Critical damping is the repeated-root case of the viscous damping model: its two characteristic roots coincide at . The solution with arbitrary initial conditions isThe extra factor 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 , , and fixed, and compare all . Every response decreases positively toward equilibrium; the critically damped responseis smaller than every overdamped response at each . It therefore reaches and stays within any specified band , , 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 . Its initial conditions are zero and . For release from rest, at ; the causal Green function of a damped oscillator is positive at positive time for . Its convolution with gives : 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 , the factor in square brackets crosses zero precisely when , atFor example, gives , crossing equilibrium at 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.
New to topics? Read the docs here!