Proiect Aerolasticitate Si Dinamica Zborului RV-7 [PDF]

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PROIECT AEROLASTICITATE SI DINAMICA ZBORULUI RV-7

Profesor coordinator: Sef lucrari dr. ing. Zaharia Sebastian Student: Cotinghiu Ionut-Sebastian Grupa: CA, 2631

TEMA DE PROIECT AEROELASTICITATEA ȘI DINAMICA STRUCTURILOR Se consideră un avion de acrobatie având caracteristicile: Capacitate: 2 locuri G max : 7995.15 daN Viteză maximă: 351 km/h la altitudinea de z=6900m Sistemul de propulsie: Lycoming O-360 Distanța maximă de zbor: 1239 km

Să studieze acest avion din punct de vedere aeroelastic, organul principal de calcul fiind aripa. Proiectul va cuprinde : 1. Preliminarii • Caracteristici avion • schița avionului (in 3 vederi), desenul de ansamblu al organului de calcul • cazul / cazurile critice de calcul (diagrama de manevra si rafală) • stabilirea numărului de secțiuni de calcul, schema necesara pentru calculul organului stabilit 2. Caracteristici elastice pentru organul de calcul • evaluarea caracteristicilor mecanice – rigidități de încovoiere si de torsiune – in secțiunile de calcul • trasarea curbelor de variație ale acestora pe anvergura 3. Determinarea funcțiilor de influenta 4. Calculul deformațiilor statice

5. Aeroelasticitatea statica Varianta I • •

Divergenta aeroelastica produsa de rasucire sub portanta Efecte aeroelastice asupra comenzilor

Varianta II • •

Divergenta aeroelastică produsă de rasucire sub portanță Divergenta aeroelastică datorata portantei si rezistentei la înaintare

6. Vibratii libere •

Se studiaza vibratiile decuplate de incovoiere– rasucire pentru organul de calcul considerat

7. Aeroelasticitate dinamica •

Calculul de flutter

8. Analiza modală a organului de calcul analizat (Ansys).

Caracteristici avion

RV-7 este un avion cu doua locuri, cu un singur motor si o aripa joasa prinsa in fuselaj. RV-7 este inlocuitorul lui RV-6 si este extreme de similar modelelor mai vechi, cu aripi lungi, rezervoare largi si un stabilizator mai mare pentru a-i imbunatatii caracteristicile in timpul zborului. RV-7 a încorporat multe schimbări ca urmare a lecțiilor învățate de-a lungul anilor în producerea a peste 2.000 de kituri RV-6. Raza aeronavelor RV-7 va accepta motoare mai mari, inclusiv Lycoming IO360, până la 200 CP (149 kW). RV-7 are de asemenea o anvergură a aripilor și o zonă de aripă peste RV-6, precum și mai mult spațiu pentru înălțime, spațiu pentru picioare și o sarcină utilă sporită. RV-7 transportă un total de 42 de galoane americane de combustibil, de la 38 de galoane americane pe RV-6.

1.3. Diagrama de manevra

Vmax_kph ≔ 351 km ―― h Vmax ≔ 97.5 m ― s Factorul de sarcinã maxim la zborul pe fata (n+): Factorul de sarcinã maxim la zborul pe spate (n-): nfata ≔ 5 nspate ≔ −2.5 Viteza de croazierã proiectatã: VC ≔ 0.9 ⋅ Vmax VC = 87.75 m ― s

Viteza de picaj cu motorul in plin:

VD ≔ 1.35 ⋅ VC m ― s

VD = 118.463

Constructia ramurilor OSA si OS'G

m ― 2 s ρ0 ≔ 1.226 kg ―― 3 m

g ≔ 9.807

Suprafata aripii: Sw ≔ 11.24 m

2

Masa aparatului: mav ≔ 815 kg

Viteza de înfundare la zborul pe fatã:

Vs1_kph ≔ 101.86

km ―― h

Vs1 ≔ 28.2944 m ― s Viteza de înfundare la zborul pe spate:

m Vs2 ≔ Vs1 + 6 = 34.294 ― s ――→ 2 ⋅ mav ⋅ g Cz_max_fata ≔ ―――― 2 ρ0 ⋅ Sw ⋅ Vs1

Cz_max_fata = 1.449 ―――→ −2 ⋅ mav ⋅ g Cz_max_spate ≔ ―――― 2 ρ0 ⋅ Sw ⋅ Vs2

Cz_max_spate = −0.986

Vitezele corespunzatoare punctelor A si G sunt:

――――――→ ‾‾‾‾‾‾‾‾‾‾‾‾‾‾ 2 ⋅ mav ⋅ nfata ⋅ g VA ≔ ―――――― ρ0 ⋅ Sw ⋅ Cz_max_fata m ― s ――――――→ ‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾ 2 ⋅ mav ⋅ nspate ⋅ g VG ≔ ―――――― ρ0 ⋅ Sw ⋅ Cz_max_spate

VA = 63.268

VG = 54.224

m ― s

1.4. Diagrama de rafala

WA ≔ 20

m ― s

WC ≔ 15

m ― s

WD ≔ 7.5

m ― s

Densitatea aerului la inaltimea de croaziera(6900 m) 4.256

ρh ≔ ρ0 ⋅ (1 − 0.0000226 ⋅ 6900) kg ρh = 0.596 ―― 3 m Clα ≔ 0.33 Gav ≔ mav ⋅ g Gav = 7.993 ⋅ 10

3

Incarcarea alara:

mav alara ≔ ―― Sw alara = 72.509

η ≔ 0.2 ⋅

4

kg ―― 2 m

‾‾‾‾‾ alara

η = 0.584 ――――――――――→ ⎛ ρh ⎞ ρ0 Sw ⋅ Clα ⋅ ⎜―⋅ VA ⋅ WA⎟ = 2.001 nr1 ≔ 1 + η ⋅ ―⋅ ―― 2 m ⎝ ρ0 ⎠

nr1

1+η

Clα ⎜ VA WA⎟ 2.001 ⎝ ρ0 ⎠ ――――――――――→ ⎞ ⎛ ρh ρ0 Sw nr2 ≔ 1 + η ⋅ ―⋅ ―― ⋅ Clα ⋅ ⎜―⋅ VC ⋅ WC⎟ = 2.042 2 mav ⎝ ρ0 ⎠ ――――――――――→ ⎛ ρh ⎞ ρ0 Sw ⋅ Clα ⋅ ⎜―⋅ VD ⋅ WD⎟ = 1.703 nr3 ≔ 1 + η ⋅ ―⋅ ―― 2 mav ⎝ ρ0 ⎠ 2

mav

Viteza corespunzatoare punctului B:

――――――→ ‾‾‾‾‾‾‾‾‾‾‾‾‾‾ 2 ⋅ mav ⋅ g ⋅ nr1 VB ≔ ―――――― = 40.027 ρ0 ⋅ Sw ⋅ Cz_max_fata

m ― s

―――――――――――→ ⎛ ρh ⎞ ρ0 Sw ⋅ Clα ⋅ ⎜―⋅ VA ⋅ ⎛⎝−WA⎞⎠⎟ = −0.001 nr1_neg ≔ 1 + η ⋅ ―⋅ ―― 2 mav ⎝ ρ0 ⎠ ―――――――――――→ ⎞ ⎛ ρh ρ0 Sw ⋅ Clα ⋅ ⎜―⋅ VC ⋅ ⎛⎝−WC⎞⎠⎟ = −0.042 nr2_neg ≔ 1 + η ⋅ ―⋅ ―― 2 mav ⎝ ρ0 ⎠ ―――――――――――→ ⎛ ρh ⎞ ρ0 Sw ⋅ Clα ⋅ ⎜―⋅ VD ⋅ ⎛⎝−WD⎞⎠⎟ = 0.297 nr3_neg ≔ 1 + η ⋅ ―⋅ ―― 2 mav ⎝ ρ0 ⎠

Stabilirea numarului de sectiuni de calcul:

Coarda da incastrare: C0 ≔ 1.46 m

Coarda la extremitate: Ce ≔ 1.46

m

Anvergura: b ≔ 7.62

m

Semianvergura: b l ≔ ―= 3.81 2

Structurã: - aripã bilongeron Lonjeron principal situat la 25% din coardã fatã de bordul de atac al aripii

Se aleg 10 sectiuni de calcul: i ≔ 0,1‥9 j ≔ 1,2‥8 y ≔0 j

l y ≔― 1 9 y

j+1

≔y +y 1

j

Coarda din fiecare sectiune este datã de relatia, aripa având forma dreaptã: ⎛ C0 − Ce ⎞ c ≔ C0 − ⎜――― ⎟ ⋅ yi i l ⎝ ⎠

Variatia cu coarda a înãltimilor : Hlp ≔ 0.096822 ⋅ c i

i

Hls ≔ 0.9096 ⋅ c i

i

Hls

i

0.9096 c

i

⎤ ⎡0 ⎢ 0.423 ⎥ ⎢ 0.847 ⎥ ⎢ 1.27 ⎥ ―→ ⎢ 1.693 ⎥ y =⎢ ⎥ 2.117 ⎥ ⎢ ⎢ 2.54 ⎥ ⎢ 2.963 ⎥ ⎢ 3.387 ⎥ ⎣ 3.81 ⎦ Lãtimea chesonului: B ≔ 0.43 ⋅ c i

i

⎤ ⎡0 ⎢ 0.423 ⎥ ⎢ 0.847 ⎥ ⎢ 1.27 ⎥ ⎢ 1.693 ⎥ y =⎢ ⎥ i 2.117 ⎥ ⎢ ⎢ 2.54 ⎥ ⎢ 2.963 ⎥ ⎢ 3.387 ⎥ ⎣ 3.81 ⎦

⎡ 1.328 ⎤ ⎢ 1.328 ⎥ ⎢ 1.328 ⎥ ⎢ 1.328 ⎥ ⎢ 1.328 ⎥ Hls = ⎢ ⎥ i 1.328 ⎥ ⎢ 1.328 ⎥ ⎢ ⎢ 1.328 ⎥ ⎢ 1.328 ⎥ ⎣ 1.328 ⎦

⎡ 1.46 ⎤ ⎢ 1.46 ⎥ ⎢ 1.46 ⎥ ⎢ 1.46 ⎥ ⎢ 1.46 ⎥ c =⎢ ⎥ i 1.46 ⎥ ⎢ ⎢ 1.46 ⎥ ⎢ 1.46 ⎥ ⎢ 1.46 ⎥ ⎣ 1.46 ⎦

m

m

m

⎡ 0.628 ⎤ ⎢ 0.628 ⎥ ⎢ 0.628 ⎥ ⎢ 0.628 ⎥ ⎢ 0.628 ⎥ B =⎢ ⎥ i 0.628 ⎥ ⎢ 0.628 ⎥ ⎢ ⎢ 0.628 ⎥ ⎢ 0.628 ⎥ ⎣ 0.628 ⎦

