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. Les figures planes : R1 0 a) Passage du repère vers R : la rotation se fait autour de l’axe Matrice de passage du repère R1 vers R0 ⎛ ⎜ x ⎜ ⎜ y ⎜ z ⎝ → 1 → 1 → 1 ⎞ ⎟ ⎛ cosψ sinψ ⎟ ⎜ ⎟ = ⎜− sinψ cosψ ⎟ ⎜ ⎝ 0 0 ⎠ P R 1→ R 0 ⎛ 0⎞⎜ x ⎟⎜ 0⎟⎜ y 1⎟⎜ ⎠ z ⎝ → 0 → 0 → 0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ → → z 1 ≡ z 0 → x 0 G ψ → → z 0 ≡ z 1 ψ → x 1 → y 1 → y 0 → → b) Passage du repère R2 vers R1 : la rotation se fait autour de l’axe x 1 ≡ x 2 Matrice de passage de R2 vers R1 → → ⎛ ⎞ ⎛ ⎞ ⎜ x2 ⎟ ⎛1 0 0 ⎞⎜ x1 ⎟ ⎜ → ⎟ ⎜ ⎟⎜ → ⎟ ⎜ y2 ⎟ = ⎜0 cosθ sinθ ⎟⎜ y1 ⎟ → ⎜ ⎟ ⎜ ⎟ → ⎜ ⎟ z ⎝0 − sinθ cosθ 2 ⎠ z1 ⎝ ⎠ ⎝ ⎠ P R 2→ R 1 → → x 1 ≡ x 2 → y 1 G θ θ → y 2 → z 2 → z 1 c) Passage du repère vers R : la rotation se fait autour de l’axe RS 2 → → z 2 ≡ z s → y s Matrice de passage de → ⎛ ⎞ ⎜ xs ⎟ ⎛ cosϕ ⎜ → ⎟ ⎜ ⎜ ys ⎟ = ⎜− sinϕ → ⎜ ⎟ ⎜ z ⎝ 0 s ⎝ ⎠ P R → s R 2 sinϕ cosϕ 0 R vers R2 s ⎛ 0⎞⎜ x ⎟⎜ 0⎟⎜ y 1⎟⎜ ⎠ z ⎝ → 2 → 2 → 2 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ → → z 2 ≡ z s G → x 2 ϕ ϕ → x s → y 2 243
UMBB Boumerdès, Faculté des sciences, Département de physique Cours exercices, Mécanique Rationnelle : TCT et LMD-ST sem :3 A.KADI 2. Vecteur rotation instantané de la sphère dans le repère R0 → → → → • → • → • → 0 2 1 0 Ω s = Ω s + Ω2 + Ω1 = ϕ z2 + θ x1 + ψ z0 → → x1 z 0 0 Exprimons et dans le repère R . D’après les matrices de passage nous avons : → → → x1 = cosψ x0 + sinψ y0 ⎛ ⎝ → → → → → → z2 = −sinθ y1+ cosθ z1 = −sinθ ⎜− sinψ x0 + cosψ y0 ⎟ + cosθ z0 → → → → z2 = sinθ sinψ x0 − sinθ cosψ y0 + cosθ z0 ⎞ ⎠ ce qui donne : → • → → → • → → • → 0 ⎛ ⎞ ⎛ ⎞ Ω s = ϕ⎜sinθ sinψ x0 − sinθ cosψ y0 + cosθ z0 ⎟ + θ ⎜cosψ x0 + sinψ y0 ⎟ + ψ z0 ⎝ ⎠ ⎝ ⎠ → • • → • • → • • → 0 ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ Ω s = ⎜ϕ sinθ sinψ + θ cosψ ⎟ x0 + ⎜− ϕ sinθ cosψ + θ sinψ ⎟ y0 + ⎜ψ + ϕ cosθ ⎟ z0 → 0 s Ω ⎝ • • ⎧ ⎪ ϕ sinθ sinψ + θ cosψ • • = ⎨−ϕ sinθ cosψ + θ sinψ • • ⎪ ⎪ ψ + ϕ cosθ R ⎩ 0 ⎠ ⎝ ⎠ ⎝ ⎠ 3. Vitesse du point de contact I de la sphère avec le plan fixe Les points G et I appartiennent à la sphère. Par la cinématique du solide, nous pouvons connaître la vitesse du point I à partir de celle de G, en effet nous avons : → 0 V → 0 → 0 s −→ ( I) = V ( G) + Ω ∧ GI Avec : • ⎧ ⎧x −→ −→ ⎪ x → 0 • ⎪ 0 d OG OG= ⎨y ⇒ V ( G) = = ⎪ R ⎩a 0 ⎪ ⎪ ⎨y dt 0 R ⎩ 0 ⎧x −→ −→ ⎪ et OI= ⎨y ⇒ GI = ⎪ R ⎩0 R 0 0 ⎧ 0 ⎪ ⎨ 0 ⎪ ⎩− a 244
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MECANIQUE RATIONNELLE

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