Nonlinear Vibrations and Chaotic Dynamics of the Laminated Composite Piezoelectric Beam

2015 ◽  
Vol 137 (1) ◽  
Author(s):  
Minghui Yao ◽  
Wei Zhang ◽  
Zhigang Yao
Keyword(s):  
Numerical Simulation ◽  
Axial Load ◽  
Nonlinear Vibrations ◽  
Dynamical Behavior ◽  
Laminated Composite ◽  
Transverse Load ◽  
Nonlinear Phenomenon ◽  
Periodic Motions ◽  
Chaotic Motions ◽  
Piezoelectric Beam

This paper investigates the complicated dynamics behavior and the evolution law of the nonlinear vibrations of the simply supported laminated composite piezoelectric beam subjected to the axial load and the transverse load. Using the third-order shear deformation theory and the Hamilton's principle, the nonlinear governing equations of motion for the laminated composite piezoelectric beam are derived. 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 principal parametric resonance and 1:9 internal resonance. From the averaged equations obtained, numerical simulation is performed to study nonlinear vibrations of the laminated composite piezoelectric beam. The axial load, the transverse load, and the piezoelectric parameter are selected as the controlling parameters to analyze the law of complicated nonlinear dynamics of the laminated composite piezoelectric beam. Based on the results of numerical simulation, it is found that there exists the complex nonlinear phenomenon in motions of the laminated composite piezoelectric beam. In summary, numerical studies suggest that periodic motions and chaotic motions exist in nonlinear vibrations of the laminated composite piezoelectric beam. In addition, it is observed that the axial load, the transverse load and the piezoelectric parameter have significant influence on the nonlinear dynamical behavior of the beam. We can control the response of the system from chaotic motions to periodic motions by changing these parameters.

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.

Experimental Study of Chaos in a Flexible Parametrically Excited Beam

1993 ◽  
Author(s):  
C. P. Baker ◽  
M. E. Genaux ◽  
T. D. Burton
Keyword(s):  
Experimental Study ◽  
Analytical Approach ◽  
Nonlinear Vibrations ◽  
Space Dimension ◽  
Equation Of Motion ◽  
Largest Lyapunov Exponent ◽  
Periodic Motions ◽  
Attractor Dimension ◽  
Chaotic Motions ◽  
Method Of Delays

Abstract Nonlinear vibrations in an axially driven limber cantilever beam are studied experimentally to determine whether the observed aperiodic motions are chaotic, and to find the attractor dimension. Using the method of delays and time series calculations, the largest Lyapunov exponent is found to be positive, indicating that the aperiodic motions are chaotic. The correlation and embedding dimensions are computed; a phase space dimension in the range of 5–7 is found for the chaotic attractor. The system is also studied analytically, using a truncated Galerkin reduction of the planar equation of motion. It is found that this analytical approach does model the near-harmonic periodic motions of this system; however, the chaotic motions and chaotic transitions are not predicted. The model does exhibit interesting chaotic responses for non-physical values of the driving parameters.

Experimental Study on Power Generation and Dynamical Behavior of the Cantilevered Piezoelectric Beam With the Unilateral Layer Separate

2017 ◽  
Author(s):  
Wei Xia ◽  
Ming Hui Yao ◽  
Wei Zhang
Keyword(s):  
Power Generation ◽  
Dynamical Behavior ◽  
Laminated Composite ◽  
Frequency Bandwidth ◽  
Effective Frequency ◽  
Generation Efficiency ◽  
Peak Voltage ◽  
Voltage Output ◽  
Piezoelectric Beam ◽  
Composite Layers

This paper investigates the complicated dynamic behavior and power generation efficiency of the cantilevered laminated composite piezoelectric beam with the unilateral layer separate. The effect of the external excitation on the voltage output, the impacts of the layered length of composite layers and the influence of the magnetic distance on the voltage output and the effective frequency bandwidth are examined. Simultaneously, the output voltage and the effective frequency bandwidth of the traditional cantilevered laminated composite piezoelectric beam are measured experimentally to verify the developed model. The amplitude of the harmonic excitation is given the certain value and is not changed. Experimental results show that the developed structure has lower natural frequency, great voltage output and great effective frequency bandwidth when the length of the separate parts between composite layers is in the range. For the different layered lengths of the developed bistable piezoelectric beam, there exist the optimal magnetic distance and an optimal layered length, respectively. The power generation efficiency of the developed bistable piezoelectric beam is better than that of the developed monostable piezoelectric beam. When the layered length of the separate parts between composite layers is optimal, the voltage output of the piezoelectric beam has four peak voltages. In addition, the power generation efficiency of the developed structure are superior to that of the traditional one. The maximum peak voltage of this structure is 6.73 times than that of the traditional piezoelectric beam, and its effective frequency bandwidth promotes 8.4 times.

scholarly journals Complex Dynamical Behavior of a Predator-Prey System with Group Defense

