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# Natural frequency and resonance

## Resonance

All objects have a natural frequency, which is the frequency at which an object would tend to oscillate if no resistive forces were applied on it (free oscillation).

Resonance is a special case of forced oscillation where the frequency of the driver force $(f)$ equals the natural frequency $(f_{0})$ of the object.

• If the system is not damped, this causes the amplitude of oscillation to increase indefinitely over time.

• If the system is damped, the amplitude of the forced oscillation under the conditions of resonance will reach a maximum value.

## Characteristics of resonance curves and applications of resonance

Oscillating systems under conditions of resonance are often modelled by graphs of amplitude $x_{0}$ vs driver frequency $f$. These are known as frequency response curves.

Increasing the amount of damping on a oscillating system under conditions of resonance decreases the amplitude of the system over all frequencies.

The frequency response curve for a more heavily damped system would display a lower maximum (or peak) amplitude than the frequency response curve of a less heavily damped system.

The driver frequency at which maximum amplitude is achieved decreases with increased damping. The peak of the frequency response curve also becomes less sharp with increased damping.

Resonance is useful in some situations such as in microwave ovens. Microwaves are used to cause water molecules to resonate and heat up food.

But resonance can also cause disaster. Uncontrolled, unwanted resonance in mechanical devices can cause them to break apart due to the stresses caused by excessive vibrations.

The Broughton suspension bridge collapsed in 1831 due to resonance induced by military troops marching in step on the bridge

Graph of amplitude vs time for a resonating oscillator (above) and graph of maximum amplitude vs driver frequency of a forced oscillator (below)