Bifurcations and Chaos of Composite Laminated Piezoelectric Rectangular Plate With One-to-Three Internal Resonance

2008 ◽  
Author(s):  
Zhi-Gang Yao ◽  
Wei Zhang
Keyword(s):  
Rectangular Plate ◽  
Multiple Scales ◽  
Plate Theory ◽  
Equations Of Motion ◽  
Governing Equations ◽  
Third Order ◽  
Chaotic Motions ◽  
Piezoelectric Layers ◽  
Parametric And External Excitations ◽  
First Time

The bifurcations and chaotic motions of a simply supported symmetric cross-ply composite laminated piezoelectric rectangular plate are analyzed for the first time, which are forced by the transverse and in-plane excitations. It is assumed that different layers of symmetric cross-ply composite laminated piezoelectric rectangular plate are perfectly bonded to each other and with piezoelectric actuator layers embedded in the plate. Based on the Reddy’s third-order shear deformation plate theory, the nonlinear governing equations of motion for the composite laminated piezoelectric rectangular plate are derived by using the Hamilton’s principle. The excitation loaded by piezoelectric layers is considered. The Galerkin’s approach is employed to discretize partial differential governing equations to a two-degree-of-freedom nonlinear system under combined the parametric and external excitations. The method of multiple scales is utilized to obtain the four-dimensional averaged equation. Numerical method is used to find the periodic and chaotic motions of the composite laminated piezoelectric rectangular plate. The numerical results show the existence of the periodic and chaotic motions in the averaged equation. It is found that the chaotic responses are especially sensitive to the forcing and the parametric excitations. The influence of the transverse, in-plane and piezoelectric excitations on the bifurcations and chaotic behaviors of the composite laminated piezoelectric rectangular plate is investigated numerically.

scholarly journals Multipulse Heteroclinic Orbits and Chaotic Dynamics of the Laminated Composite Piezoelectric Rectangular Plate

2013 ◽  
Vol 2013 ◽  
pp. 1-27 ◽  
Author(s):  
Minghui Yao ◽  
Wei Zhang
Keyword(s):  
Rectangular Plate ◽  
Chaotic Dynamics ◽  
Multiple Scales ◽  
Nonlinear Oscillations ◽  
Plate Theory ◽  
Equations Of Motion ◽  
Laminated Composite ◽  
Heteroclinic Orbits ◽  
Phase Method ◽  
Chaotic Motions

This paper investigates the multipulse global bifurcations and chaotic dynamics for the nonlinear oscillations of the laminated composite piezoelectric rectangular plate by using an energy phase method in the resonant case. Using the von Karman type equations, Reddy’s third-order shear deformation plate theory, and Hamilton’s principle, the equations of motion are derived for the laminated composite piezoelectric rectangular plate with combined parametric excitations and transverse excitation. Applying the method of multiple scales and Galerkin’s approach to the partial differential governing equation, the four-dimensional averaged equation is obtained for the case of 1 : 2 internal resonance and primary parametric resonance. The energy phase method is used for the first time to investigate the Shilnikov type multipulse heteroclinic bifurcations and chaotic dynamics of the laminated composite piezoelectric rectangular plate. The paper demonstrates how to employ the energy phase method to analyze the Shilnikov type multipulse heteroclinic bifurcations and chaotic dynamics of high-dimensional nonlinear systems in engineering applications. Numerical simulations show that for the nonlinear oscillations of the laminated composite piezoelectric rectangular plate, the Shilnikov type multipulse chaotic motions can occur. Overall, both theoretical and numerical studies suggest that chaos for the Smale horseshoe sense in motion exists.

scholarly journals Modeling and Chaotic Dynamics of the Laminated Composite Piezoelectric Rectangular Plate

2014 ◽  
Vol 2014 ◽  
pp. 1-19 ◽  
Author(s):  
Minghui Yao ◽  
Wei Zhang ◽  
D. M. Wang
Keyword(s):  
Rectangular Plate ◽  
Chaotic Dynamics ◽  
Multiple Scales ◽  
Plate Theory ◽  
Equations Of Motion ◽  
Laminated Composite ◽  
Melnikov Method ◽  
Chaotic Motions ◽  
Extended Melnikov Method ◽  
Transverse Excitation

