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 → → → 3 ( 2 3 3 3 → z2 R2 R O , x , y , z ) : repère lié à la plaque, en rotation autour de l’axe par rapport à • • on donne : ψ = Cte , θ = Cte , • ϕ = Cte 1. Matrices de passage Matrice de passage de R2 vers R1 → → ⎛ ⎞ ⎛ ⎞ ⎜ x1 ⎟ ⎛ cosθ 0 sinθ ⎞⎜ x2 ⎟ ⎜ → ⎟ ⎜ ⎟⎜ → ⎟ ⎜ y1 ⎟ = ⎜ 0 1 0 ⎟⎜ y2 ⎟ → ⎜ ⎟ ⎜ ⎟ → ⎜ ⎟ z ⎝− sinθ 0 cosθ 1 ⎠ z2 ⎝ ⎠ ⎝ ⎠ → z 1 O θ θ → z 2 → x 2 → x 1 P R 1→ R 2 → y 3 Matrice de passage de R vers R2 → → ⎛ ⎞ ⎛ ⎞ ⎜ x3 ⎟ ⎛ cosϕ sinϕ 0⎞⎜ x2 ⎟ ⎜ → ⎟ ⎜ ⎟⎜ → ⎟ ⎜ y3 ⎟ = ⎜− sinϕ cosϕ 0⎟⎜ y2 ⎟ → ⎜ ⎟ ⎜ ⎟ → ⎜ ⎟ z3 ⎝ 0 0 1⎠ z2 ⎝ ⎠ ⎝ ⎠ P R 3→ R 2 3 2b O 2 2a ϕ → x 2 ϕ → x 3 → y 2 2. Vecteur rotation instantané de par rapport à exprimé dans R ; D’après la relation de Chasles nous pouvons écrire : → → → → • → • → • → 0 2 1 0 Ω3 = Ω3 + Ω 2 + Ω1 = ϕ z2 + θ y2 + ψ z1 R3 R0 2 → z1 2 Exprimons le vecteur unitaire dans le repère R , il s’écrit : → → → z1 = −sinθ x2 + cosθ z2 D’où : → • → • → • → → • → • → • • → 0 ⎛ ⎞ ⎛ ⎞ Ω 3 = ϕ z2 + θ y2 + ψ ⎜− sinθ x2 + cosθ z2 ⎟ = −ψ sinθ x2 + θ y2 + ⎜ϕ+ ψ cosθ ⎟ z2 ⎝ ⎠ ⎝ ⎠ ⎧ → ⎪ 0 Ω3 = ⎨ ⎪ ⎪ R ⎩ 2 • −ψ sinθ • θ • • ϕ+ ψ cosθ 247
UMBB Boumerdès, Faculté des sciences, Département de physique Cours exercices, Mécanique Rationnelle : TCT et LMD-ST sem :3 A.KADI → 0 V 2 3. (O ) par dérivation et exprimée dans le repère ; → 0 2 → 0 d OO2 d OO −−→ 2 0 Par dérivation nous avons : V ( O2 ) = = + Ω 2 ∧ OO 2 dt dt −−→ −−→ R 2 Or ⎧0 −−→ ⎪ OO 2 = ⎨0 ⎪ R ⎩L 2 ⇒ d 2 −−→ OO dt 2 → = 0 ; et → 0 2 Ω → 1 2 = Ω → 0 1 + Ω • → = θ y 2 • → + ψ z 1 • ⎧ ⎪ − ψ sinθ • = ⎨ θ • ⎪ ψ cosθ ⎪ R ⎩ 2 → 0 V ( O • ⎧ ⎪ − ψ sinθ • ) = ⎨ θ • ⎪ ψ cosθ ⎪ R ⎩ 2 2 ∧ ⎧0 ⎪ ⎨0 ⎪ R ⎩L 2 = • ⎧ ⎪ Lθ • ⎨Lψ sinθ ⎪ 0 R ⎪⎩ 2 4. Vitesse du point A par rapport à exprimée dans le repère R ; Par la cinématique du solide nous pouvons écrire : R0 2 → → → −−→ 0 0 0 V ( A) V ( O2 ) + Ω3 ∧ O2 = A Le point A est dans le repère R 3 et a pour coordonnées : ⎧a ⎧a cosϕ −−→ ⎪ ⎪ O2 A= ⎨0 = ⎨asinϕ ⎪ ⎪ R ⎩0 R ⎩ 0 3 2 → 0 D’où : V ( A) = • ⎧ ⎪ Lθ • ⎨Lψ sinθ ⎪ 0 R ⎪⎩ 2 + ⎧ ⎪ ⎨ ⎪ ⎪ R ⎩ 2 • −ψ sinθ • θ • • ϕ+ ψ cosθ ∧ ⎧a cosϕ ⎪ ⎨a sinϕ ⎪ R ⎩ 0 2 → 0 V ( A) = • • • ⎧ ⎛ ⎞ ⎪ Lθ − a sinϕ⎜ ϕ+ ψ cosθ ⎟ ⎪ ⎝ ⎠ • • • ⎛ ⎞ ⎨ Lψ sinθ + a cosϕ⎜ ϕ+ ψ cosθ ⎟ ⎪ ⎝ ⎠ • • ⎪ ⎛ ⎞ − a ⎪ ⎜ψ sinϕ sinθ + θ cosϕ ⎟ R ⎩ ⎝ ⎠ 2 248
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MECANIQUE RATIONNELLE

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