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

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.


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.
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][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. 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( ) and cos( ) 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( ) [31]; by this manner, the averaging value is calculated in a new way called the weighted averaging value. 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].
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.

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]:̈+ where (̇, ) is a nonlinear function only depending on two variables of velocitẏ( ) and displacement ( ) and ℎ and 0 are constants. The equivalent linear oscillator is described by the equation as follows: The equation error between the two oscillators is taken as 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: Thus, In the formulations in (4), (6a), and (6b), the symbol ⟨•⟩ denotes the time-averaging operator in classical meaning [26]: For a -frequency function ( ), the averaging process is taken during one period ; that is, In this technique, the importance of the attended terms is considered to be the same on time scale. In fact, their roles generally differ from time to time. That may be one of the reasons causing the classical equivalent replacement to be effective only for oscillators with weak nonlinearity but normally not being good for ones with strong nonlinearity. In order to improve this shortcoming, the averaging operation with weighting functions is proposed as in the next section. This idea is introduced by Anh [31].

The Weighted
Averaging. The conventional averaging value of an integrable deterministic function ( ) on a domain : (0, ) is a constant value defined by In many cases when the function ( ) is periodic with period 2 / , the value is taken as 2 / and it leads to the averaging value of ( ) over one period: where = . Definition (9) has some deficiencies; for example, if (9) or (10) are equal to zero, the information about x(t) will be lost. For all harmonic functions such as cos( ) and sin( ), this observation is true. In [31], Anh suggested replacing the constant coefficient 1/ in (9) by a weighted coefficient function ℎ( ). Thus one gets the so-called weighted average value: (iii) Neutral Weighted Coefficients. They are denoted as ( ) and are constants.
An arbitrary weighted coefficient ℎ( ) can be obtained as summation and/or product of basic weighted coefficients. Example is where , , are constant. In this paper, a special form of weighted coefficient is introduced as follows [31,33]: where is constant. The properties of the weighted function ℎ( ) in (14) can be viewed in [31,33].
Based on the weighted coefficient (14), a new weighted averaging value is proposed: which is a linear operator. From Laplace transform, we get, for example, As -periodic functions ( ) can be expanded into Fourier series; hence we can easily calculate (15) by using (16).
In this paper, for the sake of comparison, the parameter is chosen equal to 2. In the next section, some practical examples for some strongly nonlinear vibration systems are illustrated to show the applicability, accuracy, and effectiveness of the present method.

Example 1. Consider the following Duffing equation with cubic nonlinearity:
The linearized equation of (17) is The equation error between (17) and (18) is where 2 is determined by using the mean square criterion, as follows: The periodic solution of the linearized equation (18) is Using definition (11) and weighted coefficient (14), we can calculate averaging operators in (20): Substituting (22) into (20), we get the approximate frequency of this oscillator: With chosen equal to 2, we have From (21), the approximate solution of this oscillator is Accuracy of this method for this example is shown in Table 1 and Figures 1 and 2. Table 1 shows the comparison between the approximate frequencies present obtained by this method, the ones obtained by Parameterized Perturbation 4

Journal of Applied Mathematics
Method PPM (PPM) [5], and the exact ones ; the relative error is again obtained for small and large values of the oscillation amplitude . Table 1 shows that the maximum relative error is less than 0.4733 for this method and 2.2135 for Parameterized Perturbation Method. The numerical results obtained by three different methods are illustrated in Figures  1 and 2. Numerical results validate again accuracy of this method. As it is shown in Figures 3 and 4, validity of the solution technique is guaranteed for stronger nonlinearities. Relative error versus the oscillation amplitude is illustrated in Figure 3.
The approximate frequency obtained by PPM [5] is the same as that obtained by EMB [12], as follows: And the exact frequency of this oscillator is [5] = 2

