Problem $8(20\text{'})$

A $2-$degree$-$of$-$freedom system has the following equation of motion

$\left[\begin{array}{cc}m& 0\\ 0& 2m\end{array}\right](\frac{x_1^{}}{x_2^{}})+\left[\begin{array}{cc}k& -2k\\ -2k& 4k\end{array}\right](\frac{x_1}{x_2})=(\frac{0}{0})$

Where m and k have units of kg and $\frac{N}{m},$ respectively. This system is given a set of initial condition of $x_1(0)=x_2(0)=x_1^{}(0)=0$ and $x_2^{}(0)=V,$ where $V$ has unit of $\frac{m}{sec}.$

Determine:

the natural frequency of system $(5\text{'})$

mode shapes of the system. $(5\text{'})$

$3)$ the response $x(t)$ for $t>0.(10\text{'})$