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

2012 ◽  
Vol 134 (3) ◽  
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.

Analytical Periodic Motions in a Periodically Forced, Damped Duffing Oscillator

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.

On Time-Delay Effects on Period-1 Motions in a Periodically Forced, Time-Delayed, Duffing Oscillator

2016 ◽  
Author(s):  
Albert C. J. Luo ◽  
Siyuan Xing
Keyword(s):  
Time Delay ◽  
Analytical Solutions ◽  
Analytical Method ◽  
Duffing Oscillator ◽  
Delay Systems ◽  
Harmonic Amplitude ◽  
Time Delay Systems ◽  
Periodic Motions ◽  
Delay Effects ◽  
The Stability

In recent decades, nonlinear time-delay systems were often applied in controlling nonlinear systems, and the stability of such time-delay systems was very actively discussed. Recently, one was very interested in periodic motions in nonlinear time-delay systems. Especially, the semi-analytical solutions of periodic motions in time-delay systems are of great interest. From the semi-analytical solutions, the nonlinearity and complexity of periodic motions in the time-delay systems can be discussed. In this paper, time-delay effects on periodic motions of a periodically forced, damped, hardening, Duffing oscillator are analytically discussed through a semi-analytical method. The semi-analytical method is based on discretization of the differential equation of such a Duffing oscillator to obtain the corresponding implicit discrete mappings. Through such implicit mappings and mapping structures of periodic motions, period-1 motions varying with time-delay are discussed, and the corresponding stability and bifurcation analysis of periodic motions are carried out through eigenvalue analysis. Numerical results of periodic motions are illustrated to verify analytical predictions. The corresponding harmonic amplitude spectrums and harmonic phases are presented for a better understanding of periodic motions in such a nonlinear oscillator.

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

2014 ◽  
Vol 10 (1) ◽  
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.

ASYMMETRIC PERIODIC MOTIONS WITH CHAOS IN A SOFTENING DUFFING OSCILLATOR

2013 ◽  
Vol 23 (05) ◽  
pp. 1350086 ◽  
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.

GLOBAL DYNAMICS OF A DUFFING OSCILLATOR WITH DELAYED DISPLACEMENT FEEDBACK

2004 ◽  
Vol 14 (08) ◽  
pp. 2753-2775 ◽  
Author(s):  
HUAILEI WANG ◽  
HAIYAN HU ◽  
ZAIHUA WANG
Keyword(s):  
Time Delay ◽  
Duffing Oscillator ◽  
Basin Of Attraction ◽  
Poincare Section ◽  
Periodic Motions ◽  
Stability Switches ◽  
Local Bifurcations ◽  
Section Technique ◽  
Trivial Equilibrium ◽  
The Stability

This paper presents a systematic study on the dynamics of a controlled Duffing oscillator with delayed displacement feedback, especially on the local bifurcations of periodic motions with respect to the time delay. The study begins with the analysis of the stability switches of the trivial equilibrium of the system with various parametric combinations and gives the critical values of time delay, where the trivial equilibrium may change its stability. It shows that as the time delay increases from zero to the positive infinity, the trivial equilibrium undergoes a different number of stability switches for different parametric combinations, and becomes unstable at last for all parametric combinations. Then, the method of multiple scales and the numerical computation method are jointly used to obtain a global diagram of local bifurcations of periodic motions with respect to the time delay for each type of parametric combinations. The diagrams indicate two kinds of local bifurcations. One is the saddle-node bifurcation and the other is the pitchfork bifurcation, of which the former means the sudden emerging of two periodic motions with different stability and the latter implies the Hopf bifurcation in the sense of dynamic bifurcation. A novel feature, referred to as the property of "periodicity in delay", is observed in the global diagrams of local bifurcations and used to justify the validity of infinite number of bifurcating branches in the bifurcation diagrams. The stability of the periodic motions is discussed not only from the high-order approximation of the asymptotic solution, but also from viewpoint of basin of attraction, which gives a good explanation for coexisting periodic motions and quasi-periodic motions, as well as an overall idea of basin of attraction. Afterwards, a conventional Poincaré section technique is used to reveal the abundant dynamic structures of a tori bifurcation sequence, which shows that the system will repeat similar quasi-periodic motions several times, with an increase of time delay, enroute to a chaotic motion. Finally, a new Poincaré section technique is proposed as a comparison with the conventional one, and the results show that the dynamical structures on the two kinds of Poincaré sections are topologically symmetric in rotation.

Approximate solutions of periodic motions in nonlinear systems via a generalized harmonic balance

2011 ◽  
Vol 18 (11) ◽  
pp. 1661-1674 ◽  
Author(s):  
Albert CJ Luo ◽  
Jianzhe Huang
Keyword(s):  
Harmonic Balance ◽  
Nonlinear Dynamical Systems ◽  
Duffing Oscillator ◽  
Harmonic Balance Method ◽  
Approximate Solutions ◽  
Nonlinear Damping ◽  
Balance Method ◽  
Periodic Motions ◽  
Approximate Analytical Solutions ◽  
Generalized Harmonic Balance Method

In this paper, the generalized harmonic balance method is presented for approximate, analytical solutions of periodic motions in nonlinear dynamical systems. The nonlinear damping, periodically forced, Duffing oscillator is studied as a sample problem. The approximate, analytical solution of period-1 periodic motion of such an oscillator is obtained by the generalized harmonic balance method. The stability and bifurcation analysis of the HB2 approximate solution of period-1 motions in the forced Duffing oscillator is carried out, and the parameter map for such HB2 solutions is achieved. Numerical illustrations of period-1 motions are presented. Similarly, the same ideas can be extended to period- k motions in such an oscillator. The methodology presented in this paper can be applied to other nonlinear vibration systems, which are independent of small parameters.