⎡ 0.141 ⎤ ⎢ 0.141 ⎥ ⎢ 0.141 ⎥ ⎢ 0.141 ⎥ ⎢ 0.141 ⎥ Hlp = ⎢ ⎥ i 0.141 ⎥ ⎢ ⎢ 0.141 ⎥ ⎢ 0.141 ⎥ ⎢ 0.141 ⎥ ⎣ 0.141 ⎦

m

m

II. CARACTERISTICI ELASTICE ALE ARIPII 1. Calculul rigiditatii la torsiune

Grosimea invelisului:

δ ≔ 1.5 ⋅ 10 δ = 0.002

−3

m

m

Modulul de elasticitate transversal:

Gt ≔ 2.846 ⋅ 10

10

N ― m

――――――→ 1 I ≔ ―⋅ ⎛2 ⋅ B + Hlp + Hls ⎞ i i i i δ ⎝ ⎠

Aria sectiunii chesonului:

―――――→ 1 Ω ≔ ―⋅ ⎛Hlp + Hls ⎞ ⋅ B i i 2 ⎝ i ⎠ i Rigiditatea la torsiune:

―――― → 2 4 ⋅ Gt ⋅ ⎛Ω ⎞ ⋅ c i ⎝ i⎠ GId ≔ ――――― i I i

⎡ 1.946 ⋅ 10 7 ⎢ 7 ⎢ 1.946 ⋅ 10 7 ⎢ 1.946 ⋅ 10 ⎢ 1.946 ⋅ 10 7 ⎢ 7 1.946 ⋅ 10 GId = ⎢ 7 i ⎢ 1.946 ⋅ 10 ⎢ 1.946 ⋅ 10 7 ⎢ 7 1.946 ⋅ 10 ⎢ 7 ⎢ 1.946 ⋅ 10 ⎢⎣ 1.946 ⋅ 10 7

⎡ 1.817 ⋅ 10 3 ⎢ 3 ⎢ 1.817 ⋅ 10 3 ⎢ 1.817 ⋅ 10 ⎢ 1.817 ⋅ 10 3 ⎢ 3 1.817 ⋅ 10 I =⎢ 3 i ⎢ 1.817 ⋅ 10 ⎢ 1.817 ⋅ 10 3 ⎢ 3 1.817 ⋅ 10 ⎢ 3 ⎢ 1.817 ⋅ 10 ⎢⎣ 1.817 ⋅ 10 3

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦

4⋅10⁷ 3.6⋅10⁷ 3.2⋅10⁷ 2.8⋅10⁷ 2.4⋅10⁷ 2⋅10⁷

GId

1.6⋅10⁷ 1.2⋅10⁷ 8⋅10⁶ 4⋅10⁶ 0 0

0.4

0.8

1.2

1.6

y

2

2.4

2.8

3.2

3.6

4

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦

2. Calculul rigiditatii la incovoiere

Se va calcula dupã relatia: Ix(y)=Ix_învelis(y)+Ix_lp(y)+Ix_ls(y); Asadar avem nevoie de momentele de inertie axiale în fiecare sectiune: Ix_învelis= momentul de inertie axial al învelisului în sectiunea S Ix_lp , Ix_ls = momentul de inertie axial al lonjeronului principal, respectiv secundar în sectiunea S

Lonjeronul principal: profil I, având urmãtoarele dimensiuni:

- înãltimea lonjeronului: Hlp - lãtimea tãlpii lonjeronului: Hlp

i

Blp ≔ ―― i 2.5

- grosimea profilului: Hlp δlp ≔ ―― 90

- grosimea tãlpii lonjeronului: δtp ≔ 1.2 ⋅ δlp

Aria sectiunii transversale a longeronului principal: ―――――――――→ Alp ≔ δlp ⋅ ⎛Hlp − 2 ⋅ δtp ⎞ + 2 ⋅ δtp ⋅ Blp i i i i i ⎝ i ⎠

―――――― ――――――――― ――→ 3 2 ⎛ ⎞⎞ ⎛ 3 ⎛ Hlp − δtp ⎞ ⎛⎝Hlp − 2 ⋅ δtp⎞⎠ δtp ⋅ Blp ⎜ ⎟⎟ ⎜ Ixlp0 ≔ δlp ⋅ ――――― + 2 ⋅ ―――+ ⎜――― ⎟ ⋅ Blp ⋅ δtp 12 12 2 ⎝⎝ ⎠⎠ ⎠ ⎝

Se noteazã Z_CGlp ca fiind distanta între centrul de greutate ale sectiunii lonjeronului principal, si cel al sectiunii aripii:

ZCGlp ≔ 0 ⋅ c

- momentul de inertie axial al lonjeronului principal: ――――― → 2 Ixlp ≔ Ixlp0 + ZCGlp ⋅ Alp Invelisul:

ainv_0 ≔ 0.297627 ainv_e ≔ 0.125467 Ixinv_0 ≔ ainv_0 ⋅ 10 Ixinv_e ≔ ainv_e ⋅ 10

−5

−5

pas0 ≔ Ixinv_0 − Ixinv_e ―→ pas0 pas ≔ ―― 9 ――――→ Ixinv ≔ Ixinv_0 − pas ⋅ i i

⎡ 2.976 ⋅ 10 −6 ⎤ ⎢ −6 ⎥ ⎢ 2.785 ⋅ 10 −6 ⎥ ⎢ 2.594 ⋅ 10 ⎥ ⎢ 2.402 ⋅ 10 −6 ⎥ ⎢ −6 ⎥ 2.211 ⋅ 10 ⎥ ⎢ Ixinv = −6 i ⎢ 2.02 ⋅ 10 ⎥ −6 ⎢ 1.829 ⋅ 10 ⎥ ⎢ −6 ⎥ 1.637 ⋅ 10 ⎢ ⎥ −6 ⎢ 1.446 ⋅ 10 ⎥ ⎢⎣ 1 255 10 −6 ⎥⎦

⎣

⎦

Modulul de elasticitate transversal:

E ≔ 7.53 ⋅ 10

10

Momentul de inertie axial total:

Ix ≔ Ixinv + Ixlp Rigiditatea la incovoiere:

―→ EIx ≔ E ⋅ Ix ⎡ 2.976 ⋅ 10 −6 ⎤ ⎢ −6 ⎥ ⎢ 2.785 ⋅ 10 −6 ⎥ ⎢ 2.594 ⋅ 10 ⎥ ⎢ 2.402 ⋅ 10 −6 ⎥ ⎢ −6 ⎥ 2.211 ⋅ 10 ⎥ Ixinv = ⎢ −6 i ⎥ ⎢ 2.02 ⋅ 10 ⎢ 1.829 ⋅ 10 −6 ⎥ ⎢ −6 ⎥ 1.637 ⋅ 10 ⎥ ⎢ −6 ⎢ 1.446 ⋅ 10 ⎥ ⎢⎣ 1.255 ⋅ 10 −6 ⎥⎦ ⎡ 4.354 ⋅ 10 −6 ⎤ ⎢ −6 ⎥ ⎢ 4.163 ⋅ 10 −6 ⎥ ⎢ 3.971 ⋅ 10 ⎥ ⎢ 3.78 ⋅ 10 −6 ⎥ ⎢ −6 ⎥ 3.589 ⋅ 10 ⎥ Ix = ⎢ −6 i ⎢ 3.397 ⋅ 10 ⎥ ⎢ 3.206 ⋅ 10 −6 ⎥ ⎢ −6 ⎥ 3.015 ⋅ 10 ⎥ ⎢ −6 ⎢ 2.824 ⋅ 10 ⎥ ⎢⎣ 2.632 ⋅ 10 −6 ⎥⎦

Nm

Nm

2

2

⎡ 1.378 ⋅ 10 −6 ⎤ ⎢ −6 ⎥ ⎢ 1.378 ⋅ 10 −6 ⎥ ⎢ 1.378 ⋅ 10 ⎥ ⎢ 1.378 ⋅ 10 −6 ⎥ ⎢ −6 ⎥ 1.378 ⋅ 10 ⎥ Ixlp = ⎢ −6 i ⎢ 1.378 ⋅ 10 ⎥ ⎢ 1.378 ⋅ 10 −6 ⎥ ⎢ −6 ⎥ 1.378 ⋅ 10 ⎥ ⎢ −6 ⎢ 1.378 ⋅ 10 ⎥ ⎢⎣ 1.378 ⋅ 10 −6 ⎥⎦ ⎡ 3.278 ⋅ 10 5 ⎢ 5 ⎢ 3.134 ⋅ 105 ⎢ 2.99 ⋅ 10 ⎢ 2.846 ⋅ 10 5 ⎢ 5 2.702 ⋅ 10 EIx = ⎢ 5 i ⎢ 2.558 ⋅ 10 ⎢ 2.414 ⋅ 10 5 ⎢ 5 2.27 ⋅ 10 ⎢ 5 ⎢ 2.126 ⋅ 10 ⎢⎣ 1.982 ⋅ 10 5

Nm

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦

2

Nm

2

3.3⋅10⁵ 3.15⋅10⁵ 3⋅10⁵ 2.85⋅10⁵ 2.7⋅10⁵

EIx

2.55⋅10⁵ 2.4⋅10⁵ 2.25⋅10⁵ 2.1⋅10⁵ 1.95⋅10⁵ 0

0.4

0.8

1.2

1.6

2

2.4

2.8

3.2

3.6

4

y

3. Calculul sarcinilor aerodinamice globale

Se noteaza pt. cele 5 cazuri de calcul A,B,C,D,G: - Unghiul de incidenta: - Unghiul de incidenta iav in radiani - Coeficientul de portanta al avionului Czav - Coeficientul de rezistenta la inaintare al avionului Cxav

⎡1⎤ ⎢2⎥ α≔⎢3⎥ ⎢4⎥ ⎢⎣ 5 ⎥⎦

⎡ 0.50 ⎤ ⎢ 0.51 ⎥ Czav ≔ ⎢ 0.58 ⎥ ⎢ 0.62 ⎥ ⎢⎣ 0.77 ⎥⎦

⎡ 0.016 ⎤ ⎢ 0.016 ⎥ Cxav ≔ ⎢ 0.017 ⎥ ⎢ 0.018 ⎥ ⎢⎣ 0.020 ⎥⎦

―――――――――→ Cn ≔ Czav ⋅ cos ⎛⎝iav⎞⎠ + Cxav ⋅ sin ⎛⎝iav⎞⎠ Sarcinile aerodinamice globale sunt: ―――――――――→ Ct ≔ Czav ⋅ sin ⎛⎝iav⎞⎠ + Cxav ⋅ cos ⎛⎝iav⎞⎠ ――→ ρ0 2 Na ≔ ―⋅ Sw ⋅ VD ⋅ Cn 2 ―――――→ ρ0 2 Ta ≔ ―⋅ Sw ⋅ VD ⋅ Ct 2

π ⋅α iav ≔ ―― 180

⎡ 0.017 ⎤ ⎢ 0.035 ⎥ iav = ⎢ 0.052 ⎥ ⎢ 0.07 ⎥ ⎢⎣ 0.087 ⎥⎦

Ta

2

Sw V D

⎡ 4.837 ⋅ 10 4 ⎢ 4 ⎢ 4.934 ⋅ 10 4 Na = ⎢ 5.609 ⋅ 10 ⎢ 5.992 ⋅ 10 4 ⎢ 4 ⎣ 7.434 ⋅ 10

