Free, Damped and Forced Oscillations; Resonance

So far, this whole chapter has treated S.H.M. as continuing forever, unchanged, with a constant amplitude that never dies away. This idealisation is called a FREE OSCILLATION: a system is disturbed once, from outside, and then left entirely alone to oscillate purely on its own, with no further external force acting on it at all, and with no energy lost to resistance of any kind. A freely oscillating system moves at a frequency fixed purely by its own physical properties -- $\omega_0=\sqrt{k/m}$ for a spring-mass system, or $\omega_0=\sqrt{g/L}$ for a simple pendulum -- called the NATURAL FREQUENCY of that particular system.\n\nIn reality, of course, every real oscillating system loses SOME energy as it moves, to friction, to air resistance, or to internal resistive forces of one kind or another. Oscillations that progressively lose energy this way are called DAMPED OSCILLATIONS. A damped oscillator still oscillates at very nearly its own natural frequency $\omega_0$, but its amplitude does NOT remain constant -- it decreases steadily with time as energy continually leaks away, typically dying down in something close to an exponentially-shrinking envelope, until the oscillation eventually stops altogether. A pendulum swinging in ordinary open air (rather than in an ideal vacuum) gradually and visibly comes to rest this way; and the shock absorbers fitted to a vehicle's suspension springs are deliberately designed to damp out unwanted bouncing quickly, after the vehicle has passed over a bump in the road, rather than letting it bounce on and on.\n\nA third, quite different situation arises when a system is instead driven CONTINUOUSLY by some external periodic force, applied at a chosen driving frequency $\omega_d$ that need not equal the system's own natural frequency $\omega_0$ at all -- this situation is called a FORCED OSCILLATION. After an initial, short-lived settling-in period (during which both the natural and the driving frequencies are briefly present together), a forced oscillator stops oscillating at its own natural frequency and instead locks on to oscillating steadily at the DRIVING frequency $\omega_d$ itself, with a steady-state amplitude that depends both on how close $\omega_d$ happens to be to $\omega_0$, and on how strongly the system is damped.\n\nRESONANCE is the special, and technologically very important, case of forced oscillation in which the driving frequency $\omega_d$ is equal to (or very nearly equal to) the system's own natural frequency $\omega_0$. When this condition is met, the external driving force keeps arriving in step with the system's own natural rhythm of oscillation, so it keeps adding energy to the oscillation in exactly the right phase, cycle after cycle after cycle -- and, as a direct result, the amplitude of the resulting oscillation grows very large indeed, far larger than the very same driving force could ever produce at any OTHER driving frequency. With little or no damping present, this resonant amplitude can, in principle, grow extremely large; with more damping present, the growth in amplitude is limited to some finite (though often still large) value, but a pronounced peak in amplitude right at $\omega_d=\omega_0$ still remains the hallmark of resonance.\n\nEveryday examples of resonance are easy to find: pushing a swing in rhythm with its own natur …