Multi-Pulse Heteroclinic Orbits With a Melnikov Method and Chaotic Dynamics in Motion of Parametrically Excited Viscoelastic Moving Belt

2007 ◽  
Author(s):  
Ming-Hui Yao ◽  
Wei Zhang ◽  
Dong-Xing Cao
Keyword(s):  
Normal Form ◽  
Chaotic Dynamics ◽  
Multiple Scales ◽  
Melnikov Method ◽  
Equation Of Motion ◽  
Constitutive Law ◽  
Heteroclinic Orbits ◽  
Global Dynamics ◽  
Chaotic Motions ◽  
Parametrically Excited

The multi-pulse heteroclinic orbits and chaotic dynamics of a parametrically excited viscoelastic moving belt are studied in detail. Using Kelvin-type viscoelastic constitutive law, the equation of motion for viscoelastic moving belt with the external damping and parametric excitation are determined. The four-dimensional averaged equation under the case of 1:1 internal resonance and primary parametric resonance is obtained by directly using the method of multiple scales and Galerkin’s approach to the partial differential governing equation of motion for viscoelastic moving belt. 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, an extension of the Melnikov method is utilized to analyze the multi-pulse global bifurcations and chaotic dynamics for a parametrically excited viscoelastic moving belt. 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 heteroclinic orbits of viscoelastic moving belts are found by using numerical simulation. The results obtained above mean the existence of the chaos for the Smale horseshoe sense for a parametrically excited viscoelastic moving belt.

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.

Multi-Pulse Heteroclinic Orbits and Chaotic Dynamics of Laminated Composite Piezoelectric Rectangular Plate

2009 ◽  
Author(s):  
Ming-Hui Yao ◽  
Wei Zhang ◽  
Dong-Xing Cao
Keyword(s):  
Normal Form ◽  
Chaotic Dynamics ◽  
Multiple Scales ◽  
Laminated Composite ◽  
Heteroclinic Orbits ◽  
Global Dynamics ◽  
Rectangular Plates ◽  
Internal Resonances ◽  
Chaotic Motions ◽  
Simply Supported

The multi-pulse orbits and chaotic dynamics of the simply supported laminated composite piezoelectric rectangular plates under combined parametric excitation and transverse loads are studied in detail. It is assumed that different layers are perfectly bonded to each other with piezoelectric actuator patches embedded. The nonlinear equations of motions for the laminated composite piezoelectric rectangular plates are derived from von Karman-type equation and third-order shear deformation laminate theory of Reddy. The four-dimensional averaged equation under the case of primary parametric resonance and 1:2 internal resonances is obtained by directly using the method of multiple scales and Galerkin approach to the partial differential governing equation of motion for the laminated composite piezoelectric rectangular plates. 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 extended Melnikov method is utilized to analyze the multi-pulse global bifurcations and chaotic dynamics for the laminated composite piezoelectric rectangular plates. 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 laminated composite piezoelectric rectangular plates 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 laminated composite piezoelectric rectangular plates.

MULTI-PULSE HETEROCLINIC ORBITS AND CHAOTIC MOTIONS IN A PARAMETRICALLY EXCITED VISCOELASTIC MOVING BELT

2013 ◽  
Vol 23 (01) ◽  
pp. 1350001 ◽  
Author(s):  
MINGHUI YAO ◽  
WEI ZHANG ◽  
JEAN W. ZU
Keyword(s):  
Normal Form ◽  
Numerical Simulations ◽  
Chaotic Dynamics ◽  
Dissipative System ◽  
Melnikov Method ◽  
Heteroclinic Orbits ◽  
Unperturbed System ◽  
Chaotic Motions ◽  
Parametrically Excited ◽  
Extended Melnikov Method

This paper investigates the multi-pulse global heteroclinic bifurcations and chaotic dynamics for nonlinear, nonplanar oscillations of the parametrically excited viscoelastic moving belts by using an extended Melnikov method in the resonant case. Applying the method of multiple scales, the Galerkin's approach and the theory of normal form, the explicit normal form is obtained for the case of 1:1 internal resonance and primary parametric resonance. Studies are performed for the heteroclinic bifurcations of the unperturbed system and for the characteristics of the hyperbolic dynamics of the dissipative system, respectively. The extended Melnikov method is used to investigate the Shilnikov type multi-pulse bifurcations and chaotic dynamics of the viscoelastic moving belt. Based on the investigation, the geometric structure of the multi-pulse orbits is described in the four-dimensional phase space. Numerical simulations show that the Shilnikov type multi-pulse chaotic motions can occur. Furthermore, numerical simulations lead to the discovery of the new shapes of chaotic motion. Overall, both theoretical and numerical studies suggest that chaos for the Smale horseshoe sense in motion exists.

