Problem $(20\%)$

A simplified version of the front suspension system of a motor cycle is shown in the figure. The vertical displacement $u$ of point $P$ is the input to the system. The displacements $x$ and $y$ are measured from their respective equilibrium positions before the input is given to the system. Assume $m_1,b_1,k_1$ are the front tire and shock absorber assembly and $m_2,b_2,k_2$ represent half of the body of the vehicle.

The equations of motion are given by

$m_1{x}^{}=-k_1(x-u)-b_1(x^{}-u)$

$m_2{y}^{}=-k_2(y-x)-b_2(y^{}-x^{})$

$($a$)$ Find the transfer function of the system if the output is displacement of $m_2(1.e,y)$ and the input is $u.$

$($b$)$ Let the input to the system is a unit step. We need the system to have a percentage overshoot $(\%$OS$)$ less than $45\%$ and settling time less than $10$ secs$.$ Show the region In the complex s$-$plane plane that shows the pole locations corresponding to the time domain specifications.

$($c$)$ Let $m_1=10kg,m_2=100kg,b_2=100N\frac{s}{m},k_2=200\frac{N}{m}.$ Select $b_1$ and $k_1$ so that dominant poles of the system is within and close to boundary of the feasible reglon evaluated from part $($b$).$ Show the dominant poles $($their numeric values$).$

$($d$)$ With the chosen $b_1$ and $k_1,$ plot the step response and show the $\%$OS and settling time $($their numeric values$).$

You can use MATLAB to solve parts $($c$),($d$),$ and $($e$).$ Submit your code with the solutions. Hint: some MATLAB functions that will be helpful:

tf: to define the transfer function

step: plotting the step response

stepinfo: in addition to plotting the step response, provides some time domain specifications.