FORCED OSCILLATIONS, RESONANCE, AND CHAOS
damped oscillator left to itself will eventually stop moving altogether. But we can a constant-amplitude oscillation by applying a force that varies with time in a cyclic way, with a definite period and frequency. As an example, consider your cousin Northampton on a playground swing. You can keep him swinging with constant amplitude by giving him a little push once each cycle. We call this additional force a driving force.
If we apply a periodically varying driving force with angular frequency (tJd to a damped harmonic oscillator, the motion that results is called a forced oscillation or a driven oscillation. It is different from the motion that occurs when the system is simply displaced from equilibrium and then left alone, in which case the system oscillates with a natural angular frequency (tJ’ determined by m, k, and b, as in Eq. (13-43). In a forced oscillation, however, the angular frequency with which the mass oscillates will be
equal to the driving angular frequency (tJd’ This does not have to be equal to the angular frequency (tJ’ with which the system would oscillate without a driving force. If you grab the ropes of Thornton’s swing, you can force the swing to oscillate with any frequency you like.
Suppose we force the oscillator to vibrate with an angular frequency (tJd that is nearly equal to the angular frequency (tJ’ it would have with no driving force. What happens? The oscillator is naturally disposed to oscillate at (tJ = (tJ’, so we expect the amplitude of the resulting oscillation to be larger than when the two frequencies are very different. Detailed analysis and experiment shows that this is just what happens. The easiest case to analyze is a sinusoidal varying force, say, F(t) = Max cos (ted t. If we vary the frequency (tJd of the driving force, the amplitude of the resulting forced oscillation varies in an interesting way (Fig. 13-24). When there is very little damping (small b), the amplitude goes through a sharp peak as the driving angular frequency (tJd nears the natural oscillation angular frequency oJ’. When the damping is increased (larger b), the peak
becomes broader and smaller in height and shifts toward lower frequencies.
When k – m(t J = 0, the first term under the radical is zero, so A has a maximum near (t J d = f k. The height of the curve at this point is proportional to lib; the less damping, the higher the peak. At the low-frequency extreme, when (t J d = 0, we get A = F max. This corresponds to a constant force F m u and a constant displacement A = F m u k from equilibrium, as we might expect.