Abstract
This study investigates the dynamical behavior of the composite laminated piezoelectric rectangular plate with 1 : 2 internal resonance near the singularity using the extended singularity theory method. Based on the previous four-dimensional averaged equations of polar coordinates where the partial derivative terms are not equal to zero, the universal unfolding with codimension 3 of the proposed system is given. The main material parameters that affect the dynamic behavior of the laminated piezoelectric rectangular composite plate near the singularity under transverse excitation are revealed by the transition set of universal unfolding with codimension 3. In addition, the plots of the transition set in three bifurcation parameters space are discussed. These numerical results can show that the stability near the singularity of the proposed system is better when period ratio is less than zero.
Highlights
Singularity theory has been widely applied as a qualitative bifurcation analysis method
It can use the uniform and clear method to deal with various complex bifurcation problems and establish comprehensive links between system dynamic behavior and system parameters with singularity theory
Yu et al [8] studied the torsional feedback mechanism under a galloping excitation. en, with the gradual adding of bifurcation parameters, the singularity theory has been applied to a single-degree-of-freedom system with multiparameters
Summary
Singularity theory has been widely applied as a qualitative bifurcation analysis method. Zhang and Chen [14] studied stability and bifurcation behaviors for a model of supported functionally graded materials rectangular plate subjected to the transversal and in-plane excitations by means of combination of analytical and numerical methods. Guo and Zhang [15] studied 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 the proposed system with three bifurcation parameters is a high codimensional bifurcation problem with codimension 4, and 6 forms of universal unfolding are given. E main material parameters that affect the dynamic behavior of the laminated piezoelectric rectangular composite plate near the singularity under transverse excitation are revealed by universally unfolding with codimension 3. ⎧⎪⎪⎪⎨ g1 k11a61 + k12a41a22 + k13a21a42 + k14σ1a41 + k15σ1a21a22 + σ21a21 + k16a21 + τ1 0,
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