scholarly journals Multipulse Homoclinic Orbits and Chaotic Dynamics of a Reinforced Composite Plate with Carbon Nanotubes

2020 ◽  
Vol 2020 ◽  
pp. 1-18
Author(s):  
Fengxian An ◽  
Fangqi Chen ◽  
Xiaoxia Bian ◽  
Li Zhang
Keyword(s):  
Carbon Nanotubes ◽  
Normal Form ◽  
Composite Plate ◽  
Chaotic Dynamics ◽  
Homoclinic Orbits ◽  
Phase Method ◽  
Chaotic Motions ◽  
Normal Form Theory ◽  
Reinforced Composite ◽  
Averaged Equations

The multipulse homoclinic orbits and chaotic dynamics of a reinforced composite plate with the carbon nanotubes (CNTs) under combined in-plane and transverse excitations are studied in the case of 1 : 1 internal resonance. The method of multiple scales is adopted to derive the averaged equations. From the averaged equations, the normal form theory is applied to reduce the equations to a simpler normal form associated with a double zero and a pair of pure imaginary eigenvalues. The energy-phase method proposed by Haller and Wiggins is utilized to examine the global bifurcations and chaotic dynamics of the CNT-reinforced composite plate. The analytical results demonstrate that the multipulse Shilnikov-type homoclinic orbits and chaotic motions exist in the system. Homoclinic trees are constructed to illustrate the repeated bifurcations of multipulse solutions. In order to verify the theoretical results, numerical simulations are given to show the multipulse Shilnikov-type chaotic motions in the CNT-reinforced composite plate. The results obtained here imply that the motion is chaotic in the sense of the Smale horseshoes for the CNT-reinforced composite 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.

Multipulse Orbits and Chaotic Dynamics of an Aero-Elastic FGP Plate Under Parametric and Primary Excitations

2017 ◽  
Vol 27 (04) ◽  
pp. 1750050 ◽  
Author(s):  
F. X. An ◽  
F. Q. Chen
Keyword(s):  
Normal Form ◽  
Numerical Simulations ◽  
Chaotic Dynamics ◽  
Melnikov Method ◽  
Functionally Graded ◽  
Chaotic Motions ◽  
Averaged Equations ◽  
Functionally Graded Piezoelectric ◽  
Pure Imaginary Eigenvalues ◽  
Explicit Expressions

The multipulse global bifurcations and chaotic dynamics of a simply supported Functionally Graded Piezoelectric (FGP) rectangular plate with bonded piezoelectric layer are investigated with the case of 1:2 internal resonance and primary parametric resonance. Based on the averaged equations obtained, the theory of normal form is utilized to obtain the explicit expressions of normal form with a double zero and a pair of pure imaginary eigenvalues. According to the explicit expressions of normal form, the extended Melnikov method developed by Camassa et al. is employed to study the Shilnikov-type multipulse homoclinic bifurcations and chaotic dynamics of the aero-elastic FGP plate. The analytical results indicate that there exists the Shilnikov-type multipulse chaotic dynamics for the FGP plate. Numerical simulations are presented to show that for the FGP plate, the Shilnikov-type multipulse chaotic motions can occur. The influence of the in-plane excitation and the piezoelectric voltage excitation to the system dynamic behaviors is also discussed by numerical simulations. The results obtained here imply the existence of chaos in the sense of the Smale horseshoes for the FGP plate.

Multi-Pulse Chaotic Dynamics of the Cantilevered Pipe Conveying Pulsating Fluid

2009 ◽  
Author(s):  
Ming-Hui Yao ◽  
Wei Zhang ◽  
Dong-Xing Cao
Keyword(s):  
Normal Form ◽  
Chaotic Dynamics ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Phase Method ◽  
Global Dynamics ◽  
Pipe Material ◽  
Chaotic Motions ◽  
Cantilevered Pipe ◽  
Pulsating Fluid

