= Damping ratio
{title2=$\zeta=\frac{b}{2\sqrt{mk}}$}
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>:
$$
mr^2+br+k=0,\qquad r_\pm=-\zeta\omega_0\pm\omega_0\sqrt{\zeta^2-1}.
$$
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.
\Image[/damped-oscillator-characteristic-root-paths.png]
{title=Characteristic roots of an ideal viscous oscillator and their dimensionless decay rates as the damping ratio varies}
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 $-\operatorname{Re}(r_\pm)/\omega_0$. 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.
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