# Higher-Order Solutions of Coupled Systems Using the Parameter Expansion Method

Authors: S. S. Ganji; M. G. Sfahani; S. M. Modares Tonekaboni; A. K. Moosavi; D. D. Ganji
Journal: Mathematical Problems in Engineering
 (2009)

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## Abstract

We consider periodic solution for coupled systems of mass-spring. Three practical cases of these systems are explained and introduced. An analytical technique called Parameter Expansion Method (PEM) was applied to calculate approximations to the achieved nonlinear differential oscillation equations. Comparing with exact solutions, the first approximation to the frequency of oscillation produces tolerable error 3.14% as the maximum. By the second iteration the respective error became 1/5th, as it is 0.064%. So we conclude that the first approximation of PEM is so benefit when a quick answer is required, but the higher order approximation gives a convergent precise solution when an exact solution is required.


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## Body

## 1. Introduction

Nonlinear oscillators have been widely considered in physics and engineering. Surveys of literature with numerous references, and useful bibliographies, have been given by Mickens [1], Nayfeh and Mook [2], Agarwal et al. [3], and more recently by He [4]. To solve governing nonlinear equations and because limitation of existing exact solutions is one of the most time consuming and difficult affairs, many approaches for approximating the solutions to nonlinear oscillatory systems were excogitated. The most widely studied approximation methods are perturbation methods [5]. But these methods have a main shortcoming; there is no small parameter in the equation, and no approximation could be obtained.Later, new analytical methods without depending on presence of small parameter in the equation were developed for solving these complicated nonlinear systems. These techniques include the Homotopy Perturbation [6–13], Modified Lindstedt-Poincaré [14], Parameter-Expanding [15–18], Parameterized Perturbation [19], Multiple Scale [20], Harmonic Balance [20, 21], Linearized Perturbation [22], Energy Balance [23–25], Variational Iteration [26, 27], Variational Approach [25, 28, 29], Iteration Perturbation [30], Variational Homotopy Perturbation [31] methods, and more [32]. Among these methods, Parameter Perturbation Method (PEM) is considered to be one powerful method that capable to handle strongly nonlinear behaviors. For this sake, we apply PEM to analysis of three practical cases [2, 33, 34] of nonlinear oscillatory system. Unlike the past investigations, here, it had assumed that the spring's property is nonlinear. The TDOF oscillation systems were consist of two coupled nonhomogeneous ordinary differential equations. So, we attempted to transform the equations of motion of a mechanical system which associated with the linear and nonlinear springs into a set of differential algebraic equations by introducing new variables. The analytical solutions of practical cases based on the cubic oscillation are presented by means of PEM for two iterations. Comparisons between analytical and exact solutions show that PEM can converge to an accurate periodic solution for nonlinear systems.


## 2. The Models of Nonlinear Oscillation Systems

In this section, a practical case of nonlinear oscillation system of SDOF in Case1 and two cases of TDOF systems in Cases 2 and 3 are considered.


### 2.1. Single-Degree-of-Freedom

Case 1   1 (Model of a Bulking Column).
First, we consider the system shown in Figure1. The mass m can move in the horizontal direction only. Using this model representing a column, we demonstrate how one can study its static stability by determining the nature of the singular point at u=0 of the dynamic equations. This “dynamic” approach is simpler to use, and arguments are more satisfying than the “static” approach [2]. Vito [35] analyzed the stability of vibration of a particle in a plane constrained by identical springs.Model for the bulking of a column [2].
(a)(b)Neglecting the weight of springs and columns shows that the governing equation for the motion ofm is [2]
(2.1)mü+(k1-2Pl)u+(k3-2Pl3)u3+⋯=0,
where u(0)=A, u̇(0)=0. The spring force is given by
(2.2)Fspring=k1u+k3u3+⋯.


