Problem $1[20\frac{\%}{100}\%]$

The first is a forward moving vehicle and suspension system $($see Fig. $1).$ At time $t=0,$ the

wheel hits a bump defined by the surface roughness $u(t).$

$($a$)$ What is the transfer function for $z?$

$($b$)$ What is the characteristic polynomial for this suspension system?

$($c$)$ Manually factor, using the quadratic equation, the characteristic polynomial to compute the

eigenvalues.

$($d$)$ Use the 'root' command in MATLAB to get the eigenvalues.

$($e$)$ Use the $\text{'}$tf$\text{'}$ command in MATLAB to enter the transfer function; note, MATLAB will

express the answer in terms of $\text{'}s\text{'}$ instead of $\text{'}D\text{'}.$

$($f$)$ Use the 'damp' command to get the eigenvalues of this transfer function. What is the

damping ratio? What is the undamped natural frequency? What is the damped natural

frequency? What is the time constant of this system?

$($g$)$ After hitting a bump, how long will it take for the vibration to settle out within $1\%?$

$($h$)$ The original equations used to model this suspension consist of an equation for each

component, i$.$e$.$

spring force Fs $=10000(z-u)$

damper force $Fd=1000(z-u)$

force balance on mass: $100z^{\text{'}\text{'}}+Fd+Fs=0$

Convert the $2$nd and $3$rd equations to algebraic equations using the $\text{'}D\text{'}$ operator. Re$-$write

these algebraic equations in terms of $\text{'}s\text{'}$ instead of $\text{'}D\text{'}$; in symbolic math, the use of $\text{'}D\text{'}$

probably will not work for representing algebraic equations since the use of $\text{'}D\text{'}$ implies

taking the derivative. Then, use the symbolic math command 'solve' in MATLAB to solve

for $z_{{?}_{?}}(i)$ the answer should be the transfer function times u$.$l