Resonance, explained.
Resonance is an enhanced response to forcing near a system's natural oscillation frequency, with the response shaped by damping.
Why it happens
For unit mass, stiffness and forcing, steady amplitude is 1/√((1−ω²)²+(cω)²), with phase atan2(cω,1−ω²). Compare the frequency-response curve with displacement A cos(ωt−phase) at the chosen frequency.
Resonance is an enhanced response to forcing near a system's natural oscillation frequency, with the response shaped by damping.
Read the result
The frequency-response view sweeps the drive; the oscillation view holds your chosen drive fixed. Amplitude and phase describe the settled response rather than startup motion.
A worked example
Driving a damped spring
Keep damping at 0.3 and compare amplitude below, near and above the unit natural frequency.
Increase damping while keeping the chosen drive near one. The amplification falls and the frequency-response peak changes shape.
Switch to the oscillation view to inspect the corresponding steady displacement and phase lag over time.
OPTIONAL DEEPER DETAILGo deeper: inside the model
Inside this model
For unit mass, stiffness and forcing, steady amplitude is 1/√((1−ω²)²+(cω)²), with phase atan2(cω,1−ω²). Compare the frequency-response curve with displacement A cos(ωt−phase) at the chosen frequency.
Where this idea is useful
A practical use
A machine can vibrate strongly when periodic forcing couples with a structural mode; added damping changes the response.
A common misconception
“Resonance means any repeated push causes unlimited motion.”
Frequency, damping and system structure matter. This damped linear steady-state response is finite throughout the allowed parameter range.
What this explanation leaves out
- This is a linear single-mode steady-state oscillator. It omits startup transients, nonlinearities and structural safety limits. The peak can shift with damping, and the display is not an engineering design calculation.
Why can the peak differ from the natural frequency?
For a damped forced oscillator the maximum displacement response can shift below the undamped natural frequency. Strong damping can remove a distinct resonance peak.
Which forcing frequency and dissipation mechanism would matter in a vibrating system you encounter?
Associated thinkers
Associations marked provisional are awaiting source review.
Further reading
Explore the original research or the teaching reference behind this experiment.