### 2.2. Two-Degree-of-Freedom

Case 2   2 (Two-Mass System with Three Springs).
The model of two-mass system with three springs is shown in Figure2. In this system, two equal masses m are connected with the fixed supports using spring k1. The connection between two masses makes a compact item which is a spring with nonlinear properties. The linear coefficient of spring elasticity is k2 and of the cubic nonlinearity is k3. The system has two degrees of freedom. The generalized coordinates are x and y.Figure 2
. Model of the two-mass system with three springs [34].The mathematical model of the system is [34]
(2.3)mẍ+k1x+k2(x-y)+k3(x-y)3=εf1(x,ẋ,y,ẏ),x(0)=X0,ẋ(0)=0,mÿ+k1y+k2(y-x)+k3(y-x)3=εf2(x,ẋ,y,ẏ),y(0)=Y0,ẏ(0)=0,
where εfi is small nonlinearity (i=1,2). Dividing (2.3) by mass m yields
(2.4)ẍ+k1mx+k2m(x-y)+k3m(x-y)3=εmf1(x,ẋ,y,ẏ),ÿ+k1my+k2m(y-x)+k3m(y-x)3=εmf2(x,ẋ,y,ẏ).Introducing the new variables(2.5)x=u,y-x=v.Transforming (2.4) yields
(2.6)ü+k1mu-k2mv-k3mv3=εmf1(u,u̇,v+u,v̇+u̇),(2.7)v̈+ü+k1m(v+u)+k2mv+k3mv3=εmf2(u,u̇,v+u,v̇+u̇).From (2.6), we have
(2.8)ü+k1mu=εmf1(u,u̇,v+u,v̇+u̇)+k2mv+k3mv3,Substituting (2.8) into (2.7) gives
(2.9)v̈+[k1+2k2m]v+[2k3m]v3=ζ(f2(u,u̇,v+u,v̇+u̇)-f1(u,u̇,v+u,v̇+u̇)),v(0)=y(0)-x(0)=Y0-X0=A,v̇(0)=0.Settingε=0, (2.9) can be written as
(2.10)v̈+[k1+2k2m]v+[2k3m]v3=0,v(0)=y(0)-x(0)=Y0-X0=A,v̇(0)=0.Note that the case ofk3>0 corresponds to a hardening spring while k3<0 indicates a softening one.Case 3   3 (Two-Mass System with a Connection Spring).
Similarly, the model of system with one spring is shown in Figure3. Two masses, m1 and m2, are connected with a spring in which linear coefficient of rigidity is k1, and the nonlinear coefficient is k3. The system has two degrees of freedom.Figure 3
Model of the two-mass system with spring [33].The generalized coordinates of the system arex and y. The equation of motion of the system is described by [33]:
(2.11)m1ẍ+k1(x-y)+k3(x-y)3=0,x(0)=X0,ẋ(0)=0,m2ÿ+k1(y-x)+k3(y-x)3=0,y(0)=Y0,ẏ(0)=0.Similar to the previous section, to simplify these equations, we apply the variables that was introduced in (2.5). Using these variables, (2.11) transformed to
(2.12)m1ü-k1v-k2v3=0,(2.13)m2(v̈+ü)+k1v+k2v3=0.Solving (2.12) for u yields
(2.14)ü=k1m1v+k2m1v3,Substituting (2.14) into (2.13) gives
(2.15)v̈+[k1(m1+m2)m1m2]v+[k2(m1+m2)m1m2]v3=0,v(0)=y(0)-x(0)=Y0-X0=A,v̇(0)=0.As mentioned, these models can be transformed to a cubic nonlinear differential equation in general form with different valuesα and β. The general form of cubic nonlinear differential is as follows:
(2.16)v̈+αv+βv3=0,v(0)=A,v̇(0)=0.


## 2.1. Single-Degree-of-Freedom

Case 1   1 (Model of a Bulking Column).
First, we consider the system shown in Figure1. The mass m can move in the horizontal direction only. Using this model representing a column, we demonstrate how one can study its static stability by determining the nature of the singular point at u=0 of the dynamic equations. This “dynamic” approach is simpler to use, and arguments are more satisfying than the “static” approach [2]. Vito [35] analyzed the stability of vibration of a particle in a plane constrained by identical springs.Model for the bulking of a column [2].
(a)(b)Neglecting the weight of springs and columns shows that the governing equation for the motion ofm is [2]
(2.1)mü+(k1-2Pl)u+(k3-2Pl3)u3+⋯=0,
where u(0)=A, u̇(0)=0. The spring force is given by
(2.2)Fspring=k1u+k3u3+⋯.


