Period-1 Evolutions to Chaos in a Periodically Forced Brusselator

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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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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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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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 ◽

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.

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Period-3 Motions and Bifurcation in a Softenning Duffing Oscillator

Volume 7B: 9th International Conference on Multibody Systems, Nonlinear Dynamics, and Control ◽

10.1115/detc2013-13002 ◽

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.

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

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

10.1115/imece2012-86592 ◽

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.

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Multiple Harmonic Balance Method for Aperiodic Vibration of a Piecewise-Linear System

Journal of Vibration and Acoustics ◽

10.1115/1.2893802 ◽

1998 ◽

Vol 120 (1) ◽

pp. 181-187 ◽

Cited By ~ 15

Author(s):

Y. B. Kim

Keyword(s):

Linear System ◽

Harmonic Balance ◽

Piecewise Linear ◽

Harmonic Balance Method ◽

Steady State Solution ◽

Balance Method ◽

Piecewise Linear System ◽

Higher Dimensional ◽

Computational Procedures ◽

The Stability

A multiple harmonic balance method is presented in this paper for obtaining the aperiodic steady-state solution of a piecewise-linear system. As the method utilizes general and systematic computational procedures, it can be applied to analyze the multi-tone or combination-tone responses for the higher dimensional nonlinear systems such as rotors. Moreover, it is capable of informing the stability of the obtained solution using Floquet theory. To demonstrate the systematic approach of the new method, the almost periodic forced vibration of an articulated loading platform (ALP) with a piecewise-linear stiffness is computed as an example.

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Parametric Stability of a Nonlinear Rotating Blade

Volume 4B: Dynamics, Vibration and Control ◽

10.1115/imece2013-62141 ◽

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.

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Developing a Generalized Harmonic Balance Method for Accurate Identification of Crack Depth in a Continuous Shaft-Disc System

Volume 2: Integrated System Design and Implementation; Structural Health Monitoring; Bioinspired Smart Materials and Systems; Energy Harvesting ◽

10.1115/smasis2015-8954 ◽

2015 ◽

Author(s):

Hamid Khorrami ◽

Ramin Sedaghati ◽

Subhash Rakheja

Keyword(s):

Harmonic Balance ◽

Crack Detection ◽

Crack Depth ◽

Harmonic Component ◽

Harmonic Balance Method ◽

Balance Method ◽

Vibrational Properties ◽

Accurate Identification ◽

Harmonic Components ◽

Generalized Harmonic Balance Method

In this work, the effect of a crack on the vibrational properties of a shaft-disc system has been studied applying a generalized harmonic balance method. In the reviewed literature, the reported methods to find the unbalance response of a continuous shaft-disc system provide only the first harmonic component of the response; whereas, the presented method gives the super-harmonic components as well. The shaft-disk system consists of a flexible shaft with a single rigid disc mounted on rigid short bearing supports. The shaft contains a transverse breathing crack (fatigue crack). The main concept for crack detection in vibration-based methods is basically the investigation of crack-induced changes in the selected vibrational properties. Shaft critical speeds and harmonic and super-harmonic components of the unbalance lateral response have been used as typical vibrational properties for crack detection in a rotating shaft system. A generalized harmonic balance method has been developed to efficiently investigate changes in vibrational properties due to the effect of crack properties, depth and location. The results of the developed analytical model have been compared with those obtained from the finite element model and close agreement has been observed.

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Periodicity and Stability in Transverse Motion of a Nonlinear Rotor-Bearing System Using Generalized Harmonic Balance Method

Journal of Engineering for Gas Turbines and Power ◽

10.1115/1.4034257 ◽

2016 ◽

Vol 139 (2) ◽

Cited By ~ 3

Author(s):

Zhiwei Liu ◽

Yuefang Wang

Keyword(s):

Harmonic Balance ◽

Harmonic Balance Method ◽

Iteration Scheme ◽

Balance Method ◽

Transverse Motion ◽

Harmonic Load ◽

Mass Eccentricity ◽

Rotor Bearing ◽

Generalized Harmonic Balance Method ◽

Bearing System

Many rotor assemblies of industrial turbomachines are supported by oil-lubricated bearings. It is well known that the operation safety of these machines is highly dependent on rotors whose stability is closely related to the whirling motion of lubricant oil. In this paper, the problem of transverse motion of rotor systems considering bearing nonlinearity is revisited. A symmetric, rigid Jeffcott rotor is modeled considering unbalanced mass and short bearing forces. A semi-analytical, seminumerical approach is presented based on the generalized harmonic balance method (GHBM) and the Newton–Raphson iteration scheme. The external load of the system is decomposed into a Fourier series with multiple harmonic loads. The amplitude and phase with respect to each harmonic load are solved iteratively. The stability of the motion response is analyzed through identification of eigenvalues at the fixed point mapped from the linearized system using harmonic amplitudes. The solutions of the present approach are compared to those from time-domain numerical integrations using the Runge–Kutta method, and they are found to be in good agreement for stable periodic motions. It is revealed through bifurcation analysis that evolution of the motion in the nonlinear rotor-bearing system is complicated. The Hopf bifurcation (HB) of synchronous vibration initiates oil whirl with varying mass eccentricity. The onset of oil whip is identified when the saddle-node bifurcation of subsynchronous vibration takes place at the critical value of parameter.

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Calculating periodic vibrations of nonlinear autonomous multi-degree-of-freedom systems by the incremental harmonic balance method

Vietnam Journal of Mechanics ◽

10.15625/0866-7136/26/3/5700 ◽

2004 ◽

Vol 26 (3) ◽

pp. 157-166

Author(s):

Nguyen Van Khang ◽

Thai Manh Cau

Keyword(s):

Harmonic Balance ◽

Floquet Theory ◽

Monodromy Matrix ◽

Harmonic Balance Method ◽

Degree Of Freedom ◽

Balance Method ◽

Incremental Harmonic Balance Method ◽

Van Der Pol ◽

Incremental Harmonic Balance ◽

The Stability

In this paper the incremental harmonic balance method is used to calculate periodic vibrations of nonlinear autonomous multip-degree-of-freedom systems. According to Floquet theory, the stability of a periodic solution is checked by evaluating the eigenvalues of the monodromy matrix. Using the programme MAPLE, the authors have studied the periodic vibrations of the system multi-degree van der Pol form.

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