A car suspension system can be modelled as shown in Figure Q$.2$ where $m$ is the mass of the car, whilst $k$ and c are the total effective stiffness and damping of the suspension system. The suspension system is designed to isolate the car body and passengers from unevenness in the road; The unevenness of a road surface can be modelled as a sine wave of amplitude $y_0$ and with peaks placed a distance s apart.

a$.$ If the car is travelling at a velocity $v$ show that the applied input can take the form: $[2]$

$y(t)=y_{o}sin(\frac{2v}{s})t$

b$.$ In static testing a car of mass $1000kg$ is given an initial vertical displacement and the subsequent free vibration allowed to develop. It is noted that after one complete oscillation the amplitude has reduced to $20\%$ of its initial value. Calculate the damping ratio of the system. $[4]$

c$.$ The suspension system of the car is to be designed to transmit only $35\%$ of any displacement of the wheel to the car body when travelling at $70k\frac{m}{h}r$ and with the road surface peaks $5m$ apart. Calculate the overall stiffness $k$ required. $[8]$

d$.$ If the peaks have an amplitude of $4cm$ and are $7m$ apart calculate the displacement of the car body when travelling at a velocity of $90k\frac{m}{h}r.[6]$