which case for the fixed t, q σ , q˙σ , q¨σ the vector UK is completely determined by the equations of constraints and the vector UL remains arbitrary. In other words, any vector UL is kinematically admissible. This implies that U1 − U = U1L + UK − UL − UK = ∆U λε λ ,

λ = 1, l ,

where ∆U λ are arbitrary. Using this relation for U1 in the function Z1 and taking into account relations (5.5) and (5.6), we have 2  M P λ Z1 (U1 ) = = + ∆U ε λ U− 2 M

2 M ∆U λε λ = 2 2 M ∆U λε λ > Z1 (U) , = Z1 (U) + 2

= Z1 (U) + ∇ ′′′ Z1 · ∆U λε λ +

U1 = U .

Condition Z1 (U1 ) > Z1 (U) is obtained here from equations of constrained motion (5.5). However equations (5.2)–(5.5) can also be regarded as necessary conditions of minimality for the function Z1 under constraints (4.1). Equations (5.2)–(5.5) have a structure, suggested by Appell for mechanical systems with the constraints up to be the second order. Therefore they can be called equations of Appell’s form with third-order constraints. In many works (see, for example, [193]) the equations of motion for nonholonomic system with the constraints of the form (4.1) are represented (using the notation of the work [193]) as ∂ S˙ = Q∗λ , ∂α∗λ

λ = 1, l ,

(5.7)

where

dS ˙ , Q∗ = Qσ hσ . S˙ = = MW · W λ λ dt Equations (5.7) are obtained from Appell’s equations ∂S = Qσ + Rσ , ∂ q¨σ

σ = 1, s ,

in a formal way and therefore cannot be regarded as a minimum condition for the function Z1 . This is a foundational distinction between equations (4.9), (5.2)–(5.5) and equations (5.7).

124

IV. Application of a Tangent Space

Finally, it should be specially accented that in a number of problems a formal application of generalized Gaussian principle to the analysis of motion of mechanical system can give rise to unexpected results [79]. Therefore the general theory of motion of nonholonomic systems with high-order constraints, established in Chapter V, is noteworthy.

Chapter V THE MIXED PROBLEM OF DYNAMICS. NEW CLASS OF CONTROL PROBLEMS

The law, of a motion of mechanical system, represented in the vector form, is applied to the solution of the mixed problem of dynamics. The essence of the problem is to find additional generalized forces such that the program constraints, given in the form of system of differential equations of order n  3, are satisfied. The notion of generalized control force is introduced. The fact is proved that if the number of program constraints is equal to the number of generalized control forces, then the latter can be found as the time functions from the system of differential equations in generalized coordinates and these forces. The conditions, under which this system of equations has a unique solution, are determined. The conditions are also obtained under which for the constraints of any order the motion control is realized according to Gauss’ principle. Thus, the theory is constructed with the help of which a new class of control problems can be solved. This theory is used to consider two problems connected with the dynamics of spacecraft motion. In the first problem a radial control force, providing the motion of spacecraft with modulo constant acceleration, is determined as a time function. In the second problem we seek the law, of varying in time the radial and tangential control forces, by which a smooth passage of spacecraft from one circular orbit to another occur.

§ 1. The generalized problem of P. L. Chebyshev. A new class of control problems As is well known, P. L. Chebyshev is a founder of the theory of synthesis of mechanisms. He has posed and solved the problem of constructing the machines, the concrete points of which are subject to a given motion. Among such devices we can recall, for example, the mechanisms with the stoppings of definite elements in the given positions. We generalize this Chebyshev’s problem to the case that the motion of certain points of mechanism is a solution of the given differential equations of order n  3. The following is noteworthy. As is noted by L. A. Pars [179] and V. V. Rumyantsev [199], the forces cannot depend on accelerations. However we can always choose the system of forces, realizing the required motion of mechanical system q σ = q σ (t), σ = 1, s, in which case any law of varying any of derivatives with respect to generalized coordinates can be satisfied. We can, by that, provide the vanishing of any combination with respect to derivatives of generalized coordinates of system. Therefore it is obvious that 125

126

V. The Mixed Problem of Dynamics. New Class of Control Problems

we can require such a motion of mechanical system that the system of differential equations of any order is satisfied. Taking into account the above notes, we consider the following problem. Suppose, in the generalized coordinates q σ the motion of mechanical system acted by the given generalized forces Qσ is described by Lagrange’s equations of the second kind d ∂T ∂T − σ = Qσ , σ dt ∂ q˙ ∂q σ = 1, s , α, β = 0, s ,

M gαβ q˙α q˙β , 2 q 0 = t , q˙0 = 1 ,

T =

(1.1)

where M is a mass of a whole system. It is necessary to find as functions of time the forces Rσ , which have to be added to the forces Qσ in order that the motion satisfies the following system of differential equations (n−1) (n)

(n−1)

˙ ... , q ) q σ + aκ ˙ ... , q ) = 0 , fnκ ≡ aκ nσ (t, q, q, n0 (t, q, q, σ = 1, s ,

κ = 1, k ,

(1.2)

k  s.

For n  3 we call this problem the generalized problem of P. L. Chebyshev. However, before we attack the problem, recall briefly some results, obtained for the constraints given in the form (1.2) for n = 1, 2. In this case the sought additional forces, which represented as R σ = Λκ

∂fnκ (n)

,

∂ qσ

where the Lagrange multipliers Λκ , κ = 1, k, are uniquely determined as functions of variables t, q σ , q˙σ , σ = 1, s, if  κ ∂fn ∂fnµ στ det g = 0 , (n) (n) σ τ ∂q ∂q κ, µ = 1, k . σ, τ = 1, s , Here g στ are elements of the matrix inverse to the matrix with the elements gστ . Note that for n = 1 there exists a sufficiently large class of problems, in which the forces Rσ , σ = 1, s, result from interactions of mass bodies, what leads to the occurrence of the constraints of the form (1.2). A distinctive feature here is that the generalized reactions Λκ , κ = 1, k, i. e. the forces, arising out of these interactions, are necessary and sufficient for realizing the motion that satisfy the equations of constraints. In this case the methods of nonholonomic mechanics make it possible to find this motion without determining the generalized reactions. For n = 2 we can also obtain this motion ˙ κ = 1, k, are unknown. However they cannot even if the functions Λκ (t, q, q),

127

1. The generalized problem of P. L. Chebyshev.

be called generalized reactions since at present one cannot find the interaction between mass bodies, which leads to the occurrence of reactions, providing the validity of the absolutely nonintegrable equations (1.2) for n = 2 (except for the example, considered in the work [335]). Then the values Λκ , κ = 1, k, can be regarded only as the forces, which are given by control system. It is clear that for n  3 the validity of equations (1.2) is also provided by a control system only. Therefore we introduce a notion of generalized control force. So, we assume that a control system generates a certain force such that its virtual elementary work is as follows ˙ σ, δA = Λ bσ (t, q, q)δq

σ = 1, s .

We shall say that the value Λ, entering into this relation, is a generalized control force. Suppose, the k control forces Λκ , κ = 1, k, can be generated by the control system. Then we have ˙ σ, δA = Λκ bκ σ (t, q, q)δq

κ = 1, k ,

σ = 1, s .

(1.3)

Note that a mechanism, by which the control forces arise, is, as a rule, such that in relation (1.3) the coefficients bκ σ are either constant or the functions of generalized coordinates only. From formulas (1.3) it follows that the additional generalized forces Rσ , σ = 1, s, corresponding to the generalized control forces Λκ , κ = 1, k, take the form R σ = Λ κ bκ σ . Below we show that if in the time interval t0  t  t∗ we have µ στ det[bκ σ anτ g ] = 0 ,

σ, τ = 1, s ,

(1.4)

κ, µ = 1, k ,

then for n = 1, 2, the generalized control forces Λκ , κ = 1, k, can uniquely be found as the functions of the variables t, q σ , q˙σ , If n  3, then, as will be shown below, the generalized control forces Λκ , κ = 1, k, can be found as the functions of time only. In this case the differential equation with respect to each of the functions Λκ is of order (n − 2). Thus, for n  3 the generalized control forces Λκ , κ = 1, k, and the generalized coordinates q σ , σ = 1, s, are regarded as the sought functions of time, satisfying the following initial data (n−3)

(n−3)

κ = 1, k ,

σ = 1, s .

Λκ (t0 ) = Λ0κ , Λ˙ κ (t0 ) = Λ˙ 0κ , ... , Λκ (t0 ) = Λκ 0 , q σ (t0 ) = q0σ ,

q˙σ (t0 ) = q˙0σ ,

(1.5)

In the next section we show that if in the time interval t0  t  t∗ condition (1.4) is satisfied, then in this time interval for the given initial data (1.5) the

128

V. The Mixed Problem of Dynamics. New Class of Control Problems

motion and the generalized control forces can uniquely be found such that relations (1.2) are satisfied. Thus, the generalized problem of P. L. Chebyshev can uniquely be solved. The statement of this problem has the features of as the direct, as inverse problem of dynamics. Really, on the one hand, for the given forces Qσ , σ = 1, s, we seek the motion of system and, on the other hand, we seek as the functions of time the additional forces Rσ , σ = 1, s, providing a motion such that relations (1.2) are satisfied for n  3. Therefore the academician S. S. Grigoryan calls the generalized problem of P. L. Chebyshev the mixed problem of dynamics. The solution of the problem, constructed below, permits us to find as functions of time the generalized control forces, the existence of which is a necessary and sufficient condition in order that the motion satisfies a system of equations of any order. This construction is actual since it permits us to solve a new sufficiently large class of control problems.

§ 2. A generation of a closed system of differential equations in generalized coordinates and the generalized control forces For the solution of the mixed problem of dynamics it is rational to use the notion of a tangent space, introduced in § 1 of Chapter IV. In this case if the forces Rσ are added to the forces Qσ , system of equations (1.1) can be written as one vector equation M W = Y + Λκ bκ , where

Y = Qσ eσ ,

κ = 1, k ,

σ bκ = bκ σe ,

W = (gστ q¨τ + Γσ,αβ q˙α q˙β )eσ = (¨ q σ + Γσαβ q˙α q˙β )eσ ,   1 ∂gτ α ∂gαβ ∂gτ β Γσαβ = g στ Γτ,αβ = g στ + − , 2 ∂q α ∂q β ∂q τ σ, τ = 1, s ,

(2.1)

α, β = 0, s ,

(2.2)

κ = 1, k .

σ

Here eσ and e are the vectors of the fundamental and reciprocal bases of a tangent space, respectively. From relations (2.1) and (2.2) we have q¨σ = F2σ (t, q, q, ˙ Λ) , σ, τ = 1, s ,

στ F2σ = −Γσαβ q˙α q˙β + (Qτ + Λκ bκ τ )g /M ,

α, β = 0, s ,

κ = 1, k .

(2.3)

Consider first the case n = 3. Using formulas (2.2) and taking into account that σ, τ = 1, s , α = 0, s , e˙ τ = Γστα q˙α eσ ,

129

2. A generation of a closed system of differential equations we represent system of equations (1.2) in vector form κ κ σ ˙ aκ ˙ q¨) , aκ 3 · W = χ3 (t, q, q, 3 = a3σ e ,   d σ α β κ κ κ τ τ α β σ α χ3 = −a30 + a3σ q + Γαβ q˙ q˙ )Γτ α q˙ , (Γ q˙ q˙ ) + (¨ dt αβ

σ, τ = 1, s ,

α, β = 0, s ,

(2.4)

κ = 1, k .

Differentiating in time equation (2.1), we obtain ˙ =Y ˙ + Λ˙ κ bκ + Λκ b˙ κ , MW

κ = 1, k ,

(2.5)

where κ σ α τ b˙ κ = (b˙ κ τ − bσ Γτ α q˙ )e ,

˙ = (Q˙ τ − Qσ Γσ q˙α )eτ , Y τα σ, τ = 1, s ,

α, β = 0, s ,

κ = 1, k .

Multiplying equation (2.5) scalarly by the vectors aµ3 and taking into account relations (2.4), we get µ Λ˙ κ hκµ ˙ q¨, Λ) , 3 = B3 (t, q, q, µ κ κ µ στ , hκµ 3 = b · a3 = bσ a3τ g

˙ · aµ − Λκ b˙ κ · aµ , B3µ = M χµ3 − Y 3 3 σ, τ = 1, s ,

κ, µ = 1, k .

Condition (1.4) is satisfied by assumption and therefore we have Λ˙ κ = h3κµ (t, q, q, ˙ q¨)B3µ (t, q, q, ˙ q¨, Λ) ,

κ, µ = 1, k .

(2.6)

Here h3κµ are the elements of the matrix inverse to the matrix with the elements hκµ ¨σ from the 3 . By formulas (2.3) we can eliminate the derivatives q µ 3 functions hκµ , B3 and represent the right-hand sides of equations (2.6) as Λ˙ κ = Cκ3 (t, q, q, ˙ Λ) ,

κ = 1, k .

(2.7)

For arbitrary n the functions hnκµ , Bnµ occur such that we need to eliminate (n−1)

the derivatives q¨σ , ... , q σ , from them. By (2.3) we have ...σ ∂F2σ ∂F2σ τ ∂F2σ τ ∂F2σ ˙ q = + q˙ + q¨ + Λκ , ∂t ∂q τ ∂ q˙τ ∂Λκ

σ, τ = 1, s ,

κ = 1, k . (2.8)

Using formulas (2.3), we can eliminate the derivatives q¨σ from relations (2.8) and represent them in the form ...σ ˙ , q = F3σ (t, q, q, ˙ Λ, Λ) σ = 1, s . Reasoning as above, we obtain (n−1) σ

q

(n−3)

σ ˙ ... , Λ ) , = Fn−1 (t, q, q, ˙ Λ, Λ,

σ = 1, s .

130

V. The Mixed Problem of Dynamics. New Class of Control Problems

Thus, in the general case we have (n−2)

(n−3)

˙ ... , Λ ) , Λκ = Cκn (t, q, q, ˙ Λ, Λ,

κ = 1, k ,

n  3.

(2.9)

The particular case of these equations is system (2.7). Equations (2.3) and (2.9) make up the closed system of equations with respect to the functions q σ (t) and Λκ (t). By initial data (1.5) it has a unique solution, which was to be proved. We remark that if the differential, describing the motion, depends nonlin(n−1)

early on the higher derivatives q σ , then, differentiating in time this equa(n)

tion, we obtain the equation, which depends linearly on the derivatives q σ . Therefore the theory suggested can also be applied to high-order nonlinear equations. Now we turn to the second-order equations. Representing them in vector form, we obtain κ σ aκ 2 = aσ e ,

κ ˙ , aκ 2 · W = χ2 (t, q, q)

κ κ σ α β χκ 2 = −a20 + a2σ Γαβ q˙ q˙ ,

σ = 1, s ,

α, β = 0, s ,

(2.10)

κ = 1, k .

Multiplying equation (2.1) scalarly on the vectors aµ2 , µ = 1, k, we find µ Λκ hκµ ˙ , 2 = B2 (t, q, q) µ κ κ µ στ hκµ , 2 = b · a2 = bσ a2τ g

B2µ = M χµ2 − Y · aµ2 , σ, τ = 1, s ,

κ, µ = 1, k .

By assumption, condition (1.4) is satisfied. Then ˙ 2µ (t, q, q) ˙ , Λκ = h2κµ (t, q, q)B

κ, µ = 1, k .

Thus, for n = 2 the generalized control forces can uniquely be found as the functions of time, generalized coordinates, and generalized velocities. This is also valid in the case when in equations (1.2) n = 1. In fact, differentiating them in time, we obtain equations such that κ aκ 2σ = a1σ ,

σ aκ ˙κ ˙κ 20 = a 1σ q˙ + a 10 ,

σ = 1, s ,

κ = 1, k .

The generation of control forces as the functions of the variables t, q σ (t), q˙ (t), σ = 1, s, is more complicated problem than its generation as the functions of time, which can be obtained by means of integrating a system of differential equations. Taking into account these reasonings it is rational to differentiate in time the equations of program motion for n = 1, 2, twice and once, respectively, and to reduce them, thus, to third-order equations. σ

131

3. The mixed problem of dynamics and Gaussian principle § 3. The mixed problem of dynamics and Gaussian principle

The obtained solution of the mixed problem of dynamics depends substantially on as the form of equations (1.2), as the system of the vectors bκ , in which the sought force R(t) is expanded. Consider the particular case when the coefficients aκ nσ , entering into equations (1.2), are functions of the variables t, q σ (t), q˙σ (t), σ = 1, s, only. Using these coefficients, we represent a tangential space as a direct sum of K and L–spaces. For n = 2 this was made in § 1 of Chapter IV. In this Chapter a system of linearly independent vectors ε l+κ = ∇ ′′ f2κ , l = −k , κ = 1, k , is a basis of K–space. These vectors have no the index n since for n = 0, 1, they have respestively the form ε l+κ = ∇ f0κ ,

ε l+κ = ∇ ′ f1κ ,

κ = 1, k .

For n  3 the vectors (n) κ aκ fn = aκ ˙ σ, n =∇ nσ (t, q, q)e

σ = 1, s ,

κ = 1, k ,

which are also assumed to be independent, must have the index n since for the given n, by assumption, equations (1.2) cannot, generally speaking, be obtained by means of differentiation in time of lower-order equations. The vectors aκ n , κ = 1, k, are assumed to be a basis of K–space and the vectors aλn such that aλn · aκ n = 0,

λ = 1, l ,

l = s−k,

κ = 1, k ,

are assumed to be a basis of L–space. In this case the vectors W and Y like those, considered in § 1 of Chapter IV, can be represented as W = WL + WK ,

Y = YL + YK ,

WL · WK = 0 ,

YL · Y K = 0 .

We shall show that if equations (1.2) take the form (n)

(n−1)

fnκ ≡ aκ ˙ q σ + aκ ˙ ... , q ) = 0 , nσ (t, q, q) n0 (t, q, q, σ = 1, s ,

κ = 1, k ,

(3.1)

n  3,

then for the given values of the variables t, q σ , q˙σ , σ = 1, s, these equations do not constrain the mechanical system to have the acceleration WL different from the acceleration, given by Newton’s law M WL = YL .

(3.2)

We begin the proof from the case n = 3. Representing equations (2.4) as d κ ˙κ (a · W) = χκ 3 +a 3 ·W, dt 3

κ = 1, k ,

132

V. The Mixed Problem of Dynamics. New Class of Control Problems

we obtain aκ 3

·W =

aκ 3

· W|t=t0 +

t

˙κ (χκ 3 +a 3 · W)dt ,

κ = 1, k .

(3.3)

t0

The right-hand sides of these relations cannot be found as the functions of the variables t, q σ , q˙σ , σ = 1, s, since, generally speaking, the order of equations (2.4) cannot be lowered. We shall regard these right-hand sides as the functions of time, which are equal to Ψκ 3 (t). Note that in order to find them we need to know the motion of system, satisfying Newton’s law (2.1). This law involves the unknown control forces Λκ (t), κ = 1, k, which, as is shown, are to be determined from system of equations (2.3), (2.7). Thus, the functions Ψκ 3 (t), κ = 1, k, are certain unknown functions. From equations (3.3), represented in the form ˙ · W = Ψκ aκ 3 (t, q, q) 3 (t) ,

κ = 1, k ,

it follows that for the given values of variables t, q σ , q˙σ , σ = 1, s, equations (2.4) are satisfied for any vector WL , and, in particular, for that, given by Newton’s law (3.2). For n-order constraints, equations (3.1) yields that the (n − 2)-nd time derivatives of the scalar products aκ n · W, κ = 1, k, are known functions of (n−1)

the variables t, q σ , q˙σ , . . . , q σ , σ = 1, s. Hence these products themselves can be represented in terms of definite integrals. Therefore for any n and the fixed values of the variables t, q σ , q˙σ , σ = 1, s, the mechanical system cannot be constrained by equations (3.1) to have in an L–space the acceleration different from that, given by the Newton equations (3.2). Recall that at the end of § 2 of Chapter IV the linear nonholonomic constraints of any order, by which equation (3.2) is satisfied, were called ideal. As is shown, this is the constraints, given in the form (3.1). The definition of ideality of these constraints has a geometric interpretation. For constraints, given by equations (3.1), the scalar products aκ n · W, κ = 1, k, are expressed via a definite integral. They cannot be found as the functions of the variables t, q σ , q˙σ and therefore it is necessary to regard them as the unknown functions Ψκ n (t), κ = 1, k. For the fixed values of the variables t, q σ , q˙σ , σ = 1, s, in the space of accelerations W the system of equations κ aκ n · W = Ψn (t) ,

κ = 1, k ,

(3.4)

gives an l-dimensional plane. Its position relative to the origin is determined by a system of the independent sought functions Ψκ n (t), κ = 1, k. This plane is similar to that, represented in Fig. IV. 3 of Chapter IV. Substituting into equations (3.3) the acceleration W, represented in the form W = WL + WK ,

3. The mixed problem of dynamics and Gaussian principle

133

we obtain WK = hκµ Ψµn (t)aκ n ,

κ, µ = 1, k .

Here hκµ are elements of the matrix inverse to the matrix with the elements hκµ , which are given by the relations µ hκµ = aκ n · an ,

κ, µ = 1, k .

For n = 1, 2, the vector WK , which is equal to k-dimensional perpendicular, dropped from the origin of coordinates to the introduced l-dimensional plane, is uniquely defined by equations (3.1) as a function of the variables t, q σ , q˙σ , σ = 1, s. For n  3 this vector is defined by a value assignment of the unknown functions Ψκ n (t), κ = 1, k. This is a vital difference of equations (3.1) for n  3 and the same equations for n = 1, 2. Equation (2.1) yields that the position of the considered l-dimensional plane relative to the end of the vector Y/M is given by the vector Λκ (t)bκ /M . Thus, the position of this plane is defined by a value assignment of as the functions Ψκ n (t), κ = 1, k, as the functions Λκ (t), κ = 1, k. This implies that if it is impossible, in principle, to lower the order of equations (3.1) up to n = 2 and to find the σ σ values Ψκ n (t), κ = 1, k, as the functions of the variables t, q , q˙ , σ = 1, s, then it is also impossible, in principle, to find the generalized control forces Λκ , κ = 1, k, as the functions of the same variables. Therefore in the previous section we sought them as the functions of time. In accordance with Gaussian principle the constraint measure, given by the following relation τ4 (W − Y/M )2 , Zg = 4 must be minimal. In the enunciation of this principle, which is due to Gauss itself, it is not said that the value Zg is to be regarded as the function, which is given on a set of the accelerations W admitted by constraints, and that its minimum is sought on this set. The formulation of Gauss is more general. We give it, using the treatise of P. Appell [4, Vol. II, p. 421]: "The motion of a system of mass points, which are arbitrary one another and are liable to any influence, is at every moment in as perfect as possible agreement with the motion of these points under the condition that they became free, i. e. the motion is under the minimally possible constraint if we regard as a measure of constraint, applied in an infinitely small time, a sum of the products of a mass of each point by the square value of its deviation from the position of this point under the condition that it is free". Now we apply the formulation of Gaussian principle to the case when the constraints are given by equations (3.1). Under the condition that the system considered is free, it might have the acceleration Y/M . However, really, on account of the existence of constraints its acceleration W is an element of the set that is an l-dimensional plane. The position of this plane relative to the origin of coordinates is given by a system of the independent functions Ψκ n (t), κ = 1, k. For all that, on the above-mentioned plane we can

134

V. The Mixed Problem of Dynamics. New Class of Control Problems

find the point, for which the value Zg is minimal in accordance with Gaussian principle. This point can be obtained if the generation of the generalized control κ forces is in agreement with equations (3.1) so that bκ σ = anσ , σ = 1, s, κ = 1, k. In this case equation (2.1) takes the form M W = Y + Λκ a κ n ,

κ = 1, k .

(3.5)

This implies that a k-dimensional perpendicular, dropped from the point Y/M to the considered plane, is given by the vector Λκ aκ n such that the value Zg is minimal. Thus, for bκ = aκ n , κ = 1, k, the motion control is realized in accordance with Gaussian principle. We call such control ideal. The condition that for ideal control the vector of control force R = M W − Y is orthogonal to the introduced l-dimensional plane can be represented as δA = (M W − Y) · δy = 0 .

(3.6)

Here δy is an arbitrary tangential vector, satisfying the following system of equations κ = 1, k . (3.7) aκ n · δy = 0 , Representing relations (3.6), (3.7) in scalar form and taking into account that aκ n = we obtain

∂fnκ (n) ∂ qσ

eσ ,

σ = 1, s ,

κ = 1, k ,

 d ∂T  ∂T σ − − Q σ δq = 0 , dt ∂ q˙σ ∂q σ ∂fnκ (n) ∂ qσ

δq σ = 0 ,

σ = 1, s ,

κ = 1, k .

(3.8) (3.9)

As is remarked in § 6 of Chapter II and in § 2 of Chapter IV, N. G. Chetaev makes use of the imposing of the conditions ∂f1κ σ δq = 0 , ∂ q˙σ

σ = 1, s ,

κ = 1, k ,

on the virtual displacement δq σ , σ = 1, s, for "... the D’Alembert and Gaussian principles to be saved together ..."[245, p. 68]. The equations (3.8), (3.9) are obtained under the assumption that the motion control is realized in accordance with Gaussian principle. This implies that the imposing of conditions (3.9), which is similar by its structure to the conditions of N. G. Chetaev, on the virtual displacements, entering into the D’Alembertian–Lagrange principle (3.8), makes it possible to "... save the D’Alembert and Gaussian principles together ..."

3. The mixed problem of dynamics and Gaussian principle

135

We shall show that an ideality condition of control, written in the form of equations (3.6), (3.7), can also be represented in the form of the Mangeron– Deleanu principle. In fact, computing the partial differential δ (n) of the equations of program constraints (3.1) for the fixed values of the variables (n−1)

t, q σ , q˙σ , . . . , q σ , σ = 1, s, we obtain δ (n) fnκ =

∂fnκ (n)

(n)

(n−1)

(n−1) δ (n) q σ = aκ V = 0, n ·δ

∂ qσ

(n−1)

(n)

δ (n−1) V = δ (n) q σ eσ ,

κ = 1, k .

This system is identical to system (3.7) under the assumption δy =

τ n (n−1) (n−1) δ V . n!

Thus, equation (3.6) can be represented in the form of the Mangeron–Deleanu principle [59. 1976] (n−1)

(M W − Y) · δ (n−1) V = 0 ,

(3.10)

which was to be proved. Note that the principle (3.10) can be used for proσ σ gram constraints (3.1) only if the vectors aκ n depend on the variables t, q , q˙ , σ = 1, s. From the representation of the control force R in the form ∇(n) fnκ , R = Λ κ aκ n = Λκ (t)∇

κ = 1, k ,

it follows that for ideal control, to each program constraint corresponds its generalized control force Λκ (t), κ = 1, k. Note that the idea that the forces are generated by constraints is due to G. Hertz. In the simplest case of one holonomic constraint such that the mass point is on the given surface, the ideality criterion is that this constraint can be provided by imposing on the mass point the normal reaction only. In this case this reaction is a generalized reaction, corresponding to this constraint. If it is nonlinear, i. e. the surface is rough, the normal reaction is insufficient and it is necessary still to overcome the frictional force. Thus, the constraint is ideal in the case when it is provided by its generalized reaction, which, as is noted by A. M. Lyapunov [138], is necessary and sufficient for the constraint to be ideal. This definition of ideality, as is shown above, can be extended to high-order program constraints, given in the form (3.1). In the case when the order of constraints is less than three, the forces Λκ , κ = 1, k, are known functions of the variables t, q σ , q˙σ , σ = 1, s. Therefore at initial time their values are defined by the initial values of coordinates and velocities. For program constraints (3.1) the generalized control forces

136

V. The Mixed Problem of Dynamics. New Class of Control Problems

Λκ , κ = 1, k, are unknown functions of time, satisfying the system of (n−2)order differential equations. Hence at initial time the values Λκ , κ = 1, k, must be given together with their time derivatives up to the (n − 3)-rd order. Thus, for n  3 the problem must be solved with initial data (1.5). Consider the question of initialization for our problem from another standpoints. Equation (3.5), corresponding to ideal control, can be obtained from the Mangeron–Deleanu principle. This was made in scalar (but not vector) form by M. A. Chuev [252. 1975]. We remark that the generalized Gaussian principle was proposed first in this work and in the paper [252. 1974] and then independently in the works [188, 189]. Multiplying equation (3.5) scalarly by the vectors aλn , λ = 1, l, l = s − k, such that κ = 1, k , aλn · aκ n = 0, we obtain the equations (M W − Y) · aλn = 0 ,

λ = 1, l .

(3.11)

If the order of constraints is less than three, then, joining equations (3.11) and (2.10), we obtain the closed system of differential equations, describing the motion with the given initial values of coordinates and velocities. Expressing the Newton law in L–space by equations (3.11) and adding to them, in particular, for n = 3 equation (2.4), for program constraints (3.1) we obtain a system of differential equations of general order (2s + k). This system does not involve the sought functions Λκ (t), κ = 1, k, which are, in principle, intrinsic to this problem. Therefore their elimination leads to the problem of initialization for the variables q σ , σ = 1, s. This problem is discussed, in particular, in the work of M. A. Chuev. He remarks [252, p. 69] that ". . . the Mangeron–Deleanu principle makes it possible to obtain the equations which are not contradict to the principle of superposition for the constraints of the form (3.1) only and under very strong restriction on the initial data". M. A. Chuev writes that the principle of superposition is violated in the case when ". . . the forces depend on the derivatives of coordinates, the order of which is greater than unity"[252, p. 69]. In this case he refers to the treatise of L. Pars [179], § 1.4, where it is shown that the force cannot be a function of acceleration. The validity of this assertion results from the following reasonings of V. I. Arnold. In his book [6, p. 8] he remarks: "The initial state of mechanical system (a set of all positions and velocities of points of system at any time) uniquely defines its whole motion". V. I. Arnold calls this law of nature Newton’s determinacy principle. According to this principle the position of mechanical system and its velocity V at time t defines a derivative of any order of the vector V at as this moment as all subsequent moments of time. V. I. Arnold [6, p. 12] writes "In particular, the position and velocity define acceleration. In other words, there exists the function F . . ."of the variables t, q σ , q˙σ , σ = 1, s, such that W = F(t, q, q) ˙ .

4. The motion of spacecraft with modulo constant acceleration

137

By (2.1) for the given active force Y(t, q, q) ˙ the existence of the function F results from the fact that the vector of the control force R = Λκ bκ , κ = 1, k, providing the fulfilment of program constraints (1.2) for n = 1, 2, is uniquely defined as the function of variables t, q σ , q˙σ , σ = 1, s. For n  3, the causality principle is also preserved but by the fact that the generalized control forces are sought as time functions. In this case if bκ = aκ n , κ = 1, k, then the control forces as functions of time are generated by Gaussian principle. § 4. The motion of spacecraft with modulo constant acceleration in Earth’s gravitational field Let a spacecraft move in Earth’s gravitational field along elliptical orbit. Suppose, beginning from a certain instant of time, the spacecraft has a constant acceleration. This condition we regard as a second-order nonlinear nonholonomic program constraint. The imposing of constraint can occur at any point of orbit, the additional force at this moment of time is lacking. The motion of spacecraft along elliptical orbit is described by the equation µρρ d2 ρ = − 3 , µ = γM , ρ = |ρρ| . (4.1) 2 dt ρ Here ρ is a radius–vector, connecting the geocenter and a spacecraft, γ is a gravity constant, M is Earth’s mass. The constant µ can be represented as [189] 4π 2 a3 , µ= T2 where a is a major semiaxis of the spacecraft elliptical orbit, T is a time of complete revolution. In the dimensionless variables r = xi + yj = ρ /a ,

τ = 2πt/T ,

equation (4.1) takes the form ¨r = −r/r3 ,

r = |r| .

(4.2)

A point denotes the derivative with respect to the dimensionless time τ . The integral of energy and the area integral in equation (4.2) are the following [189]  v 2 = 2/r − 1 , v = |˙r| , r2 ϕ˙ = 1 − e2 , (4.3)

respectively. Here e is an eccentricity of elliptical orbit. Suppose, at initial time, beginning from which a spacecraft must move with a constant acceleration, it is on the axis x. Without loss of generality, we can assume that in this case the initial data are the following:  ˙ = x˙ 0 = 2x0 − x20 − 1 + e2 /x0 , x(0) = x0 , x(0) (4.4)  ˙ = y˙ 0 = 1 − e2 /x0 , 1 − e  x0  1 + e . y(0) = y0 = 0 , y(0)

138

V. The Mixed Problem of Dynamics. New Class of Control Problems

The equation of constraint in the accepted notations can be represented as ¨r2 − 1/x40 = 0 .

(4.5)

This equation is satisfied, in particular, when the vector ¨r collinear to the vector r is constant in value. In this case the time derivative of the vector ¨r is orthogonal to the vector r, i. e. we have ... er · r = 0 ,

er = r/r .

(4.6)

This equation is a linear third-order nonholonomic program constraint. Thus, in this problem system of equations (2.4) is reduced to one equation (4.6). We assume that a spacecraft is equipped by the generalized control force Λ such that the vector of control force is as follows R = Λer . From equation (4.6) we conclude that for this force R the control is ideal, i. e. it is in accordance with Gaussian principle. Beginning from the instant of imposing constraint (4.6), the motion of spacecraft is described by the following equation ¨r = −

r r +Λ . r3 r

(4.7)

At tine of imposing a constraint the control force is lacking, i. e. we have Λ(0) = 0 .

(4.8)

Differentiating relation (4.7) with respect to τ , we obtain r˙ r˙ rr ˙ 3r˙ r ... r = − 3 + 4 r + Λ˙ + Λ − Λ 2 . r r r r r Multiplying this equation scalarly by r and taking into account equation of constraint (4.6) and the relation r2 = r2 , we have

r · r˙ = rr˙ ,

2˙r Λ˙ = − 3 . (4.9) r In this problem, system of equations (2.6) is reduced thus to one equation (4.9). Assuming in it dΛ Λ˙ = − r˙ , dr we find dΛ 2 =− 3. dr r

4. The motion of spacecraft with modulo constant acceleration

139

Integrating this equation and taking into account that by (4.4) and (4.8) Λ = 0 for r = x0 , we get 1 1 Λ= 2 − 2. r x0 Substituting this relation into equation (4.7), we obtain ¨r = −r/(rx20 ) .

(4.10)

Now we shall show that this equation follows directly from Gaussian principle. In fact, determining a minimum of the function Z = |¨r + r/r3 |2 on a set of the values ¨r, admitted by equations (4.5), we obtain ¨r + r/r3 + Λ∗ ¨r = 0 .

(4.11)

Here Λ∗ is a sought Lagrange multiplier. This implies that (1 + Λ∗ )2 ¨r2 = 1/r4 . Taking into account equation of constraint (4.5), we have Λ∗ = x20 /r2 − 1 . Substituting this value of the Lagrange multiplier into equation (4.11), we arrive at equation (4.10), which was to be proved. Using equation (4.10), we find a motion satisfying equation (4.5), in the case when the control force R = Λr/r, due to which this motion occurs, is unknown. However for this motion to be found really, it is necessary to know the force as a function of time. Therefore we do not eliminate a control force from equation (4.7) and consider it together with equation (4.9). Projecting the vector equation (4.7) on the unit vectors of the polar system of coordinates er = r/r and eϕ , we find 1 = Λ, r2 rϕ¨ + 2r˙ ϕ˙ = 0 .

r¨ − rϕ˙ 2 +

(4.12)

Adding these two equations to equation (4.9), we obtain the closed system of equations, from which the motion and the control force can be found. The numerical integration of system equations (4.9), (4.12) was performed with the following initial data r(0) = x0 = 1 − e , r(0) ˙ = 0 , ϕ(0) = 0 ,  2 ϕ(0) ˙ = 1 − e2 /x0 , Λ(0) = 0 .

The computation demonstrates that for any value of the eccentricity e different from zero and unity, the trajectory of spacecraft motion is a curve, which

140

V. The Mixed Problem of Dynamics. New Class of Control Problems

lies between two concentric circles. To find their radii and their dependence on the values x0 and e, we need to consider equation (4.10). The integral of energy with arbitrary initial data (4.4) is as follows v2 r 1 x˙ 2 + y˙ 02 4 − x0 + 2 = 0 + = . 2 x0 2 x0 2x0 Then using the Binet formula [189] v 2 = c2 we obtain

 du 2 dϕ

 + u2 ,

 dr 2 dϕ

=

u=

1 , r

c2 = r4 ϕ˙ 2 = 1 − e2 ,

2r4  4 − x0 r 1 − e2  . − − c2 2x0 x20 2r2

The trajectory of spacecraft makes contact with the circle of radius r at dr the point such that = 0. Therefore the sought radii r1 and r2 are positive dϕ roots of the equation 2r3 − (4 − x0 )x0 r2 + x20 (1 − e2 ) = 0 . Note that the motion between the circles of these radii is not periodic in the sense that the point never returns at initial position in integer revolutions. As an example, in Figs. V. 1, V. 2, and V. 3 are shown the results of computation in the time interval 0  t  T /2 (0  τ  π) for e = 0.4. In Fig. V. 1 the original elliptical orbit and the concentric circles of radii r1 = 0.6 and r2 = 0.754, respectively, which the solution of equation (4.10) lies between, are shown by thin lines. The solution is shown by a bold line. The hodograph of the control force R = Λ(τ )r/r, providing this solution, is shown in Fig. V. 2 by a thick line. The results shown are under the assumption that Λ  0. The graph of the function Λ(τ ) is shown in Fig. V. 3. Note that the value Λ, as follows from equations (4.1) and (4.7), is measured in terms of the fractions of the gravitational force F , where F =

µm . a2

Here m is a spacecraft mass. Consider now the solution of this problem, resulting from generalized Gaussian principles. Differentiating equation of constraints (4.5) in time, we obtain ... ¨r · r = 0 . (4.13) Determining a minimum of the function Z=

1 ... r˙ 3rr ˙ 2 r + 3 − 4  2 r r

4. The motion of spacecraft with modulo constant acceleration

141

Fig. V. 1

... on a set of the values r , admitted by equations (4.13), we arrive at the following equation r˙ 3r˙ ... r = − 3 + 4 r + Λ∗ ¨r , (4.14) r r where Λ∗ is a sought Lagrange multiplier. Relations (4.5), (4.3) yield the relation x4 3x4 r˙ Λ∗ = 30 r˙ · ¨r − 04 r · ¨r . r r The Cartesian coordinates were used for numerical integration of equation (4.14) after the substitution the value Λ∗ . Initial data (4.4) were completed by the following initial data for accelerations: x ¨(0) = x ¨0 = −1/x20 ,

y¨(0) = y¨0 = 0 .

The computation demonstrates that even for very small eccentricity (independently of x0 ) the trajectory goes at infinity. It tends asymptotically to a straight-line motion with constant acceleration. In Fig. V. 4 is shown that the tendency to a straight-line motion increases with value e. All the curves correspond to the case x0 = 1 − e. Interesting feature of this solution is that the straight-line motion with constant acceleration is obtained after approximately three revolutions for the value e ≈ 4 · 10−6 when the motion, before imposing a constraint, satisfied this constraint with great accuracy. Consider the cause of this phenomenon. In the absence of active forces and constraints Gaussian principle leads to the motion with the zero acceleration W, what is in agreement with Newton’s

142

V. The Mixed Problem of Dynamics. New Class of Control Problems

Fig. V. 2

first law. Note that the equations of dynamics can be obtained from this principle. In the case when the active forces and constraints are lacking the usage of generalized Gaussian principle does not lead to the motion with the zero acceleration W but leads to that with the zero time derivative of order k of the vector W, where k is an order of principle. Hence for k = 1 the application of

Fig. V. 3

4. The motion of spacecraft with modulo constant acceleration

143

Fig. V. 4

generalized Gaussian principle in the absence of active forces and constraints leads to a uniformly accelerated straight-line motion. A satellite (which becomes a spacecraft) tends to such a "natural"in the frame of this principle motion even for e ≈ 4 · 10−6 . It is clear that the considered problem on the motion of satellite (spacecraft) with modulo constant acceleration can have a solution such that the satellite tends asymptotically to the straight-line motion with this constant acceleration. Thus, such a solution is a result of application of generalized Gaussian first-order principle to this problem. However it should be kept in mind that a generalized Gaussian principles, unlike the usual one, does not result from the equation of dynamics and therefore its application to another problems can lead to unpredictable results. Consider an example. Suppose, on the motion of mass point on plane there is imposed the ideal holonomic constraint: x2 + y 2 = l2 or in vector form r2 = l2 . In the absence of external forces the point moves in a circle with the constant velocity v0 . Differentiating the equation of constraints in time thrice, we obtain ˙ = −3 v · w . r·w The application of the generalized principle gives ˙ = Λr . mw Introducing the polar coordinates r and ϕ, we have d ∂T1 1 ∂T1 ˙ · eϕ = − = 0, mw dt ∂ ϕ¨ 2 ∂ ϕ˙ ˙2 mw ˙ 2] m[(¨ r − rϕ˙ 2 )2 + (rϕ¨ + 2r˙ ϕ) T1 = = . 2 2

144

V. The Mixed Problem of Dynamics. New Class of Control Problems

Hence

... ϕ = ϕ˙ 3 .

