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

Authors: D. V. Hieu; N. Q. Hai; D. T. Hung
Journal: Journal of Applied Mathematics
 (2018)

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## Abstract

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.


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## Body

## 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 Lindstedt-Poincare 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. 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)u¨+u=0 and u¨+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]:(1)X¨+2hX˙+ω02X+gX˙,X=0,where g(X˙,X) is a nonlinear function only depending on two variables of velocity X˙(t) and displacement X(t) and h and ω0 are constants. The equivalent linear oscillator is described by the equation as follows:(2)X¨+2h+μX˙+ω02+λX=0.The equation error between the two oscillators is taken as(3)eX˙,X=gX˙,X-μX˙-λX.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:(4)e2X˙,X=gX˙,X-μX˙-λX2⟶minμ,λ.Thus,(5)∂∂λe2X˙,X=0∂∂μe2X˙,X=0yields(6a)λ=gXX˙2-gX˙XX˙X2X˙2-XX˙2(6b)μ=gX˙X2-gXXX˙X2X˙2-XX˙2.In the formulations in (4), (6a), and (6b), the symbol ∙ denotes the time-averaging operator in classical meaning [26]:(7)ft=limT→+∞1T∫0Tftdt.For aω-frequency function f(ωt), the averaging process is taken during one period T; that is,(8)fωt=1T∫0Tfωtdt=12π∫02πfτdτ,τ=ωt.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].


### 2.2. The Weighted Averaging

The conventional averaging value of an integrable deterministic functionx(t) on a domain D: (0,d) is a constant value defined by(9)xt=1d∫0dxtdt.In many cases when the functionx(ωt) is periodic with period 2π/ω, the value d is taken as 2π/ω and it leads to the averaging value of x(t) over one period:(10)xωt=ω2π∫02π/ωxωtdt=12π∫02πxτdτ,where τ=ωt. Definition (9) has some deficiencies; for example, if (9) or (10) are equal to zero, the information aboutx(t) will be lost. For all harmonic functions such as cos⁡(nωt) and sin⁡(nωt), this observation is true. In [31], Anh suggested replacing the constant coefficient 1/d in (9) by a weighted coefficient function h(t). Thus one gets the so-called weighted average value:(11)xt=∫0dhtxtdtwith the condition(12)∫0dhtdt=1.There are three basic weighted coefficients:(i)
Basic Optimistic Weighted Coefficients. They are increasing functions of t and denoted asO(t). Examples are αtβand αeβt,α, β > 0.(ii)
Basic Pessimistic Weighted Coefficients. They are decreasing functions of t and denoted as P(t). Examples are αtβ and αeβt,α<0,β>0; or α>0,β<0.(iii)
Neutral Weighted Coefficients. They are denoted as N(t) and are constants.An arbitrary weighted coefficienth(t) can be obtained as summation and/or product of basic weighted coefficients. Example is(13)ht=∑i=1nAiOit+BiPit+CiOitPit+Nt,where Ai,Bi,Ci are constant.In this paper, a special form of weighted coefficient is introduced as follows [31, 33]:(14)ht=s2ω2te-sωt,s>0,where s is constant.The properties of the weighted functionh(t) in (14) can be viewed in [31, 33].Based on the weighted coefficient (14), a new weighted averaging value is proposed:(15)xωt=∫0∞s2ω2te-sωtxωtdt=∫0∞s2τe-sτxτdτwhich is a linear operator. From Laplace transform, we get, for example,(16)cos⁡nωt=∫0∞s2ω2te-sωtcos⁡nωtdt=∫0∞s2τe-sτcos⁡nτdτ=s2s2-n2s2+n22sin⁡nωt=∫0∞s2ω2te-sωtsin⁡nωtdt=∫0∞s2τe-sτsin⁡nτdτ=s22sns2+n22.Asω-periodic functions x(ωt) can be expanded into Fourier series; hence we can easily calculate (15) by using (16).In this paper, for the sake of comparison, the parameters 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.


## 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]:(1)X¨+2hX˙+ω02X+gX˙,X=0,where g(X˙,X) is a nonlinear function only depending on two variables of velocity X˙(t) and displacement X(t) and h and ω0 are constants. The equivalent linear oscillator is described by the equation as follows:(2)X¨+2h+μX˙+ω02+λX=0.The equation error between the two oscillators is taken as(3)eX˙,X=gX˙,X-μX˙-λX.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:(4)e2X˙,X=gX˙,X-μX˙-λX2⟶minμ,λ.Thus,(5)∂∂λe2X˙,X=0∂∂μe2X˙,X=0yields(6a)λ=gXX˙2-gX˙XX˙X2X˙2-XX˙2(6b)μ=gX˙X2-gXXX˙X2X˙2-XX˙2.In the formulations in (4), (6a), and (6b), the symbol ∙ denotes the time-averaging operator in classical meaning [26]:(7)ft=limT→+∞1T∫0Tftdt.For aω-frequency function f(ωt), the averaging process is taken during one period T; that is,(8)fωt=1T∫0Tfωtdt=12π∫02πfτdτ,τ=ωt.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].


## 2.2. The Weighted Averaging

The conventional averaging value of an integrable deterministic functionx(t) on a domain D: (0,d) is a constant value defined by(9)xt=1d∫0dxtdt.In many cases when the functionx(ωt) is periodic with period 2π/ω, the value d is taken as 2π/ω and it leads to the averaging value of x(t) over one period:(10)xωt=ω2π∫02π/ωxωtdt=12π∫02πxτdτ,where τ=ωt. Definition (9) has some deficiencies; for example, if (9) or (10) are equal to zero, the information aboutx(t) will be lost. For all harmonic functions such as cos⁡(nωt) and sin⁡(nωt), this observation is true. In [31], Anh suggested replacing the constant coefficient 1/d in (9) by a weighted coefficient function h(t). Thus one gets the so-called weighted average value:(11)xt=∫0dhtxtdtwith the condition(12)∫0dhtdt=1.There are three basic weighted coefficients:(i)
Basic Optimistic Weighted Coefficients. They are increasing functions of t and denoted asO(t). Examples are αtβand αeβt,α, β > 0.(ii)
Basic Pessimistic Weighted Coefficients. They are decreasing functions of t and denoted as P(t). Examples are αtβ and αeβt,α<0,β>0; or α>0,β<0.(iii)
Neutral Weighted Coefficients. They are denoted as N(t) and are constants.An arbitrary weighted coefficienth(t) can be obtained as summation and/or product of basic weighted coefficients. Example is(13)ht=∑i=1nAiOit+BiPit+CiOitPit+Nt,where Ai,Bi,Ci are constant.In this paper, a special form of weighted coefficient is introduced as follows [31, 33]:(14)ht=s2ω2te-sωt,s>0,where s is constant.The properties of the weighted functionh(t) in (14) can be viewed in [31, 33].Based on the weighted coefficient (14), a new weighted averaging value is proposed:(15)xωt=∫0∞s2ω2te-sωtxωtdt=∫0∞s2τe-sτxτdτwhich is a linear operator. From Laplace transform, we get, for example,(16)cos⁡nωt=∫0∞s2ω2te-sωtcos⁡nωtdt=∫0∞s2τe-sτcos⁡nτdτ=s2s2-n2s2+n22sin⁡nωt=∫0∞s2ω2te-sωtsin⁡nωtdt=∫0∞s2τe-sτsin⁡nτdτ=s22sns2+n22.Asω-periodic functions x(ωt) can be expanded into Fourier series; hence we can easily calculate (15) by using (16).In this paper, for the sake of comparison, the parameters 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.