Analytical Solutions for Periodic Motions in a Hardening Mathieu-Duffing Oscillator

2013 ◽  
Author(s):  
Albert C. J. Luo ◽  
Dennis M. O’Connor
Keyword(s):  
Fourier Series ◽  
Bifurcation Analysis ◽  
Analytical Solutions ◽  
Duffing Oscillator ◽  
Series Solutions ◽  
Periodic Motions ◽  
Period 2 ◽  
Approximate Analytical Solutions ◽  
Stability And Bifurcation Analysis ◽  
The Stability

Analytical solutions for period-m motions in a hardening Mathieu-Duffing oscillator are obtained using the finite Fourier series solutions, and the stability and bifurcation analysis of such periodic motions are completed. To verify the approximate analytical solutions of periodic motions, numerical simulations of the hardening Mathieu-Duffing oscillator are presented. Period-1 asymmetric and period-2 symmetric motions are illustrated.

Period-1 Motions in a Time-Delayed Duffing Oscillitor With Periodic Excitation

2014 ◽  
Author(s):  
Albert C. J. Luo ◽  
Hanxiang Jin
Keyword(s):  
Fourier Series ◽  
Analytical Solutions ◽  
Duffing Oscillator ◽  
Eigenvalue Analysis ◽  
Periodic Excitation ◽  
Periodic Motions ◽  
Stability And Bifurcation ◽  
The Stability ◽  
Amplitude Characteristics

In this paper, analytical solutions of period-1 motions in a time-delayed Duffing oscillator with a periodic excitation are investigated through the Fourier series, and the stability and bifurcation of such periodic motions are discussed by eigenvalue analysis. The symmetric and asymmetric period-1 motions in such time-delayed Duffing oscillator are obtained analytically, and the frequency-amplitude characteristics of period-1 motions in such a time-delayed Duffing oscillator are investigated. Numerical illustrations of period-1 motions are given by numerical and analytical solutions.

Period-1 Evolutions to Chaos in a Periodically Forced Brusselator

2018 ◽  
Vol 28 (14) ◽  
pp. 1830046 ◽  
Author(s):  
Albert C. J. Luo ◽  
Siyu Guo
Keyword(s):  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Good Method ◽  
Balance Method ◽  
Harmonic Amplitude ◽  
Periodic Motions ◽  
Bifurcation Tree ◽  
Stable Period ◽  
The Stability ◽  
Generalized Harmonic Balance Method

In this paper, analytical solutions of periodic evolutions of the Brusselator with a harmonic diffusion are obtained through the generalized harmonic balance method. The stability and bifurcation of the periodic evolutions are determined. The bifurcation tree of period-1 to period-8 evolutions of the Brusselator is presented through frequency-amplitude characteristics. To illustrate the accuracy of the analytical periodic evolutions of the Brusselator, numerical simulations of the stable period-1 to period-8 evolutions are completed. The harmonic amplitude spectrums are presented for the accuracy of the analytical periodic evolution, and each harmonics contribution on the specific periodic evolution can be achieved. This study gives a better understanding of periodic evolutions to chaos in the slowly varying Brusselator system, and the bifurcation tree of period-1 evolution to chaos are clearly demonstrated, which can help one understand a route of periodic evolution to chaos in chemical reaction oscillators. From this study, the generalized harmonic balance method is a good method for slowly varying systems, and such a method provides very accurate solutions of periodic motions in such nonlinear systems.

Period-m Motions to Chaos in a Periodically Forced, Duffing Oscillator with a Time-Delayed Displacement

2014 ◽  
Vol 24 (10) ◽  
pp. 1450126 ◽  
Author(s):  
Albert C. J. Luo ◽  
Hanxiang Jin
Keyword(s):  
Time Delay ◽  
Analytical Solutions ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Duffing Oscillator ◽  
Initial Time ◽  
Periodic Motions ◽  
Delayed System ◽  
The Stability ◽  
Analytical Bifurcation Trees

In this paper, periodic motions in a periodically excited, Duffing oscillator with a time-delayed displacement are investigated through the Fourier series, and the stability and bifurcation of such periodic motions are discussed through eigenvalue analysis. The time-delayed displacement is from the feedback control of displacement. The analytical bifurcation trees of period-1 motions to chaos in the time-delayed Duffing oscillator are presented through asymmetric period-1 to period-4 motions. Stable and unstable periodic motions are illustrated through numerical and analytical solutions. From numerical illustrations, the analytical solutions of stable and unstable period-m motions are relatively accurate with AN/m < 10-6 compared to numerical solutions. From such analytical solutions, any complicated solutions of period-m motions can be obtained for any prescribed accuracy. Because time-delay may cause discontinuity, the appropriate time-delay inputs (or initial conditions) in the initial time-delay interval should satisfy the analytical solution of periodic motions in the time-delayed dynamical systems. Otherwise, periodic motions in such a time-delayed system cannot be obtained directly.

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