Damping Ratio Calculator — Spring-Mass-Damper System
Enter the mass, spring stiffness and viscous damping coefficient to calculate the damping ratio ζ, natural frequency, critical damping, and the free-vibration time response. Identify whether your system is underdamped, critically damped or overdamped.
kg
N/m
N·s/m
Underdamped
- 1
Natural frequency ωₙ = √(k ÷ m)
√(50 ÷ 2) = 5 rad/s - 2
Critical damping c_crit = 2 × √(m × k)
2 × √(2 × 50) = 20 N·s/m - 3
Damping ratio ζ = c ÷ c_crit
4 ÷ 20 = 0.2000ζ < 1 underdamped (oscillates); ζ = 1 critically damped; ζ > 1 overdamped (slow return).
How does this calculator work?
Damping ratio ζ = c / (2√(mk)); the natural frequency is ωₙ = √(k/m). ζ < 1 is underdamped (oscillates), ζ = 1 is critically damped (fastest non-oscillatory return), ζ > 1 is overdamped (slow return). Settling time ≈ 4/(ζ ωₙ). Enter m, k and c to get ζ and the full vibration response.
Formula
How this is calculated
A spring-mass-damper is the canonical model for any vibrating system — from a car suspension to a building in an earthquake. It is described by three parameters: mass m (kg), spring stiffness k (N/m) and viscous damping coefficient c (N·s/m). The natural frequency ωₙ = √(k/m) (rad/s) is the rate at which the undamped system oscillates freely. The critical damping coefficient c_crit = 2√(mk) is the minimum damping that prevents oscillation.
The damping ratio ζ = c / c_crit tells you which regime the system is in. When ζ < 1 (underdamped) the system oscillates while the amplitude decays exponentially; the oscillation frequency is ωd = ωₙ√(1 − ζ²). At ζ = 1 (critically damped) the system returns to equilibrium as fast as possible without oscillating. When ζ > 1 (overdamped) it returns slowly without oscillating, with two distinct exponential modes. The settling time to within 2% of equilibrium is approximately t_s ≈ 4 / (ζ ωₙ) for underdamped systems.
This model assumes a linear, time-invariant, single-degree-of-freedom system with ideal viscous damping. Real structures can exhibit non-linear stiffness, frequency-dependent damping (structural or Coulomb friction), and multiple vibration modes — none of which this simple model captures.
Frequently asked questions
Underdamped (ζ < 1): oscillates while amplitude decays — typical of vehicle suspensions or guitar strings. Critically damped (ζ = 1): returns to rest fastest without oscillating — used in door closers and some control systems. Overdamped (ζ > 1): returns slowly without oscillating — typical of very stiff, heavily damped systems.
Reinforced concrete buildings typically have ζ ≈ 0.05 (5%); steel frames ζ ≈ 0.02 (2%); timber ζ ≈ 0.05–0.10. Tuned mass dampers in skyscrapers target ζ ≈ 0.10–0.20 for the damped mode. Automotive suspensions are designed around ζ ≈ 0.25–0.35 for ride comfort.
The logarithmic decrement method measures successive peak amplitudes x₁ and x₂ from a free-vibration test. The damping ratio is ζ = δ / √(4π² + δ²), where the logarithmic decrement δ = ln(x₁/x₂). For small damping ζ ≈ δ / (2π). Frequency-domain methods using the half-power bandwidth of a resonance peak are also widely used.
TG we-Calculate Editorial Team. (2026). Damping Ratio Calculator — Spring-Mass-Damper System [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/damping-ratio-calculator
TG we-Calculate Editorial Team. "Damping Ratio Calculator — Spring-Mass-Damper System." TG we-Calculate. 2026. https://we-calculate.com/calculator/damping-ratio-calculator.
TG we-Calculate Editorial Team, "Damping Ratio Calculator — Spring-Mass-Damper System," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/damping-ratio-calculator
@misc{wecalculate_damping_ratio_calculator, title = {Damping Ratio Calculator — Spring-Mass-Damper System}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/damping-ratio-calculator}}, year = {2026}, note = {TG we-Calculate} }
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