MECANIQUE RATIONNELLE

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UMBB Boumerdès, Faculté des sciences, Département de physique Cours exercices, Mécanique Rationnelle : TCT et LMD-ST sem :3 A.KADI 1. Matrices de passage Matrice de passage de R vers R1 0 → → ⎛ ⎞ ⎛ ⎞ ⎜ x0 ⎟ ⎛cosθ − sinθ 0⎞⎜ x1 ⎟ → → ≡ O ⎜ → ⎟ ⎜ ⎟⎜ → ⎟ z 0 z = 1 ⎜ y0 ⎟ ⎜ sinθ cosθ 0⎟⎜ y1 ⎟ → ⎜ ⎟ ⎜ ⎟ → ⎜ ⎟ z0 ⎝ 0 0 1⎠ z1 ⎝ ⎠ ⎝ ⎠ θ P R 0→ R 1 → x 0 Matrice de passage de R2 vers R1 → → ⎛ ⎞ ⎛ ⎞ → → ⎜ x y 1 ≡ A 2 ⎟ ⎛cosψ 0 − sinθ ⎞⎜ x1 ⎟ y 2 ⎜ → ⎟ ⎜ ⎟⎜ → ⎟ ⎜ y2 ⎟ = ⎜ 0 1 0 ⎟⎜ y1 ⎟ → ⎜ ⎟ ⎜ ⎟ → ⎜ ⎟ ⎝ ⎠ ψ z2 sinψ 0 cosψ z1 ⎝ ⎠ ⎝ ⎠ → P R 2→ R 1 z 1 θ → x 1 ψ → z 2 → y → x 2 1 → y → x 1 0 → → → 2. Ω 0 0 0 2 puis V ( B) et γ ( B) par dérivation direct et par la cinématique du solide a) la vitesse instantanée de rotation → Ω 0 2 → 0 2 Ω → 1 2 = Ω → 0 1 + Ω • → = ψ y 1 ⎧0 • → ⎪ • 1= ⎨ψ • ⎪ R ⎩θ 1 + θ z → 0 b) V ( B) par dérivation direct et par la cinématique du solide *) par dérivation directe Nous avons : ⎧a −→ −→ −→ ⎪ OB = OA+ AB= ⎨0 ⎪ R ⎩0 1 ⎧0 ⎪ + ⎨0 ⎪ R ⎩b 2 = ⎧a ⎪ ⎨0 ⎪ R ⎩0 1 ⎧bsinψ ⎪ + ⎨ 0 ⎪ R ⎩b cosψ 1 = ⎧ a + bsinψ ⎪ ⎨ 0 ⎪ R ⎩ bcosψ 1 Par dérivation nous avons : V → 0 −→ −→ 0 1 → d OB d OB −→ 0 ( B) = = + Ω1 ∧ OB dt dt 251
UMBB Boumerdès, Faculté des sciences, Département de physique Cours exercices, Mécanique Rationnelle : TCT et LMD-ST sem :3 A.KADI → 0 V ( B) = • ⎧ b ⎪ ψ cosψ ⎨ 0 + • ⎪− bψ sinψ R ⎪⎩ 1 ⎧0 ⎪ ⎨0 ∧ • ⎪ R ⎩θ 1 ⎧ a + bsinψ ⎪ ⎨ 0 ⎪ R ⎩ bcosψ 1 = • ⎧ ⎪ bψ cosψ • ⎨( a + bsinψ ) θ • ⎪ ⎪− bψ sinψ R ⎩ 1 *) par la cinématique du solide → 0 Nous pouvons écrire : V → 0 → 0 2 −→ ( B) = V ( A) + Ω ∧ AB → 0 Nous avons : V → 0 → 0 1 −→ ( A) = V ( O) + Ω ∧ OA ⇔ → 0 V ⎧0 ⎧a ⎪ ⎪ ( A) = ⎨0∧ ⎨0 • ⎪ ⎪ R ⎩θ R ⎩0 1 1 = ⎧ 0 • ⎪ ⎨aθ ⎪ R ⎩ 0 1 → → Car V 0 ( O) = 0 Nous avons ainsi : • ⎧ ⎧ 0 ⎧0 ⎧bsinψ ⎪ bψ cosψ → • ⎪ • • 0 ⎪ ⎪ ( B) = ⎨aθ + ⎨ψ + ⎨ 0 = ⎨( a + bsinψ ) θ • • ⎪ ⎪ ⎪ ⎪ R ⎩ 0 ⎩ R ⎩b ψ R θ cos ⎪− bψ sinψ 1 1 1 R ⎩ 1 V → b) γ 0 (B) par dérivation et par la cinématique du solide *) par dérivation → 0 0 1 0 → → 0 d V ( B) d V ( B) 0 0 Par dérivation nous avons : γ ( B) = = + Ω1 ∧V ( B) dt dt → → ⎧ •• • 2 ⎪bψ cosψ − bψ sinψ → •• • • 0 ⎪ γ ( B) = ⎨ ( a + bsinψ ) θ + bθ ψ cosψ • ⎪ •• 2 ⎪− bψ sinψ − bψ cosψ R ⎩ 1 + ⎧0 ⎪ ⎨0 ⎪ R ⎩θ 1 ∧ • • ⎧ ⎪ bψ cosψ • ⎨( a + bsinψ ) θ • ⎪ ⎪− bψ sinψ R ⎩ 1 252
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CLASSES PREPARATOIRES AUX GRANDES E
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Préface Quand Ali KADI m’a amica
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UMBB Boumerdès, Faculté des scien
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MECANIQUE RATIONNELLE

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