Find the response of the damped oscillator (ddot{x}+) (2 zeta omega_{n} dot{x}+omega_{n}^{2} x=F(t) / m), with (omega_{n}^{2}=4(mathrm{rad}

Question:

Find the response of the damped oscillator $\ddot{x}+$ $2 \zeta \omega_{n} \dot{x}+\omega_{n}^{2} x=F(t) / m$, with $\omega_{n}^{2}=4(\mathrm{rad} / \mathrm{s})^{2}$, to the forces per unit mass listed, and solve for the two underdamped cases of $\zeta=0.1$ and $\zeta=0.9$ :

(a) $F(t) / m=1-e^{-t}, t \geq 0$

(b) $F(t) / m=\cos 2 t, 0 \leq t \leq \pi$

(c) $F(t) / m=\cos 2 t+3,0 \leq t \leq \pi$

(d) $F(t) / m=\cos 2 t+\cos 3 t, 0 \leq t \leq \pi$.

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