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Damping dissipates the <energy> of an <oscillation>, reducing its <amplitude>. A <linearly damped string> applies this mechanism to each <normal mode>.

Why can stronger <damping> make a return to <equilibrium> slower? A weak brake lets a <spring> exchange <potential energy> with <kinetic energy> many times; a very strong brake makes the final motion creep. <Critical damping> marks the boundary between these behaviours. The distinction depends on the <force> law: the classification below concerns a single <damped harmonic oscillator> with <viscous damping>, rather than every mechanism that removes <energy>.

Let $x(t)$ be <displacement> from <equilibrium>, $m>0$ the moving <mass>, $k>0$ the <spring constant>, and $b\geq0$ the constant coefficient of <viscous damping>. With no driving <force>, <Newton's second law> gives
$$
m\ddot x+b\dot x+kx=0,\qquad x(0)=x_0,\quad \dot x(0)=v_0.
$$
The two <initial conditions> specify <displacement> and <velocity>. Define the undamped <eigenfrequency> as an <angular frequency> $\omega_0=\sqrt{k/m}$ and the <damping ratio> as $\zeta=b/(2\sqrt{mk})$. The <ordinary differential equation> becomes $\ddot x+2\zeta\omega_0\dot x+\omega_0^2x=0$. A constant weight in a vertical <spring> merely shifts <equilibrium>; it is already absorbed in the definition of $x$.<University Physics Volume 1 by William Moebs and others/15.5 Damped Oscillations>[§15.5, equations 15.23–15.26 and the force discussion preceding 15.23]{ref}

For example, the ideal parameters $m=1\,\mathrm{kg}$ and $k=4\,\mathrm{N\,m^{-1}}$ give $\omega_0=2\,\mathrm{s^{-1}}$ and <critical damping> at $b=4\,\mathrm{kg\,s^{-1}}$. Values $b=1,4,8\,\mathrm{kg\,s^{-1}}$ give <damping ratios> $0.25,1,2$, respectively. These are model choices, not measured values. For release from rest with $x_0>0$, set $X=x/x_0$ and $\tau=\omega_0t$; then $X''+2\zeta X'+X=0$, where primes now denote <derivatives> with respect to $\tau$.

\Image[/damped-oscillator-initial-value-responses.png]
{title=Ideal viscous oscillator responses: release from rest in three damping regimes, and the effect of initial velocity at critical damping}

The left panel uses identical <initial conditions> $X(0)=1$, $X'(0)=0$. <Underdamping> produces repeated crossings of <equilibrium>; <critical damping> and <overdamping> give positive decreasing responses. The right panel keeps <critical damping> but changes initial <velocity>: $X'(0)=-2$ gives one crossing, although there is no <oscillation>. Both panels are exact model predictions.


 Damping (source code)