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.

Multi-Pulse Orbits With a Melnikov Method and Chaotic Dynamics in Motion of Composite Laminated Rectangular Thin Plate

2008 ◽  
Author(s):  
Ming-Hui Yao ◽  
Wei Zhang ◽  
Xiang-Ying Guo ◽  
Dong-Xing Cao
Keyword(s):  
Numerical Simulation ◽  
Normal Form ◽  
Differential Equations ◽  
Thin Plate ◽  
Chaotic Dynamics ◽  
Equations Of Motion ◽  
Melnikov 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 extended Melnikov 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.

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.

Periodic and Chaotic Oscillations of Composite Laminated Thin Plate With Third-Order Shear Deformation

2009 ◽  
Author(s):  
W. Zhang ◽  
X. Y. Guo
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Shear Deformation ◽  
Thin Plate ◽  
Nonlinear Oscillations ◽  
Equations Of Motion ◽  
Rectangular Thin Plate ◽  
Third Order ◽  
Partial Differential ◽  
Chaotic Motions

An analysis on the nonlinear oscillations and chaotic dynamics is presented for a simply-supported symmetric cross-ply composite laminated rectangular thin plate with parametric and forcing excitations. Based on the Reddy’s third-order shear deformation plate theory and the von Karman type equation, the nonlinear governing partial differential equations of motion for the composite laminated rectangular thin plate are derived by using the Hamilton’s principle. The Galerkin method is utilized to discretize the governing partial differential equations of motion to a three-degree-of-freedom nonlinear system including the cubic nonlinear terms. The case of 1:3:3 internal resonance and fundamental parametric resonance, 1/3 subharmonic resonance is considered. The method of multiple scales is employed to obtain the averaged equation. The stability analysis is given for the steady-state solutions of the averaged equation. The Numerical method is used to investigate the periodic and chaotic motions of the composite laminated rectangular thin plate. The results of numerical simulation demonstrate that there exist different kinds of periodic and chaotic motions of the composite laminated rectangular thin plate under certain conditions.

Nonlinear Vibration Response Analysis on a Composite Plate Reinforced With Carbon Nanotubes

2012 ◽  
Author(s):  
Xiang-Ying Guo ◽  
Wei Zhang ◽  
Qian Wang
Keyword(s):  
Carbon Nanotubes ◽  
Differential Equations ◽  
Nonlinear Vibration ◽  
Thin Plate ◽  
Composite Plate ◽  
Laminated Plates ◽  
Vibration Response ◽  
Response Analysis ◽  
Rectangular Thin Plate ◽  
Chaotic Motions

In order to compare nonlinear vibration response of the different enabled materials in the matrix of composites, the nonlinear vibrations of a composite plate reinforced with carbon nanotubes (CNT) are studied. In this paper, the carbon nanotubes are supposed to be long fibers. The nonlinear governing partial differential equations of motion for the composite rectangular thin plate are derived by using the Reddy’s third-order shear deformation plate theory, the von Karman type equation and the Hamilton’s principle. Then, the governing equations get reduced to ordinary differential equations in thickness direction with variable coefficients and these are solved by the Galerkin method. The case of 1:1 internal resonance is considered. The asymptotic perturbation method is employed to obtain the four-dimensional averaged equations. The numerical method is used to investigate the periodic and chaotic motions of the composite rectangular thin plate reinforced with carbon nanotubes. The results of numerical simulation demonstrate that there exist different kinds of periodic and chaotic motions of the composite plate under certain conditions. At last, the nonlinear vibration responses of the plate are compared with the same responses of angle-ply composite laminated plates.

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 Kind of Energy Transfer From the High-Frequency to Low-Frequency Mode in a Composite Laminated Plate

2010 ◽  
Author(s):  
Wei Zhang ◽  
Xiang-Ying Guo ◽  
Qian Wang ◽  
Cui-Cui Liu ◽  
Yun-cheng He
Keyword(s):  
Numerical Simulation ◽  
Natural Frequency ◽  
Thin Plate ◽  
High Frequency ◽  
Low Frequency ◽  
Frequency Mode ◽  
Rectangular Thin Plate ◽  
Chaotic Motions ◽  
Simply Supported ◽  
Low Frequency Mode

This paper focuses on the analysis on a new kind of nonlinear resonant motion with the low-frequency large-amplitude, which can be induced by the high-frequency small-amplitude mode through the mechanism of modulation of amplitude and phase. The system investigated is a simply supported symmetric cross-ply composite laminated rectangular thin plate subjected to parametric excitations. Experimental research has been carried out for the first time. The test plate was excited near the first natural frequency with parametric forces and the above mentioned high-to-low frequency mode has been observed, whose frequency is extremely lower than the first natural frequency. Theoretical job goes to analysis the above phenomenon accordingly. Based on the Reddy’s third-order shear deformation plate theory and the von Karman type equation, the nonlinear governing equations of the simply supported symmetric cross-ply composite laminated rectangular thin plate subjected to parametric excitations are formulated. The Galerkin method is utilized to discretize the governing partial differential equations into a two-degree-of-freedom nonlinear system. Numerical simulation is conducted to investigate this non-autonomous system subsequently. The results of numerical simulation demonstrate that there is a qualitative agreement between the experimental observation and the theoretical result. Besides, the multi-pulse chaotic motions are also reported in numerical simulations.