## 3. Some Examples and Discussions

### 3.1. Example 1

Consider the following Duffing equation with cubic nonlinearity:(17)u¨+αu+βu3=0,u0=A,u˙0=0.The linearized equation of (17) is(18)u¨+ω2u=0.The equation error between (17) and (18) is(19)eu=αu+βu3-ω2u,where ω2 is determined by using the mean square criterion, as follows:(20)ω2=αu2+βu4u2.The periodic solution of the linearized equation (18) is(21)ut=Acos⁡ωt.Using definition (11) and weighted coefficient (14), we can calculate averaging operators in (20):(22)u2=A2cos2ωt=∫0+∞A2s2ω2te-sωtcos2ωtdt=∫0+∞A2s2τe-sτcos2τdτ=A2s4+2s2+8s2+42u4=A4cos4ωt=∫0+∞A4s2ω2te-sωtcos4ωtdt=∫0+∞A4s2τe-sτcos4τdτ=A4248s4+416s2+1536+28s6+s8s2+42s2+162.Substituting (22) into (20), we get the approximate frequency of this oscillator:(23)ω=α+βA2248s4+416s2+1536+28s6+s8s4+2s2+8s2+162.Withs chosen equal to 2, we have(24)ω=α+921612800βA2=α+0.72βA2.From (21), the approximate solution of this oscillator is(25)ut=Acos⁡α+0.72βA2t.Accuracy of this method for this example is shown in Table1 and Figures 1 and 2. Table 1 shows the comparison between the approximate frequencies ωpresent obtained by this method, the ones obtained by Parameterized Perturbation Method ωPPM (PPM) [5], and the exact ones ωe; the relative error is again obtained for small and large values of the oscillation amplitude A. 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.Table 1
Comparison of the approximate frequencies with the exact frequency, Example 1.
A α β ω e ω P P M R. error (%) ω P r e s e n t R. error (%) 0.1 0.5 0.1 0.7076 0.7076 0.0000 0.7076 0.0000 0.5 0.1 2 0.6800 0.6892 1.3501 0.6782 0.2647 1 2 0.5 1.5403 1.5411 0.0520 1.5362 0.2662 2 5 2 3.2958 3.3166 0.6313 3.2802 0.4733 5 2 5 9.5818 9.7852 2.1228 9.5917 0.1033 10 1 0.5 6.0722 6.2048 2.0994 6.0828 0.1746 15 0.5 2 17.9866 18.3848 2.2135 18.0139 0.1518 20 5 1 17.0977 17.4642 2.1436 17.1172 0.1141Figure 1
Comparison of time history diagram of displacement between the present, PPM, and exact solutions atα=2,β=10,A=2, Example 1.Figure 2
Comparison of time history diagram of displacement between the present, PPM, and exact solutions atα=5,β=5,A=10, Example 1.Figure 3
Comparison of relative error between the present and PPM solutions withα=2 and β=5, Example 1.Figure 4
A simple pendulum attached to a rotating rigid frame.The approximate frequency obtained by PPM [5] is the same as that obtained by EMB [12], as follows:(26)ωPPM=ωEBM=α+34βA2.And the exact frequency of this oscillator is [5](27)ωe=2π42∫0π/2dtβA2cos2t+βA2+2α.