Ct

⎤ ⎥ ⎥ ⎥ N ⎥ ⎥ ⎦

⎡ 2.391 ⋅ 10 3 ⎢ 3 ⎢ 3.267 ⋅ 10 3 Ta = ⎢ 4.577 ⋅ 10 ⎢ 5.918 ⋅ 10 3 ⎢ 3 ⎣ 8.415 ⋅ 10

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

N

Na

Ta

7.55⋅10⁴ 7.3⋅10⁴ 7.05⋅10⁴ 6.8⋅10⁴ 6.55⋅10⁴ 6.3⋅10⁴ 6.05⋅10⁴ 5.8⋅10⁴ 5.55⋅10⁴ 5.3⋅10⁴ 5.05⋅10⁴ 4.8⋅10⁴

8.95⋅10³ 8.35⋅10³ 7.75⋅10³ 7.15⋅10³ 6.55⋅10³ 5.95⋅10³ 5.35⋅10³ 4.75⋅10³ 4.15⋅10³ 3.55⋅10³ 2.95⋅10³ 2.35⋅10³ 0 0.4 0.8 1.2 1.6 2 2.4 2.8 3.2 3.6 4

0 0.4 0.8 1.2 1.6 2 2.4 2.8 3.2 3.6 4

y

y

⎡ 0.5 ⎤ ⎢ 0.51 ⎥ Cn = ⎢ 0.58 ⎥ ⎢ 0.62 ⎥ ⎢⎣ 0.769 ⎥⎦

⎡ 0.025 ⎤ ⎢ 0.034 ⎥ Ct = ⎢ 0.047 ⎥ ⎢ 0.061 ⎥ ⎢⎣ 0.087 ⎥⎦

Cn

Ct

0.775

0.091

0.75

0.085

0.725

0.079

0.7

0.073

0.675

0.067

0.65

0.061

0.625

0.055

0.6

0.049

0.575

0.043

0.55

0.037

0.525

0.031

0.5

0.025 0 0.40.81.21.6 2 2.42.83.23.6 4

0 0.4 0.8 1.2 1.6 2 2.4 2.8 3.2 3.6 4

y

y

4. Calculul sarcinilor masice globale Pentru calculul sarcinilor masice globale avem nevoie de:

- masa aeronavei:

mav ≔ 815 kg - masa aripii:

mw ≔ 0.105 ⋅ mav = 85.575

kg

- masa combustibilui:

mcomb ≔ 159

l

- masa trenului de aterizare:

mtren ≔ 0.05 ⋅ mav = 40.75 kg - tractiunea:

T ≔ 1376

V ≔ VA 0

N

V ≔ VB 1

V ≔ VC 2

V ≔ VD 3

Valorea portantei pt.cele 5 cazuri de calcul:

――― ――→ 2 ρ0 ⋅ (V) P ≔ ―――⋅ Sw ⋅ Czav 2

Valorea rezistentei la inaintare pt.cele 5 cazuri de calcul:

―――― ――→ 2 ρ0 ⋅ (V) R ≔ ―――⋅ Sw ⋅ Cxav 2

V ≔ VG 4

Vsort ≔ V

Valoarile factorilor de sarcina:

P nz ≔ ――― mav ⋅ g

T−R nx ≔ ――― mav ⋅ g

――――――――→ nc1 ≔ ⎛⎝−nz⎞⎠ ⋅ cos ⎛⎝iav⎞⎠ + nz ⋅ sin ⎛⎝iav⎞⎠ ――――――――→ nc2 ≔ ⎛⎝nz⎞⎠ ⋅ sin ⎛⎝iav⎞⎠ + nx ⋅ cos ⎛⎝iav⎞⎠

Sarcinile masice globale se calculeaza dupã cum urmeazã:

- pt. aripa:

―――→ Nga ≔ nc1 ⋅ mw ⋅ g ―――→ Tga ≔ nc2 ⋅ mw ⋅ g - pt. combustibil:

――――→ Ngcomb ≔ nc1 ⋅ mcomb ⋅ g ――――→ Tgcomb ≔ nc2 ⋅ mcomb ⋅ g - pt. trenul de aterizare:

―――→ Ngtren ≔ nc1 ⋅ mtren ⋅ g ―――→ Tgtren ≔ nc2 ⋅ mtren ⋅ g ⎡ 1.725 ⎤ ⎢ 0.704 ⎥ nz = ⎢ 3.85 ⎥ ⎢ 7.5 ⎥ ⎢⎣ 1.952 ⎥⎦

⎡ 0.117 ⎤ ⎢ 0.15 ⎥ nx = ⎢ 0.059 ⎥ ⎢ −0.046 ⎥ ⎢⎣ 0.121 ⎥⎦

⎡ −1.422 ⋅ 10 3 ⎢ −570.166 ⎢ 3 Nga = ⎢ −3.057 ⋅ 10 3 ⎢ −5.84 ⋅ 10 ⎢⎣ −1.489 ⋅ 10 3

⎤ ⎥ ⎥ ⎥ ⎥ ⎥⎦

⎡ −2.643 ⋅ 10 3 ⎢ 3 ⎢ −1.059 ⋅ 10 3 Ngcomb = ⎢ −5.681 ⋅ 10 ⎢ −1.085 ⋅ 10 4 ⎢ 3 ⎣ −2.766 ⋅ 10

⎡ −677.367 ⎢ −271.508 3 ⎢ Ngtren = −1.456 ⋅ 10 ⎢ 3 ⎢ −2.781 ⋅ 10 ⎣ −709.017

⎡ 1.379 ⋅ 10 4 ⎢ 3 ⎢ 5.63 ⋅ 10 4 P = ⎢ 3.077 ⋅ 10 ⎢ 5.995 ⋅ 10 4 ⎢ 4 ⎣ 1.56 ⋅ 10

⎤ ⎥ ⎥ ⎥N ⎥ ⎥ ⎦

⎡ 123.401 ⎤ ⎢ 146.488 ⎥ Tga = ⎢ 218.808 ⎥ ⎢ 400.917 ⎥ ⎢⎣ 244.303 ⎥⎦

N

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

⎤ ⎥ ⎥ ⎥ ⎥ ⎦

N

N

⎡ 229.281 ⎤ ⎢ 272.178 ⎥ Tgcomb = ⎢ 406.549 ⎥ ⎢ 744.911 ⎥ ⎢⎣ 453.92 ⎥⎦

N

⎡ 58.762 ⎤ ⎢ 69.756 ⎥ Tgtren = ⎢ 104.194 ⎥ N ⎢ 190.913 ⎥ ⎢⎣ 116.335 ⎥⎦

⎡ 441.284 ⎢ 176.63 R = ⎢ 901.924 ⎢ 3 1.74 ⋅ 10 ⎢ ⎣ 405.176

⎤ ⎥ ⎥ ⎥ ⎥ ⎦

N

N

5. Calculul sarcinilor distribuite pe anvergura

Pentru calculul sarcinilor distribuite pe anvergură se presupun cunoscute: -Pozitia centrului elastic pt. fiecare sectiune:

aE ≔ 0.35 ⋅ c -Pozitia focarului pt. fiecare sectiune:

aF ≔ 0.25 ⋅ c Diferenta dintre cele doua:

ey ≔ aE − aF ――――――→ 2 ⎛ C0 − Ce ⎞ ⎞ ⎛ f (y) ≔ ⎜C0 − ⎜――― ⎟ ⋅ y⎟ l ⎝ ⎝ ⎠ ⎠ ―― ―→ 1 1 Z ≔ ―⌠ ⌡ f (y) d y l 0 Z = 0.559 Diferenta dintre cele doua:

x≔0 Pentru calculul sarcinilor distribuite se folosesc valorile fortelor concentrate de la cazul

V ≔ VD a) calculul sarcinilor aerodinamice distribuite:

Na = 5.992 ⋅ 10

4

3

――――――→ ――――――→ Na ci ⋅ eyi Ta ci ⋅ eyi 2 3 3 pa ≔ ―― ⋅ ――⋅ cos (x) ta ≔ ―― ⋅ ――⋅ cos (x) i i 2⋅l Z 2⋅l Z ――――――――→ Ta ci ⋅ eyi 3 tal ≔ ―― ⋅ ――⋅ cos (x) ⋅ sin (x) i 2⋅l Z

b) sarcini masice distribuite - aripã ――――――→ Nga c ⋅ ey 3 i i pga ≔ ――⋅ ――⋅ cos (x) i 2⋅l Z

――――――――→ Tga c ⋅ ey 3 i i tgal ≔ ―― ⋅ ――⋅ cos (x) ⋅ sin (x) i 2⋅l Z

――――――→ Tga c ⋅ ey 3 i i 2 tga ≔ ―― ⋅ ――⋅ cos (x) i 2⋅l Z

c) sarcini masice distribuite - combustibil - Z va avea o nouã expresie: Na = 5.992 ⋅ 10

4

3

ls ≔ l = 3.81 l ≔y 1

l ≔y

5

2

7

―――― ―→ l 2

1 z ≔ ――⋅ ⌠ ⌡ f (y) d y l −l l 2

1

z = 2.132

1

―――――――→ Ngcomb c ⋅ ey 3 i i ⋅ ――⋅ cos (x) pgcmb ≔ ――― i 2 ⋅ ls z ―――――――→ Tgcomb c ⋅ ey 3 i i tgcmb ≔ ――― ⋅ ――⋅ cos (x) i 2 ⋅ ls z ―――――――――→ Tgcomb c ⋅ ey 3 i i tgcmbl ≔ ――― ⋅ ――⋅ cos (x) ⋅ sin (x) i 2 ⋅ ls z

⎡ 2.996 ⋅ 10 3 ⎢ 3 ⎢ 2.996 ⋅ 10 3 ⎢ 2.996 ⋅ 10 ⎢ 2.996 ⋅ 10 3 ⎢ 3 2.996 ⋅ 10 ⎢ pa = 3 i ⎢ 2.996 ⋅ 10 ⎢ 2.996 ⋅ 10 3 ⎢ 3 2.996 ⋅ 10 ⎢ 3 ⎢ 2.996 ⋅ 10 ⎢⎣ 2.996 ⋅ 10 3

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦

⎡ 295.901 ⎤ ⎢ 295.901 ⎥ ⎢ 295.901 ⎥ ⎢ 295.901 ⎥ ⎢ 295.901 ⎥ ta = ⎢ ⎥ i 295.901 ⎥ ⎢ ⎢ 295.901 ⎥ ⎢ 295.901 ⎥ ⎢ 295.901 ⎥ ⎣ 295.901 ⎦

N ― m

3⋅10³ 2.75⋅10³ 2.5⋅10³ 2.25⋅10³ 2⋅10³

pa

1.75⋅10³ 1.5⋅10³

ta

1.25⋅10³ 1⋅10³ 750 500 250 0

0.4

0.8

1.2

1.6

y

2

2.4

2.8

3.2

3.6

4

N ― m

⎡ −292.01 ⎤ ⎢ −292.01 ⎥ ⎢ −292.01 ⎥ ⎢ −292.01 ⎥ ⎢ −292.01 ⎥ N pga = ⎢ ⎥ ― i −292.01 m ⎥ ⎢ ⎢ −292.01 ⎥ ⎢ −292.01 ⎥ ⎢ −292.01 ⎥ ⎣ −292.01 ⎦

