We have discussed several mechanical systems that have normal modes of oscillation every particle of the system oscillates with simple harmonic motion with frequency as the frequency of this mode. The systems we discussed in Sections have an infinite series of rrormal modes, but the basic concept is closely related to the simple harmonic oscillator, discussed in Chapter 13, which has only a single normal mode (that is, only one frequency at which it will oscillate after being disturbed).
Suppose we apply a periodically varying force to a system that can oscillate. The system is then forced to oscillate with a frequency equal to the frequency of the applied force (called the driving frequency). This motion is called aforced oscillation. We talked about forced oscillations of the harmonic oscillator in Section 13-9, and we suggest that you review that discussion. In particular, we described the phenomenon of mechanical resonance. A simple example of resonance is pushing Cousin Throckmorton on a swing. The swing is a pendulum; it has only a single normal mode, with a frequency determined by its length. If we push the swing periodically with this frequency, we can build up the amplitude of the motion. But if we push with a very different frequency, the swing hardly moves at all.
Resonance also occurs when a periodically varying force is applied to a system with many normal modes. An example is shown in Fig. 20-20a. An open organ pipe is placed next to a loudspeaker that is driven by an amplifier and emits pure sinusoidal sound waves of frequency f, which can be varied by adjusting the amplifier. The air in the pipe is forced to vibrate with the same frequency f as the driving force provided by the loudspeaker. In general the amplitude of this motion is relatively small, and the motion of the air inside the pipe will not be any of the normal-mode patterns shown in Fig. 20-15 .’ But if the frequency f of the force is close to one of the normal-mode frequencies, the air in the pipe will move in the normal-mode pattern for that frequency, and the ‘amplitude can become quite large. Figure 20-20b shows the amplitude of oscillation of the air in the pipe as a function of the driving frequency f The shape of this graph is called the resonance curve of the pipe; it has peaks where f equals the normal-mode frequencies of’ pipe. The detailed shape of the resonance curve depends on the geometry of the .
If the frequency of the force is precisely equal to a normal-mode .’ tern is in resonance, and the amplitude of the forced oscillari If there were no friction or other energy-dissipating mechanism, a dri\ . farce a normalmode frequency would continue to’ add energy to the system, and the amplimde would increase indefinitely. In such an idealized case the peaks in the resonance curve of Fig. 20-20b would be infinitely high. But in any real system there is always some distance of energy.
Response is a very important concept, not only in mechanical system but it all area of Physics. In Chapter 32 we will see examples of resonance in electric circuits.