(Solved): Did not completely solve the problem. Please finish the solution. Problem 2(2.0%) Two masses m1 ...

Problem $2(2.0\%)$ Two masses $m_1$ and $my$ are attached to a massless and unstretchable string with a tension $T$ as shown in Figure 2. Suppose that the natural frequencies of this system are $a_1$ and $a_2$. and the corresponding modal ratios are ${?}_1$ and ${?}_2$. Further, assume that $T$ remains unchanged when the masses $m$ t and $m$ are given the initial conditions, $y_1(0)=y_{16},{\hat{y}}_1(0)={\hat{y}}_{10},y_2(0)=y_{20}$ and ${\stackrel{?}{y}}_2(0)={\stackrel{?}{y}}_{20}$. Solve free-vibration responses of this compound pendulum. In other words, determine the parameters ${\mathrm{Y}}_1,{\mathrm{Y}}_2,?$, and ${?}_2$ of the following solution formulations using the given I.C.s. $\begin{array}{r}\left\{\begin{array}{l}{y}_1(t)\\ y_2(t)\end{array}\right\}=\left[\begin{array}{cc}{?}_1& {?}_2\\ 1& 1\end{array}\right]\left\{\begin{array}{l}{?}_1\sin \left({?}_{1}t+{?}_1\right)\\ {?}_2\sin \left({?}_{2}t+{?}_2\right)\end{array}\right\}=\left\{\begin{array}{l}0\\ 0\end{array}\right\}\\ {?}_1=\frac{{?}_1}{L}\\ {?}_2=\frac{U_3}{L}\end{array}$ Figure 2. A TDOF system consisting of un-stretchable strings and two masses.