Parametric Oscillator - Parametric Resonance

Parametric resonance is the parametrical resonance phenomenon of mechanical excitation and oscillation at certain frequencies (and the associated harmonics). This effect is different from regular resonance because it exhibits the instability phenomenon.

Parametric resonance occurs in a mechanical system when a system is parametrically excited and oscillates at one of its resonant frequencies.Parametric resonance takes place when the external excitation frequency equals to twice the natural frequency of the system. Parametric excitation differs from forcing since the action appears as a time varying modification on a system parameter. The classical example of parametric resonance is that of the vertically forced pendulum.

For small amplitudes and by linearising, the stability of the periodic solution is given by :

where is some perturbation from the periodic solution. Here the term acts as an ‘energy’ source and is said to parametrically excite the system. The Mathieu equation describes many other physical systems to a sinusoidal parametric excitation such as an LC Circuit where the capacitor plates move sinusoidally.