PROBLEM $3(25$ Points$)$:

An electric motor of mass $80$ kg operates on an elastic foundation. There is an unbalance at the rotor of amount $\backslash ($ m$\_\{0\}$ e$=0\mathrm{.}3\backslash$mathrm$\{$~kg$\}.\backslash$mathrm$\{$m$\}\backslash ).$

a$)$ When the system is displaced $15$ mm from the equilibrium position and released from rest, the amplitude of free vibration is observed to decrease to $0\mathrm{.}1$ mm after $4$ cycles. Find the damping ratio of the foundation.

b$)$ When the operation speed of the motor is $\backslash (40\backslash$mathrm$\{$rad$\}/\backslash$mathrm$\{$s$\}\backslash ),$ the amplitude of the steady$-$state vibration is observed to be $2$ mm $.$ Find the stiffness coefficient of the foundation.