Problem $2.$ An underdamped shock absorber is to be designed for a motorcycle of mass $m=200$ kg $.$ A stiffness of the spring was selected as $k=2400\frac{N}{m}$ and the damper has a damping coefficient of $c=600N*\frac{s}{m}.$ The shock absorber is tested in free$-$vibration by pushing down the mass $m($initial condition$)$ to $x(0)=0\mathrm{.}1m$ and $x^{}(0)=0\frac{m}{s}$ and letting it to freely vibrate until it stops:

a$)$ Plot its damped response $x(t)$ and its damped velocity $x^{}(t).$

b$)$ How long does it take the mass $m$ to settle down to approximately $x=0?$

c$)$ To diminish the number of oscillation should we increase or decrease the stiffness? Plot the response for a damped system with no oscillations. What was the value of $k?$

d$)$ For a value of $k=2400\frac{N}{m}$ what would be the value of damping constant $c$ to reduce the settling time to $3$ s $?$

Our python code $\text{'}$L$6\_$FreeResponseForgingHammer.py$\text{'}$ developed in the class will be very useful for this problem.