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Forced vibrations can also show another very important effect. With the swing you will find that if you push in time with the natural frequency of the swing then the oscillations build up rapidly. This last fact is an example of resonance.

All systems have their own natural frequency, and if you apply a driving force of the same frequency and in phase with the initial oscillations then resonance results, the amplitude of the oscillations gets larger and larger. Parts of a car may vibrate if you drive over a bumpy road at a speed where the vibrations transmitted to the body are at the resonant frequency of that apart. (Actually cars are designed not to do this by choosing parts with natural frequencies that are not likely to be produced by driving). Bass frequencies from stereo speakers can make a room resonate, particularly annoying if you live next door and your living room resonates due to your neighbour's music!

When I was a teenager on my way to school I used to go across a small suspension bridge and if I timed my footsteps at a particular rate I could make the bridge swing! A classic and rather unfortunate example of this was the Millennium Bridge built across the River Thames in London. However the most alarming exam- ple of the effect of resonance on a suspension bridge was when the Tacoma Narrows suspension bridge collapsed in a gale, due to resonance. The rebuilt bridge was redesigned to limit the amplitude of horizontal oscillation and is now safe!

Barton's pendulums (Figure 1) are a very good way of demonstrating forced vibrations and resonance in the laboratory. A heavy driving pendulum X is hung from a thread fixed to a cord between two retort stands. Hanging from the cord are five other pendulums, each with a light paper cone as the bob.

As X is swung all the other pendulums begin to vibrate, but pendulum C has the greatest amplitude as it has the same length and therefore the same natural frequency as X.

It is found that:
A and B are about half a period behind X
C is about a quarter of a period behind X
D and E are nearly in phase with X.

(See also: Barton's pendulums)

The effects of resonance can be demonstrated in several areas of Physics. You may find that the internal mirror in a car vibrates strongly when travelling at one particular speed over a rough road. If a cello is played near a piano, the sounding of the cello note may make a string of the same frequency on the piano resonate. The breaking of a wine glass by a loud note of a particular frequency is also due to this effect. The absorption of light by a gas depends on a resonance effect within the atom and the tuned LCR circuit resonates at one particular frequency. Microwave ovens operate at a frequency of 2.45 GHz (2.45x109 Hz) and this is NOT the resonant frequency of a water molecule. This frequency is much lower than the diatomic molecule resonant frequencies mentioned earlier. If 2.45 GHz were the resonant frequency of water molecules the microwaves would all be absorbed in the surface layer of a substance (liquid water or food) and so the interior of the food would not get cooked at all.

The variation of amplitude of the system with input frequency is shown in Figure 2.

No mechanical system will vibrate at only its resonant frequency. The actual dependence of the amplitude of the system on the frequency of the driving force varies over a range of frequencies near the resonant frequency and this variation known as a resonance curve. The amount of damping of a system affects the shape of the resonance curve. Two examples of this variation are shown in Figure 2. One is for light damping (the oscillations of a tuning fork) and the other for heavy damping (the oscillations of the air in a bottle when you blow across the top). The air can be made to oscillate over quite a large range of frequencies but the oscillations
are heavily damped and die away rapidly.

Heavily damped systems give broad resonance curves while lightly damped systems give sharply peaked resonance curves

Student investigation
This investigation enables you to study the effect of resonance in a simple mechanical system.

Set up the apparatus as shown in the diagram, and connect the vibrator to a signal generator set at a frequency of about 10 Hz. Observe the maximum amplitude of the top of the hacksaw blade.

Slowly increase the frequency and record the amplitude at intervals of 2 Hz.
Plot a graph of the amplitude against frequency. It should show resonant peaks and harmonics.

Student investigation
Study the effects of the transfer of energy between two coupled pendulums when one is displaced. Explain why there is a difference if both are displaced on opposite sides of the centre of oscillation

An alternative method for investigating resonance is to fix a springy metal strip top the top of the vibration generator and load each end with equal small masses of plasticene. The vibration generator is then turned on and a resonance frequency found.

This is similar to the case where a man walks across a field carrying a long plank on his shoulder. At each step the plank flexes a little (a) and the ends move up and down. He then starts to trot and as a result bounces up and down (b). At one particular speed resonance will occur between the motion of the man and the plank and the ends of the plank then oscillate with large amplitude.

A number of helicopter crashes were thought to be due to resonance. This occurred when hovering at a particular rotor speed. The vibrations of the helicopter were transferred to their eyeballs which resonated causing severe problems!
© Keith Gibbs 2013