## 2.2. Two-Degree-of-Freedom

Case 2   2 (Two-Mass System with Three Springs).
The model of two-mass system with three springs is shown in Figure2. In this system, two equal masses m are connected with the fixed supports using spring k1. The connection between two masses makes a compact item which is a spring with nonlinear properties. The linear coefficient of spring elasticity is k2 and of the cubic nonlinearity is k3. The system has two degrees of freedom. The generalized coordinates are x and y.Figure 2
. Model of the two-mass system with three springs [34].The mathematical model of the system is [34]
(2.3)mẍ+k1x+k2(x-y)+k3(x-y)3=εf1(x,ẋ,y,ẏ),x(0)=X0,ẋ(0)=0,mÿ+k1y+k2(y-x)+k3(y-x)3=εf2(x,ẋ,y,ẏ),y(0)=Y0,ẏ(0)=0,
where εfi is small nonlinearity (i=1,2). Dividing (2.3) by mass m yields
(2.4)ẍ+k1mx+k2m(x-y)+k3m(x-y)3=εmf1(x,ẋ,y,ẏ),ÿ+k1my+k2m(y-x)+k3m(y-x)3=εmf2(x,ẋ,y,ẏ).Introducing the new variables(2.5)x=u,y-x=v.Transforming (2.4) yields
(2.6)ü+k1mu-k2mv-k3mv3=εmf1(u,u̇,v+u,v̇+u̇),(2.7)v̈+ü+k1m(v+u)+k2mv+k3mv3=εmf2(u,u̇,v+u,v̇+u̇).From (2.6), we have
(2.8)ü+k1mu=εmf1(u,u̇,v+u,v̇+u̇)+k2mv+k3mv3,Substituting (2.8) into (2.7) gives
(2.9)v̈+[k1+2k2m]v+[2k3m]v3=ζ(f2(u,u̇,v+u,v̇+u̇)-f1(u,u̇,v+u,v̇+u̇)),v(0)=y(0)-x(0)=Y0-X0=A,v̇(0)=0.Settingε=0, (2.9) can be written as
(2.10)v̈+[k1+2k2m]v+[2k3m]v3=0,v(0)=y(0)-x(0)=Y0-X0=A,v̇(0)=0.Note that the case ofk3>0 corresponds to a hardening spring while k3<0 indicates a softening one.Case 3   3 (Two-Mass System with a Connection Spring).
Similarly, the model of system with one spring is shown in Figure3. Two masses, m1 and m2, are connected with a spring in which linear coefficient of rigidity is k1, and the nonlinear coefficient is k3. The system has two degrees of freedom.Figure 3
Model of the two-mass system with spring [33].The generalized coordinates of the system arex and y. The equation of motion of the system is described by [33]:
(2.11)m1ẍ+k1(x-y)+k3(x-y)3=0,x(0)=X0,ẋ(0)=0,m2ÿ+k1(y-x)+k3(y-x)3=0,y(0)=Y0,ẏ(0)=0.Similar to the previous section, to simplify these equations, we apply the variables that was introduced in (2.5). Using these variables, (2.11) transformed to
(2.12)m1ü-k1v-k2v3=0,(2.13)m2(v̈+ü)+k1v+k2v3=0.Solving (2.12) for u yields
(2.14)ü=k1m1v+k2m1v3,Substituting (2.14) into (2.13) gives
(2.15)v̈+[k1(m1+m2)m1m2]v+[k2(m1+m2)m1m2]v3=0,v(0)=y(0)-x(0)=Y0-X0=A,v̇(0)=0.As mentioned, these models can be transformed to a cubic nonlinear differential equation in general form with different valuesα and β. The general form of cubic nonlinear differential is as follows:
(2.16)v̈+αv+βv3=0,v(0)=A,v̇(0)=0.


