Multiwave non-linear couplings in elastic structures. Part II: two-dimensional example

Abstract

The non-linear resonance coupling in a thin plate in the Kirchhoff-Love approximation is selected as a two-dimensional example of mechanical systems exhibiting a rich range of resonant wave-like phenomena. This is originally examined by use of Whitham's average-Lagrangian method. In particular, the existence of three basic resonant triads between longitudinal, shear and bending modes is shown. Some of these necessarily enter cascade wave processes related to the instability of some of the mode components of the triad under small perturbations. A short comparison with Kolmogorov's cascades of turbulence is given.