The multi-pulse orbits and chaotic dynamics of the cantilevered pipe conveying pulsating fluid with harmonic external force are studied in detail. The nonlinear geometric deformation of the pipe and the Kelvin constitutive relation of the pipe material are considered. The nonlinear governing equations of motion for the cantilevered pipe conveying pulsating fluid are determined by using Hamilton principle. The four-dimensional averaged equation under the case of principle parameter resonance, 1/2 subharmonic resonance and 1:2 internal resonance and primary parametric resonance is obtained by directly using the method of multiple scales and Galerkin approach to the partial differential governing equation of motion for the cantilevered pipe. The system is transformed to the averaged equation. From the averaged equation, the theory of normal form is used to find the explicit formulas of normal form. Based on normal form obtained, the energy phase method is utilized to analyze the multi-pulse global bifurcations and chaotic dynamics for the cantilevered pipe conveying pulsating fluid. 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 cantilevered pipe are found by using numerical simulation. The results obtained above mean the existence of the chaos for the Smale horseshoe sense for the pulsating fluid conveying cantilevered pipe.

Chaotic Dynamics in a Higher-Dimensional Nonlinear System for a Composite Laminated Plate

2009 ◽  
Author(s):  
Wei Zhang ◽  
Mei-juan Gao
Keyword(s):  
Numerical Simulation ◽  
Nonlinear System ◽  
Normal Form ◽  
Chaotic Dynamics ◽  
Phase Method ◽  
Laminated Plate ◽  
Internal Resonances ◽  
Chaotic Motions ◽  
Composite Laminated Plate ◽  
Higher Dimensional

In this paper, we first analyze the chaotic dynamics of a higher-dimensional nonlinear system for a composite laminated plate in the case of 1:3:3 internal resonances with the theory of normal form and the energy-phase method. The theory of normal form is used to obtain the simpler normal form of the system. The energy-phase method is employed to analyze the multi-pulse chaotic dynamics of the higher-dimensional nonlinear system for a composite laminated plate. Moreover, the numerical simulation is performed to find the multi-pulse chaotic motion of the composite laminated plate. The global theory analysis and the results of numerical simulation demonstrate that the existence of the periodic motions and chaotic motions with the jumping phenomena in the composite laminated plate.

Multi-Pulse Chaotic Dynamics of Circular Mesh Antenna with 1:2 Internal Resonance

2017 ◽  
Vol 09 (04) ◽  
pp. 1750060 ◽  
Author(s):  
Y. Sun ◽  
W. Zhang ◽  
M. H. Yao
Keyword(s):  
Internal Resonance ◽  
Chaotic Dynamics ◽  
Circular Cylindrical Shell ◽  
Domain Of Attraction ◽  
Homoclinic Orbits ◽  
Energy Difference ◽  
Phase Method ◽  
Chaotic Motions ◽  
Difference Function ◽  
First Time

The multi-pulse homoclinic orbits and chaotic dynamics of an equivalent circular cylindrical shell for the circular mesh antenna are investigated in the case of 1:2 internal resonance in this paper for the first time. Applying the method of averaging, the four-dimensional averaged equation in the Cartesian form is obtained. The theory of normal form is used to reduce the averaged equation to a simpler form. Based on the simplified system, the energy phase method is employed to investigate the homoclinic bifurcations and the Shilnikov type multi-pulse chaotic dynamics. First, the energy difference function and the zeroes of the energy difference function are obtained. Then, the existence of the Shilnikov type multi-pulse orbits is determined. The homoclinic trees are depicted to describe the relationship among the layers diameter, the pulse numbers and the phase shift. Finally, we need to verify the condition which makes sure that any multi-pulse orbit departing from a slow sink comes back to the domain of attraction of one of the sinks. The results obtained here show the existence of the Shilnikov type multi-pulse chaotic motions of the circular mesh antenna. Numerical simulations are used to find multi-pulse chaotic motions. The results of the theoretical analysis are in qualitative agreement with the results obtained using numerical simulation.