### 3.2. Example 2

The motion of a simple pendulum attached to a rotating rigid frame that is shown in Figure4 has following nonlinear differential equation [6, 14]:(28)θ¨+1-Λcos⁡θsin⁡θ=0with the initial conditions:(29)θ0=A,θ˙0=0,in whichθ and t are generalized dimensionless displacements and time variables, and Λ=Ω2r/g.Equation (28) can be rewritten as follows:(30)θ¨+sin⁡θ-Λ2sin⁡2θ=0.Based on Taylor-Maclaurin expansion, the approximations ofsin⁡(θ) and sin⁡(2θ) are considered:(31)sin⁡θ≈θ-θ36+θ5120,sin⁡2θ≈2θ-4θ33+4θ515.Substituting (31) into (30) yields(32)θ¨+1-Λθ+-16+2Λ3θ3+1120-2Λ15θ5=0.The linearized equation of (32) is(33)θ¨+1-Λ+αθ=0.The equation error between (32) and (33) is(34)eθ=-16+2Λ3θ3+1120-2Λ15θ5-αθ.The unknown coefficientα is determined from the mean square error criterion:(35)∂∂αe2θ=0.It yields(36)α=-1/6+2Λ/3θ4+1/120-2Λ/15θ6θ2.The periodic solution and the frequency of (33) are(37)θt=Acos⁡ωt,ω=1-Λ+α.Similarly to Example 1, we calculate averaging operatorsθ2, θ4, and θ6:(38)θ2=A2cos2ωt=A2s4+2s2+8s2+42θ4=A4cos4ωt=A2248s4+416s2+1536+28s6+s8s2+42s2+162θ6=A6cos6ωt=A61658880+440064s2+282496s4+45712s6+3168s8+94s10+s12s2+42s2+162s2+362.Substituting averaging operators in (38) into (36), and then substituting (36) into (37), we get the approximate frequency (with s chosen equal to 2):(39)ω=1-Λ+0.72-16+2Λ3A2+0.5751120-2Λ15A4,and the approximate solution:(40)θt=Acos⁡1-Λ+0.72-16+2Λ3A2+0.5751120-2Λ15A4t.Comparisons between the results of the present method and the ones achieved by Jouybari and Ramzani [6] using Min–Max Approach (MMA), by Younesian et al. [14] using EBM, and using accurate numerical method are demonstrated in Tables 2 and 3, for two cases of Λ=0.75 and Λ=1. A very interesting agreement between the results is observed.Table 2
The results of present, approximate, obtained by Jouybari & Ramzani and by Yousesian et al., and numerical solution forθ(t) when Λ=0.75 and A=1, Example 2.
t Numerical solution [12] MMA solution [6] R. error(%) EBM solution[14] R. error(%) Present solution R. error(%) 0 1.000000000 1.000000000 0.000000000 1.000000000 0.000000000 1.000000000 0.000000000 0.5 0.939545503 0.945169971 0.598637105 0.944189348 0.494265045 0.945834709 0.669388122 1 0.769687769 0.786692549 2.20930885 0.782987048 1.727879737 0.789206596 2.535940908 1.5 0.520367974 0.541946378 4.146758655 0.534386712 2.694004762 0.547083274 5.13392471 2 0.226896964 0.237770336 4.792206916 0.226137434 0.334746656 0.245694103 8.284438306 2.5 −0.082424976 −0.092479615 12.19853434 −0.107353589 30.24400395 −0.082311253 0.137971529 3 −0.386261801 −0.412588246 6.815699852 −0.428861683 11.0287587 −0.401399783 3.919098902 3.5 −0.660436345 −0.687452426 4.090641165 −0.702499667 6.369019864 −0.677004442 2.50865918 4 −0.871990011 −0.886930535 1.713382471 −0.897723721 2.951147338 −0.879268816 0.834734906 4.5 −0.986529936 −0.989147789 0.265359712 −0.992742682 0.629757473 −0.986281489 0.025183929 5 −0.982752773 −0.982895042 0.014476581 −0.970950401 1.200950262 −0.986449715 0.376182302 5.5 −0.861382758 −0.868857967 0.867815025 −0.852109657 1.07653664 −0.879755271 2.132909305 6 −0.644890958 −0.659541879 2.271844692 −0.632155313 1.974852468 −0.677756428 5.09628327 6.5 −0.368002797 −0.377900390 2.689542873 −0.341638967 7.164029789 −0.402335838 9.329559797 7 −0.063204827 −0.054818323 13.26877139 −0.012988435 79.45024832 −0.083329972 31.84115194 7.5 0.245785975 0.274275125 11.59103972 0.317111883 29.01951912 0.244703077 0.440585758 8 0.537446333 0.573291546 6.669542762 0.611815759 13.8375539 0.546227301 1.633831596 8.5 0.78289188 0.809440784 3.391132885 0.838227903 7.068156461 0.788578403 0.726348445 9 0.946701653 0.956826699 1.069507586 0.971676667 2.638108207 0.945502349 0.12668236 9.5 0.999764591 0.999286944 0.047775947 0.995531392 0.423419577 0.999999478 0.023494231 10 0.931963943 0.932165326 0.021608454 0.908764206 2.489338474 0.946166081 1.523893505Table 3
The results of present, approximate obtained by Jouybari & Ramzani, and numerical solution forθ(t) when Λ=1 and A=1, Example 2.
t Numerical solution [2] MMA solution[6] R. error(%) EBM solution[14] R. error(%) Present solution R. error(%) 0 1.000000000 1.000000000 0.000000000 1.000000000 0.000000000 1.000000000 0.000000000 0.5 0.953968489 0.963119576 0.959265123 0.961786066 0.819479583 0.964200044 1.072525468 1 0.825442672 0.855198634 3.604849011 0.850064873 2.982908666 0.859363450 4.109404463 1.5 0.638888746 0.684197515 7.091808908 0.673375035 5.397855138 0.692996509 8.469043059 2 0.422287711 0.462729406 9.576810773 0.445220578 5.430626183 0.477011079 12.95878771 2.5 0.195637881 0.207129984 5.874170657 0.183038862 6.439969057 0.226871698 15.96511721 3 −0.032545617 −0.063747522 95.87129659 −0.093132124 186.1587291 −0.039511677 21.40398813 3.5 −0.260619945 −0.329922957 26.5915995 −0.362185221 38.97064593 −0.303066019 16.28657929 4 −0.485739543 −0.571762994 17.70978959 −0.603557273 24.25533019 −0.544920861 12.18375544 4.5 −0.696309233 −0.771429307 10.78832083 −0.79880073 14.71925003 −0.747759417 7.388984888 5 −0.869266232 −0.914194339 5.16850941 −0.93299355 7.331162267 −0.897058465 3.197206101 5.5 −0.976272484 −0.989527622 1.357729345 −0.995879662 2.008371466 −0.982128207 0.599804163 6 −0.996192580 −0.991872507 0.43365842 −0.982652815 1.35915136 −0.996877655 0.068769334 6.5 −0.924854298 −0.921056034 0.410687825 −0.894323908 3.301102678 −0.940250751 1.664743629 7 −0.776876524 −0.782301685 0.698329893 −0.737643731 5.050068034 −0.816301976 5.074867213 7.5 −0.579081909 −0.585844101 1.167743612 −0.524587017 9.410567167 −0.633906052 9.467424582 8 −0.358097172 −0.346174158 3.329547098 −0.271437236 24.20011739 −0.406122511 13.41125894 8.5 −0.130576292 −0.080970115 37.99018661 0.002457915 101.8823593 −0.149260634 14.30913814 9 0.097635936 0.190206352 94.8118283 0.276165213 182.8520157 0.118288292 21.15241257 9.5 0.325420035 0.447353037 37.46942071 0.528765792 62.48716586 0.377367786 15.96329218 10 0.548087739 0.671502582 22.51735155 0.740953929 35.1889262 0.609427780 11.19164627To illustrate and verify the accuracy of the current analytical approach, the Equivalent Linearization Method with a weighted averaging, in comparison with other well-known 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 θ(t) for different Λ are illustrated in Figures 5–7.Figure 5
A comparison between the approximate and numerical solutions withΛ=1 and A=0.5, Example 2.Figure 6
A comparison between the approximate and numerical solutions withΛ=1 and A=0.3, Example 2.Figure 7
A comparison between the approximate and numerical solutions withΛ=1 and A=1, Example 2.The approximate frequency obtained by using MMA [6] is(41)ωMMA=1-18A2-112ΛA4+12ΛA2-Λ+1192A4.The approximate frequency obtained by using EBM [15] is(42)ωEBM=2Acos⁡A2-cos⁡A-Λ4cos⁡2A+Λ4cos⁡2A.