Assuming that for t = 0 ϕ = 0, ϕ˙ = ω0 = v0 /l, ϕ¨ = 0, we obtain 1  ω0 1  t= F arccos , √ , ω0 ω 2

F (θ, k) =

θ 0

dα  . 1 − k 2 sin2 α

This solution implies that the angular velocity ω = ϕ˙ becomes infinite in a time √ t∗ = F (π/2, 1/ 2)/ω0 = 1,854/ω0 . The example considered shows that a generalized Gaussian principle should be applied very carefully. However, as is shown in studying the motion of a spacecraft with modulo constant acceleration, exactly the generalized Gaussian principle permits for one of possible motions, which cannot be obtained by usual Gaussian principle, to be found. § 5. The satellite maneuver alternative to the Homann elliptic motion In the previous section we study the spacecraft motion with modulo constant acceleration. Now we consider a more complicated problem of the passage of a spacecraft from one elliptical orbit to another close to circular. This passage, as is known, can be realized for the Homann elliptic motion by means of an instantaneous application of impulses at the beginning and at the end of orbital passage [189]. Theory, given in § § 2 and 3 of this Chapter makes it possible to realize this orbital passage in the case of a soft application of control forces. For the solution of this problem we make use of the dimensionless variables and equations, introduced in the previous section. As is known, the radial component wr of the vector of acceleration ¨r has the form [189] wr = r¨ − rϕ˙ 2 . (5.1) At the initial point of elliptical orbit this component is equal to (−1/r02 ). For a smooth passage on a circular orbit of radius r1 it is necessary for the value wr , which continuously increases (or decreases, respectively) beginning from the value (−1/r02 ), to be tended asymptotically to the value (−1/r12 ). For the description of this passage we make use of the function, which occurs when solved the problem on a longitudinal collision of bars with circled ends. English scholar Sears showed that in this problem a smooth increasing of the dimensionless force of collusion Q(t¯) from zero to unity is described by the following equation [77] dQ = Q1/3 (1 − Q) . dt¯

5. The satellite maneuver alternative to the Homann elliptic motion

145

The Sears equation is integrated in closed form but the integration leads to a complicated dependence of t¯ on Q: √ 1 1 + Q1/3 + Q2/3 √ 2Q1/3 + 1 π 3 ¯ √ t = ln + . − 3 arctan 2 6 (1 − Q1/3 )2 3 Therefore it is convenient to regard at once the Sears function as a solution of the Sears differential equation. Taking into account the properties of the Sears function, we assume that the varying of the value wr of satellite for τ  0 is described by the generalized Sears equation Q˙ = pQq (1 − Q) . (5.2) Here Q = (wr + 1/r02 )/(1/r02 − 1/r12 ) .

(5.3)

The definition of the generalized Sears function Q(τ ) involves two parameter p and q. The parameter p controls a time, beginning from which Q  1 − ε. The second parameter q defines a behavior of the function Q(τ ) in the neighborhood of zero. In Fig. V. 5 are shown the graphs of the function Q(τ ), satisfying equation (5.2). To the value p = 2 corresponds the curve with long dashes, to p = 1 the solid curve, to p = 0.5 the curve with short dashes. For all these curves we assume that q = 1/3. By (5.1) and (5.3) equation (5.2) becomes ... f3 ≡ r − r˙ ϕ˙ 2 − 2rϕ˙ ϕ¨ + (1/r12 − 1/r02 )pQq (1 − Q) = 0 .

(5.4)

This equation is a linear third-order nonholonomic program constraint. We introduce, as in the previous section, the generalized control force Λ, which the vector R = Λer = Λr/r corresponds to. In this case the program constraint (5.4) control is ideal since the vector ∇′′′ f3 = er coincides with that, entering into the relation for the

Fig. V. 5

146

V. The Mixed Problem of Dynamics. New Class of Control Problems

control force R. Then the equation of motion over the coordinate r takes the form 1 r¨ − rϕ˙ 2 + 2 = Λ . (5.5) r From relations (5.4) and (5.5) it follows that the sought generalized radial control force Λ(τ ) must satisfy the equation 2r˙ Λ˙ + 3 = (1/r02 − 1/r12 )pQq (1 − Q) = 0 . r

(5.6)

In accordance with the technique of generating the differential equation with respect to the generalized control force Λ the value Q in equation (5.6) in accordane with (5.1), (5.3), (5.5) takes the form Q = (Λ − 1/r2 + 1/r02 )/(1/r02 − 1/r12 ) . We shall regard this relation as the change of the variables Λ to the new variables Q. Then equation (5.6) is reduced to equation (5.2) and equation (5.5) becomes r¨ − rϕ˙ 2 = −1/r02 + (1/r02 − 1/r12 )Q . (5.7) Consider now the equation with respect to the coordinate ϕ: rϕ¨ + 2r˙ ϕ˙ ≡

1 d 2 ˙ = P (τ ) . (r ϕ) r dτ

(5.8)

Here P (τ ) is a sought tangential control force. It is necessary to perform a passage of spacecraft on circular orbit of radius a1 = ar1 . Find the value r2 ϕ˙ on this orbit. For this purpose we introduce new dimensionless variables, assuming that r∗ =

ρ , a1

τ1 =

2πt . T1

Using formula (4.3), we have r∗2

dϕ = 1. dτ1

Returning in this relation to the variables r=

ρ , a

τ=

2πt , T

and taking into account that [189] T2 a3 = 2, 3 a1 T1 we obtain r2 ϕ˙ =

r1 = √ r1 .

a1 , a

5. The satellite maneuver alternative to the Homann elliptic motion

147

Fig. V. 6

The function wr (τ ) = r¨ − rϕ˙ 2 and the Sears function Q are related by (5.7). Suppose, for τ  0 the function c(τ ) = r2 ϕ˙ is varied similarly, i. e. we put   √ c(τ ) = 1 − e2 + ( r1 − 1 − e2 )Q(τ ) . Then equation (5.8) can be represented as  √ ˙ . rϕ¨ + 2r˙ ϕ˙ = ( r1 − 1 − e2 ) · Q/r

(5.9)

The problem is reduced, thus, to the solution of equations (5.2), (5.7), and (5.9). Equation (5.2) has as the zero solution Q(τ ) ≡ 0, as a nonzero solution. When numerical computing we assumed that Q(0) = 0.0001 . For simplicity, the initial data for the variables r and ϕ were given as  r(0) = r0 = 1 − e , ϕ(0) = 0 , r(0) ˙ = 0 , ϕ(0) ˙ = 1 − e2 /r02 .

Fig. V. 7

148

V. The Mixed Problem of Dynamics. New Class of Control Problems

Fig. V. 8

The results of computation for e = 0.01 , r0 = 0.99 , r1 = 3 , p = 0.25 , q = 1/3 in the time interval 0  t  5T (0  τ  10π) are shown in Figs. V. 6, V. 7, V. 8. We remark that the control forces Λ and P obtained are measured in terms of the fractions of the gravitational force F = µm/a2 , where m is a satellite mass.

C h a p t e r VI APPLICATION OF THE LAGRANGE MULTIPLIERS TO THE CONSTRUCTION OF THREE NEW METHODS FOR THE STUDY OF MECHANICAL SYSTEMS

The Lagrange multipliers are used to construct three new methods for the study of mechanical systems. The first of them corresponds to the problem of determining the normal frequencies and normal forms of oscillations of elastic system, consisting of the elements with the known normal frequencies and normal forms. In this method the conditions of connection of elastic bodies to one another are regarded as holonomic constraints. Their reactions equal to the Lagrange multipliers are the forces of interaction between the bodies of system. Using the equations of constraints, the system of linear uniform equations with respect to the amplitudes of the Lagrange multipliers for normal oscillations is obtained. By the solution of this system the normal frequencies and normal forms of complete system are expressed in terms of the normal frequencies and normal forms of its elements. An approximate algorithm for determining the normal frequencies and normal forms, based on a quasistatic account of higher forms of its elements, is proposed. The second method suggested is connected with the study of the dynamics of system of rigid bodies. In this case the Lagrange multipliers are introduced for the abstract constraints taking into account that the number of introduced coordinates, by which the kinetic energy of rigid body has a simple form, is excessive. In this case the elimination of the Lagrange multipliers leads to a new special form of equations of motion of rigid body. This form is utilized to describe a motion of a dynamic stand, which lets us to imitate the state of a pilot in the cabin in extremal situations. The third method is used in the problem of vibration suppression (damping) of mechanical systems. It is shown that formulation of such problems is equivalent to imposing a high-order constraint on the motion of system. This makes it necessary to solve a mixed problem of dynamics. It turns out, that the Pontryagin maximum principle chooses from the possible class of mixed problems that one in which a control force is given by a series in natural frequencies of system. In the suggested method of vibration suppression (damping) the generalized Gauss principle is used, which makes it possible to find the control force as a polynomial in time. The computational results obtained by the Pontryagin maximum principle and the generalized Gauss principle are compared.

149

150

VI. Application Lagrange Multipliers § 1. Some remarks on the Lagrange multipliers

The quantities Λκ , entering into equations (2.22) of Chapter I, are usually called the Lagrange multipliers or generalized reactions. Note that it is rational to call them generalized reactions and to compute them, for necessity, only if equations (2.17) of Chapter I describe the constraints between the elements of system or the constraints between the elements of the system and the bodies, no entering into the system. In the first case a distinctive example is a mechanical system consisting of two mass points, connected by a weightless nonstretchable bar, and in the second case that is a one mass point on the given ideally ground surface. In the first case the generalized reaction is a stretching (pressing) force of bar and in the second one a force, holding the point on a surface and directed at right angles to it. It is also possible more general and abstract situation when equations (2.17) of Chapter I do not describe the imposing of materially realizable constraints on the system but express the fact that equations (2.17) relate the generalized coordinates, which are convenient to compute the kinetic energy, and a possible elementary work. The reasonings, given below, are used in § 7 of this Chapter. Suppose, for example, the mechanical system consists of a one element, which is a free rigid body. We shall show that its kinetic energy can be represented as Ix ˙2 Iy ˙ 2 Iz ˙ 2 M ρ˙ 2 i + j + k . + (1.1) T = 2 2 2 2 Here M is a mass of body, ρ is a radius-vector of its center of mass, i, j, k are the unit vectors of the central axes of inertia of the bodies x, y, z, respectively, and

2 2 Iy = y dm , Iz = z 2 dm . Ix = x dm , m

m

m

By definition of kinetic energy we have

2 1  1 T = ρ˙ + xi˙ + y j˙ + z k˙ dm . v 2 dm = 2 2 m

m

Since the axis x, y, z are the principal axes of inertia of body, we obtain



x dm = y dm = z dm = xy dm = yz dm = zx dm = 0 m

m

m

m

m

m

and therefore the quantity T can really be represented in the form (1.1). The unit vectors i, j, k, as is known, satisfy the relations f 1 ≡ i2 − 1 = 0 , f4 ≡ i · j = 0 ,

f 2 ≡ j2 − 1 = 0 , f5 ≡ j · k = 0 ,

f 3 ≡ k2 − 1 = 0 , f6 ≡ k · i = 0 .

(1.2)

1. Some remarks on the Lagrange multipliers

151

It is clear that these equations are of purely mathematical nature and the fact of their existence do not mean that on an rigid body some materially realizable constraints are imposed. In this case the quantities Λκ , entering into equations (2.22) of Chapter I, are auxiliary and their computation does not make a sense. Therefore it is better to call them the Lagrange multipliers but not generalized reactions. In the case of such abstract constraints the application of equations (2.22) of Chapter I is not obvious. This was shown above in a purely mathematical way. Now we shall consider this case on the example of one mass point. Represent the radius-vector r of this point as r = rn ,

r = |r| .

In this case the position of point is given by four parameters: the quantity r and three components of the unit vector n. The equation of constraint, the kinetic energy, and the virtual elementary work are the following f ≡ n2 − 1 = 0 , m T = (r˙ 2 + r2 n˙ 2 ) , 2 δA = F · δr = Qr δr + Qn · δn , where Qr = F · n ,

Qn = r F .

This implies that it is useful to represent Lagrange’s equations of the first kind (2.22) of Chapter I, corresponding to three components of the vector n, in vector form d ∂T ∂T ∂f − = Qn + Λ . (1.3) dt ∂ n˙ ∂n ∂n Here we make use of the following notations ∂ ∂ ∂ ∂ = i+ j+ k. ∂n ∂nx ∂ny ∂nz The vector form of (1.3) is very convenient since the vector n, entering into the equation of constraint, the kinetic energy, and the virtual elementary work, can be regarded formally as a usual variable. In accordance with this simple rule, the system of equations (2.22) of Chapter I is as follows m¨ r = mrn˙ 2 + F · n , ¨ + 2rr˙ n) ˙ = r F + 2Λn . m(r2 n Since by the equation of constraint we have n · n˙ = 0 ,

¨ = −n˙ 2 , n·n

(1.4)

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VI. Application Lagrange Multipliers

system of equations (1.4) gives 2Λ = −rF · n − mr2 n˙ 2 = −mr¨ r. Substituting this relation into the second equation of system (1.4), we obtain mr(¨ rn + 2r˙ n˙ + r¨ n) = mr¨r = rF , what is equivalent to Newton’s second law, represented in usual form. Thus, for one mass point the second Newton’s law in the Cartesian coordinates can be written in usual simple form, in curvilinear coordinates (cylindrical, spherical, and so on) in the form of Lagrange’s equations of the second kind, and in the excessive coordinates it can be represented in the form of Lagrange’s equations of the first kind. The considered example of applying the excessive coordinates to the description of motion of one free mass point demonstrates that for abstract constraints the reactions as the real forces are lacking. Then we cannot say that the sum of their works on virtual displacements of system is equal to zero and therefore on the base of this definition of the ideality of holonomic constraints (2.17) of Chapter I, we cannot generate equations (2.22) of this Chapter for abstract constraints. Note that the fact that equations (2.22) of Chapter I can be used for such abstract constraints is a bright demonstration of the perfect mathematical apparatus introduced in the mechanics by Lagrange. § 2. Generalized Lagrangian coordinates of elastic body In the next sections Lagrange’s equations with multipliers are applied to mechanical system consisting of elastic bodies. In this case for each elastic body a certain system of generalized Lagrangian coordinates is introduced. A rational method for their introduction is based on the notion of normal forms of oscillations of elastic body. Consider this approach, assuming, for the sake of generality, that the body can move freely. We introduce the Cartesian system of coordinates Cxyz, connected rigidly with a body before its deformation. Suppose, the axes of this system are the principal central axes of inertia of this body. Before deformation, an arbitrary point of body with the coordinates x, y, z can be displaced, firstly, by virtue of the motion of this body as the rigid one and, secondly, by the deformation of body. Both displacements are measured from the position of system Cxyz at time t = 0. We assume that these displacements are so small that the vector of displacement u(x, y, z, t) of the point with the coordinates x, y, z at time t can be represented as u(x, y, z, t) = (ξ(t) + ϕy (t)z − ϕz (t)y)i + (η(t) + ϕz (t)x − ϕx (t)z)j+ ∞  (2.1) qσ (t)uσ (x, y, z) . +(ζ(t) + ϕx (t)y − ϕy (t)x)k + σ=1

2. Generalized Lagrangian coordinates of elastic body

153

Here i, j, k are the unit vectors of the axes x, y, z, respectively, the quantities ξ, η, ζ give the displacement of the center of mass of body, and the quantities ϕx , ϕy , ϕz are the angles of rotation of body about the axes x, y, z, respectively. By assumption, these angles are so much small that the difference between the projections of the vector u on the original and rotated axes can be neglected. The functions uσ (x, y, z), entering into relation (2.1) are normal forms of oscillations. This means that if at time t = 0 all the points of body have a zero velocity and the vector of displacement is equal to Cσ uσ (x, y, z), then for t = 0 this vector takes the form u(x, y, z, t) = Cσ cosωσ t uσ (x, y, z) .

(2.2)

Here ωσ is a normal frequency, corresponding to the normal form uσ (x, y, z), and Cσ is a small constant. The system of functions uσ is complete and therefore any displacement of a point of body by virtue of its deformation can be expanded in these functions. This implies that the quantities ξ, η, ζ, ϕx , ϕy , ϕz , q1 , q2 , . . . , given at time t, define uniquely the position of all points of body at this instant of time. Hence these quantities are generalized Lagrangian coordinates. Substituting the vector of displacement given into the form (2.1) in relation for kinetic energy of elastic body, we obtain  2

∂u 1 T = ρ(x, y, z) dxdydz = 2 ∂t V

∞ Aϕ˙ 2x + B ϕ˙ 2y + C ϕ˙ 2z  Mσ q˙σ2 M ˙2 (ξ + η˙ 2 + ζ˙ 2 ) + + , = 2 2 2 σ=1

Mσ = ρ(x, y, z)u2σ (x, y, z)dxdydz .

(2.3)

V

Here ρ(x, y, z) is a density of body material, M is a mass of body, A, B, C are the principal central moments of inertia of body. The kinetic energy obtained depends on the squared generalized velocities only. Such a simple form results from that the vector of displacement u(x, y, z, t) is represented as a series in normal forms of oscillations. Find now a potential energy of deformation of elastic body. Using relation (2.1), we obtain a deformation tensor and, assuming that generalized Hooke’s law is satisfied, we compute then a stress tensor. Substituting these tensors into the relation for potential energy of deformation of elastic body, we find ∞  Mσ ωσ2 2 qσ . (2.4) Π= 2 σ=1

Such form of the potential energy of deformation follows from that the introduced generalized Lagrangian coordinates qσ are the principal coordinates. By formulas (2.1) and (2.2) these coordinates must satisfy the equations

154

VI. Application Lagrange Multipliers q¨σ + ωσ2 qσ = 0 ,

σ = 1, 2, ... .

We can obtain them from Lagrange’s equations of the second kind d ∂T ∂T ∂Π − =− , dt ∂ q˙σ ∂qσ ∂qσ

σ = 1, 2, ... ,

if the kinetic and potential energies of elastic body are represented in the form (2.3) and (2.4), respectively.

§ 3. The application of Lagrange’s equations of the first kind to the study of normal oscillations of mechanical systems with distributed parameters Developing the ideas of the works of S. A. Gershgorin and P. F. Papkovich (see, for example, the book: Collection of works on a ship vibration, L.: Sudpromgiz. 1960), we apply Lagrange’s equations of the first kind to the analysis of normal oscillations of elastic system. Let its elements be bars, rings, plates, shells, and rigid bodies, connected to each other. For any of these elements, distinguished mentally from such elastic system, their normal frequencies and normal forms of oscillations are known. In other words, for any of elements it is known the principal or normal coordinates in which its kinetic and potential energies have the simplest form. These elements make up united system since their coordinates are related by expression (2.17) of Chapter I. Thus, using these relations we can construct in analytic form the considered elastic system by its elements. Therefore they can also be used to obtain the relation between the normal frequencies and forms of this elastic system and the normal frequencies and forms of its elements. This is a base of the method to be suggested for determining the normal frequencies and normal forms of oscillations of elastic systems. For its realization it is used the approximate approach, based on the dynamic account of the first forms of oscillations of a system of elements and on a quasistatic account of the rest of them. The efficiency of a quasistatic account of higher forms of normal oscillations in the dynamical problems of the elasticity theory is shown in the works [36, 75, 77]. The method considered can be applied to any system of the connected to each other elastic bodies with distributed parameters. The essence of this method is demonstrated on the example of lateral oscillations of the bar, of the length l, possessing the weight, and such that in the sections xk , k = 1, n, the disks with the mass mk and the moments of inertia Jk are fitted on it. For the solution of this problem some methods were developed [12, 71, 107]. In particular, for determining the critical velocities of rotating loaded shafts, possessing the weight, the method of equivalent disk [258] is proposed. However all these approaches do not use the fact that for the bar without disks, its

3. The application of Lagrange’s equations of mechanical systems

155

normal forms of the oscillations Xσ (x), σ = 1, 2, . . . , which represent a complete system of functions, as a rule, are known. In this case it is useful recall that the lateral oscillations of the bar with disks can also be represented as y(x, t) =

∞ 

qσ (t)Xσ (x) .

σ=1

Suppose, uk is a displacement of the center of mass of the k-th disk in the axis y and ϕk is its angle of rotation. The quantities qσ (σ = 1, 2, . . . ), uk , ϕk (k = 1, n) we regard as generalized Lagrangian coordinates under the holonomic constraints fk ≡ fn+k ≡

∞ 

σ=1 ∞ 

qσ Xσ (xk ) − uk = 0 , (3.1) qσ Xσ′ (xk ) − ϕk = 0 ,

k = 1, n .

σ=1

Formulas (2.3) and (2.4) yield that the kinetic and potential energies of system take the form  ∞ n   Mσ q˙σ2  mk u˙ 2k Jk ϕ˙ 2k T = + + , 2 2 2 σ=1 k=1

∞  Mσ ωσ2 qσ2 , Π= 2 σ=1

Mσ =

l

ρSXσ2 dx .

0

Here ωσ are the normal frequencies of a shaft without disks, ρ is a density, and S is a cross-section area of shaft. Using Lagrange’s equations of the first kind (2.22) of Chapter I we obtain qσ + Mσ (¨

ωσ2 qσ )

n    = Λk Xσ (xk ) + Λn+k Xσ′ (xk ) , k=1

mk u ¨k = −Λk ,

Jk ϕ¨k = −Λn+k ,

σ = 1, 2, ... ,

(3.2)

k = 1, n .

From these equations it follows that the quantity Λk , k = 1, n, is equal to the force, acting on the bar due to the disk k and the quantity Λn+k , k = 1, n, is equal to the moment, applied to the bar by virtue of the k-th disk. Note that if in the equation of constraint the quantity f (t, q) is equal to displacement −−→ of the point A relative to the point B along the line AB, then the Lagrange multiplier Λ, corresponding to this constrain, is equal to the force, which is −−→ applied to the point A and acts along the line AB. A similar rule is valid for the moment of the force of reaction if the equation of constraint shows that the angles of rotation are equal.

156

VI. Application Lagrange Multipliers

In the case of the oscillations of a system with the sought normal frequency p, the quantities qσ , uk , ϕk , Λk , Λn+k are varied by the harmonic laws qσ = qσ eipt , uk = u k eipt , ϕk = ϕ k eipt ,  k eipt , Λn+k = Λ  n+k eipt , Λk = Λ σ = 1, 2, ... ,

(3.3)

k = 1, n ,

where i is the imaginary unity. From equations (3.2) we have qσ =

k Λ u k = , mk p2

n    n+k X ′ (xk ) Λk Xσ (xk ) + Λ σ

Mσ (ωσ2 − p2 )

k=1

 n+k Λ ϕ k = , Jk p2

σ = 1, 2, ... ,

, (3.4) k = 1, n .

Substituting relations (3.3) and formulas (3.4) into the equations of constraints (3.1), we obtain 2n  j=1

j = 0 , αij (p2 )Λ

i = 1, 2n ,

αij = αji .

(3.5)

Here the index i corresponds to the number of constraints and αij = βij + γij , βkl (p2 ) =

γkl (p2 ) =

 0 ,

k = l ,

1 − , k = l, m k p2

∞  Xσ (xk )Xσ (xl ) , Mσ (ωσ2 − p2 ) σ=1

γk,n+l (p2 ) =

βn+k,n+l (p2 ) =

γn+k,n+l (p2 ) =

∞  Xσ (xk )Xσ′ (xl ) , Mσ (ωσ2 − p2 ) σ=1

 0 ,

k=  l, 1 − , k = l, Jk p2

∞  Xσ′ (xk )Xσ′ (xl ) , Mσ (ωσ2 − p2 ) σ=1

k, l = 1, n .

The quantities γij (p2 ) are called dynamic compliances [14, 36]. Note that the coefficients γkl (p2 ), k, l = 1, n, are introduced first by I. M. Babakov and are called the harmonic coefficients of influence of the frequency p [12]. For p = 0 they become usual coefficients of influence.  j = 0, j = If the determinant of system (3.5) is not equal to zero, then Λ  1, 2n. However this is impossible. The quantity Λk for certain k  n is equal to zero if the point xk is a node of the sought normal form of a shaft with disks. The angle of inclination of tangent line at the node is not equal to  n+k = 0. On the contrary, if at the point xk the angle of zero and therefore Λ  k = 0. Thus, the shaft with  n+k = 0, and Λ inclination is equal to zero, then Λ

3. The application of Lagrange’s equations of mechanical systems

157

 j = 0, j = 1, 2n. Then the equation of disks has no normal forms such that Λ frequencies is as follows det[αij (p2 )] = 0 . (3.6) For the shaft, having at the points xk the lumped masses mk in place of  k , k = 1, n, can be vanished. This occurs in the case disks, all quantities Λ when the points xk , k = 1, n, are the nodes of the certain original forms Xσ (x) of a shaft without masses. For example, for the one lumped mass m1 and σπx l Xσ = sin , σ = 1, 2, ... , x1 = , l 2 the normal forms X2ν and the normal frequencies ω2ν are saved. In this case from the equation α11 (p2 ) = 0 we can find the frequencies, for which the normal forms are symmetric about the midpoint of shaft. This example demonstrates that the method suggested does not allow us to find the normal forms, of oscillations of elastic system, such that the constraints between the elements of system are satisfied if the reactions of constraint are lacking. However these forms have the frequencies from the spectrum of frequencies of its elements and, as a rule, these forms can easily be found. We now return to equations (3.5) and (3.6).  ρj , j = 1, 2n, Suppose, pρ , ρ = 1, 2, ... , are roots of equation (3.6) and Λ are the corresponding to them solutions of system (3.5). Given in formulas j = Λ  ρj , we obtain the eigenfunctions of this problem (3.4) p = pρ and Λ X∗ρ (x) =

∞  n    ρ,n+k Xσ′ (xk ) Λρk Xσ (xk ) + Λ

σ=1 k=1

Mσ (ωσ2 − p2ρ )

Xσ (x) ,

(3.7)

ρ = 1, 2, ... .

Thus, we obtain the normal forms of oscillations of a shaft with disks as series in the normal forms of oscillations of a shaft without disks. In the works [36, 75, 77] it is shown the efficiency of the approximate approach consisting in that in the relation for dynamic compliance the first N forms of normal oscillations are accounted dynamically and the rest of them quasistatically. Using this approach, we shall say that the problem considered is solved in the N -th approximation if, by assumption, the coefficients γij (p2 ) in system (3.5) and in equation (3.6) have the form N 2 γkl (p ) =

N N   Xσ (xk )Xσ (xl ) Xσ (xk )Xσ (xl ) + γ (0) − , kl 2 2 Mσ (ωσ − p ) Mσ ωσ2 σ=1 σ=1

158

VI. Application Lagrange Multipliers

N (p2 ) γn+k,n+l

N N   Xσ′ (xk )Xσ′ (xl ) Xσ′ (xk )Xσ′ (xl ) = (0)− , + γ n+k,n+l Mσ (ωσ2 − p2 ) Mσ ωσ2 σ=1 σ=1

N (p2 ) = γk,n+l

N N   Xσ (xk )Xσ′ (xl ) Xσ (xk )Xσ′ (xl ) (0) − . + γ k,n+l 2 2 Mσ (ωσ − p ) Mσ ωσ2 σ=1 σ=1

In what follows all the quantities of the N -th approximation are marked by the upper index N . Recall that the new addends γij (0) in these formulas are usual coefficients of influence and can be found in closed form. For N = 0 equation (3.6) reduced to the equation of frequency for a system with n disks on a weightless shaft [12]. It is useful to consider the curves of static bend Xρst (x) of shaft under the  ρj . They can be found in closed form by the methods generalized reactions Λ of the resistance of material and be represented in the form of infinite series: ∞  n    ρ,n+k Xσ′ (xk ) Λρk Xσ (xk ) + Λ Xρst (x) = Xσ (x) . Mσ ωσ2 σ=1 k=1

Then relations (3.7) imply that the normal forms of oscillations of a shaft with disks can be rewritten as ∞  n    ρ,n+k Xσ′ (xk ) Λρk Xσ (xk ) + Λ X∗ρ (x) = p2ρ Xσ (x) + Xρst (x) , 2 (ω 2 − p2 ) M ω σ (3.8) σ σ ρ σ=1 k=1 ρ = 1, 2, ... .

We remark that infinite series, entering into this relation, converges much faster than series (3.7). In fact, the frequencies ωσ2 grow as σ 4 and the quantities Xσ′ (xk ) as σ. Hence the series, entering into representation (3.8), converges as 1/σ 7 . Such a rapid convergence is provided by discrimination of a curve of static bend. By formulas (3.8) the normal forms in the N -th approximation are the following N (x) = X∗ρ

n N N  ′ N  Λρk Xσ (xk ) + Λ ρ,n+k Xσ (xk )  N 2   pρ Xσ (x) + Xρst,N (x) . 2 ω 2 − (pN )2 M ω σ σ σ ρ σ=1 k=1

N . Here Xρst,N (x) is a curve of static bend under the generalized reactions Λ ρj The zero approximation corresponds to a weightless shaft. Already the first approximation permits us to find with high accuracy the first normal frequency and form. E x a m p l e VI .1 . In the case of one lumped mass m1 , fixed at the midpoint of the pin-ended beam of the mass M and the length l, the equation of frequencies, as is remarked above, takes the form ∞  2 1 + = 0, α11 (p2 ) ≡ − 2 2 m1 p M (ω2ν−1 − p2 ) ν=1 (3.9) EJ 4 2 4 ω2ν−1 = π (2ν − 1) , ρSl4

3. The application of Lagrange’s equations of mechanical systems

159

where EJ is a flexural rigidity of shaft. Note that, using the representations of trigonometric and hyperbolic tangents in the form of infinite sums of simple fractions, we can show that equation (3.9) is equivalent to equation [51]  l2 p ρS 2M 2 . (3.10) = ξ(tg ξ − th ξ) , ξ = m1 4 EJ For the dynamic account of the first form and the quasistatic account of the rest of forms, equations (3.9) take the form −

1 2 l3 2 + = 0. + − 2 m1 p2 M (ω1 − p2 ) 48 EJ M ω12

(3.11)

In the whole size of changing the quotient m1 /M the frequency p1 , obtained from this square equation, differs from the first frequency, determined from the exact equation (3.10), no more than 0.1 %. Neglecting the last two addends in relation (3.11), we arrive at the following simple approximate formula for the first normal frequencies: p21 =

M ω12 . 2m1 + M

(3.12)

The error of this formula grows with the quotient m1 /M but does not attain 1 %. In this case from relations (3.8) and (3.9) it follows that the first form of oscillations has the form   11 p21 l3  πx 3πx 1 − p21 2Λ sin sin + 4 4 + ... + X∗1 (x) = 4 l l π EJ(1 − p21 ) 3 (3 − p21 )  11 x(3l2 − 4x2 ) Λ + , 48 EJ p2 l p21 = 12 , 0  x  . ω1 2 Being restricted to the first term of this rapidly convergent series and taking into account that the frequency p1 can be represented most exactly in the form (3.12), we obtain for the first eigenfunction the following approximate relation   πx π 4 m1 3x 4x3 l + − 3 , X∗1 (x) = sin 0  x  . l 48 M l l 2 Since the quantity π 4 /96 slightly differs from unity, we can assume approximately that   πx 2 m1 3x 4x3 l 0  x  . + − 3 , X∗1 (x) = sin l M l l 2

160

VI. Application Lagrange Multipliers

This implies that the form, corresponding to a shaft without the mass m1 , is taken with the weight Mequiv = M/2 equal to the equivalent mass of shaft, which is computed at the point of holding the weight, and the form for a weightless shaft is taken with the weight m1 . This approximate method to construct the first form can also be applied to another similar problems, for example, to the problem on the lateral oscillations of beam with the weight at the end. E x a m p l e VI .2 . In the case of the disk, holden at the distance of x1 from the left support of pin-ended beam, the exact equation of frequencies, which is in terms of the Krylov functions, is rather lengthy [14]. The computation demonstrates that the first normal frequency, which is found from the cubic equation obtained from determinant (3.6) for N = 1, differs from the exact value no more than 0.1 % if the parameters of system are varied within limits 0.25 M  m1  2M ,

0.05 M l2  J1  0.5 M l2 ,

0 < x1  0.5 l ,

where J1 is an equatorial moment of inertia of disk.

§ 4. Lateral vibration of a beam with immovable supports At the beginning of § 3 we remark that the method suggested can be used to consider the oscillations of different elastic systems. As is known, in studying the oscillations of bars and plates with fixed supports, it is necessary to consider the nonlinear equations [1, 250]. We shall show, for example, how the similar equation can be obtained for free lateral oscillations of pinended beam of length l and rigidity EJ in the case whith no displacement of supports. The lateral and longitudinal oscillations of bar are given by the functions y(x, t) and u(x, t), respectively. The lateral oscillations of the pin-ended beam and the longitudinal oscillations of bar with the fixed left end point and the free right end point can be represented as [227] y(x, t) = u(x, t) =

∞ 

k=1 ∞  j=1

qk sin

kπx , l

u2j−1 sin

ωk2 =

k 4 π 4 EJ , ρSl4

(2j − 1)πx (2j − 1)2 π 2 E , p22j−1 = . 2l (2l)2 ρ

Note that here the longitudinal oscillations are considered for the bar, which is assumed to be fixed at the left end point only and at the right end point the bar is assumed first to be free though at the both end points there is no displacement. Further we introduce a holonomic constraint, by which the displacement at the right end point is absent.

161

4. Lateral vibration of a beam with immovable supports

Now we write the relations for the kinetic and potential energies of bar in the case of its lateral oscillations:

l  2 ∞ ∂y M ρS 1 ρSl T1 = = , dx = mk q˙k2 , mk = 2 ∂t 2 2 2 k=1 0 (4.1)

l  2 2 ∞  ∂ y EJ 1 Π1 = dx = ωk2 mk qk2 . 2 ∂x2 2 k=1

0

In the case of longitudinal oscillations we have

l  2 ∞ ∂u ρS 1 T2 = dx = m2j−1 u˙ 22j−1 , 2 ∂t 2 j=1

m2j−1 =

M , 2

0

ES Π2 = 2

l  0

∂u ∂x

2

(4.2)

1 2 dx = p m2j−1 u22j−1 . 2 j=1 2j−1 ∞

Compute the stretch of bar in the case of lateral oscillations:

l 

l ∞ 1 π2  2 2 ′ 2 1 + (yx ) dx − l = (yx′ )2 dx = k qk , ∆= 2 4l 0

(4.3)

k=1

0

and in the case of longitudinal oscillations: ∞  u|x=l = (−1)j+1 u2j−1 .

(4.4)

j=1

In the case of the lateral oscillations of bar when the displacement of supports is absent relations (4.3) and (4.4) must coincide, what can be regarded as a nonlinear holonomic constraint ∞ ∞  π2  2 2 j+1 f≡ (−1) u2j−1 − k qk = 0 , (4.5) 4l j=1 k=1

imposed on the generalized coordinates u1 , u3 , ... and q1 , q2 , ... . Using Lagrange’s equations of the first kind (2.22) of Chapter I and also relations (4.1), (4.2), and (4.5), we obtain q¨k + ωk2 qk = −Λ

π2 k2 qk , Ml

k = 1, 2, ... ,

(4.6)

2Λ , j = 1, 2, ... . (4.7) M For the approximate solution of the problem, we account equations (4.7) quasistatically, assuming that u ¨2j−1 = 0, j = 1, 2, . . . . Then we have u ¨2j−1 + p22j−1 u2j−1 = (−1)j+1

u2j−1 = (−1)j+1

2Λ , M p22j−1

j = 1, 2, ... ,

162

VI. Application Lagrange Multipliers

and therefore u|x=l =

∞  j=1

2Λ  1 Λ = , M j=1 p22j−1 c ∞

(−1)j+1 u2j−1 =

c=

ES . l

It follows that equation of constraints (4.5) can be rewritten as ∞ Λ π2  2 2 − k qk = 0 . c 4l k=1

Saved in the infinite sum one addend only, in first-order approximation we have cπ 2  (1) 2 Λ(1) = q . 4l 1 Substituting this value of generalized reactions into the first equation of sys(1) tem (4.6), we obtain for determining the function q1 the Duffing equation  (1) 3 (1) (1) = 0, q¨1 + ω12 q1 + µ q1

µ=

Eπ 4 . 4ρl4

(4.8)

In the work [250] the same equation was generated by using another method. We obtain a second-order approximation, taking into account in system (4.7) the first equation dynamically and the rest of them quasistatically. Then (2) (2) for determining the functions q1 and u1 we have the following system of two equations π 2 (2) (2) (2) (2) q Λ , q¨1 + ω12 q1 = − Ml 1 2 (2) (2) (2) Λ , u ¨1 + p21 u1 = M where  2  π  (2) 2 ES cc1 π 2 ES (2) (2) , c1 = m1 p21 = q1 , c= . Λ = − u1 c1 − c 4l 8l l

The construction of approximate solutions of equations of nonlinear oscillations, using Gaussian principle, is considered in Appendix C. Recall that the explanation of the Rietz and Bubnov–Galerkin methods, using the integral variational principles, can be found in the work of G. Yu. Dzhanelidze and A. I. Lur’e [56].

E x a m p l e VI .3 . We study in more detail Duffing’s equation (4.8), describing the lateral oscillations of bar in the case when the longitudinal (1) displacements of supports are lacking. Here q1 is a first-order approximation of Lagrangian coordinate q1 . In studying the forced oscillations under the perturbation force P sin νt, where the constant amplitude P is taken per unit

163

4. Lateral vibration of a beam with immovable supports

mass, uniform Duffing’s equation (4.8) is replaced by the following nonuniform equation (the indices of Lagrangian coordinates and ω are omitted): q¨ + ω 2 q + µq 3 = P sin νt .

(4.9)

In addition, we consider an inelastic resistance for lateral oscillations of bar. For the energy dissipation in material in the case of oscillations to be accounted there are many propostions (see, for example, the book: Panovko Ya. G. Internal friction in oscillating elastic systems. M. 1960; Pisarenko G. S. The energy dissipation for mechanical oscillations. M. 1960; Skudrzyk E. Simple and complex vibratory systems. The Pennsylvania State University Press University Park and London. 1968; Sorokin E. S. Inelastic resistance of materials of construction in oscillating. M. 1954). Now we apply one of them. The resistance forces are usually displaced in phase by the value π/2 per unit elastic forces. The existence of elastic forces does not violate the law of sinusoidal vibrations. Therefore by the form of elastic forces we can construct the forces of inelastic resistance, replacing q(t) by the quantity q(t), ˙ what means that a phase is shifted by π/2. In addition, we multiply then the obtained relation by the coefficient ϕ = η/ν, where η is a loss coefficient. So, we assume that to the elastic force ω 2 q + µq 3 = ω 2 q(1 + µq 2 /ω 2 ) corresponds the resistance force equal to ˙ + µq 2 /ω 2 ) . ϕω 2 q(1 Since the oscillations are sinusoidal, we can replace in the latter relation the quantity q 2 by q˙2 /ν 2 . Thus, to determine the function q(t), in place of equation (4.9) we have q¨ + ω 2 q + µq 3 + ϕω 2 q˙ + ϕµq˙3 /ν 2 = P sin νt .

(4.10)

We shall seek the steady-state vibrations in the system by the Bubnov– Galerkin method, assuming (see Appendix C) that q(t) = a1 cos νt + a2 sin νt .

(4.11)

In accordance with the Bubnov–Galerkin method, the virtual work of the forces of elasticity, resistance, inertia and of the perturbing force for the virtual displacement of system δq = δa1 cos νt + δa2 sin νt in a period 2π/ν of forced oscillations must be equal to zero, i. e. by (4.10) we have δa1

2π/ν

(¨ q + ω 2 q + µq 3 + ϕω 2 q˙ + ϕµq˙3 /ν 2 − P sin νt) cos νt dt+

0

+δa2

2π/ν

(¨ q + ω 2 q + µq 3 + ϕω 2 q˙ + ϕµq˙3 /ν 2 − P sin νt) sin νt dt = 0 .

