Problem $2.$ For the translational mechanical system shown below, where $f_{vi}$ stand for viscous damping friction:

a$)$ Write the equations of motion

b$)$ Plot the free response if mass $M_2$ is displaced to the right with $x_2(0)=0\mathrm{.}5m$ assuming $f(t)=0,$ and that the viscous damping is negligible

c$)$ The maximum values $(x_1{)}_{\text{m}\text{a}\text{x}},(x_2{)}_{\text{m}\text{a}\text{x}}$ of the free response, under the same conditions as in b$)$

d$)$ Determine its natural frequencies ${}_{ni}$ where $i=1$dots$3$

Our Example $6\mathrm{.}18$ developed in the class, L$9,$ will be very useful for this problem.