Mouvements oscillatoires harmoniques
fréquence \( \class{formule} {f = \dfrac{1}{T} }\)
pulsation ω = 2 . π . f
élongation x = A . cos(ω . t + ε)
élongation angulaire θ = θm . cos(ω . t + ε)
| ressort | ||
|---|---|---|
| £Frappel = - k . x | \( \class{formule} {ω = \sqrt{\dfrac{k}{m}} }\) | Ep = k . x2 / 2 |
| pendule de torsion | |||
|---|---|---|---|
| £Mrappel = -k . £θ | \( \class{formule} { ω = \sqrt{\dfrac{k}{I}} }\) | Ep = k . θ2 / 2 | \( \class{formule} {k = \dfrac{π . η . r^4}{2 . h} }\) |
| pendule simple | |
|---|---|
| \( \class{formule} {ω = \sqrt{\dfrac{g}{l}} }\) | Ep = m . g . h |
oscillations amorties
avec Fvisc = - f . v
si \( \class{formule} {f^2 - 4 . m . k < 0 }\) ==> \( \class{formule} {x = A . exp(- \dfrac{f}{2m} . t) . cos(ω . t + ε) }\)
oscillations forcées
avec Fext = Fm . cos(ω . t + ε)
\( \class{formule} {Z = \dfrac{F_m}{v_m} = \sqrt{f^2 - (m . ω - \dfrac{k}{ω})^2} }\)
\( \class{formule} {tgε = \dfrac{(m . ω - \dfrac{k}{ω})}{f} }\)
Superposition
A2 = A12 + A22 + 2 . A1 . A2 . cosΔφ
battements: \( \class{formule} {f_b = |f_1 - f_2| }\) et \( \class{formule} {T_b = \dfrac{1}{f_b} }\)