Virtual laboratory/Physics/Oscillations/Resonance

Resonance

When a system is driven by a periodic force, its amplitude depends on the driving frequency. It reaches a maximum at a frequency close to the natural frequency f₀ — this is resonance. Then even a small force can build the system up to large amplitudes. Change the frequency of the driving force.

Resonance
Physics · Vibrations · Resonance (FIZ.DRG)
Driving frequency ff (f₀ = 1.00 fixed)0.55 ·f₀
The amplitude of forced vibrations is largest when the driving frequency:
driving f
0.55·f₀
natural f₀
1.00
amplitude A
1.42
gain
1.4×
🎯 Challenge: reach resonance
streak: 0

I will give you an amplitude-gain threshold — tune the driving frequency f to reach it (the higher it is, the closer to f₀ you must get).

What is resonance?

When the driving force has a frequency close to the system’s natural frequency (f≈f0f \approx f_0), the vibration amplitude rises sharply. Outside this narrow range the vibrations stay small.

Amplitude of forced vibrations:

A(f)=F0(f02−f2)2+(γf)2A(f) = \dfrac{F_0}{\sqrt{(f_0^2 - f^2)^2 + (\gamma f)^2}}

The denominator is smallest at f≈f0f \approx f_0, which is exactly when the amplitude reaches its maximum:

Amax at f≈f0A_{max} \text{ at } f\approx f_0

Why do a bridge or a swing build up?

Rhythmic pushing in step with the natural vibrations (tuned forcing) adds energy on every cycle. A swing builds up from small pushes, and a bridge can sway dangerously under the even step of a crowd or gusts of wind.

How do you avoid resonance?

Increase the damping (γ — it flattens and lowers the peak) or change the natural frequency f0f_0 (different mass, stiffness), so the forcing no longer hits resonance.

Why you need this for the exam:

Problems on forced vibrations boil down to one question: how close f0f_0 the forcing lies. Trick: the amplitude maximum is always at f≈f0f\approx f_0, and damping only lowers and widens the peak — it does not shift it.

Remember: resonance is a sharp rise in amplitude when the driving f hits f₀. The peak of the resonance curve is always at f₀.

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Theory & formulas

Forced vibrations

A system driven by a periodic force vibrates at the driving frequency. The amplitude depends on how close that frequency is to the natural frequency f0f_0.

Resonance

The amplitude reaches a maximum when f≈f0f\approx f_0: f0=12πkmf_0 = \dfrac{1}{2\pi}\sqrt{\dfrac{k}{m}} Then even a small force builds the system up to large amplitudes (dangerous for bridges and buildings). Damping limits the growth of the amplitude.

Symbols: ff — driving frequency, f0f_0 — natural frequency, kk — stiffness (spring constant), mm — mass.

Frequently asked questions

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What is resonance?

A sharp rise in the amplitude of forced vibrations when the driving frequency approaches the system's natural frequency f₀.

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Why can a bridge build up?

When the driving frequency (e.g. footsteps, wind) hits the bridge's natural frequency, resonance amplifies the vibrations, threatening destruction.

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How to avoid resonance?

By changing the structure's natural frequency (stiffness, mass) or adding damping, which limits the growth of the amplitude.

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