Problem $2(30$ points$)$

The response of a single degree of freedom $($mass$-$spring$-$damper system$)$ is shown in Fig. $1\mathrm{.}2.$ The time history of the free response of the system is shown in Fig. $1\mathrm{.}3.$

Figure $1\mathrm{.}2$: A single degree of freedom mass$-$spring$-$damper system.

a$)$ Is this system underdamped, critically damped or overdamped? Why?

b$)$ Estimate the damped natural frequency ${}_d$ from the figure.

c$)$ Find the time constant, $,$ or the quality factor, $Q,$ or the damping ratio, $,$ of the system. Assume ${}_n$~~${}_d$

Figure $1\mathrm{.}3$: Free response of a mass$-$spring$-$damper system.