A $2-$DOF linear elastic structure, with mass matrix $m=\left[\begin{array}{cc}4000& 0\\ 0& 5000\end{array}\right]kg,$ and stiffness

matrix $k=\left[\begin{array}{cc}300& -300\\ -300& 800\end{array}\right]{10}^3\frac{N}{m},$ external load $p={10}^3\{[30],[15]\}sin5tN.$

Questions:

$(1)$ Determine the vibration modes and natural frequency of the system;

$(2)$ Proof the orthogonality features of the vibration modes

$(3)$ Suppose the structure's response can be expressed by modal superposition method as $v=Y,Y=$

$\{[Y_1],[Y_2]\},$ where $$ is the vibration mode matrix, give the steady state response of the system $($hint:

calculate $Y_1$ and $Y_2$ first$)$