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.

scholarly journals Multipulse Chaotic Dynamics for a Laminated Composite Piezoelectric Plate

2011 ◽  
Vol 2011 ◽  
pp. 1-11 ◽  
Author(s):  
J. H. Zhang ◽  
W. Zhang
Keyword(s):  
Thin Plate ◽  
Chaotic Dynamics ◽  
Piezoelectric Plate ◽  
Equations Of Motion ◽  
Laminated Composite ◽  
Phase Portraits ◽  
Rectangular Thin Plate ◽  
Governing Equations ◽  
Chaotic Motions ◽  
Simply Supported

We investigate the global bifurcations and multipulse chaotic dynamics of a simply supported laminated composite piezoelectric rectangular thin plate under combined parametric and transverse excitations. We analyze directly the nonautonomous governing equations of motion for the laminated composite piezoelectric rectangular thin plate. The results obtained here indicate that the multipulse chaotic motions can occur in the laminated composite piezoelectric rectangular thin plate. Numerical simulations including the phase portraits and Lyapunov exponents are used to analyze the complex nonlinear dynamic behaviors of the laminated composite piezoelectric rectangular thin plate.

Disk Position Nonlinearity Effects on the Chaotic Behavior of Rotating Flexible Shaft-Disk Systems

2012 ◽  
Vol 28 (3) ◽  
pp. 513-522 ◽  
Author(s):  
H. M. Khanlo ◽  
M. Ghayour ◽  
S. Ziaei-Rad
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Dynamic Behavior ◽  
Chaotic Behavior ◽  
Equations Of Motion ◽  
Beam Theory ◽  
Disk System ◽  
Partial Differential ◽  
Rayleigh Beam ◽  
Flexible Shaft

AbstractThis study investigates the effects of disk position nonlinearities on the nonlinear dynamic behavior of a rotating flexible shaft-disk system. Displacement of the disk on the shaft causes certain nonlinear terms which appears in the equations of motion, which can in turn affect the dynamic behavior of the system. The system is modeled as a continuous shaft with a rigid disk in different locations. Also, the disk gyroscopic moment is considered. The partial differential equations of motion are extracted under the Rayleigh beam theory. The assumed modes method is used to discretize partial differential equations and the resulting equations are solved via numerical methods. The analytical methods used in this work are inclusive of time series, phase plane portrait, power spectrum, Poincaré map, bifurcation diagrams, and Lyapunov exponents. The effect of disk nonlinearities is studied for some disk positions. The results confirm that when the disk is located at mid-span of the shaft, only the regular motion (period one) is observed. However, periodic, sub-harmonic, quasi-periodic, and chaotic states can be observed for situations in which the disk is located at places other than the middle of the shaft. The results show nonlinear effects are negligible in some cases.

Nonlinear Dynamics and Application of the Four Dimensional Autonomous Hyper-Chaotic System

2014 ◽  
Author(s):  
Ge Kai ◽  
Wei Zhang
Keyword(s):  
Nonlinear Dynamics ◽  
Numerical Simulation ◽  
Differential Equations ◽  
Chaotic System ◽  
Dynamical Behavior ◽  
Three Dimensional ◽  
Phase Portraits ◽  
State Variables ◽  
Chaotic Motions ◽  
Finance System

In this paper, we establish a dynamic model of the hyper-chaotic finance system which is composed of four sub-blocks: production, money, stock and labor force. We use four first-order differential equations to describe the time variations of four state variables which are the interest rate, the investment demand, the price exponent and the average profit margin. The hyper-chaotic finance system has simplified the system of four dimensional autonomous differential equations. According to four dimensional differential equations, numerical simulations are carried out to find the nonlinear dynamics characteristic of the system. From numerical simulation, we obtain the three dimensional phase portraits that show the nonlinear response of the hyper-chaotic finance system. From the results of numerical simulation, it is found that there exist periodic motions and chaotic motions under specific conditions. In addition, it is observed that the parameter of the saving has significant influence on the nonlinear dynamical behavior of the four dimensional autonomous hyper-chaotic system.

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