Parametric Resonance ○ Entries

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Parametric Resonance ○｜Definition｜1st｜20260912181021-00-⌔

Parametric oscillator - Wikipedia#Parametric_resonance

Parametric resonance

Parametric resonance is the parametrical resonance phenomenon of mechanical perturbation 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 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, such as the botafumeiro. Parametric resonance takes place when the external excitation frequency equals twice the natural frequency of the system divided by a positive integer $n$. For a parametric excitation with small amplitude $h$ in the absence of friction, the bandwidth of the resonance is to leading order $\mathcal{O}(|h{|}^n)$.¹ The effect of friction is to introduce a finite threshold for the amplitude of parametric excitation to result in an instability.²

For small amplitudes and by linearising, the stability of the periodic solution is given by Mathieu’s equation:

• ✤ $\ddot{u}+(a+B\cos t)u=0$

where $u$ is some perturbation from the periodic solution. Here the $B\cos (t)$ 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.

Autoparametric resonance happens in a system with two coupled oscillators, such that the vibrations of one act as parametric resonance on the second. The zero point of the second oscillator becomes unstable, and thus it starts oscillating.³⁴

Printed 2026-09-12.

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Footnotes

1. Bell, M. (1957). “A note on Mathieu functions”. Glasgow Mathematical Journal. 3 (3): 132–134. doi:10.1017/S204061850003358X. ↩

2. Landau, L. D.; Lifshitz, E. M. (1976). Mechanics (3rd ed.). Pergamon Press. ISBN 0-7506-2896-0. ↩

3. Verhulst, Ferdinand (2009), “Perturbation Analysis of Parametric Resonance” (PDF), Encyclopedia of Complexity and Systems Science, New York, NY: Springer New York, pp. 6625–6639, doi:10.1007/978-0-387-30440-3_393, ISBN 978-0-387-75888-6, archived from the original on 3 Dec 2020, retrieved 2023-06-25 ↩

4. Verhulst, Ferdinand (2023-06-01). “Multiple timing and spatial scaling for bifurcations”. Nonlinear Dynamics. 111 (12): 10693–10707. doi:10.1007/s11071-023-08378-x. ISSN 1573-269X. S2CID 257593795. ↩

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