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Dynamics of Parametric Excitation

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Encyclopedia of Complexity and Systems Science

Definition of the Subject

Parametric excitation of a system differs from direct forcing in that fluctuations appear as temporal modulation of a parameter ratherthan as a direct additive term. A common paradigm is that of a pendulum hanging under gravity whose support is subjected to a verticalsinusoidal displacement. In the absence of any dissipative effects, instabilities occur to the trivial equilibrium whenever the natural frequency isa multiple of half the excitation frequency. At amplitude levels beyond the instability, further bifurcations (dynamical transitions) can lead to more complex quasi‐periodic or chaotic dynamics. In multibody mechanicalsystems, one mode of vibration can effectively act as the parametric excitation of another mode through the presence of multiplicative nonlinearity. Thusautoparametricresonance occurs when one mode's frequency is a multiple of half of the other. Other effectsinclude combination resonance, where the excitation is at a sum or difference...

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Abbreviations

Parametric excitation :

Explicit time‐dependent variation of a parameter of a dynamical system.

Parametric resonance :

An instability that is caused by a rational relationship between the frequency of parametric excitation and the natural frequency of free oscillation in the absence of the excitation. If ω is the excitation frequency, and ω0 the natural frequency, then parametric resonance occurs when \( { \omega= ({n} / {2}) \omega_0 } \) for any positive integer n. The case \( { n=1 } \) is usually the most prominent form of parametric resonance, and is sometimes called the principle subharmonic resonance.

Autoparametric resonance :

A virtual parametric resonance that occurs due to the coupling between two independent degrees of freedom within a system. The output of one degree of freedom acts like the parametric excitation of the other.

Ince–Strutt diagram :

A two‐parameter bifurcation diagram indicating stable and unstable regions, specifically plotting the instability boundaries as the required amplitude of parametric excitation against the square of the ratio of natural to excitation frequency.

Floquet theory :

The determination of the eigenvalue spectrum that governs the stability of periodic solutions to systems of ordinary differential equations.

Bifurcation :

A qualitative change in a system's dynamics as a parameter is varied. One‐parameter bifurcation diagrams often depict invariant sets of the dynamics against a single parameter, indicating stability and any bifurcation points. Two‐parameter bifurcation diagrams depict curves in a parameter plane on which one‐parameter bifurcations occur.

Modal analysis :

The study of continuum models by decomposing their spatial parts into eigenmodes of the dynamic operator. The projection of the full system onto each mode gives an infinite system of differential equations in time, one for each mode.

Monodromy matrix :

The matrix used in Floquet theory, whose eigenvalues (also known as Floquet multipliers) determine stability of a periodic solution.

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Authors and Affiliations

  1. Department of Engineering Mathematics, University of Bristol, Bristol, United Kingdom

    Alan Champneys

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  1. Alan Champneys
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Editors and Affiliations

  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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Champneys, A. (2009). Dynamics of Parametric Excitation. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_144

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