Chapter 27, Problem 26P

Consider the mass-spring system in Fig. P27.26. The frequencies for the mass vibrations can be determined by solving for the eigenvalues and by applying $M\stackrel{\mathrm{..}}{x}+kx=0$, which yields

$\left[\begin{array}{ccc}{m}_1& 0& 0\\ 0& m_2& 0\\ 0& 0& m_3\end{array}\right]\left\{\begin{array}{c}\stackrel{\mathrm{..}}{x_1}\\ \stackrel{\mathrm{..}}{x_2}\\ \stackrel{\mathrm{..}}{x_3}\end{array}\right\}+\left[\begin{array}{ccc}2k& -k& -k\\ -k& 2k& -k\\ -k& -k& 2k\end{array}\right]\left\{\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right\}=\left\{\begin{array}{c}0\\ 0\\ 0\end{array}\right\}$

Applying the guess $x=x_0{e}^{i\omega t}$ as a solution, we get the following matrix:

$\left[\begin{array}{ccc}2k-m_1{\omega }^2& -k& -k\\ -k& 2k-m_2{\omega }^2& -k\\ -k& -k& 2k-m_3{\omega }^2\end{array}\right]\left\{\begin{array}{c}{x}_{01}\\ x_{02}\\ x_{03}\end{array}\right\}{e}^{i\omega t}=\left\{\begin{array}{c}0\\ 0\\ 0\end{array}\right\}$

Use MATLAB's eig command to solve for the eigenvalues of the $k-m{\omega }^2$ matrix above. Then use these eigenvalues to solve for the frequencies $\left(\omega \right)$. Let $m_1=m_2=m_3=1\text{ kg,}$ $\text{ and }k=2\text{ N/m}$