⎡ 20.046 ⎤ ⎢ 20.046 ⎥ ⎢ 20.046 ⎥ ⎢ 20.046 ⎥ ⎢ 20.046 ⎥ N tga = ⎢ ⎥― i 20.046 m ⎥ ⎢ ⎢ 20.046 ⎥ ⎢ 20.046 ⎥ ⎢ 20.046 ⎥ ⎣ 20.046 ⎦

30 0 0

0.4

0.8

1.2

1.6

2

2.4

2.8

3.2

3.6

4

-30 -60 -90

pga

-120 -150

tga

-180 -210 -240 -270 -300

y

⎡ −142.404 ⎤ ⎢ −142.404 ⎥ ⎢ −142.404 ⎥ ⎢ −142.404 ⎥ ⎢ −142.404 ⎥ N pgcmb = ⎢ ⎥― i −142.404 m ⎥ ⎢ ⎢ −142.404 ⎥ ⎢ −142.404 ⎥ ⎢ −142.404 ⎥ ⎣ −142.404 ⎦

⎡ 9.776 ⎤ ⎢ 9.776 ⎥ ⎢ 9.776 ⎥ ⎢ 9.776 ⎥ ⎢ 9.776 ⎥ tgcmb = ⎢ ⎥ i 9.776 ⎥ ⎢ ⎢ 9.776 ⎥ ⎢ 9.776 ⎥ ⎢ 9.776 ⎥ ⎣ 9.776 ⎦

15 0 0

0.4

0.8

1.2

1.6

2

2.4

2.8

3.2

3.6

4

-15 -30 -45

pgcmb

-60 -75

tgcmb

-90 -105 -120 -135 -150

y

N ― m

6. Repartitia masei pe anvergurã

Repartitia masei pe anvergurã se calculeazã cu relatia: ―――――→ 0.7154 ⋅ mw 2 my ≔ ――――⋅ ⎛c ⎞ 1 i i ⎝ ⎠ ⌠ c dy ⎮ ⌡ i 0

⎡ 89.382 ⎤ ⎢ 89.382 ⎥ ⎢ 89.382 ⎥ ⎢ 89.382 ⎥ ⎢ 89.382 ⎥ my = ⎢ ⎥ i 89.382 ⎥ ⎢ 89.382 ⎥ ⎢ ⎢ 89.382 ⎥ ⎢ 89.382 ⎥ ⎣ 89.382 ⎦

180 160 140 120 100

my

80 60 40 20 0 0

0.4

0.8

1.2

1.6

2

2.4

2.8

3.2

3.6

4

y

Pozitia centrului de greutate pe fiecare sectiune, in functie de coarda:

aCG ≔ 0.415 ⋅ c

Momentul de inertie distribuit pe anvergura:

――→ xa ≔ aCG − aE ―――→ 2 jy ≔ my ⋅ ⎛xa ⎞ i i i ⎝ ⎠ Momentul static distribuit pe anvergura:

――→ sy ≔ m y ⋅ x a i

i

i

sy

i

my xa i

i

⎡ 89.382 ⎤ ⎢ 89.382 ⎥ ⎢ 89.382 ⎥ ⎢ 89.382 ⎥ ⎢ 89.382 ⎥ kg my = ⎢ ⎥― i 89.382 m ⎥ ⎢ ⎢ 89.382 ⎥ ⎢ 89.382 ⎥ ⎢ 89.382 ⎥ ⎣ 89.382 ⎦

⎡ 0.805 ⎤ ⎢ 0.805 ⎥ ⎢ 0.805 ⎥ ⎢ 0.805 ⎥ ⎢ 0.805 ⎥ jy = ⎢ ⎥ i 0.805 ⎥ ⎢ ⎢ 0.805 ⎥ ⎢ 0.805 ⎥ ⎢ 0.805 ⎥ ⎣ 0.805 ⎦

kg ⋅ m

⎡ 8.482 ⎤ ⎢ 8.482 ⎥ ⎢ 8.482 ⎥ ⎢ 8.482 ⎥ ⎢ 8.482 ⎥ kg sy = ⎢ ⎥ ― i 8.482 ⎥ m ⎢ 8.482 ⎥ ⎢ 8.482 ⎥ ⎢ ⎢ 8.482 ⎥ ⎣ 8.482 ⎦

2

7. Calculul momentelor de torsiune: Pentru fiecare sectiune momentul de torsiune este dat de: ―――――――――→ mt ≔ pa ⋅ ⎛⎝aE − aF⎞⎠ − pga ⋅ ⎛⎝aCG − aE⎞⎠ ⎡ 465.158 ⎤ ⎢ 465.158 ⎥ ⎢ 465.158 ⎥ ⎢ 465.158 ⎥ ⎢ 465.158 ⎥ mt = ⎢ ⎥ 465.158 ⎥ ⎢ ⎢ 465.158 ⎥ ⎢ 465.158 ⎥ ⎢ 465.158 ⎥ ⎣ 465.158 ⎦

950 855 760 665

Nm

570 475

mt

380 285 190 95 0 0

0.4

0.8

1.2

1.6

y

2

2.4

2.8

3.2

3.6

4

III. DETERMINAREA FUNCTIILOR DE INFLUENTĂ

Functiile de influenta Gww, Gw, Gw, G, vor fi determinate în sectiunile de calcul alese anterior.

j≔0‥9

k≔0‥9

η ≔y k

k

1. Incovoiere (Gww)

―η―――――→ k

⌠ η −λ ⎮ k G1ww ≔ ――⋅ ⎛y − λ⎞ d λ j,k ⎮ EIx ⎝ j ⎠ j ⎮ ⌡ 0

―y―――――→ j

⌠ η −λ ⎮ k G2ww ≔ ――⋅ ⎛y − λ⎞ d λ j,k ⎮ EIx ⎝ j ⎠ j ⎮ ⌡ 0

Gww

j,k

≔ if ⎛η < y , G1ww , G2ww ⎞ j j,k j,k ⎝ k ⎠

1. Incovoiere+Torsiune

― ――→ η k

⌠ η −λ ⎮ k G1wθ ≔ ―― d λ j,k ⎮ EIx j ⎮ ⌡ 0

― ――→ y j

G

⌠ η −λ ⎮ k

dλ

η ⎮ k G2wθ ≔ ―― d λ j,k ⎮ EIx j ⎮ ⌡ 0

――――――――→ ≔ if ⎛η < y , G1wθ , G2wθ ⎞ j,k j j,k j,k ⎝ k ⎠

Gwθ

― ――→ η k

⌠ y −λ ⎮ j G1wθ ≔ ――d λ j,k ⎮ EIx j ⎮ ⌡ 0

― ――→ y j

⌠ y −λ ⎮ j G2wθ ≔ ――d λ j,k ⎮ EIx j ⎮ ⌡ 0

Gθw

j,k

――――――――→ ≔ if ⎛η < y , G1wθ , G2wθ ⎞ j j,k j,k ⎝ k ⎠

3.Torsiune (G θθ ) ―η――→ k

⌠ 1 G1θθ ≔ ⎮ ―― dλ j,k GI d ⎮ j ⌡ 0

―y――→ j

⌠ 1 dλ G2θθ ≔ ⎮ ―― j,k ⎮ GIdj ⌡ 0

――――――――→ ≔ if ⎛η < y , G1θθ , G2θθ ⎞ j,k j j,k j,k ⎝ k ⎠

Gθθ

λ≔y I0

0

o≔1‥9 ―――――――――――→ λ −λ o o−1 ⎛ 1 1 ⎞ I0 I0

I0 ≔ 0 o

I0 ≔ I0 o

o o−1 ⎛ 1 1 ⎞ + ―――⋅ ⎜――+ ――― ⎟ o−1 EIx EIx 2 o o − 1 ⎝ ⎠

I1 ≔ 0

―――――――――――→ ⎛ λ ⎞ λ −λ λ o o−1 o o−1 ⎜ ⎟ I1 ≔ I1 + ―――⋅ ――+ ――― o o−1 2 EIx ⎜ EIx ⎟ o o−1⎠ ⎝

I2 ≔ 0

――――――――2―――→ 2 ⎛ ⎛λ ⎞ ⎛λ ⎞ ⎞ λ −λ o o − 1 ⎜ ⎝ o⎠ ⎝ o − 1⎠ ⎟ I2 ≔ I2 + ―――⋅ ――+ ――― o o−1 ⎜ ⎟ 2 EIx EIx o o−1 ⎠ ⎝

Id0 ≔ 0

―――――――――――→ λ −λ o o−1 ⎛ 1 1 ⎞ + ―― Id0 ≔ Id0 + ―――⋅ ⎜――― o o−1 GId GId ⎟ 2 o−1 o⎟ ⎜⎝ ⎠

o

o

o

tabel1 ≔ augment (I0 , I1 , I2 , Id0)

⎡0 ⎢0 ⎢ 0 ⎢ ⎢0 ⎢0 Gww = ⎢0 ⎢0 ⎢ 0 ⎢ ⎢0 ⎣0

0 0 0 0 0 0 0 0 0 ⎤ −8 −7 −7 −7 −7 −7 −7 −7 −6 8.068 ⋅ 10 2.017 ⋅ 10 3.227 ⋅ 10 4.437 ⋅ 10 5.648 ⋅ 10 6.858 ⋅ 10 8.068 ⋅ 10 9.278 ⋅ 10 1.049 ⋅ 10 ⎥ −7 −7 −6 −6 −6 −6 −6 −6 −6 ⎥ 2.114 ⋅ 10 6.765 ⋅ 10 1.184 ⋅ 10 1.691 ⋅ 10 2.199 ⋅ 10 2.706 ⋅ 10 3.214 ⋅ 10 3.721 ⋅ 10 4.228 ⋅ 10 −7 −6 −6 −6 −6 −6 −6 −6 −6 ⎥ 3.554 ⋅ 10 1.244 ⋅ 10 2.399 ⋅ 10 3.598 ⋅ 10 4.798 ⋅ 10 5.997 ⋅ 10 7.197 ⋅ 10 8.396 ⋅ 10 9.595 ⋅ 10 ⎥ −7 −6 −6 −6 −6 −5 −5 −5 −5 5.147 ⋅ 10 1.872 ⋅ 10 3.79 ⋅ 10 5.989 ⋅ 10 8.235 ⋅ 10 1.048 ⋅ 10 1.273 ⋅ 10 1.497 ⋅ 10 1.722 ⋅ 10 ⎥ −7 −6 −6 −6 −5 −5 −5 −5 −5 6.92 ⋅ 10 2.57 ⋅ 10 5.338 ⋅ 10 8.699 ⋅ 10 1.236 ⋅ 10 1.606 ⋅ 10 1.977 ⋅ 10 2.348 ⋅ 10 2.718 ⋅ 10 ⎥ −7 −6 −6 −5 −5 −5 −5 −5 −5 8.904 ⋅ 10 3.352 ⋅ 10 7.071 ⋅ 10 1.173 ⋅ 10 1.702 ⋅ 10 2.263 ⋅ 10 2.828 ⋅ 10 3.394 ⋅ 10 3.959 ⋅ 10 ⎥ −6 −6 −6 −5 −5 −5 −5 −5 −5 ⎥ 1.114 ⋅ 10 4.233 ⋅ 10 9.023 ⋅ 10 1.515 ⋅ 10 2.228 ⋅ 10 3.008 ⋅ 10 3.821 ⋅ 10 4.64 ⋅ 10 5.458 ⋅ 10 ⎥ −6 −6 −5 −5 −5 −5 −5 −5 −5 1.368 ⋅ 10 5.233 ⋅ 10 1.124 ⋅ 10 1.903 ⋅ 10 2.825 ⋅ 10 3.854 ⋅ 10 4.954 ⋅ 10 6.09 ⋅ 10 7.232 ⋅ 10 ⎥ −6 −6 −5 −5 −5 −5 −5 −5 −5 1.659 ⋅ 10 6.379 ⋅ 10 1.378 ⋅ 10 2.348 ⋅ 10 3.509 ⋅ 10 4.823 ⋅ 10 6.252 ⋅ 10 7.757 ⋅ 10 9.301 ⋅ 10 ⎦

