= Solution
For <underdamping>, the general response is a free transient plus a particular driven response. Use the paper's $e^{i\omega t}$ convention. The steady displacement <complex amplitude> is
$$
x_0=\frac{F_0}{k-m\omega^2+ib\omega}.
$$
Thus
$$
\boxed{x(t)=e^{-\gamma t}[C\cos(\omega_dt)+D\sin(\omega_dt)]
+\operatorname{Re}\!\left[\frac{F_0e^{i\omega t}}{k-m\omega^2+ib\omega}\right].}
$$
Here $\gamma=b/(2m)$, $\omega_d=\sqrt{k/m-\gamma^2}$ and the real constants $C,D$ enforce the initial conditions. For a real positive $F_0$, the steady term is $|x_0|\cos(\omega t-\phi)$, where
$$
|x_0|=\frac{F_0}{\sqrt{(k-m\omega^2)^2+b^2\omega^2}},\qquad
\phi=\operatorname{atan2}(b\omega,k-m\omega^2).
$$
For light <damping>, $\omega_d\simeq\omega_0$ and the transient decays slowly compared with an oscillation. The two-argument phase convention makes the displacement lag evolve from approximately zero below resonance through $\pi/2$ near $\omega_0$ to approximately $\pi$ far above it.
Back to article page
