## 自由振動系統的特徵值與特徵向量

$f(x)=-kx$

$m\ddot{x}=-kx$

$x(t)=\alpha_1e^{i\omega t}+\alpha_2e^{-i\omega t}$

\displaystyle\begin{aligned} x(t)&=\frac{1}{2}(c_1-ic_2)e^{i\omega t}+\frac{1}{2}(c_1+ic_2)e^{-i\omega t}\\ &=\frac{1}{2}(c_1-ic_2)(\cos(\omega t)+i\sin(\omega t))+\frac{1}{2}(c_1+ic_2)(\cos(\omega t)-i\sin(\omega t))\\ &=c_1\cos(\omega t)+c_2\sin(\omega t)\\ &=\hat{c}\cos(\omega t-\phi),\end{aligned}

$\displaystyle c_1=x(0),~~c_2=\frac{\dot{x}(0)}{\omega}$

\begin{aligned} m\ddot{x}_1&=-k_1x_1+k_2(x_2-x_1)=-(k_1+k_2)x_1+k_2x_2\\ m\ddot{x}_2&=-k_2(x_2-x_1)+k_1(-x_2)=k_2x_1-(k_1+k_2)x_2.\end{aligned}

$\begin{bmatrix} \ddot{x}_1\\ \ddot{x}_2 \end{bmatrix}+\begin{bmatrix} \frac{k_1+k_2}{m}&-\frac{k_2}{m}\\[0.5em] -\frac{k_2}{m}&\frac{k_1+k_2}{m} \end{bmatrix}\begin{bmatrix} x_1\\ x_2 \end{bmatrix}=\begin{bmatrix} 0\\ 0 \end{bmatrix}$

$\ddot{\mathbf{x}}+A\mathbf{x}=\mathbf{0}$

$A\mathbf{x}=\omega^2\mathbf{x}$

$\det(A-\lambda I)=\begin{vmatrix} a-\lambda&-b\\ -b&a-\lambda \end{vmatrix}=(a-\lambda)^2-b^2$

$\displaystyle \lambda_1=a+b=\frac{k_1+2k_2}{m},~~\lambda_2=a-b=\frac{k_1}{m}$

$\mathbf{u}_1=\left[\!\!\begin{array}{r} 1\\ -1 \end{array}\!\!\right],~~\mathbf{u}_2=\begin{bmatrix} 1\\ 1 \end{bmatrix}$

$\mathbf{x}(t)=\alpha_{1}\mathbf{u}_1e^{i\omega_1t}+\alpha_{2}\mathbf{u}_1e^{-i\omega_1t}+\beta_1\mathbf{u}_2e^{i\omega_2t}+\beta_2\mathbf{u}_2e^{-i\omega_2t}$

$\mathbf{x}(t)=c_{11}\mathbf{u}_1\cos(\omega_1 t)+c_{12}\mathbf{u}_1\sin(\omega_1t)+c_{21}\mathbf{u}_2\cos(\omega_2t)+c_{22}\mathbf{u}_2\sin(\omega_2t)$

\begin{aligned} \mathbf{x}(0)&=c_{11}\mathbf{u}_1+c_{21}\mathbf{u}_2,\\ \dot{\mathbf{x}}(0)&=\omega_1c_{12}\mathbf{u}_1+\omega_2c_{22}\mathbf{u}_2,\end{aligned}

$\displaystyle c_{11}=\frac{1}{2}(x_1(0)-x_2(0)),~~c_{21}=\frac{1}{2}(x_1(0)+x_2(0)),$

$\displaystyle c_{12}=\frac{1}{2\omega_1}(\dot{x}_1(0)-\dot{x}_2(0)),~~c_{22}=\frac{1}{2\omega_2}(\dot{x}_1(0)+\dot{x}_2(0))$

$\mathbf{x}(t)=c_{11}\mathbf{u}_1\cos(\omega_1 t)+c_{21}\mathbf{u}_2\cos(\omega_2t)$

$x_1(0)=1$$x_2(0)=1$，則 $x_1(t)=x_2(t)=\cos(t)$，第二振型被激發，但第一振型則未被激發。圖四(b)顯示初始值為 $x_1(0)=1$$x_2(0)=0.8$ 的振動軌跡。

[1] 維基百科：塔科馬海峽吊橋
[2] 維基百科：卡門渦街

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### 4 Responses to 自由振動系統的特徵值與特徵向量

1. leonyo says:

文中 “如何幫大樓抗風防震？淺談台北101大樓阻尼器” 的超連結好像已經失效
搜尋到的新連結如下
latex \displaystyle\begin{aligned} x(t)&=\frac{1}{2}(c_1-ic_2)e^{i\omega t}+\frac{1}{2}(c_1+ic_2)e^{-i\omega t}\\ &=\frac{1}{2}(c_1-ic_2)(\cos(\omega t)+i\sin(\omega t))+\frac{1}{2}(c_1+ic_2)(\cos(\omega t)-i\sin(\omega t))\\ &=c_1\cos(\omega t)+c_2\sin(\omega t)\\ &=\hat{c}\cos(\omega t-\phi),\end{aligned}&fg=000000