Critically damped

Definition

A critically damped system returns to equilibrium as quickly as possible without oscillating. It occurs when the damping coefficient is equal to the critical damping value.

5 Must Know Facts For Your Next Test

1. Critical damping occurs when the damping coefficient $\gamma$ equals $2 \sqrt{km}$, where $k$ is the spring constant and $m$ is the mass.
2. In a critically damped system, the system does not oscillate but returns to equilibrium in the shortest possible time.
3. The characteristic equation of a critically damped system has a repeated real root.
4. Critical damping is often desired in systems like automotive shock absorbers and door closers to avoid oscillations.
5. The general solution for a critically damped harmonic oscillator can be written as $x(t) = (A + Bt)e^{-\gamma t / 2m}$.

Review Questions

• What is the condition required for critical damping in terms of the damping coefficient, spring constant, and mass?
• Why is critical damping preferred in certain practical applications like automotive shock absorbers?
• Write down the general solution for the displacement of a critically damped harmonic oscillator.

Related terms

Underdamped: An underdamped system oscillates with a gradually decreasing amplitude over time due to non-zero but insufficient damping.

Overdamped: An overdamped system returns to equilibrium without oscillating but more slowly than a critically damped system due to excessive damping.

Damping Ratio: $\zeta$, defined as $\frac{c}{2 \sqrt{km}}$, where $c$ is the damping coefficient. Critical damping occurs at $\zeta = 1$.