5.14 Response and damping
You have learned so far in this chapter that when a musician plays an instrument, they force the primary vibrator to vibrate. If the primary vibrator is driven at one of its resonance frequencies, the normal mode of vibration corresponding to that resonance frequency will be excited. Now, in practice it is also true to say that even if the primary vibrator is driven at a frequency close to the resonance frequency, the normal mode will still be excited, but just to a lesser degree. In other words, there is a range of frequencies over which the normal mode will be excited.
This is depicted graphically in Figure 22, which shows a frequency-response curve, also sometimes referred to as a resonance curve, for the first mode of vibration of an instrument. As the frequency at which the primary vibrator is driven is increased, the resulting standing wave increases in amplitude then decreases. The amplitude of the standing wave is largest when the primary vibrator is forced to vibrate at exactly the resonance frequency fR.
The sharpness of the frequency response is often expressed in terms of the range of frequencies over which the amplitude of vibration is greater than half the amplitude at resonance, i.e. the width of the resonance peak at half its height. For example, in Figure 22, the sharpness of the frequency response is equal to f″−f′.
If energy is supplied to the instrument in a short burst, the primary vibrator will be set vibrating but the vibrations will gradually die away because of damping. The damping is a measure of how rapidly the system loses energy through friction or by radiating the energy away as sound, etc. If the damping is light, the vibrations will continue for a long time. It turns out that lightly damped primary vibrators have a narrow frequency response. If the damping is heavy, on the other hand, the vibrations will die away quickly, and heavily damped primary vibrators have a broad frequency response. Figure 23 shows frequency-response curves for the first mode of vibration of a lightly damped primary vibrator (solid line) and of a heavily damped primary vibrator (dashed line).
The curves shown in Figure 22 and Figure 23 show the frequency response of only a single mode of vibration. If the frequency range is extended upwards to take in more modes of vibration, the frequency-response curve will contain several peaks – each peak being at the frequency of a mode of vibration (see Figure 24).