Assuming that the phase angle is zero, show that the response \(x(t)\) of an underdamped singledegree-of-freedom system reaches a maximum value when

\[\sin \omega_{d} t=\sqrt{1-\zeta^{2}}\]

and a minimum value when

\[\sin \omega_{d} t=-\sqrt{1-\zeta^{2}}\]

Also show that the equations of the curves passing through the maximum and minimum values of \(x(t)\) are given, respectively, by

\[x=\sqrt{1-\zeta^{2}} X e^{-\zeta \omega_{n} t}\]

and

\[x=-\sqrt{1-\zeta^{2}} X e^{-\zeta \omega_{n} t}\]