Damped Oscillation Calculator

Compute the displacement at a given time, the damping regime, and the Q factor from the natural angular frequency and damping coefficient.

Input

Enter the natural angular frequency ω0, damping coefficient γ, initial amplitude A, initial phase φ, and time t to get the displacement at that time and the damping behavior.

Natural angular frequency ω0

rad/s

Angular frequency without damping, equal to √(k/m).

Damping coefficient γ

1/s

Strength of damping per unit time, equal to c/(2m).

Initial amplitude A

units

Amplitude at t = 0, in arbitrary units.

Initial phase φ

rad

Phase in radians.

Time t

Time in seconds at which to evaluate the displacement.

Result

Displacement at t = 0.5 s

0.15742units

Damping regime

Underdamped

Damping ratio ζ

0.1

Damped angular frequency ω''

9.949874 rad/s

Q factor

Damped period

0.631484 s

Time constant τ

1 s

For underdamping x(t) = A e^(−γt) cos(ω''t + φ) with ω'' = √(ω0² − γ²). Q = ω0 / (2γ) and τ = 1/γ. Critical and overdamped cases assume zero initial velocity.

How it works

Solves the equation of motion x'' + 2γx' + ω0²x = 0 for a damped oscillator, using the natural angular frequency ω0, the damping coefficient γ, and the damping ratio ζ = γ / ω0.
For underdamping (ζ below 1) the system oscillates at the damped angular frequency ω' = √(ω0² − γ²), with displacement x(t) = A e^(−γt) cos(ω't + φ).
Critical damping (ζ = 1) does not oscillate and returns to rest as fast as possible, with displacement of the form x(t) = (A + Bt) e^(−γt). This tool assumes zero initial velocity.
Overdamping (ζ above 1) gives monotonic decay set by the two real roots −γ ± √(γ² − ω0²). This tool assumes zero initial velocity.
The quality factor is Q = ω0 / (2γ), and the time constant for the amplitude to fall to 1/e is τ = 1/γ.
Angles and phase are in radians, time and the time constant in seconds, and angular frequency in rad/s. The amplitude is in arbitrary units.

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