scholarly journals Iterative Homotopy Harmonic Balance Approach for Determining the Periodic Solution of a Strongly Nonlinear Oscillator

2015 ◽  
Vol 2015 ◽  
pp. 1-8 ◽  
Author(s):  
Huaxiong Chen ◽  
Mingkang Ni
Keyword(s):  
Periodic Solution ◽  
Harmonic Balance ◽  
Nonlinear Oscillator ◽  
Order Approximation ◽  
Analytical Approximation ◽  
Higher Order ◽  
Strongly Nonlinear ◽  
Novel Approach ◽  
Approximate Frequency ◽  
Strongly Nonlinear Oscillator

A novel approach about iterative homotopy harmonic balancing is presented to determine the periodic solution for a strongly nonlinear oscillator. This approach does not depend upon the small/large parameter assumption and incorporates the salient features of both methods of the parameter-expansion and the harmonic balance. Importantly, in obtaining the higher-order analytical approximation, all the residual errors are considered in the process of every order approximation to improve the accuracy. With this procedure, the higher-order approximate frequency and corresponding periodic solution can be obtained easily. Comparison of the obtained results with those of the exact solutions shows the high accuracy, simplicity, and efficiency of the approach. The approach can be extended to other nonlinear oscillators in engineering and physics.

scholarly journals Global Residue Harmonic Balance Method for a System of Strongly Nonlinear Oscillator

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Huaxiong Chen ◽  
Wei Liu
Keyword(s):  
Harmonic Balance ◽  
Nonlinear Oscillator ◽  
Order Approximation ◽  
Harmonic Balance Method ◽  
High Accuracy ◽  
Balance Method ◽  
Strongly Nonlinear ◽  
Improve Accuracy ◽  
Residual Errors ◽  
Strongly Nonlinear Oscillator

In this paper, the global residue harmonic balance method is applied to obtain the approximate periodic solution and frequency for a well-known system of strongly nonlinear oscillator in engineering. This method can improve accuracy by considering all the residual errors in deriving each order approximation. With this procedure, the expressions of the higher-order approximate solution and corresponding frequency for the considered system can be determined easily. The comparison of the obtained results with previously existing and corresponding exact solutions shows the high accuracy and efficiency of the method.

scholarly journals Approximate Potentials with Applications to Strongly Nonlinear Oscillators with Slowly Varying Parameters

2003 ◽  
Vol 10 (5-6) ◽  
pp. 379-386 ◽  
Author(s):  
Jianping Cai ◽  
Y.P. Li
Keyword(s):  
Present Method ◽  
Nonlinear Oscillator ◽  
Elliptic Functions ◽  
Nonlinear Oscillators ◽  
Oscillatory System ◽  
Varying Parameters ◽  
Strongly Nonlinear ◽  
Elapsed Time ◽  
Strongly Nonlinear Oscillators ◽  
Strongly Nonlinear Oscillator

A method of approximate potential is presented for the study of certain kinds of strongly nonlinear oscillators. This method is to express the potential for an oscillatory system by a polynomial of degree four such that the leading approximation may be derived in terms of elliptic functions. The advantage of present method is that it is valid for relatively large oscillations. As an application, the elapsed time of periodic motion of a strongly nonlinear oscillator with slowly varying parameters is studied in detail. Comparisons are made with other methods to assess the accuracy of the present method.

Motion Confinement in a Linear Chain of Coupled Oscillators With a Strongly Nonlinear End Attachment

2003 ◽  
Author(s):  
Leonid Manevitch ◽  
Oleg Gendelman ◽  
Andrey I. Musienko ◽  
Alexander F. Vakakis ◽  
Lawrence Bergman
Keyword(s):  
Coupled Oscillators ◽  
Nonlinear Oscillator ◽  
Linear Chain ◽  
Standing Waves ◽  
Strongly Nonlinear ◽  
Resonant Interactions ◽  
Energy Pumping ◽  
Weakly Coupled ◽  
Strongly Nonlinear Oscillator ◽  
Infinite Linear

We study the dynamics of a semi-infinite linear chain of particles that is weakly coupled to a strongly nonlinear oscillator at its free end. We analyze families of localized standing waves situated inside the lower or upper attenuation zones of the linear chain, corresponding to energy predominantly confined in the nonlinear oscillator. These families of standing waves are generated due to resonant interactions between the chain and the nonlinear attachment. A scenario for the realization of energy pumping phenomena in the system under consideration is discussed, and is confirmed by direct numerical simulations of the chain-attachment dynamic interaction.

Subharmonic Orbits of a Strongly Nonlinear Oscillator Forced by Closely Spaced Harmonics

2010 ◽  
Vol 6 (1) ◽  
Author(s):  
Themistoklis P. Sapsis ◽  
Alexander F. Vakakis
Keyword(s):  
Dynamical System ◽  
Nonlinear Oscillator ◽  
Resonance Condition ◽  
Asymptotic Results ◽  
Strongly Nonlinear ◽  
Singular Perturbation Analysis ◽  
Local Bifurcation Analysis ◽  
The Family ◽  
Subharmonic Orbits ◽  
Strongly Nonlinear Oscillator

We study asymptotically the family of subharmonic responses of an essentially nonlinear oscillator forced by two closely spaced harmonics. By expressing the original oscillator in action-angle form, we reduce it to a dynamical system with three frequencies (two fast and one slow), which is amenable to a singular perturbation analysis. We then restrict the dynamics in neighborhoods of resonance manifolds and perform local bifurcation analysis of the forced subharmonic orbits. We find increased complexity in the dynamics as the frequency detuning between the forcing harmonics decreases or as the order of a secondary resonance condition increases. Moreover, we validate our asymptotic results by comparing them to direct numerical simulations of the original dynamical system. The method developed in this work can be applied to study the dynamics of strongly nonlinear (nonlinearizable) oscillators forced by multiple closely spaced harmonics; in addition, the formulation can be extended to the case of transient excitations.

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