A Comparison of Henon-Inspired Runge-Kutta and Harmonic Balance Methods for Capturing Chaotic Behavior in a Forced Duffing Oscillator System With Freeplay Nonlinearity

Hysteresis, quasi-periodicity and chaoticity in a nonlinear dissipative hybrid oscillator are studied. The modified Rayleigh-Duffing oscillator is considered. We simultaneously take into account the new nonlinear cubic, pure quadratic and hybrid dissipative terms which modify the classical Rayleigh-Duffing oscillator. The influence of each of these new parameters on the dynamics of the oscillator has been seriously studied and interesting results are obtained. It is clear that each of these new dissipation terms can be used to control the dynamics of this oscillator. Some may be used to reduce or eliminate hysteresis, amplitude jump and resonance phenomena; others may accentuate them. Similarly, these new parameters can be used to impose on the systems modeled by this oscillator, a regular, quasi-periodic or even chaotic behavior according to their field of application. Thus, one of the original results obtained is the equation of the curve delimiting the zone of instabilities of the amplitudes of harmonic oscillations. This equation thus makes it possible to know the zone of amplitude permitted or of the amplitude jump for the systems and thus to control and predict the loss or gain of energy during this jump. Finally, the second stability of the oscillations of the system is studied as well as the influence of the dissipation parameters on this stability. It should be noted that the influence of some of these parameters depends on the simultaneous presence of these parameters.

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Nonlinear Transverse Vibration of a Hyperelastic Beam Under Harmonically Varying Axial Loading

Journal of Computational and Nonlinear Dynamics ◽

10.1115/1.4049562 ◽

2021 ◽

Vol 16 (3) ◽

Author(s):

Yuanbin Wang ◽

Weidong Zhu

Keyword(s):

Governing Equation ◽

Transverse Vibration ◽

Harmonic Balance ◽

Fourier Transforms ◽

Multiple Scales ◽

Equations Of Motion ◽

Multiple Scales Method ◽

Frequency Responses ◽

Runge Kutta ◽

Nonlinear Transverse Vibration

Abstract Nonlinear transverse vibration of a hyperelastic beam under a harmonically varying axial load is analyzed in this work. Equations of motion of the beam are derived via the extended Hamilton's principle, where transverse vibration is coupled with longitudinal vibration. The governing equation of nonlinear transverse vibration of the beam is obtained by decoupling the equations of motion. By applying the Galerkin method, the governing equation transforms to a series of nonlinear ordinary differential equations (ODEs). Response of the beam is obtained via three different methods: the Runge–Kutta method, multiple scales method, and harmonic balance method. Time histories, phase-plane portraits, fast Fourier transforms (FFTs), and amplitude–frequency responses of nonlinear transverse vibration of the beam are obtained. Comparison of results from the three methods is made. Results from the multiple scales method are in good agreement with those from the harmonic balance and Runge–Kutta methods when the amplitude of vibration is small. Effects of the material parameter and geometrical parameter of the beam on its amplitude–frequency responses are analyzed.

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Observe-Based Projective Synchronization of Chaotic Complex Modified Van Der Pol-Duffing Oscillator With Application to Secure Communication

Journal of Computational and Nonlinear Dynamics ◽

10.1115/1.4029715 ◽

2015 ◽

Vol 10 (5) ◽

Cited By ~ 9

Author(s):

Ping Liu ◽

Hongjun Song ◽

Xiang Li

Keyword(s):

Secure Communication ◽

Chaotic Behavior ◽

Design Method ◽

Duffing Oscillator ◽

Scaling Factor ◽

Observer Design ◽

Nonlinear Observer ◽

Projective Synchronization ◽

Chaotic Oscillator ◽

Van Der Pol

This paper addresses the projective synchronization (PS) of the complex modified Van der Pol-Duffing (MVDPD for short) chaotic oscillator by using the nonlinear observer control and also discusses its applications to secure communication in theory. First, we construct the complex MVDPD oscillator and analysis its chaotic behavior. Moreover, an observer design method is applied to achieve PS of two identical MVDPD chaotic oscillators with complex offset terms, which are synchronized to the desired scale factor. The unpredictability of the scaling factor could further enhance the security of the communication. Finally, numerical simulations are given to demonstrate the effectiveness and feasibility of the proposed synchronization approach and also verify the success application to the proposed scheme’s in the secure communication.

