What this calculator does
Critical damping is the exact amount of damping at which a disturbed mass-spring-damper system returns to rest as quickly as possible without overshooting or oscillating past its resting point. Less damping than this and the system swings back and forth before settling, more and it creeps back to rest more slowly than it needs to.
The critical damping coefficient depends only on the mass and the spring constant of the system, c_critical = 2√(m×k). Comparing an actual damping coefficient against this value, as a ratio called the damping ratio (ζ), tells you which regime a real system sits in: underdamped (ζ less than 1, it oscillates), critically damped (ζ equal to 1) or overdamped (ζ greater than 1, it is sluggish but does not overshoot).
The formula
The critical damping coefficient comes from c_critical = 2√(m×k), where m is the mass and k is the spring constant. If an actual damping coefficient is also entered, the damping ratio ζ = c ÷ c_critical shows how that real system compares: ζ below 1 means the system will oscillate before settling, ζ above 1 means it settles without oscillating but more slowly than the fastest possible non-oscillating return.
| Term | Meaning |
|---|---|
| Mass (m) | The moving mass in the system, such as the suspended weight on a spring. |
| Spring constant (k) | The stiffness of the spring, in newtons per metre. |
| Critical damping coefficient | The exact damping needed for the fastest return to rest with no oscillation, c_critical = 2√(m×k). |
| Damping ratio (ζ) | The actual damping coefficient divided by the critical damping coefficient, used to classify the system. |
The inputs explained
| Field | What to enter |
|---|---|
| Mass (kg) | The mass attached to the spring. |
| Spring constant (N/m) | The spring constant, the stiffness of the spring. |
| Actual damping coefficient (0 to skip) (N·s/m) | An actual damping coefficient to compare against the critical value, if known. Leave at zero to skip this comparison. |
When to use it
Designing a suspension or shock absorber
A suspension is deliberately built to sit close to critical damping, or slightly underdamped, so that a wheel returns to its resting position quickly after a bump without bouncing repeatedly.
Tuning a door closer or hinge damper
A door closer that is underdamped will swing shut and bounce back open; one that is heavily overdamped closes so slowly it barely latches. Critical damping is the target for closing quickly without a rebound.
Checking a measurement or control instrument
Needle gauges and some control systems are damped close to critical so a reading settles on the correct value quickly rather than oscillating around it or drifting into place too slowly.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How the critical damping coefficient changes with mass and spring stiffness
The critical damping coefficient for a fixed spring constant across a range of masses.
| Mass | Critical damping coefficient |
|---|---|
| 0.1 kg | 2.828 N·s/m |
| 0.25 kg | 4.472 N·s/m |
| 0.5 kg | 6.325 N·s/m |
| 1 kg | 8.944 N·s/m |
| 2 kg | 12.649 N·s/m |
| 5 kg | 20.000 N·s/m |
Questions
What is the difference between underdamped and overdamped?
An underdamped system (damping ratio below 1) oscillates one or more times before settling. An overdamped system (damping ratio above 1) never oscillates but takes longer than necessary to settle. Critically damped (ratio of exactly 1) is the fastest possible return with no oscillation.
Does critical damping depend on the amplitude of the disturbance?
No. The critical damping coefficient depends only on the mass and spring constant of the system, not on how far it was displaced or how hard it was disturbed.
How does this relate to the mass-spring oscillator period?
The mass-spring oscillator period calculator assumes no damping at all, giving the period of an idealised, undamped oscillation. This calculator instead looks at how much damping is needed to remove that oscillation entirely.
Can a real system ever be exactly critically damped?
In practice, exact critical damping is difficult to hold precisely because component tolerances and temperature can shift the effective damping slightly. Most designs that want a fast, non-oscillating response aim slightly on the overdamped side rather than risk drifting into underdamped oscillation.
For the undamped oscillation period of the same kind of system, see the mass-spring oscillator period calculator.