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 Calculons d’abord les moments cinétiques des solides ( S 2 ) et ) en G en utilisant la formule de transport : → → −−−→ → G1 G G 0 σ ( S2 / R0 ) = σ 2 ( S2 / R0 ) + G1G2 ∧ m2 V ( 2 ) ⎛ ⎜ 0 ⎞ ⎟ ⎛ ⎜ ( S 3 1 → • → • • • → 2 2 2 G1 ( S 2 / R0 ) = A2 α x0 + ⎜ bsinα ⎟ ∧ m2 ⎜ bα cosα ⎟ = ⎜ A2 α+ m2b α(cos α − sin α) ⎟ x0 • σ ⎛ ⎝ ⎜ ⎟ ⎝− bcosα ⎠ → • • → 2 G1( S 2 / R0 ) = ⎜ A2 α + m2b α cos 2α ⎟ x0 σ → → −−−→ → G1 G G ⎞ ⎠ 0 ⎞ ⎟ ⎜ ⎟ ⎝− bα sinα ⎠ 0 σ ( S3 / R0 ) = σ 3 ( S3 / R0 ) + G1G3 ∧ m3 V ( 3) ⎛ 0 ⎞ ⎜ ⎟ ⎟ ⎜ ⎟ ⎝− 2b cosα ⎠ ⎛ ⎜ → • → • • • → 2 2 G1( S3 / R0 ) = −A3 β x0 + ⎜ 2b sinα ∧ m3⎜2bα cosα ⎟ = ⎜− A3 β + 4m3b α cos α ⎟ x0 σ ⎛ ⎝ → • • → 2 2 G1( S3 / R0 ) = ⎜− A3 β + 4m3b α cos α ⎟ x0 σ ⎞ ⎠ ⎜ ⎝ Les moments dynamiques se déduisent facilement par dérivation des deux expressions: → d 0 σ 1( S 2 / R0 ) ⎛ ⎞ G dt ⎝ α ⎠ → •• •• • → 2 2 = ⎜ A2 α+ m2b ( α cos 2α − 2α sin 2 ) ⎟ x0 d 0 σ 1( S3 / R0 ) ⎛ ⎞ G dt ⎝ α ⎠ •• •• • → 2 2 2 2 = ⎜− A3 β + 4m3b α cos α − 8m3b α sinα cos ⎟ x0 ⎛ •• •• • → 2 2 2 2 = ⎜− A3 β + 4m3b α cos α − 4m3b α sin 2α ⎟ x0 ⎝ Le moment dynamique du système est la somme des deux expressions : → δ G1 ( ∑ / R 0 ⎛ •• ) = ⎜ A2 α+ m2b ⎝ ⎛ •• + ⎜− A3 β + 4m3b ⎝ ⎛ 2 •• • 2 ⎞ ( α cos 2α − 2α sin 2α ) ⎟ x ⎠ •• 2 2 α cos α − 4m b 3 2 0 0 → ⎞ ⎟ ⎟ ⎠ 0 ⎛ ⎝ ⎛ ⎝ • 2 ⎞ α sin 2α ⎟ x ⎠ → •• •• •• • •• • → 2 2 2 2 2 G1 ( ∑ / R0 ) = ⎜ A2 α− A3 β + m2b ( α cos 2α − 2α sin 2α ) + 4m3b ( α cos α −α sin 2α ) ⎟ x0 δ ⎝ → 0 ⎞ ⎠ ⎞ ⎠ ⎞ ⎠ ⎞ ⎠ 314
UMBB Boumerdès, Faculté des sciences, Département de physique Cours exercices, Mécanique Rationnelle : TCT et LMD-ST sem :3 A.KADI 5. Energie cinétique du système E c Σ / R ) par rapport à . ( 0 R 0 E ∑ ( / R0 ) = EC ( S1 / R0 ) + EC ( S2 / R0 ) EC ( S3 / R0 ) C + E C 2 → → • 1 ⎛ 0 ⎞ 1 0T 0 1 2 2 ( S1 / R0 ) = m1 ⎜V ( G1 ) ⎟ + Ω1 . I G1. Ω1 = m1 4b α sin 2 ⎝ ⎠ 2 2 → 2 α E C 2 → 0T Ω 2 → → • 1 ⎛ 0 ⎞ 1 0 1 2 2 1 ( S 2 / R0 ) = m2 ⎜V ( G2 ) ⎟ + . I G2. Ω 2 = m2b α + A2 α 2 ⎝ ⎠ 2 2 2 • 2 E C 2 → 0T Ω3 → → • 1 ⎛ 0 ⎞ 1 0 1 2 2 2 1 ( S3 / R0 ) = m3⎜V ( G3) ⎟ + . I G3. Ω3 = m3 4b α cos α + A3 β 2 ⎝ ⎠ 2 2 2 • 2 • • • • • 1 ⎛ 2 2 2 2 2 2 2 2 E C ( ∑ / R0 ) = ⎜b (4m1 α sin α + m2 α + 4m3 α cos α) + A2 α + A3 β 2 ⎝ ⎞ ⎟ ⎠ 315
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MECANIQUE RATIONNELLE

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