This paper investigates the multipulse heteroclinic bifurcations and chaotic dynamics of a laminated composite piezoelectric rectangular plate by using an extended Melnikov method in the resonant case. According to the von Karman type equations, Reddy’s third-order shear deformation plate theory, and Hamilton’s principle, the equations of motion are derived for the laminated composite piezoelectric rectangular plate with combined parametric excitations and transverse excitation. The method of multiple scales and Galerkin’s approach are applied to the partial differential governing equation. Then, the four-dimensional averaged equation is obtained for the case of 1 : 3 internal resonance and primary parametric resonance. The extended Melnikov method is used to study the Shilnikov type multipulse heteroclinic bifurcations and chaotic dynamics of the laminated composite piezoelectric rectangular plate. The necessary conditions of the existence for the Shilnikov type multipulse chaotic dynamics are analytically obtained. From the investigation, the geometric structure of the multipulse orbits is described in the four-dimensional phase space. Numerical simulations show that the Shilnikov type multipulse chaotic motions can occur. To sum up, both theoretical and numerical studies suggest that chaos for the Smale horseshoe sense in motion exists for the laminated composite piezoelectric rectangular plate.

Aeroelastic Flutter Analysis of Thick Porous Plates in Supersonic Flow

2019 ◽  
Vol 11 (10) ◽  
pp. 1950096 ◽  
Author(s):  
Reza Bahaadini ◽  
Ali Reza Saidi ◽  
Kazem Majidi-Mozafari
Keyword(s):  
Supersonic Flow ◽  
Plate Theory ◽  
Equations Of Motion ◽  
Functionally Graded ◽  
Governing Equations ◽  
Aerodynamic Pressure ◽  
Aeroelastic Flutter ◽  
Piezoelectric Layers ◽  
Flutter Analysis ◽  
Porosity Coefficient

The aeroelastic flutter analysis of thick porous plates surrounded with piezoelectric layers in supersonic flow is studied. In order to aeroelastic analysis of the thick porous-cellular plate, Reddy’s third-order shear deformation plate theory and first-order piston theory are used. Furthermore, the plate is composed of two face piezoelectric layers and three functionally graded porous distributions core. Applying the extended Hamilton’s principle and Maxwell’s equation, the governing equations of motion are obtained. The partial differential governing equations are transformed into a set of ordinary differential equations by applying Galerkin’s approach. The effects of porosity coefficient, porosity distributions, piezoelectric layers, geometric dimensions, electrical and mechanical boundary conditions on the flutter aerodynamic pressure and natural frequencies of porous-cellular plates are investigated.

Chaos and Bifurcation of Composite Laminated Cantilever Rectangular Plate

2012 ◽  
Author(s):  
Wei Zhang ◽  
Minghui Zhao ◽  
Xiangying Guo
Keyword(s):  
Rectangular Plate ◽  
Deformation Theory ◽  
Complex Dynamics ◽  
Equations Of Motion ◽  
Plate Boundary ◽  
Shear Deformation Theory ◽  
Governing Equations ◽  
Chaotic Motions ◽  
Cantilever Rectangular Plate ◽  
Two Degree Of Freedom

According to the Reddy’s high-order shear deformation theory and the von-Karman type equations for the geometric nonlinearity, the chaos and bifurcation of a composite laminated cantilever rectangular plate subjected to the in-plane and moment excitations are investigated with the case of 1:2 internal resonance. A new expression of displacement functions which can satisfy the cantilever plate boundary conditions are used to make the nonlinear partial differential governing equations of motion discretized into a two-degree-of-freedom nonlinear system under combined parametric and forcing excitations, representing the evolution of the amplitudes and phases exhibiting complex dynamics. The results of numerical simulation demonstrate that there exist the periodic and chaotic motions of the composite laminated cantilever rectangular plate. Finally, the influence of the forcing excitations on the bifurcations and chaotic behaviors of the system is investigated numerically.

