Problem $3(40$ points$)$:

Consider the $2-$DOF system shown in Figure $3$ and answer the following questions:

$1-$ The equations of motion of this system are given by:

$\{[{}_1^{}],[{}_2^{}]\}+\left[\begin{array}{cc}{}^2+{}^2{}^2& -{}^2{}^2\\ -{}^2{}^2& {}^2+{}^2{}^2\end{array}\right]\{[{}_1],[{}_2]\}=\frac{1}{m{l}^2}\{[T_1(t)],[T_2(t)]\}$

where $=\sqrt[\phantom{2}]{\frac{g}{l}}$ and $=\sqrt[\phantom{2}]{\frac{k}{m}}$ are the two nominal natural frequencies for a simple pendulum of length $l$ and a mass$-$spring system, respectively; and $=\frac{a}{l}$ is the length ratio. Then, calculate the two natural frequencies and corresponding modal vectors with depiction of mode shapes.

$2-$ Determine the value of a when the pendulum on the right is used as a vibration absorber for the harmonic torques $T_1(t)=m{l}^2{T}_{0}cost$ and $T_2(t)=0$ such that the steady$-$state amplitude of ${}_1(t)=0.$ What is the resulting steady$-$state amplitude of ${}_2(t)?$

Figure $3$