Multiwave nonlinear couplings in elastic structures

D. A. Kovriguine, G. A. Maugin and A. I. Potapov

Mathematical Problems in Engineering, 2006, vol. 2006, 1-11

Abstract:

This short contribution considers the essentials of nonlinear waveproperties in typical mechanical systems such as an infinitestraight bar, a circular ring, and a flat plate. It is found thatnonlinear resonance is experienced in all the systems exhibitingcontinuous and discrete spectra, respectively. Multiwaveinteractions and the stability of coupled modes with respect tosmall perturbations are discussed. The emphasis is placed onmechanical phenomena, for example, stress amplification, althoughsome analogies with some nonlinear optical systems are alsoobvious. The nonlinear resonance coupling in a plate within theKirchhoff-Love approximation is selected as a two-dimensionalexample exhibiting a rich range of resonant wave phenomena. Thisis originally examined by use of Whitham's averaged Lagrangianmethod. In particular, the existence of three basic resonanttriads between longitudinal, shear, and bending modes is shown.Some of these necessarily enter cascade wave processes related tothe instability of some mode components of the triad under smallperturbations.

Date: 2006
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jnlmpe:076041

DOI: 10.1155/MPE/2006/76041

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