MPEMathematical Problems in Engineering1563-51471024-123XHindawi Publishing Corporation32746210.1155/2009/327462327462Research ArticleHigher-Order Solutions of Coupled Systems Using the Parameter Expansion MethodGanjiS. S.1SfahaniM. G.2Modares TonekaboniS. M.3Moosavi A. K.2GanjiD. D.2WarminskiJerzy1Department of Civil and Transportation EngineeringIslamic Azad UniversityScience and Research Branch Campus4716695814 TehranIraniau.ae2Department of Civil and Mechanical EngineeringBabol University of TechnologyP.O. Box 4844714871167 BabolIrannit.ac.ir3Department of Civil EngineeringKhajeh Nasir University of Technology1996715433 TehranIrankntu.ac.ir20092107200920092901200927042009070620092009Copyright © 2009This 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.

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.

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 , Nayfeh and Mook , Agarwal et al. , and more recently by He . 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 . 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 , Modified Lindstedt-Poincaré , Parameter-Expanding , Parameterized Perturbation , Multiple Scale , Harmonic Balance [20, 21], Linearized Perturbation , Energy Balance , Variational Iteration [26, 27], Variational Approach [25, 28, 29], Iteration Perturbation , Variational Homotopy Perturbation  methods, and more . 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 Case 1 and two cases of TDOF systems in Cases 2 and 3 are considered.

2.1. Single-Degree-of-FreedomCase 1   1 (Model of a Bulking Column).

First, we consider the system shown in Figure 1. 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 . Vito  analyzed the stability of vibration of a particle in a plane constrained by identical springs.

Model for the bulking of a column .

Neglecting the weight of springs and columns shows that the governing equation for the motion of m is  mü+(k1-2Pl)u+(k3-2Pl3)u3+=0, where u(0)=A, u̇(0)=0. The spring force is given by Fspring=k1u+k3u3+.

2.2. Two-Degree-of-FreedomCase 2   2 (Two-Mass System with Three Springs).

The model of two-mass system with three springs is shown in Figure 2. 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.

. Model of the two-mass system with three springs .

The mathematical model of the system is  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 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 x=u,y-x=v.

Transforming (2.4) yields ü+k1mu-k2mv-k3mv3=εmf1(u,u̇,v+u,v̇+u̇),v̈+ü+k1m(v+u)+k2mv+k3mv3=εmf2(u,u̇,v+u,v̇+u̇).

From (2.6), we have ü+k1mu=εmf1(u,u̇,v+u,v̇+u̇)+k2mv+k3mv3,

Substituting (2.8) into (2.7) gives 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.

Setting ε=0, (2.9) can be written as v̈+[k1+2k2m]v+[2k3m]v3=0,        v(0)=y(0)-x(0)=Y0-X0=A,        v̇(0)=0.

Note that the case of k3>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 Figure 3. 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.

Model of the two-mass system with spring .

The generalized coordinates of the system are x and y. The equation of motion of the system is described by : 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 m1ü-k1v-k2v3=0,m2(v̈+ü)+k1v+k2v3=0.

Solving (2.12) for u yields ü=k1m1v+k2m1v3,

Substituting (2.14) into (2.13) gives 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 values α and β. The general form of cubic nonlinear differential is as follows: 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 : 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: v=v0+pv1+p2v2+,α=ω2+pγ1+p2γ2+,1=pδ1+p2δ2+.

Substituting (3.2)–(3.4) into (3.1) and equating the terms with the identical powers of p, we have p0:v̈0+ω2v0=0,p1:v̈1+ω2v1+γ1v0+δ1N(v0,t)=0

Considering the initial conditions v0(0)=A and v̇0(0)=0, the solution of (3.5) is v0=Acos(ωt). Substituting v0 into (3.6), we obtain p1:v̈1+ω2v1+γ1Acos(ωt)+δ1N(Acos(ωt),t)=0.

For achieving the secular term, we use Fourier expansion series as follows: N(Acos(ωt),t)=n=0b2n+1cos[(2n+1)ωt].

Substituting (3.8) into (3.7) yields p1:v̈1+ω2v1+(γ1A+δ1b1)cos(ωt)=0.

For avoiding secular term, we have (γ1A+δ1b1)=0.

Setting p=1 in (3.3) and (3.4), we have: γ1=α-ω2,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: 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 p0:v̈0+ω2v0=0,p1:v̈1+ω2v1+(αδ1+γ1)v0+βv03=0,p2:v̈1+ω2v1+(δ2α+γ2)v0+(γ1+δ1α)v1+δ2βv03+3δ1βv02v1=0.

Considering the initial conditions v(0)=A and v̇(0)=0, the solution of (4.2) is v0=Acos(ωt). Substituting u0 into (4.3), we obtain p1:v̈1+ω2v1+γ1Acos(ωt)+δ1βA3cos3(ωt)=0.

It is possible to perform the following Fourier series expansion:

βA3cos3(ωt)=n=0b2n+1cos[(2n+1)ωt]=b1cos(ωt)+n=1a2n+1cos[(2n+1)ωt]=(4πβA30π/2(cos4(φ))dφ)cos(ωt)+n=1a2n+1cos[(2n+1)ωt]=3A3β4cos(ωt)+n=1a2n+1cos[(2n+1)ωt].

