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.

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 Vibrations of a Hinged Beam Including Nonlinear Inertia Effects

1973 ◽  
Vol 40 (1) ◽  
pp. 121-126 ◽  
Author(s):  
S. Atluri
Keyword(s):  
Nonlinear Equation ◽  
Nonlinear Vibrations ◽  
Initial Conditions ◽  
Equation Of Motion ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Nonlinear Ordinary Differential Equations ◽  
Inertia Effects ◽  
Modal Expansion ◽  
General Response

This investigation treats the large amplitude transverse vibration of a hinged beam with no axial restraints and which has arbitrary initial conditions of motion. Nonlinear elasticity terms arising from moderately large curvatures, and nonlinear inertia terms arising from longitudinal and rotary inertia of the beam are included in the nonlinear equation of motion. Using a Galerkin variational method and a modal expansion, the problem is reduced to a system of coupled nonlinear ordinary differential equations which are solved for arbitrary initial conditions, using the perturbation procedure of multiple-time scales. The general response and frequency-amplitude relations are derived theoretically. Comparison with previously published results is made.

Dynamics of a Continuous System With Clearance and Motion Limiting Stops

1999 ◽  
Author(s):  
P. Metallidis ◽  
S. Natsiavas
Keyword(s):  
Piecewise Linear ◽  
Research Work ◽  
Continuous System ◽  
Continuous Model ◽  
Equation Of Motion ◽  
Model System ◽  
Periodic Motions ◽  
System Parameters ◽  
Rigid Rods ◽  
The Stability

Abstract The present study generalises previous research work on the dynamics of discrete oscillators with piecewise linear characteristics and investigates the response of a continuous model system with clearance and motion-limiting constraints. More specifically, in the first part of this work, an analysis is presented for determining exact periodic response of a periodically excited deformable rod, whose motion is constrained by a flexible obstacle. This methodology is based on the exact solution form obtained within response intervals where the system parameters remain constant and its behavior is governed by a linear equation of motion. The unknowns of the problem are subsequently determined by imposing an appropriate set of periodicity and matching conditions. The analytical part is complemented by a suitable method for determining the stability properties of the located periodic motions. In the second part of the study, the analysis is applied to several cases in order to investigate the effect of the system parameters on its dynamics. Special emphasis is placed on comparing these results with results obtained for similar but rigid rods. Finally, direct integration of the equation of motion in selected areas reveals the existence of motions, which are more complicated than the periodic motions determined analytically.

Dynamics of a Simplified van der Pol Oscillator Revisited

2007 ◽  
Author(s):  
Albert C. J. Luo ◽  
Arun Rajendran
Keyword(s):  
Bifurcation Analysis ◽  
Local Stability ◽  
Van Der Pol Oscillator ◽  
Nonsmooth Dynamics ◽  
Mapping Technique ◽  
Periodic Motions ◽  
Van Der Pol ◽  
Chaotic Motions ◽  
Mapping Structures ◽  
Stability And Bifurcation Analysis

In this paper, the dynamic characteristics of a simplified van der Pol oscillator are investigated. From the theory of nonsmooth dynamics, the structures of periodic and chaotic motions for such an oscillator are developed via the mapping technique. The periodic motions with a certain mapping structures are predicted analytically for m-cycles with n-periods. Local stability and bifurcation analysis for such motions are carried out. The (m:n)-periodic motions are illustrated. The further investigation of the stable and unstable periodic motions in such a system should be completed. The chaotic motion based on the Levinson donuts should be further discussed.

Bifurcation Analysis of a Fermi-Acceleration Oscillator Under Different Excitations

2011 ◽  
Author(s):  
Albert C. J. Luo ◽  
Yu Guo
Keyword(s):  
Dynamical Systems ◽  
Bifurcation Analysis ◽  
Fermi Acceleration ◽  
Periodic Motions ◽  
Chaotic Motions

In this paper, switchability and bifurcation of motions in a double excited Fermi acceleration oscillator is discussed using the theory of discontinuous dynamical systems. The two oscillators are chosen to have different excitation and parameters. The analytical conditions for motion switching in such a Fermi-oscillator are presented. Bifurcation scenario for periodic and chaotic motions is presented, and the analytical predictions of periodic motions are presented. Finally, different motions in such an oscillator are illustrated.

