Multi-Pulse Chaotic Dynamics of a Cantilever Plate by Using Extended Melnikov Method

2011 ◽  
Author(s):  
Wei Zhang ◽  
Yutong Huang
Keyword(s):  
Multiple Scales ◽  
Equations Of Motion ◽  
Shear Deformation Theory ◽  
Laminated Composite ◽  
Nonlinear Behavior ◽  
Melnikov Method ◽  
Transverse Displacement ◽  
Cantilever Plate ◽  
Chaotic Motions ◽  
Extended Melnikov Method

The nonlinear dynamic behavior of a laminated composite cantilever plate is investigated in this paper. The extended Melnikov method is employed to predict the multi-pulse chaotic motions of the cantilever plate. The model is based on the wing flutter of the airplane. The cantilever plate is considered to be subjected to the in-plane and transversal excitations. The Reddy’s high-order shear deformation theory as well as von Ka´rma´n type equations are used to establish the equation of motion for the cantilever plate. Applying the Galerkin procedure to the partial differential governing equations of motion for the system, we obtain equations of transverse displacement. Then the method of multiple scales is used to obtain the averaged equations. Finally, the extended Melnikov method is used to analyze the nonlinear behavior in the cantilever plate system. The theoretical result shows that there exists multi-pulse jumping movement. The numerical results also reveal such chaotic phenomenon.

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.

Multi-Pulse Chaotic Dynamics of Eccentric Rotating Composite Laminated Circular Cylindrical Shell by Using the Extended Melnikov Method

2019 ◽  
Author(s):  
Yan Zheng ◽  
Wei Zhang ◽  
Tao Liu
Keyword(s):  
Cylindrical Shell ◽  
Normal Form ◽  
Chaotic Dynamics ◽  
Multiple Scales ◽  
Circular Cylindrical Shell ◽  
Melnikov Method ◽  
Global Dynamics ◽  
Chaotic Motions ◽  
Extended Melnikov Method ◽  
Averaged Equations

Abstract The researches of global bifurcations and chaotic dynamics for high-dimensional nonlinear systems are extremely challenging. In this paper, we study the multi-pulse orbits and chaotic dynamics of an eccentric rotating composite laminated circular cylindrical shell. The four-dimensional averaged equations are obtained by directly using the multiple scales method under the case of the 1:2 internal resonance and principal parametric resonance-1/2 subharmonic resonance. The system is transformed to the averaged equations. From the averaged equation, the theory of normal form is used to find the explicit formulas of normal form. Based on the normal form obtained, the extended Melnikov method is utilized to analyze the multi-pulse global homoclinic bifurcations and chaotic dynamics for the eccentric rotating composite laminated circular cylindrical shell. The analysis of global dynamics indicates that there exist the multi-pulse jumping orbits in the perturbed phase space of the averaged equation. From the averaged equations obtained, the chaotic motions and the Shilnikov type multi-pulse orbits of the eccentric rotating composite laminated circular cylindrical shell are found by using numerical simulation. The results obtained above mean the existence of the chaos for the Smale horseshoe sense for the eccentric rotating composite laminated circular cylindrical shell.

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.

Multipulse Chaotic Dynamics of Six-Dimensional Nonautonomous Nonlinear System for a Honeycomb Sandwich Plate

2014 ◽  
Vol 24 (11) ◽  
pp. 1450138 ◽  
Author(s):  
W. L. Hao ◽  
W. Zhang ◽  
M. H. Yao
Keyword(s):  
Nonlinear System ◽  
Rectangular Plate ◽  
Chaotic Dynamics ◽  
Equations Of Motion ◽  
Melnikov Method ◽  
Global Bifurcations ◽  
Chaotic Motions ◽  
Simply Supported ◽  
Extended Melnikov Method ◽  
Honeycomb Sandwich