Example 2.
The motion of a simple pendulum attached to a rotating rigid frame that is shown in Figure 4 has following nonlinear differential equation [6,14]: with the initial conditions: in which and are generalized dimensionless displacements and time variables, and Λ = Ω 2 / .
To illustrate and verify the accuracy of the current analytical approach, the Equivalent Linearization Method with a weighted averaging, in comparison with other wellknown techniques, we performed a comparison between the results obtained by this method and outcomes achieved by Jouybari and Ramzani [6], by Younesian et al. [15], and by using the Runge-Kutta 4th-Order Method. Variations of ( ) for different Λ are illustrated in Figures 5-7.
The approximate frequency obtained by using MMA [6] is The approximate frequency obtained by using EBM [15] where is a parameter, 0 < ≤ 1.
Equation (43) can be written as follows: The linearized equation of (44) is The equation error between (44) and (45) is where 2 is determined by using the mean square criterion, as follows: The periodic solution of the linearized equation (45) is Using (48)   Numerical solution [12] MMA solution [6] . error (%) EBM solution [14] .
Thus, the approximate period of this oscillator is For = 0.1, = 0.5, = 0.75, and = 0.95, comparison of the approximate periods in (56), the approximate Variational Iteration Method (VIM) periods VIM [5] in (57), and the Energy Balance Method (EBM) periods EBM [16] in (58) with exact periods ex in (59) is tabulated in Table 4. From Table 4, the accuracy of this proposed method can be observed.
The approximate period obtained by Variational Iteration Method is as follows [5]: (57) The approximate period obtained by Energy Balance Method is as follows [16]: .
The exact period of this oscillator is as follows [18]: Comparisons of the approximate solutions obtained by two analytical methods with the exact ones is presented in Figures 8 and 9. And the relative error versus the oscillation amplitude is illustrated in Figure 10.

Example 4.
Consider the Duffing oscillator with discontinuity [3]:̈+ The linearized equation of (60) is The equation error between (60) and (61) is The unknown coefficient 2 is determined from the mean square error criterion, as follows: The periodic solution of (61) is Similar to Examples 1-3, we calculate averaging operators ⟨ 2 ⟩ and ⟨ 2 | |⟩; then, substituting these operators into (63) with note that the parameter is chosen equal to 2, we have the approximate frequency of this oscillator: Thus, the approximate solution is Accuracy of the approach for this example is shown in Figures 11 and 12. We performed a comparison between the results obtained by this method, the ones obtained by He [3] using the Homotopy Perturbation Method, and outcomes achieved using the Runge-Kutta 4th-Order Method for different values of , , and .

Example 5.
Consider the nonlinear oscillator equation of the form [23,24,34] where is a positive integer. The linearized equation of (67) is The equation error between (67) and (68) is The unknown coefficient 2 is determined from the mean square error criterion, as follows:   The periodic solution of (68) is Using definition (11) and weighted coefficient (14), we calculate averaging operators in (70): With chosen equal to 2, from (72) and (73), we have Substituting (71) and (72) into (70), we get the approximate frequency of this oscillator: For = 0 (the harmonic oscillator case) it follows from (77) and (76) that is equal to the well-known value 1. And, for → ∞, is equal to 0.97979 −1/2 . For other values of , the integral in (77) can be calculated numerically by using the formula manipulation package Maple. For some values of approximations of the frequency are given in Table 5. Table 5 presents the comparison of the approximate frequencies present in (76), the approximate frequencies app obtained by van Horssen [34], and the approximate frequencies HBM obtained by Mickens [23]. As can be seen from this table the fractional errors of the approximations obtained in [34] for ≥ 1 range from approximately 2 to approximately 11 percent, while the fractional errors of the approximations obtained in this paper range from 2 to 10 percent. And, from Table 5, when increases, the relative error of the approximate frequencies app obtained by van Horssen increases, while the relative error of the approximate frequencies app in this paper decreases.
Comparison of the approximate solutions obtained by three methods is present in Figures 13 and 14.

Conclusions
In this paper, the Equivalent Linearization Method with a weighted averaging is applied to analyze five undamped  [23] app [34] . error (%) nonlinear oscillation systems. This method is proposed by Anh in 2015. Instead of using the classical averaging process in (9) and (10), a special form of weighted coefficient ℎ( ) is introduced in (14), whereby a new weighted average value is computed by (15). The proposed weighted function is a combination of two basic coefficient functions (the optimistic weighted coefficient and the pessimistic weighted coefficient − ). The frequency-amplitude relationship was analytically obtained by this method. The frequency depends not only on the initial amplitude of the oscillator but also on the parameter in the expression of weighted coefficient ℎ( ). Accuracy of this method is investigated by five nonlinear oscillation systems. The results show that this method is useful in obtaining analytical solutions for oscillators and vibration problems with nonlinearities. The approximate solutions obtained by this method are valid not only for small parameters but also for very large parameters. And the results indicate that the solution procedure is easy and provides a remarkable accuracy. In this paper, is chosen equal to 2 for comparison and convenient calculating purposes. However, the value of the parameter in the expression of the weighted coefficient ℎ( ) should be chosen to give better results and the best solution still requires further investigation.