0

(4.12)

164

VI. Application Lagrange Multipliers

The variations δa1 and δa2 are independent and therefore the coefficients of them must vanish. Substituting into formula (4.12) the law of motion (4.11) and integrating, we obtain the following nonlinear system of algebraic equations in unknowns a1 and a2 : 3 3 3 3 a1 (ω 2 −ν 2 )+ µa31 + µa1 a22 +ϕω 2 νa2 + ϕµνa21 a2 + ϕµνa32 = 0 , 4 4 4 4 3 3 3 3 3 2 2 2 2 2 a2 (ω −ν ) + µa2 + µa2 a1 −ϕω νa1 − ϕµνa2 a1 − ϕµνa31 = P . 4 4 4 4 Hence

9 2 3 µ (1 + ϕ2 ν 2 ) a6 + µ(ω 2 − ν 2 + ϕ2 ν 2 ω 2 ) a4 + 16 2 +[(ω 2 − ν 2 )2 + ϕ2 ω 4 ν 2 ] a2 = P 2 ,

(4.13)

 where a = a21 + a22 . The amplitude-frequency characteristics of system, given by equation (4.13), are computed numerically and are shown in Fig. VI. 1 by solid lines. We observe that in this case the form of amplitude-frequency characteristic is very sensitive to the value of η. The computation are performed for ω 2 = 0.7172 · 106 s−2 , µ = 0.1414 · 106 cm−2 ·s−2 , P = 0.083 ν 2 cm·s−2 . This corresponds to the steel bar with built-in ends of length 78 cm, thickness 0.42 cm, width 10 cm, initial axial stress 0.2760·103 N·cm−2 . The amplitude of oscillations of supports is assumed to be equal to 0.05 cm.

Fig. VI. 1

5. Normal frequencies and oscillation modes of system of bars

165

The curves constructed demonstrate that, in fact, the form of amplitudefrequency characteristic responds highly actively to a small change of a loss coefficient, in which case the amplitude break occurs for small values of η only. In this case for η = 0.07 the amplitude break yet occurs but for η = 0.1 it is already absent. A cubic term in the resistance force has a great influence on amplitudefrequency characteristics. Neglecting this term, in place of equation (4.13) we obtain 9 2 6 3 µ a + µ(ω 2 − ν 2 ) a4 + [(ω 2 − ν 2 )2 + ϕ2 ω 4 ν 2 ] a2 = P 2 . 16 2

(4.14)

The amplitude-frequency characteristics, computed by formula (4.14) for the same values of ω 2 , µ, P, are shown in Fig. VI. 1 by dotted lines. Note that the amplitude-frequency characteristics for the former values of η have more sharp breaks and jumps of amplitudes, in which case the distance between the points of break and jump is larger than that for the curves, constructed by means of equation (4.13). Thus, the Duffing equation with linear resistance describes the solution of the formulated problem in the first approximation. When studying a similar equation numerically, P. E. Tovstik and T. M. Tovstik [425] have found, depending on the excitation level, possible arising of strange attractors and periodical solutions with a period multiple to the period of excitation. On these interesting peculiarities of the Duffing equation see in detail Appendix G. The theory of strange attractors is presented, for example, in the treatise by G. A. Leonov [426].

§ 5. The application of Lagrange’s equations of the first kind to the determination of normal frequencies and oscillation modes of system of bars In § 3 of this Chapter a new method for the study of normal oscillations of mechanical systems with distributed parameters, which is based on the application of Lagrange’s equations of the first kind, was suggested. This method is the most effective one for applying to the elastic systems, consisting of lumped masses, bars, rings, and plates, which can be connected rigidly to each other or by means of linear compliances. As an additional example we make use of the suggested method to study the oscillations of a system of bars with lateral and longitudinal oscillations. Considering this example, we show how a holonomic, i. e. rigid, constraint between elements of system becomes elastic. In Fig. VI. 2 is shown an elastic system, consisting of three uniform straight bars and one linear compliance δ = 1/c. We assume that the bars lie in the same plane and for small oscillations of system, bar 1 has the longitudinal oscillations and bars 2 and 3 the bending oscillations.

166

VI. Application Lagrange Multipliers

Fig. VI. 2

From the principle of releasability of system it follows that by relation (2.1) the oscillations of bars can be represented as u(x1 , t) = y2 (x2 , t) =

∞ 

σ=1 ∞ 

q1σ (t)X1σ (x1 ) ,

X1σ (x1 ) = sin

(2σ − 1)πx1 , 2l1

q2σ (t)X2σ (x2 ) ,

(5.1)

σ=1 ∞  l3  ϕ(t) + q3σ (t)X3σ (x3 ) , y3 (x3 , t) = η(t) + x3 − 2 σ=1



0  xi  li ,

i = 1, 2, 3 .

Here X2σ (x2 ) and X3σ (x3 ) are the beam functions of a cantilever and a free bar, respectively [12, 227]. The first two addends in relation for y3 (x3 , t) correspond to the motion of bar 3 as rigid body. The quantity η is equal to a displacement of the center of mass C of bar 3 in the axis y3 and ϕ is an angle of its rotation. We consider also the displacement ξ of the center of mass C of bar 3 in the axis x3 and regard the quantities ξ, η, ϕ, qνσ (ν = 1, 2, 3; σ = 1, 2, . . .) as generalized Lagrangian coordinates. Let be δ = 0. Then all constraints between the coordinates introduced are holonomic and are given by the following equations f1 = u(l1 , t) − y2 (a2 , t) =

∞ 

q1σ X1σ (l1 ) −

σ=1

f2 = y2 (l2 , t) − ξ =

∞ 

q2σ X2σ (a2 ) = 0 ,

σ=1 ∞ 

q2σ X2σ (l2 ) − ξ = 0 ,

σ=1



∞  l3  ϕ+ q3σ X3σ (a3 ) = 0 , f3 = y3 (a3 , t) = η + a3 − 2 σ=1

(5.2)

5. Normal frequencies and oscillation modes of system of bars

167

  ∞ ∞   ∂y3  ∂y2  ′ ′ f4 = + = q X (l ) + ϕ+ q3σ X3σ (a3 ) = 0 . 2σ 2 2σ ∂x2 x2 =l2 ∂x3 x3 =a3 σ=1 σ=1

By relations (2.3) and (2.4) the kinetic energy of bars and potential energy of their deformation can be represented as [12, 227] n

T =

2 m3 (ξ˙2 + η˙ 2 ) m3 l32 ϕ˙ 2   Mνσ q˙νσ + + , 2 24 2 ν=1 σ=1

Π=

n  ∞ 2 2  Mνσ ωνσ qνσ , 2 ν=1 σ=1

Mµσ =

mµ lµ

lµ

2 Xµσ (x) dx =

∞

M1σ =

m1 , 2

(5.3)

2 (lµ ) mµ Xµσ , 4

0

n = 3,

σ = 1, 2, . . . ,

ν = 1, 2, 3 ,

µ = 2, 3 .

In these formulas ωνσ are normal frequencies of bars when the constraints are absent, mν are their masses. We shall use further Lagrange’s equations of the first kind in the generalized coordinates: k

 ∂fi ∂L d ∂L − = Λi , dt ∂ q˙ρ ∂qρ ∂qρ i=1

L = T − Π.

(5.4)

Here k is the number of constraints and q1 , q2 , . . . is a system of all Lagrangian coordinates introduced above. Equations (5.4) give  m3 l32 l3  + Λ4 , ϕ¨ = Λ3 a3 − 12 2 2 M1σ (¨ q1σ + ω1σ q1σ ) = Λ1 X1σ (l1 ) ,

m3 ξ¨ = −Λ2 ,

m3 η¨ = Λ3 ,

(5.5)

2 ′ q2σ + ω2σ q2σ ) = −Λ1 X2σ (a2 ) + Λ2 X2σ (l2 ) + Λ4 X2σ (l2 ) , M2σ (¨ 2 ′ q3σ + ω3σ q3σ ) = Λ3 X3σ (a3 ) + Λ4 X3σ (a3 ) . M3σ (¨

The generalized reactions Λ1 , Λ2 , Λ3 are equal to the forces of interaction of bars at the points of their connection to each other and Λ4 is equal to the moment of interaction between bars 2 and 3. From the principle of releasability from constraints, it follows that equations (5.5) can also be regarded as the equations of dynamics for the elements of system acted by the forces Λ1 , Λ2 , Λ3 and the moment Λ4 that are due to constraints no matter what the constraints are rigid or elastic. Then we make use of Lagrange’s equations of the first kind (5.4) not only in the case of rigid constraints but also in the case when all or certain constraints are elastic. For this purpose we assume first that all constraints are rigid and write then equations (5.4). In this case in the equations of the elastic constraints the reactions are introduced corresponding to these constraints. In the considered example the first constraint

168

VI. Application Lagrange Multipliers

is elastic. The reaction Λ1 is equal to the force of stretching (contraction) of spring with the compliance δ, in which case Λ1 > 0 if the spring is stretched. Therefore the first equation of system (5.2) takes the form f1 =

∞ 

q1σ X1σ (l1 ) −

σ=1

∞ 

q2σ X2σ (a2 ) + Λ1 δ = 0 .

(5.6)

σ=1

Note that if we multiply this relation by minus unity, i. e. represent it in the form of f1∗ = −f1 = 0, then the new Lagrange multiplier Λ∗1 is, obviously, such that Λ∗1 = −Λ1 . Hence the quantity Λ∗1 δ enters into the equation f1∗ = 0 with plus sign. Thus, if the i-th constraint is elastic and its compliance is equal to δi , then, assuming first that it is holonomic, we proceed to an elastic constraint by adding the quantity Λi δi . Suppose, the elastic system considered oscillates with the sought normal frequency p. Then the reactions Λi and the coordinates qρ can be represented as  i cos(p t + α) , Λi = Λ qρ = qρ cos(p t + α) . (5.7) Then, taking into account equations (5.5), we obtain 2 Λ ξ = , m3 p2

3 Λ , m3 p2 3 + Λ 4 12 (a3 − l3 /2)Λ ϕ =− 2 , 2 l3 m3 p 1 X1σ (l1 )Λ q1σ = , 2 M1σ (ω1σ − p2 )  1 + X2σ (l2 )Λ  2 + X ′ (l2 )Λ 4 −X2σ (a2 )Λ 2σ q2σ = , 2 − p2 ) M2σ (ω2σ  3 + X ′ (a3 )Λ 4 X3σ (a3 )Λ 3σ q3σ = . 2 2 M3σ (ω3σ − p ) η = −

(5.8)

Substituting relations (5.7) and formulas (5.8) into the equations of constraints (5.2), (5.6), we find 4  j=1

j = 0 , αij (p2 )Λ

αij = αji ,

i = 1, 4 .

(5.9)

Here index i corresponds to the constraint number.  j , j = 1, 4, only for the normal This elastic system has zero values of Λ form, of oscillations with the frequency p, for which the forces of interaction between all elements of system are absent. The system considered has no such forms of oscillations. Therefore in accordance with system (5.9) all its normal frequencies must satisfy the following equation det[αij (p2 )] = 0 .

(5.10)

169

5. Normal frequencies and oscillation modes of system of bars It is useful to represent the coefficients αij as αii = δi + βii + γii , i = j . αij = βij + γij ,

(5.11)

Here δi is a compliance of the i-th constraint in the case when it is elastic. The quantities βij , which are inversely proportional to the quantity p2 , can be called the coefficients of compliance of inertia forces. For this problem we have β11 = β12 = β13 = β14 = β23 = β24 = 0 , βij = βji , β22 = −

1 , m3 p2

β34 = −

β33 = −

12(a3 − l3 /2) , m3 l32 p2

1 + 12(a3 − l3 /2)2 /l32 , m3 p2 β44 = −

12 . m3 l32 p2

These relations are used to construct the functions αij (p2 ) (i, j = 1, 4) with the help of formulas (5.11). The coefficients γij = γji are infinite sums of simple fractions: γ11 (p2 ) =

∞ 

2 2  X1σ (l1 ) X2σ (a2 ) + 2 2 − p2 ) , 2 M (ω − p ) M (ω 1σ 2σ 1σ 2σ σ=1 σ=1 ∞

∞  X2σ (a2 )X2σ (l2 ) γ12 (p ) = − 2 − p2 ) , M2σ (ω2σ σ=1 2

γ13 (p2 ) = 0 ,

∞ ′  X2σ (a2 )X2σ (l2 ) γ14 (p ) = − , 2 2) M (ω − p 2σ 2σ σ=1 2

2

γ22 (p ) =

∞ 

2 X2σ (l2 ) 2 − p2 ) , M (ω 2σ 2σ σ=1

∞ ′  X2σ (l2 )X2σ (l2 ) γ24 (p ) = , 2 M (ω − p2 ) 2σ 2σ σ=1 2

γ23 (p2 ) = 0 , 2

γ33 (p ) =

(5.12)

∞ 

2 X3σ (a3 ) 2 − p2 ) , M (ω 3σ 3σ σ=1

∞ ′  X3σ (a3 )X3σ (a3 ) γ34 (p ) = , 2 M (ω − p2 ) 3σ 3σ σ=1 2

 ′  ′ 2 2 ∞  X2σ (l2 ) X3σ (a3 ) γ44 (p ) = 2 − p2 ) + 2 − p2 ) . M2σ (ω2σ M3σ (ω3σ σ=1 σ=1 2

∞ 

For p2 = 0 the quantities γij take the form γij (0) =

∂2Π . ∂Λi ∂Λj

(5.13)

170

VI. Application Lagrange Multipliers

Here Π is a total potential energy of the deformation of elements of system under the generalized reactions Λi . To verify the validity of formula (5.13), we consider relation (5.3) for the potential energy of deformation of bars and Lagrange’s equations (5.5). For quasistatic account of all forms of normal oscillations of elements of system, i. e. in the case when q¨νσ = 0 (ν = 1, 2, 3; σ = 1, 2, . . .) we have 2 ∞  1  Λ1 X1σ (l1 ) Π= + 2 2 σ=1 M1σ ω1σ 2 ∞  ′ (l2 ) 1  − Λ1 X2σ (a2 ) + Λ2 X2σ (l2 ) + Λ4 X2σ (5.14) + + 2 2 σ=1 M2σ ω2σ 2 ∞  ′ (a3 ) 1  Λ3 X3σ (a3 ) + Λ4 X3σ + . 2 2 σ=1 M3σ ω3σ Taking into account the above relations and formulas (5.12), we conclude that relations (5.13) are valid. The substantial fact is that the potential energy of deformation of bars can be represented not only in the form of infinite series (5.14) but also in closed form: Λ21 l1 Π = Π1 + Π2 + Π3 , Π1 = , 2E1 S1 (5.15)

lµ 2 Mµ (x) dx 1 Πµ = , µ = 2, 3 . 2 Eµ Jµ 0

Here E is Young’s modulus, J is a moment of inertia of a cross section of bar, S is a cross section area. The bending moments M2 (x), M3 (x) are linear functions of the generalized reactions Λi . We remak that for numerical computation of M3 (x) the force Λ3 and the moment Λ4 , applied to bar 3, are counteracted in the case of its quasistatic deformation by the inertia forces of translational and angular motions. Using formulas (5.13) and (5.15), we obtain γ11 (0) =

l1 a32 + , E1 S1 3E2 J2

γ12 (0) = −

γ13 (0) = 0 ,

γ14 (0) = −

γ23 (0) = 0 ,

γ24 (0) =

γ34 (0) = f33 =

f34 (z)l32 , E3 J3

a22 , 2E2 J2

l22 , 2E2 J2 γ44 (0) =

a22 (3l2 − a2 ) , 6E2 J2

γ22 (0) = γ33 (0) =

l23 , 3E2 J2

f33 (z)l33 , E3 J3

f44 (z)l3 l2 + , E3 J3 E2 J2

1 11z 13z 2 z3 + z4 3z 5 − z 6 − + − + , 105 105 35 3 5

(5.16)

5. Normal frequencies and oscillation modes of system of bars

171

13z z2 2z 3 3z 4 3z 5 11 + − − + − , 210 35 2 3 2 5 13 a3 = − z + 2z 3 − z 4 , z= . 35 l3

f34 = − f44

For approximate numerical computation of frequencies with provision for equation (5.10), we make use of the approximate approach stated in § 3. Recall that it is based on the dynamic account of N first normal forms of oscillations of elements of system and on the quasistatic account of the rest of normal forms. According to this approach the relations γij (p2 ) can approximately be computed by formulas γ11 (p2 ) =

N 

N

2 2  X1σ (l1 ) X2σ (a2 ) + 2 2 − p2 ) + 2 M (ω − p ) M (ω 1σ 2σ 1σ 2σ σ=1 σ=1

+γ11 (0) −

γ12 (p2 ) = −

+

+

γ22 (p2 ) =

−

γ13 (p2 ) = 0 ,

N ′  X2σ (a2 )X2σ (l2 ) + γ14 (0)+ 2 M2σ (ω2σ − p2 ) σ=1 N ′  X2σ (a2 )X2σ (l2 ) , 2 M ω 2σ 2σ σ=1

N 

2 X2σ (l2 ) + γ22 (0)− 2 M2σ (ω2σ − p2 ) σ=1

N 2  X2σ (l2 ) 2 , M ω2σ 2σ σ=1

γ23 (p2 ) = 0 ,

N N ′ ′   X2σ (l2 )X2σ (l2 ) X2σ (l2 )X2σ (l2 ) + γ (0) − , 24 2 2 2) M (ω − p M ω 2σ 2σ 2σ 2σ σ=1 σ=1

γ33 (p2 ) =

γ34 (p2 ) =

N  X2σ (a2 )X2σ (l2 ) 2 − p2 ) + γ12 (0)+ M2σ (ω2σ σ=1

N  X2σ (a2 )X2σ (l2 ) , 2 M2σ ω2σ σ=1

γ14 (p2 ) = −

γ24 (p2 ) =

N N 2 2  X1σ (l1 )  X2σ (a2 ) − 2 2 , M1σ ω1σ σ=1 M2σ ω2σ σ=1

N 

N 2 2  X3σ (a3 ) X3σ (a3 ) + γ (0) − 33 2 2 , 2 M3σ (ω3σ − p ) M3σ ω3σ σ=1 σ=1

N N ′ ′   X3σ (a3 )X3σ (a3 ) X3σ (a3 )X3σ (a3 ) + γ (0) − , 34 2 2 2 M (ω − p ) M ω 3σ 3σ 3σ 3σ σ=1 σ=1

(5.17)

172

VI. Application Lagrange Multipliers 2 2  ′  ′ N  X2σ (l2 ) X3σ (a3 ) γ44 (p ) = 2 − p2 ) + 2 − p2 ) + M2σ (ω2σ M3σ (ω3σ σ=1 σ=1 2

N 

2 2 N  ′ N  ′   X2σ (l2 ) X3σ (a3 ) − . +γ44 (0) − 2 2 M2σ ω2σ M3σ ω3σ σ=1 σ=1

Recall that the static coefficients γij (0) are given by formulas (5.16). Using relations (5.17) and formulas (5.11), we can obtain the approximate representations of the functions αij (p2 ) (i, j = 1, 4).  j , sat ρj the quantities Λ Now we compute eigenfunctions. Denote by Λ isfying system (5.9) for the normal frequencies pρ . From relations (5.1) and (5.8) it follows that the normal forms of oscillations of the system considered are described by the functions uρ (x1 ) =

yρ2 (x2 ) =

∞ 

 ρ1 X1σ (l1 )Λ X1σ (x1 ) , 2 M1σ (ω1σ − p2ρ ) σ=1

0  x1  l1 ,

∞   ρ1 + X2σ (l2 )Λ  ρ2 + X ′ (l2 )Λ  ρ4 −X2σ (a2 )Λ 2σ X2σ (x2 ) , 2 2 M2σ (ω2σ − pρ ) σ=1

(5.18)

0  x2  l2 ,

yρ3 (x3 ) =

∞ 

 ρ3 + X ′ (a3 )Λ  ρ4 X3σ (a3 )Λ 3σ X3σ (x3 ) , 2 2 M3σ (ω3σ − pρ ) σ=1

0  x3  l3 .

Thus, we find the representation of normal forms of oscillations of the original compound elastic system by the normal forms of its unit cells (separate elements). st It is convenient to consider the functions ust ρ (x1 ), yρµ (xµ ) (µ = 2, 3) defining the deformation of bars in the quasistatics under the generalized reac ρj . These functions can be found in closed form by the methods of tions Λ strength of materials and can be represented by formulas (5.1) and (5.8) in the form of infinite series ust ρ (x1 ) =

st yρ2 (x2 ) =

∞   ρ1 X1σ (l1 )Λ X1σ (x1 ) , 2 M1σ ω1σ σ=1

0  x1  l1 ,

∞   ρ1 + X2σ (l2 )Λ  ρ2 + X ′ (l2 )Λ  ρ4 −X2σ (a2 )Λ 2σ X2σ (x2 ) , 2 M2σ ω2σ σ=1

0  x2  l2 ,

st (x3 ) = yρ3

∞   ρ3 + X ′ (a3 )Λ  ρ4 X3σ (a3 )Λ 3σ X3σ (x3 ) , 2 M3σ ω3σ σ=1

0  x3  l3 .

173

6. Transformation of the frequency equation

Then relations (5.18) imply that the sought forms of oscillations can be represented as uρ (x1 ) = ust ρ (x1 ) +

 ρ1 p2ρ X1σ (l1 )Λ 2 − p2 )ω 2 X1σ (x1 ) , M1σ (ω1σ ρ 1σ σ=1 ∞ 

st (x2 )+ yρ2 (x2 ) = yρ2   ∞  ρ1 + X2σ (l2 )Λ  ρ2 + X ′ (l2 )Λ  ρ4 p2  − X2σ (a2 )Λ ρ 2σ + X2σ (x2 ) , 2 − p2 )ω 2 M (ω 2σ ρ 2σ 2σ σ=1

yρ3 (x3 ) =

st yρ3 (x3 )

 ∞   ρ3 + X ′ (a3 )Λ  ρ4 p2  X3σ (a3 )Λ ρ 3σ + X3σ (x3 ) , 2 − p2 )ω 2 M (ω 3σ ρ 3σ 3σ σ=1 0  xi  li ,

(5.19)

i = 1, 2, 3 .

The frequencies of longitudinal oscillations ω1σ increase with σ and the frequencies of lateral oscillations ω2σ and ω3σ increase with σ 2 . The quantities ′ ′ (l2 ) and X3σ (a3 ) increase with σ. Therefore the series, entering into the X2σ first formula of (5.19), converges as 1/σ 4 and the other two sums as 1/σ 7 . Such a fast convergence of series is explained by that in solution (5.19) there are separated the quasistatic forms of deformations of elements of system. § 6. Transformation of the frequency equation to a dimensionless form and determination of minimal number of parameters governing a natural frequency spectrum of the system For numerical implementation of the suggested new method for determination of a natural frequency spectrum of the system of bars it is necessary to transform the frequency equation (5.10) to a dimensionless form. For this purpose it is necessary first of all to choose the central bar of this system. In the example shown in Fig. VI. 2, it is the second bar. Natural frequencies of the second bar in the case of its disconnection to other bars are as follows [12]: 2 = λ42σ k22 , ω2σ

k22 =

1 E2 J2 , m2 l23

(6.1)

where λ2σ are roots of the equation ch λ cos λ = −1 .

(6.2)

We have λ21 = 1.875 ,

λ22 = 4.694 ,

λ25 = 14.137 ,

λ2σ

λ23 = 7.855 , π = (2σ − 1) , 2

λ24 = 10.996 , σ > 5.

174

VI. Application Lagrange Multipliers

The quantities λ22σ = ω2σ /k2 give the dimensionless natural frequency spectrum of the considered cantilever. Now we shall find the natural frequency spectrum p∗ of the mechanical three-bar system under consideration in the form p∗ =

p . k2

(6.3)

If we mentally separate the first and the third bars from the system, then their natural frequencies can be represented as [12]: 2 ω1σ = λ41σ k12 ,

k12 =

1 E1 J1 , m1 l 1

2 = λ43σ k32 , ω3σ

k32 =

λ21σ =

(2σ − 1)π , 2

1 E3 J3 . m3 l33

(6.4)

Here λ3σ are roots of the equation ch λ cos λ = 1 , where

(6.5)

λ31 = 4.7300 ,

λ32 = 7.8532 , λ33 = 10.9956 , π λ3σ = (2σ + 1) , σ > 4 . λ34 = 14.137 , 2 Natural vibration modes of the bars of this system are as follows [12]: X1σ (x1 ) = sin

(2σ − 1)π ξ, 2

ξ=

x1 , l1

X2σ (x2 ) = sin λ2σ ξ − sh λ2σ ξ + A2σ (ch λ2σ ξ − cos λ2σ ξ) , X3σ (x3 ) = sin λ3σ ξ + sh λ3σ ξ − A3σ (cos λ3σ ξ + ch λ3σ ξ) ,

x2 , (6.6) l2 x3 ξ= . l3 ξ=

Here sh λ2σ + sin λ2σ sh λ3σ − sin λ3σ , A3σ = , ch λ2σ + cos λ2σ ch λ3σ − cos λ3σ = 1.3622 , A22 = 0.98187 , A23 = 1.000777 ,

A2σ = A21

A24 = 0.999965 , A25 = 1.0000015 , A31 = 1.0178 , A32 = 0.999223 , A34 = 0.9999986 ,

A2σ = 1 , σ > 5 , A33 = 1.0000335 ,

A35 = 1.0000001 ,

A2σ = 1 ,

(6.7)

σ > 5.

By using expressions (6.6), (6.7) and equations (6.2) and (6.5), we shall represent the reduced masses Mµσ , µ = 1, 2, 3 , specified by the formulas (5.3), in the form Mµσ = mµ A2µσ ,

µ = 1, 2, 3 ,

A21σ =

1 . 2

(6.8)

175

6. Transformation of the frequency equation

2 Formulas (6.1), (6.3), (6.4), (6.8) allow us to write the quantities Mµσ (ωµσ − p2 ), entering the expressions (5.12), in the following manner 2 2 Mµσ (ωµσ − p2 ) = mµ A2µσ (ωµσ − p2 ) =

= m2 mµ2 A2µσ k22 (kµ2 λ4µσ − p2∗ ) , Here mµ2 =

mµ , m2

kµ2 =

kµ2 , k22

µ = 1, 2, 3 .

µ = 1, 2, 3 .

(6.9)

(6.10)

From expressions (5.12) and (6.9) it follows that all quantities γij (p2∗ ), i, j = 1, 4, contain the factor 1 l3 = 2 . 2 m2 k2 E2 J2 Thus we shall multiply all elements of determinant (5.10) by m2 k22 . As this takes place the quantities α ij (p2∗ ) =

E2 J2 αij (p2 ) , l23

i, j = 1, 2, 3 ,

become dimensionless, because the functions Xµσ are dimensionless. ′ ′ (l2 ), X3σ (a3 ), and The expressions γi4 , i = 1, 2, 3, contain derivatives X2σ γ44 contains those derivatives squared. We have:   1 dX2σ  dX2σ (x2 )  1 ′ ′ X2σ (l2 ) = = = Xξ,2σ (1) , dx2 x2 =l2 l2 dξ ξ=1 l2    (6.11) a3 1 ′ 1 dX3σ  ′ X3σ . = (a3 ) = X l3 dξ  a3 l3 ξ,3σ l3 ξ= l

3

Therefore we shall multiply the fourth row and the fourth column of determinant (5.10) by l2 , in this case all α ij will become dimensionless. As follows from formulas (6.9), (6.10), the quantities α ij (p2∗ ), i, j = 1, 4, depend on four dimensionless parameters m12 =

m1 , m2

m32 =

m3 , m2

k12 =

k12 , k22

k32 =

k32 . k22

Other four dimensionless parameters will be added to them: l2 , l3

a2 , l2

a3 , l3

δ , δ∗

δ∗ =

l23 . E2 J2

Hence we have eight dimensionless parameters in total.

176

VI. Application Lagrange Multipliers

The relations l2 /l3 , a3 /l3 , δ/δ∗ appear in the quantities α 11 , β33 , β34 , β44 in the following manner: α 11 = m2 k22 (δ + γ11 ) = δ E2 J2 = 3 δ + m2 k22 γ11 = + m2 k22 γ11 , l2 δ∗   2  a3 1 m2 1 2  1 + 12 − , β33 = m2 k2 β33 = − m3 l3 2 p2∗   m2 l2 a3 1 1 − , β34 = m2 k22 l2 β34 = −12 m3 l3 l3 2 p2∗  2 m 2 l2 1 . β44 = m2 k22 l22 β44 = −12 m3 l3 p2∗

The parameters a2 /l2 and a3 /l3 will also enter determinant (5.10) in terms ′ (a3 ). We note that the quantities of the functions X2σ (a2 ), X3σ (a3 ) and X3σ γ 34 and γ 44 , as it follows from formulas (5.12), (6.11), are proportional to l2 /l3 and (l2 /l3 )2 respectively. When using the N th approximation we should bear in mind that the quantities γ ij (0) are the functions of the parameters intriduced above. parameters. It is necessary to express γ ij (0) in terms of them. As this takes place we should have in view that m1 k12 =

Then we have:

E1 S1 , l1

m2 k22 =

E2 J2 , l23

m3 k32 =

E3 J3 . l33

m2 k22 E2 J2 l1 1 = 3 · = , m1 k12 l2 E1 S 1 m12 k12  3 1 E2 J2 l3 m2 k22 = = . m3 k32 m32 k32 E3 J3 l2

Having taken those expressions into account, we obtain: 1 a2 a3 1 3 ζ , ζ= , z= , m12 k12 3 l2 l3 a2 (3l2 − a2 ) ζ 2 (3 − ζ) γ 12 (0) = − 2 , = − 6l23 6 1 1 1 24 (0) = , 22 (0) = , γ γ 14 (0) = − ζ 2 , γ 2 3  2 f33 (z) f34 (z) l2 , γ 34 (0) = γ 33 (0) = , m32 k32 m32 k32 l3  2 f44 (z) l2 + 1. γ 44 (0) = m32 k32 l3

γ 11 (0) =

177

6. Transformation of the frequency equation When considering the N th approximation we shall have: N 2  2 X2σ (a2 ) + + 4 2 4 2 m12 (k12 λ1σ − p∗ ) σ=1 A2σ (λ2σ − p2∗ ) σ=1     N N 2 1 3  X2σ 1 1 (a2 ) 8  1− 2 + , ζ − + m12 k12 π σ=1 (2σ − 1)2 3 A22σ λ42σ σ=1 N γ 11 (p2∗ ) =

N γ 12 (p2∗ ) = −

N N  X2σ (a2 )X2σ (l2 ) ζ 2 (3 − ζ)  X2σ (a2 )X2σ (l2 ) − + , A22σ (λ42σ − p2∗ ) 6 A22σ λ42σ σ=1 σ=1

N γ 14 (p2∗ ) = −

N (p2∗ ) γ 24

N 

N N ′ ′  X2σ (a2 )Xξ,2σ (1) ζ 2  X2σ (a2 )Xξ,2σ (1) − + , 2 4 2 4 2 A2σ (λ2σ − p∗ ) 2 A2σ λ2σ σ=1 σ=1

N γ 22 (p2∗ ) =

N 

N

1  4 4 + − , 4 2 (λ2σ − p∗ ) 3 σ=1 λ42σ σ=1

N N ′ ′  X2σ (l2 )Xξ,2σ (1) 1  X2σ (l2 )Xξ,2σ (1) + − = , 2 4 2 4 2) A (λ − p 2 A λ ∗ 2σ 2σ 2σ 2σ σ=1 σ=1

N 

N

2 2  f33 (z) X3σ (a3 ) X3σ (a3 ) + − 2 4 2 k λ4 , 2 m A (k λ − p ) m k m A 32 32 32 32 32 ∗ 3σ 2σ 3σ 32 2σ σ=1 σ=1   N N ′ ′  X3σ (z)Xξ,3σ (z) X3σ (z)Xξ,3σ (z) f34 (z) N l, + γ 34 (p2∗ ) = − m32 A23σ (k32 λ43σ − p2∗ ) m32 k32 σ=1 m32 A232 k32 λ43σ σ=1 N γ 33 (p2∗ ) =

N γ 44 (p2∗ ) =

+

N N ′ ′   (Xξ,2σ (1))2 (Xξ,3σ (z))2 2 + l + A2 (λ4 − p2∗ ) m32 A23σ (k32 λ43σ − p2∗ ) σ=1 2σ 2σ σ=1

N N ′ ′   (Xξ,2σ (1))2 (Xξ,3σ (z))2 f44 (z) 2 2 l +1− − l . 2 4 2 m32 k32 A2σ λ2σ m32 A3σ k32 λ43σ σ=1 σ=1

Here: l = l2 /l3 , X2σ (a2 ) = sin(λ2σ ζ) − sh(λ2σ ζ) + A2σ (ch(λ2σ ζ) − cos(λ2σ ζ)) , ζ = a2 /l2 , X2σ (l2 ) = sin λ2σ − sh λ2σ + A2σ (ch λ2σ − cos λ2σ ) , ′ Xξ,2σ (1) = λ2σ (cos λ2σ − ch λ2σ + A2σ (sh λ2σ + sin λ2σ )) , X3σ (a3 ) = sin(λ3σ z) + sh(λ3σ z) − A3σ (cos(λ3σ z) + ch(λ3σ z)) , z = a3 /l3 , ′ ′ Xξ,3σ (z) = Xz,3σ (z) = λ3σ (cos(λ3σ z) + ch(λ3σ z) − A3σ (sh(λ3σ z) − sin(λ3σ z)) . We note that when using the given formulas one should remember that the values of X2σ (l2 ) have changing (alternating) signs (the plus sign — for odd σ and the minus sign — for even σ), and    X2σ (l2 )   = A2σ , A2σ > 0 .    2

178

VI. Application Lagrange Multipliers So the frequency equation (5.10) can be written in the dimensionless form det[ αij (p2∗ )] = 0 .

Its coefficients

(6.12)

α ii = δi + βii + γ ii ,  α ij = βij + γ ij , i = j ,

depend on the eight dimensionless parameters m1 , m2

m3 , m2

k12 , k22

k32 , k22

l2 , l3

a2 , l2

a3 , l3

δ . δ∗

The dimensionless frequencies p∗ of the system under investigation related to the required dimension frequencies p by formula (6.3) may be found from equations (6.12). Further development of ideas of the method presented in § 3, § 5, and § 6 is given in Appendix F. The method suggested there allows us to determine the first frequency of elastic systems to a high accuracy. This can be used for testing the complex programs used for analysis of vibration of the elastic systems. § 7. A special form of equations of the dynamics of system of rigid bodies For many bodies, the equations of motion for a system of rigid bodies, represented in the form of Lagrange’s equations of the second kind, are most complicated [120, 406] and very difficult not only for their integration but even for their writing. Therefore the question is actual how to represent these equations in the form convenient for computer calculation, what, in turn, is reduced to finding new forms of representation for equations of motion of one body. As is known, the kinetic energy of free rigid body with six degrees of freedom, cannot be represented as the sum, involving only the squares of generalized velocities multiplied by the constant values. In the independent generalized coordinates its kinetic energy has a rather complicated form. This explains the difficulties, connected with the application of Lagrange’s equations of the second kind even to one rigid body. Taking into account this fact, we make use of Lagrange’s equations of the first kind (2.22) of Chapter I since the kinetic energy of body in dependent coordinates has very simple form (1.1). Suppose, the active forces Fν are applied to the body at the points Nν = (xν , yν , zν ). Then the possible elementary work is as follows  δA = Fν · (δρρ + xν δi + yν δj + zν δk) = (7.1) ν = Qρ · δρρ + Qi · δi + Qj · δj + Qk · δk ,

179

7. A special form of equations of the dynamics where



Qρ =

Fν ,

Qi =

ν

Qj =



yν Fν ,



xν Fν ,

ν

Qk =

ν



(7.2) zν Fν .

ν

The form of kinetic energy (1.1), equations of constraint (1.2), and possible elementary work (7.1) result in that in this case in order to write equations (2.22) of Chapter I, it is convenient to use the vector form of Lagrange’s equations of the first kind (1.3). Using the rules of application of this formula described in § 1, we have ∂T d ∂T − = Qρ , dt ∂ρ˙ ∂ρρ

κ = 1, 6 ,

∂T d ∂T ∂f κ = Qi + Λκ ≡ Qi + 2Λ1 i + Λ4 j + Λ6 k , − dt ∂ i˙ ∂i ∂i ∂f κ d ∂T ∂T − = Qj + Λκ ≡ Qj + 2Λ2 j + Λ5 k + Λ4 i , dt ∂ j˙ ∂j ∂j ∂T ∂f κ d ∂T = Qk + Λκ ≡ Qk + 2Λ3 k + Λ6 i + Λ5 j . − dt ∂ k˙ ∂k ∂k In this case from relations (1.1) and (7.2) it follows that for the rigid body vector Lagrange’s equations of the first kind take the form  M ρ¨ = Fν , ν

Ix¨i =



xν Fν + 2Λ1 i + Λ4 j + Λ6 k ,

(7.3)

ν

Iy¨j =



yν Fν + 2Λ2 j + Λ5 k + Λ4 i ,

ν

¨= Iz k



zν Fν + 2Λ3 k + Λ6 i + Λ5 j .

ν

We eliminate the unknown multipliers Λκ , κ = 1, 6, from vector Lagrange’s equations of the first kind. For this purpose we differentiate twice in time the equations of constraint (1.2): ¨ · k, i˙2 = −¨i · i , j˙ 2 = −¨j · j , k˙ 2 = −k ¨ =0 , 2i˙ · j˙ + ¨i · j + i · ¨j = 0 , 2j˙ · k˙ + ¨j · k + j · k ¨ · i + k · ¨i = 0 . 2k˙ · i˙ + k

(7.4)

180

VI. Application Lagrange Multipliers

Substituting into these relations the second derivatives in time from equations (7.3), we obtain the formulas for the Lagrange multipliers Λκ :  xν Fν · i − Ix i˙2 , 2Λ1 = − ν

2Λ2 = −



yν Fν · j − Iy j˙ 2 ,

ν

2Λ3 = −



zν Fν · k − Iz k˙ 2 ,

ν

Iy  2Ix Iy ˙ ˙ Ix  Λ4 = − i·j− xν Fν · j − yν Fν · i , Ix + Iy Ix + Iy ν Ix + Iy ν 2Iy Iz ˙ ˙ Iy  Iz  yν Fν · k − zν Fν · j , j·k− Λ5 = − Iy + Iz Iy + Iz ν Iy + Iz ν Ix  2Iz Ix ˙ ˙ Iz  Λ6 = − k·i− zν Fν · i − xν Fν · k . Iz + Ix Iz + I x ν Iz + Ix ν

(7.5)

Substituting then relations (7.5) into system (7.3), we find  Fν , M ρ¨ = ν

¨i = −i˙2 i −

2Iy ˙ ˙ 2Iz ˙ + Lz j − Ly k , (i · j)j − (k˙ · i)k Ix + Iy Iz + Ix Ix + Iy Iz + Ix

2Iz ˙ ˙ 2Ix ˙ ˙ Lx Lz (j · k)k − (i · j)i + k− i, Iy + Iz Ix + Iy Iy + Iz Ix + Iy (7.6) 2Ix 2Iy ˙ ˙ Ly Lx 2 ˙ ˙ ¨ ˙ (k · i)i − (j · k)j + i− j. k = −k k − Iz + I x Iy + Iz Iz + Ix Iy + Iz ¨j = −j˙ 2 j −

Here Lx , Ly , Lz are a projection of the principal moment of active forces  L= (xν i + yν j + zν k) × Fν . (7.7) ν

E x a m p l e VI .4 . We shall show that equations (7.6) result in dynamic Euler’s equations with respect to the projections p, q, r of the vector of angle velocity ω on the axes x, y, z. We have i˙ = ω × i = rj − qk , i˙ · j˙ = −p q , −i˙2 i = −(q 2 + r2 )i ,

k˙ · i˙ = −r p ,

¨i = ω˙ × i + ω × i˙ = rj ˙ − qk ˙ − (q 2 + r2 )i + p qj + p rk . Then the projection of the second equation of system (7.6) on x-axis gives the identity and on y-axis the following relation r˙ + p q =

Lz 2Iy pq + . Ix + Iy Ix + Iy

8. The study of certain problems of robotics

181

Taking into account that A = Iy + Iz , B = Iz + Ix , C = Ix + Iy , we obtain the third dynamic equation C r˙ − (A − B) p q = Lz . The projecting of the same vector equation on z-axis gives the second Euler’s equation B q˙ − (C − A) r p = Ly . Similarly, from the third equation of system (7.6) we can obtain the third and first Euler’s equations and from the fourth equation of system (7.6) the second and first dynamic Euler’s equations. Note, that if on the vectors i, j, k constraints (1.2) are not imposed then a body considered is pseudorigid according to the terminology of J. Casey [417. 2004]. Basing on the apparatus of the monograph by Truesdell [418], J. Casey describes the dynamics of this continuum by the Lagrange equations the number of which is equal to twelve. The theory of vector Lagrange equations of the first kind is used in the next section when studying some problems of robotics.