### 3.3. Example 3

The equation of motion of a mass attached to the center of a stretched elastic wire in dimensionless form is as follows [5, 16, 18]:(43)u¨+u-ηu1+u2=0,u0=A,u˙0=0,where η is a parameter, 0<η≤1.Equation (43) can be written as follows:(44)1+u2u¨+1+u2u-ηu=0.The linearized equation of (44) is(45)u¨+ω2u=0.The equation error between (44) and (45) is(46)eu=1+εu2u¨+1+εu2u-ηu-u¨-ω2u,where ω2 is determined by using the mean square criterion, as follows: (47)ω2=1+u2u¨u+u21+u2-ηu2-uu¨u2.The periodic solution of the linearized equation (45) is(48)ut=Acos⁡ωt.Using (48), we have(49)uu¨=-Acos⁡ωtAω2cos⁡ωt=-A2ω2cos2ωt=-ω2u2.Thus(50)1+u2u¨u=-ω2u21+u2,uu¨=-ω2u2.Substituting (49) and (50) into (47), we get(51)ω2=1-ω2u21+u2u2-η+ω2.Using definition (11) and weighted coefficient (14), we calculate averaging operators in (51):(52)u2=A2cos2ωt=∫0+∞A2s2ω2te-sωtcos2ωtdt=∫0+∞A2s2τe-sτcos2τdτ=A2s4+2s2+8s2+42u21+u2=A2cos2ωt1+A2cos2ωt=∫0+∞A2s2ω2te-sωtcos2ωt1+A2cos2ωtdt=∫0+∞A2s2τe-sτcos2τ1+A2cos2τdτ.Withs chosen equal to 2, substituting (52) into (51), we get the approximate frequency of this oscillator:(53)ω=1-ηA22Iand the approximate solution of this oscillator:(54)ut=Acos⁡1-ηA22It,where(55)I=∫0+∞4A2τe-2τcos2τ1+A2cos2τdτ.Thus, the approximate period of this oscillator is(56)T=2πω=2π1-ηA2/2I.Forη=0.1, η=0.5, η=0.75, and η=0.95, comparison of the approximate periods T in (56), the approximate Variational Iteration Method (VIM) periods TVIM [5] in (57), and the Energy Balance Method (EBM) periods TEBM [16] in (58) with exact periods Tex in (59) is tabulated in Table 4. From Table 4, the accuracy of this proposed method can be observed.Table 4
Comparison of the approximate periods with the exact periods, Example 3.
A η T e x [18] T V I M [5] R. error (%) T P r e s e n t R. error (%) T E B M [16] R. error (%) 0.1 0.1 6.62168 6.621237 0.00669 6.62174 0.00091 6.62169 0.00011 1 0.1 6.537508 6.517854 0.30063 6.53806 0.00844 6.53573 0.02723 10 0.1 6.322938 6.314678 0.13064 6.32097 0.03112 6.32006 0.04558 100 0.1 6.287182 6.286329 0.01357 6.28696 0.00353 6.28687 0.00498 0.1 0.5 8.869257 8.863794 0.06159 8.86990 0.00725 8.86925 0.0001 1 0.5 7.992133 7.814722 2.21982 7.99408 0.02436 7.97282 0.2416 10 0.5 6.490208 6.445572 0.68774 6.47925 0.16884 6.47431 0.24498 100 0.5 6.303280 6.298952 0.06866 6.30214 0.01809 6.30167 0.02557 0.1 0.75 12.49673 12.47385 0.18309 12.49943 0.02161 12.4967 0.00041 1 0.75 9.625404 9.168186 4.75012 9.61936 0.06279 9.56406 0.63733 10 0.75 6.602092 6.531632 1.06724 6.58442 0.26767 6.57665 0.38536 100 0.75 6.313404 6.306879 0.10335 6.31168 0.02731 6.31097 0.03854 0.1 0.95 27.15679 26.86136 1.08787 27.18975 0.12137 27.1540 0.01043 1 0.95 12.075277 10.96676 9.18005 12.00887 0.54994 11.8737 1.66932 10 0.95 6.696117 6.60302 1.39031 6.67235 0.35494 6.66211 0.50781 100 0.95 6.321539 6.313243 0.13123 6.31934 0.03479 6.31844 0.04897The approximate period obtained by Variational Iteration Method is as follows [5]:(57)TVIM=2π1+A21+A2-1+A2η.The approximate period obtained by Energy Balance Method is as follows [16]:(58)TEBM=2π2/AA2/4-η-A2/2+1+A2+1.The exact period of this oscillator is as follows [18]:(59)Tex=4∫0π/21-2η1+A2sin2θ+1+A2-1/2dθ.Comparisons of the approximate solutions obtained by two analytical methods with the exact ones is presented in Figures8 and 9. And the relative error versus the oscillation amplitude is illustrated in Figure 10.Figure 8
Comparison of time history diagram of displacement between the present, VIM, and exact solutions atη=0.9,A=1,Example 3.Figure 9
Comparison of time history diagram of displacement between the present, VIM, and exact solutions atη=0.7,A=10, Example 3.Figure 10
Comparison of relative error between the present, VIM, and EBM solutions withη=0.75, Example 3.


### 3.4. Example 4

Consider the Duffing oscillator with discontinuity [3]:(60)u¨+αu+βuu=0,u0=A,u˙0=0.The linearized equation of (60) is(61)u¨+ω2u=0.The equation error between (60) and (61) is(62)eu=αu+βuu-ω2u.The unknown coefficientω2 is determined from the mean square error criterion, as follows:(63)ω2=α+βu2uu2.The periodic solution of (61) is(64)ut=Acos⁡ωt.Similar to Examples 1–3, we calculate averaging operatorsu2 and u2u; then, substituting these operators into (63) with note that the parameter s is chosen equal to 2, we have the approximate frequency of this oscillator:(65)ω=α+0.8324βA.Thus, the approximate solution is(66)ut=Acos⁡α+0.8324εAt.Accuracy of the approach for this example is shown in Figures11 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 A, α, and β.Figure 11
Comparison of time history diagram of displacement between the present, HPM, and numerical solutions atα=0,β=10,A=0.1, Example 4.Figure 12
Comparison of time history diagram of displacement between the Present, HPM, and numerical solutions atα=10,β=100,A=2, Example 4.


