Consider the following system described below $($parameters and inputs are in SI units$)$:

$I_m=0\mathrm{.}4,R$:$=0\mathrm{.}18$

$I_p=0\mathrm{.}2,k=30$

$m_r$:$=6,b$:$=20$

$T(t)=2,$

Note: b is the damping coefficient between the mass and the ground $($it is shown as c in the diagram, but we will use b$).$ Complete the following:

a$)$ Find an equation of motion for the simplified equivalent mass system. Assume that the shaft between the inertias is rigid.

b$)$ Find the equations of motion in state variable form for this system. You should end up with $2$ state variable equations.

c$)$ Numerically solve for the state variables and plot each state variable versus time. Assume, that the mass is initially displaced a distance of $0\mathrm{.}1$ in the positive direction and the mass velocity is initially zero. Also plot the spring force and damping force.

d$)$ This system is required to have a maximum displacement overshoot that is less than $15\%$ of the steady state value to avoid the rack impacting another portion of the machine. Determine whether this system meets that requirement. If not, what parameters do you recommend changing to cause this system to meet the overshoot design requirement?

Verify that your results are reasonable by comparing the initial and steady state values with your expectations. Use python to solve this problem.