Consider the mass$-$spring$-$damper system as shown in the accompanying figure,

which represents a single$-$degree$-$of$-$freedom model for a vehicle driving over a

road. The road surface is described by$,$

$y(t)=$Ysin$({}_{}t)$

$,$ where $Y=0\mathrm{.}01m.$ The car has a mass of $mkg$ while the suspension system

provides an equivalent stiffness $kkN{m}^{-1}$ and equivalent damping $c.$

$($a$)$ Find the differential equation governing the motion of the vehicle.

$($b$)$ Find the frequency of the base excitation ${}_b$ in terms of the velocity $v$ of the car.

Note: $v$ is in $km{h}^{-1}.$

$($c$)$ State the effect of velocity $v$ on the amplitude ratio $r.$

$($d$)$ Find the natural frequency ${}_n$ and the amplitude ratio $r$ if the car's velocity is $v.$

$($e$)$ The amplitude of the steady$-$state displacement of the vehicle is given by the

equation below. Find the equivalent damping $c$ such that the maximum

displacement is $x.$

$x=Y\sqrt[\phantom{2}]{\frac{1+(2r{)}^2}{(1-r^2{)}^2+(2r{)}^2}}$