For the step response \(x_{s}(t)\) governed by the equation

\[ \ddot{x}_{s}+2 \zeta \omega_{n} \dot{x}_{s}+\omega_{n}^{2} x_{s}=u(t) \]

where \(u(t)\) is the unit step function, evaluate rise time \(t_{r}\), peak time \(t_{p}\), overshoot, and settling time where \(\delta=0.02\) for the following cases:

(a) \(\zeta=0.01, \omega_{n}=1 \mathrm{rad} / \mathrm{s}\),

(b) \(\zeta=0.01, \omega_{n}=\) \(2 \mathrm{rad} / \mathrm{s}\),

(c) \(\zeta=0.01, \omega_{n}=0.5 \mathrm{rad} / \mathrm{s}\),

(d) \(\zeta=0.1\), \(\omega_{n}=1 \mathrm{rad} / \mathrm{s}\),

(e) \(\zeta=0.1, \omega_{n}=2 \mathrm{rad} / \mathrm{s}\),

(f) \(\zeta=0.1\), \(\omega_{n}=0.5 \mathrm{rad} / \mathrm{s},(\mathrm{g}) \zeta=0.5, \omega_{n}=1 \mathrm{rad} / \mathrm{s}\), (h) \(\zeta=0.5, \omega_{n}=2 \mathrm{rad} / \mathrm{s}\), (i) \(\zeta=0.5, \omega_{n}=0.5 \mathrm{rad} / \mathrm{s}\).