### 3.5. Example 5

Consider the nonlinear oscillator equation of the form [23, 24, 34](67)u¨+u1/2n+1=0,u0=A,u˙0=0,where n is a positive integer.The linearized equation of (67) is(68)u¨+ω2u=0.The equation error between (67) and (68) is(69)eu=u1/2n+1-ω2u.The unknown coefficientω2 is determined from the mean square error criterion, as follows:(70)ω2=uu1/2n+1u2.The periodic solution of (68) is(71)ut=Acos⁡ωt.Using definition (11) and weighted coefficient (14), we calculate averaging operators in (70):(72)u2=A2cos2ωt=∫0+∞A2s2ω2te-sωtcos2ωtdt=∫0+∞A2s2τe-sτcos2τdτ=A2s4+2s2+8s2+42(73)uu1/2n+1=Acos⁡ωtA1/2n+1cos⁡ωt1/2n+1=∫0+∞AA1/2n+1s2ω2te-sωtcos⁡ωtcos⁡ωt1/2n+1dt=∫0+∞AA1/2n+1s2τe-sτcos⁡τcos⁡τ1/2n+1dτ.Withs chosen equal to 2, from (72) and (73), we have(74)u2=A22(75)uu1/2n+1=∫0+∞A2n+2/2n+14τe-2τcos⁡τcos⁡τ1/2n+1dτ.Substituting (71) and (72) into (70), we get the approximate frequency of this oscillator:(76)ω=A-n/2n+12In,where(77)In=∫0+∞4τe-2τcos⁡τcos⁡τ1/2n+1dτ.Forn=0 (the harmonic oscillator case) it follows from (77) and (76) that ω is equal to the well-known value 1. And, for n→∞, ω is equal to 0.97979A-1/2. For other values of n, the integral in (77) can be calculated numerically by using the formula manipulation package Maple. For some values of n approximations of the frequency are given in Table 5. Table 5 presents the comparison of the approximate frequencies ωpresent in (76), the approximate frequencies ωapp obtained by van Horssen [34], and the approximate frequencies ωHBM obtained by Mickens [23]. As can be seen from this table the fractional errors of the approximations obtained in [34] for n≥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 n increases, the relative error of the approximate frequencies ωapp obtained by van Horssen increases, while the relative error of the approximate frequencies ωapp in this paper decreases.Table 5
Comparison of the approximate frequencies, Example 5.
n ω H B M [23] ω a p p [34] R. error (%) ω p r e s e n t R. error (%) 0 1.00000 1.000000 0.00000 1.000000 0.00000 1 A - 1 / 3 1.04912 A - 1 / 3 1.07045 2.03313 A - 1 / 3 0.94307 10.10847 2 A - 2 / 5 1.04812 A - 2 / 5 1.08613 3.62649 A - 2 / 5 0.95714 8.680304 3 A - 3 / 7 1.04405 A - 3 / 7 1.09302 4.69039 A - 3 / 7 0.96342 7.72281 4 A - 4 / 9 1.04017 A - 4 / 9 1.09689 5.45295 A - 4 / 9 0.96697 7.037311 5 A - 5 / 11 1.03684 A - 5 / 11 1.09938 6.03179 A - 5 / 11 0.96926 6.517881 6 A - 6 / 13 1.03403 A - 6 / 13 1.10110 6.48627 A - 6 / 13 0.97085 6.110074 7 A - 7 / 15 1.03164 A - 7 / 15 1.10238 6.85704 A - 7 / 15 0.97202 5.779148 8 A - 8 / 17 1.02960 A - 8 / 17 1.10335 7.16298 A - 8 / 17 0.97293 5.504079 9 A - 9 / 19 1.02783 A - 9 / 19 1.10412 7.42243 A - 9 / 19 0.97364 5.272273 10 A - 10 / 21 1.02628 A - 10 / 21 1.10474 7.64509 A - 10 / 21 0.97422 5.07269 50 A - 50 / 101 1.00919 A - 50 / 101 1.10947 9.93668 A - 50 / 101 0.97863 3.028171 500 A - 500 / 1001 1.00149 A - 500 / 1001 1.11059 10.8938 A - 500 / 1001 0.97968 2.17776 ∞ A - 1 / 2 1.00000 A - 1 / 2 π / 2 2 11.07200 A - 1 / 2 0.97979 2.02100Comparison of the approximate solutions obtained by three methods is present in Figures13 and 14.Figure 13
Comparison of time history diagram of displacement between the present, HBM [23], and App [34] solutions with n=10, Example 5.Figure 14
Comparison of time history diagram of displacement between the Present, HBM [23] and App [34] solutions with n=50, Example 5.


## 3.1. Example 1

Consider the following Duffing equation with cubic nonlinearity:(17)u¨+αu+βu3=0,u0=A,u˙0=0.The linearized equation of (17) is(18)u¨+ω2u=0.The equation error between (17) and (18) is(19)eu=αu+βu3-ω2u,where ω2 is determined by using the mean square criterion, as follows:(20)ω2=αu2+βu4u2.The periodic solution of the linearized equation (18) is(21)ut=Acos⁡ωt.Using definition (11) and weighted coefficient (14), we can calculate averaging operators in (20):(22)u2=A2cos2ωt=∫0+∞A2s2ω2te-sωtcos2ωtdt=∫0+∞A2s2τe-sτcos2τdτ=A2s4+2s2+8s2+42u4=A4cos4ωt=∫0+∞A4s2ω2te-sωtcos4ωtdt=∫0+∞A4s2τe-sτcos4τdτ=A4248s4+416s2+1536+28s6+s8s2+42s2+162.Substituting (22) into (20), we get the approximate frequency of this oscillator:(23)ω=α+βA2248s4+416s2+1536+28s6+s8s4+2s2+8s2+162.Withs chosen equal to 2, we have(24)ω=α+921612800βA2=α+0.72βA2.From (21), the approximate solution of this oscillator is(25)ut=Acos⁡α+0.72βA2t.Accuracy of this method for this example is shown in Table1 and Figures 1 and 2. Table 1 shows the comparison between the approximate frequencies ωpresent obtained by this method, the ones obtained by Parameterized Perturbation Method ωPPM (PPM) [5], and the exact ones ωe; the relative error is again obtained for small and large values of the oscillation amplitude A. 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.Table 1
Comparison of the approximate frequencies with the exact frequency, Example 1.
A α β ω e ω P P M R. error (%) ω P r e s e n t R. error (%) 0.1 0.5 0.1 0.7076 0.7076 0.0000 0.7076 0.0000 0.5 0.1 2 0.6800 0.6892 1.3501 0.6782 0.2647 1 2 0.5 1.5403 1.5411 0.0520 1.5362 0.2662 2 5 2 3.2958 3.3166 0.6313 3.2802 0.4733 5 2 5 9.5818 9.7852 2.1228 9.5917 0.1033 10 1 0.5 6.0722 6.2048 2.0994 6.0828 0.1746 15 0.5 2 17.9866 18.3848 2.2135 18.0139 0.1518 20 5 1 17.0977 17.4642 2.1436 17.1172 0.1141Figure 1
Comparison of time history diagram of displacement between the present, PPM, and exact solutions atα=2,β=10,A=2, Example 1.Figure 2
Comparison of time history diagram of displacement between the present, PPM, and exact solutions atα=5,β=5,A=10, Example 1.Figure 3
Comparison of relative error between the present and PPM solutions withα=2 and β=5, Example 1.Figure 4
A simple pendulum attached to a rotating rigid frame.The approximate frequency obtained by PPM [5] is the same as that obtained by EMB [12], as follows:(26)ωPPM=ωEBM=α+34βA2.And the exact frequency of this oscillator is [5](27)ωe=2π42∫0π/2dtβA2cos2t+βA2+2α.