This paper studies the global bifurcations and multipulse chaotic dynamics of a four-edge simply supported honeycomb sandwich rectangular plate under combined in-plane and transverse excitations. Based on the von Karman type equation for the geometric nonlinearity and Reddy's third-order shear deformation theory, the governing equations of motion are derived for the four-edge simply supported honeycomb sandwich rectangular plate. The Galerkin method is employed to discretize the partial differential equations of motion to a three-degree-of-freedom nonlinear system. The six-dimensional nonautonomous nonlinear system is simplified to a three-order standard form by using the normal form method. The extended Melnikov method is improved to investigate the six-dimensional nonautonomous nonlinear dynamical system in a mixed coordinate. The global bifurcations and multipulse chaotic dynamics of the four-edge simply supported honeycomb sandwich rectangular plate are studied by using the improved extended Melnikov method. The multipulse chaotic motions of the system are found by using numerical simulation, which further verifies the result of theoretical analysis.

The Extended Melnikov Method for Non-Autonomous Higher Dimensional Nonlinear Systems and Multi-Pulse Chaotic Dynamics of Laminated Composite Piezoelectric Plate

2009 ◽  
Author(s):  
Wei Zhang ◽  
Jun-Hua Zhang
Keyword(s):  
Thin Plate ◽  
Chaotic Dynamics ◽  
Plate Theory ◽  
Nonlinear Dynamical System ◽  
Laminated Composite ◽  
Melnikov Method ◽  
Type Equation ◽  
Rectangular Thin Plate ◽  
Chaotic Motions ◽  
Extended Melnikov Method

The global bifurcations and multi-pulse chaotic dynamics of a simply supported laminated composite piezoelectric rectangular thin plate under combined parametric and transverse excitations are investigated in this paper for the first time. The formulas of the laminated composite piezoelectric rectangular plate are derived by using the von Karman-type equation, the Reddy’s third-order shear deformation plate theory 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 laminated composite piezoelectric rectangular thin plate. The results obtained here indicate that the multi-pulse chaotic motions can occur in the laminated composite piezoelectric rectangular thin plate. Numerical simulation is also employed to find the multi-pulse chaotic motions of the laminated composite piezoelectric rectangular thin plate.

FURTHER STUDIES ON NONLINEAR OSCILLATIONS AND CHAOS OF A SYMMETRIC CROSS-PLY LAMINATED THIN PLATE UNDER PARAMETRIC EXCITATION

2006 ◽  
Vol 16 (02) ◽  
pp. 325-347 ◽  
Author(s):  
WEI ZHANG ◽  
CHUNZHI SONG ◽  
MIN YE
Keyword(s):  
Thin Plate ◽  
Parametric Excitation ◽  
Multiple Scales ◽  
Nonlinear Oscillations ◽  
Equations Of Motion ◽  
Laminated Composite ◽  
Chaotic Motions ◽  
Parametrically Excited ◽  
Simply Supported ◽  
Analysis Of Stability

In this paper, the nonlinear oscillations and chaotic dynamics of a parametrically excited simply supported symmetric cross-ply laminated composite rectangular thin plate are further investigated. Considering geometric nonlinearity and nonlinear damping, a two-degree-of-freedom nonlinear system under parametric excitation is obtained to give the nonlinear governing equations of motion for laminated composite plate subjected to in-plane load. The method of multiple scales is utilized to obtain the averaged equations that are numerically solved to obtain the steady bifurcation responses and analysis of stability for laminated composite thin plate. It is illustrated that under certain conditions laminated composite thin plate may have the multiple steady bifurcation solutions and jumping may occur. The chaotic motion of rectangular symmetric cross-ply laminated composite thin plate is also found by using numerical simulation. It is found that the occurrence of the periodic, quasi-periodic and chaotic motions for a parametrically excited four-edges simply supported rectangular symmetric cross-ply laminated composite thin plate depends on the parametric excitation.

