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
| Field | What to enter |
|---|---|
| Spring constant k | Units: N/m, N/mm, kN/m, lbf/in, lbf/ft, kgf/mm |
| Mass m | Units: kg, g, mg, t, lb, oz |
| Actual 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) |
|---|---|
| 10,000 | 3,162.2777 |
| 15,000 | 3,872.9833 |
| 20,000 | 4,472.136 |
| 25,000 | 5,000 |
| 30,000 | 5,477.2256 |
| 40,000 | 6,324.5553 |
Formula
c_c = 2·√(k·m) = 2·m·ω₀, ω₀ = √(k/m), ζ = c/c_c
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.
Frequently asked questions
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.