大学物理III-6简谐振动
简谐振动的判据
\[f = -kx\] \[\frac{\mathrm{d}^2 x}{\mathrm{d}t^2} + \omega^2 x = 0\] \[x = A\cos (\omega t + \varphi)\]弹簧振子
\[\frac{k}{m} = \omega^2\] \[T = 2\pi\sqrt{\frac{m}{k}}\]单摆
\[\omega^2 = \frac{g}{l}\] \[T = 2\pi\sqrt{\frac{l}{g}}\]复摆
其中
\[M = -mgl\sin \theta = J\alpha = J \frac{\mathrm{d}^2 \theta}{\mathrm{d}t^2}\]应用近似 $\sin\theta \approx \theta$
\[-mgl\theta = J\frac{\mathrm{d}^2\theta}{\mathrm{d}t^2}\]套用形式
\[\frac{\mathrm{d}^2 \theta}{\mathrm{d} t} + \omega^2\theta = 0\]即
\[\omega = \sqrt{\frac{mgl}{J}}\]扭摆
\[M_z = -D\theta\] \[J_z \frac{\mathrm{d}^2 \theta}{\mathrm{d}t^2} = -D \theta\]即
\[\omega = \sqrt{\frac{D}{J_z}}\]同方向同频率简谐振动的合成
\[x_1 = A_1\cos(\omega t + \varphi_1)\] \[x_2 = A_2\cos(\omega t + \varphi_2)\] \[A = \sqrt{A_1^2 + A_2^2 + 2 A_1 A_2 \cos(\varphi_2 - \varphi_1)}\] \[\tan \varphi = \frac{A_1 \sin \varphi_1 + A_2 \sin \varphi_2}{A_1 \cos \varphi_1 + A_2 \cos \varphi_2}\]两个同方向不同频率的简谐振动的合成
运用和差化积公式
\[x = \left(2A_1 \cos \frac{\omega_1 - \omega_2}{2}t\right)\cos \frac{\omega_1 + \omega_2}{2} t\]