= Critical damping
{title2=$\zeta=1,\quad b_{\mathrm{crit}}=2\sqrt{mk}$}
= Critically damped
{synonym}
<Critical damping> is the repeated-root case of the <viscous damping> model: its two <characteristic roots> coincide at $r=-\omega_0$. The solution with arbitrary <initial conditions> is
$$
x(t)=\bigl[x_0+(v_0+\omega_0x_0)t\bigr]e^{-\omega_0t}.
$$
The extra factor $t$ is essential: one <exponential function> alone cannot fit arbitrary initial <displacement> and <velocity> when the <characteristic equation> has a double <polynomial root>. Substitution, or taking the limit of the <underdamped> formula as its <angular frequency> tends to zero, gives the displayed expression.
The precise fastest-return statement is this: hold $m$, $k$, $x_0>0$ and $v_0=0$ fixed, and compare all $\zeta\geq1$. Every response decreases positively toward <equilibrium>; the <critically damped> response
$$
x_{\mathrm c}(t)=x_0(1+\omega_0t)e^{-\omega_0t}
$$
is smaller than every <overdamped> response at each $t>0$. It therefore reaches and stays within any specified band $|x|\leq\epsilon x_0$, $0<\epsilon<1$, sooner than those responses. Exact arrival at rest takes infinite <time>. This is a comparison among nonoscillatory regimes for these particular <initial conditions>, not a universal minimum of every possible settling-time criterion.
One short comparison argument uses $q=\partial X/\partial\zeta$. Its <initial conditions> are zero and $q''+2\zeta q'+q=-2X'$. For release from rest, $X'<0$ at $\tau>0$; the <causal Green function of a damped oscillator> is positive at positive <time> for $\zeta\geq1$. Its <convolution> with $-2X'$ gives $q>0$: increasing <damping> beyond the critical value increases the remaining <displacement>.
There is no repeated <oscillation> at <critical damping>, but there can be overshoot once. For $x_0>0$, the factor in square brackets crosses zero precisely when $v_0<-\omega_0x_0$, at
$$
t_*=-\frac{x_0}{v_0+\omega_0x_0}>0.
$$
For example, $v_0=-2\omega_0x_0$ gives $x=x_0(1-\omega_0t)e^{-\omega_0t}$, crossing <equilibrium> at $t=1/\omega_0$ as shown above. An <overdamped> response can also cross once for suitable <initial conditions>. Conversely, specially chosen <initial conditions> can remove its slow <exponential decay> entirely. Thus the generic slow-root argument must not be applied to a response whose slow-root coefficient is zero.
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