## 3.2. Example 2

The motion of a simple pendulum attached to a rotating rigid frame that is shown in Figure4 has following nonlinear differential equation [6, 14]:(28)θ¨+1-Λcos⁡θsin⁡θ=0with the initial conditions:(29)θ0=A,θ˙0=0,in whichθ and t are generalized dimensionless displacements and time variables, and Λ=Ω2r/g.Equation (28) can be rewritten as follows:(30)θ¨+sin⁡θ-Λ2sin⁡2θ=0.Based on Taylor-Maclaurin expansion, the approximations ofsin⁡(θ) and sin⁡(2θ) are considered:(31)sin⁡θ≈θ-θ36+θ5120,sin⁡2θ≈2θ-4θ33+4θ515.Substituting (31) into (30) yields(32)θ¨+1-Λθ+-16+2Λ3θ3+1120-2Λ15θ5=0.The linearized equation of (32) is(33)θ¨+1-Λ+αθ=0.The equation error between (32) and (33) is(34)eθ=-16+2Λ3θ3+1120-2Λ15θ5-αθ.The unknown coefficientα is determined from the mean square error criterion:(35)∂∂αe2θ=0.It yields(36)α=-1/6+2Λ/3θ4+1/120-2Λ/15θ6θ2.The periodic solution and the frequency of (33) are(37)θt=Acos⁡ωt,ω=1-Λ+α.Similarly to Example 1, we calculate averaging operatorsθ2, θ4, and θ6:(38)θ2=A2cos2ωt=A2s4+2s2+8s2+42θ4=A4cos4ωt=A2248s4+416s2+1536+28s6+s8s2+42s2+162θ6=A6cos6ωt=A61658880+440064s2+282496s4+45712s6+3168s8+94s10+s12s2+42s2+162s2+362.Substituting averaging operators in (38) into (36), and then substituting (36) into (37), we get the approximate frequency (with s chosen equal to 2):(39)ω=1-Λ+0.72-16+2Λ3A2+0.5751120-2Λ15A4,and the approximate solution:(40)θt=Acos⁡1-Λ+0.72-16+2Λ3A2+0.5751120-2Λ15A4t.Comparisons between the results of the present method and the ones achieved by Jouybari and Ramzani [6] using Min–Max Approach (MMA), by Younesian et al. [14] using EBM, and using accurate numerical method are demonstrated in Tables 2 and 3, for two cases of Λ=0.75 and Λ=1. A very interesting agreement between the results is observed.Table 2
The results of present, approximate, obtained by Jouybari & Ramzani and by Yousesian et al., and numerical solution forθ(t) when Λ=0.75 and A=1, Example 2.
t Numerical solution [12] MMA solution [6] R. error(%) EBM solution[14] R. error(%) Present solution R. error(%) 0 1.000000000 1.000000000 0.000000000 1.000000000 0.000000000 1.000000000 0.000000000 0.5 0.939545503 0.945169971 0.598637105 0.944189348 0.494265045 0.945834709 0.669388122 1 0.769687769 0.786692549 2.20930885 0.782987048 1.727879737 0.789206596 2.535940908 1.5 0.520367974 0.541946378 4.146758655 0.534386712 2.694004762 0.547083274 5.13392471 2 0.226896964 0.237770336 4.792206916 0.226137434 0.334746656 0.245694103 8.284438306 2.5 −0.082424976 −0.092479615 12.19853434 −0.107353589 30.24400395 −0.082311253 0.137971529 3 −0.386261801 −0.412588246 6.815699852 −0.428861683 11.0287587 −0.401399783 3.919098902 3.5 −0.660436345 −0.687452426 4.090641165 −0.702499667 6.369019864 −0.677004442 2.50865918 4 −0.871990011 −0.886930535 1.713382471 −0.897723721 2.951147338 −0.879268816 0.834734906 4.5 −0.986529936 −0.989147789 0.265359712 −0.992742682 0.629757473 −0.986281489 0.025183929 5 −0.982752773 −0.982895042 0.014476581 −0.970950401 1.200950262 −0.986449715 0.376182302 5.5 −0.861382758 −0.868857967 0.867815025 −0.852109657 1.07653664 −0.879755271 2.132909305 6 −0.644890958 −0.659541879 2.271844692 −0.632155313 1.974852468 −0.677756428 5.09628327 6.5 −0.368002797 −0.377900390 2.689542873 −0.341638967 7.164029789 −0.402335838 9.329559797 7 −0.063204827 −0.054818323 13.26877139 −0.012988435 79.45024832 −0.083329972 31.84115194 7.5 0.245785975 0.274275125 11.59103972 0.317111883 29.01951912 0.244703077 0.440585758 8 0.537446333 0.573291546 6.669542762 0.611815759 13.8375539 0.546227301 1.633831596 8.5 0.78289188 0.809440784 3.391132885 0.838227903 7.068156461 0.788578403 0.726348445 9 0.946701653 0.956826699 1.069507586 0.971676667 2.638108207 0.945502349 0.12668236 9.5 0.999764591 0.999286944 0.047775947 0.995531392 0.423419577 0.999999478 0.023494231 10 0.931963943 0.932165326 0.021608454 0.908764206 2.489338474 0.946166081 1.523893505Table 3
The results of present, approximate obtained by Jouybari & Ramzani, and numerical solution forθ(t) when Λ=1 and A=1, Example 2.
t Numerical solution [2] MMA solution[6] R. error(%) EBM solution[14] R. error(%) Present solution R. error(%) 0 1.000000000 1.000000000 0.000000000 1.000000000 0.000000000 1.000000000 0.000000000 0.5 0.953968489 0.963119576 0.959265123 0.961786066 0.819479583 0.964200044 1.072525468 1 0.825442672 0.855198634 3.604849011 0.850064873 2.982908666 0.859363450 4.109404463 1.5 0.638888746 0.684197515 7.091808908 0.673375035 5.397855138 0.692996509 8.469043059 2 0.422287711 0.462729406 9.576810773 0.445220578 5.430626183 0.477011079 12.95878771 2.5 0.195637881 0.207129984 5.874170657 0.183038862 6.439969057 0.226871698 15.96511721 3 −0.032545617 −0.063747522 95.87129659 −0.093132124 186.1587291 −0.039511677 21.40398813 3.5 −0.260619945 −0.329922957 26.5915995 −0.362185221 38.97064593 −0.303066019 16.28657929 4 −0.485739543 −0.571762994 17.70978959 −0.603557273 24.25533019 −0.544920861 12.18375544 4.5 −0.696309233 −0.771429307 10.78832083 −0.79880073 14.71925003 −0.747759417 7.388984888 5 −0.869266232 −0.914194339 5.16850941 −0.93299355 7.331162267 −0.897058465 3.197206101 5.5 −0.976272484 −0.989527622 1.357729345 −0.995879662 2.008371466 −0.982128207 0.599804163 6 −0.996192580 −0.991872507 0.43365842 −0.982652815 1.35915136 −0.996877655 0.068769334 6.5 −0.924854298 −0.921056034 0.410687825 −0.894323908 3.301102678 −0.940250751 1.664743629 7 −0.776876524 −0.782301685 0.698329893 −0.737643731 5.050068034 −0.816301976 5.074867213 7.5 −0.579081909 −0.585844101 1.167743612 −0.524587017 9.410567167 −0.633906052 9.467424582 8 −0.358097172 −0.346174158 3.329547098 −0.271437236 24.20011739 −0.406122511 13.41125894 8.5 −0.130576292 −0.080970115 37.99018661 0.002457915 101.8823593 −0.149260634 14.30913814 9 0.097635936 0.190206352 94.8118283 0.276165213 182.8520157 0.118288292 21.15241257 9.5 0.325420035 0.447353037 37.46942071 0.528765792 62.48716586 0.377367786 15.96329218 10 0.548087739 0.671502582 22.51735155 0.740953929 35.1889262 0.609427780 11.19164627To illustrate and verify the accuracy of the current analytical approach, the Equivalent Linearization Method with a weighted averaging, in comparison with other well-known 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 θ(t) for different Λ are illustrated in Figures 5–7.Figure 5
A comparison between the approximate and numerical solutions withΛ=1 and A=0.5, Example 2.Figure 6
A comparison between the approximate and numerical solutions withΛ=1 and A=0.3, Example 2.Figure 7
A comparison between the approximate and numerical solutions withΛ=1 and A=1, Example 2.The approximate frequency obtained by using MMA [6] is(41)ωMMA=1-18A2-112ΛA4+12ΛA2-Λ+1192A4.The approximate frequency obtained by using EBM [15] is(42)ωEBM=2Acos⁡A2-cos⁡A-Λ4cos⁡2A+Λ4cos⁡2A.