Multi-Pulse Chaotic Motion for a Non-Autonomous Buckled Plate by Using the Extended Melnikov Method

2007 ◽  
Author(s):  
Wei Zhang ◽  
Jun-Hua Zhang ◽  
Ming-Hui Yao
Keyword(s):  
Thin Plate ◽  
Chaotic Dynamics ◽  
Melnikov Method ◽  
Type Equation ◽  
Rectangular Thin Plate ◽  
Governing Equations ◽  
Chaotic Motions ◽  
Simply Supported ◽  
Extended Melnikov Method ◽  
Two Degree Of Freedom

The multi-pulse Shilnikov orbits and chaotic dynamics for a parametrically excited, simply supported rectangular buckled thin plate are studied by using the extended Melnikov method. Based on von Karman type equation and the Galerkin’s approach, two-degree-of-freedom nonlinear system is obtained for the rectangular thin plate. The extended Melnikov method is directly applied to the non-autonomous governing equations of the thin plate. The results obtained here show that the multipulse chaotic motions can occur in the thin plate.

Bifurcations and Chaos in Forced, Coupled Pendula

2001 ◽  
Author(s):  
Kazuyuki Yagasaki
Keyword(s):  
Averaging Method ◽  
Nonlinear Behavior ◽  
Melnikov Method ◽  
Original System ◽  
Chaotic Motions ◽  
Small Perturbations ◽  
Small Oscillations ◽  
Codimension One ◽  
Averaged Systems ◽  
Direct Numerical Integration

Abstract We consider forced, coupled pendula and show that they exhibit very complicated dynamics using the averaging method and Melnikov-type techniques. First, the averaged system for small oscillations of the pendula near the hanging state is analyzed. Codimension-one and -two local bifurcations at which several non-synchronized periodic orbits and quasiperiodic orbits are born in the original system are detected. The validity of the theoretical results is demonstrated by comparison with direct numerical integration results. Moreover, chaotic motions, which result from the Shilnikov type phenomena in the averaged systems, are observed in numerical simulations. Second, the second-order averaging method is applied to small perturbations of rotary orbits with no damping and external forcing. Analyzing the averaged system, we can describe nonlinear behavior in the original system. Finally, using a generalization of Melnikov method, we prove the occurrence of many other homo-clinic phenomena, which also yield chaotic dynamics.

Analysis of Laminated Composite Plates by Local Inverse Multiquadrics Collocation Method

2014 ◽  
Vol 607 ◽  
pp. 731-734
Author(s):  
Song Xiang
Keyword(s):  
Collocation Method ◽  
Present Method ◽  
Shear Deformation Theory ◽  
Laminated Composite ◽  
Composite Plates ◽  
Shear Stresses ◽  
Transverse Displacement ◽  
Laminated Composite Plates ◽  
Sinusoidal Load ◽  
Support Domain

In present paper, deflection and stress of laminated composite plates are analyzed by a meshless local collocation method based on inverse multiquadrics radial basis function. This method approximates the governing equations based on first-order shear deformation theory using the nodes in the support domain of any data center. Transverse displacement, normal stresses, and shear stresses of the simply supported laminated composite plates under sinusoidal load are computed by the present method. The convergence characteristics are studied by several numerical examples. The present results are compared with available published results which demonstrate the accuracy and efficiency of present method.

Theoretical and Experimental Investigation of an L-Shape Beam Structure

2008 ◽  
Author(s):  
Dong-Xing Cao ◽  
Wei Zhang ◽  
Ming-Hui Yao
Keyword(s):  
Nonlinear Vibrations ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Space Station ◽  
Dynamic Responses ◽  
Flexible Beam ◽  
Beam Structure ◽  
Large Space ◽  
Chaotic Motions ◽  
Experimental Apparatus

Flexible multi-beam structures are significant components of large space station, architecture engineering and other structural systems. The understanding of the dynamic characteristics of these structures is essential for their design and control of vibrations. In this paper, the planar nonlinear vibrations and chaotic dynamics of an L-shape flexible beam structure will be investigated using theoretical and experimental methods. The L-shape beam structure considered here is composed of two flexible beams with right-angle. The governing equations of motion for the L-shape beam structure are established firstly. Then, the method of multiple scales is utilized to obtain a four-dimensional averaged equation. Numerical method is used to analyze the nonlinear dynamic responses and chaotic motions. Finally, The experimental apparatus and schemes for measuring the amplitude of nonlinear vibrations for the L-shape beam structure are introduced briefly. Then, the detailed analysis for experimental data and signals which represent the nonlinear responses of the beam structure are given.

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