Solve for the damped response beginning with the equation,

\(x(t)=\frac{1}{m \omega_{d}} \int_{0}^{t} \sin \omega(t-\tau) \exp \left(-\zeta \omega_{n} \tau\right) \sin \omega_{d} \tau d \tau\).

In the same graph, plot the results for the following cases:

\[ \omega=1.0,1.5,2.0,2.5,3.0 \mathrm{rad} / \mathrm{s} \]

where \(\zeta=0.1, \omega_{n}=2 \mathrm{rad} / \mathrm{s}\), and \(m=1 \mathrm{~kg}\). Discuss.