Abstract
In this paper, the authors study the bifurcation problems of the composite laminated piezoelectric rectangular plate structure with three bifurcation parameters by singularity theory in the case of 1:2 internal resonance. The sign function is employed to the universal unfolding of bifurcation equations in this system. The proposed approach can ensure the nondegenerate conditions of the universal unfolding of bifurcation equations in this system to be satisfied. The study presents that the proposed system with three bifurcation parameters is a high codimensional bifurcation problem with codimension 4, and 6 forms of universal unfolding are given. Numerical results show that the whole parametric plane can be divided into several persistent regions by the transition set, and the bifurcation diagrams in different persistent regions are obtained.
Highlights
The bifurcation equations with multiple variables can be reduced to the bifurcation equation with single state variable by elimination method, but the work [1] showed that some bifurcation characteristics of system would be vanished
When an original steady-state system is subjected to small perturbations, all possible bifurcation behaviors can be revealed from the universal unfolding of bifurcation equations
We know that the universal unfolding of the bifurcation equations can reveal all possible bifurcation behaviors when the original system is subject to small perturbations
Summary
The bifurcation equations with multiple variables can be reduced to the bifurcation equation with single state variable by elimination method, but the work [1] showed that some bifurcation characteristics of system would be vanished. Zhang et al [27] investigated the nonlinear oscillations and chaotic dynamics of a parametrically excited supported symmetric crossply laminated composite rectangular thin plate with the geometric nonlinearity and nonlinear damping. Zhang et al [30] studied the motion equations of supported flexible beam with five nonlinear terms under parametric excitation and analyzed the different forms of bifurcated response curve when the parameters are located in different regions. The singularity theory of asymmetry dynamical systems with multiple state variables and single bifurcation parameter had been well developed by Golubitsky and Schaeffer. The sign function is introduced to solve this problem, which leads to the fact that nondegenerate conditions of universal unfolding of the bifurcation equations are satisfied It may determine the summation of the number of sign functions and the number of auxiliary parameters is equal to codimension. The transition sets in the parameters plane are calculated and the bifurcation diagrams are depicted
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