§ 8. The application of special form of equations of dynamics to the study of certain problems of robotics Consider a motion control of a platform of dynamic stand [66]. We assume that the motion are controlled by the six bars of variable lengths (hydraulic cylinders). One end point of each bar is connected by spherical joints with a fixed point and the other end point of each bar with one of the points Nν , ν = 1, 6, of platform. Such large dynamic stands are employed by the leading air construction companies, in which case one stand is constructed approximately for ten aeroplanes. The cabin of aeroplane and the platform of stand are fixed and, using the control devices of aeroplane, the pilot moves a stand by varying the lengths of bars. In this case the pilot has a full illusion of a real motion in space together with aeroplane. The stands are used to train pilots, including the training of an accurate behavior in extremal situations and the practice of landing the aeroplane in concrete wold’s airports, to maintain good flying form, and so on. We introduce the fixed system of coordinates Oξηζ and the system Cxyz with the unit vectors i, j, k, which is rigidly fixed with the platform of stand −−→ and directed in the lines of its principal central axes of inertia. Let be ρ = OC. Then the position of platform as a rigid body is defined by the vectors ρ , i, j, k. The assumption that the body is rigid, as is remarked in § 1, can be regarded as the imposing of ideal constraints. These constraints are given by

182

VI. Application Lagrange Multipliers

equations (1.2). Then vector Lagrange’s equations of the first kind, describing the motion of the platform of stand, can be represented in the form of equations (7.3). The force Fν , applied to the platform by virtue of the bar, can be represented as Fν = Fν lν /|lν | , lν = ρ + xν i + yν j + zν k , where Fν = uν is a control parameter. In this case in system (7.3) the summing is over all ν from 1 to 6. Equations of constraint (1.2) yield relations (7.4). By these relations, from equations (7.3) we can obtain formulas (7.5) for the coefficients Λ1 , Λ2 , . . . , Λ6 . Eliminating the Lagrange multipliers from system (7.3), we arrive to differential system (7.6) with respect to the vectors ρ , i, j, k, which involves six control parameters uν = Fν , ν = 1, 6, being the strains of bars. In this system Lx , Ly , Lz are the projections of principal moment of active forces (7.7). We remark that system (7.6) found can be computed since it is solvable for the second time derivatives of unknown vectors. Considering the problem of dynamic stand, we need to pay attention to the following fact, which is not connected directly with the previous contents. The given law of varying the lengths of bars (hydraulic cylinders) lν (t) we shall regard as nonstationary constraints f ν ≡ l2ν − lν2 (t) = 0, ν = 1, 6, i. e. we shall study the motion control by constraints. We introduce the Lagrange  ν . In this case the reaction of the ν-th constraint has the form multipliers Λ 2

It follows that

 ν lν ≡ Fν .  ν ∂lν = 2Λ Rlν = Λ ∂lν ν = 2Λ

Fν uν = . lν lν

If the position of body is given by six generalized coordinates being the lengths of bars, then the theorem of the holonomic mechanics of Chapter III becomes descriptive. According to this theorem the motion such that one of generalized coordinates is a given function of time can be obtained if we introduce one additional force corresponding to this coordinate. Consider now the system of rigid bodies, connected sequentially to each other by spherical joints. Similar mechanical systems often occur in the robotics. Suppose, the number of joints s is equal to the number of moving bodies. The friction in joints is assumed to be negligible, i. e., by assumption, the constraints are ideal. Suppose also that the joint with number σ connects the bodies (σ − 1) and σ. Then the equations of constraint have the form ρ σ + xσσ iσ + yσσ jσ + zσσ kσ − σ σ −ρρσ−1 − xσσ−1 iσ−1 − yσ−1 jσ−1 − zσ−1 kσ−1 = 0 ,

σ = 1, s .

(8.1)

9. Application of the generalized Gaussian principle

183

Here the vectors ρ σ , iσ , jσ , kσ , corresponding to the body σ, have the same sense as above; xσρ , yρσ , zρσ are the coordinates of the joint with number σ in the system Cρ xρ yρ zρ . A fixed body is assumed to be zero. Denote by Rσ the force caused by the body (σ − 1) and applied by means of the joint to the body σ. We now make use of the releasability principle. In this case the equation of motion of body σ involves the reactions Rσ and Rσ+1 . Note that the body s is under one reaction Rs . We differentiate twice equations (8.1) in time and eliminate then the second derivatives, using for each bodies the obtained special form of equations of its motion. In this case we obtain the system of s equations with respect to s unknown reactions Rσ . The equation corresponding to arbitrary σ, which is not equal to 1 and s, involves the reactions Rσ−1 , Rσ , Rσ+1 . For σ = 1 and σ = s we have the equations in unknowns R1 , R2 and Rs−1 , Rs , respectively. This implies that this system of equations has a structure convenient for solving it with the help of the computer by the method of sequential elimination of the sought reactions. Determining these reactions and substituting them into the equations of motion, we obtain a system of differential equations of motion of the considered chain of bodies. This system is solvable for the second derivatives, i. e. it can be used for numerical integration by computer.

§ 9. Application of the generalized Gaussian principle to the problem of suppression of mechanical systems oscillations Introduction. This Section shows possibility and expedience of the employment of the generalized Gaussian principle presented in § 3 of Chapter IV, for studying the problems of control of mechanical systems oscillations. Similar problems have been analyzed in studies [419. 1980] in details, where the fundamental approach to their solution is the method that is based on the minimization of the functional of control force squared. This Section shows that application of the Gaussian generalized principle here proves to be very effective. Consideration is being given to the mechanical system with finite quantity of degrees of freedom. The presence of one control force acting within some time interval is assumed. This force, which is necessary to provide the movement of the system from a specified position to another one within the finite time, is being determined. In particular, if the final position of the system is to be the one of stable equilibrium then the problem under consideration becomes the problem of vibration suppression. It is shown that this problem can be solved with the help of Gaussian generalized principle. For a certain mechanical system with two degrees of freedom the problem was solved in the studies [419. 1980] using the minimization of the functional of control force squared. It turns out that the solution developed in [419. 1980] within a

184

VI. Application Lagrange Multipliers

certain range of variation of system dimensionless parameters slightly differs from the solution obtained by using the method proposed. Oscillation suppression of a trolley with a pendulum. The problem statement. Let us analyze the following problem [419. 1980]. The load of mass m2 is suspended with the cable of length l attached to the crane trolley of mass m1 running on horizontal rails (see Fig. VI. 3). F

x

ϕ

Fig. VI. 3

It is required to move the suspended load by the given distance a from a state of rest to another state of rest in the fixed time T by choosing the horizontal force F (t), applied to the trolley. The equations of motion of the system under consideration for smallamplitude oscillations will take the form (m1 + m2 )¨ x − m2 lϕ¨ = F , x ¨ − lϕ¨ = gϕ .

(9.1)

In order to provide termination of free oscillation of the load at the time t = T, the control force F (t) should be such that the boundary conditions ϕ(0) = ϕ(T) = 0 ,

ϕ(0) ˙ = ϕ( ˙ T) = 0 , x(0) = x(0) ˙ = x( ˙ T) = 0 , x(T) = a

(9.2)

are satisfied. Let us introduce the principal coordinates and go to dimensionless variables ϕ, ξ, τ, γ, u by the formulas m1 + m 2  m2 lϕ  , τ = γt , x− m1 l m 1 + m2 (m1 + m2 )g F γ2 = , u= 2 . m1 l γ m1 l

ξ=

Here ξ is a dimensionless displacement of the system center of mass, γ is a natural frequency, u is a control. Now instead of the differential equations system (9.1) we obtain two independent equations ϕ¨ + ϕ = u ,

ξ¨ = u ,

(9.3)

185

9. Application of the generalized Gaussian principle

in which the derivatives correspond to the dimensionless time τ . For the sake of simplicity the letter t will stand for the dimensionless time τ too. Boundary conditions (9.2) will be rewritten as ϕ(0) = ϕ(T ) = 0 , ˙ ˙ ) = 0, ξ(0) = ξ(0) = ξ(T

ϕ(0) ˙ = ϕ(T ˙ ) = 0, ξ(T ) = 1 ,

T = γ T .

(9.4)

The system of equations (9.1) is linear. Hence, the solution of the boundary problem (9.1), (9.2) will depend linearly on the quantity a. Therefore, when examining the boundary problem (9.3), (9.4) for the sake of simplicity the value of a can be accepted such that ξ(T ) = 1. Analysis of the results following from the method that is based on the minimization of the functional of control force squared. To solve the formulated problem (9.3), (9.4) it is necessary to add one condition more. It should express the principle, which forms the basis for choosing the force F (t) from the entire set of the forces such that this problem has a solution. In the treatise [419. 1980] it is shown that if the choice of the control u is subject to the condition of minimality of the functional

T J= u2 (t)dt , (9.5) 0

and the maximum principle of Pontryagin [420] is used, then the control u will be as follows [419. 1980, p. 328] u(t) = C1 + C2 t + C3 sin t + C4 cos t .

(9.6)

Here Ck , k = 1, 4, are arbitrary constants. By choosing these constants so that boundary conditions (9.4) are satisfied, we shall uniquely determine the required control u(t). From the form of expression (9.6) it follows that the function u(t) is a general solution of the differential equation .... u +u ¨ = 0, which can be represented in the form   d2 d2 + 1 u = 0, dt2 dt2 which is directly connected with the initial system (9.3). Note, that if we considered the system of equations x ¨σ + ωσ2 xσ = u ,

σ = 1, s ,

(9.7)

then the control u, minimizing the functional (9.5), would satisfy the differential equation  2  2   2  d d d 2 2 2 . . . (9.8) + ω + ω + ω 1 2 s u = 0, dt2 dt2 dt2

186

VI. Application Lagrange Multipliers

The solution of this equation takes the form u(t) =

s 

(Aσ cos ωσ t + Bσ sin ωσ t) .

(9.9)

σ=1

The problem of suppression of small-amplitude oscillations of the mechanical system in the time T , i. e. the following boundary problem xσ (0) = x0σ , x˙ σ (0) = x˙ 0σ , xσ (T ) = x˙ σ (T ) = 0 ,

(9.10)

σ = 1, s , may be solved by the choice of arbitrary constants Aσ and Bσ . Hence, as follows from expression (9.9), the minimization of functional (9.5) is achieved by search of the required control u(t) in the form of the series in resonance frequencies. Oscillation suppression by the minimization of the functional of control force squared as an example of the mixed problem of mechanics. The system of equations (9.7), representing small-amplitude oscillations of the mechanical system under the action of control force u(t), is written in principal coordinates xσ , σ = 1, s. In initial coordinates q σ , σ = 1, s, this system will take the form s 

(aστ q¨τ + cστ q σ ) = bσ u(t) ,

σ = 1, s .

(9.11)

τ =1

Here aστ , cστ , bσ , σ, τ = 1, s, are given constant values. These constants are such that when going to the principal coordinates the system (9.11) takes (9.7). In system (9.11) any of coefficients bσ may always be supposed equal to one. Let us assume that, for example, b1 = 1. Having substituted the control force u(t), set by the first equation of system (9.11), into equation (9.8), we obtain the differential equation of the order (2s + 2) relative to generalized coordinates q σ , σ = 1, s. Let us represent this equation in the form s 

(2s+2)

(2s)

(a2s+2,σ q σ + a2s,σ q σ + · · · + a0,σ q σ ) = 0 ,

(9.12)

σ=1

where a2n,σ , n = 0, s + 1, σ = 1, s, are constants that are found in the process of calculation. Hence, as applied to system (9.11) the minimization of functional (9.5) means the obedience of oscillations of the system to constraint equation (9.12). Let us assume that system (9.11) and the constraint (9.12) are specified. The problem of determining the control force u(t), providing satisfaction of

187

9. Application of the generalized Gaussian principle

the constraint (9.12), is a particular case of the so called mixed dynamic problem. Thus, the presence of constraint (9.12) following from the minimization of functional (9.5) makes it possible to consider the problem of determining the control force u(t), which provides oscillation suppression, as a certain mixed dynamic problem. Application of the generalized Gaussian principle to the problem of oscillation suppression. The theory of mixed dynamic problems applied to system (9.11) implies the following fact. Let the motion of the system considered be subject to the constraint given in the form s 

a2s+2,σ (t, q, q, ˙ ... ,

(2s+1) (2s+2) σ

) q

q

+ a2s+2,0 (t, q, q, ˙ ... ,

(2s+1)

q

) = 0,

σ=1

σ = 1, s ,

κ = 1, k ,

(9.13)

k  s,

where a2s+2,σ , σ = 0, s, are some functions of variables indicated. Then for certain restrictions to the relation between coefficients a2s+2,σ and bσ , σ = 0, s, a 2s-order differential equation with respect to the control force u(t) can be derived in accordance with the algorithm developed in Chapter V. Note that in the general case this equation includes the time, generalized coordinates, and generalized velocities. It is essential that when minimizing functional (9.5), from the set of the equations of form (9.13) there is chosen that subset to all elements of which one single equation (9.8) corresponds. Note specially that its structure is defined only by the spectrum of natural frequencies of the system and doesn’t depend on the choice of generalized coordinates. The following question naturally arises: if the equation of constraints (9.13) could be subject to another condition that also entails one single 2s-order equation with constant coefficients. This alternative condition can be found as follows. As in § 1 of Chapter IV, let us introduce the tangent space and represent the system of equations (9.11) therein as one vector equation M W = Y + u(t)b , where MW =

s 

σ,τ =1

τ σ

aστ q¨ e ,

Y=−

s 

σ,τ =1

τ σ

cστ q e ,

b=

s 

bσ eσ .

σ=1

The vectors of a reciprocal basis in this case do not depend on time and coordinates q σ , σ = 1, s, i. e. are constant. In accordance with the generalized Gaussian principle the constraint is ideal if the quantity  (2s) (2s)2 (2s) − →2 R 2s = M W − Y = ( u b)2 (9.14)

188

VI. Application Lagrange Multipliers

is minimal. We choose from constraints (9.13) that subset, for elements of − →2 which the quantity R 2s is equal to its lower boundary, which equals zero. As follows from expression (9.14), all these elements are described by one single equation (2s)

u = 0.

(9.15)

Therefore, the alternative solution of the problem of vibration suppression with one control force can be derived on the basis of the generalized Gauss principle. The general solution of equation (9.15) is u(t) =

2s 

Ck tk−1 .

(9.16)

k=1

Contrary to the control u(t), set by formula (9.9), the control that is found in the polynomial form (9.16), will not have oscillations corresponding to all natural system frequencies. The sought function u(t) will be sufficiently smooth, which is its definite advantage. Vibration suppression of n physical pendulums. As an example let us consider the following boundary problem x ¨1 = u , x ¨j +

xi (0) = x˙ i (0) = x˙ i (T ) = 0 , i = 1, s , s = n + 1 ,

2 ωj−1 xj

= u,

x1 (T ) = a , xj (T ) = 0 , j = 2, s .

(9.17)

Natural frequencies of the system are assumed to be such that ω1 = 1 ,

1 < ω2 < ... < ωn .

Note that problem (9.17) is, in particular, the problem of vibration suppression of n physical pendulums suspended to a trolley, whis is to be moved with the sought acceleration u(t) in time T by the given distance a. The system is supposed to be in the state of equilibrium at the initial and terminal time instants [419. 1980, p. 340]. Formulas (9.16) and (9.17) imply that functions xi (t), i = 1, s, can be represented as 2s  xi (t) = Ck ξik (t) , i = 1, s , k=1

where

ξ1k (t) = ξjk (t) =

0

t

τ k−1 (t − τ )dt ,

k = 1, 2s ,

0

t

τ k−1 sin ωj−1 (t − τ ) dt , ωj−1

j = 2, s .

189

9. Application of the generalized Gaussian principle

Constants Ck for different frequencies ωk , k = 1, n, are uniquely determined from the solution of the system of equations 2s 

ajk Ck = aδj1 ,

j = 1, 2s ,

(9.18)

k=1

where δj1 = 1 , j = 1 , ajk = ξjk (T ) , j = 1, s ,

δj1 = 0 , j = 1 , as+j,k = ξ˙jk (T ) , k = 1, 2s .

Let us show that the functions u(t) and xi (t), i = 1, s, in problem (9.17) are such that u(t) = −u(T − t) , x1 (t) = a − x1 (T − t) , xj (t) = −xj (T − t) .

(9.19)

Introduce into consideration the following functions x 1 (t) = x1 (T − t) − a , x j (t) = xj (T − t) ,

(9.20)

j = 2, s .

They are such that

¨ x i (t) = x ¨i (T − t) ,

i = 1, s .

This and expressions (9.20) imply that problem (9.17) appears in new functions as

Here

¨ (t) , x 1 = u

x i (0) = x ˙ i (0) = x ˙ i (T ) = 0 , i = 1, s ,

2 ¨ x j = u (t) , x j + ωj−1

x j (T ) = 0 , j = 2, s , u (t) = u(T − t) .

x 1 (T ) = −a .

(9.21)

(9.22)

The solution of the system of linear algebraic equations is proportional to the value of a, hence, comparing (9.17) with (9.21) results in x i (t) = −xi (t) , i = 1, s , u (t) = −u(t) .

It follows from the latter formulas and expressions (9.20) and (9.22) that relations (9.19) are really satisfied. When minimizing functional (9.5) the control of this problem will be sought as u(t) =

n 

(Ck cos ωk t + Cn+k sin ωk t) + C2n+1 + C2n+2 t .

k=1

190

VI. Application Lagrange Multipliers

In this case constants Ck , k = 1, 2s, s = n+1, are also found from the system of form (9.18), therefore, relations (9.19) remain. Computing was done for n = 2 and a = 1. In this case the solution depends on two parameters T , T2

T2 , T1

T1 = 2π ,

T2 =

2π . ω2

The results of calculations are shown in Fig. VI. 4 and VI. 5. The Figures show diagrams obtained by the generalized Gaussian principle by solid curves and those obtained by the minimization of functional (9.5) — by dashed curves. Thus, it is shown that in the problem of transfer of the system from one phase state to another one with one control force the generalized Gauss principle can be invoked along with the minimization of the functional of the control force squared. Comparision of these two approaches when studying the motion of a trolley with two pendulums shows that for some values of dimensionless parameters the solutions obtained by these two methods practically coincide (see Fig. VI. 4). But there are also such ranges of dimensionless parameters for which these solutions essentially differ from one another (see Fig. VI. 5). The presence of large-amplitude oscillations in the solution constructed by the minimization of the functional of the control force squared can be explained by the fact that in this case the control is sought as a sum of harmonics tending to cause a resonance of the system. In contrast,

Fig. VI. 4

9. Application of the generalized Gaussian principle

191

Fig. VI. 5

when using the generalized Gauss principle the control force is sought as a polynomial, which provides a smooth change of all functions required. The spatial motion of a load on a cable attached to a controlled trolley is also considered in the works by B. Simeon [434] and M. A. Chuev [252. 2008].

C h a p t e r VII EQUATIONS OF MOTION IN QUASICOORDINATES In the present chapter it is shown that all known types of equations of motion of nonholonomic systems are equivalent since they can be obtained from the invariant vector form of the law of motion of mechanical system with ideal constraints. The nonholonomicity of constraints, which does not allow for the equations of motion to be represented in the form of Lagrange’s equations of the second kind, turns out to be most clearly if the equations of motion of nonholonomic system are written in quasicoordinates. In the case of linear constraints these equations are generated here by three different methods. This permits us to consider the problem of nonholonomicity from three different points of view. § 1. The equivalence of different forms of equations of motion of nonholonomic systems Vector equation (3.10) of Chapter IV gives the law of motion of both holonomic and nonholonomic systems such that for the ideal constraints the generalized accelerations q¨σ , σ = 1, s, satisfy system of equations (1.5) and the vectors ε l+κ , κ = 1, k, satisfy the conditions (1.10) of Chapter IV. The important fact is that this equation has a vector form invariant under the choice of the system of coordinates, in which the motion is described and the equations of constraints are given. Therefore in the present section we obtain from this equation all main types of the equations of motion of nonholonomic systems and show thus their equivalence. Projecting equation (3.10) of Chapter IV on a system of the vectors ε λ , λ = 1, l, which make up the basis of L–space, we obtain the following system of scalar equations M W · ελ = Y · ελ ,

λ = 1, l .

(1.1)

Let the vectors ε λ , λ = 1, l, be the functions of the variables t, q, q˙σ , σ = 1, s. Then, supplementing equations (1.1) by equations (1.6) of Chapter IV, we obtain the closed system of equations such that the law of motion takes the form W = F(t, q, q) ˙ . V. S. Novoselov writes [169, p. 28] that the reduction of the problem to this equation can be regarded as "the reduction of the problem of nonholonomic mechanics to the conditional problem of mechanics of holonomic systems". A concrete form of equations (1.1) depends on both the representation of system of the vectors ε λ , λ = 1, l, and the form of expansion of the scalar products M W · ε λ , λ = 1, l. Consider the main forms of equations (1.1). 193

194

VII. Equations of Motion in Quasicoordinates

We assume that the integralable differential constraints and the linear first-order nonholonomic constraints are the special case of the constraints, given by the following equations ˙ = 0, f1κ (t, q, q)

κ = 1, k .

By assumption, the vectors ε l+κ = ∇ ′ f1κ , κ = 1, k, satisfy condition (1.10) of Chapter IV and therefore the equations of constraints imply that for the given values of the variables t and q σ , σ = 1, s, the generalized velocities q˙σ , σ = 1, s, can be expressed in terms of the independent variables v∗λ , λ = 1, l. In the works of V. S. Novoselov [169] they are called kinematical characteristics and in the works [149, 203, 229, 247, 248], devoted to the Poincar´e–Chetaev equations, the Poincar´e parameters. The variables v∗λ , λ = 1, l, are given by the functions v∗λ = f∗λ (t, q, q) ˙ , λ = 1, l . Supplementing them by the relations v∗l+κ = f∗l+κ (t, q, q) ˙ = f1κ (t, q, q) ˙ ,

κ = 1, k ,

we obtain q˙σ = q˙σ (t, q, v∗ ) ,

σ = 1, s .

(1.2)

f∗σ

Suppose that at least one of the relations dt, σ = 1, s, is not a total differential and cannot be reduced to it. In this case, as is known, the variables σ

π =

t

v∗σ (t) dt ,

σ = 1, s ,

t0

cannot be regarded as a new system of Lagrangian coordinates. Therefore they are called quasicoordinates and the quantities π˙ σ = v∗σ , σ = 1, s, quasivelocities. For linear constraints the generalized velocities and quasivelocities are related as v∗ρ = aρσ (t, q) q˙σ + aρ0 (t, q) ,

q˙σ = bστ (t, q) v∗τ + bσ0 (t, q) ,

ρ, σ, τ = 1, s , or in short form β v∗α = aα β q˙ , 0

q = t,

β q˙α = bα β v∗ , 0

q˙ =

v∗0

= 1,

α, β = 0, s , a0β

= b0β = δβ0 .

(1.3)

Index "1"of the coefficients al+κ α , κ = 1, k, α = 0, s, entering into the equations of constraint, is omitted for short. Applying the variables v∗ρ , ρ = 1, s, we can introduce the vectors ερ =

∂v∗ρ σ e , ∂ q˙σ

ετ =

∂ q˙σ eσ , ∂v∗τ

ρ, σ, τ = 1, s ,

195

1. The equivalence of different forms of equations such that ε ρ · ε τ = δτρ ,

ρ, τ = 1, s .

(1.4)

In this case the system of the vectors ε λ , λ = 1, l, makes up the basis of L–space since we have ε l+κ =

∂v∗l+κ σ e = ∇ ′ f1κ , ∂ q˙σ

ε l+κ · ε λ = 0 ,

κ = 1, k ,

λ = 1, l .

The account of the equations of constraints, based on the representation of generalized velocities in the form q˙σ = F σ (t, q, v∗1 , . . . , v∗l ) ,

σ = 1, s ,

means, as V. V. Rumyantsev [203, p. 3] writes, that "there is performed the parametrization of constraints imposed on the system . . . ". In this case the basis of L–space is known and given by the following relation ελ =

∂F σ eσ , ∂v∗λ

λ = 1, l ,

σ = 1, s .

Thus, the partition of tangent space into the subspaces K and L by the equations of constraints can be made using their parametrization. In this case the basis, of L–space, necessary for the passage to the concrete form of equations (1.1) is known. If the constraints are linear, then by relations (1.3) their parametrization can take the form q˙σ = bσλ (t, q) v∗λ + bσ0 (t, q) ,

σ = 1, s ,

λ = 1, l .

(1.5)

From formulas (1.5) we have ε λ = bσλ (t, q) eσ . Relations (3.1) and (3.2) of Chapter IV yield that the vectors M W, entering into equations (1.1), can be represented as MW =

d(M V) , dt

MV =

∂T σ e . ∂ q˙σ

where

Since eσ =

∂ q˙σ ρ ε , ∂v∗ρ

we get MV =

ρ, σ = 1, s ,

∂T ∂ q˙σ ρ ε . ∂ q˙σ ∂v∗ρ

(1.6)

196

VII. Equations of Motion in Quasicoordinates

The generalized velocities q˙σ are regarded as the functions of all variables v∗ρ , ρ, σ = 1, s, and only in the final relations, taking into account the equations of constraints, we assume that v∗l+κ = 0, κ = 1, k. Then we obtain MV =

∂T ∗ ρ ∂T ∂ q˙σ ρ ε , ρ ε = σ ∂ q˙ ∂v∗ ∂v∗ρ

(1.7)

where T ∗ = T ∗ (t, q, v∗ ) is a function of variables t, q σ , v∗σ , σ = 1, s, which is found by means of the substitution of relations (1.2) into the function T = T (t, q, q). ˙ Relations (1.6), (1.7) give   d ∂T ∗ ∂T ∗ ρ ε˙ . ερ + MW = ρ dt ∂v∗ ∂v∗ρ Then M W · ελ =

d ∂T ∗ ∂T ∗ ρ ε˙ · ε λ . + dt ∂v∗λ ∂v∗ρ

Taking into account relations (1.4), we obtain ε˙ ρ · ε λ = −εερ · ε˙ λ , and therefore M W · ελ =

d ∂T ∗ − M V · ε˙ λ , dt ∂v∗λ

Since ε˙ λ =



d ∂ q˙σ dt ∂v∗λ



eσ +

λ = 1, l .

(1.8)

∂ q˙σ e˙ σ ∂v∗λ

and accordance with relations (3.3) and (3.4) of Chapter IV eσ = we have

∂V , ∂ q˙σ

∂T M V · ε˙ λ = σ ∂ q˙



e˙ σ = d ∂ q˙σ dt ∂v∗λ



∂V , ∂q σ

+

∂T ∂ q˙τ . ∂q τ ∂v∗λ

Taking into account that ∂T ∗ ∂T ∂T ∂ q˙σ = τ + σ , τ ∂q ∂q ∂ q˙ ∂q τ we obtain M V · ε˙ λ =

∂T ∂ q˙σ



d ∂ q˙σ ∂ q˙τ ∂ q˙σ − λ λ dt ∂v∗ ∂v∗ ∂q τ



+

∂ q˙τ ∂T ∗ , ∂v∗λ ∂q τ

λ = 1, l .

(1.9)

197

1. The equivalence of different forms of equations

From relations (1.8) and (1.9) it follows that equations (1.1) take the form d ∂T ∗ ∂T ∗ ∂T λ , − − σ Tλσ = Q λ dt ∂v∗ ∂π λ ∂ q˙

λ = 1, l .

(1.10)

Here the following quantities

σ

 λ = Qσ ∂ q˙ , Q ∂v∗λ

λ = 1, l ,

σ = 1, s ,

are generalized forces, corresponding to the Poincar´e parameters (quasivelocities) v∗λ , λ = 1, l, and d ∂ q˙σ ∂ q˙σ Tλσ = − . dt ∂v∗λ ∂π λ Here and in equations (1.10) we use the notation ∂ q˙τ ∂ ∂ = . λ ∂π ∂v∗λ ∂q τ Taking into account that ∂T ∗ ∂v∗ρ ∂T = σ, ρ σ ∂v∗ ∂ q˙ ∂ q˙ equations (1.10) take the form

Here

∂T ∗ ∂T ∗ ρ d ∂T ∗ λ , − + W =Q λ λ dt ∂v∗ ∂π ∂v∗ρ λ Wλρ = −

∂v∗ρ σ T , ∂ q˙σ λ

λ = 1, l

λ = 1, l ,

ρ = 1, s .

(1.11)

ρ, σ = 1, s .

Equations (1.10) and (1.11), as follows from their generation, can be used for holonomic and nonholonomic systems with the linear and nonlinear in velocities ideal constraints. In the case when a time enters, in explicit form, into neither kinetic energy, nor the equations of constraints, equations (1.10) and (1.11) were obtained by G. Hamel in 1938 [314] and in the general case by V. S. Novoselov [169] in 1957. Therefore in § 2 of Chapter II equations (1.11) are called the Hamel–Novoselov equations. For their generation in the works [169] there are used general equation of mechanics (2.7) of Chapter IV and the definition of variations of coordinates δq σ =

∂ q˙σ ρ δπ , ∂v∗ρ

ρ, σ = 1, s ,

(1.12)

based on the analysis of the postulate of N. G. Chetaev (2.6) of Chapter IV. V. S. Novoselov calls equations (1.10) the equations of the type of S. A. Chaplygin since under the assumptions introduced by S. A. Chaplygin, they

198

VII. Equations of Motion in Quasicoordinates

result in the Chaplygin’s equations. For the same reason equations (1.11) are called by V. S. Novoselov the equations of the type of Voronets–Hamel and the coefficients Wλρ are called the Voronets–Hamel coefficients of the first kind. It is shown that these coefficients can be transformed into the relations   ∂ q˙σ d ∂v∗ρ ∂v∗ρ ρ Wλ = − σ , (1.13) λ = 1, l , ρ, σ = 1, s . ∂v∗λ dt ∂ q˙σ ∂q It follows that the quasivelocity is a true velocity in the case when the application of the Lagrange operator to the function v∗ρ (t, q, q) ˙ gives a zero. In the case of linear uniform stationary constraints from relations (1.3) and (1.13) we conclude that the coefficients Wλρ , λ = 1, l, ρ = 1, s, take the form   ρ ∂aσ ∂aρτ bσλ bτµ , − Wλρ = cρλµ v∗µ , cρλµ = ∂q τ ∂q σ (1.14) ρ, σ, τ = 1, s . λ, µ = 1, l , Then equations (1.11) are the following ∗ d ∂T ∗ ∂T ∗ ρ µ ∂T λ , − + c v =Q ρ ∗ λµ dt ∂v∗λ ∂π λ ∂v∗

λ, µ = 1, l ,

(1.15)

ρ = 1, s .

For l = s these equations and the relations for the coefficients cρστ , ρ, σ, τ = 1, s, as V. S. Novoselov [169, p. 55] remarks, "are obtained first by P. V. Voronets in 1901 [41] and then by G. Hamel [313] in 1904". Further V. S. Novoselov writes: "It should be remarked that before the work of Voronets was published, in 1901 in "Comptes rendus"the note of Poincar´e [373] was printed, where he obtains the equations highly close to equation"(1.15). The Poincar´e equations correspond to the case when in equations (1.15) for l = s the coefficients cρστ , ρ, σ, τ = 1, s, are constant and the forces are expressed via the forcing function U :  τ = bστ ∂U , Q ∂q σ

σ, τ = 1, s .

Thus, equations (1.15) can be represented in the form proposed by Poincar´e [149]: ∗ d ∂L∗ ρ σ ∂L = c v + Xτ L∗ , ρ στ ∗ dt ∂v∗τ ∂v∗

ρ, σ, τ = 1, s .

(1.16)

Here L∗ (q, v∗ ) = T ∗ + U is the Lagrange function and the quantities Xτ = bστ

∂ , ∂q σ

σ, τ = 1, s ,

(1.17)

199

1. The equivalence of different forms of equations

are linear differential operators. They, as L. M. Markhashov [149, p. 43] writes, ". . . make up the basis of a certain s-dimensional Lie algebra . . . "with the commutator [Xσ , Xτ ] = Xσ Xτ − Xτ Xσ = cρστ Xρ ,

ρ, σ, τ = 1, s .

(1.18)

The coefficients cρστ , ρ, σ, τ = 1, s, which appeared in the commutator, are called the structural constants of Lie algebra. In the same work on the next page L. M. Markhashov remarks: "the arbitrary chosen system, of s operators acting in s-dimensional space, for which only the condition det [bστ (q)] = 0 is valid, does not make up the Lie algebra . . . "since in this case in relations (1.18) the coefficients cρστ are the functions of q σ , σ = 1, s. Using a tangent space and the vectors ε ρ , ε τ , ρ, τ = 1, s, in it, we can represent relations (1.17) and (1.18), respectively, as Xτ = ε τ · ∇ ,   Xσ , Xτ = ε σ · ∇ (εετ · ∇ ) − ε τ · ∇ (εεσ · ∇ ) = cρστ ε ρ · ∇ = cρστ Xρ .

From this representation of the operators Xτ and their commutator it follows that they make up the closed system of operators [203] even if the coefficients cρστ are variable. Consider the contravariant components of the vector δy relative to the basis {εετ }, denoting them by δ ′ v∗ρ , i. e. assuming that δ ′ v∗ρ = δy · ερ . In this case we have δy = δ ′ v∗τ ε τ = δ ′ v∗τ bστ eσ = δq σ eσ , and therefore δq σ = bστ δ ′ v∗τ ,

σ, τ = 1, s .

Comparing these relations with relations (1.12), one can observe that in the works of V. S. Novoselov the quantities δ ′ v∗τ are denoted by δπ τ while in the works, devoted to the Poincar´e–Chetaev equations (see, for example, [203]), we have δ ′ v∗τ = ωτ , τ = 1, s . Let r (t, q) be a radius-vector of arbitrary point of mechanical system. Then we have δr =

∂r ∂r ′ τ δq σ = bστ δ v∗ = δ ′ v∗τ Xτ r . σ ∂q ∂q σ

(1.19)

Thus, using the operators Xτ , we can represent the virtual displacements δr, entering into the general equation of mechanics, in the form (1.19). Poincar´e made extraordinary discovery. He states that there exist mechanical systems such that their tangential space has a remarkable property. The introduced

200

VII. Equations of Motion in Quasicoordinates

in this space basis ε τ = bστ eσ , corresponding to quasivelocities, is given by the functions bστ of generalized coordinates, for which in commutator (1.18) the coefficients cρστ are constant. As is remarked earlier, in this case the operators Xτ make up the basis of the Lie algebra. A distinctive example of mechanical system with such remarkable property of tangent space is a rigid body, which goes round a fixed point. In this case the Poincar´e parameters are, in particular, the projections of the vector of instantaneous angle velocity on the principal axes of inertia of body and Poincar´e equations (1.16) become the dynamic Euler equations (see, for example, [203]). Consider now the case when linear transformations (1.3) are nonuniform and nonstationary. In this case from relations (1.13) and (1.14) it follows that equations (1.11) under both potential and nonpotential forces have the form ∗ ∂L∗ ∂L∗ d ∂L∗ ρ ∂L  − = cρµλ v∗µ ρ + c0λ ρ + Qλ , λ λ dt ∂v∗ ∂π ∂v∗ ∂v∗

λ, µ = 1, l ,

Here

ρ, σ = 1, s .

   ρ  ∂aρβ ∂aγ ∂bσ ∂bσ cρατ = aρσ bβα τβ − bβτ αβ = − bγα bβτ , ∂q ∂q ∂q β ∂q γ α, β, γ = 0, s ,

(1.20)

(1.21)

ρ, σ, τ = 1, s .

Two different representations of the coefficients cρατ result from that aργ bγβ = δβρ . Equations (1.20) are called the equations of nonholonomic systems in the Poincar´e–Chetaev variables [149, 203, 229] or the equations of motion of nonholonomic systems in quasicoordinates [28, 166]. N. G. Chetaev generalized Poincar´e equations (1.16) to the case when the number of Lagrangian coordinates exceeded the number of the independent Poincar´e parameters, i. e., making use of the Poincar´e approach, he obtained equations (1.15) provided that the coefficients cl+κ λµ = 0, κ = 1, k, and the ν coefficients cλµ , λ, µ, ν = 1, l, were constant. He remarked, however, that the equations found make a sense also for the variable coefficients cνλµ , λ, µ, ν = 1, l [248]. This generalization of the Poincar´e equations can be found also in the works of L. M. Markhashov, V. V. Rumyantsev, and Fam Guen [149, 203, 229]. Finally, we consider two simplest forms of expansion of scalar products in equations (1.1), proposed by Appell and Maggi. Introducing the Appell function T1 =

M W2 , 2

we have M W = M Wσ eσ =

∂T1 σ ∂T1∗ ρ e = ε . σ ∂ q¨ ∂ v˙ ∗ρ

201

2. The Poincar´e–Chetaev–Rumyantsev approach Then, using equations (1.1), we obtain Appell’s equations ∂T1∗ λ , =Q ∂ v˙ ∗σ

λ = 1, l .

Maggi’s equations   σ d ∂T ∂ q˙ ∂T − − Q = 0, σ dt ∂ q˙σ ∂q σ ∂v∗λ

λ = 1, l ,

are obtained from equations (1.1) if we take into account (1.4) and the following relation ∂ q˙σ ελ = eσ , λ = 1, l . ∂v∗λ The connection of Maggi’s equations with the Poincar´e–Chetaev ones is considered in the work of L. M. Markhashov [149]. In this work he writes (p. 46): "Poincar´e equations are obtained almost together with the main forms of equations of motion of nonholonomic systems. In spite of a great likeness for a long time the both theories were developed independently. The generalized Poincar´e–Chetaev equations proper for both holonomic and nonholonomic systems are obtained in the work . . . " [229]. In the work of V. V. Rumyantsev [203] the application of the Poincar´e–Chetaev equations to nonholonomic dynamics is considered from the new point of view. We especially stress that in the work [203] V. V. Rumyantsev extends first the approach of Poincar´e–Chetaev to nonlinear nonholonomic constraints and therefore equations (1.20) should be called the equations of Poincar´e–Chetaev– Rumyantsev. Recall that these equations and equations (1.10) and (1.11) in § 3 of Chapter II are obtained from Maggi’s equations. § 2. The Poincar´ e–Chetaev–Rumyantsev approach to the generation of equations of motion of nonholonomic systems In the previous section the Poincar´e–Chetaev–Rumyantsev equations (1.20) are obtained on the base of the vector form of the law of motion of mechanical systems with ideal constraints. Thus, there was given their geometric interpretation. However, the particular approach, used by the authors for generating these equations, was not brought to light. This approach deserves additional attention since it permits us to explain from the new point of view the reason why the equations of motion of nonholonomic systems cannot be represented in the form of Lagrange’s equations of the second kind without multipliers. Consider briefly this approach. Let on the motion of mechanical system the linear nonholonomic constraints be imposed which are given by the equations l+κ σ (t, q) = 0 , f1κ ≡ al+κ σ (t, q) q˙ + a0

κ = 1, k ,

σ = 1, s ,

l = s−k.

(2.1)

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VII. Equations of Motion in Quasicoordinates

Suppose, equations (2.1) are such that, using them for the introduction of the quasivelocities v∗ρ , ρ = 1, s, by formulas v∗λ = aλσ (t, q) q˙σ + aλ0 (t, q) , v∗l+κ

=

σ al+κ σ (t, q) q˙

+

λ = 1, l ,

al+κ (t, q) , 0

κ = 1, k ,

σ = 1, s ,

we obtain q˙σ = bστ (t, q) v∗τ + bσ0 (t, q) ,

σ, τ = 1, s ,

(2.2)

or in compact form β v∗α = aα β (t, q) q˙ ,

q0 = t ,

v∗0 = q˙0 = 1 ,

q˙β = bβα (t, q) v∗α , a0β = b0β = δβ0 ,

α, β = 0, s .

(2.3)

Here δβ0 are the Kronecker symbols. To generate the equations of motion of nonholonomic system we apply the generalized D’Alembert–Lagrange principle   ∂T d ∂T σ − − Q σ = 1, s , σ δq = 0 , dt ∂ q˙σ ∂q σ in which the quantities δq σ under constraints (2.1) must satisfy the conditions of N. G. Chetaev σ al+κ σ δq = 0 ,

κ = 1, k ,

σ = 1, s .

Making use of the through numeration µ = 1, 2, 3, ... for the notations of the Cartesian coordinates of the points of system and the projections of active forces applied to these points, we have ¨µ mµ x

d ∂T ∂T ∂xµ = − σ, ∂q σ dt ∂ q˙σ ∂q

Qσ = Xµ

∂xµ , ∂q σ

σ = 1, s .

This implies that the generalized D’Alembert–Lagrange principle can be represented as ∂xµ (mµ x ¨µ − Xµ ) σ δq σ = 0 . (2.4) ∂q We remark that if in the system there exist the rigid and elastic bodies, then the summing over all µ goes to integration. The quantities v∗ρ , ρ = 1, s, introduced by formulas (2.3), are called the Poincar´e–Chetaev parameters. These parameters in differential form are introduced in the following way δ ′ v∗ρ = aρσ δq σ ,

δq σ = bσρ δ ′ v∗ρ ,

ρ, σ = 1, s .