## 3.3. Example 3

The equation of motion of a mass attached to the center of a stretched elastic wire in dimensionless form is as follows [5, 16, 18]:(43)u¨+u-ηu1+u2=0,u0=A,u˙0=0,where η is a parameter, 0<η≤1.Equation (43) can be written as follows:(44)1+u2u¨+1+u2u-ηu=0.The linearized equation of (44) is(45)u¨+ω2u=0.The equation error between (44) and (45) is(46)eu=1+εu2u¨+1+εu2u-ηu-u¨-ω2u,where ω2 is determined by using the mean square criterion, as follows: (47)ω2=1+u2u¨u+u21+u2-ηu2-uu¨u2.The periodic solution of the linearized equation (45) is(48)ut=Acos⁡ωt.Using (48), we have(49)uu¨=-Acos⁡ωtAω2cos⁡ωt=-A2ω2cos2ωt=-ω2u2.Thus(50)1+u2u¨u=-ω2u21+u2,uu¨=-ω2u2.Substituting (49) and (50) into (47), we get(51)ω2=1-ω2u21+u2u2-η+ω2.Using definition (11) and weighted coefficient (14), we calculate averaging operators in (51):(52)u2=A2cos2ωt=∫0+∞A2s2ω2te-sωtcos2ωtdt=∫0+∞A2s2τe-sτcos2τdτ=A2s4+2s2+8s2+42u21+u2=A2cos2ωt1+A2cos2ωt=∫0+∞A2s2ω2te-sωtcos2ωt1+A2cos2ωtdt=∫0+∞A2s2τe-sτcos2τ1+A2cos2τdτ.Withs chosen equal to 2, substituting (52) into (51), we get the approximate frequency of this oscillator:(53)ω=1-ηA22Iand the approximate solution of this oscillator:(54)ut=Acos⁡1-ηA22It,where(55)I=∫0+∞4A2τe-2τcos2τ1+A2cos2τdτ.Thus, the approximate period of this oscillator is(56)T=2πω=2π1-ηA2/2I.Forη=0.1, η=0.5, η=0.75, and η=0.95, comparison of the approximate periods T in (56), the approximate Variational Iteration Method (VIM) periods TVIM [5] in (57), and the Energy Balance Method (EBM) periods TEBM [16] in (58) with exact periods Tex in (59) is tabulated in Table 4. From Table 4, the accuracy of this proposed method can be observed.Table 4
Comparison of the approximate periods with the exact periods, Example 3.
A η T e x [18] T V I M [5] R. error (%) T P r e s e n t R. error (%) T E B M [16] R. error (%) 0.1 0.1 6.62168 6.621237 0.00669 6.62174 0.00091 6.62169 0.00011 1 0.1 6.537508 6.517854 0.30063 6.53806 0.00844 6.53573 0.02723 10 0.1 6.322938 6.314678 0.13064 6.32097 0.03112 6.32006 0.04558 100 0.1 6.287182 6.286329 0.01357 6.28696 0.00353 6.28687 0.00498 0.1 0.5 8.869257 8.863794 0.06159 8.86990 0.00725 8.86925 0.0001 1 0.5 7.992133 7.814722 2.21982 7.99408 0.02436 7.97282 0.2416 10 0.5 6.490208 6.445572 0.68774 6.47925 0.16884 6.47431 0.24498 100 0.5 6.303280 6.298952 0.06866 6.30214 0.01809 6.30167 0.02557 0.1 0.75 12.49673 12.47385 0.18309 12.49943 0.02161 12.4967 0.00041 1 0.75 9.625404 9.168186 4.75012 9.61936 0.06279 9.56406 0.63733 10 0.75 6.602092 6.531632 1.06724 6.58442 0.26767 6.57665 0.38536 100 0.75 6.313404 6.306879 0.10335 6.31168 0.02731 6.31097 0.03854 0.1 0.95 27.15679 26.86136 1.08787 27.18975 0.12137 27.1540 0.01043 1 0.95 12.075277 10.96676 9.18005 12.00887 0.54994 11.8737 1.66932 10 0.95 6.696117 6.60302 1.39031 6.67235 0.35494 6.66211 0.50781 100 0.95 6.321539 6.313243 0.13123 6.31934 0.03479 6.31844 0.04897The approximate period obtained by Variational Iteration Method is as follows [5]:(57)TVIM=2π1+A21+A2-1+A2η.The approximate period obtained by Energy Balance Method is as follows [16]:(58)TEBM=2π2/AA2/4-η-A2/2+1+A2+1.The exact period of this oscillator is as follows [18]:(59)Tex=4∫0π/21-2η1+A2sin2θ+1+A2-1/2dθ.Comparisons of the approximate solutions obtained by two analytical methods with the exact ones is presented in Figures8 and 9. And the relative error versus the oscillation amplitude is illustrated in Figure 10.Figure 8
Comparison of time history diagram of displacement between the present, VIM, and exact solutions atη=0.9,A=1,Example 3.Figure 9
Comparison of time history diagram of displacement between the present, VIM, and exact solutions atη=0.7,A=10, Example 3.Figure 10
Comparison of relative error between the present, VIM, and EBM solutions withη=0.75, Example 3.


