For a single degree of freedom spring-mass-damper system with \(m=1 \mathrm{~kg}\), \(k=4 \times 10^{4} \mathrm{~N} / \mathrm{m}\), and \(c=10 \mathrm{~N}-\mathrm{s} / \mathrm{m}\), complete the following for the case of free vibration.
(a) Calculate the natural frequency (in rad/s), damping ratio, and damped natural frequency (in \(\mathrm{rad} / \mathrm{s}\) ).
(b) Given an initial displacement of \(5 \mathrm{~mm}\) and zero initial velocity, write the expression for the underdamped, free vibration in the form \(x(t)=e^{-\zeta \omega_{n} t}\left(A \cos \left(\omega_{d} t\right)+B \sin \left(\omega_{d} t\right)\right) \mathrm{mm}\).
(c) Plot the first ten cycles of motion.
(d) Calculate the viscous damping value, \(c\) (in \(\mathrm{N}-\mathrm{s} / \mathrm{m}\) ), to give the critically damped case for this system.