Physics · Oscillations

Damping / Resonance

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Wrap-up

What you learned

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Key takeaway

Damping removes energy, so one free motion loses amplitude over time without moving the equilibrium line.
The response graph compares steady-state amplitude across driving frequencies, not seconds in one run.
Resonance is strongest near the natural frequency, and damping lowers and broadens the peak instead of simply deleting it.

Common misconception

A tall response peak is not an unlimited time spike; it is the finite steady-state response of a damped system driven near its natural frequency.

In a real damped system, energy is lost every cycle, so the amplitude stays finite even near resonance.

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Useful comparisonMechanics

Projectile Motion

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Review the foundationOscillations

Simple Harmonic Motion

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Next in Energy in SHMOscillations

Oscillation Energy

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Reference and support

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Equation snapshotEnergy in, energy out · Natural frequencyShow 3 equations

Keep the damping term, natural frequency, and response-amplitude formula visible while you compare the time trace with the resonance curve.

Energy in, energy out
$m \overset{x}{¨} + b \overset{x}{˙} + k x = F_{0} sin (ω_{d} t)$
The restoring term, damping term, and driving term act on the same oscillator. The motion you see comes from the balance between energy lost and energy added.
Natural frequency
$ω_{0} = \frac{k}{m}$
This is the frequency the system prefers when it is displaced and released without driving. Resonance is judged by how close the driver is to this value.
Response amplitude
$A (ω_{d}) \approx \frac{F _{0} / m}{( ω _{0}^{2} - ω _{d}^{2} ) ^{2} + ( 2 β ω _{d} ) ^{2}}$
This gives the steady-state amplitude for each driving frequency. Lower damping makes the peak taller and sharper, while higher damping makes it lower and broader.

Why it behaves this way

Explanation

Damping and resonance are two different ideas that shape how a real oscillator behaves. Damping removes energy from the motion, so left on its own the oscillation fades. A driver adds energy at a chosen frequency, so the motion can be sustained or amplified.

Resonance is the case where the driver transfers energy especially efficiently because its frequency is close to the oscillator's natural frequency. This module keeps that story clear by separating two views: a time-history view, where you watch one motion fade or settle, and a response view, where you compare steady-state amplitude across many driving frequencies.

Key ideas

01Larger damping removes energy faster, so the transient motion dies away more quickly.

02In driven motion, the steady-state response is largest when the driving frequency is close to the natural frequency.

03Damping does not remove resonance completely. It lowers the peak and makes the response less sharply concentrated around one frequency.

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Accessibility

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The simulation alternates between two views of the same damped, driven oscillator: a time-based motion view and a frequency-response view. In the motion view, the displacement changes through time and can decay or settle. In the response view, the horizontal axis becomes driving frequency, and the graph shows steady-state amplitude instead of a time history.

The controls make the physical meaning explicit, so it stays clear whether you are changing damping, the natural frequency, or the rate and strength of the external driver.

Graph summary

The transient graph shows one oscillator through time and makes damping visible as a shrinking envelope.

The response graph is not a later-time continuation of that motion. It shows how steady-state amplitude changes across driving frequencies and how damping changes the height and width of the resonance peak.

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Step 3 of 3

Oscillations and Energy

Damping / Resonance appears later in this track, so it is cleaner to start from the beginning first.

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