⎡0 ⎢0 ⎢ 0 ⎢ ⎢0 ⎢0 Gwθ = ⎢0 ⎢0 ⎢ 0 ⎢ ⎢0 ⎣0

0 0 0 0 0 0 0 0 0 ⎤ −7 −7 −6 −6 −6 −6 −6 −6 −6 ⎥ 2.859 ⋅ 10 8.576 ⋅ 10 1.429 ⋅ 10 2.001 ⋅ 10 2.573 ⋅ 10 3.145 ⋅ 10 3.716 ⋅ 10 4.288 ⋅ 10 4.86 ⋅ 10 −7 −6 −6 −6 −6 −6 −6 −6 −6 ⎥ 2.996 ⋅ 10 1.199 ⋅ 10 2.397 ⋅ 10 3.596 ⋅ 10 4.794 ⋅ 10 5.993 ⋅ 10 7.191 ⋅ 10 8.39 ⋅ 10 9.589 ⋅ 10 −7 −6 −6 −6 −6 −6 −5 −5 −5 ⎥ 3.148 ⋅ 10 1.259 ⋅ 10 2.833 ⋅ 10 4.722 ⋅ 10 6.611 ⋅ 10 8.5 ⋅ 10 1.039 ⋅ 10 1.228 ⋅ 10 1.417 ⋅ 10 ⎥ −7 −6 −6 −6 −6 −5 −5 −5 −5 3.316 ⋅ 10 1.326 ⋅ 10 2.984 ⋅ 10 5.305 ⋅ 10 7.958 ⋅ 10 1.061 ⋅ 10 1.326 ⋅ 10 1.592 ⋅ 10 1.857 ⋅ 10 ⎥ −7 −6 −6 −6 −6 −5 −5 −5 −5 3.503 ⋅ 10 1.401 ⋅ 10 3.152 ⋅ 10 5.604 ⋅ 10 8.756 ⋅ 10 1.226 ⋅ 10 1.576 ⋅ 10 1.926 ⋅ 10 2.277 ⋅ 10 ⎥ −7 −6 −6 −6 −6 −5 −5 −5 −5 3.712 ⋅ 10 1.485 ⋅ 10 3.34 ⋅ 10 5.938 ⋅ 10 9.279 ⋅ 10 1.336 ⋅ 10 1.782 ⋅ 10 2.227 ⋅ 10 2.672 ⋅ 10 ⎥ −7 −6 −6 −6 −6 −5 −5 −5 −5 ⎥ 3.947 ⋅ 10 1.579 ⋅ 10 3.552 ⋅ 10 6.315 ⋅ 10 9.868 ⋅ 10 1.421 ⋅ 10 1.934 ⋅ 10 2.487 ⋅ 10 3.039 ⋅ 10 ⎥ −7 −6 −6 −6 −5 −5 −5 −5 −5 4.214 ⋅ 10 1.686 ⋅ 10 3.793 ⋅ 10 6.743 ⋅ 10 1.054 ⋅ 10 1.517 ⋅ 10 2.065 ⋅ 10 2.697 ⋅ 10 3.372 ⋅ 10 ⎥ −7 −6 −6 −6 −5 −5 −5 −5 −5 4.521 ⋅ 10 1.808 ⋅ 10 4.069 ⋅ 10 7.233 ⋅ 10 1.13 ⋅ 10 1.627 ⋅ 10 2.215 ⋅ 10 2.893 ⋅ 10 3.662 ⋅ 10 ⎦

⎡0 ⎢ 1.321 ⋅ 10 −6 ⎢ −6 ⎢ 2.704 ⋅ 10 −6 ⎢ 4.156 ⋅ 10 ⎢ 5.682 ⋅ 10 −6 tabel1 = ⎢ −6 ⎢ 7.293 ⋅ 10 −6 ⎢ 8.997 ⋅ 10 ⎢ 1.081 ⋅ 10 −5 ⎢ −5 ⎢ 1.273 ⋅ 10 −5 ⎣ 1.48 ⋅ 10

tabel1

0 2.859 ⋅ 10 1.171 ⋅ 10 2.715 ⋅ 10 4.986 ⋅ 10 8.063 ⋅ 10 1.204 ⋅ 10 1.703 ⋅ 10 2.317 ⋅ 10 3.061 ⋅ 10

0 −7 −6 −6 −6 −6 −5 −5 −5 −5

1.21 ⋅ 10

⎤

0 −7

7.494 ⋅ 10 2.456 ⋅ 10 5.902 ⋅ 10 1.185 ⋅ 10 2.122 ⋅ 10 3.506 ⋅ 10 5.467 ⋅ 10 8.159 ⋅ 10

−7 −6 −6 −5 −5 −5 −5 −5

2.175 ⋅ 10

−8 ⎥

⎥ ⎥ −8 6.525 ⋅ 10 ⎥ −8 ⎥ 8.7 ⋅ 10 −7 ⎥ 1.087 ⋅ 10 ⎥ −7 1.305 ⋅ 10 ⎥ −7 1.522 ⋅ 10 ⎥ −7 ⎥ 1.74 ⋅ 10 ⎥ −7 1.957 ⋅ 10 ⎦ 4.35 ⋅ 10

−8

IV. CALCULUL DEFORMATIILOR STATICE

Δy ≔ y − y 1

Δy = 0.423

0

⎡1 0 ⎢― 2 ⎢0 1 ⎢ 0 0 ⎢ ⎢0 0 0 0 W ≔ Δy ⋅ ⎢ ⎢0 0 ⎢0 0 ⎢0 0 ⎢0 0 ⎢ ⎢⎣ 0 0 ⎡ 0.212 ⎢0 ⎢0 ⎢0 ⎢0 W=⎢ 0 ⎢ ⎢0 ⎢0 ⎢0 ⎣0

⎤ ⎥ 0⎥ ⎥ 0 ⎥ 0⎥ 0⎥ 0⎥ 0⎥ 0⎥ 0⎥ 1⎥ ―⎥ 2⎦

0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0

0 0 1 0 0 0 0 0

0 0 0 1 0 0 0 0

0 0 0 0 1 0 0 0

0 0 0 0 0 1 0 0

0 0 0 0 0 0 1 0

0 0 0 0 0 0 0 1

0 0 0 0 0 0 0

0 0.423 0 0 0 0 0 0 0 0

0 0 0.423 0 0 0 0 0 0 0

0 0 0 0.423 0 0 0 0 0 0

0 0 0 0 0.423 0 0 0 0 0

⎡0⎤ ⎢0⎥ ⎢0⎥ ⎢0⎥ ⎢0⎥ N≔⎢ ⎥ 0 ⎢ ⎥ ⎢0⎥ ⎢0⎥ ⎢0⎥ ⎣0⎦

0 0 0 0 0 0.423 0 0 0 0

0 0 0 0 0 0 0.423 0 0 0

p ≔ pa + pga + pgcmb

w ≔ Gww ⋅ W ⋅ p + Gww ⋅ N + Gwθ ⋅ W ⋅ mt + Gwθ ⋅ Mt θ ≔ Gθw ⋅ W ⋅ p + Gθw ⋅ N + Gθθ ⋅ W ⋅ mt + Gθθ ⋅ Mt

⎡0⎤ ⎢0⎥ ⎢0⎥ ⎢0⎥ ⎢0⎥ Mt ≔ ⎢ ⎥ 0 ⎢ ⎥ ⎢0⎥ ⎢0⎥ ⎢0⎥ ⎣0⎦

0 0 0 0 0 0 0 0.423 0 0

0 0 0 0 0 0 0 0 0.423 0

⎤ 0 ⎥ 0 ⎥ 0 ⎥ 0 ⎥ 0 ⎥ 0 ⎥ 0 ⎥ 0 ⎥ 0 ⎥ 0.212 ⎦

θ

Gθw W p + Gθw N + Gθθ W mt + Gθθ Mt

⎤ ⎡0 ⎢ 0.009 ⎥ ⎢ 0.027 ⎥ ⎢ 0.053 ⎥ ⎢ 0.086 ⎥ w=⎢ ⎥ 0.127 ⎥ ⎢ ⎢ 0.174 ⎥ ⎢ 0.229 ⎥ ⎢ 0.292 ⎥ ⎣ 0.364 ⎦

⎤ ⎡0 ⎢ 0.003 ⎥ ⎢ 0.011 ⎥ ⎢ 0.025 ⎥ ⎢ 0.044 ⎥ θ=⎢ ⎥ 0.069 ⎥ ⎢ ⎢ 0.101 ⎥ ⎢ 0.139 ⎥ ⎢ 0.185 ⎥ ⎣ 0.238 ⎦

m

0.385 0.35 0.315 0.28 0.245 0.21

w

0.175 0.14 0.105 0.07 0.035 0 0

0.4

0.8

1.2

1.6

2

2.4

2.8

3.2

3.6

4

y

0.25 0.225 0.2 0.175 0.15 0.125

θ

0.1 0.075 0.05 0.025 0 0

0.4

0.8

1.2

1.6

y

2

2.4

2.8

3.2

3.6

4

rad

V. AEROELASTICITATEA STATICA 5.1 Divergenta aeroelastica produsa de rasucire sub portanta

Matricea unitate:

Czlocal ――― Czglobal MU ≔ 1 j

――→ 2 ρ0 ⋅ VD q ≔ ――― 2

q = 8.602 ⋅ 10

3

⎡ 0.143 ⎤ ⎢ 0.220 ⎥ ⎢ 0.297 ⎥ ⎢ 0.373 ⎥ ⎢ 0.449 ⎥ czglob ≔ ⎢ ⎥ 0.525 ⎥ ⎢ ⎢ 0.600 ⎥ ⎢ 0.674 ⎥ ⎢ 0.748 ⎥ ⎣ 0.821 ⎦

⎡ 0.390 ⎤ ⎢ 0.471 ⎥ ⎢ 0.555 ⎥ ⎢ 0.639 ⎥ ⎢ 0.726 ⎥ czloc ≔ ⎢ ⎥ 0.813 ⎥ ⎢ ⎢ 0.899 ⎥ ⎢ 0.981 ⎥ ⎢ 1.066 ⎥ ⎣ 1.136 ⎦

