Structural devices

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# 6.5 Q-value

The rate at which the mass–spring system loses energy to its surroundings is referred to as the Q-value for the oscillator. The Q-value is defined as:

ΔE/E is the fractional energy loss per cycle of the oscillation. This can also be expressed in terms of angular frequency as:

or frequency as:

where Δω and Δf are the width of the peak at its halfway point. This energy loss is referred to as damping and is due to the frictional losses I have already mentioned. A large value of Q equates to a very small energy loss. Q stands for quality; in systems where oscillations are desired, such as the design of a bell, the larger the value of Q the longer the bell will ring.

Figure 30 shows two typical plots of amplitude vs angular frequency (A0 vs ω) for two driven resonators, one with a high Q and the other with a lower Q-value.

Figure 30 Plots of amplitude vs frequency for two resonators. Plot (a) is for a high Q-value, plot (b) is for a lower Q

You can see that the amplitude of plot (a) is far greater than that of (b), but also that it falls away much more rapidly either side of the peak. This sharpness is characteristic of an undamped oscillator with a high Q and it exactly mirrors the amplitude vs frequency behaviour shown in Figure 28.

## SAQ 15

Should I look for high or low Q-values in the following cases? Why?

• (a) A clock pendulum;

• (b) car suspension;

• (c) a bell.

• (a) We want the clock to run for as long as possible and at an extremely precise frequency; so we look for the highest possible Q-value.

• (b) Car suspension is damped by shock absorbers; these reduce resonance effects which might otherwise prove disastrous. A very low Q-value is appropriate here.

• (c) As I have already mentioned we need a high Q-value in order to ensure that the bell will ring for some time. However, a high Q-value would lead to a quiet bell and so a compromise has to be sought between volume and longevity.

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