4.1 Lecture Thursday April 4, 2013

Multidegree freedom system, free vibration, no damping

$$\begin{array}{}\text{(1)}& \left[m\right]\left\{\ddot{v}\left(t\right)\right\}+\left[k\right]\left\{v\left(t\right)\right\}=\left\{0\right\}\end{array}$$

Assume $$\left\{v\left(t\right)\right\}=\left\{\hat{v}\right\}\sin (\omega t+\theta )$$

where $\left\{\hat{v}\right\}$ is an amplitude vector of constants. Acts like shape function. Hence $\left\{\ddot{v}\left(t\right)\right\}=-{\omega }^2\left\{\hat{v}\right\}\sin (\omega t+\theta )$. Substituting into Eq ??

$$\begin{array}{rl}\left[m\right](-{\omega }^2\left\{\hat{v}\right\}\sin (\omega t+\theta ))+\left[k\right]\left\{\hat{v}\right\}\sin (\omega t+\theta )& =\left\{0\right\}\\ (-\left[m\right]{\omega }^2+\left[k\right])\left\{\hat{v}\right\}& =\left\{0\right\}\end{array}$$

This is an eigenvalue problem. Hence $$det(\left[k\right]-\left[m\right]{\omega }^2)=0$$

We obtain $n$ unique eigenvalues ${\omega }_i$ and corresponding $n$ independent mode shapes ${\left\{\hat{v}\right\}}_i$

$$(-\left[m\right]{\omega }_i^2+\left[k\right]){\left\{\hat{v}\right\}}_i=\left\{0\right\}$$

Where ${\left\{\hat{v}\right\}}_i={\left\{\begin{array}{c}1\\ v_2\\ ⋮\\ v_n\end{array}\right\}}_i$ or ${\left\{\hat{v}\right\}}_i={\left\{\begin{array}{c}{\phi }_1\\ {\phi }_2\\ ⋮\\ {\phi }_n\end{array}\right\}}_i$. Hence for each ${\omega }_i$ we get different shape function vector ${\left\{\hat{v}\right\}}_i.$ Let the mode shape matrix $\left[\mathrm{\Phi }\right]$ be

$$\begin{array}{rl}\left[\mathrm{\Phi }\right]& =[{\left\{\hat{v}\right\}}_1,{\left\{\hat{v}\right\}}_2,\cdots ,{\left\{\hat{v}\right\}}_n]\\ & =\left\{\begin{array}{cccc}{\phi }_{11}& {\phi }_{12}& \cdots & {\phi }_{1n}\\ {\phi }_{21}& {\phi }_{22}& \cdots & {\phi }_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ {\phi }_{n1}& {\phi }_{n2}& \cdots & {\phi }_{nn}\end{array}\right\}\end{array}$$