17.2.2 Resonance

Resonance

• In order to sustain oscillations in a simple harmonic system, a periodic force must be applied to replace the energy lost in damping
• This periodic force does work on the resistive force decreasing the oscillations
• These are known as forced oscillations, and are defined as:

Periodic forces which are applied in order to sustain oscillations

• For example, when a child is on a swing, they will be pushed at one end after each cycle in order to keep swinging and prevent air resistance from damping the oscillations
• These extra pushes are the forced oscillations, without them, the child will eventually come to a stop
• The frequency of forced oscillations is referred to as the driving frequency (f)
• All oscillating systems have a natural frequency (f0), this is defined as:

The frequency of an oscillation when the oscillating system is allowed to oscillate freely

• Oscillating systems can exhibit a property known as resonance
• When resonance is achieved, a maximum amplitude of oscillations can be observed
• Resonance is defined as:

When the driving frequency applied to an oscillating system is equal to its natural frequency, the amplitude of the resulting oscillations increases significantly

• For example, when a child is pushed on a swing:
• The swing plus the child has a fixed natural frequency
• A small push after each cycle increases the amplitude of the oscillations to swing the child higher
• If the driving frequency does not quite match the natural frequency, the amplitude will increase but not to the same extent at when resonance is achieved
• This is because at resonance, energy is transferred from the driver to the oscillating system most efficiently
• Therefore, at resonance, the system will be transferring the maximum kinetic energy possible
• A graph of driving frequency f against amplitude a of oscillations is called a resonance curve. It has the following key features:
• When f < f0, the amplitude of oscillations increases
• At the peak where f = f0, the amplitude is at its maximum. This is resonance
• When f > f0, the amplitude of oscillations starts to decrease
• Damping reduces the amplitude of resonance vibrations
• The height and shape of the resonance curve will therefore change slightly depending on the degree of damping
• Note: the natural frequency f0 will remain the same
• As the degree of damping is increased, the resonance graph is altered in the following ways:
• The amplitude of resonance vibrations decrease, meaning the peak of the curve lowers