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.

A New Multi-Pulse Chaotic Motion for Parametrically Excited Viscoelastic Axially Moving String

2005 ◽  
Author(s):  
Wei Zhang ◽  
Ming-Hui Yao ◽  
Li-Lai Bai
Keyword(s):  
Chaotic Dynamics ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Three Dimensional ◽  
Constitutive Law ◽  
Chaotic Motions ◽  
Parametrically Excited ◽  
Dimensional Phase Space ◽  
Axially Moving ◽  
First Time

In this paper, the Shilnikov type multi-pulse orbits and chaotic dynamics of parametrically excited viscoelastic moving string are studied in detail. Using Kelvin-type viscoelastic constitutive law, the equation of motion for viscoelastic moving string with the external damping and parametric excitation is given. The four-dimensional averaged equation under primary parametric resonance is obtained by directly using the method of multiple scales and Galerkin’s approach to the partial differential governing equation of viscoelastic moving string. The Shilnikov type multi-pulse chaotic motions of viscoelastic moving string are also found by using numerical simulation. A new phenomenon on the multi-pulse jumping orbits and a new strange attractor are observed from three-dimensional phase space for the first time.

Multi-Pulse Chaotic Dynamics of Laminated Composite Piezoelectric Rectangular Plate

2009 ◽  
Author(s):  
Wei Zhang ◽  
Ming-Hui Yao ◽  
Dong-Xing Cao
Keyword(s):  
Normal Form ◽  
Chaotic Dynamics ◽  
Multiple Scales ◽  
Laminated Composite ◽  
Type Equation ◽  
Global Dynamics ◽  
Rectangular Plates ◽  
Internal Resonances ◽  
Chaotic Motions ◽  
Simply Supported

The multi-pulse orbits and chaotic dynamics of the simply supported laminated composite piezoelectric rectangular plates under combined parametric excitation and transverse loads are studied in detail. It is assumed that different layers are perfectly bonded to each other with piezoelectric actuator patches embedded. The nonlinear equations of motions for the laminated composite piezoelectric rectangular plates are derived from von Karman-type equation and third-order shear deformation laminate theory of Reddy. The four-dimensional averaged equation under the case of primary parametric resonance and 1:2 internal resonances is obtained by directly using the method of multiple scales and Galerkin approach to the partial differential governing equation of motion for the laminated composite piezoelectric rectangular plates. 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 extended Melnikov method is utilized to analyze the multi-pulse global bifurcations and chaotic dynamics for the laminated composite piezoelectric rectangular plates. 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 laminated composite piezoelectric rectangular plates 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 laminated composite piezoelectric rectangular plates.

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.

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.

Global Dynamics of an Elliptically Excited Pendulum Model

2020 ◽  
Vol 30 (12) ◽  
pp. 2050166
Author(s):  
Liangqiang Zhou ◽  
Fangqi Chen
Keyword(s):  
System Parameter ◽  
Melnikov Method ◽  
Dynamic Responses ◽  
Heteroclinic Orbits ◽  
Global Dynamics ◽  
Uniqueness Of Solutions ◽  
Chaotic Motions ◽  
Text Type ◽  
Subharmonic Bifurcations ◽  
Pendulum Model

Using both analytical and numerical methods on the global dynamics, including the existence and uniqueness of solutions, subharmonic bifurcations and dynamic responses, of an elliptically excited pendulum model are investigated in this paper. The heteroclinic orbits, as well as periodic orbits with [Formula: see text] and [Formula: see text] types of unperturbed systems are obtained analytically. Chaotic vibrations arising from heteroclinic intersections are studied by means of the Melnikov method. The critical curves separating the chaotic and nonchaotic regions are plotted for different system parameters. The chaotic feature on the system parameter [Formula: see text], named the ratio between the horizontal and the vertical diameter of the upright ellipse traced out by the pivot during each period, is discussed in detail. The conditions for subharmonic bifurcations with the [Formula: see text] type or the [Formula: see text] type are also presented with the subharmonic Melnikov method. It is proved rigorously that the system can undergo chaotic motions through finite subharmonic bifurcations with the [Formula: see text] type. In addition, chaotic motions can occur through infinite subharmonic bifurcations with the [Formula: see text] type. An interesting dynamical phenomenon, i.e. “controllable frequency”, which decreases monotonically with the system parameter [Formula: see text], is presented. A number of related numerical simulations are given to confirm the analytical results.

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.

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