Dynamics · Vibration
Damping Calculator
Size damping ratio, transient metrics, and decay time for a spring-mass-damper — watch the response curve track ζ as you tune mass, stiffness, and dissipation.
Final-to-initial amplitude (0.1 = 10% of initial) for decay time.
Live decay preview
Result
System parameters
ζ = c / (2√(km))
Enter mass and spring stiffness — the decay curve updates with your damping level.
SDOF transient metrics — damping ratio, time-domain response, and experimental δ.
Underdamped (ζ < 1)
Oscillatory decay — vehicle suspensions, instrument mounts, and most structural modes.
Critically damped (ζ = 1)
Fastest non-oscillatory return — door closers, dial indicators, and control systems tuned for no overshoot.
Overdamped (ζ > 1)
Slow, aperiodic recovery — sometimes intentional for comfort, slower than critical.
Damping ratio
ζ = c / (2√(km))
c = damping coefficient, k = stiffness, m = mass
Logarithmic decrement
δ = ln(x₁/x₂) = 2πζ/√(1−ζ²)
Successive peaks on underdamped free decay
Settling time (2%)
Ts = 4/(ζωn)
Time to stay within 2% of final value
Peak overshoot
OS% = 100·e^(−πζ/√(1−ζ²))
Step response — underdamped only
- Vibration control:Isolate sensitive equipment from floor or machinery excitation.
- Structural dynamics:Size damping for seismic or wind response and fatigue.
- Vehicle suspensions:Balance ride comfort (ζ) against handling and wheel hop.
- Precision instruments:Limit ring-down so measurements settle before sampling.
- Shock absorbers:Dissipate impact energy through velocity-proportional damping.
- Control systems:Target settling time, overshoot, and rise time in closed-loop design.
- Record free vibration after an impulse or initial displacement.
- Measure successive peak amplitudes x₁ and x₂ one cycle apart.
- Compute δ = ln(x₁/x₂).
- Obtain ζ = δ/√(4π² + δ²) and compare to analytical c from m and k.
Related calculator
For harmonic forcing and frequency-response amplification, use the Resonance Calculator.
Worked example: Damping ratio for m = 1 kg, k = 100 N/m, c = 5 N·s/m
- Critical damping cc = 2√(km) = 2√(100) = 20 N·s/m
- ζ = c / cc = 5 / 20
ζ = 0.25 (underdamped)
What is the damping ratio?
The damping ratio ζ = c/(2√(km)) compares the actual damping to critical damping. ζ < 1 is underdamped (oscillates), ζ = 1 is critically damped (fastest return without overshoot), and ζ > 1 is overdamped.
How are settling and rise time defined?
Settling time is when the response stays within 2% of its final value, Ts = 4/(ζωn). Rise time is the 0–100% time for an underdamped step response, Tr = (π − acos ζ)/ωd, and is only defined for underdamped systems.
What is the logarithmic decrement?
The logarithmic decrement δ = ln(x₁/x₂) = 2πζ/√(1−ζ²) is the natural log of the ratio of successive oscillation peaks. It is the easiest way to measure damping experimentally and back-calculate ζ.
What is peak overshoot?
Peak overshoot is how far an underdamped response exceeds its final value, PO = e^(−πζ/√(1−ζ²)) × 100%. For example, ζ = 0.5 gives about 16% overshoot; critically and over-damped systems have none.