In this case from equations of constraint (2.1) we have σ δ ′ v∗l+κ = al+κ σ δq = 0 ,

κ = 1, k ,

203

2. The Poincar´e–Chetaev–Rumyantsev approach which are the conditions of N. G. Chetaev. These conditions imply that δq σ = bσλ δ ′ v∗λ ,

σ = 1, s ,

λ = 1, l .

Substituting the above relations into equation (2.4) and taking into account that the quantities δ ′ v∗λ , λ = 1, l, are arbitrary, we obtain (mµ x ¨ µ − Xµ )

∂xµ σ b = 0, ∂q σ λ

λ = 1, l .

In the case when there exist both potential and nonpotential forces these equations take the form   ∂U ∂xµ σ ¨ µ − Xµ − b = 0, λ = 1, l . (2.5) mµ x ∂xµ ∂q σ λ Here U is a forcing function. Introduce the following notations ∂ ∂ = Xλ = bσλ σ , λ ∂π ∂q

 λ = Xµ ∂xµ bσ , Q ∂q σ λ

σ = 1, s ,

λ = 1, l ,

and represent equations (2.5) as ¨µ mµ x

∂U ∂xµ λ , = +Q ∂π λ ∂π λ

λ = 1, l .

(2.6)

Compute the time derivative of the function f (t, q) according to formulas (2.2), (2.3). We obtain ∂f df ∂f ∂f = + σ q˙σ = v∗α Xα f = v∗α α , dt ∂t ∂q ∂π π0 = q0 = t , σ = 1, s , α = 0, s , where

(2.7)

∂ ∂ = Xα = bβα β , ∂π α ∂q

α, β = 0, s .

(2.8)

∂ ∂ = Xτ = bστ , τ ∂π ∂q σ

σ, τ = 1, s ,

(2.9)

We remark that

since b0τ = 0, τ = 1, s. Relations (2.7) yield, in particular, that x˙ µ = v∗α Xα xµ = v∗α

∂xµ , ∂π α

α = 0, s ,

(2.10)

204

VII. Equations of Motion in Quasicoordinates

and therefore

∂ x˙ µ ∂xµ = Xρ xµ , ρ = ∂v∗ ∂π ρ

ρ = 1, s .

(2.11)

Substituting the velocities x˙ µ , expressed via the Poincar´e–Chetaev parameters, into the relation for the kinetic energy of system mµ x˙ 2µ /2, we obtain the function T ∗ of variables t, q σ , v∗σ , σ = 1, s. This function is such that ∂T ∗ ∂ x˙ µ ∂xµ = mµ x˙ µ ρ = mµ x˙ µ ρ , ∂v∗ρ ∂v∗ ∂π

ρ = 1, s ,

(2.12)

∂T ∗ ∂ x˙ µ = mµ x˙ µ λ , λ = 1, l . (2.13) λ ∂π ∂π Taking into account relations (2.12), the left-hand side of equations (2.6) has the form   d ∂xµ ∂xµ d ∂xµ ¨µ λ = = mµ x mµ x˙ µ λ − mµ x˙ µ ∂π dt ∂π dt ∂π λ (2.14) d ∂xµ d ∂T ∗ = − mµ x˙ µ . dt ∂v∗λ dt ∂π λ Below we shall show that ∂ x˙ µ ∂ x˙ µ d ∂xµ = + cρατ v∗α , τ τ dt ∂π ∂π ∂v∗ρ

ρ, τ = 1, s ,

α = 0, s .

(2.15)

Here cρατ are certain unknown functions of the variables t and q σ , σ = 1, s. Relations (2.13)–(2.15) imply that, finally, equations (2.6) are the following ∗ ∗ d ∂L∗ ∂L∗ ρ ρ ∂L µ ∂L λ , − = c v + c +Q ρ ρ ∗ µλ 0λ dt ∂v∗λ ∂π λ ∂v∗ ∂v∗

L∗ = T ∗ + U ,

λ, µ = 1, l ,

(2.16)

ρ, σ = 1, s .

We shall show that relations (2.15) are valid and find the entering into them coefficients cρατ , α = 0, s, ρ, τ = 1, s. From relations (2.7), (2.10), and (2.11) we conclude that relations (2.15) are valid if ∂ 2 xµ ∂xµ ∂ 2 xµ = + cρατ , ∂π α ∂π τ ∂π τ ∂π α ∂π ρ i. e. in the case when   Xα , Xτ xµ = Xα Xτ xµ − Xτ Xα xµ = cρατ Xρ xµ , α = 0, s ,

ρ, τ = 1, s .

By formulas (2.8) and (2.9), we obtain   ∂ ∂xµ ∂ β ∂xµ Xα , Xτ xµ = bβα β bστ − bστ b . σ ∂q ∂q ∂q σ α ∂q β

(2.17)

205

2. The Poincar´e–Chetaev–Rumyantsev approach Since 2 ∂ 2 xµ σ β ∂ xµ = b b , τ α ∂q β ∂q σ ∂q σ ∂q β ∂b0α ∂δα0 = = 0, α, β = 0, s , σ, τ = 1, s , ∂q σ ∂q σ

bβα bστ

b0τ = 0 , we have

  Xα , Xτ xµ =



bβα

∂bστ ∂bσ − bβτ αβ β ∂q ∂q



∂xµ . ∂q σ

(2.18)

Represent the coefficients of ∂xµ /∂q σ in relation (2.18) as bβα

∂bστ ∂bσ − bβτ αβ = cρατ bσρ . β ∂q ∂q

(2.19)

Then from relations (2.18) and (2.9) we obtain that relations (2.17) and, therefore, relations (2.15) are valid. The coefficients aρσ are the elements of a matrix inverse to the matrix with the elements bσρ , ρ, σ = 1, s. Therefore from (2.19) we have   ∂bσ ∂bσ cρατ = aρσ bβα τβ − bβτ αβ , ∂q ∂q

ρ, σ, τ = 1, s ,

α, β = 0, s .

(2.20)

By relations (2.3), we obtain aρσ bστ = δτρ , Then aρσ

aργ bγα = δαρ ,

ρ, σ, τ = 1, s ,

∂bστ ∂aρσ = −bστ , γ ∂q ∂q γ ρ, σ, τ = 1, s ,

α, γ = 0, s .

∂aργ ∂bγα γ = −b , α ∂q β ∂q β α, γ = 0, s . aργ

Taking into account that ∂b0α = b0τ = 0 , ∂q β we have aρσ

α, β = 0, s ,

∂aρβ ∂bστ β = −bτ , ∂q γ ∂q γ ρ, σ, τ = 1, s ,

τ = 1, s ,

∂aργ ∂bσα γ = −b , α ∂q β ∂q β α, β, γ = 0, s . aρσ

(2.21)

Replacing in formulas (2.20) in the first double sum over all σ and β the dummy index β by γ and applying then relations (2.21), we obtain cρατ

=



∂aρβ ∂aργ − ∂q β ∂q γ



bγα bβτ ,

ρ, τ = 1, s ,

α, β, γ = 0, s .

(2.22)

206

VII. Equations of Motion in Quasicoordinates

Comparing equations (1.20) with equations (2.16) and relations (1.21) with relations (2.20) and (2.22), we conclude that equations (2.16) coincide with the Poincar´e–Chetaev–Rumyantsev equations from the previous section. In addition, they are obtained here with the usage of the technique suggested by these authors. The basic formulas for generating equations (2.16) are expressions (2.15), connected directly with commutator (2.17) introduced by Poincar´e. As was shown by Lagrange, in the case when the quantities π τ are the true coordinates q∗τ , τ = 1, s, the following relations ∂ x˙ µ d ∂xµ = , τ dt ∂q∗ ∂q∗τ

τ = 1, s ,

are satisfied. In the case of quasicoordinates these Lagrange identities are violated and it appears a correction, which is accounted by means of the coefficients cρατ . For their computation Yu. I. Neimark and N. A. Fufaev apply the so-called permutable relations [163, 166]. We assume, following the works of V. V. Dobronravov, V. S. Novoselov and Yu. I. Neimark, N. A. Fufaev, that δ ′ v∗ρ = δπ ρ ,

δ ′ q˙σ = δq σ ,

ρ, σ = 1, s .

Then, using relations (2.3), represented in differential form, we obtain δq σ = bσρ δπ ρ ,

δπ ρ = aρσ δq σ ,

ρ, σ = 1, s .

By definition, we have π0 = q0 = t ,

δπ 0 = δq 0 = δt = 0 , ρ = 1, s ,

dπ 0 = dq 0 = dt ,

dπ ρ = aργ dq γ ,

γ = 0, s ,

and δ dπ ρ =

∂aργ β γ δq dq + aργ δ dq γ , ∂q β ρ = 1, s ,

d δπ ρ =

∂aρβ

∂q γ β, γ = 0, s .

dq γ δq β + aρβ d δq β ,

Consider the difference δ dπ ρ − d δπ ρ ,

ρ = 1, s ,

and substitute into it the quantities δq β and dq γ , given in the form δq β = bβτ δπ τ ,

dq γ = bγα dπ α ,

τ = 1, s ,

α, β, γ = 0, s .

Then we obtain the following relations δ dπ ρ − d δπ ρ = cρατ dπ α δπ τ + aργ δ dq γ − aρβ d δq β , ρ, τ = 1, s ,

α, γ = 0, s .

(2.23)

207

3. The approach of J. Papastavridis

Here the quantities cρατ are given by formulas (2.22). Note that Yu. I. Neimark and N. A. Fufaev believe that the procedure of computation of the coefficients cρατ by means of the generation of permutable relations (2.23) is more simple than their direct computation by formulas (2.20) or (2.22).

§ 3. The approach of J. Papastavridis to the generation of equations of motion of nonholonomic systems At present, J. Papastavridis is one of the leading specialists in the field of nonholonomic mechanics. In his works [370] a new original approach to generating the equations of motion of nonholonomic systems is proposed. Consider briefly this approach, assuming, for the sake of generality, that all or certain equations of constraints ˙ = 0, ϕκ (t, q, q)

κ = 1, k ,

depend nonlinearly on velocities. We introduce quasivelocities by formulas v∗λ = ϕλ∗ (t, q, q) ˙ , v∗l+κ Assuming that

=

ϕl+κ (t, q, q) ˙ ∗

λ = 1, l , = ϕ (t, q, q) ˙ , κ



∂v∗ρ det  0, = ∂ q˙σ

l = s−k, κ = 1, k .

ρ, σ = 1, s ,

we have q˙σ = q˙σ (t, q, v∗ ) ,

σ = 1, s .

The reasoning of J. Papastavridis, as that of the other scholars, is based on the D’Alembert–Lagrange principle (2.4) and the conditions of N. G. Chetaev ∂ϕκ σ δq = 0 , ∂ q˙σ

σ = 1, s ,

σ = 1, s .

In accordance with the conditions of N. G. Chetaev the relation between quasivelocities and the generalized velocities, represented in differential form, is as follows δ ′ v∗ρ =

∂ϕρ∗ σ δq , ∂ q˙σ

δq σ =

∂ q˙σ ′ ρ δ v∗ , ∂v∗ρ

ρ, σ = 1, s .

In this case from the equations of constraints we have δq σ =

∂ q˙σ ′ λ δ v∗ , ∂v∗λ

λ = 1, l ,

σ = 1, s .

208

VII. Equations of Motion in Quasicoordinates

Substituting these relations into the generalized D’Alembert–Lagrange principle in the form (2.4) and taking into account that the quantities δ ′ v∗λ , λ = 1, l, are arbitrary, we obtain   ∂xµ ∂ q˙σ mµ x ¨ µ − Xµ = 0, ∂q σ ∂v∗λ

λ = 1, l ,

σ = 1, s .

(3.1)

In these equations the quantities x3ν−2 , x3ν−1 , x3ν are the Cartesian coordinates of the point, the position of which is given by the radius-vector rν = x3ν−2 i1 + x3ν−1 i2 + x3ν i3 . This point has the mass mν = mµ , µ = 3ν − 2, 3ν − 1, 3ν , and the active force, applied to it, is as follows Fν = X3ν−2 i1 + X3ν−1 i2 + X3ν i3 . Taking into account the above relation, equations (3.1) become   ∂rν ∂ q˙σ mν ¨rν − Fν · σ = 0, ∂q ∂v∗λ

λ = 1, l ,

σ = 1, s .

Replacing the summing over ν by the integration, we obtain

  ∂r ∂ q˙σ ¨r dm − dF · σ = 0, λ = 1, l , σ = 1, s . ∂q ∂v∗λ

(3.2)

Here r = r(t, q) is a radius-vector of elementary mass dm, acted by the active force dF. We remark that in equations (3.2) the notations follow to the surveys of J. Papastavridis [370. 1998]. Following this work, we introduce the vectors ∂r ∂ q˙σ ετ =   eσ = σ ,  eσ , σ, τ = 1, s , (3.3) ∂q ∂v∗λ which belong to not tangential space but to the usual Euclidean space, in which the motion of the mechanical system is considered. Introduce the notation ∂ q˙σ ∂ ∂ = , τ ∂π ∂v∗τ ∂q σ and represent the vectors  ε τ as  ετ =

∂r , ∂π τ

σ, τ = 1, s ,

τ = 1, s .

Then equations (3.2) take the form

∂r λ , ¨r · dm = Q ∂π λ

λ = 1, l ,

(3.4)

209

3. The approach of J. Papastavridis where λ = Q

∂r ∂ q˙σ · dF = Qσ λ , λ ∂π ∂v∗

Qσ =

∂r · dF , ∂q σ

λ = 1, l ,

σ = 1, s .

The functions under the integral in equations (3.4) can be represented in the following way ∂r  d  ∂r d ∂r ˙ ¨r · r · = , λ = 1, l . (3.5) − r˙ · ∂π λ dt ∂π λ dt ∂π λ

Taking into account that r˙ = we obtain

∂r α q˙ = q˙α  eα , ∂q α

q0 = t ,

 e0 =

∂r , ∂t

α = 0, s ,

∂ r˙ ∂ r˙ ∂ eα  , e˙ σ = σ = q˙α σ , ∂ q˙σ ∂q ∂q ∂ r˙ ∂ q˙σ ∂ q˙ρ ∂ q˙σ ˙ ∂ q˙σ ∂ eα ∂ q˙σ    = q˙α σ + ρ eσ = eσ , eσ + λ λ λ λ ∂π ∂q ∂v∗ ∂q ∂v∗ ∂v∗ ∂π λ  eσ =

α = 0, s ,

σ = 1, s ,

λ = 1, l .

On the other hand, we have     d ∂ q˙σ d ∂ q˙σ ∂ q˙σ ˙ d ∂r    = eσ = eσ , eσ + dt ∂π λ dt ∂v∗λ ∂v∗λ dt ∂v∗λ

λ = 1, l ,

∂ r˙ d ∂r ∂ r˙ = + Tλσ σ , λ λ dt ∂π ∂π ∂ q˙

σ = 1, s ,

σ = 1, s .

Therefore

where Tλσ =

λ = 1, l ,

d ∂ q˙σ ∂ q˙σ − . dt ∂v∗λ ∂π λ

(3.6)

Then, from the relations ∂ r˙ ∂ q˙σ ∂r ∂r = = , λ ∂v∗ ∂v∗λ ∂q σ ∂π λ

λ = 1, l ,

it follows that relations (3.5) have the form ¨r ·

∂(˙r2 /2) d ∂(˙r2 /2) ∂(˙r2 /2) ∂r = − − Tλσ , λ λ λ ∂π dt ∂v∗ ∂π ∂ q˙σ λ = 1, l ,

(3.7)

σ = 1, s .

Substituting these relations into equations (3.4), we obtain ∂T ∗ ∂T d ∂T ∗ λ , − − σ Tλσ = Q λ dt ∂v∗ ∂π λ ∂ q˙

λ = 1, l ,

σ = 1, s .

(3.8)

210

VII. Equations of Motion in Quasicoordinates

Here T ∗ is a kinetic energy of system expressed in terms of quasivelocities. Taking into account relations (3.3), we represent the sum Tλσ in the following way

∂ r˙ eσ = Tλσ  ∂ q˙σ

Tλσ  eσ = Tλσ

Since

we obtain

∂v∗ρ  ερ . ∂ q˙σ

∂ q˙σ ∂r ∂ q˙σ ∂ r˙  = ερ , eσ =  ρ = ρ ∂v∗ ∂v∗ ∂q σ ∂v∗ρ Tλσ  eσ = −Wλρ

Here

Wλρ = −

∂ r˙ . ∂v∗ρ

∂v∗ρ σ T . ∂ q˙σ λ

(3.9)

Finally, relations (3.7) are the following: ¨r ·

∂r d ∂(˙r2 /2) ∂(˙r2 /2) ∂(˙r2 /2) = − + Wλρ , λ λ λ ∂π dt ∂v∗ ∂π ∂v∗ρ

λ = 1, l ,

ρ = 1, s ,

and equations (3.4) take the form d ∂T ∗ ∂T ∗ ∂T ∗ ρ λ , − + W =Q λ λ dt ∂v∗ ∂π ∂v∗ρ λ

λ = 1, l ,

ρ = 1, s .

We shall show that relations (3.9) can be represented as   ∂ q˙σ d ∂v∗ρ ∂v∗ρ ρ − σ . Wλ = ∂v∗λ dt ∂ q˙σ ∂q Formulas (3.9) and (3.6) yield the relation   ∂v∗ρ d ∂ q˙σ ∂ q˙σ . − Wλρ = − σ ∂ q˙ dt ∂v∗λ ∂π λ Since

we obtain

(3.10)

(3.11)

(3.12)

∂v∗ρ ∂ q˙σ = δλρ , ∂ q˙σ ∂v∗λ 

d ∂v∗ρ dt ∂ q˙σ



∂v∗ρ d ∂ q˙σ ∂ q˙σ = − . ∂v∗λ ∂ q˙σ dt ∂v∗λ

(3.13)

Since the function v∗ρ (t, q, q(t, ˙ q, v∗ )) is identically equal to v∗ρ , we find ∂v∗ρ ∂ q˙τ ∂v∗ρ + τ = 0, σ ∂q ∂ q˙ ∂q σ

ρ, σ, τ = 1, s ,

3. The approach of J. Papastavridis

211

and therefore ∂v∗ρ ∂ q˙σ ∂ q˙τ ∂v∗ρ ∂ q˙τ ∂ q˙σ ∂v∗ρ ∂ q˙σ ∂v∗ρ ∂ q˙σ = = = − . ∂ q˙σ ∂π λ ∂ q˙σ ∂q τ ∂v∗λ ∂ q˙τ ∂q σ ∂v∗λ ∂q σ ∂v∗λ

(3.14)

From relations (3.12)–(3.14) it follows that, in fact, the coefficients Wλρ can be represented in the form (3.11). Equations (3.8) and (3.10) coincide with equations (1.10) and (1.11), respectively. Recall that in the case when a time does not enter, in explicit form, into both the kinetic energy and the equations of constraints equations (1.10) and (1.11) have been obtained by G. Hamel [314] in 1938 and in the general case by V. S. Novoselov [169] in 1957. In 1998 V. V. Rumyantsev [203] obtained these equations, having generalized the Poincar´e and Chetaev equations. He states [203, p. 57] that these equations ". . . can be regarded as the general equations of the classical mechanics, involving as special cases all known equations of motion". Equations (1.10) and (1.11) pass to one another and they are represented in the first and second forms in quasicoordinates. Therefore they can be called equations of motion in quasicoordinates. For linear nonholonomic constraints and for potential and nonpotential forces, these equations, as was shown above, pass to the Poincar´e–Chetaev–Rumyantsev equations (1.20). Since equations (1.10), (1.11) coincide with equations (3.8), (3.10), respectively, and equations (1.20) coincide with equations (2.16), we can say that in the present chapter the Poincar´e–Chetaev–Rumyantsev equations (1.20) were obtained by three different methods. Their generation is based on the vector form of the law of motion with ideal constraints in § 1 and on the D’Alembert–Lagrange principle and the conditions of N. G. Chetaev in § § 2 and 3. In addition, in § 2 the technique of Poincar´e–Chetaev–Rumyantsev was used and in § 3 the technique of Papastavridis.

APPENDICES

APPENDIX

A

THE METHOD OF CURVILINEAR COORDINATES In Appendix A the kinematics of point in curvilinear coordinates is considered. The formulas obtained are extended to the motion of any of mechanical systems. The theory, given in the Appendix, is widely used in studying the base material of the monograph.

§ 1. The curvilinear coordinates of point. Reciprocal bases Suppose, the position of the point M in three-dimensional space is defined by the radius-vector r = r(q 1 , q 2 , q 3 ), i. e. the Cartesian coordinates of point x1 , x2 , x3 are uniquely represented via the quantities q 1 , q 2 , q 3 : xk = Fk (q 1 , q 2 , q 3 ) , If

  ∂x1  1  ∂q D(x1 , x2 , x3 )  ∂x2 = 1  ∂q D(q 1 , q 2 , q 3 )  ∂x3   ∂q 1

k = 1, 2, 3 . ∂x1 ∂q 2 ∂x2 ∂q 2 ∂x3 ∂q 2

(A.1)

 ∂x1   ∂q 3  ∂x2   = 0 , ∂q 3  ∂x3  ∂q 3 

then system of equations (A.1) is solvable for q 1 , q 2 , q 3 : q σ = f σ (x1 , x2 , x3 ) ,

σ = 1, 2, 3 ,

(A.2)

and the quantities q 1 , q 2 , q 3 are called the curvilinear coordinates of point in space. From relations (A.2) it follows directly that, equating any curvilinear coordinate q σ to the constant quantity Cσ , we obtain the equation of coordinate surface f σ (x1 , x2 , x3 ) = Cσ , σ = 1, 2, 3 . The crossing of two coordinate surfaces gives a coordinate line, along which one coordinate is varied only. For example, the crossing of the coordinate surfaces q 1 = f 1 (x1 , x2 , x3 ) = C1 and q 2 = f 2 (x1 , x2 , x3 ) = C2 gives a coordinate line, which the coordinate q 3 varies along (Fig. A. 1). 213

214

Appendix A

Fig. A. 1

The crossing of the coordinate lines q 1 , q 2 , q 3 is the point M . If through this point we construct the tangents to the coordinate lines in ascending order of the quantities q 1 , q 2 , q 3 , then we obtain the axes of curvilinear coordinates, which can make up as orthogonal (for example, the axes of spherical or cylindrical coordinates) as nonorthogonal systems. For the motion to be given in curvilinear coordinates it is necessary that the quantities q 1 , q 2 , q 3 are given as time functions: q σ = q σ (t) , σ = 1, 3 . (A.3) These functions are called equations of motion of point. Taking into account that the radius-vector r = r(q 1 , q 2 , q 3 ) of the point M is a differentiable function, we obtain dr =

3  ∂r σ dq . σ ∂q σ=1

Denoting eσ =

∂r , ∂q σ

we have dr =

3 

σ = 1, 3 ,

eσ dq σ .

(A.4)

(A.5)

σ=1

Note that |∂r/∂q σ | = |eσ | = Hσ , where Hσ are scale factors Lam´e. Using formulas (A.4), we obtain  2  2  2 ∂x1 ∂x2 ∂x3 Hσ = + + , σ = 1, 3 . (A.6) ∂q σ ∂q σ ∂q σ Formula (A.5) gives a decomposition of the vector dr using the axes of the curvilinear system of coordinates {q σ } with the basis {eσ }. The quantities

215

Appendix A

dq σ from relations (A.5) are called contravariant components of the vector dr. A set of the vectors {eσ } is called a natural or fundamental basis of the curvilinear system of coordinates {q σ } at the point M . The tangential planes to the coordinate surfaces at the point M are called coordinate planes. They pass through the corresponding vectors of basis. For example, the tangential plane to the surface q 3 = C3 passes through the vectors e1 and e2 . Denote by eτ a certain vector collinear to a vector of normal to the coordinate surface q τ = Cτ at the point M . Obviously, the system of all vectors {eτ } also makes up a certain basis. For the definition of that the basis {eτ } is unique we need  1, σ=τ, τ τ (A.7) e · eσ = δσ = 0, σ = τ . Here δστ are the Kronecker symbols. The basis {eτ } is called a reciprocal or dual basis relative to the fundamental one. The reciprocal basis can also be introduced by using the gradient operation (see the next section). Note that for any fundamental basis there exists a unique reciprocal basis and if the fundamental basis is orthonormal, then the reciprocal basis coincides with the fundamental one.

§ 2. The relation between a reciprocal basis and gradients of scalar functions We assume that we have the certain function f (x1 , x2 , x3 ) in Cartesian coordinates of a point and this function can be represented in the curvilinear coordinates: f (q 1 , q 2 , q 3 ). The differential of this function in Cartesian coordinates is as follows 3  ∂f dxk df = ∂xk

(A.8)

k=1

and in curvilinear coordinates 3  ∂f σ df = dq . σ ∂q σ=1

The gradient of the function f is the vector grad f =

3  ∂f ik . ∂xk

k=1

(A.9)

216

Appendix A

If we introduce Hamiltonian operator nabla ∇=

3  ∂ ik , ∂xk

(A.10)

k=1

then the gradient of the function f  takes the form grad f = ∇ f . 3 Taking into account that dr = k=1 dxk ik , relation (A.8) can be represented as the scalar product df = ∇ f · dr .

(A.11)

The question arises how in place of the formulae (A.10), which is valid for Cartesian coordinates, to find a relation for the vector ∇ in curvilinear coordinates in such a way that the derivative df can be represented in the form (A.11) with dr in the form (A.5)? Substituting relation (A.5) into (A.11) and comparing with (A.9), we obtain the relation ∂f (A.12) ∇ f · eσ = σ . ∂q It is easily checked that relation (A.12) is valid if ∇f =

3  ∂f τ e . ∂q τ τ =1

(A.13)

Representation (A.13) is convenient to obtain the vectors of reciprocal basis. Really, using the concrete coordinate surface of the form (A.2) and taking into account relation (A.13), we have ∇ f σ = grad f σ =

3  ∂f σ

τ =1

∂q τ

eτ = eσ .

§ 3. Covariant and contravariant components of vector To the curvilinear system of coordinates q σ , σ = 1, 2, 3, correspond both the fundamental basis eσ = ∂r/∂q σ , σ = 1, 2, 3, and the reciprocal basis eτ = ∇ f τ , τ = 1, 2, 3. Any of the vectors a can be decomposed into as fundamental as reciprocal bases, i. e. can be represented in the form a=

3 

aσ eσ ,

σ=1

a=

3 

aτ eτ .

(A.14)

τ =1

Here aσ are contravariant components of the vector a and aτ are covariant components of the vector a in the basis {eτ }. Further, we make use of the rule of dummy index, summation of repeated indices in the corresponding limits is implied. Then from formulas (A.14) and (A.7) we obtain a · eσ = aτ eτ · eσ = aτ δτσ = aσ ,

a · eσ = aτ eτ · eσ = aτ δστ = aσ .

217

Appendix A

Thus, we find the simple formulas to obtain the components of the arbitrary vector a decomposed into considered bases, respectively: aσ = a · eσ ,

aσ = a · eσ .

(A.15)

Relations (A.14) and (A.15) yield the rules of raising an index and missing an index: aσ = a · eσ = aτ eτ · eσ = g τ σ aτ ,

aσ = a · eσ = aτ eτ · eσ = gτ σ aτ . (A.16)

Here gστ = gτ σ = eσ · eτ , σ, τ = 1, 2, 3, are elements of basic metric form or basic metric tensor and g στ = g τ σ = eσ · eτ , σ, τ = 1, 2, 3, are components of complementary metric form or complementary metric tensor. Applying the metric tensor, it is not difficult to obtain the transition formulas from the fundamental basis to the reciprocal one and vice versa: eσ = (eσ · eτ )eτ = gστ eτ ,

eσ = (eσ · eτ )eτ = g στ eτ .

Note that if two vectors, represented in the same bases, are multiplied scalarly by each other, then the obtained relation turns out rather lengthy (nine addends): a · b = gστ aσ bτ = g στ aσ bτ . If the vectors are decomposed into the different bases, then the scalar product involves only three addends: a · b = aσ bτ eσ · eτ = aσ bτ δστ = aσ bσ = aσ bσ . From the chain of relations aσ = a · eσ = |a||eσ | cos ϕ = |eσ |preσ a , aτ = a · eτ = |a||eτ | cos ψ = |eτ |preτ a we obtain the formulas for computing the projections of the vector a on the vectors of fundamental and reciprocal bases: preσ a =

aσ , |eσ |

preτ a =

aτ . |eτ |

(A.17)

§ 4. Covariant and contravariant components of velocity vector Let now the motion of the point M be considered in the curvilinear system of coordinates, in which case equations of motion (A.3) are known. By definition, the velocity is given by the vector v = dr/dt and therefore we have v=

∂r dr = σ q˙σ = q˙σ eσ . dt ∂q

218

Appendix A

At the same time the velocity vector in fundamental basis can be represented as v = v σ eσ . Then, comparing with the previous formula, we obtain the following representations for the contravariant components of velocity vector: v σ = q˙σ ,

σ = 1, 2, 3 .

(A.18)

For the orthogonal fundamental basis {eσ } the relation for the modulus of velocity vector has the form (A.19) v = |v| = (v σ eσ )2 = (H1 q˙1 )2 + (H2 q˙2 )2 + (H3 q˙3 )2 , where, as we have shown earlier, Hσ are scale factors. According to formulas (A.16) and (A.18) the covariant components of velocity vector are the following vσ = gστ v τ = gστ q˙τ ,

σ = 1, 2, 3 .

(A.20)

Consider another possible representation of the components vσ . For this purpose it is convenient to introduce the function T1 =

v2 , 2

(A.21)

which can be regarded as the kinetic energy of point with unit mass, what is marked by index "1". Function (A.21) can be rewritten as T1 =

1 1 1 v · v = q˙σ eσ · q˙τ eτ = gστ q˙σ q˙τ . 2 2 2

(A.22)

By relation (A.22), formula (A.20) can be represented now in the form vσ =

∂T1 . ∂ q˙σ

(A.23)

As will be shown below, the function T1 , given by formula (A.21), plays an important role in computing the covariant components of acceleration vector of point. § 5. Christoffel symbols In § 1 we introduce the vectors of fundamental basis eσ =

∂r , ∂q σ

σ = 1, 2, 3 ,

which show the changes of radius-vector of point versus the changes of generalized coordinates. Let us study now the effect of coordinates q τ , τ = 1, 2, 3, on the vector eσ , σ = 1, 2, 3. For this purpose we consider the derivatives ∂eσ , ∂q τ

σ, τ = 1, 2, 3 .

(A.24)

219

Appendix A

Since the vector can be represented in one of the forms of (A.14), where the covariant and contravariant components are computed by formulas (A.15), for the sought vectors we obtain     ∂eσ ∂eσ ρ ∂eσ ∂eσ ρ e eρ , = · e , = · e ρ ∂q τ ∂q τ ∂q τ ∂q τ ρ, σ, τ = 1, 2, 3 .

The covariant and contravariant components of vectors (A.24) in the above relation are called the Christoffel symbols of the first and second kinds and are denoted by Γρ,στ and Γρστ , respectively. Thus, we have Γρ,στ =

∂eσ · eρ , ∂q τ

Γρστ =

∂eσ ρ ·e . ∂q τ

Finally, the previous relations take the form: ∂eσ ∂eσ = Γρ,στ eρ , = Γρστ eρ , ∂q τ ∂q τ ρ, σ, τ = 1, 2, 3 . Obviously, by formulas (A.16) the Christoffel symbols are related as Γρστ = g ρπ Γπ,στ ,

Γρ,στ = gρπ Γπστ ,

π, ρ, σ, τ = 1, 2, 3 .

(A.25)

Represent the Christoffel symbols of the first kind via the elements of basic metric tensor. Assuming that the mixed second derivatives are continuous in the coordinates of radius-vector of point, we obtain the chain of relations ∂eσ ∂2r ∂2r ∂eτ = = = σ. ∂q τ ∂q σ ∂q τ ∂q τ ∂q σ ∂q Applying this formula twice, we can perform the following transformations:   ∂eσ 1 ∂eσ ∂eτ Γρ,στ = · e = · e + · e = ρ ρ ρ ∂q τ 2 ∂q τ ∂q σ   ∂eρ 1 ∂(eσ · eρ ) ∂(eτ · eρ ) ∂eρ + − τ · eσ − σ · eτ = = 2 ∂q τ ∂q σ ∂q ∂q   ∂eσ 1 ∂(eσ · eρ ) ∂(eτ · eρ ) ∂eτ + − ρ · eσ − ρ · eτ . = 2 ∂q τ ∂q σ ∂q ∂q This implies the formula for computing the Christoffel coefficients of the first kind:   1 ∂gρσ ∂gρτ ∂gστ . (A.26) + − Γρ,στ = 2 ∂q τ ∂q σ ∂q ρ

220

Appendix A

The Christoffel coefficients of the second kind can be determined, in turn, by formulas (A.25).

§ 6. Covariant and contravariant components of acceleration vector. The Lagrange operator Represent the acceleration vector in the following way: w=

deρ dv d = (v ρ eρ ) = v˙ ρ eρ + v ρ . dt dt dt

We have

∂eρ deρ (q) = σ q˙σ , dt ∂q

and therefore this formula can be represented as w = v˙ ρ eρ + v ρ v σ

∂eρ . ∂q σ

(A.27)

Multiplying scalarly the above relation by the vectors eπ , we obtain the contravariant components of acceleration vector: wπ = q¨π + Γπρσ q˙ρ q˙σ . Multiplying scalarly (A.27) by the vectors eπ , we obtain the covariant components of acceleration vector: wπ = gπρ q¨ρ + Γπ,ρσ q˙ρ q˙σ .

(A.28)

Now we proceed to the obtaining of the second representation of covariant components of acceleration. We write the acceleration vector as: w=

dv d deτ = (vτ eτ ) = v˙ τ eτ + vτ . dt dt dt

However, since

deτ (q) ∂eτ = σ q˙σ , dt ∂q

this formula takes the form w = v˙ τ eτ + vτ v σ

∂eτ . ∂q σ

Multiplying scalarly this relation on the vectors eρ , we obtain wρ = v˙ ρ + vτ v σ

∂eτ · eρ . ∂q σ

(A.29)

221

Appendix A

Consider the last scalar product in formula (A.29). By the property of the vectors of reciprocal bases (A.7), we have eτ · eρ = δρτ = const . Then

∂(eτ · eρ ) = 0. ∂q σ

Hence we obtain

∂eτ ∂eρ · eρ = − σ · eτ = −Γτρσ . ∂q σ ∂q

It follows that formula (A.29) can be rewritten as wρ = v˙ ρ − Γτρσ vτ v σ .

(A.30)

However, by the rule of raising an index (A.16) we have vτ = gτ π v π ,

Γπ,ρσ = gπτ Γτρσ .

Therefore formula (A.30) becomes wρ = v˙ ρ − Γπ,ρσ v π v σ .

(A.31)

Relations (A.26) give Γπ,ρσ v π v σ =

  1 ∂gπρ ∂gπσ ∂gρσ π σ + − v v . 2 ∂q σ ∂q ρ ∂q π

If in the right-hand side of this relation in the first double sum we interchange the summation indices π and σ, then this sum coincides with the last double sum, given with the minus sign. In this case, collecting terms, we obtain Γπ,ρσ v π v σ =

1 ∂gπσ π σ ∂T1 v v = , 2 ∂q ρ ∂q ρ

(A.32)

where T1 is a function, introduced by formula (A.21). Recall that according to (A.23) the covariant components of velocity vector are also represented by this function. Therefore we have v˙ ρ =

d ∂T1 . dt ∂ q˙ρ

(A.33)

Using relations (A.32) and (A.33), from formula (A.31) we find the final second representation of a covariant component of acceleration vector of point, namely d ∂T1 ∂T1 − ρ. (A.34) wρ = dt ∂ q˙ρ ∂q

222

Appendix A

Fig. A. 2

Introducing the Lagrange operator Lρ =

d ∂ ∂ − ρ, ρ dt ∂ q˙ ∂q

we rewrite representation (A.34) as wρ = Lρ (T1 ) . The projections of acceleration on the vectors of fundamental basis can be found by formulas (A.17): preρ w =

Lρ (T1 ) . Hρ

§ 7. The case of cylindrical system of coordinates As an example of application of the found formulas we consider the cylindrical system of coordinates q 1 = ρ, q 2 = ψ, q 3 = z (Fig. A. 2). The Cartesian coordinates of point x, y, z are represented via the cylindrical coordinates in the following way: x = ρ cos ψ ,

y = ρ sin ψ ,

z = z.

(A.35)

To the constant values of the generalized coordinates ρ = C1 ,

ψ = C2 ,

z = C3

correspond the coordinate surfaces, passing through the point M (C1 , C2 , C3 ) (Fig. A. 2): a vertical cylinder of radius ρ; a vertical plane, which makes the

223

Appendix A

angle ψ with the plane Oxz; a horizontal plane, raised off Oxy by z. The crossing of these coordinate surfaces at the point M gives the coordinate lines: the horizontal straight line O1 M , the vertical straight line N M , and the circle of radius ρ centered at the point O1 . The vectors of fundamental basis, which in accordance with formulas (A.4) and (A.35) have the form e1 = eρ = e2 = eψ =

∂r = cos ψi + sin ψj , ∂ρ

∂r = −ρ sin ψi + ρ cos ψj , ∂ψ

(A.36)

∂r = k, ∂z are directed along the tangents to the coordinate curves in ascending order of the corresponding curvilinear coordinates. These vectors, as is shown in Fig. A. 2, make up an orthogonal but unnormalized system since by formulas (A.6) and (A.35) we have e3 = ez =

Hρ = 1 ,

Hψ = ρ, ,

Hz = 1 .

(A.37)

The orthogonality of fundamental basis can also be established analytically. Really, formulas (A.36) imply that e1 · e2 = −ρ cos ψ sin ψ + ρ sin ψ cos ψ = 0 , e1 · e3 = 0, e2 · e3 = 0 . The reciprocal basis eρ ,eψ ,ez coincides in directions with the fundamental one and has the lengths |eρ | = 1 ,

|eψ | =

1 , ρ

The found bases permit us to construct  1 0 (gστ ) = 0 ρ2 0 0

|ez | = 1 . the matrix of basic metric tensor  0 0 , (A.38) 1

and the matrix of complementary metric tensor   1 0 0 (g στ ) = 0 1/ρ2 0 . 0 0 1

(A.39)

It is easily seen that the product of matrix (A.38) by matrix (A.39) is the unit matrix. For the computation of the Christoffel symbols of the first kind

224

Appendix A

we make use of formula (A.26). Since in matrix (A.38) the variable element is g22 = gρρ = ρ2 only, then the only nonzero symbols are the following Γ2,21 = Γ2,12 = −Γ1,22 = ρ .

(A.40)

The Christoffel symbols of the second kind can be computed now by formulas (A.25), using the elements of matrix (A.39). Formulas (A.18) and (A.19) give v 2 = v ψ = ψ˙ , v 3 = v z = z˙ ,  ˙ 2 + z˙ 2 , v = |v| = ρ˙ 2 + (ρψ)

v 1 = v ρ = ρ˙ ,

therefore

(A.41)

1 2 ˙ 2 + z˙ 2 ) . (ρ˙ + (ρψ) (A.42) 2 If contravariant components of velocity (A.41) are known, the covariant components can be obtained by formulas (A.20): T1 =

v1 = g1τ v τ = v 1 = ρ˙ , v2 = g2τ v τ = ρ2 v 2 = ρ2 ψ˙ ,

(A.43)

v3 = g3τ v τ = v 3 = z˙ . The covariant components of accelerations can be found by means of representations (A.28). Since for the cylindrical system of coordinates the nonzero Christoffel symbols of the first kind are the symbols, given by formulas (A.40) only, we have w1 = g1σ q¨σ + Γ1,στ q˙σ q˙τ = q¨1 + Γ1,22 q˙2 q˙2 = ρ¨ − ρψ˙ 2 , w2 = g2σ q¨σ + Γ2,στ q˙σ q˙τ = ρ2 q¨2 + Γ2,12 q˙1 q˙2 + Γ2,21 q˙2 q˙1 = ρ2 ψ¨ + 2ρψ˙ ρ˙ , w3 = g3σ q¨σ + Γ3,στ q˙σ q˙τ = q¨3 = z¨ . (A.44) Note that it is rather convenient to determine the covariant components of velocity and acceleration, applying the functions T1 . Really, using formulas (A.23) with provision for (A.42) we can compute at once the covariant components of velocity (A.43) obtained above: ∂T1 = ρ˙ , ∂ ρ˙ ∂T1 v2 = vψ = = ρ2 ψ˙ , ∂ ψ˙ v 1 = vρ =

v 3 = vz =

∂T1 = z˙ , ∂ z˙

225

Appendix A

and by formulas (A.34) the covariant components of acceleration, which coincide with relations (A.44): ∂T1 d d ∂T1 − = ρ˙ − ρψ˙ 2 = ρ¨ − ρψ˙ 2 , dt ∂ ρ˙ ∂ρ dt ∂T1 d d ∂T1 ˙ = ρ2 ψ¨ + 2ρψ˙ ρ˙ , − = (ρ2 ψ) w2 = wψ = ˙ dt ∂ ψ ∂ψ dt ∂T1 d ∂T1 w3 = wz = − = z¨ . dt ∂ z˙ ∂z w1 = wρ =

The projections of velocity and acceleration are obtained by formulas (A.17), taking into account lengths (A.37) of the vectors of fundamental basis: preρ v = ρ˙ , ˙2

preρ w = ρ¨ − ρψ ,

preψ v = ρψ˙ ,

prez v = z˙ ,

preψ w = ρψ¨ + 2ψ˙ ρ˙ ,

prez w = z¨ .

