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

Hindawi Journal of Applied Mathematics Volume 2018, Article ID 7487851, 15 pages https://doi.org/10.1155/2018/7487851 Research Article The Equivalent Linearization Method with a Weighted Averaging for Solving Undamped Nonlinear Oscillators D. V. Hieu ,1 N. Q. Hai,2 and D. T. Hung1 1 Thai Nguyen University of Technology, Thai Nguyen, Vietnam Hanoi Architectural University, Ha Noi, Vietnam 2 Correspondence should be addressed to D. V. Hieu; Received 27 June 2017; Accepted 5 March 2018; Published 18 April 2018 Academic Editor: Peter G. L. Leach Copyright © 2018 D. V. Hieu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The Equivalent Linearization Method (ELM) with a weighted averaging is applied to analyze five undamped oscillator systems with nonlinearities. The results obtained via this method are compared with the ones achieved by Parameterized Perturbation Method (PPM), Min–Max Approach (MMA), Variational Iteration Method (VIM), Homotopy Perturbation Method (HPM), Energy Balance Method (EBM), Harmonic Balance Method (HBM), 4th-Order Runge-Kutta Method, and the exact ones. The obtained results demonstrate that this method is very convenient for solving nonlinear equations and also can be successfully applied to a lot of practical engineering and physical problems. 1. Introduction Nonlinear oscillation problem is very important in the physical science, mechanical structures, and other kinds of mathematical sciences. Most of real systems are modeled by nonlinear differential equations which are important issues in mechanical structures, mathematical physics, and engineering. In most cases, it is difficult to solve such equations, especially analytically; and in addition, the most important information, such as the natural circular frequency of a nonlinear oscillation which depends on the initial conditions, will be lost during the procedure of numerical simulation. Recently, nonlinear oscillator models have been widely considered in physics and engineering. It is obvious that there are many nonlinear equations in the study of different branches of science which do not have analytical solutions. Due to the limitation of existing exact solutions, many analytical approaches have been investigated. Many researchers have been working on various analytical methods for solving nonlinear oscillation systems in the last decades, such as Homotopy Perturbation Method (HPM) [1– 5], Max–Min Approach (MMA) [5–10], Variational Iteration Method (VIM) [2, 5], Energy Balance Method (EBM) [5, 11– 16], Amplitude-Frequency Formulation (AFF) [5–7, 15, 17, 18], Improved Amplitude-Frequency Formulation (IAFF) [5], Parameter Expansion Method (PEM) [5, 7, 18–20], Homotopy Analysis Method (HAM) [5, 21], Modified Homotopy Perturbation Method (MHPM) [5, 6], Modified LindstedtPoincare Method [22], Harmonic Balance Method [23, 24], and combined Newton’s Method with the Harmonic Balance Method [25]. The Equivalent Linearization Method is one of the common approaches to approximate analysis of dynamical systems. The original linearization for deterministic systems was proposed by Krylov and Bogoliuboff [26]. Then Caughey expanded the method for stochastic systems [27]. Thenceforward, there have been some extended versions of the Equivalent Linearization Method [28–32]. It has been shown that the Equivalent Linearization Method is presently the simplest tool widely used for analyzing nonlinear stochastic problems. Nevertheless, the accuracy of the Equivalent Linearization Method with conventional averaging normally reduces for middle or strong nonlinear systems. 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. Recently, Anh proposed a new way for determining averaging values: instead of using conventional averaging process the author introduced weighted coefficient functions ℎ(𝑡) [31]; by this manner, the averaging value is calculated in a new way called the weighted averaging value. 2 Journal of Applied Mathematics 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 + 𝜀𝑢2 )𝑢̈ + 𝑢 = 0 and 𝑢̈ + 𝑢3 /(1 + 𝑢2 ) = 0, and the cubic Duffing with discontinuity [33]. In this paper, the Equivalent Linearization Method with a weighted averaging continues to be applied to analyze five undamped nonlinear oscillators: the Duffing oscillator with cubic nonlinearity (Example 1); the motion of simple pendulum attached to a rotating rigid frame (Example 2); the motion of a mass attached to the center of a stretched elastic wire (Example 3); the Duffing oscillator with discontinuity (Example 4); and the nonlinear oscillator with fractional elastic force (Example 5). It should be emphasized that the generalized forms of the Duffing equation (in Example 1) and the oscillator (in Example 3) have been solved by Energy Balance Method [12, 16], and the generalized form of the nonlinear oscillator with fractional elastic force (in Example 5) has been solved by Energy Balance Method and Variational Approach [14]; these solutions are achieved by Younesian et al. [12, 14, 16]. The frequency-amplitude relationship is analytically obtained by using this proposed method. Accuracy and validity of results are then presented by comparing the results with the ones obtained by the other well-known techniques and the exact and numerical ones. 2. The Equivalent Linearization Method with a Weighted Averaging 2.1. The Equivalent Linearization Method. In order to introduce the general idea of the Equivalent Linearization Method, we consider a nonlinear oscillator governed by the following equation [33]: 𝑋̈ + 2ℎ𝑋̇ + 𝜔02 𝑋 + 𝑔 (𝑋,̇ 𝑋) = 0, (1) where 𝑔(𝑋,̇ 𝑋) is a nonlinear function only depending on two ̇ and displacement 𝑋(𝑡) and ℎ and 𝜔0 variables of velocity 𝑋(𝑡) are constants. The equivalent linear oscillator is described by the equation as follows: 𝑋̈ + (2ℎ + 𝜇) 𝑋̇ + (𝜔02 + 𝜆) 𝑋 = 0. (2) The equation error between the two oscillators is taken as 𝑒 (𝑋,̇ 𝑋) = 𝑔 (𝑋,̇ 𝑋) − 𝜇𝑋̇ − 𝜆𝑋. (3) The coefficients of linearization in the linearized equation (2) are found from an optimal criterion. There are some criteria for determining these coefficients. The most common criterion is the mean square error criterion which requires that the mean square of equation error be minimum: 2 ⟨𝑒2 (𝑋,̇ 𝑋)⟩ = ⟨(𝑔 (𝑋,̇ 𝑋) − 𝜇𝑋̇ − 𝜆𝑋) ⟩ 󳨀→ min. 𝜇,𝜆 Thus, 𝜕 ⟨𝑒2 (𝑋,̇ 𝑋)⟩ = 0 𝜕𝜆 yields 𝜆= 𝜇= ̇ ⟨𝑋𝑋⟩ ̇ ⟨𝑔𝑋⟩ ⟨𝑋̇ 2 ⟩ − ⟨𝑔𝑋⟩ ̇ ⟨𝑋2 ⟩ ⟨𝑋̇ 2 ⟩ − ⟨𝑋𝑋⟩ (6a) 2 ̇ ⟨𝑋2 ⟩ − ⟨𝑔𝑋⟩ ⟨𝑋𝑋⟩ ̇ ⟨𝑔𝑋⟩ ̇ 2 ⟨𝑋2 ⟩ ⟨𝑋̇ 2 ⟩ − ⟨𝑋𝑋⟩ . (6b) In the formulations in (4), (6a), and (6b), the symbol ⟨∙⟩ deno (...truncated)