Double Hopf bifurcation of composite laminated piezoelectric plate subjected to external and internal excitations

doi:10.1007/s10483-017-2196-9

Applied Mathematics and Mechanics (English Edition) ›› 2017, Vol. 38 ›› Issue (5): 689-706.doi: https://doi.org/10.1007/s10483-017-2196-9

Double Hopf bifurcation of composite laminated piezoelectric plate subjected to external and internal excitations

Yan ZHOU¹, Wei ZHANG²

1. Mathematics Science College, Inner Mongolia Normal University, Hohhot 010022, China;
2. Beijing Key Laboratory on Nonlinear Vibrations and Strength of Mechanical Structures, College of Mechanical Engineering, Beijing University of Technology, Beijing 100124, China

收稿日期:2016-08-16 修回日期:2016-09-23 出版日期:2017-05-01 发布日期:2017-05-01
通讯作者: Yan ZHOU E-mail:yanzhou0924@163.com
基金资助:
Project supported by the National Natural Science Foundation of China (Nos. 11402127, 11290152 and 11072008)

Double Hopf bifurcation of composite laminated piezoelectric plate subjected to external and internal excitations

Yan ZHOU¹, Wei ZHANG²

1. Mathematics Science College, Inner Mongolia Normal University, Hohhot 010022, China;
2. Beijing Key Laboratory on Nonlinear Vibrations and Strength of Mechanical Structures, College of Mechanical Engineering, Beijing University of Technology, Beijing 100124, China

Received:2016-08-16 Revised:2016-09-23 Online:2017-05-01 Published:2017-05-01
Contact: Yan ZHOU E-mail:yanzhou0924@163.com
Supported by:
Project supported by the National Natural Science Foundation of China (Nos. 11402127, 11290152 and 11072008)

摘要/Abstract

摘要：

The double Hopf bifurcation of a composite laminated piezoelectric plate with combined external and internal excitations is studied. Using a multiple scale method, the average equations are obtained in two coordinates. The bifurcation response equations of the composite laminated piezoelectric plate with the primary parameter resonance, i.e., 1:3 internal resonance, are achieved. Then, the bifurcation feature of bifurcation equations is considered using the singularity theory. A bifurcation diagram is obtained on the parameter plane. Different steady state solutions of the average equations are analyzed. By numerical simulation, periodic vibration and quasi-periodic vibration responses of the composite laminated piezoelectric plate are obtained.

关键词: normal cone, increasing operator, positive solution for integral equation, composite laminated piezoelectric plate, double Hopf bifurcation, periodic solution, quasi-periodic solution

Abstract:

Key words: normal cone, increasing operator, positive solution for integral equation, double Hopf bifurcation, composite laminated piezoelectric plate, periodic solution, quasi-periodic solution

中图分类号:

O193

Yan ZHOU, Wei ZHANG. Double Hopf bifurcation of composite laminated piezoelectric plate subjected to external and internal excitations[J]. Applied Mathematics and Mechanics (English Edition), 2017, 38(5): 689-706.

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Double Hopf bifurcation of composite laminated piezoelectric plate subjected to external and internal excitations

Double Hopf bifurcation of composite laminated piezoelectric plate subjected to external and internal excitations

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