§ 8. Covariant components of acceleration vector for nonstationary basis Consider now a more general case when the radius-vector r depends not only on q = (q 1 , q 2 , q 3 ) but on time t, i. e. it is the function of the form r = r(t, q). In particular, this is possible in the case when the curvilinear coordinates q σ give the position of points relative to the system of coordinates Ox1 x2 x3 , which has a given motion relative to the stationary (absolute) system of coordinates O1 ξ1 ξ2 ξ3 . In this case even for the fixed values of q σ the radius-vector r varies in time in virtue of the translational motion of system Ox1 x2 x3 . The absolute velocity v is computed by formula v = r˙ =

∂r ∂r + σ q˙σ . ∂t ∂q

(A.45)

Introducing, for short, the notation q 0 = t (therefore q˙0 = 1), we can express velocity (A.45) in the following way: v=

∂r α q˙ , ∂q α

α = 0, 3 .

(A.46)

We emphasize that such a representation is introduced only for short and therefore we do not need to consider the problem in four-dimensional space. The coordinate vectors are, as before, only the vectors eσ (t, q) =

∂r , ∂q σ

σ = 1, 3 .

226

Appendix A

Thus, the nonstationary basis varies not only with the change from point to point but at each point in a time. Compute a covariant component of the acceleration w:   dv ∂r d d ∂r ∂r · π = . (A.47) wπ = w · eπ = v· π −v· dt ∂q dt ∂q dt ∂q π Differentiating first relation (A.45) with respect to q˙π and then with respect to q π (π = 1, 2, 3), we have ∂v ∂r = π, ∂ q˙π ∂q

∂v ∂2r ∂2r d ∂r = + σ π q˙σ = . π π ∂q ∂t∂q ∂q ∂q dt ∂q π

It follows that the addends, entering into relation (A.47), take the form ∂v 1 ∂v 2 ∂T1 ∂r =v· π = = π, π ∂q ∂ q˙ 2 ∂ q˙π ∂ q˙ ∂v 1 ∂v 2 ∂T1 d ∂r v· =v· π = = π. π π dt ∂q ∂q 2 ∂q ∂q v·

Finally, for wπ we obtain wπ =

d ∂T1 ∂T1 − π, dt ∂ q˙π ∂q

T1 =

v2 , 2

π = 1, 3 .

(A.48)

Thus, Lagrange’s form of representation of the covariant component wπ does not change also in the case of nonstationary basis. According to representation (A.46) the kinetic energy T1 of a point with unit mass is as follows T1 =

v2 ∂r α β 1 1 ∂r · q˙ q˙ = gαβ q˙α q˙β , = 2 2 ∂q α ∂q β 2

α, β = 0, 3 .

(A.49)

If in relation (A.49) we discriminate the addends, involving explicitly ∂r/∂q 0 = ∂r/∂t, then we have (2)

(1)

(0)

T1 = T1 + T1 + T 1 , 1 ∂r ∂r ρ σ 1 (2) · σ q˙ q˙ = gρσ q˙ρ q˙σ , T1 = ρ 2 ∂q ∂q 2 ∂r ∂r σ (1) · T1 = q˙ = g0σ q˙σ , ∂t ∂q σ   1 ∂r 1 (0) T1 = = g00 . 2 ∂t 2

(A.50)

Note that in formulas (A.50) the metric coefficients are the quantities gρσ , (2) ρ, σ = 1, 3, entering into the relation T1 only.

227

Appendix A

By formulas (A.48), (A.49) the covariant components of acceleration vector in expanded form are the following wπ = gπρ q¨ρ + Γπ,αβ q˙α q˙β ,

π, ρ = 1, 3 ,

α, β = 0, 3 .

(A.51)

This formula is the extension of the first representation of covariant component of acceleration (A.28) to the case of nonstationary basis. We emphasize that like the previous remark, in formula (A.51) the Christoffel symbols ourselves are only the following Γπ,ρσ =

∂eρ · eπ , ∂q σ

π, ρ, σ = 1, 3 ,

and by the use of the vector e0 = ∂r/∂t, the quantities Γπ,ρ0 , Γπ,00 denote only the functions ∂eρ ∂2r · eπ = ρ · eπ , ∂t ∂q ∂t ∂e0 ∂2r · eπ = 2 · eπ , = ∂t ∂t π, ρ = 1, 3 .

Γπ,ρ0 = Γπ,00

They are introduced here for brevity of notation and allow us to obtain in the case of nonstationary basis the formulas similar to those in the stationary case. § 9. Covariant components of a derivative of vector In Chapter IV the relations for the covariant components of derivatives of vector are used. We obtain here the corresponding formulas for the vector a of arbitrary physical structure. Recall that in § 6 of this Appendix they already have been obtained as a result of the differentiation of velocity vector. Consider the representation of the vector a in reciprocal basis: a = aτ eτ . Find the vector b, which is a derivative of the vector a: b = a˙ = a˙ τ eτ + aτ

deτ . dt

Since we have

deτ (t, q) ∂eτ = α q˙α , dt ∂q the previous formula takes the form b = a˙ τ eτ + aτ q˙α

∂eτ . ∂q α

228

Appendix A

Multiplying this relation scalarly by the vectors eρ , we get bρ = a˙ ρ + aτ q˙α

∂eτ · eρ . ∂q α

Arguing as in § 6, we find ∂eτ · eρ = −Γτρα , ∂q α and therefore finally we have bρ = a˙ ρ − Γτρα aτ q˙α .

(A.52)

The particular case of this formula is relation (A.30). Formula (A.52) is often used in Chapter IV. Note that in Chapter IV it is also obtained more general formulas. The formulas, found above, can be used to describe motion of representation point in the curvilinear coordinates q = (q 1 , ... , q s ). In this case the indices π, ρ, σ, τ are varied from 1 to s = 3N and for nonstationary system, α and β from 0 to s = 3N . In Chapter IV, using a tangent space, the formulas of this Appendix are extended to mechanical systems, consisting of not only the mass points but the rigid and elastic bodies. In this case the covariant and contravariant components of the velocity vectors v and the acceleration vectors w of mechanical system, as well as for one point, are represented by the following function T1 =

1 T = gαβ q˙α q˙β , M 2

α, β = 0, s ,

where M is a mass of total system and T is its kinetic energy.

APPENDIX

B

STABILITY AND BIFURCATION OF STEADY MOTIONS OF NONHOLONOMIC SYSTEMS Appendix B contain a brief survey of the works, devoted to questions of the existence, stability, and branching of a steady motion of conservative nonholonomic systems. This Appendix is the plenary report of A. V. Karapetyan with the same title, which was spoken in the International science conference on mechanics "The third Polyakhov readings" (St.Petersburg, February 4-6, 2003). In studying the questions of existence, stability, and branching of steady motions of conservative nonholonomic systems two approaches [97, 98, 101, 333, 334] are usually applied. In the general case when steady motions of conservative nonholonomic systems correspond to the symmetries, to which the linear first integrals do not correspond (unlike the conservative holonomic systems), the methods of Lyapunov–Malkin and Andronov–Hopf (see [91, 94, 97, 99. 1985, 101, 333]) are used. These methods are based on the analysis of equations of perturbed motion and on the characteristic equation of the linearized equations of perturbed motion. The latter has always zero roots, the number of which is not so less as the dimension of a family of steady motions, which unperturbed steady motion belongs to. If the number of zero roots is equal to the above-mentioned dimension and the rest of roots have negative real parts, then the unperturbed motion is stable, in which case any perturbed motion sufficiently close to the unperturbed one tends asymptotically to a steady motion of the considered family but, generally speaking, not unperturbed motion (according to the Lyapunov–Malkin theory). On the boundary of domain of stability (in the space of parameters of problem) the characteristic equation has either zero root, either a pair of pure imaginary roots. In the first case another families of steady motions are branched off unperturbed steady motion and in the second case the families of periodic motions (the Andronov–Hopf bifurcation occurs). The described approach to the study of steady motions of conservative nonholonomic systems is also applied in the case when the nonholonomic constraints have a so-called "dissipative" effect [94, 99. 1981, 1985]. The second approach to the study of questions of existence, stability, and branching of steady motions of nonholonomic systems is based on the modified theory of Routh–Salvadori, Poincar´e–Chetaev, and Smale (see [97, 98, 99. 1983, 100. 1994, 2000, 101, 333, 334]). It can be applied to the cases when to the symmetries of system correspond not only steady motions but also the linear first integrals. Consider this case in more detail. At first we consider the case when the linear integrals, corresponding to the symmetries of system, are given in explicit form. 229

230

Appendix B Let H = H (v, r) =

1 (A (r) v · v) + (a (r) · v) + a (r) = h 2

(B.1)

be a total mechanical energy of system and K = K (v; r) = BT (r) v + b (r) = k = const

(B.2)

be a k-dimensional vector of linear integrals (the sign "T" means a transposition). Here v is an n-dimensional vector of quasivelocities (in particular, of impulses or generalized velocities), r ∈ M is an m-dimensional vector of determining coordinates such that the n × n-matrix A (r) of positive definite quadratic form, the n-dimensional vector a (r), and the scalar function a (r), entering together into a total mechanical energy, and also the n × k-matrix B (r) and the k-dimensional vector b (r) of the coefficients of the first integrals depend on these coordinates. Denote by M a configuration space of the system dim M  n. According to the Routh theory, on the fixed levels of the first integrals K = k to the critical points of the functions H correspond steady motions, in which case to the minimum points correspond stable steady motions. Taking into account a structure of function (B.1) and the first integrals (B.2), the problem of obtaining the critical points of this function on the fixed levels of these integrals can be solved in two stages. At the first stage we determine a single minimum of the function H on the fixed levels k of the first integrals K = const with respect to the variables v (in this case the variables r are regarded as parameters):     min H  = H vk (r); r , v

K=k

   1  H vk (r); r = a(r) + C(r)ck · ck − 2  −1  − A (r)a(r) · a(r) = Wk (r) ,

(B.3)

ck = ck (r) = k − b(r) + BT (r)A(r)a(r) , −1  , C(r) = BT (r)A−1 (r)B(r)

vk (r) = A−1 (r)B(r)C(r)ck − A−1 (r)a(r) . Here and below we assume that rankB (r) = k, ∀ r ∈ M, i. e. the integrals are independent of a whole configuration space. The function Wk (r) is called an effective potential, which depends, obviously, on the variables r ∈ M and the parameters k ∈ Rk . Then the problem of study of steady motions of system is reduced to the problem of the analysis of effective potential.

231

Appendix B

Theorem 1. If the effective potential takes a nondegenerate stationary value at the point r0 ∈ M, then the relation r = r0 ,

v = v0 = vk (r0 )

describes a steady motion. The point r0 , at which the effective potential has a stationary value, depends on the constants k of the first integrals. This means that the points r0 (k), which are stationary in configuration space, make up k-parametric   k , r ∈ M . The same families in the space families in the space k ∈ R   k n k ∈ R , r ∈ M , v ∈ R make up the points r = r0 (k), v = v0 = vk (r0 ), which are stationary in the phase space, i. e. make up steady motions. Even for the fixed values of the constants k, the effective potential Wk (r) can take stationary values not only at the point r0 but, generally speaking, at the certain another points r1 , r2 , . . . . These points also depend on the constants k. In the general case for certain values of k∗ the families r0 (k), r1 (k), r2 (k), . . . can have common points. Such values of k∗ are called bifurcational by Poincar´e. Obviously, the corresponding steady motions r = r0 (k), v = v0 = vk (r0 ) have common points if and only if the families r0 (k), r1 (k), r2 (k), . . . have common points (see (B.3)). In addition, by construction of effective potential, for indices we have the following relation ind δ 2 H (v0 , r0 ) |(2) = ind δ 2 Wk (r0 ) . The latter permits us to simplify substantially the construction of the bifurcational diagrams of Poincar´e–Chetaev and to restrict ourself by constructing the families r0 (k) ∪ r1 (k) ∪ r2 (k) ∪ . . . in the space {k, r} only. Consider the set   Σh,k = h ∈ R, k ∈ Rk : h = hs (k) , s = 0, 1, 2, ... (B.4) of the space {h, k}, where

h = hs (k) = H (vk (r) ; r) ; r = rs (k) , s = 0, 1, 2, ... . Set (B.4) is called bifurcational by Smale: in this set we have the crossplottings of topological types of domains, of motions in configuration space, which are defined by the relation Wk (r)  h, r ∈ M. Theorem 2. If the effective potential takes locally a strictly minimal sta 0 0 tionary value for the fixed values k k , of the constants k at the point r 0  0  0 then r = r0 k , v = v0 k is a stable steady motion. Theorem 3. Ifthe index of the second   variation  of  effective potential is odd at the point r0 k0 , then r = r0 k0 , v = v0 k0 is an unstable steady motion. Theorems 1 – 3 follow from the Routh–Salvadori theory [97, 98, 334] and correspond to the special form of the first integrals (B.1), (B.2).

232

Appendix B

Remark. If for the certain r0 ∈ M we have rankB (r) < k, then to construct an effective potential in the neighborhood of the point r0 , we need in additional consideration [332]. The results considered are applied to the study of questions of existence, stability, and branching of steady motions of a heavy nonuniform dynamically symmetric ball on absolutely roughened horizontal plane [100. 1994, 127. 1999]. Note that the approach described is applied to the analysis of steady motions of the conservative nonholonomic systems of Chaplygin, which allow the linear integrals, as known as unknown, [99. 1983, 100. 2000] in explicit form. The point is that in the case when the k-parametric groups of symmetry exist and the "dissipative" effect is lacking (otherwise the linear integrals do not exist in abstracto) the equations of motions of such systems can be reduced to the form ∂T DW d ∂T = + G˙r − , dt ∂ r˙ ∂r Dr

p˙ = Γ˙r .

(B.5)

Here r is an m-dimensional vector of determining coordinates, p is an kdimensional vector of impulses of pseudocyclic coordinates, 2T = (D˙r · r˙ ), where D = D (r) is a symmetric m × m-matrix of positive definite quadratic form, G = G (r, r˙ , p) is an alternate m × m-matrix, W = W (r, p) is an "effective" potential, Γ = Γ (r, p) is a k × m-matrix, depending linearly on p, D ∂ ∂ = + ΓT . Dr ∂r ∂p It is easily seen that equations (B.5) allow the generalized integral of energy H = T + W = const . (B.6) Suppose, γα = γα (r, p) is an m-dimensional vector, composed of the elements of the α-th row of the matrix Γ, where α = 1, k. If 

Dγα Dr

T

=

Dγα , Dr

α = 1, k ,

then the system of km equations in partial derivatives ∂p = Γ (r, p) ∂r is completely integrable and has the family of solutions p = Φ (r) k, which depends on the k arbitrary constants k, and the determinant of the k × kmatrix Φ (r) is not equal to zero. The latter means that system (B.5) except for generalized integral of energy (B.6) allows the k linear integrals K = Φ−1 (r) p = k = const .

(B.7)

233

Appendix B

Though the explicit form of these integrals is unknown the general theory of Routh–Salvadori permits us to assert that the stationary values of integral (B.6) on fixed levels of integrals (B.7) correspond to the steady motions r = r0 ,

r˙ = 0,

p = p0

(B.8)

of system (B.5), in which case the locally strictly minimal values correspond to the stable steady motions. Obviously, steady motions (B.8) make up a k-parametric family since the k + m constants r0 and p0 in (B.8) satisfy the system of m equations DW = 0. (B.9) Dr The function H has a minimum on steady motion (B.8) under conditions (B.7) if the function W under these conditions has a minimum at the point (r0 , p0 ). The latter occurs if all the eigenvalues of the matrix D2 W Dr2

(B.10)

are positive at the point (r0 , p0 ). If the determinant of matrix (B.10) is negative at the point (r0 , p0 ), then steady motion (B.8) is unstable. Obviously, to generate equations (B.9) and matrix (B.10) it is not required to know an explicit form of the first integrals (B.7). The existence and structure of these integrals permit us to make use of the Routh–Salvadori theory and to affirm reasonably that steady motion (B.8) is stable for all positive eigenvalues of matrix (B.10), which is symmetric under the condition that the matrices Dγα , Dr

α = 1, k ,

are symmetric. However for the bifurcational diagrams of Poincar´e–Chetaev and Smale to be constructed it is necessary to know the solution of system (B.7)–(B.9) in the form of r = r0 (k) and the quantities h = h (k) = W (r0 (k) , p0 (k)), respectively, i. e. it is necessary to obtain the first integrals (B.7) in explicit form (though in terms of special functions, not necessarily in terms of the elementary ones as in the problem on the motion of dynamically symmetric ball on absolutely roughened plane). In the problem on the motion of circular disk on absolutely roughened plane these first integrals are known in the form of hypergeometric Gaussian series. This makes it possible [127. 1999, 2001] to study completely the problem on a steady rolling of disk on horizontal plane.

APPENDIX

C

THE CONSTRUCTION OF APPROXIMATE SOLUTIONS FOR EQUATIONS OF NONLINEAR OSCILLATIONS WITH THE USAGE OF THE GAUSS PRINCIPLE The Gauss principle is applied to the construction of approximate solutions of equations of nonlinear oscillations, in particular, of the solution by the Bubnov–Galerkin method. If the motion of mechanical system is incompletely defined, then it is rational to construct the equations, which permit us to determine completely this motion, using Gauss’ principle represented in the form δ ′′ Z = 0 ,

(C.1)

where the function Z is given by formula (3.8) of Chapter IV. Two accents of the symbol δ are used to emphasize that the second time derivatives of generalized coordinates are varied only. We make use of this principle to find the approximate solutions of the nonlinear equation m¨ x = F (t, x, x) ˙ , (C.2) where m is a mass of mass point, x is its coordinate in the case of linear motion, F is a projection of the force acting on the point. Suppose, we seek the motion of mass point in the interval [0, τ ] in the form n  aν f ν (t) , (C.3) x(t) = ν=1

ν

where f (t) are linearly independent functions, aν are the sought parameters. The function x(t), given in the form (C.3), does not satisfy, generally speaking, differential equation (C.2) and therefore, substituting it into this equation, we obtain m¨ x − F (t, x, x) ˙ = R, (C.4)

where R is a residual. From the mechanical point of view this residual is regarded as a force, under which the motion of point exactly satisfies the law (C.3). We shall assume that the motion in the form (C.3) is incompletely given in the sense that the parameters aν are not known. In order to find these parameters, it is necessary that the average value of the square force R in the interval [0, τ ] is minimal by virtue of the varying of the accelerations only (as in the Gauss principle) i. e. the following relation δ

′′

τ 0



2 m¨ x − F (t, x, x) ˙ dt = 0 235

236

Appendix C

is satisfied. In other words, we shall seek the coefficients aν under the assumption that the error of mean square on the interval [0, τ ] is minimum. Taking into account that in the Gauss principle the accelerations are varied only, we have τ 0



 m¨ x − F (t, x, x) ˙ δx ¨dt = 0 .

Substituting relation (C.3) into this equation, we obtain    τ   n n n n   ν ν ν m f¨ν dt = 0 . δaν aν f¨ − F t, aν f , aν f˙ ν=1

0

ν=1

ν=1

(C.5)

ν=1

The quantities δaν are arbitrary and independent. Therefore from equation (C.5) it follows that    τ   n n n  aν f¨ν − F t, aν f ν , aν f˙ν ν = 1, n . (C.6) f¨ν dt = 0 , m 0

ν=1

ν=1

ν=1

The conditions, under which this system of algebraic equations has solutions different from zero, depend on the form of as the function F (t, x, x), ˙ as the functions f ν (t), ν = 1, n. The quantity R, introduced by formula (C.4), was regarded above as a force. Now we regard it as an error, which occurs for the function x(t), given in the form (C.3), to be satisfied equation (C.2). According to this approach the system of algebraic equations (C.6) with respect to the parameters aν becomes a system, which under certain assumptions permit us to find a partial approximate solution of equations (C.2) in the form (C.3). We apply now this method to determine the approximate periodic solutions of equation (C.2). For the sake of simplicity, we seek the periodic solutions in the form x(t) = a1 cos ωt + a2 sin ωt .

(C.7)

Then system (C.6), in which the time τ is assumed to be equal to the period 2π/ω, can be represented in the following way: 2π/ω  0



− mω 2 (a1 cos ωt + a2 sin ωt)−

 −F (t, a1 cos ωt + a2 sin ωt , −a1 ω sin ωt + a2 ω cos ωt) cos ωtdt = 0 , 2π/ω  0



− mω 2 (a1 cos ωt + a2 sin ωt)−

 −F (t, a1 cos ωt + a2 sin ωt , −a1 ω sin ωt + a2 ω cos ωt) sin ωtdt = 0 .

(C.8)

237

Appendix C

Equations (C.8) are used to construct approximately the solution of equation (C.2) in the form (C.7) by the Bubnov–Galerkin method. They are usually deduced from the fundamental equation of dynamics. Recall that, as is shown for Example VI. 3 considered in § 4 of Chapter VI, the clarification of the methods of Ritz and Bubnov–Galerkin by means of the integral variational principles, can be found in the work of G. Yu. Dzhanelidze and A. I. Lur’e [56]. The above method for obtaining the approximate solutions of equations (C.2) can easily be extended to the case of arbitrary mechanical system with s degrees of freedom. In this case the Gauss principle (C.1) is used in integral form, i. e. we assume that τ

(M W − Y)δ ′′ Wdt = 0 .

(C.9)

0

Recall that we have MW − Y =



 d ∂T ∂T − σ − Qσ eσ , dt ∂ q˙σ ∂q

δ ′′ W = δ ′′ q¨σ eσ .

Here T is a kinetic energy of system, Qσ is a generalized force, corresponding to the generalized coordinate q σ , eσ and eσ are the vectors of fundamental and reciprocal bases, respectively. Therefore equation (C.9) can be rewritten as  τ  ∂T d ∂T − σ − Qσ δ ′′ q¨σ dt = 0 . dt ∂ q˙σ ∂q 0

It follows that the functions q σ (t), given as q σ (t) =

n 

aσν f ν (t) ,

σ = 1, s ,

(C.10)

ν=1

can be regarded as an approximate solution of Lagrange’s equations if the parameters aσν satisfy the following equations τ  0

 d ∂T ∂T − σ − Qσ f¨ν dt = 0 , dt ∂ q˙σ ∂q

σ = 1, s ,

ν = 1, n .

(C.11)

Here the functions q σ (t) are assumed to be given in the form (C.10). The applying of formulas (C.11) to the solution of nonlinear system of differential equations, which describes the steady-state oscillations of a certain electromechanical system, using the Bubnov–Galerkin method, can be found in the work [262].

APPENDIX

D

THE MOTION OF NONHOLONOMIC SYSTEM WITH OUT REACTIONS OF NONHOLONOMIC CONSTRAINTS In Appendix D the motion of nonholonomic systems in the case when the reactions of constraints are lacking is considered. By the Mei Fengxiang terminology such a motion is called a free motion of nonholonomic system. The free motion of the Chaplygin sledge is studied. A realizing of the free motion of nonholonomic systems acted by external forces is discussed.

§ 1. Existence conditions for "free motion" of nonholonomic system The motion of nonholonomic system is defined by forces, constraints, and the initial data. In the work of Mei Fengxiang [362. 1994] the notion of free motion of nonholonomic system, which is regarded as a motion under zero values of reactions of nonholonomic constraints, is introduced. In this work, in particular, the free motion of the Chaplygin sledge is considered. In the work [362] existence conditions for a free motion of nonholonomic system are given. Below we obtain these conditions. The motion of mechanical system with the ideal nonholonomic constraints ˙ = 0, ϕκ (t, q, q)

q = (q 1 , . . . , q s ) ,

κ = 1, k ,

(D.1)

is described by Lagrange’s equations of the first kind in curvilinear coordinates d ∂T ∂T − σ = Qσ + Rσ , σ = 1, s . (D.2) σ dt ∂ q˙ ∂q Here the generalized reactions of nonholonomic constraints take the form Rσ = Λκ

∂ϕκ , ∂ q˙σ

σ = 1, s ,

κ = 1, k .

Equations (D.2) can be written by using the Christoffel symbols of the first kind ∂ϕκ M (gστ q¨τ + Γσ,αβ q˙α q˙β ) = Qσ + Λκ σ , ∂ q˙ 0 α, β = 0, s , q = t, q˙0 = 1 . σ, τ = 1, s , This system can be solved as the algebraic system with respect to q¨τ , τ = 1, s: q¨τ =

 ∆στ  ∂ϕκ Qσ + Λκ σ − M Γσ,αβ q˙α q˙β . ∆ ∂ q˙ 239

(D.3)

240

Appendix D

Here ∆ is a determinant of the matrix (M gστ ), ∆στ is an algebraic complement with (σ, τ ) number. We differentiate equations of constraints (D.1) with respect to time ∂ϕκ ∂ϕκ τ ∂ϕκ τ dϕκ ≡ + q˙ + q¨ = 0 , dt ∂t ∂q τ ∂ q˙τ

κ = 1, k ,

τ = 1, s ,

(D.4)

and substitute solutions (D.3) into formulas (D.4). Then we obtain  ∂ϕκ ∂ϕκ ∂ϕκ τ ∆στ ∂ϕκ  Q + q ˙ + + Λ − M Γσ,αβ q˙α q˙β = 0 , σ κ ∂t ∂q τ ∆ ∂ q˙τ ∂ q˙σ κ = 1, k , σ, τ = 1, s , α, β = 0, s .

(D.5)

The Lagrange multipliers Λκ , κ = 1, k, can be determined from this system if the corresponding determinant is not equal to zero. Assuming Λκ = 0, κ = 1, k, from relations (D.5) we obtain necessary and sufficient conditions for the existence of free motion of nonholonomic system. In the case when the constraints are stationary and the kinetic energy is independent of time they have the form  ∂ϕκ τ ∆στ ∂ϕκ  Qσ − M Γσ,ρτ q˙ρ q˙τ = 0 , q˙ + τ τ ∂q ∆ ∂ q˙ κ = 1, k , ρ, σ, τ = 1, s . Just these conditions under (7) number are given in the work [362]. If in place of Lagrange’s equations of the first kind we take Maggi’s equations   σ d ∂T ∂ q˙ ∂T − − Q = 0, λ = 1, l , σ = 1, s , (D.6) σ dt ∂ q˙σ ∂q σ ∂v∗λ then in the second group of equations for a free motion of nonholonomic system we have zeros:   ∂ q˙σ d ∂T ∂T − − Q = 0, κ = 1, k , σ = 1, s . (D.7) σ dt ∂ q˙σ ∂q σ ∂v∗l+κ

§ 2. Free motion of the Chaplygin sledge Consider the case when the center of mass of the Chaplygin sledge is situated above a runner. Let x, y be coordinates of the center of mass C of sledge in a horizontal plane and θ be an angle of its rotation. Then the kinetic energy of system is as follows T =

J M 2 (x˙ + y˙ 2 ) + θ˙2 , 2 2

241

Appendix D

where M is a mass of sledge, J is a moment of inertia of sledge about the vertical axis, passing through the center of mass. On the motion of sledge it is imposed the nonholonomic constraint ϕ ≡ x˙ sin θ − y˙ cos θ = 0 .

(D.8)

Denote q1 = x ,

q2 = θ ,

q3 = y

and introduce the quasivelocities v∗2 = θ˙ ,

v∗1 = x˙ ,

v∗3 = x˙ sin θ − y˙ cos θ .

This implies that x˙ = v∗1 ,

θ˙ = v∗2 ,

y˙ = v∗1 tg θ −

v∗3 . cos θ

We generate the Maggi’s equations: Mx ¨ − Qx + (M y¨ − Qy ) tg θ = 0 , J θ¨ − Qθ = 0 ,  1  (M y¨ − Qy ) − = Λ, cos θ

(D.9)

where Qx , Qy , Qθ are generalized exterior forces. Condition (D.7) takes the form  1  = 0. (D.10) (M y¨ − Qy ) − cos θ Differentiating equation of constraint (D.8) in time, we obtain ˙ x˙ ctg θ + y) x ¨ − y¨ ctg θ = −θ( ˙ .

(D.11)

Consider the motion of the Chaplygin sledge under the conditions Qx = Qy = Qθ = 0 .

(D.12)

Then the first equation of system (D.9) is as follows x ¨ = −¨ y tg θ .

(D.13)

Substituting (D.13) into relation (D.11), we have ˙ x˙ + y˙ tg θ) cos2 θ . y¨ = θ( Now we represent condition (D.10) (assuming Qy = 0) as ˙ x˙ + y˙ tg θ) cos θ = 0 . θ(

(D.14)

242

Appendix D

Taking into account the equation of constraints y˙ = x˙ tg θ and assuming that cos θ = 0, equation (D.14) becomes θ˙x˙ = 0 .

(D.15)

We notice that the obtained condition (D.15) of a free motion of the Chaplygin sledge imposes a restriction on the choice of initial data. Really, if    x˙ t=0 = x˙ 0 , y˙ t=0 = y˙ 0 , θ˙t=0 = θ˙0 ,

then, according to formula (D.15), the following relation θ˙0 x˙ 0 = 0

(D.16)

is satisfied. Restriction (D.16) allows the following choice of initial data: x˙ 0 = 0 ,

θ˙0 = 0 .

(D.17)

Since nonholonomic constraint (D.8) are satisfied, this implies the initial condition for y: ˙ y˙ 0 = 0 . (D.18) To the initial data (D.17), (D.18) corresponds a motion such that the center of mass of sledge rests and the sledge uniformly rotates round it. In this case the force, preventing the displacement of sledge in transverse direction relative to a runner, is lacking and we say that with initial data (D.17), (D.18) the sledge moves (rotates) to be free. Condition (D.16) allows another choice of the initial data: θ˙0 = 0 ,

x˙ 0 = 0 ,

y˙ 0 = 0 ,

x˙ 0 sin θ0 − y˙ 0 cos θ0 = 0 .

In this case the center of mass of sledge has a linear and uniform motion along the initial orientation of runner and the sledge does not rotate. Condition (D.16) also allows the following obvious choice of the initial data: x˙ 0 = y˙ 0 = θ˙0 = 0 . This corresponds to the rest of sledge if the exterior forces are lacking. If the equation of constraint is obtained under condition (D.14), then in place of relations (D.15) we have θ˙y˙ = 0 .

(D.19)

The investigation of possible motions under condition (D.19) leads to the same three free motions of the Chaplygin sledge. Thus, if conditions (D.12) and (D.16) are satisfied, then the Chaplygin sledge moves free in the above-mentioned sense. If not the sledge has a standard motion proper for nonholonomic system. In this case we need to find a

243

Appendix D

reaction of constraint in order to check wether this nonholonomic constraint is nonretaining.

§ 3. The possibility of free motion of nonholonomic system under active forces By nonholonomic constraints (D.1) the law of system motion in L–space and in K–space can be represented, respectively, as M WL = YL + RL ,

M WK = YK + RK .

(D.20)

In the case of ideal constraints RL = 0 and in studying a free motion of nonholonomic system we have, in addition, RK = 0. Therefore equations (D.20) take the form M WL = YL ,

M WK = Y K .

(D.21)

Equations (D.21) imply that in the case of free motion the component WK of the vector of acceleration of system W is the function of variables t, q, q, ˙ given in the following way WK =

YK (t, q, q) ˙ . M

(D.22)

On the other hand, as is shown in § 1 of Chapter IV, the vector-function WK (t, q, q) ˙ is uniquely defined by equations of constraint (D.1). Then the ˙ entering into relation (D.22), can be called the control force force YK (t, q, q), K , under which the incomplete program of motion, given in the form Ycontrol (D.1), is realized. Thus, by formulas (D.21) a free motion of nonholonomic system can be regarded as a motion such that the active force Y has a component, belonging to L–space only, and the control force Ycontrol belonging to K–space only. Applying this approach to a free motion of nonholonomic system we obtain that in accordance with the theory of constrained motion K K the control force Ycontrol = Ycontrol has the form Ycontrol = Λcontrol ∇′ ϕκ . κ is a generalized control force, under which the constraint with Here Λcontrol κ κ number is realized. Note that the same approach is usedused in Chapter III to solve the problem of flight dynamics on the directing of a mass point on a target by the curve of pursuit. The concept of study of free motion of nonholonomic systems, developed in the work of Mei Fengxiang [362. 1994], can also be of another practical importance. For example, in the treatise [226] it is considered a controllable motion of nonholonomic systems. The control is chosen from the condition that the nonholonomic system has a given program motion. In this case the control forces also provide generation of forces equal to the corresponding reactions of nonholonomic constraints. Under these control forces the reactions

244

Appendix D

of nonholonomic constraints are equal to zero and therefore by the terminology of the work [362. 1994] such a controllable motion is a free motion of nonholonomic system. In the book [226] the possibility of small deviations of the obtained generalized coordinates and velocities from the required ones is taken into account. This is gone along with the occurrence of small reactions of nonholonomic constraints, which are regarded in the considered problem as disturbances. Finally, the original problem is reduced to the conditional problem of adaptive control with unknown disturbances. The algorithms of control and also the estimation of so far as a program motion of system is realized for a given accuracy of stabilization are given.

APPENDIX

E

THE TURNING MOVEMENT OF A CAR AS A NONHOLONOMIC PROBLEM WITH NONRETAINING CONSTRAINTS The turning movement of a car with its possible sideslip is considered as a nonholonomic problem with nonretaining constraints. The four possible types of the car motion are studied.

§ 1. General remarks The complete theory of the motion of a car with deformable wheels is developed by N. A. Fufaev and detailed in his book [130]. The treatise by V. F. Zhuravlev and N. A. Fufaev [72] is devoted to mechanics of systems with nonretaining constraints. In this treatise the Boltzmann–Hamel equations are used for studying the motion of nonholonomic systems, and the possibility of restoring nonholonomic constraints is investigated on the basis of behaviour of solution curves in the common space of generalized coordinates and quasivelocities. In this Appendix E the Maggi equations, which make it possible to easily determine the generalized reaction forces of nonholonomic constraints, are applied; the beginning and stop of wheels sideslip being determined by these constraint forces. We now return to examples II. 4 and II. 5 considered in Chapter II. Pay attention that the Boltzmann-Hamel and Maggi equations are formed there for the realized constraints (4.16). In this case the turning of a car is studied, saying figuratively, under the "dynamic control when the turning moment ˙ and restoring moment L3 (θ) are applied to L1 (t), resisting moment L2 (θ), the rotating front axle (see Fig. II. 4). This scheme required introducing four generalized coordinates ϕ, θ, ξC , ηC , that was reasonable from the methodical point of view, for in this case we get an example, in which with two constraints (4.16) we have to obtain two Boltzmann-Hamel’s equations or two Maggi’s equations. This mathematical model can be of interest in studying the motion of wheeled robot vehicles, development of which is given much attention at present (see, for ex., works by V. N. Belotelov, V. I. Kalyonova, A. V. Karapetyan, A. I. Kobrin, A. V. Lenskii, Yu. G. Martynenko, V. M. Morozov, D. E. Okhotsimskii, M. A. Salmina [146-148, 423]). Let us go to the "kinematic control under which the turn of the front axle is determined by a driver as a certain time function θ = θ(t). In such a scheme a turning car has three degrees of freedom. In this case, we shall consider nonholonomic constraints ϕ1 ≡ −ξ˙C sin ϕ + η˙ C cos ϕ − l2 ϕ˙ = 0 ,

(E.1)

ϕ2 ≡ −ξ˙C sin(ϕ + θ) + η˙ C cos(ϕ + θ) + l1 ϕ˙ cos θ = 0 ,

(E.2)

245

246

Appendix E

which should be satisfied by the car motion, as nonretaining. The active forces F1 (t) and F2 (t) have the same meaning as in examples II. 4 and II. 5, of Chapter II.

§ 2. The turning movement of a car with retaining (bilateral) constraints We shall study the car motion in the horizontal plane with respect to the fixed system of coordinates Oξηζ (see Fig. E. 1). We shall set the position of a car by generalized coordinates q 1 = ϕ (the angle between the longitudinal Cx-axis of the car and the Oξ-axis ), q 2 = ξC , q 3 = ηC (the coordinates of point C). The angle θ is equal to the angle between the front axle and a perpendicular to the Cx-axis . It is a given time function: θ = θ(t) . Two nonholonmic constraints (E.1) and (E.2), expressing the absence of side slipping of the front and rear axles of the car are imposed on the car motion. The kinetic energy of the system consists of the kinetic energies of the car body and front axle and is calculated according to the formula: 2

˙ ξ˙C sin ϕ + η˙ C cos ϕ) , 2T = M ∗ (ξ˙C + η˙C 2 )+J ∗ ϕ˙ 2 +J2 θ˙2 + 2J2 ϕ˙ θ˙ + 2M2 l1 ϕ(− M ∗ = M1 + M 2 ,

J ∗ = J1 + J2 + M2 l12 . (E.3)

Using the expression for virtual elementary work δA = Qϕ δϕ + QξC δξC + QηC δηC , we shall find the generalized forces acting on the car, as was done in examples II. 4, II. 5, Chapter II. For the rear drive car we obtain the following

Fig. E. 1

247

Appendix E expressions: Q1 ≡ Qϕ = 0 , Q2 ≡ QξC = F1 (t) cos ϕ − F2 (vC )ξ˙C /vC , Q3 ≡ QηC = F1 (t) sin ϕ − F2 (vC )η˙ C /vC ,  2 vC = ξ˙C + η˙C 2 .

(E.4)

In order to form the Maggi equations describing the vehicle motion we introduce new nonholonomic variables by to the formulas: v∗2 = −l2 ϕ˙ − ξ˙C sin ϕ + η˙ C cos ϕ , v∗3 = l1 ϕ˙ cos θ − ξ˙C sin(ϕ + θ) + η˙ C cos(ϕ + θ) , v∗1 = ϕ, ˙

and write the reverse transformation q˙1 ≡ ϕ˙ = v∗1 ,

q˙2 ≡ ξ˙C = β12 v∗1 + β22 v∗2 + β32 v∗3 ,

q˙3 ≡ η˙ C = β13 v∗1 + β23 v∗2 + β33 v∗3 ,

(E.5)

where

l1 cos ϕ cos θ + l2 cos(ϕ + θ) , sin θ cos(ϕ + θ) cos ϕ β22 = , β32 = − , sin θ sin θ (E.6) l1 sin ϕ cos θ + l2 sin(ϕ + θ) 3 β1 = , sin θ sin(ϕ + θ) sin ϕ β23 = , β33 = − . sin θ sin θ The first group of the Maggi equations in this case consists of a single equation β12 =

(M W1 − Q1 )

∂ q˙1 ∂ q˙2 ∂ q˙3 + (M W − Q ) + (M W − Q ) = 0. 2 2 3 3 ∂v∗1 ∂v∗1 ∂v∗1

(E.7)

The expressions M Wσ may be calculated using kinetic energy by the formulas M Wσ =

d ∂T ∂T − σ, σ dt ∂ q˙ ∂q

σ = 1, 3 .

Finally, using expressions (E.3), (E.4), (E.5), (E.6), we represent the equation of motion (E.7) in the following expanded form: J ∗ ϕ¨ + J2 θ¨ + M2 l1 (−ξ¨C sin ϕ + η¨C cos ϕ)+ +β12 (M ∗ ξ¨C − M2 l1 (ϕ¨ sin ϕ + ϕ˙ 2 cos ϕ) − F1 (t) cos ϕ + F2 (vC )ξ˙C /vC )+ +β13 (M ∗ η¨C + M2 l1 (ϕ¨ cos ϕ − ϕ˙ 2 sin ϕ) − F1 (t) sin ϕ + F2 (vC )η˙ C /vC ) = 0 . (E.8) The equations of constraints (E.1) and (E.2) should be added to this equation.