SHILNIKOV-TYPE MULTIPULSE ORBITS AND CHAOTIC DYNAMICS OF A PARAMETRICALLY AND EXTERNALLY EXCITED RECTANGULAR THIN PLATE

2007 ◽  
Vol 17 (03) ◽  
pp. 851-875 ◽  
Author(s):  
M. H. YAO ◽  
W. ZHANG
Keyword(s):  
Numerical Simulation ◽  
Normal Form ◽  
Thin Plate ◽  
Chaotic Dynamics ◽  
Multiple Scales ◽  
Dynamical Analysis ◽  
Phase Method ◽  
Rectangular Thin Plate ◽  
Chaotic Motions ◽  
Parametric And External Excitations

The Shilnikov-type multipulse orbits and chaotic dynamics for a simply supported rectangular thin plate under combined parametric and external excitations are studied in this paper for the first time. The rectangular thin plate is subjected to spatially and temporally varying transversal and in-plane excitations, simultaneously. The formulas of the rectangular thin plate are derived from the von Kármán equation and Galerkin's method. The method of multiple scales is used to find the averaged equation in the case of 1:2 internal resonance. Based on the averaged equation, the theory of normal form is used to obtain the explicit expressions of normal form associated with a double zero and a pair of purely imaginary eigenvalues using the Maple program. The dissipative version of the energy-phase method is utilized to analyze the multipulse global bifurcations and chaotic dynamics of a parametrically and externally excited rectangular thin plate. The global dynamical analysis indicates that there exist the multipulse jumping orbits in the perturbed phase space of the averaged equation for a parametrically and externally excited rectangular thin plate. These results show that the chaotic motions of the multipulse Shilnikov-type can occur for a parametrically and externally excited rectangular thin plate. Numerical simulation results are presented to verify the analytical predictions. It is also found from the results of numerical simulation that the Shilnikov-type multipulse orbits exist for a parametrically and externally excited thin plate.

Shilnikov Type Multi-Pulse Orbits of Functionally Graded Materials Rectangular Plate

2010 ◽  
Author(s):  
Wei Zhang ◽  
Ming-Hui Yao ◽  
Dong-Xing Cao
Keyword(s):  
Normal Form ◽  
Rectangular Plate ◽  
Functionally Graded Materials ◽  
Chaotic Dynamics ◽  
Functionally Graded ◽  
Phase Method ◽  
Global Dynamics ◽  
Graded Materials ◽  
Chaotic Motions ◽  
Simply Supported

Multi-pulse chaotic dynamics of a simply supported functionally graded materials (FGMs) rectangular plate is investigated in this paper. The FGMs rectangular plate is subjected to the transversal and in-plane excitations. The properties of material are graded in the direction of thickness. Based on Reddy’s third-order shear deformation plate theory, the nonlinear governing equations of motion for the FGMs plate are derived by using the Hamilton’s principle. The four-dimensional averaged equation under the case of 1:2 internal resonance, primary parametric resonance and 1/2-subharmonic resonance is obtained by directly using the asymptotic perturbation method and Galerkin approach to the partial differential governing equation of motion for the FGMs rectangular plate. The system is transformed to the averaged equation. From the averaged equation, the theory of normal form is used to find the explicit formulas of normal form. Based on normal form obtained, the energy phase method is utilized to analyze the multi-pulse global bifurcations and chaotic dynamics for the FGMs rectangular plate. 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 FGMs rectangular plate are found by using numerical simulation. The results obtained above mean the existence of the chaos for the Smale horseshoe sense for the simply supported FGMs rectangular plate.

MULTIPULSE SHILNIKOV ORBITS AND CHAOTIC DYNAMICS FOR NONLINEAR NONPLANAR MOTION OF A CANTILEVER BEAM

2005 ◽  
Vol 15 (12) ◽  
pp. 3923-3952 ◽  
Author(s):  
MINGHUI YAO ◽  
WEI ZHANG
Keyword(s):  
Phase Space ◽  
Normal Form ◽  
Cantilever Beam ◽  
Parametric Resonance ◽  
Chaotic Dynamics ◽  
Phase Portraits ◽  
Phase Method ◽  
Global Dynamics ◽  
Governing Equations ◽  
Chaotic Motions

