Determination of natural frequencies by coupled method of homotopy perturbation and variational method for strongly nonlinear oscillators

Abstract

In this paper a new approach combining the features of the homotopy concept with variational approach is proposed to find accurate analytical solutions for nonlinear oscillators with and without a fractional power restoring force. Since the first-order approximation leads to very accurate results, comparisons with other results are presented to show the effectiveness of this method. The validity of the method is independent of whether or not there exist small or large parameters in the considered nonlinear equations; the obtained results prove the validity and efficiency of the method, which can be easily extended to other strongly nonlinear problems. At the end we compare our procedure with the optimal homotopy perturbation method.

On the variational homotopy perturbation method for nonlinear oscillators

“…Although the combination of the homotopy perturbation method and the variational principle proposed by Akbarzade and Langari [4] led to reasonably approximate frequencies, one cannot take the approach seriously because of its inconsistencies. First, the first-order corrections to the solutions do not satisfy one of the boundary conditions proposed by the authors and, second, in most of the cases the resulting corrections are trivial (that is to say, they vanish identically).…”

“…where f is a nonlinear function of u ′′ , u ′ and u, and considered the "variational functional" [4] (and references therein)…”

“…There has recently been great interest in developing simple solutions to textbook models of nonlinear oscillators [1][2][3][4] (and references therein). However, some of them are of questionable utility as shown, for example, by Rajendran et al [5] who concluded that He's calculations of the limit cycle of the van der Pol oscillator [1] "contain several errors which once rectified make the method inapplicable to it".…”

Numerical Solution of Duffing Equation by Using an Improved Taylor Matrix Method

“…Thus, methods of solution for these equations are of great importance to engineers and scientists. Many new techniques have appeared in the open literature such as homotopy perturbation transform method [8], variational iteration method [9], energy balance method [10], Hamiltonian approach [11], coupled homotopy-variational formulation [12], variational approach [13], and amplitude-frequency formulation [14].…”

Dynamic Analysis of Nonlinear Oscillator Equation Arising in Double-Sided Driven Clamped Microbeam-Based Electromechanical Resonator

“…Perturbation methods are based on the existence of small parameters, the so-called perturbation quantity. Many nonlinear problems do not contain such perturbation quantity, so to overcome the shortcomings, many new techniques have appeared in open literature such as: variational iteration method [9 -13], energy balance method [14 -19], Hamiltonian approach [20 -23], coupled homotopy-variational formulation [24,25], variational approach [26 -28], amplitude-frequency formulation [29,30], and other classical methods [31 -45].…”