## 3.4. Example 4

Consider the Duffing oscillator with discontinuity [3]:(60)u¨+αu+βuu=0,u0=A,u˙0=0.The linearized equation of (60) is(61)u¨+ω2u=0.The equation error between (60) and (61) is(62)eu=αu+βuu-ω2u.The unknown coefficientω2 is determined from the mean square error criterion, as follows:(63)ω2=α+βu2uu2.The periodic solution of (61) is(64)ut=Acos⁡ωt.Similar to Examples 1–3, we calculate averaging operatorsu2 and u2u; then, substituting these operators into (63) with note that the parameter s is chosen equal to 2, we have the approximate frequency of this oscillator:(65)ω=α+0.8324βA.Thus, the approximate solution is(66)ut=Acos⁡α+0.8324εAt.Accuracy of the approach for this example is shown in Figures11 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 A, α, and β.Figure 11
Comparison of time history diagram of displacement between the present, HPM, and numerical solutions atα=0,β=10,A=0.1, Example 4.Figure 12
Comparison of time history diagram of displacement between the Present, HPM, and numerical solutions atα=10,β=100,A=2, Example 4.


## 3.5. Example 5

Consider the nonlinear oscillator equation of the form [23, 24, 34](67)u¨+u1/2n+1=0,u0=A,u˙0=0,where n is a positive integer.The linearized equation of (67) is(68)u¨+ω2u=0.The equation error between (67) and (68) is(69)eu=u1/2n+1-ω2u.The unknown coefficientω2 is determined from the mean square error criterion, as follows:(70)ω2=uu1/2n+1u2.The periodic solution of (68) is(71)ut=Acos⁡ωt.Using definition (11) and weighted coefficient (14), we calculate averaging operators in (70):(72)u2=A2cos2ωt=∫0+∞A2s2ω2te-sωtcos2ωtdt=∫0+∞A2s2τe-sτcos2τdτ=A2s4+2s2+8s2+42(73)uu1/2n+1=Acos⁡ωtA1/2n+1cos⁡ωt1/2n+1=∫0+∞AA1/2n+1s2ω2te-sωtcos⁡ωtcos⁡ωt1/2n+1dt=∫0+∞AA1/2n+1s2τe-sτcos⁡τcos⁡τ1/2n+1dτ.Withs chosen equal to 2, from (72) and (73), we have(74)u2=A22(75)uu1/2n+1=∫0+∞A2n+2/2n+14τe-2τcos⁡τcos⁡τ1/2n+1dτ.Substituting (71) and (72) into (70), we get the approximate frequency of this oscillator:(76)ω=A-n/2n+12In,where(77)In=∫0+∞4τe-2τcos⁡τcos⁡τ1/2n+1dτ.Forn=0 (the harmonic oscillator case) it follows from (77) and (76) that ω is equal to the well-known value 1. And, for n→∞, ω is equal to 0.97979A-1/2. For other values of n, the integral in (77) can be calculated numerically by using the formula manipulation package Maple. For some values of n approximations of the frequency are given in Table 5. Table 5 presents the comparison of the approximate frequencies ωpresent in (76), the approximate frequencies ωapp obtained by van Horssen [34], and the approximate frequencies ωHBM obtained by Mickens [23]. As can be seen from this table the fractional errors of the approximations obtained in [34] for n≥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 n increases, the relative error of the approximate frequencies ωapp obtained by van Horssen increases, while the relative error of the approximate frequencies ωapp in this paper decreases.Table 5
Comparison of the approximate frequencies, Example 5.
n ω H B M [23] ω a p p [34] R. error (%) ω p r e s e n t R. error (%) 0 1.00000 1.000000 0.00000 1.000000 0.00000 1 A - 1 / 3 1.04912 A - 1 / 3 1.07045 2.03313 A - 1 / 3 0.94307 10.10847 2 A - 2 / 5 1.04812 A - 2 / 5 1.08613 3.62649 A - 2 / 5 0.95714 8.680304 3 A - 3 / 7 1.04405 A - 3 / 7 1.09302 4.69039 A - 3 / 7 0.96342 7.72281 4 A - 4 / 9 1.04017 A - 4 / 9 1.09689 5.45295 A - 4 / 9 0.96697 7.037311 5 A - 5 / 11 1.03684 A - 5 / 11 1.09938 6.03179 A - 5 / 11 0.96926 6.517881 6 A - 6 / 13 1.03403 A - 6 / 13 1.10110 6.48627 A - 6 / 13 0.97085 6.110074 7 A - 7 / 15 1.03164 A - 7 / 15 1.10238 6.85704 A - 7 / 15 0.97202 5.779148 8 A - 8 / 17 1.02960 A - 8 / 17 1.10335 7.16298 A - 8 / 17 0.97293 5.504079 9 A - 9 / 19 1.02783 A - 9 / 19 1.10412 7.42243 A - 9 / 19 0.97364 5.272273 10 A - 10 / 21 1.02628 A - 10 / 21 1.10474 7.64509 A - 10 / 21 0.97422 5.07269 50 A - 50 / 101 1.00919 A - 50 / 101 1.10947 9.93668 A - 50 / 101 0.97863 3.028171 500 A - 500 / 1001 1.00149 A - 500 / 1001 1.11059 10.8938 A - 500 / 1001 0.97968 2.17776 ∞ A - 1 / 2 1.00000 A - 1 / 2 π / 2 2 11.07200 A - 1 / 2 0.97979 2.02100Comparison of the approximate solutions obtained by three methods is present in Figures13 and 14.Figure 13
Comparison of time history diagram of displacement between the present, HBM [23], and App [34] solutions with n=10, Example 5.Figure 14
Comparison of time history diagram of displacement between the Present, HBM [23] and App [34] solutions with n=50, Example 5.


## 4. Conclusions

In this paper, the Equivalent Linearization Method with a weighted averaging is applied to analyze five undamped 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(t) 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 t and the pessimistic weighted coefficient e-sωt). 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 s in the expression of weighted coefficient h(t). 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, s is chosen equal to 2 for comparison and convenient calculating purposes. However, the value of the parameter s in the expression of the weighted coefficient h(t) should be chosen to give better results and the best solution still requires further investigation.



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