Sympathetic Vibratory Physics

A brief introduction to parametric resonance

Let it be sufficient to say that since 1883, when Lord Rayleigh published his paper "On maintained vibrations", Philosophical Magazine, vol. 15, pages 229-235, a body of research has been produced on the topic of parametric resonance. Ironically, most of it has been dealing with how to prevent instability in mechanical systems and electric circuits. The latest research around parametric resonance is visible in many scientific disciplines, from biology to quantum physics, pointing to the fact that parametric resonance is an often encountered and yet to be fully understood and utilized natural phenomenon.

Now let’s review some well known definitions related to parametric resonance and oscillators. The statements in items 1 through 4 are taken verbatim from Wikipedia’s pages on harmonic and parametric oscillators.

1. “A parametric oscillator is a simple harmonic oscillator whose parameters (its resonance frequency w and damping ß) vary in time. Another intuitive way of understanding a parametric oscillator is as follows: a parametric oscillator is a device that oscillates when one of its "parameters" (a physical entity, like capacitance) is changed.”

2. “Remarkably, if the parameters vary at roughly twice the natural frequency of the oscillator, the oscillator phase-locks to the parametric variation and absorbs energy at a rate proportional to the energy it already has. Without a compensating energy-loss mechanism, the oscillation amplitude grows exponentially. (This phenomenon is called parametric excitation, parametric resonance or parametric pumping.) However, if the initial amplitude is zero, it will remain so; this distinguishes it from the non-parametric resonance of driven simple harmonic oscillators, in which the amplitude grows linearly in time regardless of the initial state.”

3. Capacitance in a parallel RLC electric circuit is equivalent to mass in a translational mechanical system, or to moment of inertia in a rotational system.

NOTE: A familiar experience of parametric oscillation is playing on a swing. By alternately raising and lowering their center of mass (and thereby changing their moment of inertia, and thus the resonance frequency) at key points in the swing, children can quickly reach large amplitudes provided that they have some amplitude to start with (e.g., get a push). Doing so at rest, however, goes nowhere.

4. "The problem of the simple harmonic oscillator occurs frequently in physics because a mass at equilibrium under the influence of any conservative force, in the limit of small motions, will behave as a simple harmonic oscillator. A conservative force is one that has a potential energy function.”

NOTE: For example, G force acting on a pendulum is considered a conservative force.

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Parametric Resonance