What is the damping ratio?

The damping ratio ζ (zeta) = c / (2√(km)) compares actual damping to the critical damping value. ζ 1 is overdamped (slow return, no oscillation).

What is the natural frequency versus the damped frequency?

The undamped natural frequency ωₙ = √(k/m) is the frequency at which the system would oscillate with no damping. The damped natural frequency ωd = ωₙ√(1−ζ²) is the actual oscillation frequency when damping is present. For underdamped systems ωd < ωₙ; for ζ ≥ 1 there is no oscillation.

What is the time constant τ?

The time constant τ = 2m/c = 1/(ζωₙ) describes how quickly the amplitude envelope decays. After one time constant the amplitude falls to about 36.8% of its initial value. Smaller τ means faster energy dissipation.

How is settling time calculated?

Settling time (2% criterion) is approximately 4τ = 4/(ζωₙ). It estimates how long the system takes to stay within 2% of its final value. For overdamped systems the dominant pole determines the effective time constant.

What is critical damping?

Critical damping (ζ = 1) occurs when c = 2√(km). It produces the fastest return to equilibrium without oscillation. Car suspensions and door closers are often tuned near critical damping for smooth, responsive behavior.

Can I use this calculator for electrical RLC circuits?

Yes — the mathematics is identical. Map inductance L→m, capacitance C→1/k, and resistance R→c. The same formulas give resonant frequency, damping ratio, and time constant for series RLC circuits.

Damped Oscillation Calculator

Enter spring constant, mass, and damping coefficient to get damping ratio, damped frequency, time constant, and settling time instantly.

System Parameters

Spring Constant (k)

N/m

Mass (m)

Damping Coefficient (c)

N·s/m

Example presets:

Results

Enter values and click Calculate

Copied!

Summary

A damped oscillator is any mass-spring system where friction or viscosity dissipates energy over time. The behavior depends on the damping ratio ζ (zeta): underdamped (ζ < 1) systems oscillate with decaying amplitude, critically damped (ζ = 1) return to rest as fast as possible without oscillating, and overdamped (ζ > 1) return slowly without oscillating. This calculator derives all key parameters from the physical quantities — spring constant k, mass m, and damping coefficient c.

How it works

Enter the spring constant k in N/m (stiffness of the spring).
Enter the mass m in kilograms attached to the spring.
Enter the damping coefficient c in N·s/m (viscous resistance).
Click Calculate to compute all damped oscillation parameters.
The tool applies standard second-order system formulas: ωₙ = √(k/m), ζ = c/(2√(km)), ωd = ωₙ√(1−ζ²).
The damping type (underdamped, critical, overdamped) is identified automatically.

Use cases

Analyze mechanical vibrations in automotive suspension systems.
Design damping for precision instruments and sensor mounts.
Verify settling time for robotic arms and servo-controlled platforms.
Understand oscillatory behavior in structural engineering applications.
Tune shock absorbers for desired ride comfort or performance.
Study second-order system dynamics in control engineering courses.
Calculate decay rates for resonance suppression in bridges and buildings.
Check if a spring-mass system is overdamped or underdamped before prototyping.

Frequently Asked Questions

Last updated: 2026-06-13 · Reviewed by Nham Vu