Non-Linear vibrations and chaos in harmonically excited rectangular plates with one-to-one internal resonance

Abstract

Nonlinear flexural vibrations of a rectangular plate with uniform stretching are studied for the case when it is harmonically excited with forces acting normal to the midplane of the plate. The physical phenomena of interest here arise when the plate has two distinct linear modes of vibration with nearly the same natural frequency. It is shown that, depending on the spatial distribution of the external forces, the plate can undergo harmonic motions either in one of the two individual modes or in a mixed-mode. Stable single-mode and mixed-mode solutions can also coexist over a wide range in the amplitudes and frequency of excitation. For low damping levels, the presence of Hopf bifurcations in the mixed-mode response leads to complicated amplitude-modulated dynamics including period doubling bifurcations, chaos, coexistence of multiple chaotic motions, and crisis, whereby the chaotic attractors suddenly disappear and the plate resumes small amplitude harmonic motions in a single-mode. Numerical results are presented specifically for 1:1 resonance in the (1, 2) and (3, 1) plate modes.

Asymmetric Non-Linear Forced Vibrations of Free-Edge Circular Plates. Part 1: Theory

“…Dynamical system (40) exhibits a number of different possible behaviours. For example, limit cycles and chaotic behaviour were found in numerical simulations of equation (40), for specific values of the coefficients [32,36]. A complete study of the bifurcation set will not be conducted in this paper.…”

“…The major contributions have been brought in the study of the vibrations of rectangular plates, where degenerated modes are observed, i.e., two modes having different mode shapes but identical natural frequencies. Yasuda and Asano [31] and Chang et al [32] performed a thorough analysis of the different behaviours exhibited by a system of form (31). The first authors were primarily interested in the case of an equivalent forcing on the two modes ðQ 1 ' Q 2 Þ: Chang et al conducted an exhaustive study of the bifurcations.…”

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“…Mode coupling and energy exchange between internally resonant modes are a second common feature, now well established in the literature [4][5][6][7][8][9][10][11].…”

“…For two functions F(x, y, t) and G(x, y, t) representing general scalar/vector/tensor fields, an L 2 spatial inner product over is defined as 6) where is the standard product in the case of scalar fields, the standard dot-product in the case of vector fields and F G = tr F T G in the case of tensor fields (tr(·) indicates the trace operation and T the transpose operation). The norm of such a field F is defined as…”

Nonlinear stability analysis of a composite laminated piezoelectric rectangular plate with multi-parametric and external excitations

“…Chang et al [6] studied the bifurcations and chaos of a rectangular thin plate with 1:1 internal resonance. Zhang [7] investigated the local and global bifurcations of a rectangular thin plate under parametrical excitation by the analytical and numerical approaches when the averaged equations have one non-semisimple double zero and a pair of pure imaginary eigenvalues The method of multiple scales is used to obtain the averaged equations.…”