A modified harmonic balance method to obtain higher-order approximations to strongly nonlinear oscillators

A new approach, namely the global residue harmonic balance, was developed to determine the accurately approximate periodic solution of a class of nonlinear Jerk equation containing velocity times acceleration-squared and velocity. Unlike other improved harmonic balance methods, all the forward harmonic residuals were considered in the present approximation to improve the accuracy. Comparison of the results obtained using this approach with the exact one and the existing results reveals that the high accuracy, simplicity and efficiency of the presented solution procedure. The method can be easily extended to other strongly nonlinear oscillators.

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Higher-order approximate steady-state solutions for strongly nonlinear systems by the improved incremental harmonic balance method

Journal of Vibration and Control ◽

10.1177/1077546317710160 ◽

2017 ◽

Vol 24 (16) ◽

pp. 3744-3757 ◽

Cited By ~ 3

Author(s):

Jiangchuan Niu ◽

Yongjun Shen ◽

Shaopu Yang ◽

Sujuan Li

Keyword(s):

Nonlinear Systems ◽

Harmonic Balance ◽

Duffing Oscillator ◽

Harmonic Balance Method ◽

Higher Order ◽

Balance Method ◽

Incremental Harmonic Balance Method ◽

Strongly Nonlinear ◽

Incremental Harmonic Balance ◽

Steady State Solutions

Combining the harmonic balance method with the incremental harmonic balance approach, an improved incremental harmonic balance method is presented to obtain the higher-order approximate steady-state solutions for strongly nonlinear systems, which can simplify the calculation process for high-order nonlinear terms. Taking a strongly nonlinear Duffing oscillator with cubic nonlinearity and a strongly nonlinear Duffing oscillator with quintic nonlinearity as examples, the forced vibrations under harmonic excitation are investigated. Based on the first-order approximate analytical solutions obtained by the harmonic balance method, the higher-order approximate solutions are obtained by the improved incremental harmonic balance method. The correctness of the approximate analytical results is verified by the numerical results. The comparison results show that the approximations obtained by the improved incremental harmonic balance method agree with the numerical solutions well, and the improved method is effective to analyze the dynamical response for strongly nonlinear systems.

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The Spreading Residue Harmonic Balance Method for Strongly Nonlinear Vibrations of a Restrained Cantilever Beam

Advances in Mathematical Physics ◽

10.1155/2017/5214616 ◽

2017 ◽

Vol 2017 ◽

pp. 1-8 ◽

Cited By ~ 5

Author(s):

Y. H. Qian ◽

J. L. Pan ◽

S. P. Chen ◽

M. H. Yao

Keyword(s):

Exact Solutions ◽

Nonlinear Vibration ◽

Harmonic Balance ◽

Nonlinear Dynamical Systems ◽

Harmonic Balance Method ◽

Time History ◽

Approximate Solutions ◽

Balance Method ◽

Lumped Mass ◽

Strongly Nonlinear

The exact solutions of the nonlinear vibration systems are extremely complicated to be received, so it is crucial to analyze their approximate solutions. This paper employs the spreading residue harmonic balance method (SRHBM) to derive analytical approximate solutions for the fifth-order nonlinear problem, which corresponds to the strongly nonlinear vibration of an elastically restrained beam with a lumped mass. When the SRHBM is used, the residual terms are added to improve the accuracy of approximate solutions. Illustrative examples are provided along with verifying the accuracy of the present method and are compared with the HAM solutions, the EBM solutions, and exact solutions in tables. At the same time, the phase diagrams and time history curves are drawn by the mathematical software. Through analysis and discussion, the results obtained here demonstrate that the SRHBM is an effective and robust technique for nonlinear dynamical systems. In addition, the SRHBM can be widely applied to a variety of nonlinear dynamic systems.

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A split-frequency harmonic balance method for nonlinear oscillators with multi-harmonic forcing

Journal of Sound and Vibration ◽

10.1016/j.jsv.2006.01.050 ◽

2006 ◽

Vol 295 (3-5) ◽

pp. 939-963 ◽

Cited By ~ 14

Author(s):

J.F. Dunne ◽

P. Hayward

Keyword(s):

Harmonic Balance ◽

Harmonic Balance Method ◽

Nonlinear Oscillators ◽

Balance Method ◽

Harmonic Forcing ◽

Split Frequency ◽

Frequency Harmonic

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NEIMARK BIFURCATIONS OF A GENERALIZED DUFFING–VAN DER POL OSCILLATOR WITH NONLINEAR FRACTIONAL ORDER DAMPING

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127413501770 ◽

2013 ◽

Vol 23 (11) ◽

pp. 1350177 ◽

Cited By ~ 1

Author(s):

A. Y. T. LEUNG ◽

H. X. YANG ◽

P. ZHU

Keyword(s):

Fractional Order ◽

Harmonic Balance ◽

Harmonic Balance Method ◽

Viscoelastic Behavior ◽

Higher Order ◽

Balance Method ◽

Van Der Pol Oscillator ◽

Homotopy Continuation ◽

Van Der Pol ◽

Steady State Responses

A generalized Duffing–van der Pol oscillator with nonlinear fractional order damping is introduced and investigated by the residue harmonic homotopy. The cubic displacement involved in fractional operator is used to describe the higher-order viscoelastic behavior of materials and of aerodynamic damping. The residue harmonic balance method is employed to analytically generate higher-order approximations for the steady state responses of an autonomous system. Nonlinear dynamic behaviors of the harmonically forced oscillator are further explored by the harmonic balance method along with the polynomial homotopy continuation technique. A parametric investigation is carried out to analyze the effects of fractional order of damping and the effect of the magnitude of imposed excitation on the system using amplitude-frequency curves. Jump avoidance conditions are addressed. Neimark bifurcations are captured to delineate regions of instability. The existence of even harmonics in the Fourier expansions implies symmetry-breaking bifurcation in certain combinations of system parameters. Numerical simulations are given by comparing with analytical solutions for validation purpose. We find that all Neimark bifurcation points in the response diagram always exist along a straight line.

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A Simplified Lindstedt-Poincaré Method for Saving Computational Cost to Determine Higher Order Nonlinear Free Vibrations

Mathematics ◽

10.3390/math9233070 ◽

2021 ◽

Vol 9 (23) ◽

pp. 3070

Author(s):

Chein-Shan Liu ◽

Yung-Wei Chen

Keyword(s):

Nonlinear Term ◽

Computational Cost ◽

Nonlinear Oscillators ◽

Analytic Solutions ◽

Free Vibrations ◽

Higher Order ◽

Van Der Pol ◽

Strongly Nonlinear ◽

Two Sides ◽

Strongly Nonlinear Oscillators

In order to improve the Lindstedt-Poincaré method to raise the accuracy and the performance for the application to strongly nonlinear oscillators, a new analytic method by engaging in advance a linearization technique in the nonlinear differential equation is developed, which is realized in terms of a weight factor to decompose the nonlinear term into two sides. We expand the constant preceding the displacement in powers of the introduced parameter so that the coefficients can be determined to avoid the appearance of secular solutions. The present linearized Lindstedt-Poincaré method is easily implemented to provide accurate higher order analytic solutions of nonlinear oscillators, such as Duffing and van Der Pol nonlinear oscillators. The accuracy of analytic solutions is evaluated by comparing to the numerical results obtained from the fourth-order Runge-Kotta method. The major novelty is that we can simplify the Lindstedt-Poincaré method to solve strongly a nonlinear oscillator with a large vibration amplitude.

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