Multi-Pulse Chaotic Dynamics for Composite Laminated Plates

2010 ◽  
Author(s):  
Jun-Hua Zhang ◽  
Wei Zhang ◽  
Qian Wang
Keyword(s):  
Rectangular Plate ◽  
Chaotic Dynamics ◽  
Equations Of Motion ◽  
Nonlinear Dynamical System ◽  
Melnikov Method ◽  
Laminated Plates ◽  
Chaotic Motions ◽  
Nonlinear Dynamical ◽  
Simply Supported ◽  
First Time

The heteroclinic bifurcation and multi-pulse chaotic dynamics of a simply-supported symmetric cross-ply composite laminated rectangular plate with parametric and forcing excitations are investigated in this paper for the first time. The formulas of the simply-supported composite laminated rectangular plate are derived by using Hamilton’s principle and the Galerkin’s approach. The extended Melnikov method is improved to enable us to analyze directly the non-autonomous nonlinear dynamical system, which is applied to the non-autonomous governing equations of motion for the simply-supported composite laminated rectangular plate. The results obtained here indicate that the multi-pulse chaotic motions can occur in the simply-supported composite laminated rectangular plate. Numerical simulation is also employed to find the multipulse chaotic motions of the simply-supported composite laminated rectangular plate.

Nonlinear Dynamics of a Cantilever Beam Actuated by Piezoelectric Layers

2001 ◽  
Author(s):  
K. Wolf ◽  
O. Gottlieb
Keyword(s):  
Cantilever Beam ◽  
Nonlinear Equations ◽  
Evolution Equations ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Third Order ◽  
Nonlinear Properties ◽  
Silicon Cantilever ◽  
Piezoelectric Layers ◽  
Nonlinear Equations Of Motion

Abstract The nonlinear equations of motion for a silicon cantilever beam, covered symmetrically by piezoelectric ZnO layers are derived for voltage excitation. Starting with the nonlinear description of the strain distribution in the beam for finite displacement, the Lagrangeian of the system is obtained from the electric enthalpy density. By application of Hamilton’s principle, the nonlinear equations of motion are consistently derived and truncated to third order for perturbation analysis. The evolution equations are obtained by the multiple scales method and periodic solutions to the equations of motion are determined and discussed with respect to the influence of geometric nonlinearities and nonlinear properties of the piezoelectric layer.

Nonlinear Vibration of a Cantilever FGM Rectangular Plate with Piezoelectric Layers Based on Third-Order Plate Theory

2011 ◽  
Vol 415-417 ◽  
pp. 2151-2155
Author(s):  
Ying Wang ◽  
Yu Xin Hao ◽  
Jian Hua Wang
Keyword(s):  
Differential Equations ◽  
Nonlinear Vibration ◽  
Rectangular Plate ◽  
Initial Conditions ◽  
Plate Theory ◽  
Nonlinear Differential Equations ◽  
Nonlinear Dynamic Analysis ◽  
Control Voltage ◽  
Third Order ◽  
Piezoelectric Layers

This paper deals with nonlinear dynamic analysis of a cantilever FGM rectangular plate with piezoelectric layers subjected to the transversal excitation in thermo-electro-mechanical environment. The material properties of plate and piezoelectric layers are assumed to be temperature-dependent. The governing equations of the functionally graded plate are based on the Reddy’s third-order shear deformation plate theory that includes thermo-piezoelectric effects and Hamilton’s principle. The governing nonlinear partial differential equations are transformed into ordinary nonlinear differential equations using the Galerkin method and the nonlinear and linear frequencies obtained using the Runge–Kutta method. The effects played by control voltage and system initial conditions on the nonlinear vibration of the plate are studied.

