Forced Oscillation Amplitude and Resonance Calculator

Compute the steady-state amplitude, phase lag, resonance angular frequency, and Q factor of a driven damped oscillator, with a resonance curve.

Input

Calculates the steady-state response when a periodic driving force acts on a damped oscillator. Enter the natural angular frequency, damping, driving angular frequency, and force amplitude.

Natural angular frequency ω0

rad/s

Angular frequency of the undamped oscillator in rad/s

Damping coefficient γ

rad/s

Strength of damping in rad/s. Zero means no damping

Driving angular frequency ω

rad/s

Angular frequency of the external driving force in rad/s

Force amplitude F0/m

m/s²

Driving force amplitude per unit mass in m/s squared

Result

Steady-state amplitude A

0.237826m

Resonance angular frequency

9.974969 rad/s

Phase lag

25.346176 deg

Q factor

ratio ω/ω0

0.9

At the entered driving angular frequency the steady-state amplitude is 0.237826 m.

Computes the amplitude A and phase lag φ of the steady-state solution x(t) = A cos(ωt − φ).

How it works

Models the equation of motion x'' + 2γx' + ω0²x = (F0/m)cos(ωt) with steady-state solution x(t) = A cos(ωt − φ).
The steady-state amplitude is A = (F0/m) / √((ω0² − ω²)² + (2γω)²) in meters.
The phase lag is φ = atan2(2γω, ω0² − ω²) in radians, the lag of the displacement behind the driving force.
The resonance angular frequency, when ω0² > 2γ², is ω_res = √(ω0² − 2γ²) rad/s, where the amplitude is maximal. With no such ω_res there is no resonance peak.
The Q factor (sharpness of resonance) is Q = ω0 / (2γ); smaller damping gives larger Q.
ω0, γ, and ω are all angular frequencies in rad/s. They relate to ordinary frequency f in Hz by ω = 2πf.
The driving force is entered as the amplitude per unit mass F0/m in m/s².
The resonance curve plots ω/ω0 on the horizontal axis and steady amplitude on the vertical axis; the green dot marks the response at the entered driving angular frequency.

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