248

Appendix E

If the initial conditions and analytic representation of the functions F1 (t), F2 (vC ) are given, then after numerical integrating the nonlinear system of differential equations (E.1), (E.2), (E.8) we shall find the law of the car motion: ϕ = ϕ(t), ξC = ξC (t), ηC = ηC (t). (E.9) Now we can determine the generalized reaction forces. The second group of Maggi’s equations will be written as follows: Λ1 = (M W1 − Q1 )

∂ q˙1 ∂ q˙2 ∂ q˙3 + (M W − Q ) + (M W − Q ) , 2 2 3 3 ∂v∗2 ∂v∗2 ∂v∗2

Λ2 = (M W1 − Q1 )

∂ q˙1 ∂ q˙2 ∂ q˙3 + (M W2 − Q2 ) 3 + (M W3 − Q3 ) 3 , 3 ∂v∗ ∂v∗ ∂v∗

or in the extended form for the rear drive vehicle: Λ1 = β22 (M ∗ ξ¨C − M2 l1 (ϕ¨ sin ϕ + ϕ˙ 2 cos ϕ) − F1 (t) cos ϕ + F2 (vC )ξ˙C /vC )+ +β23 (M ∗ η¨C + M2 l1 (ϕ¨ cos ϕ − ϕ˙ 2 sin ϕ) − F1 (t) sin ϕ+

(E.10)

+F2 (vC )η˙ C /vC ) , Λ2 =

β32 (M ∗ ξ¨C − M2 l1 (ϕ¨ sin ϕ + ϕ˙ 2 cos ϕ) − F1 (t) cos ϕ + F2 (vC )ξ˙C /vC )+ (E.11) +β33 (M ∗ η¨C + M2 l1 (ϕ¨ cos ϕ − ϕ˙ 2 sin ϕ) − F1 (t) sin ϕ+ +F2 (vC )η˙ C /vC ) .

After inserting expressions (E.9) into these formulas we find the law of varying the generalized reaction forces Λi = Λi (t),

i = 1, 2 .

These functions allow us to investigate the possibility of realizing the nonholonomic constraints (E.1), (E.2). If the reaction forces appear to exceed the forces provided by Coulomb’s frictional forces, then these constraints will not be realized and the vehicle will begin to slip along the axles to which the wheels are fastened. In order to write the conditions of the beginning of side slipping of the wheels in the analytical form, it is necessary to establish the relation between the determined generalized reactions Λ1 , Λ2 and reaction forces RB , RA applied to the wheels from the road (see Fig. E. 1). This is a question of principal importance, so let us consider the relation between the generalized reaction force Λ of the nonholonomic constraint and the reaction force R for the following quite general case. Assume that the equation of the nonholonomic constraint sets the condition of the fact that for the plane motion the velocity v of a point of mechanical system along the direction of the unit vector n is equal to zero, i. e. assume that constraint equation written in vectorial form is as follows: ϕn = v · n = 0 .

249

Appendix E This equation in a scalar form appears as ϕn = xn ˙ x + yn ˙ y = 0.

If the constraint is ideal, then the reaction force R can be represented as R = Rx i + Ry j =  ∂ϕn ∂ϕn  =Λ i+ j = Λn , ∂ x˙ ∂ y˙

where i and j are unit vectors in x− and y− directions. Hence, the generalized reaction force Λ is equal to the projection of the constraint reaction force R onto the direction of vector n. It is clear that this representation of the vector R in the form Λn can be extended also to the constraints (E.1), (E.2). Writing these constraints in the vector form (E.12) ϕ1 = vB · j = 0 , ϕ2 = vA · j1 = vA · (−i sin θ + j cos θ) = 0 ,

(E.13)

where j1 is the unit vector of the ordinate axis of the movable frame Ax1 y1 of the car front axle, we obtain RB = Λ1 j ,

(E.14)

RA = Λ2 (−i sin θ + j cos θ) .

(E.15)

Remark that if the constraints (E.12) and (E.13) are violated, then nonzero values of ϕ1 and ϕ2 are equal to projections of velocities of the points B and A onto the vectors j and j1 , correspondingly. In this case the resulting friction forces applied to the wheels may be represented as fr 1 Rfr B = −Λ1 sign(ϕ )j , fr 2 Rfr A = −Λ2 sign(ϕ )(−i sin θ + j cos θ) . fr Finding the positive quantities Λfr 1 and Λ2 will be reported below.

§ 3. The turning movement of a rear-drive car with nonretaining constraints General remarks. Let us return to the question considered in the previoius paragraphs. Note that the Maggi equation (E.8) was obtained for the satisfied constraints (E.1), (E.2), i. e. when these nonholonomic constraints were retaining (bilateral). Let us study the vehicle motion in the case when the constraints (E.1), (E.2) may be nonretaining, i. e. when side slipping of the front or rear wheels

250

Appendix E

(or both front and rear wheels simultaneously) begins. The dynamic conditions of realizing the kinematic constraints (E.1), (E.2) is the requirement that the forces of interaction between the wheels and the road should not exceed the corresponding Coulomb’s friction forces. For the driven front wheels in accordance with formula (E.15) this is expressed by inequality: |Λ2 | < F2fr = k2 N2 ,

(E.16)

where F2fr , k2 are the frictional force and the coefficient of friction between front wheels and the road, respectively, N2 is the normal pressure of the front axle. When considering the rear driving wheels it is necessary to take into account that the value of this wheels-road interaction force FB is determined by the vector sum of the driving force F1 and side reaction force RB given by formula (E.14) (see Fig. E. 2). To provide the absence of side slipping of the rear axle, the following condition should be satisfied (the introduced notation is analogous to the notation used for the front axle): (E.17) FB = (F1 )2 + (Λ1 )2 < F1fr = k1 N1 .

According to Fig. E. 2 this means that the end of the force vector FB should not go beyond the circle of radius F1fr . Otherwise the road will not be able to develop such reaction value |Λ1 | that is required for realization of the nonholonomic constraint (E.1). Thus, this constraint becomes nonretaining, the side velocity component of driven wheels appears, and Coulomb’s friction force F1fr starts acting to them from the road. This Coulomb’s friction force F1fr arises from simultaneous action of the driving force F1 and side friction force Λfr 1 , so that 2 (FB )2 ≡ (F1fr )2 ≡ (k1 N1 )2 = (F1 )2 + (Λfr 1) .

Fig. E. 2

(E.18)

251

Appendix E Note that at the beginning of side slipping the driver sets F1 = 0 .

Possible types of the car motion. We shall explain possible different types of motion of the mechanical model of a car. In Fig. E. 3 in the phase space of variables q σ , q˙σ , σ = 1, 3, we see the representation of two hypersurfaces. The first one corresponds to the constraint given by equation (E.12), and the second one corresponds to the constraint given by equation (E.13). In an explicit form these constraints are presented by formulas (E.1), (E.2). For simultaneous realization of nonholonomic constraints (E.1) and (E.2) the point of the phase space should be located in the line of intersection of these hypersurfaces. This corresponds to the I-st type of the car motion (bold curve I in Fig. E. 3). If the first constraint is violated (FB = F1fr ) and the second constraint is satisfied, then the representation point is located at the hypersurface ϕ2 = 0 (II-nd type of motion). If the second constraint is violated, but the first constraint is fulfilled ϕ1 = 0, then the representation point belongs to hypersurface ϕ1 = 0 (III-rd type of motion). In the case if both constraints are violated, the representation point does not belong to hypersurfaces, as this takes place the vehicle moves in the presence of side friction forces acting on the front and rear axles (IV-th type of motion). From any type of motion the representation point can change to any other type of motion. For example, in the I-st type of motion, if inequality (E.17) is not satisfied , the vehicle becomes released of the constraint (E.1). If in this case inequality (E.16) is still satisfied, then constraint (E.2) keeps on working, thus, the representation point can move only over the hypersurface ϕ2 = 0 (the car changes to the II-nd type of motion). Here two cases of possible restoring the I-st type of motion should be distinguished. In some area G1 the solution curves go through the curve I, without stop-

Fig. E. 3

252

Appendix E

ping there (see Fig. E. 3). This instantaneous realization of the constraint (E.1) corresponds to the stop of side motion of the rear axle in one direction and change of the same axle to the side motion in the reverse direction. In contrast to this the behaviour of solution curves within the area G2 characterizes restoration of the constraint ϕ1 = 0 and change from the II-nd type of motion to the I-st one. Without preliminary studies of behaviour of solution curves in the common space of generalized coordinates and quasivelocities [72], it is possible to find out in which area G1 or G2 the equation ϕ1 = 0 turned out to be satisfied, in the following manner. By the values of phase variables, such that the constraint (E.1) is fulfilled, let us calculate the reaction Λ1 by the formula (E.10). If for the obtained value of Λ1 the inequality (E.17) is satisfied, then the constraint ϕ1 = 0 becomes retaining (bilateral) (the solution curve is within the area G2 ), otherwise this constraint is not restored (the solution curve is within the area G1 ). In investigating the II-nd type of motion it is necessary also to ensure that inequality (E.16) is satisfied, for if it is violated the vehicle will change to the IV-th type of motion. If constraint (E.1) is restored and constraint (E.2) is violated at the same time, then the III-rd type of motion will occur. Note that for the sake of simplicity, it was assumed in the foregoing that the static and dynamic coefficicients of Coulomb’s friction force are equal to each other. The difference of these quantities could be taken into account in a similar way as it has been done in § 4 of Chapter I, when studying accelerated motion of a car with the possible slipping of its driving wheels. Let us write out the equations of motion for the turning car four types of motion cosidered. I-st type of motion. For this motion both constraints (E.1) и (E.2) are fulfilled : ϕ2 = 0 . ϕ1 = 0 , Maggi’s equation for a rear-wheel drive vehicle takes the form (E.8), which should be integrated together with the equations of constraints (E.1) and (E.2). Having obtained the law of motion ϕ = ϕ(t) ,

ξC = ξC (t) ,

ηC = ηC (t) ,

the generalized reactions can be found from (E.6), (E.10), (E.11) Λ1 = Λ1 (t) ,

Λ2 = Λ2 (t) .

By these values, fulfillment of inequelities (E.16) and (E.17) is being checked. When one of them is violated, the vehicle changes to the II-nd or III-rd type of motion, and when both of them are violated simultaneously it changes to the IV-th type. II-nd type of motion. For this type of motion only the second constraint is fulfilled: ϕ2 = 0 . ϕ1 = 0 ,

253

Appendix E

The rear axle of the vehicle executes lateral motion, therefore the lateral frictional force Λfr 1 , calculated by formula (E.18) is applied to it. As this takes place, if ϕ1 > 0, then according to formula (E.12) the rear wheels sideslip in the positive direction of the y-axis. Therefore, the lateral frictional force is opposed to the y-axis, and if ϕ1 < 0, it is aligned with the y-axis (see Fig. E. 1). Let us obtain Maggi’s equations in the presence of one constraint (E.2). Let us go to quasivelocities by formulas: v 2 = ξ˙C , v 1 = ϕ˙ , ∗

∗

= −ξ˙C sin(ϕ + θ) + η˙ C cos(ϕ + θ) + l1 ϕ˙ cos θ . Let us find the inverse transformation: ϕ˙ = v 1 , ξ˙C = v 2 , η˙ C = β 3 v 1 + β 3 v 2 + β 3 v 3 , v∗3

∗

∗

1 ∗

2 ∗

3 ∗

where β13 = −l1 cos θ/ cos(ϕ + θ) ,

β23 = tg(ϕ + θ) ,

β33 = 1/ cos(ϕ + θ) . (E.19)

Now we may get two Maggi’s equations for the rear-wheel drive vehicle J ∗ ϕ¨ + J2 θ¨ + M2 l1 (−ξ¨C sin ϕ + η¨C cos ϕ) − Λfr sign(ϕ1 )l2 + 1

+β13 (M ∗ η¨C

2

+ M2 l1 (ϕ¨ cos ϕ − ϕ˙ sin ϕ)−

1 −F1 (t) sin ϕ + F2 (vC )η˙ C /vC + Λfr 1 sign(ϕ ) cos ϕ) = 0 , M ∗ ξ¨C − M2 l1 (ϕ¨ sin ϕ + ϕ˙ 2 cos ϕ) − F1 (t) cos ϕ + F2 (vC )ξ˙C /vC − 1 3 ∗ −Λfr ¨C + M2 l1 (ϕ¨ cos ϕ − ϕ˙ 2 sin ϕ) − F1 (t) sin ϕ+ 1 sign(ϕ ) sin ϕ + β2 (M η 1 +F2 (vC )η˙ C /vC + Λfr 1 sign(ϕ ) cos ϕ) = 0 .

(E.20) From the second group of Maggi’s equations there remains one equation for determination of the generalized reaction Λ2 . For the vehicle with driving rear wheels it is as follows: Λ2 = β33 (M ∗ η¨C + M2 l1 (ϕ¨ cos ϕ − ϕ˙ 2 sin ϕ) − F1 (t) sin ϕ+ (E.21) 1 +F2 (vC )η˙ C /vC + Λfr 1 sign(ϕ ) cos ϕ) . The equations of motion (E.20) are integrated together with the constraint equation (E.2). If the dynamic condition (E.16) for the constraint (E.2) to be realized holds for the obtained value of Λ2 , then the II-nd type of motion continues. If the condition (E.16) is violated, then the vehicle will change to IV-th type of motion. In the course of checking inequality (E.16) it is necessary to keep watching if the constraint ϕ1 = 0 begins to hold. If this constraint is realized under certain obtained values of t, q σ , q˙σ , σ = 1, 3, and if inequality (E.17)

254

Appendix E

holds for the value Λ1 calculated by formula (E.10), then the constraint ϕ1 = 0 is restored, the rear axle ceases to execute lateral motion and the car changes to the I-st type of motion. If inequality (E.17) is not fulfilled for the value Λ1 calculated by formula (E.10), then the car keeps the II-nd type of motion (rear axle begins lateral motion in the opposite direction). Theoretically the car may change from the II-nd type of motion to the III-rd one: for this purpose, at a certain time instant inequality (E.16) must cease to hold and simultaneously the constraint ϕ1 = 0 must be restored. III-rd type of motion. This motion is studied in a similar way to the II-nd type. Now the following should be fulfilled: ϕ1 = 0 ,

ϕ2 = 0 .

Due to side slipping of the front axle of the car this front axle is acted upon by the side friction force (E.22) Λfr 2 = k2 N2 . In order to form Maggi’s equations for this nonholonomic problem with one constraint (E.1) let us change to quasivelocities by using the formulas: v 1 = ϕ˙ , v 2 = ξ˙C , ∗

∗

= −ξ˙C sin ϕ + η˙ C cos ϕ − l2 ϕ˙ . This corresponds to the reverse transformation: ϕ˙ = v 1 , ξ˙C = v 2 , η˙ C = β 3 v 1 + β 3 v 2 + β 3 v 3 , v∗3

∗

1 ∗

∗

2 ∗

3 ∗

where β13 = l2 / cos ϕ ,

β23 = tg ϕ ,

β33 = 1/ cos ϕ .

(E.23)

Two Maggi equations for the car with driving rear wheels have the form: J ∗ ϕ¨ + J2 θ¨ + M2 l1 (−ξ¨C sin ϕ + η¨C cos ϕ) + Λfr sign(ϕ2 )l1 cos θ+ 2

+β13 (M ∗ η¨C + M2 l1 (ϕ¨ cos ϕ − ϕ˙ 2 sin ϕ)− 2 −F1 (t) sin ϕ + F2 (vC )η˙ C /vC + Λfr (E.24) 2 sign(ϕ ) cos(ϕ + θ)) = 0 , ∗¨ 2 ˙ M ξC − M2 l1 (ϕ¨ sin ϕ + ϕ˙ cos ϕ) − F1 (t) cos ϕ + F2 (vC )ξC /vC − 2 3 ∗ ¨C + M2 l1 (ϕ¨ cos ϕ − ϕ˙ 2 sin ϕ)− −Λfr 2 sign(ϕ ) sin(ϕ + θ) + β2 (M η 2 −F1 (t) sin ϕ + F2 (vC )η˙ C /vC + Λfr 2 sign(ϕ ) cos(ϕ + θ)) = 0 .

The generalized reaction Λ1 is expressed as Λ1 = β33 (M ∗ η¨C + M2 l1 (ϕ¨ cos ϕ − ϕ˙ 2 sin ϕ) − F1 (t) sin ϕ+ 2 +F2 (vC )η˙ C /vC + Λfr 2 sign(ϕ ) cos(ϕ + θ)) .

(E.25)

255

Appendix E

The equations of motion (E.24) are integrated together with the constraint equation (E.1). If the dynamic condition (E.17) for realizing the constraint (E.1) is satisfied for the value of Λ1 obtained by formula (E.25), then the III-rd type of motion continues. If the condition (E.17) is violated, then the car changes to the IV-th type of motion. In the course of checking inequality (E.17) it is necessary to keep watching if the constraint ϕ2 = 0 begins to be realized. If this constraint is realized under certain calculated values t, q σ , q˙σ , σ = 1, 3, then these values of the variables should be substituted in formula (E.11). If for the obtained Λ2 the inequality (E.16) is satisfied, then the constraint ϕ2 = 0 is restored, the front axles ceases to execute lateral motion, and the car changes to the I-st type of motion. If for the calculated value of Λ2 inequality (E.16) is not satisfied, then the car continues the III-rd type of motion (the front axle begins lateral motion in the opposite direction). Theoretically the III-rd type of motion can change to the II-nd one: for this purpose, at a certain instant inequality (E.17) must cease to hold, and at the same time the constraint ϕ2 = 0 must be restored. IV-th type of motion. For such motion the following must take place: ϕ1 = 0 ,

ϕ2 = 0 .

This means that the car moves as a holonomic system when its wheels are fr acted upon by side frictional forces Λfr 1 and Λ2 set by formulas (E.18) and (E.22). The motion of the rear wheel-drive car is determined by the following Lagrange equations of the second kind: J ∗ ϕ¨ + J2 θ¨ + M2 l1 (−ξ¨C sin ϕ + η¨C cos ϕ)− 1 fr 2 −Λfr 1 sign(ϕ )l2 + Λ2 sign(ϕ )l1 cos θ = 0 ,

M ∗ ξ¨C − M2 l1 (ϕ¨ sin ϕ + ϕ˙ 2 cos ϕ) − F1 (t) cos ϕ + F2 (vC )ξ˙C /vC − 1 fr 2 −Λfr 1 sign(ϕ ) sin ϕ − Λ2 sign(ϕ ) sin(ϕ + θ) = 0 , ∗

(E.26)

2

M η¨C + M2 l1 (ϕ¨ cos ϕ − ϕ˙ sin ϕ) − F1 (t) sin ϕ + F2 (vC )η˙ C /vC + 1 fr 2 +Λfr 1 sign(ϕ ) cos ϕ + Λ2 sign(ϕ ) cos(ϕ + θ) = 0 .

In course of calculation of motion by equations (E.26) it is necessary to keep watching if either function ϕ1 or ϕ2 vanishes, or both functions ϕ1 and ϕ2 do so for the current values of t, q σ , q˙σ ,

σ = 1, 3 .

(E.27)

If ϕ1 = 0 holds for the values (E.27), then Λ1 should be calculated for these values of variables by formula (E.25). If for this value of Λ1 inequality (E.17) is satisfied, then the car changes to the III-rd type of motion, otherwise it keeps the motion of the IV-th type.

256

Appendix E

If it turns out that ϕ2 = 0 for the values (E.27), then for these values of variables Λ2 should be calculated by formula (E.21). If the inequality (E.16) is satisfied for this value of Λ2 , then the car changes to the II-nd type of motion, otherwise it keeps the motion of the IV-th type. If it turns out that for the values (E.27) the both functions ϕ1 and ϕ2 vanish simultaneously, then Λ1 and Λ2 should be found from formulas (E.10), (E.11). If both inequalities (E.16) and (E.17) are fulfilled for these values, then the car changes to the I-st type of motion. If only inequality (E.16) is satisfied, then the II-nd type of motion begins. If only inequality (E.17) is fulfilled, then from this point on the car will execute the III-rd type of motion.

§ 4. Equations of motion of a turning front-drive car with non-retaining constraints Consider a motion of a front-drive car. All necessary changes in the equations of motion are caused by the fact that the application point of the force F1 (t) changes. Now the force is applied to the point A and aligned with the axis Ax1 (see Fig. E. 1). So, for a front-drive car the expressions of generalized forces appear as Q1 ≡ Qϕ = l1 F1 (t) sin θ , Q2 ≡ QξC = F1 (t) cos(ϕ + θ) − F2 (vC )ξ˙C /vC , Q3 ≡ QηC = F1 (t) sin(ϕ + θ) − F2 (vC )η˙ C /vC ,  2 vC = ξ˙C + η˙C 2 .

Below we present the equations of motion for the four types of motion. I-st type of motion. The car motion without slipping. In this case the Maggi equations have the form J ∗ ϕ¨ + J2 θ¨ + M2 l1 (−ξ¨C sin ϕ + η¨C cos ϕ) − l1 F1 (t) sin θ +β 2 (M ∗ ξ¨C − M2 l1 (ϕ¨ sin ϕ + ϕ˙ 2 cos ϕ) − F1 (t) cos(ϕ + θ) 1

+F2 (vC )ξ˙C /vC ) + β13 (M ∗ η¨C + M2 l1 (ϕ¨ cos ϕ − ϕ˙ 2 sin ϕ) − F1 (t) sin(ϕ + θ) +F2 (vC )η˙ C /vC ) = 0 . (E.28) As this takes place, the generalized constraint reaction forces are expressed as Λ1 = β22 (M ∗ ξ¨C − M2 l1 (ϕ¨ sin ϕ + ϕ˙ 2 cos ϕ) − F1 (t) cos(ϕ + θ) +F2 (vC )ξ˙C /vC ) + β 3 (M ∗ η¨C + M2 l1 (ϕ¨ cos ϕ − ϕ˙ 2 sin ϕ) 2

−F1 (t) sin(ϕ + θ) + F2 (vC )η˙ C /vC ) ,

(E.29)

257

Appendix E Λ2 = β32 (M ∗ ξ¨C − M2 l1 (ϕ¨ sin ϕ + ϕ˙ 2 cos ϕ) − F1 (t) cos(ϕ + θ) +F2 (vC )ξ˙C /vC ) + β 3 (M ∗ η¨C + M2 l1 (ϕ¨ cos ϕ − ϕ˙ 2 sin ϕ) 3

(E.30)

−F1 (t) sin(ϕ + θ) + F2 (vC )η˙ C /vC ) . In equation (E.28) and relations (E.29), (E.30) the quantities β12 , β13 , β22 , β23 , β32 , β33 should be calculated according to formulae (E.6). In the case of a front-drive car, inequalities, the fulfillment of which means the realization of constraints (E.1) and (E.2), appear as |Λ1 | < F1fr = k1 N1 , FA =

(F1 )2 + (Λ2 )2 < F2fr = k2 N2 ,

(E.31) (E.32)

for the driven rear wheels and driving front wheels, correspondingly. If inequation (E.31) is violated, it means that side slipping of the rear axle begins (the violation of the constraint ϕ1 = 0, the change to the II-nd type of motion). If inequation (E.32) is violated, it means that side slipping of the front axle begins (the violation of the constraint ϕ2 = 0, the change to the III-rd type of motion). II-nd type of motion. The car rear axle executes side (lateral) motion. For a front-drive car two Maggi’s equations have the form J ∗ ϕ¨ + J2 θ¨ + M2 l1 (−ξ¨C sin ϕ + η¨C cos ϕ) − l1 F1 (t) sin θ 1 3 ∗ +Λfr ¨C + M2 l1 (ϕ¨ cos ϕ − ϕ˙ 2 sin ϕ) 1 sign(ϕ )l2 + β1 (M η 1 −F1 (t) sin(ϕ + θ) + F2 (vC )η˙ C /vC + Λfr 1 sign(ϕ ) cos ϕ) = 0 , M ∗ ξ¨C − M2 l1 (ϕ¨ sin ϕ + ϕ˙ 2 cos ϕ) − F1 (t) cos(ϕ + θ) + F2 (vC )ξ˙C /vC 1 3 ∗ −Λfr ¨C + M2 l1 (ϕ¨ cos ϕ − ϕ˙ 2 sin ϕ) 1 sign(ϕ ) sin ϕ + β2 (M η 1 −F1 (t) sin(ϕ + θ) + F2 (vC )η˙ C /vC + Λfr 1 sign(ϕ ) cos ϕ) = 0 ,

where coefficients β13 , β23 , β33 are determined from (E.19), and the quantity Λfr 1 is defined as (E.33) Λfr 1 = k1 N1 . The relation for determining the generalized reaction Λ2 is Λ2 = β33 (M ∗ η¨C + M2 l1 (ϕ¨ cos ϕ − ϕ˙ 2 sin ϕ) − F1 (t) sin(ϕ + θ)+ 1 +F2 (vC )η˙ C /vC + Λfr 1 sign(ϕ ) cos ϕ) .

III-rd type of motion. The car front axle executes side (lateral) motion. In this case the, for a front-drive car the two Maggi equations take the form J ∗ ϕ¨ + J2 θ¨ + M2 l1 (−ξ¨C sin ϕ + η¨C cos ϕ) − l1 F1 (t) sin θ+ 2 3 ∗ ¨C + M2 l1 (ϕ¨ cos ϕ − ϕ˙ 2 sin ϕ)− +Λfr 2 sign(ϕ )l1 cos θ + β1 (M η

258

Appendix E

Fig. E. 4

2 −F1 (t) sin(ϕ + θ) + F2 (vC )η˙ C /vC + Λfr 2 sign(ϕ ) cos(ϕ + θ)) = 0 ,

M ∗ ξ¨C − M2 l1 (ϕ¨ sin ϕ + ϕ˙ 2 cos ϕ) − F1 (t) cos(ϕ + θ) + F2 (vC )ξ˙C /vC − 2 3 ∗ −Λfr ¨C + M2 l1 (ϕ¨ cos ϕ − ϕ˙ 2 sin ϕ)− 2 sign(ϕ ) sin(ϕ + θ) + β2 (M η 2 −F1 (t) sin(ϕ + θ) + F2 (vC )η˙ C /vC + Λfr 2 sign(ϕ ) cos(ϕ + θ)) = 0 .

Here the quantities β13 , β23 , β33 are defined from (E.23), and the friction force Λfr 2 is expressed as 2 (k2 N2 )2 = (F1 )2 + (Λfr (E.34) 2) . In this case the generalized constraint reaction force Λ1 is Λ1 = β33 (M ∗ η¨C + M2 l1 (ϕ¨ cos ϕ − ϕ˙ 2 sin ϕ) − F1 (t) sin(ϕ + θ)+ 2 +F2 (vC )η˙ C /vC + Λfr 2 sign(ϕ ) cos(ϕ + θ)) .

IV-th type of motion. Both car axles execute side motion. Equations of motion of a front-drive car slipping on the horizontal surface (level) are J ∗ ϕ¨ + J2 θ¨ + M2 l1 (−ξ¨C sin ϕ + η¨C cos ϕ) − l1 F1 (t) sin θ+ 1 fr 2 +Λfr 1 sign(ϕ )l2 + Λ2 sign(ϕ )l1 cos θ = 0 ,

M ∗ ξ¨C − M2 l1 (ϕ¨ sin ϕ + ϕ˙ 2 cos ϕ) − F1 (t) cos(ϕ + θ) + F2 (vC )ξ˙C /vC − 1 fr 2 −Λfr 1 sign(ϕ ) sin ϕ − Λ2 sign(ϕ ) sin(ϕ + θ) = 0 ,

M ∗ η¨C + M2 l1 (ϕ¨ cos ϕ − ϕ˙ 2 sin ϕ) − F1 (t) sin(ϕ + θ) + F2 (vC )η˙ C /vC + 1 fr 2 +Λfr 1 sign(ϕ ) cos ϕ + Λ2 sign(ϕ ) cos(ϕ + θ) = 0 . fr The values of side friction forces Λfr 1 and Λ2 are defined from (E.33) and (E.34).

Taking into account conditions (E.31), (E.32) of constraints realization and formulae for defining the values of friction forces (E.33), (E.34), the logic

259

Appendix E

of change from one type of motion to another is the same as in the case of a rear-drive car (see § 3).

§ 5. Calculation of motion of a certain car As an example, let us consider the motion of a hypothetical compact motor car with M1 = 1000 kg ; M2 = 110 kg ; J1 = 1500 kg·m2 ; J2 = 30 kg·m2 ; l1 = 0.75 m ; l2 = 1.65 m ; k1fr = 0.4 ; k2fr = 0.4 for the power characteristics: k2 = 100 N·s·m−1 . F2 (vC ) = k2 vC N ; The following car motion is studied. In the beginning the vehicle moves rectilinearly (the planes of the front and rear wheels are parallel) during eight seconds, in this case ϕ = π/6. During this time the function F1 (t) changes by the law F1 (t) = 200t (F1 is measured in Newtons, t is measured in seconds), i. e. at the initial time F1 (0) = 0, and at the end of rectilinear motion F1 (8) = 1600. Graphs of dependences of coordinates on time are presented in Fig. E. 4. After eight seconds of rectilinear motion the driver starts to turn the steering wheel at a smooth manner at the angle θ = π(t − 8)/8 , that is, in two seconds the angle θ is equal to π/4 . For this motion F1 (t) = 1600 . By the computed values of constraint reactions Λ1 and Λ2 we get graphs shown in Fig. E. 5 . It follows from the graphs that inequality (E.17) is satisfied, but condition (E.16) is violated when t1 = 9.5147 , θ(t1 ) = 0.5948 . Thus, when 8 < t < 9.5147, the car moves by the I-st type, but after t1 = 9.5147 it changes to the III-rd one. After occurrence of the III-rd type of motion a driver tries to eliminate the side slipping of front wheels of the car, by the way of setting F1 = 0 and changing a turning angle of the front axle according to the law θ = −10(t − t1 ) + θ1 . Let us calculate the constraint reaction |Λ1 | and check if dynamic condition (E.17) is satisfied. As we can see from Fig. E. 6, in case the car keeps moving by the III-rd of motion, then the force FB does not exceed the friction force at least during the time interval 9.5147 < t < 13, that is the dynamic condition (E.17) of realizing the constraint (E.1) is satisfied. At

Fig. E. 5

260

Appendix E

the same time we check if the condition ϕ2 = 0 is fulfilled. As follows from Fig. E. 6 , it starts to be satisfied at the moment t2 = 9.8415; in this case as we can see from calculations, the constraint reaction force |Λ2 | becomes close in value to the friction force between wheels and the road, and the front axle stops moving in side directions.

Fig. E. 6

Fig. E. 7

Fig. E. 8

Appendix E

261

Thus, for t1 < t < t2 the car moves according to the III-rd type, but after t2 = 9.8415 it returns to the I-st type of motion. Now suppose that for t1 < t < 14 the driving force is varied by the law F1 = 200(t − t2 )/(2 − t2 ) . In this case, according to Fig. E. 7 dynamic conditions (E.16) and (E.17) are satisfied, that is, the restored constraint ϕ2 = 0 will be realized further. So, the car is in the I-st type of motion. In Fig. E. 8 graphs of functions during all the time interval of the car motion are given.

§ 6. Reasonable choice of quasivelocities Previously, when studying possible types of the car motion, we had to use different forms of the equations of motion (E.8), (E.20), (E.24), (E.26). This makes certain difficulties, especially when numerically integrating the given systems of differential equations with the help of computer. For similar problems with nonholonomic nonretaining constraints N. A. Fufayev [72] suggests to use a single form of Boltzmann-Hamel’s equations. Let us see, how this idea may be applied in the case of using Maggi’s equations in analogous problems. (We notice that for solving similar problems the equations of motion of nonholonomic systems with variable kinematic structure can be effective [221]). Quite different forms of the equations of motion (E.8), (E.20), (E.24), (E.26) were obtained due to the fact that for different types of the car motion new transition formulas for quasivelocities were chosen every time, or the generalized coordinates were used directly to get the Lagrange equations of the second kind. Now we shall use the form of Maggi’s equations for all the four types of motion, the generalized velocities being always expressed in terms of quasivelocities by the same formulas (E.5). In these formulas the quasivelocities have a certain physical meaning: v∗1 is the angular velocity of rotation of the car body, v∗2 and v∗3 are, according to formulas (E.12) and (E.13), the side velocities of the rear and front axles, correspondingly, aligned with the vectors j and j1 . If the nonholonomic constraints (E.1), (E.2) are realized, quasivelocities v∗2 and v∗3 vanish, and if these constraints turn out to be nonretaining, then these quasivelocities have real nonzero values (except for the instant stops of axles in their side motion). For the motion of the I-st type we still use the equation of motion (E.8) (or (E.21)) and the formulas for determination of the generalized reactions (E.10), (E.11). For the II-nd type of motion, if the constraint ϕ2 = 0 holds, then the generalized reaction Λ2 calculated by the formula (E.11) arises. In this case the equation of motion (E.8) should be completed with the differential equation (E.10), where Λ1 is changed for the projection of the side friction force 1 (−Λfr 1 sign(ϕ )) acting on the rear axle during its side slipping. It is necessary to add the constraint equation (E.2) to these differential equations.

262

Appendix E

For the III-rd type of motion Λ1 is calculated in the same way by formula (E.10), and equation (E.11), in which the reaction Λ2 is replaced with the 2 projection of the side friction force (−Λfr 2 sign(ϕ )), is added to equation (E.8). The constraint equation (E.1) is added to these differential equations. Maggi’s equations are linear combinations of the Lagrange equations of the second kind, therefore, in order to keep the uniformity of differential equations and for the IV-th type of motion corresponding to the holonomic problem, it is convenient to use the form of Maggi’s equations. Eventually the equations of motion will take the form (E.8), (E.10), (E.11), where Λ1 1 fr 2 and Λ2 are replaced with (−Λfr 1 sign(ϕ )) and (−Λ2 sign(ϕ )). The logic of change from one type of motion to another is the same as in § 3. Pay attention that the obtained equations of motion have a singularity at θ = 0. Therefore, the difficulties may occur in calculations, when turning begins with rectilinear motion. In this case, instead of some possible modifications of the system of differential equations, which we used in the calculations given above, we can advise to change initially to the special system of curvelinear coordinates suggested in works [423].

APPENDIX

F

CONSIDERATION OF REACTION FORCES OF HOLONOMIC CONSTRAINTS AS GENERALIZED COORDINATES IN APPROXIMATE DETERMINATION OF LOWER FREQUENCIES OF ELASTIC SYSTEMS A new method for determination of lower frequencies of mechanical systems consisting of elastic bodies connected to each other is offered. The conditions of connection of bodies are written as holonomic constraints, the reactions of which are considered as generalized coordinates. Therefore the number of degrees of freedom proves to be equal to the number of constraints.

On the possibility of introducing generalized reaction forces as Lagrangean coordinates. This Appendix presents a development of the method suggested in the Chapter VI. The equation of frequencies (6.12) of this chapter makes it possible, if necessary, to determine any number of the system’s natural frequencies for a reasonably great number N of dynamically considered oscillation modes of the system elements. However, as a rule, it is necessary to know only several first frequencies and modes. When calculating them one can use the following approximate approach to this problem. The potential energy of the system consisting of elastic bodies connected to each other can be represented as a positively defined quadratic form of the generalized constraint reactions introduced n 1  cij Λi Λj , Π= 2 i,j=1

(F.1)

when considering all the natural vibration modes of the system’s elements quasi-statically. Recall that the coefficients of this form are calculated by formulas (5.15), (5.13) of Chapter VI. In quasistatics the deformed state of all system elements is uniquely determined by setting the quantities Λi , i = 1, n. The given elastic system comes to this state as a result of the fact that its points have obtained displacements, which can be found as linear functions of the reactions Λi , i = 1, n. Hence the position of all points of the system at the time t is uniquely determined by setting the quantities Λi , i = 1, n. Therefore, they can be considered as the generalized Lagrange coordinates; and the kinetic energy of the system can be represented in the form T =

n 1  aij Λ˙ i Λ˙ j . 2 i,j=1

263

(F.2)

264

Appendix F

Here aij , i, j = 1, n, are some constants, the calculation procedure for which will be shown below through a number of examples. Lagrange’s equations of the second kind corresponding to expressions (F.1) and (F.2) are n 

¨ j + cij Λj ) = 0 , (aij Λ

i = 1, n .

j=1

By assuming as in § 5 of Chapter VI ˜ i cos(p t + α) , Λi = Λ

i = 1, n ,

we come to the following equation of frequencies: det[cij − p2 aij ] = 0 .

(F.3)

When calculating the factors aij and cij of this determinant, one need not know the natural frequencies and natural modes of oscillation of the system’s elements. It is essential that these factors can be determined rather simply for the bars of variable section too. Let us start analyzing this approach with solving the problem of approximate determining the first natural frequency and mode of bending oscillations of the cantilever of variable cross-section. Bending oscillations of the cantilever of variable cross-section. Let us assume that at the end x = l the bar is rigidly clamped and that the area of cross-section and the moment of inertia of this section are defined correspondingly as follows: S(x) = A(ξ)S(l) ,

J(x) = B(ξ)J(l) ,

ξ=

x , l

0  ξ  1.

(F.4)

Here A(ξ) and B(ξ) are some prescribed functions. Note that they may be step functions too. Let us introduce into consideration the deflection of neutral layer of the cantilever y(x, t). As the bar is rigidly clamped at the end x = l, then  ∂y  y(l, t) = 0 , = 0. (F.5) ∂x x=l

We shall consider these two conditions as holonomic constraints imposed on the motion of a free bar. The constraint reaction forces are the bending moment M = Λ1 and the lateral force Q = Λ2 applied to the end x = l of the free bar (see Fig. F. 1). The motion of the free bar under the action of these forces can be represented as, first, translational motion (motion of the center of mass C), secondly, rotation about the center of mass and, third, bending. This bending deformation in quasistatics can be found in the following manner.

265

Appendix F

Fig. F. 1

The acceleration of the center of mass Wc and the angular acceleration ϕ¨ at the time t are Wc =

Λ2 (t) , l ρ 0 S(x)dx

ϕ¨ =

Λ1 (t) + (l − xc )Λ2 (t) . l ρ 0 S(x)(xc − x)2 dx

(F.6)

Here ρ is the density, and xc is the coordinate of the center of mass. The intensity of inertia forces caused by translational and rotation motion of the bar appears as q(x, t) = −ρ(Wc + ϕ(x ¨ − xc ))S(x) .

(F.7)

The bending moment in section x, corresponding to the load q(x, t), is equal to  x q(x1 , t)(x − x1 )dx1 . (F.8) M (x, t) = 0

The deflection caused by the action of the bending moment M (x, t) satisfies the equation ∂2y EJ(x) 2 = M (x, t) . ∂x This equation in dimensionless variables y¯ =

y , l

ξ=

x , l

B(ξ)

∂ 2 y¯ = L(ξ, t) . ∂ξ 2

takes the form:

L(ξ, t) =

M (x, t)l EJ(l)

(F.9)

(F.10)

Formulas (F.4), (F.6)–(F.9) imply that the dimensionless moment L(ξ, t) is equal to ¯ 1 (t)f1 (ξ) + Λ ¯ 2 (t)f2 (ξ) . L(ξ, t) = Λ (F.11) Here

2 ¯ 2 (t) = Λ2 (t)l , ¯ 1 (t) = Λ1 (t)l , Λ Λ EJ(l) EJ(l)  ξ A(η)(c − η) f1 (ξ) = (ξ − η) dη , a 0

266

Appendix F 

ξ



(c − η)A(η)(1 − c) A(η) f2 (ξ) = − a b 0  1  1 A(ξ)(c − ξ)2 dξ; b = A(ξ) dξ , a= 0



0

(ξ − η) dη , 1 c= b



(F.12)

1

ξA(ξ) dξ .

0

Integrating (F.10) and taking into account the constraint equation (F.5) produce  1 2  fk (η)(η − ξ) ¯ k (t)hk (ξ) , hk = y¯(ξ, t) = Λ dη . (F.13) B(η) ξ k=1

The potential energy of the bar  1 l M 2 (x, t) Π= dx 2 0 EJ(x) can be represented by using the formulas (F.4), (F.9), (F.11) as Π=

2 EJ(l)  ¯ iΛ ¯j , c¯ij Λ 2l i,j=1

where c¯ij =



0

1

(F.14)

fi (ξ)fj (ξ) dξ . B(ξ)

The kinetic energy of the system  2  ρ l ∂y T = S(x) dx , 2 0 ∂t as follows from formulas (F.4), (F.9), (F.13), is T =

2  1 ¯˙ i Λ ¯˙ j , ρS(l) l3 a ¯ij Λ 2 i,j=1

a ¯ij =



1

A(ξ)hi (ξ)hj (ξ) dξ .

(F.15)

0

Equation (F.3) and expressions (F.14), (F.15) imply that the dimensionless frequencies p∗ related to the required frequencies p as  1 EJ(l) p = p∗ 2 (F.16) l ρS(l) are the roots of the equation det [¯ cij − p2∗ a ¯ij ] = 0 ,

i, j = 1, 2 .

