Damped Oscillation Calculator
A mass on a spring with a resistive damping force loses energy as it oscillates, so its amplitude decays. The mass, the spring stiffness, and the damping coefficient together set the natural frequency, the damping ratio, and the slower frequency at which it actually oscillates. This calculator uses the standard mass-spring-damper relations, taking the mass, stiffness, and damping coefficient, and returns the natural angular frequency, the damping ratio, the damped frequency in hertz, and the period of the damped oscillation.
Damped oscillation formula
Natural angular freq omega_n = sqrt(k / m)
Damping ratio zeta = c / (2 * sqrt(m * k))
Damped angular freq omega_d = omega_n * sqrt(1 - zeta^2)
Damped frequency f = omega_d / (2 * pi); Period = 2 * pi / omega_d
This applies when the system is underdamped (damping ratio below 1). At or above a damping ratio of 1 the system does not oscillate, so the damped frequency and period are not defined.
Damping facts
- A damping ratio below 1 is underdamped and oscillates with decaying amplitude.
- A damping ratio of exactly 1 is critically damped and settles fastest without overshoot.
- A damping ratio above 1 is overdamped and returns slowly without oscillating.
- The damped frequency is always at or below the natural frequency.
- Car suspensions are tuned near critical damping.
Damped oscillation: frequently asked questions
What is a damped oscillation?
A damped oscillation is the motion of a mass on a spring with a resistive force, such as friction or a dashpot, that removes energy. The amplitude decays over time. The behaviour is set by the mass, the spring stiffness, and the damping coefficient, summarised by the damping ratio.
What is the damping ratio?
The damping ratio zeta is the damping coefficient divided by twice the square root of the mass times the stiffness. Below 1 the system is underdamped and oscillates with decaying amplitude; at exactly 1 it is critically damped; above 1 it is overdamped and returns without oscillating.
How is the damped frequency calculated?
The undamped natural angular frequency is the square root of stiffness over mass. The damped angular frequency is the natural frequency times the square root of (1 minus the damping ratio squared). For an underdamped system this is slightly lower than the natural frequency.
What does critical damping mean?
Critical damping (damping ratio equal to 1) is the smallest damping that returns the system to rest without overshooting and oscillating. Car suspensions and door closers are tuned near critical damping for a fast, smooth settling without bounce.
What units should I use?
Use SI units: mass in kilograms, stiffness in newtons per metre, and the damping coefficient in newton-seconds per metre. The natural and damped frequencies come out in radians per second and hertz, and the period in seconds.
Official sources
- NIST: Physical Measurement Laboratory: oscillation and damping.
- NASA: NASA Glenn Research Center: vibration and damping.
Reviewed by the CalculatorHub team, edited by James Graham, 16 June 2026. See our methodology.