= Underdamping
= Underdamped
{synonym}
For $0<\zeta<1$, define the damped <angular frequency> $\omega_{\mathrm d}=\omega_0\sqrt{1-\zeta^2}$. The <initial value problem> has solution
$$
x(t)=e^{-\zeta\omega_0t}\left[x_0\cos(\omega_{\mathrm d}t)+\frac{v_0+\zeta\omega_0x_0}{\omega_{\mathrm d}}\sin(\omega_{\mathrm d}t)\right].
$$
Every 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 $e^{-\zeta\omega_0t}$, 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.
Back to article page