⎡ 2.727 ⎤ ⎢ 2.141 ⎥ ⎢ 1.869 ⎥ ⎢ 1.713 ⎥ czloc ⎢ 1.617 ⎥ ⎥ ――= ⎢ czglob ⎢ 1.549 ⎥ ⎢ 1.498 ⎥ ⎢ 1.455 ⎥ ⎢ 1.425 ⎥ ⎣ 1.384 ⎦

⎡ 1.46 ⎤ ⎢ 1.46 ⎥ ⎢ 1.46 ⎥ ⎢ 1.46 ⎥ ⎢ 1.46 ⎥ c≔⎢ ⎥ 1.46 ⎥ ⎢ ⎢ 1.46 ⎥ ⎢ 1.46 ⎥ ⎢ 1.46 ⎥ ⎣ 1.46 ⎦

cm0 ≔ −0.21 7 13

――→ 2 ρ0 ⋅ VD q ≔ ――― 2

cz_di ≔ 7.13 q = 8.602 ⋅ 10

3

Greutatea avionului Gav = 7.993 ⋅ 10

3

Determinarea matricii [A] 1 ≔ ――――― j,j czloc

A

j

c ⋅ ――⋅ cz_di j c zglob j

⎡ 0.035 ⎢0 ⎢0 ⎢0 ⎢0 A=⎢ 0 ⎢ ⎢0 ⎢0 ⎢0 ⎣0

0 0.045 0 0 0 0 0 0 0 0

0 0 0.051 0 0 0 0 0 0 0

0 0 0 0.056 0 0 0 0 0 0

0 0 0 0 0.059 0 0 0 0 0

0 0 0 0 0 0.062 0 0 0 0

0 0 0 0 0 0 0.064 0 0 0

0 0 0 0 0 0 0 0.066 0 0

0 0 0 0 0 0 0 0 0.067 0

⎤ 0 ⎥ 0 ⎥ 0 ⎥ 0 ⎥ 0 ⎥ 0 ⎥ 0 ⎥ 0 ⎥ 0 ⎥ 0.069 ⎦

Stabilirea lui [e] si [d]

d ≔ aCG − aE ⎡ 1.46 ⎤ ⎢ 1.46 ⎥ ⎢ 1.46 ⎥ ⎢ 1.46 ⎥ ⎢ 1.46 ⎥ c =⎢ ⎥ j 1.46 ⎥ ⎢ ⎢ 1.46 ⎥ ⎢ 1.46 ⎥ ⎢ 1.46 ⎥ ⎣ 1.46 ⎦

ematr

j,j

⎡ 0.146 ⎤ ⎢ 0.146 ⎥ ⎢ 0.146 ⎥ ⎢ 0.146 ⎥ ⎢ 0.146 ⎥ ey = ⎢ ⎥ j 0.146 ⎥ ⎢ ⎢ 0.146 ⎥ ⎢ 0.146 ⎥ ⎢ 0.146 ⎥ ⎣ 0.146 ⎦

≔ ey

j

dmatr

⎡ 0.095 ⎤ ⎢ 0.095 ⎥ ⎢ 0.095 ⎥ ⎢ 0.095 ⎥ ⎢ 0.095 ⎥ d=⎢ ⎥ 0.095 ⎥ ⎢ ⎢ 0.095 ⎥ ⎢ 0.095 ⎥ ⎢ 0.095 ⎥ ⎣ 0.095 ⎦

j,j

≔d ⎡ 2.562 ⋅ 10 3 ⎢ 3 ⎢ 2.562 ⋅ 10 3 ⎢ 2.562 ⋅ 10 ⎢ 2.562 ⋅ 10 3 ⎢ 3 2.562 ⋅ 10 p=⎢ 3 ⎢ 2.562 ⋅ 10 ⎢ 2.562 ⋅ 10 3 ⎢ 3 2.562 ⋅ 10 ⎢ 3 ⎢ 2 562 ⋅ 10

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

⎡ 2.727 ⎤ ⎢ 2.141 ⎥ ⎢ 1.869 ⎥ ⎢ 1.713 ⎥ czloc ⎢ 1.617 ⎥ ⎥ ――= ⎢ czglob ⎢ 1.549 ⎥ ⎢ 1.498 ⎥ ⎢ 1.455 ⎥ ⎢ 1.425 ⎥ ⎣ 1.384 ⎦

⎢ ⎥ ⎢⎣ 2.562 ⋅ 10 3 ⎥⎦

Determinarea matricilor [E], [F]

E ≔ ⎛⎝Gθw + Gθθ ⋅ ematr⎞⎠ ⋅ W 2 F ≔ Gθθ ⋅ W ⋅ ⎛⎝c ⋅ cm0⎞⎠

Nk ≔ N Mtk ≔ Mt ⎡0 ⎢0 ⎢ 0 ⎢ ⎢0 ⎢0 E= ⎢0 ⎢0 ⎢ 0 ⎢ ⎢0 ⎣0

0 0 0 0 0 0 0 0 0 ⎤ −7 −7 −7 −7 −7 −7 −7 −7 −8 1.224 ⋅ 10 1.224 ⋅ 10 1.224 ⋅ 10 1.224 ⋅ 10 1.224 ⋅ 10 1.224 ⋅ 10 1.224 ⋅ 10 1.224 ⋅ 10 6.118 ⋅ 10 ⎥ −7 −7 −7 −7 −7 −7 −7 −7 −7 ⎥ 3.819 ⋅ 10 5.101 ⋅ 10 5.101 ⋅ 10 5.101 ⋅ 10 5.101 ⋅ 10 5.101 ⋅ 10 5.101 ⋅ 10 5.101 ⋅ 10 2.55 ⋅ 10 −7 −6 −6 −6 −6 −6 −6 −6 −7 ⎥ 6.677 ⋅ 10 1.069 ⋅ 10 1.203 ⋅ 10 1.203 ⋅ 10 1.203 ⋅ 10 1.203 ⋅ 10 1.203 ⋅ 10 1.203 ⋅ 10 6.017 ⋅ 10 ⎥ −7 −6 −6 −6 −6 −6 −6 −6 −6 9.84 ⋅ 10 1.687 ⋅ 10 2.11 ⋅ 10 2.251 ⋅ 10 2.251 ⋅ 10 2.251 ⋅ 10 2.251 ⋅ 10 2.251 ⋅ 10 1.126 ⋅ 10 ⎥ −6 −6 −6 −6 −6 −6 −6 −6 −6 1.336 ⋅ 10 2.375 ⋅ 10 3.118 ⋅ 10 3.564 ⋅ 10 3.714 ⋅ 10 3.714 ⋅ 10 3.714 ⋅ 10 3.714 ⋅ 10 1.857 ⋅ 10 ⎥ −6 −6 −6 −6 −6 −6 −6 −6 −6 1.73 ⋅ 10 3.145 ⋅ 10 4.246 ⋅ 10 5.033 ⋅ 10 5.506 ⋅ 10 5.664 ⋅ 10 5.664 ⋅ 10 5.664 ⋅ 10 2.832 ⋅ 10 ⎥ −6 −6 −6 −6 −6 −6 −6 −6 −6 ⎥ 2.174 ⋅ 10 4.013 ⋅ 10 5.518 ⋅ 10 6.689 ⋅ 10 7.526 ⋅ 10 8.028 ⋅ 10 8.197 ⋅ 10 8.197 ⋅ 10 4.098 ⋅ 10 ⎥ −6 −6 −6 −6 −6 −5 −5 −5 −6 2.678 ⋅ 10 4.998 ⋅ 10 6.962 ⋅ 10 8.569 ⋅ 10 9.819 ⋅ 10 1.071 ⋅ 10 1.125 ⋅ 10 1.143 ⋅ 10 5.715 ⋅ 10 ⎥ −6 −6 −6 −5 −5 −5 −5 −5 −6 3.255 ⋅ 10 6.127 ⋅ 10 8.616 ⋅ 10 1.072 ⋅ 10 1.245 ⋅ 10 1.379 ⋅ 10 1.475 ⋅ 10 1.532 ⋅ 10 7.757 ⋅ 10 ⎦

M≔A ⎡0 ⎢0 ⎢ 0 ⎢ ⎢0 ⎢0 M= ⎢0 ⎢0 ⎢ 0 ⎢ ⎢0 ⎣0

−1

⋅E

0 0 0 0 0 0 0 0 0 ⎤ −6 −6 −6 −6 −6 −6 −6 −6 −6 2.727 ⋅ 10 2.727 ⋅ 10 2.727 ⋅ 10 2.727 ⋅ 10 2.727 ⋅ 10 2.727 ⋅ 10 2.727 ⋅ 10 2.727 ⋅ 10 1.364 ⋅ 10 ⎥ −6 −6 −6 −6 −6 −6 −6 −6 −6 ⎥ 7.429 ⋅ 10 9.923 ⋅ 10 9.923 ⋅ 10 9.923 ⋅ 10 9.923 ⋅ 10 9.923 ⋅ 10 9.923 ⋅ 10 9.923 ⋅ 10 4.961 ⋅ 10 −5 −5 −5 −5 −5 −5 −5 −5 −5 ⎥ 1.191 ⋅ 10 1.906 ⋅ 10 2.146 ⋅ 10 2.146 ⋅ 10 2.146 ⋅ 10 2.146 ⋅ 10 2.146 ⋅ 10 2.146 ⋅ 10 1.073 ⋅ 10 ⎥ −5 −5 −5 −5 −5 −5 −5 −5 −5 1.656 ⋅ 10 2.84 ⋅ 10 3.551 ⋅ 10 3.789 ⋅ 10 3.789 ⋅ 10 3.789 ⋅ 10 3.789 ⋅ 10 3.789 ⋅ 10 1.895 ⋅ 10 ⎥ −5 −5 −5 −5 −5 −5 −5 −5 −5 2.153 ⋅ 10 3.829 ⋅ 10 5.026 ⋅ 10 5.745 ⋅ 10 5.986 ⋅ 10 5.986 ⋅ 10 5.986 ⋅ 10 5.986 ⋅ 10 2.993 ⋅ 10 ⎥ −5 −5 −5 −5 −5 −5 −5 −5 −5 2.698 ⋅ 10 4.906 ⋅ 10 6.623 ⋅ 10 7.851 ⋅ 10 8.588 ⋅ 10 8.835 ⋅ 10 8.835 ⋅ 10 8.835 ⋅ 10 4.418 ⋅ 10 ⎥ −5 −5 −5 −4 −4 −4 −4 −4 −5 ⎥ 3.293 ⋅ 10 6.08 ⋅ 10 8.361 ⋅ 10 1.013 ⋅ 10 1.14 ⋅ 10 1.216 ⋅ 10 1.242 ⋅ 10 1.242 ⋅ 10 6.21 ⋅ 10 ⎥ −5 −5 −4 −4 −4 −4 −4 −4 −5 3.972 ⋅ 10 7.415 ⋅ 10 1.033 ⋅ 10 1.271 ⋅ 10 1.457 ⋅ 10 1.589 ⋅ 10 1.669 ⋅ 10 1.696 ⋅ 10 8.478 ⋅ 10 ⎥ −5 −5 −4 −4 −4 −4 −4 −4 −4 4.688 ⋅ 10 8.825 ⋅ 10 1.241 ⋅ 10 1.544 ⋅ 10 1.793 ⋅ 10 1.986 ⋅ 10 2.124 ⋅ 10 2.207 ⋅ 10 1.117 ⋅ 10 ⎦

Se urmareste determinarea vitezei de divergenta VDmin. Pentru aceasta avem nevoie de λmax,valoarea maxima din sirul de valori proprii ale matricii M.