NONLINEAR RESPONSES OF A STRING-BEAM COUPLED SYSTEM WITH FOUR-DEGREES-OF-FREEDOM AND MULTIPLE PARAMETRIC EXCITATIONS

2009 ◽  
Vol 19 (01) ◽  
pp. 225-243 ◽  
Author(s):  
D. X. CAO ◽  
W. ZHANG
Keyword(s):  
Nonlinear Dynamic ◽  
Internal Resonance ◽  
Degrees Of Freedom ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Primary Resonance ◽  
Coupled System ◽  
Dynamic Responses ◽  
Governing Equations ◽  
Chaotic Motions

The nonlinear dynamic responses of a string-beam coupled system subjected to harmonic external and parametric excitations are studied in this work in the case of 1:2 internal resonance between the modes of the beam and string. First, the nonlinear governing equations of motion for the string-beam coupled system are established. Then, the Galerkin's method is used to simplify the nonlinear governing equations to a set of ordinary differential equations with four-degrees-of-freedom. Utilizing the method of multiple scales, the eight-dimensional averaged equation is obtained. The case of 1:2 internal resonance between the modes of the beam and string — principal parametric resonance-1/2 subharmonic resonance for the beam and primary resonance for the string — is considered. Finally, nonlinear dynamic characteristics of the string-beam coupled system are studied through a numerical method based on the averaged equation. The phase portrait, Poincare map and power spectrum are plotted to demonstrate that the periodic and chaotic motions exist in the string-beam coupled system under certain conditions.

Periodic and Chaotic Oscillations of an Axially Moving Viscoelastic Belt in Three-Dimensional Space

2007 ◽  
Author(s):  
Wei Zhang ◽  
Yan-Qi Liu ◽  
Li-Hua Chen ◽  
Ming-Hui Yao
Keyword(s):  
Internal Resonance ◽  
Multiple Scales ◽  
Dimensional Space ◽  
Equations Of Motion ◽  
Viscoelastic Model ◽  
Three Dimensional ◽  
Governing Equations ◽  
Chaotic Motions ◽  
Out Of Plane ◽  
Axially Moving

Periodic and chaotic space oscillations of an axially moving viscoelastic belt with one-to-one internal resonance are investigated for the first time. The Kelvin viscoelastic model is introduced to describe the viscoelastic property of the belt material. The external damping and internal damping of the material for the axially moving viscoelastic belt are considered simultaneously. The nonlinear governing equations of motion of the axially moving viscoelastic belt for the in-plane and out-of-plane are derived by the extended Hamilton’s principle. The method of multiple scales and Galerkin’s approach are applied directly to the partial differential governing equations of motion to obtain four-dimensional averaged equation under the case of 1:1 internal resonance and primary parametric resonance of the first order modes for the in-plane and out-of-plane oscillations. Numerical method is used to investigate periodic and chaotic space motions of the axially moving viscoelastic belt. The results of numerical simulation demonstrate that there exist periodic, period-2, period-3, period-4, period-6, quasiperiodic and chaotic motions of the axially moving viscoelastic belt.

Chaotic Motion of a Functionally Graded Materials Square Thin Plate

2012 ◽  
Vol 531 ◽  
pp. 593-596
Author(s):  
Shuang Bao Li ◽  
Yu Xin Hao
Keyword(s):  
Thin Plate ◽  
Functionally Graded Materials ◽  
Chaotic Motion ◽  
Plate Theory ◽  
Equations Of Motion ◽  
Functionally Graded ◽  
Third Order ◽  
Graded Materials ◽  
Steady State Heat ◽  
Steady State Heat Transfer

Chaotic motion of a simply supported functionally graded materials (FGM) square thin plate under one-to-two internal resonance is studied in this paper. The FGM plate is subjected to the transversal and in-plane excitations. Material properties are assumed to be temperature-dependent and change continuously throughout the thickness of the plate. The temperature variation is assumed to occur in the thickness direction only and satisfy the steady-state heat transfer equation. Based on the Reddy’s third-order plate theory and Hamilton’s principle, the nonlinear governing equations of motion for the FGM plate are derived by using the Galerkin’s method to describe the transverse oscillation in the first two modes Numerical simulations illustrate that there exist chaotic motion for the FGM rectangular plate.

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