Analyzing of Nonlinear Generalized Duffing Oscillators Using the Equivalent Linearization Method with a Weighted Averaging

Abstract

This work was carried out in collaboration between both authors. Author DVH designed the study, performed the statistical analysis, wrote the protocol, wrote the first draft of the manuscript, and managed the analyses of the study. Author NQH managed the literature searches. Both authors read and approved the final manuscript.

Analysis of Nonlinear Vibration of Euler-Bernoulli Beams Subjected to Compressive Axial Forcevia the Equivalent Linearization Method with a Weighted Averaging

“…Based on the equivalent linearization method proposed Caughey [7]; in refs. [20,21], Anh et al had developed this method by using the weighted averaging instead of the convenial (classical) averaging [22]. The equivalent linearization method with a weighted averaging has been used to analyze responses of strong nonlinear oscillation problems and this method gives the solutions with much better accuracy than the classical method [21,[23][24][25].…”

“…where ω 2 is determined by using the mean square error criterion which requires that [7]: (13) The symbol in Eqs. (12) and (13) denotes the time-averaging operator in classical meaning [22]:…”

An Approximate Solution for a Nonlinear Duffing – Harmonic Oscillator

“…With this new proposal, accuracy of the ELM has been greatly improved. The new proposed method has been applied very effectively in analysis of deterministic strong nonlinear oscillations [22][23][24][25]. In this paper, the proposed method is further extended in analysis of a Duffing-harmonic oscillator.…”

“…The averaging value of function f(ωt) does not depend on the time t but depends on the parameter s (in expression of the weighted coefficient function h(t)). The parameter s is called the adjustment parameter, accuracy of the obtained solution depends on the choice of value of the parameter s. The choice of value of the parameter s has been examined by authors in many cases [22][23][24][25], the obtained results will be very accurate when s = 2.…”

“…=1, the approximate frequency of oscillation given in Eq. (25) was achieved by using the Parameter Expansion method (PEM) [27], and the exact frequency of oscillation was given in ref. [29].…”