## 3. Basic Idea of PEM

In order to use the PEM, we rewrite the general form of Duffing equation in the following form [7]:
(3.1)v̈+αv+1·N(v,t)=0,
where N(v,t) includes the nonlinear term. Expanding the solution v, α as a coefficient of v, and 1 as a coefficient of N(v,t), the series of p can be introduced as follows:
(3.2)v=v0+pv1+p2v2+⋯,(3.3)α=ω2+pγ1+p2γ2+⋯,(3.4)1=pδ1+p2δ2+⋯.Substituting (3.2)–(3.4) into (3.1) and equating the terms with the identical powers of p, we have
(3.5)p0:v̈0+ω2v0=0,(3.6)p1:v̈1+ω2v1+γ1v0+δ1N(v0,t)=0⋮Considering the initial conditionsv0(0)=A and v̇0(0)=0, the solution of (3.5) is v0=Acos(ωt). Substituting v0 into (3.6), we obtain
(3.7)p1:v̈1+ω2v1+γ1Acos(ωt)+δ1N(Acos(ωt),t)=0.For achieving the secular term, we use Fourier expansion series as follows:(3.8)N(Acos(ωt),t)=∑n=0∞b2n+1cos[(2n+1)ωt].Substituting (3.8) into (3.7) yields
(3.9)p1:v̈1+ω2v1+(γ1A+δ1b1)cos(ωt)=0.For avoiding secular term, we have(3.10)(γ1A+δ1b1)=0.Settingp=1 in (3.3) and (3.4), we have:
(3.11)γ1=α-ω2,(3.12)δ1=1.Substituting (3.11) and (3.12) into (3.10), we will achieve the first-order approximation frequency (2.10). Note that, from (3.4) and (3.12), we can find that δi=0, for all i=2,3,4,…. In the following section we will describe the second-order of modified PEM solution in details for solving the cubic nonlinear differential equation.


## 4. Application of PEM to Cubic Equation

In order to use the PEM, we rewrite (2.16) as follows:
(4.1)v̈+αv+1·(βv3)=0,v(0)=A,v̇(0)=0.Substituting (3.2) and (3.4) into (4.1) and equating the terms with the identical powers of p, yields
(4.2)p0:v̈0+ω2v0=0,(4.3)p1:v̈1+ω2v1+(αδ1+γ1)v0+βv03=0,(4.4)p2:v̈1+ω2v1+(δ2α+γ2)v0+(γ1+δ1α)v1+δ2βv03+3δ1βv02v1=0.⋮Considering the initial conditionsv(0)=A and v̇(0)=0, the solution of (4.2) is v0=Acos(ωt). Substituting  u0 into (4.3), we obtain
(4.5)p1:v̈1+ω2v1+γ1Acos(ωt)+δ1βA3cos3(ωt)=0.It is possible to perform the following Fourier series expansion:(4.6)βA3cos3(ωt)=∑n=0∞b2n+1cos[(2n+1)ωt]=b1cos(ωt)+∑n=1∞a2n+1cos[(2n+1)ωt]=(4πβA3∫0π/2(cos4(φ))dφ)cos(ωt)+∑n=1∞a2n+1cos[(2n+1)ωt]=3A3β4cos(ωt)+∑n=1∞a2n+1cos[(2n+1)ωt].Substituting (4.6) into (4.5) gives
(4.7)v̈1+ω2v1+(γ1A+3δ1A3β4)cos(ωt)+δ1∑n=1∞a2n+1cos[(2n+1)ωt]=0.No secular term inv1 requires that
(4.8)γ1=-3δ1A2β4.Settingp=1 in (3.3) and (3.4) gives
(4.9)γ1=α-ω2,(4.10)δ1=1.Substituting (4.9) and (4.10) into (4.8), we obtain
(4.11)ω1=α+34βA2,(4.12)T1=4π4α+3βA2.From (4.7) and (4.8), then (4.9) can be rewritten in the following form:
(4.13)v̈1+ω2v1=-∑n=1∞ζ2n+1cos[(2n+1)ωt],v1(0)=0,v̇1(0)=0.The periodic solution of (4.13) can be written [19]
(4.14)v1=∑n=0∞λ2n+1cos[(2n+1)ωt].Substituting (4.14) into (4.13) gives
(4.15)-ω2∑n=0∞4n(n+1)λ2n+1cos[(2n+1)ωt]=-∑n=1∞ζ2n+1cos[(2n+1)ωt].From (4.15), the coefficients λ2n+1  (for n≥1) can be written as follows:
(4.16)λ2n+1=ζ2n+14n(n+1),Taking into account thatv1(0)=0, (4.14) yields
(4.17)λ1=-∑n=1∞λ2n+1.To determine the second-order approximate solution, it is necessary to substitute (4.14) into (4.4). Then secular term is eliminated, and parameter γ2 can be calculated. v1(t) has an infinite series; however, to simplify the solution procedure, we can truncate the series expansion of (4.14) and (4.17) and write an approximate equation v1(t) in the following form:
(4.18)v1(t)=λ3(cos3ωt-cosωt).Substitutingδ2=0 and (4.8) and (4.18) into (4.4) gives
(4.19)v̈2+ω2v2-3λ3βA2cos3(ωt)+βA2λ3cos(3ωt)(3cos2(ωt)-34)+(34λ3βA2+Aγ2)cos(ωt)=0.It is possible to do the following Fourier series expansion:(4.20)3λ3βA2cos2(ωt)(cos(3ωt)-cos(ωt))=∑n=0∞η2n+1cos[(2n+1)ωt]=η1cos(ωt)+∑n=1∞η2n+1cos[(2n+1)ωt]≅(12β2λ3A3π∫0π/2(cos(3φ)-cos(φ))cos3φdφ)×cos(ωt)+∑n=1∞η2n+1cos[(2n+1)ωt]=-34βA2λ3cos(ωt)+∑n=1∞η2n+1cos[(2n+1)ωt].Substituting (4.20) into (4.19) and collecting, we have
(4.21)v̈2+ω2v2-(34βA2λ3-Aγ2)cos(ωt)-34λ3βA2cos(3ωt)+∑n=1∞η2n+1cos[(2n+1)ωt]=0.The secular term in the solution forv2(t) can be eliminated if
(4.22)34βA2λ3-Aγ2=0.Solving (4.22) gives
(4.23)γ2=34βAλ3.On the other hand, From (4.16), the following expression for the coefficient λ3 is obtained:
(4.24)λ3=ζ38ω2=((4/π)βA3∫0π/2cos3φcos3φdφ)8ω2=βA332ω2.Then, we can obtain(4.25)γ2=3β2A4128ω2.From (3.3), (3.4), and (4.8), and taking p=1 and considering δ1=1, we have
(4.26)γ2=α-ω2-3A2β4.Comparing the right hands of (4.25) and (4.26), one can easily obtain the following expression for the second-order approximate frequency and period
(4.27)ω2=8α+6βA2+64α2+96αβA2+30β2A44,(4.28)T2=8π8α+6βA2+64α2+96αβA2+30β2A4.


