Problem $3$

A $2$DOF spring$-$mass system excited by two forces $f_1(t)$ and $f_2(t)$ and has the following equation of motion

$\left[\begin{array}{cc}3& 0\mathrm{.}1\\ 0\mathrm{.}1& 2\end{array}\right]\left[\begin{array}{c}{x}_1^{}\\ x_2^{}\end{array}\right]+\left[\begin{array}{cc}40& -10\\ -10& 10\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]=\left[\begin{array}{cc}0\mathrm{.}2& 0\mathrm{.}4\\ 0\mathrm{.}5& 0\mathrm{.}3\end{array}\right]\left[\begin{array}{c}{f}_1(t)\\ f_2(t)\end{array}\right]$

a$.$ Show that the eigenvalues and eigenvectors for this system are:

$=3\mathrm{.}2400,15\mathrm{.}4578,=\left[\begin{array}{cc}0\mathrm{.}3410& 1\mathrm{.}0\\ 1\mathrm{.}0& -0\mathrm{.}5520\end{array}\right]$

b$.$ Calculate the damping matrix in original coordinates such that the damping ratios in the first and second modes for this system are $1\%$ and $5\%,$ respectively.

c$.$ Determine the equations of motion in modal coordinates.

d$.$ Using the equations of motion in modal coordinates determined in part $($c$),$ check the natural frequencies of the system are consistent by the eigenvalues given in part $($a$).$

e$.$ Determine the response $x_2(t)$ to the applied force $f_2(t)=6U(t),$ where $U(t)$ is the unit step function. Assume zero initial conditions and use the damping matrix found in part $($b$).$

f$.$ Determine the steady state response $x_1^{}(t)$ to the force $f_2(t)=sin6t.$

g$.$ Determine the steady state response $x_2(t)$ to the simultaneously applied forces $f_1(t)=sin4t$ and $f_2(t)=6U(t),$ where $U(t)$ is the unit step function. Use the damping matrix found in part $($b$).$