Substituting (4.6) into (4.5) gives v̈1+ω2v1+(γ1A+3δ1A3β4)cos(ωt)+δ1n=1a2n+1cos[(2n+1)ωt]=0.

No secular term in v1 requires that γ1=-3δ1A2β4.

Setting p=1 in (3.3) and (3.4) gives γ1=α-ω2,δ1=1.

Substituting (4.9) and (4.10) into (4.8), we obtain ω1=α+34βA2,T1=4π4α+3βA2.

From (4.7) and (4.8), then (4.9) can be rewritten in the following form: v̈1+ω2v1=-n=1ζ2n+1cos[(2n+1)ωt],v1(0)=0,v̇1(0)=0.

The periodic solution of (4.13) can be written  v1=n=0λ2n+1cos[(2n+1)ωt].

Substituting (4.14) into (4.13) gives -ω2n=04n(n+1)λ2n+1cos[(2n+1)ωt]=-n=1ζ2n+1cos[(2n+1)ωt].

From (4.15), the coefficients λ2n+1 (for n1) can be written as follows: λ2n+1=ζ2n+14n(n+1),

Taking into account that v1(0)=0, (4.14) yields λ1=-n=1λ2n+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: v1(t)=λ3(cos3ωt-cosωt).

Substituting δ2=0 and (4.8) and (4.18) into (4.4) gives v̈2+ω2v2-3λ3βA2cos3(ωt)+βA2λ3cos(3ωt)(3cos2(ωt)-34)+(34λ3βA2+Aγ2)cos(ωt)=0.

It is possible to do the following Fourier series expansion: 3λ3βA2cos2(ωt)(cos(3ωt)-cos(ωt))=n=0η2n+1cos[(2n+1)ωt]=η1cos(ωt)+n=1η2n+1cos[(2n+1)ωt](12β2λ3A3π0π/2(cos(3φ)-cos(φ))cos3φ  dφ)×cos(ωt)+n=1η2n+1cos[(2n+1)ωt]=-34βA2λ3cos(ωt)+n=1η2n+1cos[(2n+1)ωt].

Substituting (4.20) into (4.19) and collecting, we have v̈2+ω2v2-(34βA2λ3-Aγ2)cos(ωt)-34λ3βA2cos(3ωt)+n=1η2n+1cos[(2n+1)ωt]=0.

The secular term in the solution for v2(t) can be eliminated if 34βA2λ3-Aγ2=0.

Solving (4.22) gives γ2=34βAλ3.

On the other hand, From (4.16), the following expression for the coefficient λ3 is obtained: λ3=ζ38ω2=((4/π)βA30π/2cos3φ  cos3φ  dφ)8ω2=βA332ω2.

Then, we can obtain γ2=3β2A4128ω2.

From (3.3), (3.4), and (4.8), and taking p=1 and considering δ1=1, we have 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 ω2=8α+6βA2+64α2+96αβA2+30β2A44,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: 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 Case 2, by substituting α=(k1+2k2)/m and β=2k3/m into (4.11), (4.12), (4.27), and (4.28): 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, for α=k1(m1+m2)/m1m2 and β=k2(m1+m2)/m1m2, we obtain the following frequency and period values for Case 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 Tables 1, 2, and 3. According to the appendix, the exact frequency, ωex, of nonlinear differential equation in the cubic form is ωex(A)=πα+βA22(0π/2dt1-δsin2t)-1,δ=βA22(α+βA2).

Comparison of approximate and exact periods for Case  1.

 Constant parameters Approximate solutions Exact solution |T-Tex|/Tex m l p k1 k3 A T1 T2 Te T=T1 T=T2 1 1 1 10 5 1 1.96254 1.96451 1.96451 0.101% 0.000% 5 1.5 5 5 6 3 3.23743 3.32518 3.32368 2.664% 0.045% 10 10 10 10 50 10 0.32418 0.33145 0.33143 2.210% 0.031% 50 25 40 30 100 20 0.25640 0.26216 0.26208 2.216% 0.031% 70 20 −30 50 100 10 0.60486 0.61827 0.61809 2.187% 0.030% 100 50 150 70 20 100 0.16221 0.16586 0.16580 2.218% 0.031% 500 150 220 120 500 0.5 9.67637 9.71682 9.71672 0.417% 0.001% 1000 500 1000 500 500 1 6.73241 6.75877 6.75871 0.391% 0.001%

Comparison of approximate and “Exact” frequencies for Case  2.