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Analytical Periodic Motions in a Periodically Forced, Damped Duffing Oscillator

Volume 4: Dynamics, Control and Uncertainty, Parts A and B ◽

10.1115/imece2012-86077 ◽

2012 ◽

Author(s):

Albert C. J. Luo ◽

Jianzhe Huang

Keyword(s):

Numerical Simulations ◽

Analytical Solutions ◽

Harmonic Balance ◽

Duffing Oscillator ◽

Harmonic Balance Method ◽

Approximate Solutions ◽

Balance Method ◽

Periodic Motions ◽

Conditions Of Stability ◽

Generalized Harmonic Balance Method

The analytical solutions of the period-1 motions for a hardening Duffing oscillator are presented through the generalized harmonic balance method. The conditions of stability and bifurcation of the approximate solutions in the oscillator are discussed. Numerical simulations for period-1 motions for the damped Duffing oscillator are carried out.

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Dynamics of an Electromechanical System With Angular and Ferroresonant Nonlinearities

Journal of Vibration and Acoustics ◽

10.1115/1.4004938 ◽

2011 ◽

Vol 133 (6) ◽

Cited By ~ 7

Author(s):

D. O. Tcheutchoua Fossi ◽

P. Woafo

Keyword(s):

Numerical Simulation ◽

Lyapunov Exponent ◽

Harmonic Balance ◽

Chaotic Behavior ◽

Harmonic Balance Method ◽

Electromechanical System ◽

Bifurcation Diagrams ◽

Oscillatory Solutions ◽

Torsion Bar ◽

Hysteresis Phenomena

The purpose of this paper is to study the dynamics of an electromechanical system consisting of a torsion-bar or two mechanical pumps activated by an electromotor. Oscillatory solutions showing the jump and hysteresis phenomena are obtained using the harmonic balance method and direct numerical simulation. Chaotic behavior is presented via the bifurcation diagrams and corresponding Lyapunov exponent. Some implications of the results on the applications of the devices are discussed.

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Harmonic Balance-Based Approach for Quasi-Periodic Motions and Stability Analysis

Journal of Vibration and Acoustics ◽

10.1115/1.4005823 ◽

2012 ◽

Vol 134 (3) ◽

Cited By ~ 20

Author(s):

Mikhail Guskov ◽

Fabrice Thouverez

Keyword(s):

Harmonic Balance ◽

Nonlinear Response ◽

Duffing Oscillator ◽

Point Of View ◽

Periodic Motions ◽

Computational Costs ◽

Nonlinear Responses ◽

Time Marching ◽

The Stability ◽

Good Agreement

Quasi-periodic motions and their stability are addressed from the point of view of different harmonic balance-based approaches. Two numerical methods are used: a generalized multidimensional version of harmonic balance and a modification of a classical solution by harmonic balance. The application to the case of a nonlinear response of a Duffing oscillator under a bi-periodic excitation has allowed a comparison of computational costs and stability evaluation results. The solutions issued from both methods are close to one another and time marching tests showing a good agreement with the harmonic balance results confirm these nonlinear responses. Besides the overall adequacy verification, the observation comparisons would underline the fact that while the 2D approach features better performance in resolution cost, the stability computation turns out to be of more interest to be conducted by the modified 1D approach.

