 Damping ratio

The damping ratio is the dimensionless ratio $b/b_{\mathrm{crit}}$. Substituting the exponential function $x=e^{rt}$ into the viscous damping model gives the characteristic equation and its characteristic roots:For $0<\zeta <1$, underdamping gives a pair of complex conjugate roots with negative real part. For $\zeta =1$, critical damping gives the repeated negative real root. For $\zeta >1$, overdamping gives two distinct negative real roots. At $\zeta =0$ the roots are purely imaginary and the harmonic oscillator is undamped; this boundary case has no energy loss. With $m,k>0$, negative $b$ would inject energy and lies outside the dissipative model.

The left panel locates $r/{\omega }_0$ in the complex plane; the imaginary part gives the damped angular frequency, while the negative real part gives the exponential decay rate. The right panel shows the positive rates $-\mathrm{Re}(r_{\pm })/{\omega }_0$. Beyond critical damping, one root moves toward zero, explaining slow late motion for generic initial conditions. The repeated-root solution also has a polynomial prefactor, so these rates alone do not describe its complete response.