无阻尼系统响应
$m \ddot{x} + kx = F_0 \sin \omega t$
$x(t) = C_1 \cos \omega t + C_2 \sin \omega t + \mu \frac{F_0}{k} \sin \omega t$
$\mu = \frac{1}{1 - \tilde{\omega}^2}$, $\Theta = \arctan{\frac{c \omega}{k - m \omega^2}} = 0$
等效粘性阻尼
$W = \int_0^T F(t) \dot{x} dt = \pi X_0 F_0 \sin \Theta$ (外力做功)
$E = \oint F_d \dot{x} dt = \pi c \omega X_0^2$ (阻尼力耗能)
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结构阻尼: $E_I = 2 \pi X_0^2$ => $E_I = E$ => 粘性阻尼 $C_e = \frac{E}{\pi \omega X_0^2}$ , $h = \frac{a}{\omega}$ => $C_e = \frac{h}{\omega}$
损耗因子 $\eta = \frac{a}{k}$, $\Phi = \arctan \frac{h}{k - m \omega^2} = \arctan{\frac{\eta}{1-\omega^2}}$
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摩擦阻尼: $E = 4 X_0 |F_f|$ => $C_e = \frac{4 |F_f|}{\pi \omega X_0}$
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等效粘性阻尼: $\frac{E_I}{E} = E = C_e \omega X_0^2$ => $C_e = \frac{E_I}{\pi \omega X_0^2}$
机械阻抗基本概念
$m \ddot{x} + c \dot{x} + kx = F_0 e^{j \omega t}$
$X = H(\omega) F_0$ => $H(\omega) = \frac{1}{k - m \omega^2 + j c \omega}$
$Z_M(\omega) = \frac{F_0}{X} = \frac{1}{H(\omega)} = k - m \omega^2 + j c \omega$
$|Z_M(\omega)| = k \sqrt{(1 - r^2)^2 + (2\xi r)^2}$, $\varphi = \arctan{\frac{2 \xi r}{1 - r^2}}$
速度导纳
$V(\omega) = \frac{\dot{x}(\omega)}{F_0 e^{j \omega t}} = \frac{1}{j m \omega + c + \frac{k}{j \omega}}$