Period Motions and Limit Cycle in a Periodically Forced, Plunged Galloping Oscillator

2017 ◽  
Author(s):  
Bo Yu ◽  
Albert C. J. Luo
Keyword(s):  
Hopf Bifurcation ◽  
Numerical Simulations ◽  
Limit Cycle ◽  
Analytical Solutions ◽  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Hopf Bifurcations ◽  
Periodic Motions ◽  
Stability And Bifurcations ◽  
Generalized Harmonic Balance Method

In this paper, periodic motions of a periodically forced, plunged galloping oscillator are investigated. The analytical solutions of stable and unstable periodic motions are obtained by the generalized harmonic balance method. Stability and bifurcations of the periodic motions are discussed through the eigenvalue analysis. The saddle-node and Hopf bifurcations of periodic motions are presented through frequency-amplitude curves. The Hopf bifurcation generates the quasiperiodic motions. Numerical simulations of stable and unstable periodic motions are illustrated.

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.

Period-3 Motions and Bifurcation in a Softenning Duffing Oscillator

2013 ◽  
Author(s):  
Albert C. J. Luo ◽  
Jianzhe Huang
Keyword(s):  
Hopf Bifurcation ◽  
Numerical Simulations ◽  
Analytical Solutions ◽  
Harmonic Balance ◽  
Duffing Oscillator ◽  
Harmonic Balance Method ◽  
Balance Method ◽  
Harmonic Amplitude ◽  
Stable Period

In this paper, analytical solutions for period-3 motions in the periodically forced softening Duffing oscillator are developed by the harmonic balance method. From the Hopf bifurcation of period-3 motions, period-6 motions can be analytically determined. Numerical simulations for stable period-3 motions are completed and the corresponding harmonic amplitude spectrums are presented.

Period-3 Motions to Chaos in a Softening Duffing Oscillator

2014 ◽  
Vol 24 (03) ◽  
pp. 1430010 ◽  
Author(s):  
Albert C. J. Luo ◽  
Jianzhe Huang
Keyword(s):  
Hopf Bifurcation ◽  
Analytical Solutions ◽  
Harmonic Balance ◽  
Duffing Oscillator ◽  
Harmonic Balance Method ◽  
Chaotic Motions ◽  
Saddle Node Bifurcation ◽  
Bifurcation Tree ◽  
Motion Period ◽  
Generalized Harmonic Balance Method

In this paper, period-3 motions to chaos in the periodically forced, softening Duffing oscillator are investigated analytically. The analytical solutions for period-3 and period-6 motions are approximated through the generalized harmonic balance method. The bifurcation trees of period-3 motions to chaos are presented analytically. The symmetric and asymmetric period-3 motions are found. The symmetric to asymmetric period-3 motions experience the saddle-node bifurcation. From the Hopf bifurcation of the asymmetric period-3 motion, period-6 motions are determined analytically from the bifurcation tree of period-3 motions. Such an investigation provides a complete picture of period-3 motions to period-6 motions. With such bifurcation tree, the chaotic motions relative to period-3 motions in such a softening Duffing oscillator can be determined analytically. In a similar fashion, other bifurcation trees of period-m motions to chaos can be determined analytically.

Coexisting Asymmetric and Symmetric Periodic Motions in the Mathieu-Duffing Oscillator

2012 ◽  
Author(s):  
Dennis O’Connor ◽  
Albert C. J. Luo
Keyword(s):  
Numerical Simulations ◽  
Periodic Solutions ◽  
Analytical Solutions ◽  
Harmonic Balance ◽  
Duffing Oscillator ◽  
Harmonic Balance Method ◽  
Balance Method ◽  
Complete Picture ◽  
Periodic Motions ◽  
Approximate Analytical Solutions

In this paper, periodic motions in the Mathieu-Duffing oscillator are analytically predicted through the harmonic balance method. The approximate, analytical solutions of periodic motions are achieved, and the corresponding stability analyses of the stable and unstable periodic solutions are completed. Numerical simulations are provided for a complete picture of coexisting periodic motions.

Analytical Period-1 Motions in a Periodically Forced Oscillator With Quadratic Nonlinearity

2012 ◽  
Author(s):  
Albert C. J. Luo ◽  
Bo Yu
Keyword(s):  
Analytical Solutions ◽  
Harmonic Balance ◽  
Nonlinear Oscillator ◽  
Harmonic Balance Method ◽  
Series Expansions ◽  
Periodic Motions ◽  
Approximate Analytical Solutions ◽  
Forced Oscillator ◽  
The Stability ◽  
Generalized Harmonic Balance Method

In this paper, period-1 motions in a quadratic nonlinear oscillator under excitation are investigated by the generalized harmonic balance method. The analytical solutions of period-1 motion for such an oscillator are presented by the Fourier series expansions. The stability and bifurcation analysis of period-1 motion is carried out via eigenvalue analysis. To verify the approximate analytical solutions, numerical simulations are performed for a better understanding of the parameter characteristics of the period-1 solutions, and the stable and unstable periodic motions are illustrated. The analytical period-1 solutions are different from the perturbation analysis.

Parametric Stability of a Nonlinear Rotating Blade

2013 ◽  
Author(s):  
Fengxia Wang ◽  
Albert C. J. Luo
Keyword(s):  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Parametric Stability ◽  
Rotating Blade ◽  
Geometric Term ◽  
Stability And Bifurcations ◽  
The Stability ◽  
The Galerkin Method ◽  
Gyroscopic Systems ◽  
Generalized Harmonic Balance Method

The stability of period-1 motions of a rotating blade with geometric nonlinearity is studied. The roles of cubic stiffening geometric term are considered in the study of nonlinear periodic motions and its stability and bifurcations of a rotating blade. The nonlinear model of a rotating blade is reduced to the ordinary differential equations through the Galerkin method, and the gyroscopic systems with parametric excitations are obtained. The generalized harmonic balance method is employed to determine the period-1 solutions and the corresponding stability and bifurcations.

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.

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.

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.

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