Nonlinear plate vibrations: A modal approach with application to cymbals and gongs

Michele Ducceschi , Cyril Touzé , Stefan Bilbao

Geometrically nonlinear vibrations of plates are investigated with the purpose of sound synthesis and application to cymbals, gongs and thunder plates. Von-Kármán equations are used to describe the motion of the plate when the amplitude is comparable to the plate thickness. In this contribution, the case of a rectangular plate is considered, and a modal approach is selected. A general procedure, adapted from a paper by Li [JSV, 2004], is proposed in order to deal with various case of boundary conditions, for which no analytical solutions are available. After discretisation, one is left with a system of ordinary differential equations for the temporal modal coordinates, with cubic coupling terms. The results are validated by comparing frequency-response curves with other either found in literature, or computed with a finite difference approach, showing good agreement. The nonlinear dynamics of a simply supported plate is investigated by computing its frequency-energy plot (FEP), displaying variations of eigenfrequencies with amplitude and highlighting their interactions via internal resonance. The proposed method provides a possible strategy for modelling the shimmering sounds of musical instruments such as gongs and cymbals, as it allows for an accurate description of modal viscous damping. Finally, impulsive sounds will be shown and compared to those obtained with a finite difference approach.

Mots clés: nonlinear plate vibrations; modal approach; cymbals and gongs; bifurcation

Domaines: Acoustique [physics.class-ph]

Michele Ducceschi, Cyril Touzé, Stefan Bilbao. Nonlinear plate vibrations: A modal approach with application to cymbals and gongs. Acoustics 2012, Apr 2012, Nantes, France. ⟨hal-00810591⟩