Dynamic Mechanism of a Train Suspension System

2010 ◽  
Author(s):  
Albert C. J. Luo ◽  
Dennis O’Connor
Keyword(s):  
Numerical Simulations ◽  
System Parameter ◽  
Suspension System ◽  
Dynamic Mechanism ◽  
Impact Model ◽  
Periodic Motions ◽  
Chaotic Motions ◽  
Nonlinear Dynamical ◽  
Dynamical Behaviors ◽  
Mapping Structures

Nonlinear dynamical behaviors of a train suspension system with impacts are investigated. The suspension system is modelled through an impact model with possible stick between a bolster and two wedges. Based on the mapping structures, periodic motions of such a system are described. To understand the global dynamical behaviors of the train suspension system, system parameter maps are developed. Numerical simulations for periodic and chaotic motions are performed from the parameter maps.

Analytical Dynamics of a Ball Bouncing on a Vibrating Table

2012 ◽  
Author(s):  
Yu Guo ◽  
Albert C. J. Luo
Keyword(s):  
Dynamical Systems ◽  
Numerical Simulations ◽  
Analytical Dynamics ◽  
Periodic Motions ◽  
Chaotic Motions ◽  
Theory Of Flow ◽  
Mapping Structures ◽  
Ball Bouncing ◽  
Vibrating Table

In this paper, the theory of flow switchability for discontinuous dynamical systems is applied. Domains and boundaries for such a discontinuous problem are defined and analytical conditions for motion switching are developed. The conditions explain the important role of switching phase on the motion switchability in such a system. To describe different motions, the generic mappings and mapping structures are introduced. Bifurcation scenarios for periodic and chaotic motions are presented for different motions and switchability. Numerical simulations are provided for periodic motions with impacts only and with impact chatter to stick in the system.

Chaos and Quasi-Periodic Motions on the Homoclinic Surface of Nonlinear Hamiltonian Systems With Two Degrees of Freedom

2005 ◽  
Vol 1 (2) ◽  
pp. 135-142 ◽  
Author(s):  
Albert C. J. Luo
Keyword(s):  
Energy Spectrum ◽  
Hamiltonian System ◽  
Kinetic Energy ◽  
Degrees Of Freedom ◽  
Weak Interactions ◽  
Kam Theorem ◽  
Periodic Motions ◽  
Chaotic Motions ◽  
Nonlinear Hamiltonian System ◽  
Two Degree Of Freedom

The numerical prediction of chaos and quasi-periodic motion on the homoclinic surface of a two-degree-of-freedom (2-DOF) nonlinear Hamiltonian system is presented through the energy spectrum method. For weak interactions, the analytical conditions for chaotic motion in such a Hamiltonian system are presented through the incremental energy approach. The Poincaré mapping surfaces of chaotic motions for this specific nonlinear Hamiltonian system are illustrated. The chaotic and quasi-periodic motions on the phase planes, displacement subspace (or potential domains), and the velocity subspace (or kinetic energy domains) are illustrated for a better understanding of motion behaviors on the homoclinic surface. Through this investigation, it is observed that the chaotic and quasi-periodic motions almost fill on the homoclinic surface of the 2-DOF nonlinear Hamiltonian system. The resonant-periodic motions for such a system are theoretically countable but numerically inaccessible. Such conclusions are similar to the ones in the KAM theorem even though the KAM theorem is based on the small perturbation.

Study of complex nonlinear vibrations by means of accurate analytical approach

2014 ◽  
Vol 17 (5) ◽  
pp. 721-734 ◽  
Author(s):  
Mahmoud Bayat ◽  
Iman Pakar ◽  
Mahdi Bayat
Keyword(s):  
Analytical Approach ◽  
Nonlinear Vibrations

The Dynamic Response of a System With Preloaded Compliance

1988 ◽  
Vol 110 (3) ◽  
pp. 278-283 ◽  
Author(s):  
S. W. Shaw ◽  
P. C. Tung
Keyword(s):  
Dynamic Response ◽  
Piecewise Linear ◽  
Harmonic Excitation ◽  
Periodic Motions ◽  
Chaotic Motions ◽  
Linear Features ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
Stable Periodic ◽  
Model Equations

We consider the dynamic response of a single degree of freedom system with preloaded, or “setup,” springs. This is a simple model for systems where preload is used to suppress vibrations. The springs are taken to be linear and harmonic excitation is applied; damping is assumed to be of linear viscous type. Using the piecewise linear features of the model equations we determine the amplitude and stability of the periodic responses and carry out a bifurcation analysis for these motions. Some parameter regions which contain no simple stable periodic motions are shown to possess chaotic motions.

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