MECANIQUE RATIONNELLE

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UMBB Boumerdès, Faculté des sciences, Département de physique Cours exercices, Mécanique Rationnelle : TCT et LMD-ST sem :3 ⎧ •• • • 2 2 ⎪bψ cosψ − bψ sinψ − ( a + bsinψ ) θ → •• • • 0 ⎪ γ ( B) = ⎨( a + bsinψ ) θ + 2bθ ψ cosψ • ⎪ •• 2 ⎪ − bψ sinψ − bψ cosψ R ⎩ 1 A.KADI *) par la cinématique du solide Nous pouvons écrire : → 0 2 → → 0 0 0 d Ω γ ( B) = γ ( A) + dt −→ ∧ AB + → → 0 ⎛ Ω 2 ∧ ⎜Ω ⎝ 0 2 −−→⎞ ∧ AB⎟ ⎠ Calculons d’abord : → 0 1 → → 0 0 0 d Ω γ ( A) = γ ( O) + dt −→ ∧ OA + → → ⎛ −−→ 0 0 ⎞ Ω1 ∧ ⎜Ω1 ∧ OA⎟ ⎝ ⎠ → Sachant que γ 0 ( O) = 0 , on obtient : → → 0 γ ( A ) = ⎧0 ⎪ ⎨0 ∧ •• ⎪ R ⎩θ 1 ⎧a ⎪ ⎨0 ⎪ R ⎩0 1 + ⎧0 ⎧0 ⎪ ⎪ ⎨0 ∧ ⎨0 ∧ • • ⎪ ⎪ R ⎩θ R ⎩θ 1 1 • ⎧ ⎪ − 2 ⎧a aθ •• ⎪ ⎨0= ⎨ aθ ⎪ ⎪ R ⎩0 0 1 ⎪ R ⎩ 1 → 0 2 0 d Ω dt = → 0 2 1 d Ω dt • • ⎧ ⎧0 ⎧0 ⎧0 ⎪ θ ψ → → ⎪ •• ⎪ ⎪ • •• 0 0 + Ω1 ∧ Ω 2 = ⎨ψ + ⎨0∧ ⎨ψ = ⎨ ψ •• • • •• ⎪ ⎪ ⎪ ⎪ ⎩θ ⎩θ R R ⎩θ ⎪ θ 1 1 R1 R ⎩ 1 → 0 2 0 d Ω dt • • •• ⎧ ⎧ ⎪ θ ψ ⎧bsinψ ⎪ bψ cosψ −→ •• •• • • ⎪ ∧ AB= ⎨ ψ ∧ ⎨ 0 = ⎨bθ sinψ + bθ ψ cosψ •• •• ⎪ ⎪ ⎪ ⎪ θ R ⎩b cosψ ⎪− bψ cosψ 1 R ⎩ R ⎩ 1 1 253
UMBB Boumerdès, Faculté des sciences, Département de physique Cours exercices, Mécanique Rationnelle : TCT et LMD-ST sem :3 → → 0 ⎛ Ω 2 ∧ ⎜Ω ⎝ 0 2 • ⎧ ⎧0 ⎧0 ⎧bsinψ ⎧0 ⎪ bψ cosψ −→ ⎞ ⎪ • ⎪ • ⎪ ⎪ • • ∧ AB ⎟= ⎨ψ ∧ ⎨ψ ∧ ⎨ 0 = ⎨ψ ∧ ⎨ bθ sinψ ⎠ • • • • ⎪ ⎪ ⎪ ⎪ ⎪ ⎩θ ⎩θ ⎩b cosψ R ⎩θ ⎪− bψ sinψ 1 R R 1 1 R1 R ⎩ 1 A.KADI • • ⎧ 2 2 ⎪ − bψ sinψ − bθ sinψ • • ⎪ = ⎨ bθ ψ cosψ • ⎪ 2 ⎪− bψ cosψ R ⎩ 1 En faisant la somme des trois termes nous obtenons : • •• ⎧ ⎧ ⎪ − 2 aθ ⎪ bψ cosψ → •• •• • • 0 γ ( B) = ⎨ aθ + ⎨bθ sinψ + bθ ψ cosψ •• ⎪ 0 ⎪ ⎪ ⎪− bψ cosψ R ⎩ R ⎩ 1 1 • • ⎧ 2 2 ⎪ − bψ sinψ − bθ sinψ • • ⎪ + ⎨ bθ ψ cosψ • ⎪ 2 ⎪− bψ cosψ R ⎩ 1 ⎧ •• • • 2 2 ⎪bψ cosψ − bψ sinψ − ( a + bsinψ ) θ → •• • • 0 ⎪ γ ( B) = ⎨( a + bsinψ ) θ + 2bθ ψ cosψ • ⎪ •• 2 ⎪ − bψ sinψ − bψ cosψ R ⎩ 1 Exercice 05 : Une barre homogène mince de longueur AC = 2L et de centre G , repose sans frottement sur un petit rouleau fixe au point B et s’appuie contre un mur lisse au point A . projection. 1- Déterminer la vitesse de glissement en A et en B ; R 0 : est le repère de 2- Déterminer les coordonnées du C.I.R. (centre instantanée de rotation) géométriquement et analytiquement. 254
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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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