第三章
无阻尼自由振动
静平衡: $mg = k \Delta$
初始扰动: $m \ddot{x} = \sum F_x = mg - k(x + \Delta) = -kx$
自由振动微分方程: $m \ddot{x} + kx = 0$
固有角频率: $\omega_n = \sqrt{\frac{k}{m}}$
通解: $x(t) = C_1 \cos(\omega_n t) + C_2 \sin(\omega_n t) = A \cos(\omega_n t - \varphi)$, $A = \sqrt{C_1^2 + C_2^2}$, $\varphi = \arctan \frac{C_2}{C_1}$
t时刻的自由振动解: $x(t) = x(t) \cos \omega_n (t - \tau) + \frac{\dot{x}}{\omega_n} \sin \omega_n (t - \tau)$
零初始条件下的自由振动: $x(t) = x_0 \cos \omega_n t + \frac{\dot{x_0}}{\omega_n} \sin \omega_n t = A \cos (\omega_n t - \varphi)$
固有频率计算另一方式: $mg = k \Delta \Rightarrow \omega_n = \sqrt{\frac{g}{\Delta}}$
圆盘的自由振动: $\omega_n = \sqrt{\frac{J}{I}}$, $J \ddot{\theta} + k \theta = 0 \Rightarrow \ddot{\theta} + \omega_n^2 \theta = 0$
单摆振动: $\ddot{\theta} + (\frac{g}{l})\theta = 0 \Leftarrow ml\ddot{\theta} + mgl \sin\theta = 0$
能量法
无阻尼系统能量守恒: $T + U = \text{const} \Rightarrow \frac{d}{dt}(T + U) = 0$
等效质量: 动能系统能量相等
均质梁: 均布载荷下的位移曲线 $y=y_0 [\frac{3x}{l} - 4(\frac{x}{l})^3]$, $x = \frac{1}{2} l$
$T = 2 \cdot \frac{1}{2} \int_0^l \rho A \dot{y}^2 dx$
均质杆: $T = \frac{1}{2} \int_0^l \rho A \dot{u}^2 dx$, $ms = \rho A l$, $v = \frac{i}{\omega}$
转动件: $\frac{1}{2} J_e \omega_e^2 = \frac{1}{2} J_i \omega_i^2 \Rightarrow J_e = J_i (\frac{\omega_i}{\omega_e})^2 = \frac{J_i}{i^2}$, $i = \frac{\omega_e}{\omega_i}$
等效刚度：
并联弹簧: $k_e = k_1 + k_2$
串联弹簧: $\frac{1}{k_e} = \frac{1}{k_1} + \frac{1}{k_2}$
传动轴: $\frac{1}{2} k_e \theta_e^2 = \frac{1}{2} k_i \theta_i^2 \Rightarrow k_e = \frac{k_i}{i^2} \Rightarrow \omega_n = \sqrt{\frac{k_i / i^2}{J}} = \sqrt{\frac{k}{J_i + J_1 i^2}}$ (并联弹簧)
有阻尼自由振动: $m \ddot{x} + c \dot{x} + kx = 0$ $\ddot{x} + 2 \xi \omega_0 \dot{x} + \omega_n^2 x = 0$
临界阻尼: $c_c = 2 \sqrt{km} = 2m \omega_n$
临界阻尼比: $\xi = \frac{c}{c_c} = \frac{c}{2 m \omega_n} = \frac{c}{2\sqrt{km}}$
粘性阻尼: $s_{1,2} = -\frac{c}{2m} \pm \sqrt{\frac{c^2}{4m^2} - \frac{k}{m}}$ , $x = ce^{s_1 t} + c_2 e^{s_2 t}$ $m \ddot{x} + c \dot{x} + kx = 0$
动力学方程: $\ddot{x} + 2 \xi \omega_0 \dot{x} + \omega_n^2 x = 0$, 固有频率 $\omega_n = \sqrt{\frac{k}{m}}$, 阻尼比 $\xi = \frac{c}{2m\omega_n}$
欠阻尼 $\xi < 1$: 阻尼固有频率 $\omega_d = \omega_n \sqrt{1 - \xi^2}$， 阻尼自由振荡周期 $T_d = \frac{2 \pi}{\omega_d} = \frac{2 \pi}{\omega_n \sqrt{1-\xi^2}} = \frac{T}{\sqrt{1 - \xi^2}}$
欠阻尼是一种振幅逐渐衰减的振动
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过阻尼 $\xi > 1$: 一种按指数规律衰减的非周期蠕动，没有振动发生
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临界阻尼 $\xi = 1$: 一种按指数规律衰减的非周期蠕动， 比过阻尼衰减快些