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Jianglin Zhao ◽  
Min Zhao ◽  
Hengguo Yu
Keyword(s):  
Numerical Simulation ◽  
Theoretical Basis ◽  
Dynamical Behavior ◽  
Turing Instability ◽  
Equilibrium Density ◽  
Prey Refuge ◽  
Predator Prey ◽  
Prey System ◽  
Group Defense ◽  
Complex Dynamical Behavior

A diffusive predator-prey system with prey refuge is studied analytically and numerically. The Turing bifurcation is analyzed in detail, which in turn provides a theoretical basis for the numerical simulation. The influence of prey refuge and group defense on the equilibrium density and patterns of species under the condition of Turing instability is explored by numerical simulations, and this shows that the prey refuge and group defense have an important effect on the equilibrium density and patterns of species. Moreover, it can be obtained that the distributions of species are more sensitive to group defense than prey refuge. These results are expected to be of significance in exploration for the spatiotemporal dynamics of ecosystems.

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.

Vibration and Stability of an Elastic Beam Subjected to a Periodic Axial Load With Time-Dependent Displacement Excitation at Both Ends

1991 ◽  
Author(s):  
Xiao-Feng Wu ◽  
Adnan Akay
Keyword(s):  
Axial Load ◽  
Perturbation Technique ◽  
Harmonic Balance Method ◽  
Elastic Beam ◽  
Time Dependent ◽  
Transverse Load ◽  
Order Partial Differential Equation ◽  
Weighted Residual Method ◽  
Hill’S Equation ◽  
Hill's Equation

Abstract This paper concerns the transverse vibrations and stabilities of an elastic beam simultaneously subjected to a periodic axial load, a distributed transverse load, and time-dependent displacement excitations at both ends. The equation of motion derived from Bernoulli-Euler beam theory is a fourth-order partial differential equation with periodic coefficients. To obtain approximate solutions, the method of assumed-modes is used. The unknown time-dependent function in the assumed-modes method is determined by a generalized inhomogeneous Hill’s equation. The instability regions possessed by this generalized Hill’s equation are obtained by both the perturbation technique up to the second order and the harmonic balance method. The dynamic response and the corresponding spectrum of the transversely oscillating elastic beam are calculated by the weighted-residual method.

Nonlinear Vibrations of Plates: An Update of Recent Research Developments

1996 ◽  
Vol 49 (10S) ◽  
pp. S55-S62 ◽  
Author(s):  
M. Sathyamoorthy
Keyword(s):  
Nonlinear Vibrations ◽  
Laminated Composite ◽  
Composite Plates ◽  
Future Research ◽  
Research Directions ◽  
Inertia Effects ◽  
Vibration Behavior ◽  
Recent Developments ◽  
Complicating Factors ◽  
Future Research Directions

This paper comprises a survey on the nonlinear vibration analysis of plates, with emphasis on research carried out since 1987. Most of the research reviewed here deals with the effects of geometric nonlinearity on the vibration behavior of plates. Complicating factors such as material nonlinearity, geometric imperfections, transverse shear and rotatory inertia effects, and magnetic fields on the vibration behavior are included. Recent developments in the analytical and numerical methods of solution of isotropic, orthotropic as well as laminated, composite plates are presented. Experimental, analytical, and numerical investigations are included for all the cases reviewed and some general remarks are presented along with suggestions for future research directions.

Analytical Study of a Piezoelectric Frequency Up-Conversion Harvester Under Sawtooth Wave Excitation

2018 ◽  
Author(s):  
Saeed Onsorynezhad ◽  
Amin Abedini ◽  
Fengxia Wang
Keyword(s):  
Numerical Solutions ◽  
Excitation Frequency ◽  
Dynamics Analysis ◽  
Periodic Motions ◽  
Lumped Model ◽  
Sawtooth Wave ◽  
Piezoelectric Beam ◽  
Period 2 ◽  
Mass Spring ◽  
The Impact

In this work, an impact based frequency up-conversion mechanism is studied via discontinuous dynamics analysis. The mechanism consists of a moving stopper and a piezoelectric beam. The repeated free vibration of the piezoelectric beam achieved through the impaction between the stopper and the beam, With the stopper excited by a sawtooth wave. Due to the impact, the system contains complex discontinuous dynamics, hence to better understand the energy harvesting performance of the piezoelectric beam, we seek the simple periodic motions of the system. As the system parameter varies, the output voltage and power of the piezoelectric beam with periodic motions is obtained. These results were also compared with those obtained when the piezoelectric beam is directly subjected to the same sawtooth wave. The piezoelectric beam was modeled as a mass-spring-damper system, and the linear piezoelectric constitutive equations have been used to obtain the lumped model of the piezoelectric beam. In this study, numerical solutions of the generated power and voltage were obtained via discontinuous dynamics analysis. When the excitation frequency is low, the effect of frequency-up-conversion is demonstrated by comparing the generated power of two cases: piezoelectric beam excited via impact and beam directly subject to the sawtooth wave. The stable and unstable periodic motions and bifurcation trees of the impact parameters are predicted analytically versus varying excitation frequency for period-1 and period-2.

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