# Critical Damping Calculator

URL: https://speedycalculator.com/calculators/physics/critical-damping/
Last updated: 2026-10-30

> The critical damping calculator finds the critical damping coefficient and natural angular frequency of a mass–spring system, and the damping ratio for a given actual damping.

## Inputs

- Spring constant k (N/m, N/mm, kN/m, lbf/in, lbf/ft, kgf/mm)
- Mass m (kg, g, mg, t, lb, oz)
- Actual damping coefficient c (N·s/m)

## Results

- Critical damping coefficient c_c
- Natural angular frequency ω₀
- Natural frequency
- Damping ratio ζ
- System

## Formula

```
c_c = 2·√(k·m) = 2·m·ω₀,  ω₀ = √(k/m),  ζ = c/c_c
```

## Explanation

A critically damped system returns to rest as fast as possible without oscillating – the goal for car suspensions, door closers and measuring instruments. Less damping causes ringing; more makes the return sluggish.

### How to use the Critical Damping Calculator

- **Spring constant k**
- **Mass m**
- **Actual damping coefficient c (N·s/m)**

Results update as you type, every field has its own unit menu, and everything is calculated in your browser.

### How it works

Enter the spring constant and mass. Optionally enter the actual damping coefficient to see the damping ratio.

### Inputs explained

FieldWhat to enterSpring constant kUnits: N/m, N/mm, kN/m, lbf/in, lbf/ft, kgf/mmMass mUnits: kg, g, mg, t, lb, ozActual damping coefficient c (N·s/m)A number

### What each result means

- **Critical damping coefficient c_c** = 2×sqrt([Spring constant k]×[Mass m])
- **Natural angular frequency ω₀** = sqrt([Spring constant k]÷[Mass m])
- **Natural frequency** = sqrt([Spring constant k]÷[Mass m])÷(2×pi)
- **Damping ratio ζ** = [Actual damping coefficient c (N·s÷m)]÷[Critical damping coefficient c_c]

### Example values

How the main result changes when **Spring constant k** changes and everything else stays at the example values above:

Spring constant k (N/m)Critical damping coefficient c_c (N·s/m)10,0003,162.277715,0003,872.983320,0004,472.13625,0005,00030,0005,477.225640,0006,324.5553

## Worked example

Inputs — Spring constant k: 20,000 N/m; Mass m: 250 kg; Actual damping coefficient c (N·s/m): 2,000.

Results — **Critical damping coefficient c_c = 4,472.13595 N·s/m**; Natural angular frequency ω₀ = 8.94427 rad/s; Natural frequency = 1.42353 Hz; Damping ratio ζ = 0.4472; System = Underdamped (oscillates).

## Key terms

- **Damping coefficient**: Force per unit velocity from a damper.

## FAQ

**What does critically damped mean?**
The fastest return to equilibrium without overshoot.

**What damping do car suspensions use?**
Usually ζ ≈ 0.2–0.4 – slightly underdamped for comfort.

**What is the unit of the damping coefficient?**
N·s/m (kg/s).

**Is critical damping always best?**
Not always – a little overshoot can give a faster settling time.


## Sources

- [OpenStax – University Physics Volume 1](https://openstax.org/details/books/university-physics-volume-1)
- [HyperPhysics (Georgia State University)](http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html)