Orice valoare proprieλ j se va gasi cel putin in discul de raza r:

i≔0‥9

9

r ≔ ∑ |M | i i , j| j=0 | Asadar λmax va fi:

λ ≔M i

i,i

+r

i

λmax ≔ max (λ)

2 VDmin ≔ ―――― ‾‾‾‾‾‾ ρ0 ⋅ λmax

λmax = 0.001

VDmin = 47.467

Concluzie: deoarece viteza maxima admisa este VD= 47.467 m/s => viteza de divergenta nu se poate atinge in timpul zborului.

5.3. Efecte aeroelastice asupra comenzilor

dCm0β ≔ 0.45

dCzrβ ≔ 0.25

dCzrp ≔ 0.82

t≔0‥4

B ≔ MU ⋅ y ⋅ W z ≔ i ⋅ 10 − 3 i

x ≔0

y1 ≔ 0.394

x ≔ 73.188

y1 ≔ 0.027

x ≔ 78.2

y1 ≔ 0

0

1

2

0

1

2

T 1 Czβ ≔ ――⋅ B ⋅ c ⋅ dCzrβ Sw ⋅ l T 1 Czp ≔ ――⋅ B ⋅ c ⋅ dCzrp Sw ⋅ l −1 T q dβp1 ≔ Czβ + ――⋅ B ⋅ (A − q ⋅ E) ⋅ E ⋅ ⎛⎝c ⋅ dCzrβ⎞⎠ + F ⋅ dCm0β Sw ⋅ l −1 T q dβp2 ≔ Czp + ――⋅ B ⋅ (A − q ⋅ E) ⋅ E ⋅ ⎛⎝c ⋅ dCzrp⎞⎠ Sw ⋅ l

l ≔ 3.775

S ≔ cspline (x , y1)

dβp1

dβp ≔ interp (S , x , y1 , z)

Vl ≔ sort ⎛⎝Vsort⎞⎠

V' ≔ i ⋅ 10 − 3

i

i

dβp ≔ ―― i dβp2

⎡ −3 ⎤ ⎢ 7⎥ ⎢ 17 ⎥ ⎢ 27 ⎥ ⎢ 37 ⎥ V' = ⎢ ⎥ i 47 ⎥ ⎢ ⎢ 57 ⎥ ⎢ 67 ⎥ ⎢ 77 ⎥ ⎣ 87 ⎦

i

⎡ 40.027 ⎤ ⎢ 54.224 ⎥ Vl = ⎢ 63.268 ⎥ t ⎢ 87.75 ⎥ ⎢⎣ 118.463 ⎥⎦

⎡ 0.305 ⎤ ⎢ 0.305 ⎥ ⎢ 0.305 ⎥ ⎢ 0.305 ⎥ ⎢ 0.305 ⎥ dβp = ⎢ ⎥ i 0.305 ⎥ ⎢ ⎢ 0.305 ⎥ ⎢ 0.305 ⎥ ⎢ 0.305 ⎥ ⎣ 0.305 ⎦

ics ≔ cspline ⎛⎝V' , dβp⎞⎠ ⎡ 0.305 ⎤ ⎢ 0.305 ⎥ dβp1 = ⎢ 0.305 ⎥ ⎢ 0.305 ⎥ ⎢⎣ 0.305 ⎥⎦

dβp1 ≔ interp ⎛⎝ics , V' , dβp , Vl⎞⎠

0.305 0.305 0.305 0.305 0.305 0.305

dβp1

0.305 0.305

dβp

0.305 0.305 0.305 0.305 -15

0

15

30

45

Vl

60

75

90

105

120

V'

VI. Vibratii libere VI.1 Vibratii de rasucire

Se considera aripa, o bara de sectiune constanta, incastrata la un capat si libera la celalalt. Bara se considera avand o rigiditate GId, si un moment de inertie J. Momentul de inertie este considerat fata de axa centrala a barei.

GId ≔ 1.946 ⋅ 10

7

i ≔ 1 ‥ 12

j ≔ 1 , 2 ‥ 12

y ≔ 0.1

y ≔0

0

j

⎡ 416.458 ⎤ ⎢ 416.458 ⎥ ⎢ 416.458 ⎥ ⎢ 416.458 ⎥ ⎢ 416.458 ⎥ ⎥ ⎢ 416.458 ⎥ ⎢ 416.458 ⎥ kg ⋅ m 2 J≔⎢ ⎢ 416.458 ⎥ ⎢ 416.458 ⎥ ⎢ 416.458 ⎥ ⎢ 416.458 ⎥ ⎢ 416.458 ⎥ ⎢ 416.458 ⎥ ⎢⎣ 416.458 ⎥⎦

i ≔ 0 ‥ 13

l y ≔― 1 13

y

j+1

≔y +y

moment obtinut din constructia la scara 1:1 a aripi in Solid Works, si folosirea functiei Mass Properties. Moment de inertie in raport cu axa centrala a aripii.

j ≔ 0 ‥ 13

1

j

⎡ 0.1 ⎤ ⎢ 0.29 ⎥ ⎢ 0.581 ⎥ ⎢ 0.871 ⎥ ⎢ 1.162 ⎥ ⎥ ⎢ 1.452 ⎥ ⎢ 1.742 ⎥ ⎢ m y= ⎢ 2.033 ⎥ ⎢ 2.323 ⎥ ⎢ 2.613 ⎥ ⎢ 2.904 ⎥ ⎢ 3.194 ⎥ ⎢ 3.485 ⎥ ⎢⎣ 3.775 ⎥⎦

⎡ 1 ⎤ ⎢ 2 ⎥ ⎢ 3 ⎥ ⎢ 4 ⎥ ⎢ 5 ⎥ ⎢ ⎥ 6 ⎢ ⎥ 7 n≔⎢ ⎥ ⎢ 8 ⎥ ⎢ 9 ⎥ ⎢ 10 ⎥ ⎢ 11 ⎥ ⎢ 12 ⎥ ⎢ 13 ⎥ ⎢⎣ 14 ⎥⎦

Frecventele proprii:

π al ≔ ⎛2 ⋅ n − 1⎞ ⋅ ― j j ⎝ ⎠ 2 ――――――→ 2⋅n −1 ‾‾‾‾ j GId π ωw ≔ ―――⋅ ―⋅ ―― j l 2 J

rad ―― s

j

0.170 al = j

Φ

⎡ 0.267 ⎤ ⎢ 0.801 ⎥ ⎢ 1.335 ⎥ ⎢ 1.869 ⎥ ⎢ 2.403 ⎥ ⎥ ⎢ 2.937 ⎥ ⎢ ⎢ 3.471 ⎥ ⎢ 4.006 ⎥ ⎢ 4.54 ⎥ ⎢ 5.074 ⎥ ⎢ 5.608 ⎥ ⎢ 6.142 ⎥ ⎢⎣ ⋮ ⎥⎦ ⎛2⋅n −1⎞ j ⎟ i ⎜

ωw = j

⎡ 89.947 ⎢ 269.842 ⎢ 449.737 ⎢ 629.632 ⎢ 809.527 ⎢ 989.422 ⎢ 3 ⎢ 1.169 ⋅ 10 ⎢ 1.349 ⋅ 10 3 3 ⎢ 1.529 ⋅ 10 ⎢ 3 ⎢ 1.709 ⋅ 10 ⎢ 1.889 ⋅ 10 3 ⎢⎣ ⋮

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ rad ⎥ ―― s ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦

⎛ ⎞ j ⎜ ⎟⋅π⋅y Φ ≔ ωw ⋅ sin ――― i,j j j 2 ⎝ ⎠

⎡ 13.548 ⎢ ⎢ 13.548 ⎢ 13.548 ⎢ 13.548 ⎢ ⎢ 13.548

Φ=

⎢ 13.548 ⎢ 13.548 ⎢ ⎢ 13.548 ⎢ 13.548 ⎢ 13.548 ⎢ ⎢ 13.548 ⎢ 13.548 ⎢ ⎢ 13.548 ⎣

245.552 491.084

−604.464

−2.888 ⋅ 10

245.552 491.084

−604.464

−2.888 ⋅ 10

245.552 491.084

−604.464

−2.888 ⋅ 10

245.552 491.084

−604.464

−2.888 ⋅ 10

245.552 491.084 245.552 491.084 245.552 491.084

−604.464 −604.464 −604.464

−2.888 ⋅ 10 −2.888 ⋅ 10 −2.888 ⋅ 10

245.552 491.084

−604.464

−2.888 ⋅ 10

245.552 491.084

−604.464

−2.888 ⋅ 10

245.552 491.084

−604.464

−2.888 ⋅ 10

245.552 491.084

−604.464

−2.888 ⋅ 10

245.552 491.084

−604.464

−2.888 ⋅ 10

245.552 491.084

−604.464

−2.888 ⋅ 10

3 3 3 3 3 3 3 3 3 3 3 3 3

−3.184 ⋅ 10 −3.184 ⋅ 10 −3.184 ⋅ 10 −3.184 ⋅ 10 −3.184 ⋅ 10 −3.184 ⋅ 10 −3.184 ⋅ 10 −3.184 ⋅ 10 −3.184 ⋅ 10 −3.184 ⋅ 10 −3.184 ⋅ 10 −3.184 ⋅ 10 −3.184 ⋅ 10

3 3 3 3 3 3 3 3 3 3 3 3 3

2.7⋅10⁴ 2.25⋅10⁴

Φ

1.8⋅10⁴

1,j

1.35⋅10⁴ 9⋅10³

Φ

4.5⋅10³

2,j

0 0.1 0.45 0.8 1.15 1.5 1.85 2.2 2.55 2.9 3.25 3.6 3.95 -4.5⋅10³

Φ

3,j

-9⋅10³ -1.35⋅10⁴

Φ

-1.8⋅10⁴

4,j

-2.25⋅10⁴

Φ y

j

5,j

1.377 ⋅ 10 1.377 ⋅ 10 1.377 ⋅ 10 1.377 ⋅ 10 1.377 ⋅ 10 1.377 ⋅ 10 1.377 ⋅ 10 1.377 ⋅ 10 1.377 ⋅ 10 1.377 ⋅ 10 1.377 ⋅ 10 1.377 ⋅ 10 1.377 ⋅ 10

3 3 3 3 3 3 3 3 3 3 3 3 3

8.082 ⋅ 10 8.082 ⋅ 10 8.082 ⋅ 10 8.082 ⋅ 10 8.082 ⋅ 10 8.082 ⋅ 10 8.082 ⋅ 10 8.082 ⋅ 10 8.082 ⋅ 10 8.082 ⋅ 10 8.082 ⋅ 10 8.082 ⋅ 10 8.082 ⋅ 10

