Damped Oscillation Calculator
Work out damped oscillation instantly with clear inputs, formula shown and shareable results.
About Damped Oscillation Calculator
Light damping leaves the oscillation sinusoidal but shrinks its envelope exponentially, A(t) = A0 exp(-bt/2m). With b = 0.8 kg/s on a 2 kg mass the decay constant is 0.2 per second, so amplitude halves every 3.47 s and after 3 s only 55% remains. Energy falls twice as fast as amplitude because it goes with the square.
Formulas
Amplitude envelope
A(t) = A0 x exp(-b t / 2m)
Amplitude half-life
t(half) = ln2 x 2m / b
Worked example
With the default values below, Damped Oscillation Calculator returns Amplitude At That Time of 5.48812 × 10⁻². Change any input above and every figure here is recalculated in your browser.
Inputs used
- Initial amplitudeminimum 0 m
- 0.1 m
- Damping coefficient bminimum 0 kg/s
- 0.8 kg/s
- Oscillating massminimum 0 kg
- 2 kg
- Time elapsedminimum 0 s
- 3 s
Results
- Amplitude At That Time
- 5.48812 × 10⁻² m
- Decay Constant
- 0.2000 1/s
- Time For The Amplitude To Halve
- 3.4657 s
- Amplitude Remaining
- 54.88%
Step by step
- 1Values used
- 2Amplitude envelope
- 3Amplitude half-life
- 4Amplitude At That Time= 0.05488 m
- 5Decay Constant= 0.2000 1/s
- 6Time For The Amplitude To Halve= 3.4657 s
- 7Amplitude Remaining= 54.88
Assumptions and limits
- The result is derived only from the 4 inputs listed above. Anything not among them is not part of the calculation.
- In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
- Figures are rounded for display to 5 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.
How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.
Frequently Asked Questions
What is critical damping?
The value of b that just prevents oscillation, b = 2 sqrt(km). Below it the system rings; above it the return is sluggish.
Why does energy decay faster than amplitude?
Because energy is proportional to amplitude squared, so its decay constant is twice as large.