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

Abstract

In 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. The results obtained via this method are compared with the ones achieved by the Min-Max Approach (MMA), the Modified Lindstedt -Poincare Method (MLPM), the Parameter -Expansion Method (PEM), the Homotopy Perturbation Method (HPM) and 4 th order RungeKutta method. The obtained results demonstrate that this method is very convenient for solving nonlinear equations and also can be successfully exerted to a lot of practical engineering and physical problems.

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

“…In order to introduce the general idea of the Equivalent Linearization Method, we consider a nonlinear oscillator governed by the following equation [33]:̈+…”

“…This proposed method has been applied effectively to analyze some strongly nonlinear oscillations such as the nonlinear Duffing oscillator with third, fifth, and seventh powers of the amplitude, the strongly nonlinear oscillators in forms (1 + 2 )̈+ = 0 and̈+ 3 /(1 + 2 ) = 0, and the cubic Duffing with discontinuity [33].…”

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].…”

“…14is called the classical/conventional averaging value which often leads to unacceptable errors for strongly nonlinear problems. In this paper, the weighted averaging value proposed by Anh et al [20,21] is used to calculate averaging values in Eq. (13).…”

An Approximate Solution for a Nonlinear Duffing – Harmonic Oscillator

“…To improve the accuracy of ELM, some extensions of the mean-square criterion was proposed such as a dual criterion [18,19] and local mean square error criterion [20]. Recently, Anh et al [21,22] introduced a new development of ELM based on the weighted averaging value. Instead of using the conventional/classical averaging value [12] in which the averaging value is calculated in one period of oscillation for harmonic functions, Anh et al proposed a new method for estimating the averaging value of deterministic functions by introducing a weighted coefficient function, the new averaging value is called the weighted one.…”