Question $1(20$ points$)$: The displacement of a vibrating spring$-$mass$-$damper system is recorded and plotted in the following figure. The vertical coordinate is the displacement $($unit: cm $)$ and the horizontal coordinate is time $($unit: second$).$

Please find:

$(1)$ Damping ratio

$(2)$ Natural frequency

$(3)$ Damped natural frequency

$(4)$ Equation of motion

$(5)$ Initial displacement

$(6)$ Initial velocity From Question $1(4),$ you have found the equation of motion. Use Matlab to simulate the system's displacement and velocity responses up to $20$ secs$,$ when using the following initial conditions

Condition $1$: Initial displacement equals to the initial displacement from question $1(5)$ Initial velocity equals to the initial velocity from question $1(6)$

Condition $2$: Initial displacement equals to the initial displacement from question $1(5)$ Initial velocity doubles the initial velocity from question $1(6)$

You need to plot the displacement responses of conditions $1$ and $2$ in the same graph for comparison.