(F.17)

For oscillations with the frequencies pk , k = 1, 2, in accordance with expression (F.13) we obtain: ˜ ˜ ¯ k1 h1 (ξ) + Λ ¯ k2 h2 (ξ)) cos(pk t + α) , y¯k (ξ, t) = (Λ

k = 1, 2 .

267

Appendix F ˜ , Λ ˜ , k = 1, 2, satisfy the equations ¯ ¯ The quantities Λ k1 k2 ˜ ˜ ¯ k1 + (¯ ¯ k2 = 0 , (¯ c21 − p2∗k a ¯21 )Λ c22 − p2∗k a ¯22 )Λ

k = 1, 2 .

This yields that the fist two modes of oscillation of the cantilever can be approximately represented as Yk (ξ) =

Xk (ξ) , Xk (0.5)

Xk (ξ) = h1 (ξ) −

c¯12 − p2∗k a ¯12 h2 (ξ) , c¯22 − p2∗k a ¯22

k = 1, 2 .

The exact solutions for the cantilever of wedge and cone shape were obtained by Kirchhoff in 1879. These solutions are given in many books, in particular, in the reference book by E. Kamke [421] (Chapter IV, paragraphs 4.22, 4.24). For the wedge, where B(ξ) = ξ 3 ,

A(ξ) = ξ ,

natural frequencies p∗ are the roots of the equation J1 (κ)I0 (κ) = I1 (κ)J0 (κ) ,

√ κ = 2 p∗ .

Here J0 (κ) and J1 (κ) are Bessel’s functions of the first kind, and I0 (κ) and I1 (κ) are modified Bessel’s functions of the first kind. Natural modes corresponding to the natural frequencies p∗ are as follows: √ √ J0 (κ)I1 (κ ξ) − I0 (κ)J1 (κ ξ) X(ξ) √ Y (ξ) = . , X(ξ) = X(0.5) ξ In the case of a cone, where A(ξ) = ξ 2 ,

B(ξ) = ξ 4 ,

the equation of frequencies, and functions X(ξ) take the form: κ(J0 (κ)I1 (κ) + I0 (κ)J1 (κ)) = 4J1 (κ)I1 (κ) , √ √ √ √ √ √ I1 (κ)[J1 (κ ξ) − κ 2 ξ J0 (κ ξ)] J1 (κ)[I1 (κ ξ) − κ 2 ξ I0 (κ ξ)] √ √ X(ξ) = + . ξ ξ ξ ξ

By using the suggested approximate approach we obtain — for the wedge: p∗2 = 17.3006 , p∗1 = 5.3187 , h1 (ξ) = 1 −

5ξ ξ3 + 2ξ 2 − , 2 2

h2 (ξ) =

ξ2 ξ3 1 ξ − + − , 6 2 2 6

— for the cone: p∗1 = 8.73521 , 2

h1 (ξ) =

p∗2 = 25.1813 ,

3

7 5ξ 2ξ − 3ξ + − , 6 2 3

h2 (ξ) =

1 ξ ξ2 ξ3 − + − . 6 2 2 6

268

Appendix F

The exact values of the first two frequencies are as follows: — for the wedge: p∗1 = 5.3151 , p∗2 = 15.2072 , — for the cone: p∗1 = 8.71926 , p∗2 = 21.1457 . The second frequency error for the wedge as well as for the cone is great enough. Therefore this approximate method can be used only for determination of the first frequency and the first mode for the cantilever of variable cross-section. The first natural modes for the wedge and the cone are shown in Fig. F. 2. Solid curves correspond to the approximate solution, and dashed curves over them correspond to the exact solution. For visualization of differences between the depicted curves, the deflection at ξ = 1/2 is taken as a unit of measurement for each of them. The cone is a more flexible bar than the wedge and thus the first natural mode of the cone for ξ < 1/2 is located higher than the corresponding curve for the wedge. For the cone the mass per unit of length decreases while approaching to the end by the quadratic law, and for the wedge the linear law is applied. For the bar of constant cross-section the mass per unit of length is constant. Pay attention to the following fact. The first frequency error for the cone is equal to 0.2%, for the wedge it is equal to 0.07%. For the bar of constant cross-section we have: p∗2 = 22.7125 , — approximately: p∗1 = 3.516035 , — exactly: p∗1 = 3.516015 , p∗2 = 22.0345 . Thus, the first frequency error makes up only 5.7 · 10−4 % . Upon comparison of given above errors for the cone, the wedge and the bar of constant cross-section we can expect that for the bar of constant cross-section with the mass localized at the end the approximate solution will become practically

Y1 6 5 4 3 2 1

0.2

0.4

0.6 Fig. F. 2

0.8

1

ξ

269

Appendix F the exact one. Really, in this case we obtain: A(ξ) = 1 + γδ(ξ) ,

B(ξ) = 1 ,

γ=

m2 . m1

Here δ(ξ) is the Dirac delta-function, m2 is the load mass, m1 is the bar mass. For γ = 1 we have: p∗2 = 16.6203 , — approximately: p∗1 = 1.5572990 , — exactly: p∗1 = 1.5572976 , p∗2 = 16.2501 . We see that the first frequency error decreased six times relative to the case when γ = 0. For the cantilever with the disk at its end we obtain quite accurate solution if we consider the presence of disk at the end as the third and the forth holonomic constraints. This system with four degrees of freedom makes it possible to determine to a rather high accuracy not only the first frequency but the second and the third ones. So let’s analyze the following problem. Determination of the lower natural frequencies of bending oscillations of the cantilever of variable cross-section with a disk at its end. In the rotor dynamics, the urgent problem is accurate determination of first two critical critical speeds of the cantilever shaft with a disk at its end. We remind that the values of these critical speeds are proportional to natural frequencies of the cantilever with a disk. Actually, as for instance in the case of marine screw (water propeller) or airscrew, there is not a disk at the shaft end but a body of rather complicated shape. There are methods allowing us to determine the moment of inertia of this body relative to the axis that is perpendicular to the shaft axis. Let us assume that this moment is set in the form I = m2 R 2 , where m2 = γρlS(l) is mass of the body, and R = rl is its radius of inertia. Notice that with given functions A(ξ) and B(ξ) the required natural frequencies p∗ will depend on two parameters γ and r. In the case of the bar of constant cross-section the exact values of the frequencies p∗ are found from the equation   √ V (x) + γxU (x) S(x) + γxV (x) det = 0, x = p∗ . (F.18) S(x) − γr2 x3 T (x) T (x) − γr2 x3 U (x) Here

1 1 (ch x + cos x) , T (x) = (sh x + sin x) , 2 2 1 1 U (x) = (ch x − cos x) , V (x) = (sh x − sin x) 2 2 are the Krylov functions. In approximate determination of the frequencies p∗ we shall consider the conditions of rigid fixing (F.5) as two holonomic constraints as before. We S(x) =

270

Appendix F

shall denote now their reaction forces: the bending moment M (t) and lateral force Q(t) by Λ1 (t) and Λ2 (t) correspondingly. The condition that the deflection y(0, t) is equal to the displacement of mass m2 , and the angle of rotation of the bar’s end  ∂y  ϕ= ∂x x=0

is equal to the angle of the body rotation will be considered as two holonomic constraints imposed on motion of the free bar. The reaction forces of these constraints are the lateral force Λ3 (t) and bending moment Λ4 (t). They are applied to the bar at the cross-section x = 0. Positive directions of reactions, applied to the bar are shown in Fig. F. 3. Formulas (F.6) in this case will take the form: Wc =

Λ2 (t) + Λ3 (t) , l ρ 0 S(x)dx

ϕ¨ =

Λ1 (t) − Λ4 (t) + (l − xc )Λ2 (t) − xc Λ3 (t) . l ρ 0 S(x)(xc − x)2 dx

The intensity of inertial forces q(x, t) will be calculated by formula (F.7) as before; formula (F.8) will take the form:  x M (x, t) = Λ4 (t) + xΛ3 (t) + q(x1 , t)(x − x1 ) dx1 . 0

When going to dimensionless variables we obtain: L(ξ, t) =

4 

¯ k (t)fk (ξ) . Λ

k=1

Here

¯ 1 (t) = Λ1 (t)l , Λ EJ(l) 2 ¯ 3 (t) = Λ3 (t)l , Λ EJ(l)

2 ¯ 2 (t) = Λ2 (t)l , Λ EJ(l) ¯ 4 (t) = Λ4 (t)l . Λ EJ(l)

The functions f1 (ξ) and f2 (ξ) are set by formulas (F.12), and the functions f3 (ξ) and f4 (ξ) are as follows:   ξ (η − c)cA(η) A(η) − f3 (ξ) = ξ + (ξ − η) dη , a b 0

Fig. F. 3

271

Appendix F f4 (ξ) = 1 +



ξ

0

A(η)(η − c) (ξ − η) dη . a

Formulas (F.13), (F.14), (F.17) remain valid, but their indices i, j and k run now from 1 to 4. When calculating the kinetic energy it is necessary to take into account the kinetic energy of the disk, therefore the factors a ¯ij of determinant (F.17) in this case are as follows:  1 a ¯ij = A(ξ)hi (ξ)hj (ξ) dξ + γhi (0)hj (0) + γr2 ϕi (0)ϕj (0) , i, j = 1, 4 . 0

(F.19)

Here

dhi (ξ) , i = 1, 4 . dξ In the case of the bar of constant cross-section the equation (F.18) allows us to calculate the natural frequencies exactly and so to estimate an error of this approximate method. The radius of inertia for the thin disk R is equal to R1 /2, where R1 is the radius of the disk and therefore R1 = 2lr. If the shaft of radius r1 and the disk of thickness h are made of the same material then for r1 = l/20 and h = R1 /20 we obtain ϕi (ξ) =

γ = 160 r3 .

(F.20)

Assuming that γ and r are related to each other with this expression and r varies within the range from 0 to 1/2, let us follow the change of error for the first, second and third frequencies. Upon calculations we obtain the following values for the error in percentage terms (%): r = 0.000

1.5 · 10−4

r = 0.125

3.7 · 10−5

9/5 · 10−2

0.85

r = 0.250

1.4 · 10−6

3.7 · 10−4

0.40

r = 0.500

−6

−5

0.35

− 9.0 · 10

0.56

2.67

3.9 · 10

The first column corresponds to the first frequency, the second column corresponds to the second frequency and the third one corresponds to the third frequency. We see that the higher frequency, the greater error. For r  0.125 the error for the first frequency is close to the limits of accuracy which is provided by the software package "Mathematica 5.2". In this regard one can say that this method permits to determine the first frequency exactly. Therefore it may be used both for the rotor dynamics and for testing the programs for analysis of complicated mechanical systems. In rotor engineering it is important to have the analytical dependence of the first natural frequency on the system’s parameters. This method based on consideration of four holonomic constraints does not allow us to do that

272

Appendix F

as it leads to the solution of algebraic equation of the fourth order. But if we limit ourselves to consideration of only two constraints at the end where the disk is located, then the required first frequency will be determined in analytical form as a root of biquadratic equation. Let us prove, that this simple solution also makes it possible to find the first frequency accurately enough. When getting this solution it is reasonable to measure the coordinate of the bar cross-section not from the free end but from the end that is rigidly clamped. Formulas (F.4) and (F.17) remain valid, but now S(l) and J(l) will correspond not to the rigidly clamped end, but to the place of disk fixation. The bending moment Λ1 (t) and lateral force Λ2 (t), applied to the end x = l, are constraint reactions and considered in this problem as the generalized coordinates. Their positive directions, as well as the positive direction of the moment M (x, t) applied to the cross-section x, are shown in Fig. F. 4. The dimensionless bending moment L(ξ, t) introduced by formula (F.9) is equal in this case to ¯ 2 (t)f2 (ξ) , ¯ 1 (t)f1 (ξ) + Λ L(ξ, t) = Λ f1 (ξ) = 1 ,

f2 (ξ) = 1 − ξ ,

¯ 1 (t) = Λ1 (t)l , Λ EJ(l)

2 (F.21) ¯ 2 (t) = Λ2 (t)l . Λ EJ(l)

Expression (F.11), as seen, survives and therefore the potential energy will be written in the form (F.14). Integrating equation (F.10) and taking into account that  ∂ y¯  = 0, y¯(0, t) = ∂ξ ξ=0 imply expression (F.13), where now hk (ξ) =



0

ξ

fk (η)(η − ξ) dη , B(η)

k = 1, 2 .

(F.22)

As the deflection is represented in the same form (F.13), the kinetic energy will be written in the same form (F.15) too. The factors a ¯ij in this case should

Fig. F. 4

273

Appendix F

be calculated by formulas (F.19), but now hi (0) should be replaced with hi (1), and ϕi (0) should be replaced with ϕi (1). When calculating for the bar of constant cross-section the error of the first and second frequencies in percentage terms (%) for the same relation (F.20) between γ and r, we obtain: r r r r

= 0.000 = 0.125 = 0.250 = 0.500

0.47 8.4 · 10−2 2.6 · 10−3 1.1 · 10−5

58 15.6 0.21 1.1 · 10−3

For r  0.25 we can say that for the first frequency we obtain the exact value. Notice, however that for r = 0.25 the disk diameter is equal to the shaft length, and for r = 0.5 it is two times greater. For such relation between these quantities for the assumed values r1 = l/20 and h = R1 /20 this disk can not be regarded as a perfectly rigid body. It is necessary to take into account the influence of its compliance on the natural frequencies of the system. It is feasible but it will require additional calculations, the basic framework of which will be shown through the example of the cantilever with a flexible bar at its end. This example will require no new mathematical apparatus. It is reduced to the same calculations as above. Determination of the first three frequencies of the cantilever with a flexible bar at its end. Let us analyze the problem, when the bar executing longitudinal oscillations in the mechanical system depicted in Fig. VI. 2 is absent (see Fig. F. 5). Within the frames of such problem we have three constraints and three reaction forces correspondingly. The bending moment Λ1 (t) and the lateral force Λ2 (t) are applied to the cantilever as is shown in Fig. F. 4. The third reaction force is the lateral force Λ3 (t) applied to the bar which is perpendicular to the cantilever. Both kinetic and potential energy of the cantilever are determined by the formulas given above. Therefore it is necessary to take into account only the second bar. When released from the constraints it becomes free and similar to the bar shown in Fig. F. 1, but now the bending moment M (t) = Λ1 (t) and lateral force Q(t) = Λ3 (t) are applied not to the end of the bar but to the cross-section x∗ = zl. Therefore, the constraint equations will be written in the form  ∂y  y(x∗ , t) = 0 , = 0. (F.23) ∂x x=x∗

274

Appendix F

We shall not provide the parameters of the second bar with indices when considering the question how the deflection curve will change depending on the place of application of the reactions. We shall do that upon obtaining expressions for the potential energy of its deflection and for the deflection curve. Formulas (F.6) in this case will appear as Wc =

Λ3 (t) , l ρ 0 S(x) dx

ϕ¨ =

Λ1 (t) + (x∗ − xc )Λ3 (t) , l ρ 0 S(x)(xc − x)2 dx

and formula (F.7) remains valid. The bending moment M (x, t) applied to the left of the cross-section x = x∗ is set by expression (F.8), and the bending moment applied to the right of cross-section takes the form  l M (x, t) = q(x1 , t)(x1 − x) dx1 , x∗ < x < l . x

Hence the bar is divided into two sections and the deflections of its left and right parts have to be calculated independently. Denoting the bending moment M (x, t) for 0 < x < x∗ by M1 (x, t), and for x∗ < x < l by M2 (x, t), and going to dimensionless variables (F.9), we obtain: ¯ (2) (t)f1n (ξ) + Λ ¯ (2) (t)f3n (ξ) , Ln (ξ, t) = Λ 1 3

Fig. F. 5

n = 1, 2 .

275

Appendix F Here

2 ¯ ¯ (2) (t) = Λ1 (t)l , Λ ¯ (2) (t) = Λ3 (t)l , Λ 1 3 EJ(l) EJ(l)  ξ (c − η)A(η) f11 (ξ) = (ξ − η) dη , 0  ξ  z , a 0   ξ (c − η)(z − c)A(η) A(η) f31 (ξ) = − (ξ − η) dη , 0  ξ  z , a b 0  1 A(η)(η − c) (ξ − η) dη , z  ξ  1 , f12 (ξ) = a ξ   1 (η − c)(z − c)A(η) A(η) + f32 (ξ) = (ξ − η) dη , z  ξ  1 . a b ξ (F.24) We remind that the values a, b, c, included in these expressions are calculated ¯ (2) (t) means ¯ (2) (t) and Λ by formulas (F.12). The index "2"of the quantities Λ 1 3 that transition to the dimensionless variables corresponds to the parameters l, E and J(l) of the second bar (see Fig. 5). The functions A(ξ), B(ξ) and the values a, b, c, should be also provided with index "2"hereinafter, but for the sake of simplicity they are omitted. Integrating equation (F.10) for L(ξ, t) = L1 (ξ, t), and then for L(ξ, t) = L2 (ξ, t), and taking into account the constraint equations (F.23) imply (2)

(2)

¯ (t)h31 (ξ) , ¯ (t)h11 (ξ) + Λ y¯(ξ, t) = Λ 1 3 y¯(ξ, t) =

¯ (2) (t)h12 (ξ) Λ 1

+

¯ (2) (t)h32 (ξ) , Λ 3

0  ξ  z, z  ξ  1,

(F.25)

where fk1 (ξ) =



ξ

fk2 (ξ) =



z

ξ

z

fk1 (η)(η − ξ) dη , B(η)

fk2 (η)(ξ − η) dη , B(η)

0  ξ  z,

z  ξ  1,

k = 1, 3 .

By using the unit function U (x) =



1, 0,

x  0, x < 0,

we represent expressions (F.25) as ¯ (2) (t)h1 (ξ) + Λ ¯ (2) (t)h3 (ξ) , y¯(ξ, t) = Λ 1 3

0  ξ  1.

(F.26)

Here hk (ξ) = hk1 (ξ)U (z − ξ) + hk2 (ξ)U (ξ − z) .

(F.27)

276

Appendix F

The potential energy of deformation of the second bar has to be calculated independently for its right and left sections. Calculating and summing these energies produce Π=

EJ(l) (2) ¯ (2) (2) ¯ (2) (2) ¯ (2) 2 ¯ (2) (¯ c11 (Λ1 (t))2 + 2¯ c13 Λ ¯33 (Λ 1 (t)Λ3 (t) + c 3 (t)) ) , 2l

where (2)

c¯kk =



z

0

(2)

c¯13 =



2 fk1 (ξ) dξ + B(ξ)

z

0



(F.28)

1

2 fk2 (ξ) dξ , k = 1, 3 , B(ξ) z  1 f11 (ξ)f31 (ξ) f12 (ξ)f32 (ξ) dξ + dξ . B(ξ) B(ξ) z

Adding the potential energy of bending of the first bar to potential energy (F.28), we represent their sum in the form 3 E1 J1 (l1 )  ¯ (1) Λ ¯ (1) . Π= c¯ij Λ i j 2l1 i,j=1

(F.29)

Here index "1" means that this quantity corresponds to the first bar. The ¯ (1) , i = 1, 3, are introduced by the formulas: dimensionless variables Λ i ¯ (1) (t) = Λ1 (t)l1 , Λ 1 E1 J1 (l1 )

2 ¯ (1) (t) = Λk (t)l1 , Λ k E1 J1 (l1 )

k = 2, 3 .

Note that in these formulas J1 (l1 ) corresponds not to the place of rigid fixing, as it was in the beginning of this Appendix, but to the point where the first bar is connected to the second one (see Fig. F 5). In formulas (F.24) and (F.28) all quantities refer to the second bar. Introduction of the parameters E1 J1 (l1 )l23 , E2 J2 (l2 )l13

α=

β=

l2 l1

allows us to represent the potential energy (F.28) of the second bar as Π2 = α

E1 J1 (l) (2) ¯ (1) (2) ¯ (1) (2) ¯ (1) −1 2 ¯ (1) (¯ c11 (Λ1 (t))2 β −2 + 2¯ c13 Λ + c¯33 (Λ 1 (t)Λ3 (t)β 3 (t)) ) . 2l1

This implies that the factors c¯ij in expression (F.29) are as follows: (1)

(2)

c¯11 = c¯11 + αβ −2 c¯11 , (2)

c¯13 = αβ −1 c¯13 ,

(1)

c¯22 = c¯22 ,

(1)

c¯12 = c¯12 , c¯23 = 0 ,

(2)

c¯33 = α¯ c33 .

Here in accordance with formulas (F.14), (F.21)  1  1  1 dξ (1 − ξ) dξ (1 − ξ)2 dξ (1) (1) (1) , c¯12 = , c¯22 = . c¯11 = B1 (ξ) B1 (ξ) 0 B1 (ξ) 0 0

277

Appendix F

The kinetic energy of the first bar will be represented by using expressions (F.15), (F.21), (F.22) in the form T1 = (1)

a ¯ij (1) h1 (ξ)

=



0

2  1 (1) ¯ ˙ (1) Λ ¯˙ (1) , ρ1 S1 (l1 ) l13 a ¯ij Λ i j 2 i,j=1  1 (1) (1) = A1 (ξ)hi (ξ)hj (ξ) dξ , 0

ξ

(ξ − η) dη , B1 (η)

(1) h2 (ξ)

=



ξ

0

(1 − η)(ξ − η) dη . B1 (η)

Let us calculate the kinetic energy of the second bar now. The assumption that amplitude of oscillations of the bars under consideration is small allows us, as was noted in § 2 of Chapter VI, to calculate the kinetic energy of translational motion of the second bar along the axis independently from the kinetic energy of its motion in the direction that is perpendicular to its axis. The kinetic energy of translational motion of the second bar is m2 l12 (1) ¯˙ (1) (1) ¯˙ (1) )2 , (h1 (1)Λ1 + h2 (1)Λ 2 2  1 m2 = ρ2 S2 (l2 ) l2 A2 (ξ) dξ .

T21 =

0

Displacements of the cross-sections of the second bar in the direction perpendicular to the bar axis are caused, first, by rotation of the bar about the cross-section x∗ = zl2 , and, secondly, by the deflection defined by expression (F.26). Therefore we have: ¯ (2) (t)h(2) (ξ) + Λ ¯ (2) (t)h(2) (ξ)) . y2 (ξ, t) = l2 (ψ(t)(z − ξ) + Λ 1 1 3 3 Here (1)  dhk  ϕk (1) = , dξ ξ=1

¯ (1) (t) + ϕ2 (1)Λ ¯ (1) (t) , ψ(t) = ϕ1 (1)Λ 1 2 (2)

(2)

k = 1, 2 .

Index "2" of the functions h1 (ξ) and h3 (ξ) means that these functions defined by expressions (F.27) are calculated for the parameters of the second bar. Taking into account that ¯ (1) , ¯ (2) = αβ −2 Λ Λ 1 1

¯ (2) = αβ −1 Λ ¯ (1) , Λ 3 3

the kinetic energy T22 =

1 ρ2 2



0

l2

S2 (x)



∂y2 ∂t

2

dx ,

278

Appendix F

will be represented as T22 =

 1 1 ¯˙ (1) + ϕ2 (1)Λ ¯˙ (1) )(z − ξ)+ A2 (ξ)((ϕ1 (1)Λ ρ2 S2 (l2 )l23 1 2 2 0 ¯˙ (1) h(2) (ξ) + αβ −1 Λ ¯˙ (1) h(2) (ξ))2 dξ . +αβ −2 Λ 1

1

3

3

Introducing into consideration the third parameter γ=

ρ2 S2 (l2 )l2 , ρ1 S1 (l1 )l1

the total kinetic energy of the second bar appears as follows: T2 =

3  γ (2) ¯ ˙ (1) Λ ¯˙ (1) . ρ1 S1 (l1 ) l13 a ¯ij Λ i j 2 i,j=1 (2)

Analytic expressions for the factors aij , dependent on the functions A2 (ξ) and parameters α and β, are rather intricate and thus not given here. Note that they are easily found with the software package "Mathematica 5.2". The kinetic energy of both bars is T =

3  1 ¯˙ (1) Λ ¯˙ (1) , ρ1 S1 (l1 ) l13 a ¯ij Λ i j 2 i,j=1

(1)

(2)

a ¯ij = a ¯ij + γ¯ aij .

We’ll find the required natural frequencies p∗ by solving equation (F.17). Notice that in formula (F.16) of transition to dimensional frequencies all quantities correspond to the first bar at the point of its connection to the second bar. Comparison with the bars of constant cross-section. The problem of bars of constant cross-section has been solved exactly by the methods of mathematical physics. As this takes place the equation of frequencies is obtained by equating the determinant of sixth order to zero. Its elements are the Krylov functions, the arguments of which depend on the parameters α, γ and z. This intricate transcendental equation, the computational solution of which was a matter of some difficulty even for modern computers, was used for testing the method suggested in § 3 of Chapter VI. Note that calculation of first three frequencies by using this suggested method does not create any difficulties. As stated above, the problem under investigation is a particular case of the problem discussed in §§ 5 and 6 of Chapter VI. When the bar executing longitudinal vibration is absent, the frequency determinant is a determinant of third order. Comparing the roots of the transcendental equation with the roots of the frequency equation shows that the first frequency is determined with four valid significant digits in the second approximation, the second frequency is determined with the same accuracy in the fourth approximation,

279

Appendix F

and the third one is obtained with the same accuracy in the sixth approximation. If the first and the second bars are made of the same material and have the same cross-sections, then at z = 1/2 the solution depends on a single parameter β = l2 /l1 , for in this case γ = β and α = β 3 . The calculations show that for the first frequency the error decreases as β rises, and for β = 1/8, 1/4, 1/2, 2 it is equal to 0.22, 0.12, 0.056, 0.0022 percent (%) correspondingly. Note that in this example in case β  0.25, it is reasonable to consider the second bar as a concentrated mass located at the end of the cantilever beam and to use the method presented in the beginning of this Appendix. Let us discuss briefly the errors of the method under consideration for the second and third frequencies. Let us examine this problem through the example of the bars, differing only in length. For α = β = γ = 1 and z = 1/2 the exact and approximate values of first three dimensionless frequencies p∗ are as follows: 1.44851 ,

6.20782 ,

14.0641 ,

1.44876 ,

6.24235 ,

14.1204 .

The errors in percentage terms (%) are equal correspondingly to 0.017 ,

0.56 ,

0.40 .

If the second bar is symmetrically positioned in relation to the first one, there exists a mode of oscillations such that the first bar does not oscillate, and both halves of the second bar oscillate like a cantilever of length l = βl1 /2. The first frequency of the cantilever oscillation in the dimensionless variables is 4 (F.30) p∗ = 3.516 2 . β This frequency in the series of frequencies of the system consisting of two bars has the number n. This number increases as β decreases. For example, for β = 1/4 it will be the ninth frequency, and the third root of equation (F.17) will correspond to it. Let us find this root in the explicit form. When the second bar does not oscillate, then the bending moment Λ1 and the lateral force Λ2 applied to the end of the first bar vanish. Therefore in this oscillation mode only the lateral force Λ3 applied to the middle of the second bar is not equal to zero. Under the action of this force the second bar moves translationally and bends so that the application point of the force Λ3 is immovable. In quasistatics the intensity of inertial forces is constant in this case, therefore either the second or the third root of equation (F.17) at z = 1/2 is equal to   1  ξ c 4 2 p∗ = , c= f (ξ) dξ , f (ξ) = (ξ − η) dη , a β2 0 0  1  1 a= h2 (ξ) dξ , h(ξ) = f (η) (ξ − η) dη . 0

ξ

280

Appendix F

When calculating we obtain p∗ = 3.530

4 . β2

(F.31)

This frequency exceeds its exact value obtained by formula (F.30) by 0.40 %. For β = 1/2 the frequency approximately defined by expression (F.31) corresponds to the exact value of the fourth frequency, for β = 1 and β = 2 it corresponds to the third frequency, and for β = 4 it corresponds to the second frequency. Hence this approximate method makes it possible to determine the first frequency for any values of the system parameters with a rather high degree of accuracy, and for some values of the parameters it allows us to define the second and the third frequencies as well.

APPENDIX

G

THE DUFFING EQUATION AND STRANGE ATTRACTOR The nonhomogeneous Duffing equation with a linear resistance is studied. In this case the possibility of arising of strange attractors and periodical solutions with a period multiple to the period of excitation, depending on the excitation level, is investigated by a numerical method. The Appendix presents the first part of the paper by P. E. Tovstik and T. M. Tovstik [425]. The probability properties of a strange attractor considered in the second part of this paper are not covered in the Appendix. A more completed table of solution properties is given. The relationship of strange attractors with the classical theory of motion stability is presented in the monograph by G. A. Leonov [426].

In § 4 of Chapter VI the nonhomogeneous Duffing equation (4.9) has been obtained for describing the lateral vibration of a beam with the supports fixed in the logitudinal directiion. Taking into consideration a linear resistance and some changes in notation, in dimensionless variables it appears as dx d2 x + x + x3 = b cos ωt , +c dt2 dt where ω=

 π 2 L



EJ Ω, ρS

b=

(G.1)

2L4 √ f0 . π 4 E JS

Here Ω and f0 are the frequency and the amlitude of the disturbed force, respectively, c is a coefficient characterizing damping, E and ρ are the coefficient of elasticity and density of beam material, respectively, L is the length of beam, S is the cross-section area of the beam, J is the moment of inertia of cross section of the beam with respect to zero line. Equation (G.1) includes three parameters — c, b, and ω. Let us fix two of them (ω = 1, c = 0.25) and vary parameter b in a wide range 0  b  100. Equation (G.1) has been integrated numerically under the arbitrary given initial conditions x(0), x(0) ˙ belonging to the domain of ([−5.0, 5.0] × [−5.0, 5.0]). It has been determined, how the existance of strange attractors and limiting solutions with a period multiple to the period of excitation depends on the value of b and the initial conditions. To this end, the range 0  b  100 has been partitioned with a step 0.1, and for each bi = 0.1i the qualitative character of a limiting trajectory has been defined. The neighbour values of bi with the same qualitative characteristics have been united in intervals. When subsequently varying i from 0 up to 1000, 55 intervals with different qualitative behaviour of solutions have been found in total. 281

282

Appendix G Таблица G. 1. b

0.0 − 2.9 3.0 − 9.6 9.7 − 11.9 12.0 − 14.8 14.9 − 22.9 23.0 − 35.5 35.6 − 38.6 38.7 − 38.9 39.0 39.1 − 39.2 39.3 − 39.39 39.4 39.5 39.6 39.7 − 40.6 40.7 − 41.2 41.3 41.4 − 41.6 41.7 − 44.3 44.4 − 48.6 48.7 − 49.1 49.2 49.3 49.4 49.5 − 50.6 50.7 50.8 50.9 − 52.0

n k 1 2 1 2 1 2 2 2 4 2 2 4 2 4 2 1 3 1 2 1 2 2 3 2 1 2 2 1

1 1, 1 1 1, 1 1 1, 1 2, 2 4, 4 4, 4, 3, 3 4, 4 8, 8 A, A, 8, 8 A, A A, A, 10, 10 A, A A A, 5, 5 A A, 3 3 3, 3 6, 6 A, 6, 6 A, A A 4, 4 8, 8 A

b

n

k

52.1 52.2 − 52.7 52.8 − 53.1 53.2 53.3 − 54.3 54.3 − 54.7 54.8 − 54.9 55.0 55.1 − 57.9 58.0 58.1 − 58.2 58.3 58.4 58.5 − 58.7 58.8 58.9 − 59.0 59.1 − 60.8 60.9 − 62.0 62.1 62.2 − 62.3 62.4 − 62.7 62.8 − 67.4 67.5 − 77.3 77.4 − 91.5 91.6 91.7 − 92.7 92.8 − 100.0

3 2 2 2 1 2 3 8 2 1 2 3 3 3 4 3 3 2 3 4 3 2 1 2 4 3 2

A, 2, 2 2, 2 4, 4 A, A A A, 1 A, 5, 1 A, 15, 15, 15, 10, 10, 5, 1 A, 1 1 A, 1 A, A, 1 4, 4, 1 2, 2, 1 3, 3, 2, 1 2, 2, 1 1, 1, 1 1, 1 A, 1, 1 3, 3, 1, 1 3, 1, 1 1, 1 1 1, 1 1, 1, 1, 1 1, 1, 1 1, 1

Table G. 1 contains the results. In it, for corresponding values of b there are: — the number n of different limiting solutions, which can be obtained when changing the initial conditions, — the multiplicity k of a period kT of the limiting solution, the number of values of k given through a comma being equal to n, — in the case, when a periodical solution is not kT -periodical, but a strange attractor, the number k is substituted in Table G. 1 for the letter A. For example, for b = 54.9 there are three (n = 3) different stable limiting solutions: a strange attractor, a 5T -periodical solution, and a T -periodical solution. If we compare this table with the results obtained in the paper [425], then we see that this table only complements them. A greater number

283

Appendix G

of intervals b has been got as a result of decreasing a partition step. This demonstrates that a more detailed study of the variation interval of b can lead to arising of new intervals that are qualitatively different from the considered ones. It follows from the table, that strange attractors occur in the range 39.4  b  62.1. Note that for b > 100 strange attractors also occur, but in this case the vibration amplitude is so great that this vibration can hardly be modelled by the Duffing equation. We illustrate the dependence of limiting solutions on the initial conditions through the example b = 4.0. As follows from Table G. 1, for this value of b there are two stable T -periodical solutions. In Fig. G. 1 on the part of the plane −5  x(0), x(0) ˙  5 the domains of initial conditions that lead to the first or second solutions are given. These solutions are presented in Fig. G. 2. The part of the plane −50  x(0), x(0) ˙  50, which is a hundred times greater, has been also regarded, in this case it turned out that the structure of the domain is more complicated than in Fig. G. 1. It is impossible to describe here all the considered variation range of b. We limit ourselves to four consecutive intervals in the range 40.0  b  45.0 and regard four successive values of b. The results are given in the form of the Poincar´e diagrams (see Fig. G. 3), on which points with the coordinates x(mT ), x(mT ˙ ) for integer m are dotted. The Poincar´e diagrams (or crosssections) are the powerful means, which make it possible to determine the qualitative character of solution behaviour and find out bifurcations, i. e. the transfers from one qualitative state to another.

Fig. G. 1

284

Appendix G x 2

2

1 1

π –1

–2 Fig. G. 2

Fig. G. 3

2π

t

Appendix G

285

If a limiting solution is kT -periodical, then a diagram has k different dots. The number of dots for a strange attractor depends on duration of integration. In Fig. G. 3 each of the strange attractors contains 800 dots. Consider consecutively the diagrams shown in Fig. G. 3. On the first of them (for b = 40.0) two strange attractors 1 and 2 are depicted, which can be obtained under certain initial conditions. As b increases, attractors approach to each other, and for b = 41.0 there is only one attractor under any initial conditions. For b = 43.0 we have two stable limiting solutions: a strange attractor 1 and a 3T -periodical solution 2 depicted by three dots on the Poincar´e diagram. As b grows further, a strange attractor vanishes, and there remains only one 3T -periodical solution depicted by three dots in Fig. G. 3 for b = 45.0.

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INDEX

Abstract constraints, 151, 152 Acceleration vector for arbitrary mechanical system, 107 Amplitude-frequency characteristics, 164 Andronov–Hopf bifurcation, 229 Appell function, 200 Appell’s equations, 101, 123 Appell’s form equations with third-order constraints, 123 Approximate periodic solutions, 236 Approximate solution of Lagrange’s equations, 237 Axes of curvilinear coordinates, 214

Contravariant components, 215, 216 of tangent vector, 105 of velocity vector, 218 Coordinate line, 213 plane, 215 surface, 213 Covariant components, 216 of velocity vector, 218 Curves of static bend (deflection), 158 Curvilinear coordinates, 213 Cylindrical system of coordinates, 222 D’Alembert–Lagrange principle, 13, 207 Dynamic compliances, 156 Dynamic control of the motion of a car, 245 Dynamic Euler equations, 200

Basic metric form, 217 Basic metric tensor, 217 Basis of the Lie algebra, 200 Basis of s-dimensional Lie algebra with the commutator, 199 Bending oscillations of the cantilever of variable cross-section, 264 Bubnov–Galerkin method, 237 Chaplygin’s equations, 30, 39 in quasicoordinates, 43 Chaplygin’s type equations, 32, 42, 197 Chetaev’s postulate, 76, 92–93 Chetaev’s type constraints, 76 Christoffel symbols of first kind, 219 of second kind, 219 Coefficients of influence, 156 Complementary metric form, 217 Complementary metric tensor, 217 Condition of a free motion of the Chaplygin sledge, 242 Condition of the ideality of constraints, 110 Conditions of N. G. Chetaev, 202, 203, 207 Configuration space of the system, 230 Constraints completely defined by their analytic representations, 9

Effective potential, 230 Elastic constraints, 165, 167, 168 Equations of motion, 214 of nonholonomic systems in quasicoordinates, 200 in quasicoordinates, 211 Equations of noncomplete program of motion, 96 Equations of nonholonomic systems in the Poincar´e–Chetaev variables, 200 Equations, represented in Maggi’s form, for third-order constraints, 121 Euclidean structure of the tangent space, 106 Formula for computing the Christoffel coefficients of the first kind, 219 Free (unconstrained) motion of nonholonomic system, 239 Gaussian function, 116 Gaussian principle, 103, 109 generalized, 119, 183 General (fundamental) equation of dynamics, 110–111

327

328 Generalized control force, 127 Generalized D’Alembert–Lagrange principle, 202 Generalized forces, 6 corresponding to quasivelocities, 94 Generalized impulses, 7 Generalized operator Appell, 114 Lagrange, 114 Generalized problem of P.L. Chebyshev, 126 Generalized reactions, 82, 150 Gradient of the function, 215 Hamel–Boltzmann equations, 31, 44 Hamel–Novoselov equations, 32, 43, 197 Hamiltonian nabla operator, 216 Harmonic coefficients of influence, 156 High-order program constraints, 135 Ideal constraints holonomic, 3, 4, 81 nonholonomic, 29, 81 Ideal control, 134 Ideality condition of control, 135 Introducing generalized reaction forces as Lagrangean coordinates, 263 Kinematical characteristics, 194 Kinematic control of the motion of a car, 245 Kronecker symbols, 215 Lagrange multipliers, 82, 150 Lagrange operator, 6, 222 Lagrange’s equations of first kind, 4, 7 of first kind in generalized coordinates, 11 for nonholonomic systems, 33 of second kind with multipliers, 11 undetermined, 33 Lam´e factors, 214 Linear transformation of forces, 94

Index Maggi’s equations, 30, 201 second group, 30, 101 Mangeron–Deleanu principle, 135 Manifold of positions of the mechanical system, 105 Maximum principle of Pontryagin, 185 Metric tensor, 106 Mixed problem of dynamics, 128 Motion of dynamically symmetric ball on absolutely roughened plane, 233 Natural (fundamental) basis, 215 Necessary and sufficient conditions for the existence of free motion of nonholonomic systems, 240 New class of control problems, 128 Newton’s determinacy principle, 136 Noncomplete program of motion, 100 Nonholonomic bases, 28 Nonlinear second-order nonholonomic constraints, 116 Normal (natural) forms (modes) of oscillations, 153 Normal (natural) frequency, 153 Objects of nonholonomicity, 31 Parametrization of constraints, 195 Permutable relations, 206, 207 Poincar´e–Chetaev equations, 34 Poincar´e–Chetaev parameters, 202 Poincar´e–Chetaev–Rumyantsev equations, 44, 201, 206 Poincar´e diagrams, 283 Poincar´e equations, 44, 200 Poincar´e parameters, 194 Possible types of the car motion, 251 Principle of virtual accelerations, 110 Principle of virtual displacements, 111 Principle of virtual velocities, 110 Program constraints, 116, 119, 135 Quasicoordinates, 41 Quasivelocities, 41 Reciprocal (dual) basis, 107, 215 Representation point, 1, 2

329

Index Residual, 235 Rule of dummy index, 216 Rules of raising and missing an index, 217 Series in resonance frequencies, 186 Set, bifurcational by Smale, 231 Steady motions, 231 of conservative nonholonomic systems, 229 Steady rolling of disk on horizontal plane, 233 Strange attractors, 281 Structural constants of Lie algebra, 199 Subspace of motions, 8 of reactions, 8 Suslov–Jourdain principle, 68

Tangent space, 106 Udwadia–Kalaba equations, 34 Variation of the generalized velocity, 67 Variations of coordinates, 12, 106 Vector of generalized impulse, 113 Velocity vector of mechanical system, 113 Virtual displacements, 12, 106, 112 Virtual elementary work, 7, 84, 106 Virtual velocity, 112 Voronets equations, 40 Voronets–Hamel coefficients of first kind, 198 Voronets–Hamel equations, 44 Voronets–Hamel type equations, 198