For a single degree of freedom spring-mass-damper system under free vibration, the following information is known: \(m=2 \mathrm{~kg}, k=1 \times 10^{6} \mathrm{~N} / \mathrm{m}, c=\) \(500 \mathrm{~N}-\mathrm{s} / \mathrm{m}, x_{0}=4 \mathrm{~mm}\), and \(\dot{x}_{0}=0 \mathrm{~mm} / \mathrm{s}\).
Determine the corresponding expression for velocity \((\mathrm{mm} / \mathrm{s})\) if position is given in the form \(x(t)=A e^{-\zeta \omega_{n} t} \cos \left(\omega_{d} t+\varphi_{c}\right)\). Numerically evaluate all coefficients and constant terms in your final expression.