Damped Oscillator Calculator (Mass–Spring–Damper)
Damping ratio, natural and damped frequency and the position at any moment of a mass on a spring with a damper, with the motion drawn over time.
Results
The dashed line marks the value you entered.
What this tool does
Springs, suspensions, guitar strings and swaying buildings all follow this one equation of mass, stiffness and damping. The damping ratio ζ decides the character: below 1 the system rings with a shrinking sine, at exactly 1 it returns as fast as possible without overshooting, and above 1 it creeps back slowly. Door closers aim close to 1, and many measuring instruments sit near 0.7, which settles fastest to within a few per cent. The motion shown starts from the given displacement, released from rest.
Formula
m x″ + c x′ + k x = 0 · ωₙ = √(k ÷ m) · ζ = c ÷ (2√(k m)) · ω_d = ωₙ √(1 − ζ²)
Variables
| Symbol | Meaning | Unit |
|---|---|---|
m | Mass | kg |
k | Spring constant | N/m |
c | Damping coefficient c | N·s/m |
x0 | Starting displacement (released from rest) | mm |
t | Time | s |
ZT | Damping ratio ζ | — |
RG | Behaviour | — |
FN | Natural frequency | Hz |
FD | Damped frequency | Hz |
X | Position at that time | mm |
AK | Amplitude left after each cycle | % |
CC | Critical damping coefficient | N·s/m |
Worked example
- Mass1 kg
- Spring constant100 N/m
- Damping coefficient c2 N·s/m
- Starting displacement (released from rest)50 mm
- Time0.5 s
- Damping ratio ζ0.1000
- BehaviourUnderdamped — oscillates as it settles
- Natural frequency1.59155 Hz
- Damped frequency1.58357 Hz
- Position at that time4.92753 mm
- Amplitude left after each cycle53.18 %
- Critical damping coefficient20 N·s/m
Limitations
- The formula assumes ideal conditions: no friction losses, no air resistance and no efficiency losses unless you enter them.
- Mixing units is the most common source of error. Convert every input to the units shown next to each field before calculating.