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 Solution : 1. Figures planes des trois rotations et les matrices de passages correspondantes ; a) Rotation du bras Nous avons : OB = R et x , y , z ) un repère fixe. R : étant le repère de projection on exprimera toute les données dans ce repère. → → → 1( 1 1 1 → → → R0 ( 0 0 0 2 → → → → x0 , x1 ) = ( y0 , y1 R x , y , z ) : en rotation / à tel que : z et ( ) = ψ sens positif R 0 → → 0 ≡ z 1 Matrice de passage de R vers R1 → → ⎛ ⎞ ⎛ ⎞ ⎜ x0 ⎟ ⎛cosψ − sinψ 0⎞⎜ x1 ⎜ → ⎟ ⎜ ⎟⎜ → ⎜ y0 ⎟ = ⎜ sinψ cosψ 0⎟⎜ y1 → ⎜ ⎟ ⎜ ⎟ → ⎜ z ⎝ 0 0 1⎠ ⎟ ⎟⎟⎟⎟ 0 z1 ⎝ ⎠ ⎝ ⎠ P R 0→ R 1 0 → → z 0 ≡ z 1 O → x 0 ψ ψ → x 1 → y 1 → y 0 a) Rotation du cockpit → → → 2 ( 2 2 2 R 1 → → 1 ≡ x 2 → → → → ( 1 2 1 z2 R B, x , y , z ) : en rotation / tel que x et y , y ) = ( z , ) = θ sens positif ; Matrice de passage de R1 vers R2 → → ⎛ ⎞ ⎛ ⎞ ⎜ x1 ⎟ ⎛1 0 0 ⎞⎜ x2 ⎜ → ⎟ ⎜ ⎟⎜ → ⎜ y1 ⎟ = ⎜0 cosθ − sinθ ⎟⎜ y2 → ⎜ ⎟ ⎜ ⎟ → ⎜ z ⎝0 sinθ cosθ ⎠ ⎟ ⎟⎟⎟⎟ 1 z2 ⎝ ⎠ ⎝ ⎠ → → x 1 ≡ x 2 B θ θ → z 2 → z 1 P R 1→ R 2 → y 1 → y 2 a) Rotation du siège pilote → → → 3 ( 3 3 3 → → 2 ≡ y 3 → → → → ( 2 3 2 z3 R B, x , y , z ) en rotation / tel que : y et x , x ) = ( z , ) = ϕ sens positif. Matrice de passage de R vers R2 → → ⎛ ⎞ ⎛ ⎞ ⎜ x3 ⎟ ⎛cosϕ 0 − sinϕ ⎞⎜ x2 ⎜ → ⎟ ⎜ ⎟⎜ → ⎜ y3 ⎟ = ⎜ 0 1 0 ⎟⎜ y2 → ⎜ ⎟ ⎜ ⎟ → ⎜ z ⎝ sinϕ 0 cosϕ ⎠ ⎟ ⎟⎟⎟⎟ 3 z2 ⎝ ⎠ ⎝ ⎠ P R 3→ R 2 3 → → y 2 ≡ y 3 → z 2 B ϕ ϕ → z 3 → x 3 → x 2 257
UMBB Boumerdès, Faculté des sciences, Département de physique Cours exercices, Mécanique Rationnelle : TCT et LMD-ST sem :3 A.KADI 2. Vecteur position du point T par rapport à R0 exprimé dans R2 Nous avons : −→ OT −→ −→ = OB+ BT , sachant que −→ BT → = L z3 ⎧R −→ ⎪ OB= ⎨0 ⎪ R R ⎩0 1 , 2 ⎧0 ⎧Lsinϕ −→ ⎪ ⎪ ; BT = ⎨0 = ⎨ 0 d’où : ⎪ ⎪ R ⎩L R ⎩L cosϕ 3 2 ⎧ R + Lsinϕ −→ ⎪ OT = ⎨ 0 ⎪ R ⎩ L cosϕ 2 Vecteur rotation du siège pilote : → → → → • → • → • → 0 2 1 0 Ω3 = Ω3 + Ω 2 + Ω1 = y2 + θ x2 + ψ z1 ϕ ; Par la matrice de passage de vers R le vecteur ‘écrit : R1 2 → z 1 → → → z1 = sinθ y2 + cosθ z2 → • → • → • → → • → • • → • → 0 ⎛ ⎞ ⎛ ⎞ Ω 3 = ϕ y2 + θ x2 + ψ ⎜sinθ y2 + cosθ z2 ⎟ = θ x2 + ⎜ϕ + ψ sinθ ⎟ y2 + ψ cosθ z2 → 0 3 Ω • ⎧ ⎪ θ • • = ⎨ϕ + ψ sinθ • ⎪ R ⎪ψ cosθ ⎩ 2 ⎝ ⎠ ⎝ ⎠ 3. Vecteur vitesse du point T 3.1. Par composition de mouvement → V abs → 0 2 0 = V + V ⇔ V ( T ) = V ( T ) + V ( T ) rel → ent → → → 2 La vitesse relative est donnée par : → 2 V −→ 2 d BT ( T ) = dt • ⎧ ⎪ Lϕ cosϕ ⎨ 0 • ⎪− Lϕ sinϕ R ⎪⎩ 2 La vitesse relative s’écrit : → → 0 0 V2 ( T ) V ( O) → 0 2 −→ = + Ω ∧ OT • ⎧ ⎪ θ → • 0 V 2 ( T ) = ⎨ψ sinθ • ⎪ ⎪ψ cosθ R ⎩ 2 ∧ ⎧ R + Lsinϕ ⎪ ⎨ 0 = ⎪ R ⎩ L cosϕ 2 • ⎧ ⎪ Lψ sinθ cosϕ • • ⎨− Lθ cosϕ + ψ • ⎪ ⎪ −ψ sinθ R ⎩ 2 cosθ ( R + Lsinϕ ) ( R + Lsinϕ ) 258
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CLASSES PREPARATOIRES AUX GRANDES E
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Préface Quand Ali KADI m’a amica
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MECANIQUE RATIONNELLE

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