## 5. Analytical Solution of Practical Cases

In this section, we present the first and second approximate frequency and period values of (2.16) for different values of α and β. Substituting α=(k1+2P/l)/m and β=(k3+2P/l3)/m in (4.10), (4.11), (4.26), and (4.27) gives the following results for first- and second-order approximations of the model of nonlinear SDOF Bucking Column system in Case 1:
(5.1)ω1=124k1l3-8Pl2+3k3A2l3-6PA2ml3,T1=4πml34k1l3-8Pl2+3k3A2l3-6PA2,ω2=14ml3(+30A4k32l6-120A4Pk3l3+120A4P2)1/28l3k1-16l2P+6A2k3l3-12A2P+(64l6k12-256l5k1P+256l4P2+96A2l6k1k3-192A2l3k1P-192A2l5k3P+384A2l2P2+30A4k32l6-120A4Pk3l3+120A4P2)1/2)1/2,T2=8πml3/(+30A4k32l6-120A4Pk3l3+120A4P2)1/28l3k1-16l2P+6A2k3l3-12A2P+(64l6k12-256l5k1P+256l4P2+96A2l6k1k3-192A2l3k1P-192A2l5k3P+384A2l2P2+30A4k32l6-120A4Pk3l3+120A4P2)1/2)1/2,Also, we can obtain the first and second-order approximations solutions for Case2, by substituting α=(k1+2k2)/m and β=2k3/m into (4.11), (4.12), (4.27), and (4.28):
(5.2)ω1=k1+2k2+1.5k3A2m,T1=π8m2k1+4k2+3A2k3,ω2=148(k1+2k2)+12k3A2+64(k1+2k2)2+192(k1+2k2)k3A2+120k32A4m,T2=4πm8(k1+2k2)+12k3A2+64(k1+2k2)2+192(k1+2k2)k3A2+120k32A4.Similarly, forα=k1(m1+m2)/m1m2 and β=k2(m1+m2)/m1m2, we obtain the following frequency and period values for Case 3:
(5.3)ω1=(m1+m2)m1m2(k1+34A2k2),T1=2πm1m2(m1+m2)(k1+(3/4)A2k2),ω2=14(m1+m2)m1m2(8k1+6A2k2+(64k12+96A2k1k2+30A4k22)),T2=8π(1/4)((m1+m2)/m1m2)(8k1+6A2k2+(64k12+96A2k1k2+30A4k22)).


