Abstract
This short contribution considers the essentials of nonlinear wave properties in typical mechanical systems such as an infinite straight bar, a circular ring, and a flat plate. It is found that nonlinear resonance is experienced in all the systems exhibiting continuous and discrete spectra, respectively. Multiwave interactions and the stability of coupled modes with respect to small perturbations are discussed. The emphasis is placed on mechanical phenomena, for example, stress amplification, although some analogies with some nonlinear optical systems are also obvious. The nonlinear resonance coupling in a plate within the Kirchhoff-Love approximation is selected as a two-dimensional example exhibiting a rich range of resonant wave phenomena. This is originally examined by use of Whitham's averaged 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 mode components of the triad under small perturbations.
Highlights
The main aim of the first part of this short contribution is to display the essentials of nonlinear waves properties in mechanical systems of engineering origin
Nonlinear resonance is examined in two one-dimensional examples, an infinite straight elastic bar and a thin elastic circular ring, exhibiting continuous and discrete spectra, respectively
Three-wave and four-wave interactions and the stability of coupled modes with respect perturbations are discussed, the emphasis being placed on mechanical phenomena, analogies with some nonlinear optical systems are obvious
Summary
The main aim of the first part of this short contribution is to display the essentials of nonlinear waves properties in mechanical systems of engineering origin (structural members). Nonlinear resonance is examined in two one-dimensional examples, an infinite straight elastic bar and a thin elastic circular ring, exhibiting continuous and discrete spectra, respectively. In the present paper we focus attention on the nonlinear wave couplings in engineering elastic structures, more in this contribution, in one-dimensional examples one related to an elastic infinitely long straight bar and the other to a thin closed circular ring. These two structures are chosen because they exhibit a continuous spectrum and a discrete one (due to the circular periodicity), respectively. A two-dimensional example (elastic plate) is sketched out in Section 4 where the notion of cascade wave process is evoked
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