Dual approach to averaged values of functions: A form for weighting coefficient

Abstract

Averaged values play major roles in the study of dynamic processes. The use of those values allows transforming varying processes to some constant characteristics that are much easier to be investigated. In order to extend the use of averaged values one may apply the dual approach which suggests a consideration of two different aspects of a problem in question. In this short communication the main idea of the dual conception is further extended to suggest a new form for weighting coefficient and then a new averaged value of functions. This new averaged value depends on the parameter \(s\) and contains the classical averaged value when \(s=0\). In the example of Duffing oscillator it is shown that the parameter \(s\) can be chosen as \(s=n/(2\pi)\) and for \(n=4\) one gets the solution that is much accurate than the conventional one obtained by the classical criterion of equivalent linearization.

The Equivalent Linearization Method with a Weighted Averaging for Analyzing of Nonlinear Vibrating Systems

“…In order to improve this shortcoming, the averaging operation with weighting functions is proposed in the next section. This idea is introduced by Anh, N. D (Anh, 2015).…”

“…In recent years, many powerful methods are used to find approximate solution as well as the amplitude- frequency relationship to the nonlinear differential equations. Some of these methods are Homotopy Perturbation Method (HPM) (He, 1999;He, 2004a;He, 2004b;He, 2004c;Turgut et al, 2007;Bayat et al, 2012), Max-Min Approach (MMA) (He, 2008;Chen et al, 2011;Dumaz et al, 2011;Yazdi et al, 2012;Bayat et al, 2012), Variational Iteration Method (VIM) (Bayat et al, 2012), Energy Balance Method (EBM) (Ganji et al, 2009;Khah et al, 2010;Younesian et al, 2010;Bayat et al, 2012), Amplitude-Frequency Formulation (AFF) (Chen et al, 2011;Jouyburi et al, 2014;Bayat et al, 2012), Parameter Expansion Method (PEM) (Kayaa et al, 2009;Dumaz et al, 2011;Darvishia et al, 2008;Zhao, 2009;Bayat et al, 2012 ), Homotopy Analysis Method (HAM) (He, 2004c;Bayat et al, 2012, Shahram Shahlaei-Far et al, 2016, Modified Homotopy Perturbation Method (MHPM) (Jouybari et al, 2014), Equivalent linearization Method (ELM) (Krylov et al, 1943;Caughey, 1959;Iyengar, 1988;Anh et al, 1995;Anh et al, 1997;Elishakoff et al, 2009;Anh, 2015) and combining Newton's Method with the Harmonic Balance Method (Lim et al, 2006).…”

“…A reason is that some terms will vanish in the averaging process, for example the averaging value of the functions sin(t) and cos(t) over one period is equal to zero. Anh N. D. (Anh, 2015) proposed a new way for determining averaging values, instead of using conventional averaging process author introduced weighted coefficient functions.…”

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AbstractIn this paper, the Equivalent Linearization Method (ELM) with a weighted averaging, which is proposed by Anh (Anh, 2015), is applied to analyze some vibrating systems with nonlinearities. The strongly nonlinear Duffing oscillator with third, fifth, and seventh powers of the amplitude, the other strongly nonlinear oscillators and the cubic Duffing with discontinuity are considered.

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The Equivalent Linearization Method with a Weighted Averaging for Solving Undamped Nonlinear Oscillators

“…Then Caughey expanded the method for stochastic systems [27]. Thenceforward, there have been some extended versions of the Equivalent Linearization Method [28][29][30][31][32]. It has been shown that the Equivalent Linearization Method is presently the simplest tool widely used for analyzing nonlinear stochastic problems.…”