对数衰减率 (对数衰减系数) $\delta$
$\delta = \frac{2\pi \xi}{\sqrt{1 - \xi^2}}$ 对于小阻尼 $\delta = 2\pi \xi$
任意 m 个整数周期: $\delta = \frac{1}{m} \ln(\frac{x_n}{x_{n+m}})$
干摩擦系数下的自由振动 F = μN
振动方程: $m \ddot{x} + kx = \pm \mu N$
摩擦力小于弹簧回复力运动停止, 即 $x_n \leq \frac{\mu N}{k}$
在此之前发生的半周期个数 r 满足 $x_0 - r \frac{4 \mu N}{k} \leq \frac{\mu N}{k}$
一整个周期里运动振幅衰减 $X_m = X_{m-1} - \frac{4 \mu N}{k}$
振幅包络线斜率 = $-\frac{4 \mu N \omega_n}{2 \pi} = -\frac{2 \mu N \omega_n}{\pi k}$
相平面法
振动响应: $x(t) = A \cos(\omega_n t - \varphi)$ $\dot{x}(t) = -\omega_n A \sin(\omega_n t - \varphi)$
$\Rightarrow \frac{x(t)^2}{A^2} + \frac{\dot{x}(t)^2}{\omega_n^2 A^2} = 1$
振动微分方程: $m \ddot{x} + kx = 0$ $\frac{d \dot{x}}{dt} = -\frac{k}{m} x$, $\frac{dx}{dt} = \dot{x}$ $\Rightarrow \frac{d \dot{x}}{dx} = -\frac{k}{m} \frac{x}{\dot{x}}$
$\Rightarrow \dot{x}^2 + \frac{k}{m} x = C$