Nonlinear dynamics of rectangular plates: investigation of modal interaction in free and forced vibrations

Michele Ducceschi, Cyril Touze, Stefan Bilbao, Craig Webb

Research output: Contribution to journalArticlepeer-review


Nonlinear vibrations of thin rectangular plates are considered, using the von kármán equations in order to take into account the effect of geometric nonlinearities. Solutions are derived through an expansion over the linear eigenmodes of the system for both the transverse displacement and the Airy stress function, resulting in a series of coupled oscillators with cubic nonlinearities, where the coupling coefficients are functions of the linear eigenmodes. A general strategy for the calculation of these coefficients is outlined, and the particular case of a simply supported plate with movable edges is thoroughly investigated. To this extent, a numerical method based on a new series expansion is derived to compute the Airy stress function modes, for which an analytical solution is not available. It is shown that this strategy allows the calculation of the nonlinear coupling coefficients with arbitrary precision, and several numerical examples are provided. Symmetry properties are derived to speed up the calculation process and to allow a substantial reduction in memory requirements. Finally, analysis by continuation allows an investigation of the nonlinear dynamics of the first two modes both in the free and forced regimes. Modal interactions through internal resonances are highlighted, and their activation in the forced case is discussed, allowing to compare the nonlinear normal modes (NNMs) of the undamped system with the observable periodic orbits of the forced and damped structure.
Original languageEnglish
Pages (from-to)213-232
Number of pages20
JournalActa Mechanica
Issue number1
Publication statusPublished - Jan 2014


Dive into the research topics of 'Nonlinear dynamics of rectangular plates: investigation of modal interaction in free and forced vibrations'. Together they form a unique fingerprint.

Cite this