The multipulse Shilnikov orbits and chaotic dynamics for the nonlinear nonplanar oscillations of a cantilever beam are studied in this paper for the first time. The cantilever beam studied here is subjected to a harmonic axial excitation and two transverse excitations at the free end. The nonlinear governing equations of nonplanar motion with parametric and external excitations are obtained. The Galerkin procedure is applied to the partial differential governing equations to obtain a two-degree-of-freedom nonlinear system under combined parametric and forcing excitations. The resonant case considered here is principal parametric resonance-1/2 subharmonic resonance for the first mode and fundamental parametric resonance-primary resonance for the second mode. The parametrically and externally excited system is transformed to the averaged equation by using the method of multiple scales. From the averaged equation, the theory of normal form is used to find their explicit formulas. Based on normal form obtained above, the dissipative version of the energy-phase method is utilized to analyze the multipulse global bifurcations and chaotic dynamics for the nonlinear nonplanar oscillations of the cantilever beam. The energy-phase method is further improved to ensure the equivalence of topological structure for the phase portraits. The analysis of global dynamics indicates that there exist the multipulse jumping orbits in the perturbed phase space of the averaged equation for the nonlinear nonplanar oscillations of the cantilever beam. These results show that the multipulse Shilnikov type chaotic motions can occur for the nonlinear nonplanar oscillations of the cantilever beam. Numerical simulations are given to verify the analytical predictions. It is also found from the results of numerical simulation in three-dimensional phase space that the multipulse Shilnikov type orbits exist for the nonlinear nonplanar oscillations of the cantilever beam.

Mulit-Pulse Orbits and Chaotic Motion of Composite Laminated Plates

2008 ◽  
Author(s):  
Xiangying Guo ◽  
Wei Zhang ◽  
Ming-Hui Yao
Keyword(s):  
Numerical Simulation ◽  
Normal Form ◽  
Differential Equations ◽  
Thin Plate ◽  
Equations Of Motion ◽  
Laminated Plates ◽  
Phase Method ◽  
Rectangular Thin Plate ◽  
Partial Differential ◽  
Chaotic Motions

This paper presents an analysis on the nonlinear dynamics and multi-pulse chaotic motions of a simply-supported symmetric cross-ply composite laminated rectangular thin plate with the parametric and forcing excitations. Firstly, based on the Reddy’s three-order shear deformation plate theory and the model of the von Karman type geometric nonlinearity, the nonlinear governing partial differential equations of motion for the composite laminated rectangular thin plate are derived by using the Hamilton’s principle. Then, using the second-order Galerkin discretization approach, the partial differential governing equations of motion are transformed to nonlinear ordinary differential equations. The case of the primary parametric resonance and 1:1 internal resonance is considered. Four-dimensional averaged equation is obtained by using the method of multiple scales. From the averaged equation obtained here, the theory of normal form is used to give the explicit expressions of normal form. Based on normal form, the energy phase method is utilized to analyze the global bifurcations and multi-pulse chaotic dynamics of the composite laminated rectangular thin plate. The results obtained above illustrate the existence of the chaos for the Smale horseshoe sense in a parametrical and forcing excited composite laminated thin plate. The chaotic motions of the composite laminated rectangular thin plate are also found by using numerical simulation. The results of numerical simulation also indicate that there exist different shapes of the multi-pulse chaotic motions for the composite laminated rectangular thin plate.

A New Approach for Computing the Normal Forms of High Dimensional Nonlinear Systems

2010 ◽  
Author(s):  
Shuping Chen ◽  
Wei Zhang ◽  
Minghui Yao
Keyword(s):  
Nonlinear Systems ◽  
Normal Form ◽  
Cantilever Beam ◽  
Normal Forms ◽  
Nonlinear Differential Equations ◽  
Real Life ◽  
High Dimensional ◽  
Computation Method ◽  
Normal Form Theory ◽  
Averaged Equations

Normal form theory is very useful for direct bifurcation and stability analysis of nonlinear differential equations modeled in real life. This paper develops a new computation method for obtaining a significant refinement of the normal forms for high dimensional nonlinear systems. The method developed here uses the lower order nonlinear terms in the normal form for the simplifications of higher order terms. In the theoretical model for the nonplanar nonlinear oscillation of a cantilever beam, the computation method is applied to compute the coefficients of the normal forms for the case of two non-semisimple double zero eigenvalues. The normal forms of the averaged equations and their coefficients for non-planar non-linear oscillations of the cantilever beam under combined parametric and forcing excitations are calculated.

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