 Constant parameters Approximate solutions Exact solution |ω-ωex|/ωex m k1 k2 k3 X0 Y0 ω1 ω2 ωex ω=ω1 ω=ω2 1 1 1 1 5 1 5.1962 5.1068 5.1078 1.73% 0.0185% 2 1 3 5 8 10 4.3012 4.2401 4.2406 1.43% 0.0185% 5 10 20 30 −10 10 60.08328 58.7677 58.7856 2.21% 0.0305% 10 50 70 90 20 −40 220.4972 215.6448 215.7113 2.22% 0.0308% 10 25 20 0.5 −10 10 6.0415 5.9533 5.9541 1.47% 0.0132% 100 200 300 400 −50 50 244.9653 239.5715 239.6455 2.22% 0.0309%

Comparison of approximate and “Exact” frequencies for Case  3.

 Constant parameters Approximate solutions Exact solution |ω-ωex|/ωex m1 m2 k1 k2 X0 Y0 ω1 ω2 ωex ω=ω1 ω=ω2 1 2 5 1 −4 1 5.9687 5.8885 5.8892 1.35% 0.011% 3 5 2 5 5 −5 14.1798 13.8710 13.8752 2.20% 0.011% 1 5 5 1 5 −5 9.7980 9.6096 9.6119 1.94% 0.023% 10 5 10 10 20 30 15.0997 14.7763 14.7806 2.16% 0.029% 5 10 50 −0.01 −20 40 2.6268 2.5452 2.5468 3.14% 0.064% 100 1 10 5 20 25 10.2366 10.0545 10.0564 1.79% 0.020% 50 100 50 100 100 25 112.5067 110.0293 110.0633 2.22% 0.031% 1000 100 200 300 400 200 314.6461 307.7164 307.8115 2.17% 0.031%

Substituting α=(k1+2P/l)/m and β=(k3+2P/l3)/m into (6.1) gives the exact frequency for Case 1: ωex(A)=π2k1l3-2Pl2+A2k3l3-2A2Pml3(0π/2dt1-δsin2t)-1,δ=(l3k3-2P)A22(k1l3-2Pl2+A2k3l3-2A2P).

Substituting α=k1(m1+m2)/m1m2 and β=k2(m1+m2)/m1m2 into (6.1), the exact solution of Case 2 is ωex(A)=π2(k1+2k2)+2A2k3m(0π/2dt1-δsin2t)-1,δ=2k3A22(k1+2k2)+2k3A2.

Using (2.8) and ε=0, we can obtain ü+k1mu=k2mv+k3mv3.

The first- and second-order analytical approximation for u(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, substituting α=k1(m1+m2)/m1m2 and β=k2(m1+m2)/m1m2 into (6.1), the exact solution of Case 3 is: 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 obtaining u(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 that ωex 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 + . 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) k3>-k1A2+2Pl(1A2+1l2),k3>-k1+2k22A2,k2>-k1A2, where k1,k2,k3,lR+ and A,PR.

To illustrate and verify accuracy of this analytical approach, comparisons of analytical and exact results for the practical cases are presented in Tables 13 and Figures 410. 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.

Comparison of approximate periodic solutions of Bucking of a Column equation (Case  1) with the exact one for m=1.0, l=1.5, P=5.0, k1=5.0, and k3=6.0 with u(0)=3.0.

Comparison of approximate periodic solutions of Bucking of a Column equation (Case  1) with the exact one for m=l=P=k1=10.0, and k3=50.0 with u(0)=10.0.

Comparison of approximate periodic solutions of Bucking of a Column equation (Case  1) with the exact one for m=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 for m=k1=k2=k3=1.0 with x(0)=5.0 and y(0)=1.0 (Case  2).

(x-t) diagram

(y-t) diagram

Comparison of the first- and second-order analytical approximate solutions with the exact solution for m=2.0, k1=1.0, k2=3.0, k3=5.0 with x(0)=8.0 and y(0)=10.0 (Case  2).

(x-t) diagram

(y-t) diagram

Comparison of the first- and second-order analytical approximate solutions with the exact solution for m1=1.0, m2=2.0, k1=5.0, k2=1.0 with x(0)=-4.0 and y(0)=1.0 (Case  3).

(x-t) diagram

(y-t) diagram

Comparison of the first- and second-order analytical approximate solutions with the exact solution for m1=3.0, m2=5.0, k1=2.0, k2=5.0 with x(0)=5.0 and y(0)=-5.0 (Case  3).

(x-t) diagram

(y-t) diagram

Figures 46, 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.

Figures 7 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 frequency ω2 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.

Appendix

The exact solution of cubic nonlinear differential equation can be obtained by integrating the governing differential equation as follows: 12u̇2+α2u2+β2u4=C,t, where C is a constant. Imposing initial conditions u(0)=A, u̇(0)=0 yields C=α2A2+β4A4,

Equating (A.1) and (A.2) yields 12v̇2+α2v2+β4v4=α2A2+β4A4, or equivalently dt=dvα(A2-v2)+(β/2)(A4-v4).

Integrating (A.4), the period of oscillation Te is T(A)=4  0Advα(A2-v2)+(β/2)  (A4-v4).

Substituting v=Acost into (A.5) and integrating T(A)=4α+βA20π/2dt1-δsin2t, where δ=βA22(α+βA2).

The exact frequency ωex is also a function of A and can be obtained from the period of the oscillation as ωex(A)=πα+βA22(0π/2dt1-δsin2t)-1.

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