## 6. Results and Discussions

To illustrate and verify accuracy of PEM, comparisons with the exact solution are given in Tables1, 2, and 3. According to the appendix, the exact frequency, ωex, of nonlinear differential equation in the cubic form is
(6.1)ωex(A)=πα+βA22(∫0π/2dt1-δsin2t)-1,δ=βA22(α+βA2).Table 1
Comparison of approximate and exact periods for Case  1.
Constant parametersApproximate solutionsExact solution|T-Tex|/Texmlpk1k3AT1T2TeT=T1T=T211110511.962541.964511.964510.101%0.000%51.555633.237433.325183.323682.664%0.045%1010101050100.324180.331450.331432.210%0.031%50254030100200.256400.262160.262082.216%0.031%7020−3050100100.604860.618270.618092.187%0.030%1005015070201000.162210.165860.165802.218%0.031%5001502201205000.59.676379.716829.716720.417%0.001%1000500100050050016.732416.758776.758710.391%0.001%Table 2
Comparison of approximate and “Exact” frequencies for Case  2.
Constant parametersApproximate solutionsExact solution|ω-ωex|/ωexmk1k2k3X0Y0ω1ω2ωexω=ω1ω=ω21111515.19625.10685.10781.73%0.0185%21358104.30124.24014.24061.43%0.0185%5102030−101060.0832858.767758.78562.21%0.0305%1050709020−40220.4972215.6448215.71132.22%0.0308%1025200.5−10106.04155.95335.95411.47%0.0132%100200300400−5050244.9653239.5715239.64552.22%0.0309%Table 3
Comparison of approximate and “Exact” frequencies for  Case  3.
Constant parametersApproximate solutionsExact solution|ω-ωex|/ωexm1m2k1k2X0Y0ω1ω2ωexω=ω1ω=ω21251−415.96875.88855.88921.35%0.011%35255−514.179813.871013.87522.20%0.011%15515−59.79809.60969.61191.94%0.023%1051010203015.099714.776314.78062.16%0.029%51050−0.01−20402.62682.54522.54683.14%0.064%1001105202510.236610.054510.05641.79%0.020%501005010010025112.5067110.0293110.06332.22%0.031%1000100200300400200314.6461307.7164307.81152.17%0.031%Substitutingα=(k1+2P/l)/m and β=(k3+2P/l3)/m into (6.1) gives the exact frequency for Case 1:
(6.2)ωex(A)=π2k1l3-2Pl2+A2k3l3-2A2Pml3(∫0π/2dt1-δsin2t)-1,δ=(l3k3-2P)A22(k1l3-2Pl2+A2k3l3-2A2P).Substitutingα=k1(m1+m2)/m1m2 and β=k2(m1+m2)/m1m2 into (6.1), the exact solution of Case 2 is
(6.3)ωex(A)=π2(k1+2k2)+2A2k3m(∫0π/2dt1-δsin2t)-1,δ=2k3A22(k1+2k2)+2k3A2.Using (2.8) and ε=0, we can obtain
(6.4)ü+k1mu=k2mv+k3mv3.The first- and second-order analytical approximation foru(t) is obtained using (6.4) and therefore, the first and second-order analytically approximates displacements x(t) and y(t) obtained using (2.5).Similarly, substitutingα=k1(m1+m2)/m1m2 and β=k2(m1+m2)/m1m2 into (6.1), the exact solution of Case 3 is:
(6.5)ωex(A)=π2(m1+m2)m1m2(k1+k2A2)(∫0π/2dt1-δsin2t)-1,δ=k2(m1+m2)A22(k1(m1+m2)+k2(m1+m2)A2).After obtainingu(t)from (2.14), the first- and second-order analytically approximates displacements x(t) and y(t) obtained using (2.5).It should be noted thatωex  contains an integral which could only be solved numerically in general. The limitation of amplitude, A, in the cubic oscillation equation satisfies βA2+α>0; the Duffing equation has a heteroclinic orbit with period +∞ [36]. Hence, in order to avoid the heteroclinic orbit with period +∞ for the Duffing equation in (2.16), the value of  k3 in the first two cases and k2 for the third case should, respectively, satisfy in (6.4), (6.5), and (6.6)
(6.6)k3>-k1A2+2Pl(1A2+1l2),(6.7)k3>-k1+2k22A2,(6.8)k2>-k1A2,
where k1,k2,k3,l∈R+ and A,P∈R.To illustrate and verify accuracy of this analytical approach, comparisons of analytical and exact results for the practical cases are presented in Tables1–3 and Figures 4–10. For this reason, we use the following specific parameter and initial values: Case  1: m, P, l, k1, k3, A, Case  2: m, k1, k2, k3, X0, Y0, and Case  3: m1, m2, k1, k2, X0, Y0.Figure 4