3 3 3 3 3 3 3 3 3 3 3 3 3

8.911 ⋅ 10 8.911 ⋅ 10 8.911 ⋅ 10 8.911 ⋅ 10 8.911 ⋅ 10 8.911 ⋅ 10 8.911 ⋅ 10 8.911 ⋅ 10 8.911 ⋅ 10 8.911 ⋅ 10 8.911 ⋅ 10 8.911 ⋅ 10 8.911 ⋅ 10

3 3 3 3 3 3 3 3 3 3 3 3 3

−1.054 ⋅ 10 −1.054 ⋅ 10 −1.054 ⋅ 10 −1.054 ⋅ 10 −1.054 ⋅ 10 −1.054 ⋅ 10 −1.054 ⋅ 10 −1.054 ⋅ 10 −1.054 ⋅ 10 −1.054 ⋅ 10 −1.054 ⋅ 10 −1.054 ⋅ 10 −1.054 ⋅ 10

3 3 3 3 3 3 3 3 3 3 3 3 3

−1.516 ⋅ 10 −1.516 ⋅ 10 −1.516 ⋅ 10 −1.516 ⋅ 10 −1.516 ⋅ 10 −1.516 ⋅ 10 −1.516 ⋅ 10 −1.516 ⋅ 10 −1.516 ⋅ 10 −1.516 ⋅ 10 −1.516 ⋅ 10 −1.516 ⋅ 10 −1.516 ⋅ 10

4 4 4 4 4 4 4 4 4 4 4 4 4

−1.817 ⋅ 10 −1.817 ⋅ 10 −1.817 ⋅ 10 −1.817 ⋅ 10 −1.817 ⋅ 10 −1.817 ⋅ 10 −1.817 ⋅ 10 −1.817 ⋅ 10 −1.817 ⋅ 10 −1.817 ⋅ 10 −1.817 ⋅ 10 −1.817 ⋅ 10 −1.817 ⋅ 10

4 4 4 4 4 4 4 4 4 4 4 4 4

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⋱ ⎦

VI.2 Vibratii de incovoiere

EIx ≔ 2.53 ⋅ 10

5

Nm

2

⎛2⋅n −1⎞ j ⎟⋅π al ≔ ⎜――― j 2 ⎠ ⎝ 140 m ≔ ―― 14 ―― ―2――→ 2 ⎛n ⎞ ⋅ π ‾‾‾‾ EIx ⎝ j⎠ ⋅ ―― ωθ ≔ ―――― 2 j m l

al = j

⎡ 1.571 ⎤ ⎢ 4.712 ⎥ ⎢ 7.854 ⎥ ⎢ 10.996 ⎥ ⎢ 14.137 ⎥ ⎥ ⎢ 17.279 ⎥ ⎢ ⎢ 20.42 ⎥ ⎢ 23.562 ⎥ ⎢ 26.704 ⎥ ⎢ 29.845 ⎥ ⎢ 32.987 ⎥ ⎢ 36.128 ⎥ ⎢⎣ ⋮ ⎥⎦

⎡ 110.161 ⎢ 440.642 ⎢ 991.445 ⎢ 3 1.763 ⋅ 10 ⎢ 3 ⎢ 2.754 ⋅ 10 ⎢ 3.966 ⋅ 10 3 ⎢ 3 5.398 ⋅ 10 ⎢ 3 ωθ = j ⎢ 7.05 ⋅ 10 ⎢ 8.923 ⋅ 10 3 4 ⎢ 1.102 ⋅ 10 ⎢ 4 ⎢ 1.333 ⋅ 10 ⎢ 1.586 ⋅ 10 4 4 ⎢ 1.862 ⋅ 10 ⎢ 4 ⎣ 2.159 ⋅ 10

⎛2⋅n −1⎞ j ⎟⋅π⋅y Φ1 ≔ ωθ ⋅ sin ⎜――― i,j j j 2 ⎠ ⎝

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

⎡ 16.592 ⎢ ⎢ 16.592 ⎢ 16.592 ⎢ 16.592 ⎢ ⎢ 16.592 ⎢ 16.592 ⎢ 16.592 ⎢ ⎢ 16.592 ⎢ 16.592 ⎢ 16.592 ⎢ ⎢ 16.592 ⎢ 16.592 ⎢⎣

Φ1 =

400.978

1.083 ⋅ 10

400.978

1.083 ⋅ 10

400.978

1.083 ⋅ 10

400.978

1.083 ⋅ 10

400.978 400.978 400.978

1.083 ⋅ 10 1.083 ⋅ 10 1.083 ⋅ 10

400.978

1.083 ⋅ 10

400.978

1.083 ⋅ 10

400.978

1.083 ⋅ 10

400.978

1.083 ⋅ 10

400.978

1.083 ⋅ 10

3 3 3 3 3 3 3 3 3 3 3 3

−1.692 ⋅ 10 −1.692 ⋅ 10 −1.692 ⋅ 10 −1.692 ⋅ 10 −1.692 ⋅ 10 −1.692 ⋅ 10 −1.692 ⋅ 10 −1.692 ⋅ 10 −1.692 ⋅ 10 −1.692 ⋅ 10 −1.692 ⋅ 10 −1.692 ⋅ 10

3 3 3 3 3 3 3 3 3 3 3 3

−9.824 ⋅ 10 −9.824 ⋅ 10 −9.824 ⋅ 10 −9.824 ⋅ 10 −9.824 ⋅ 10 −9.824 ⋅ 10 −9.824 ⋅ 10 −9.824 ⋅ 10 −9.824 ⋅ 10 −9.824 ⋅ 10 −9.824 ⋅ 10 −9.824 ⋅ 10

3 3 3 3 3 3 3 3 3 3 3 3

−1.276 ⋅ 10 −1.276 ⋅ 10 −1.276 ⋅ 10 −1.276 ⋅ 10 −1.276 ⋅ 10 −1.276 ⋅ 10 −1.276 ⋅ 10 −1.276 ⋅ 10 −1.276 ⋅ 10 −1.276 ⋅ 10 −1.276 ⋅ 10 −1.276 ⋅ 10

4 4 4 4 4 4 4 4 4 4 4 4

2.1⋅10⁵ 1.75⋅10⁵

Φ1

1.4⋅10⁵

1,j

1.05⋅10⁵

Φ1

7⋅10⁴

2,j

3.5⋅10⁴ 0 0.1 0.45 0.8 1.15 1.5 1.85 2.2 2.55 2.9 3.25 3.6 3.95

Φ1

-3.5⋅10⁴ -7⋅10⁴

Φ1

-1.05⋅10⁵

3,j

4,j

-1.4⋅10⁵

Φ1 y

j

5,j

6.356 ⋅ 10 6.356 ⋅ 10 6.356 ⋅ 10 6.356 ⋅ 10 6.356 ⋅ 10 6.356 ⋅ 10 6.356 ⋅ 10 6.356 ⋅ 10 6.356 ⋅ 10 6.356 ⋅ 10 6.356 ⋅ 10 6.356 ⋅ 10

3 3 3 3 3 3 3 3 3 3 3 3

4.223 ⋅ 10 4.223 ⋅ 10 4.223 ⋅ 10 4.223 ⋅ 10 4.223 ⋅ 10 4.223 ⋅ 10 4.223 ⋅ 10 4.223 ⋅ 10 4.223 ⋅ 10 4.223 ⋅ 10 4.223 ⋅ 10 4.223 ⋅ 10

4 4 4 4 4 4 4 4 4 4 4 4

5.2 ⋅ 10 5.2 ⋅ 10 5.2 ⋅ 10 5.2 ⋅ 10 5.2 ⋅ 10 5.2 ⋅ 10 5.2 ⋅ 10 5.2 ⋅ 10 5.2 ⋅ 10 5.2 ⋅ 10 5.2 ⋅ 10 5.2 ⋅ 10

4 4 4 4 4 4 4 4 4 4 4 4

−6.797 ⋅ 10 −6.797 ⋅ 10 −6.797 ⋅ 10 −6.797 ⋅ 10 −6.797 ⋅ 10 −6.797 ⋅ 10 −6.797 ⋅ 10 −6.797 ⋅ 10 −6.797 ⋅ 10 −6.797 ⋅ 10 −6.797 ⋅ 10 −6.797 ⋅ 10

3 3 3 3 3 3 3 3 3 3 3 3

−1.07 ⋅ 10 −1.07 ⋅ 10 −1.07 ⋅ 10 −1.07 ⋅ 10 −1.07 ⋅ 10 −1.07 ⋅ 10 −1.07 ⋅ 10 −1.07 ⋅ 10 −1.07 ⋅ 10 −1.07 ⋅ 10 −1.07 ⋅ 10 −1.07 ⋅ 10

5 5 5 5 5 5 5 5 5 5 5 5

−1.394 ⋅ 10 −1.394 ⋅ 10 −1.394 ⋅ 10 −1.394 ⋅ 10 −1.394 ⋅ 10 −1.394 ⋅ 10 −1.394 ⋅ 10 −1.394 ⋅ 10 −1.394 ⋅ 10 −1.394 ⋅ 10 −1.394 ⋅ 10 −1.394 ⋅ 10

5 5 5 5 5 5 5 5 5 5 5 5

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⋱ ⎥⎦

VII. Fluturarea binara sub efectul portantei Se considera o suprafata portanta rigida, suspendata elastic. Se da o deplasare initiala w si o rotire θ ambele marimi fiind considerate fata de CG al suprafetei portante.

w ≔ 0.170 m π ϕ ≔ 6 ⋅ ―― 180 ϕ = 0.105 Cy ≔ 1.46 e1 ≔ 0.112

m

i ≔ 0 ‥ 13 e ≔ 0.16

distanta intre CE si CP moment obtinut din constructia la scara 1:1 a aripi 2 kg ⋅ m in SolidWorks, si folosirea functiei Mass Properties. Moment de inertie in raport cu centrul de greut al aripii.

m

J ≔ 416.458

M ≔ 80.23 kg S ≔ 11.2

m

masa suprafetei portante

2

v ≔ VC = 87.75

m ― s

ρh = 0.596 2

ρh ⋅ v q ≔ ――― 2

q = 2.294 ⋅ 10

3

180 z ≔ 0.077 ⋅ ―― π

z = 4.412

‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾ 2 2 2 2 2 2 2 2 2 2 1 1 ‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾ ―⋅ ⎛⎝ωw + ωθ + ωw ⋅ e1 ⎞⎠ − ― ⎛⎝ωw + ωθ + ωw ⋅ e1 ⎞⎠ − 4 ⋅ ωw 2 2

ω1 ≔

‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾ 2 2 2 2 2⎞ 2 2 2 2⎞ 2 1 ⎛ 1 ‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾ ⎛ ω2 ≔ ―⋅ ⎝ωw + ωθ + ωw ⋅ e1 ⎠ + ― ⎝ωw + ωθ + ωw ⋅ e1 ⎠ − 4 ⋅ ωw 2 2

rad ―― s rad ―― s

Se observa ca se verificaω1