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Excitation-Induced Stability in a Bistable Duffing Oscillator: Analysis and Experiments

Journal of Computational and Nonlinear Dynamics ◽

10.1115/1.4026974 ◽

2014 ◽

Vol 10 (1) ◽

Cited By ~ 11

Author(s):

Z. Wu ◽

R. L. Harne ◽

K. W. Wang

Keyword(s):

Perturbation Theory ◽

Harmonic Balance ◽

Duffing Oscillator ◽

Excitation Level ◽

Frequency Range ◽

Numerical Studies ◽

Method Of Harmonic Balance ◽

Mathieu’S Equation ◽

Excitation Conditions ◽

Good Agreement

The excitation-induced stability (EIS) phenomenon in a harmonically excited bistable Duffing oscillator is studied in this paper. Criteria to predict system and excitation conditions necessary to maintain EIS are derived through a combination of the method of harmonic balance, perturbation theory, and stability theory for Mathieu's equation. Accuracy of the criteria is verified by analytical and numerical studies. We demonstrate that damping primarily determines the likelihood of attaining EIS response when several dynamics coexist while excitation level governs both the existence and frequency range of the EIS region, providing comprehensive guidance for realizing or avoiding EIS dynamics. Experimental results are in good agreement regarding the comprehensive influence of excitation conditions on the inducement of EIS.

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Bifurcation Analysis of a Power System Model with Three Machines and Four Buses

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127416500826 ◽

2016 ◽

Vol 26 (05) ◽

pp. 1650082 ◽

Cited By ~ 3

Author(s):

Yu Chang ◽

Xiaoli Wang ◽

Dashun Xu

Keyword(s):

Power Systems ◽

Power System ◽

Periodic Solutions ◽

Harmonic Balance ◽

Chaotic Behavior ◽

Harmonic Balance Method ◽

Period Doubling ◽

Bifurcation Phenomena ◽

Fold Bifurcation ◽

Approximate Expressions

The bifurcation phenomena in a power system with three machines and four buses are investigated by applying bifurcation theory and harmonic balance method. The existence of saddle-node bifurcation and Hopf bifurcation is analyzed in time domain and in frequency domain, respectively. The approach of the fourth-order harmonic balance is then applied to derive the approximate expressions of periodic solutions bifurcated from Hopf bifurcations and predict their frequencies and amplitudes. Since the approach is valid only in some neighborhood of a bifurcation point, numerical simulations and the software Auto2007 are utilized to verify the predictions and further study bifurcations of these periodic solutions. It is shown that the power system may have various types of bifurcations, including period-doubling bifurcation, torus bifurcation, cyclic fold bifurcation, and complex dynamical behaviors, including quasi-periodic oscillations and chaotic behavior. These findings help to better understand the dynamics of the power system and may provide insight into the instability of power systems.

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ASYMMETRIC PERIODIC MOTIONS WITH CHAOS IN A SOFTENING DUFFING OSCILLATOR

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127413500867 ◽

2013 ◽

Vol 23 (05) ◽

pp. 1350086 ◽

Cited By ~ 9

Author(s):

ALBERT C. J. LUO ◽

JIANZHE HUANG

Keyword(s):

Harmonic Balance ◽

Duffing Oscillator ◽

Harmonic Balance Method ◽

Balance Method ◽

Periodic Motions ◽

Complex Motion ◽

Bifurcation Tree ◽

Generalized Harmonic Balance Method

In this paper, asymmetric periodic motions in a periodically forced, softening Duffing oscillator are presented analytically through the generalized harmonic balance method. For the softening Duffing oscillator, the symmetric periodic motions with jumping phenomena were understood very well. However, asymmetric periodic motions in the softening Duffing oscillators are not investigated analytically yet, and such asymmetric periodic motions possess much richer dynamics than the symmetric motions in the softening Duffing oscillator. For asymmetric motions, the bifurcation tree from asymmetric period-1 motions to chaos is discussed comprehensively. The corresponding, unstable and stable, asymmetric and symmetric, periodic motions in the softening Duffing oscillator are presented, and numerical illustrations of stable and unstable periodic motions are completed. This investigation provides a better picture of complex motion in the softening Duffing oscillator.

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