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

Forced Oscillator ◽

The Stability ◽

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.

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 ◽

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.

An Approximate Solution for Period-m Motions in a Periodically Forced Oscillator With Quadratic Nonlinearity

Volume 4B: Dynamics, Vibration and Control ◽

10.1115/imece2013-62418 ◽

2013 ◽

Author(s):

Albert C. J. Luo ◽

Bo Yu

Keyword(s):

Approximate Solution ◽

Numerical Simulations ◽

Analytical Solutions ◽

Periodic Motion ◽

Nonlinear Oscillator ◽

Approximate Solutions ◽

Bifurcation Tree ◽

Forced Oscillator ◽

The Stability

The approximate analytical solutions of the period-m motions for a periodically forced, quadratic nonlinear oscillator are presented. The stability and bifurcation of such approximate solutions in the quadratic nonlinear oscillator are discussed. The bifurcation tree of period-1 to chaos is presented. Numerical simulations for period-1 to period-4 motions in such quadratic oscillator are carried out for comparison of approximate analytical solutions. Such an investigation provides how to analytically determine bifurcation of periodic motion to chaos.

Volume 6: 13th International Conference on Multibody Systems, Nonlinear Dynamics, and Control ◽

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

10.1115/detc2017-67323 ◽

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 ◽

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.

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

Journal of Vibration and Control ◽

10.1177/1077546311421053 ◽

2011 ◽

Vol 18 (11) ◽

pp. 1661-1674 ◽

Cited By ~ 58

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 ◽

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

International Journal of Bifurcation and Chaos ◽

10.1142/s021812741830046x ◽

2018 ◽

Vol 28 (14) ◽

pp. 1830046 ◽

Cited By ~ 3

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 ◽

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.

Volume 6: 11th International Conference on Multibody Systems, Nonlinear Dynamics, and Control ◽

Period-1 Motions in a Two-Degree-of-Freedom Nonlinear Oscillator With Periodic Excitation

10.1115/detc2015-48103 ◽

2015 ◽

Author(s):

Albert C. J. Luo ◽

Bo Yu

Keyword(s):

Analytical Solutions ◽

Nonlinear Oscillator ◽

Degrees Of Freedom ◽

Harmonic Amplitude ◽

Periodic Excitation ◽

Periodic Motions ◽

Two Degrees Of Freedom ◽

The Stability ◽

Two Degree Of Freedom

In this paper, analytical solutions for period-1 motions in a periodically forced, two-degrees-of-freedom system with a nonlinear spring are developed. The stability and bifurcation of the periodic motions are completed by the eigenvalue analysis. Both symmetric and asymmetric periodic motions are found in the system. Analytical solutions of both stable and unstable period-1 are presented. Finally, numerical simulations of stable and unstable motions in the two degrees of freedom systems are presented. The harmonic amplitude spectrums show the harmonic effects on periodic motions, and the corresponding accuracy of approximate analytical solutions can be observed.

Volume 2: 16th International Conference on Multibody Systems, Nonlinear Dynamics, and Control (MSNDC) ◽

Vibration Suppression of a Harmonically Forced Oscillator Using a Passive Nonlinear Vibration Absorber

10.1115/detc2020-22715 ◽

2020 ◽

Author(s):

Bo Yu

Keyword(s):

Nonlinear Vibration ◽

Analytical Solutions ◽

Vibration Suppression ◽

Harmonic Balance Method ◽

Vibration Absorber ◽

Harmonic Amplitude ◽

Periodic Motions ◽

Nonlinear Vibration Absorber ◽

Forced Oscillator

Abstract In this paper, the performance of a nonlinear vibration absorber with different nonlinearity is studied. The analytical solutions of periodic motions are obtained using the general harmonic balance method. As the nonlinear strength is weak, the effectiveness of the absorber is discussed. For strong nonlinearities, unstable parodic motions can be obtained and stabilities of the periodic motions are determined through the eigenvalue analysis. The Hopf and saddle bifurcations are observed. Numerical simulations are illustrated for both masses at the resonance peaks. The harmonic amplitude spectrums show the harmonic effects on periodic motions, and the corresponding accuracy of approximate analytical solutions.

Analytical Solutions of a First-Order Quadratic Nonlinear System With Parametrical and Periodical Excitation

Volume 4B: Dynamics, Vibration, and Control ◽

10.1115/imece2016-66812 ◽

2016 ◽

Author(s):

Guopeng Zhou ◽

Albert C. J. Luo ◽

Naiding Zhou ◽

Feng Liang

Keyword(s):

Dynamical System ◽

Analytical Solutions ◽

Harmonic Balance ◽

Numerical Solutions ◽

Excitation Frequency ◽

Nonlinear Dynamical System ◽

Harmonic Balance Method ◽

Nonlinear Dynamical ◽

In this paper, a quadratic nonlinear dynamical system with two periodic excitation forces is discussed. Analytical period-1 motions of such dynamical system are obtained by using generalized harmonic balance method. Stability analysis is carried out via eigenvalues analysis. To verify approximate analytical solutions, numerical simulations are completed to compare analytic and numerical solutions of the dynamical system, the approximate precision is guaranteed with appropriate harmonic balance terms. More harmonic terms should be employed to guarantee good approximation of periodic motions if excitation frequency is small. Furthermore, infinite harmonic balance terms must be introduced for chaotic systems.

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

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

10.1115/imece2012-86849 ◽

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.