Comparison of approximate periodic solutions of Bucking of a Column equation (Case  1) with the exact one form=1.0, l=1.5, P=5.0, k1=5.0, and k3=6.0 with u(0)=3.0.Figure 5
Comparison of approximate periodic solutions of Bucking of a Column equation (Case  1) with the exact one form=l=P=k1=10.0,  and k3=50.0 with u(0)=10.0.Figure 6
Comparison of approximate periodic solutions of Bucking of a Column equation (Case  1) with the exact one form=100.0, l=50.0,  P=150.0, k1=70.0, and k3=20.0 with u(0)=100.0.Comparison of the first- and second-order analytical approximate solutions with the exact solution form=k1=k2=k3=1.0 with x(0)=5.0 and y(0)=1.0 (Case  2).
(a)
(x-t) diagram(b)
(y-t) diagramComparison of the first- and second-order analytical approximate solutions with the exact solution form=2.0, k1=1.0, k2=3.0, k3=5.0 with x(0)=8.0 and y(0)=10.0 (Case  2).
(a)
(x-t) diagram(b)
(y-t) diagramComparison of the first- and second-order analytical approximate solutions with the exact solution form1=1.0, m2=2.0, k1=5.0, k2=1.0 with x(0)=-4.0 and y(0)=1.0 (Case  3).
(a)
(x-t) diagram(b)
(y-t) diagramComparison of the first- and second-order analytical approximate solutions with the exact solution form1=3.0, m2=5.0, k1=2.0, k2=5.0 with x(0)=5.0 and y(0)=-5.0 (Case  3).
(a)
(x-t) diagram(b)
(y-t) diagramFigures4–6, which are correspond to Case  1, indicate the comparison of this analytical method for different parameter with initial values m=1.0, l=1.5, P=5.0, k1=5.0 and k3=6.0, A=3.0 and m=l=P=k1=10.0, and k3=50.0, A=10.0 and m=100.0, l=50.0, P=150.0, k1=70.0, k3=20.0  and A=100.0 which are in an excellent agreement with exact solutions.Figures7 and 8 represent the x-t and y-t diagrams which obtained analytically and exactly solving of Case  2 with different parameter and initial values m=k1=k2=k3=1.0 with X0=5.0, Y0=1.0 and m=2.0, k1=1.0, k2=3.0, k3=5.0, X0=8.0, Y0=10.0. Also, the corresponding diagrams (x-t, y-t) of Case  3 are plotted in Figures 9 and 10. The different parameters and initial values that used in plotting diagrams of Case  3 are: m1=1.0, m2=2.0, k1=5.0, k2=1.0, X0=-4.0, Y0=1.0 and m1=3.0, m2=5.0, k1=2.0, k2=5.0, X0=5.0, Y0=-5.0.According to these tables and figures, the difference between analytical and exact solutions is negligible. In other words, the first-order approximate results of PEM are accurate, but we significantly improve the percentage error from lower-order to second-order analytical approximations. We did it using modified PEM in second iteration for different parameters and initial amplitudes. Hence, it is concluded and provides an excellent agreement with the exact solutions.


## 7. Conclusions

The parameter expansion method (PEM) has been used to obtain the first- and second-order approximate frequencies and periods for Single- and Two-Degrees-Of-Freedom (SDOF and TDOF) systems. Excellent agreements between approximate frequencies and the exact one have been demonstrated and discussed, and the discrepancy of the second-order approximate frequencyω2 with respect to the exact one is as low as 0.064%. In general, we conclude that this method is efficient for calculating periodic solutions for nonlinear oscillatory systems, and we think that the method has a great potential and could be applied to other strongly nonlinear oscillators.



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