Underdamping

For $0<\zeta <1$, define the damped angular frequency ${\omega }_{\mathrm{d}}={\omega }_0\sqrt{1-{\zeta }^2}$. The initial value problem has solutionEvery nonzero underdamped response crosses equilibrium repeatedly. Its oscillation period is $T_{\mathrm{d}}=2\pi /{\omega }_{\mathrm{d}}$, not $2\pi /{\omega }_0$. The amplitude envelope decays as $e^{-\zeta {\omega }_{0}t}$, but the actual displacement also changes sign. The envelope coefficient need not equal $|x_0|$: initial velocity fixes the oscillation phase as well as that coefficient.