Research Article  Open Access
S. S. Ganji, M. G. Sfahani, S. M. Modares Tonekaboni, A. K. Moosavi, D. D. Ganji, "HigherOrder Solutions of Coupled Systems Using the Parameter Expansion Method", Mathematical Problems in Engineering, vol. 2009, Article ID 327462, 20 pages, 2009. https://doi.org/10.1155/2009/327462
HigherOrder Solutions of Coupled Systems Using the Parameter Expansion Method
Abstract
We consider periodic solution for coupled systems of massspring. 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 [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 LindstedtPoincaré [14], ParameterExpanding [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 Case 1 and two cases of TDOF systems in Cases 2 and 3 are considered.
2.1. SingleDegreeofFreedom
Case 1 1 (Model of a Bulking Column). First, we consider the system shown in Figure 1. The mass 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 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.
(a)
(b)
Neglecting the weight of springs and columns shows that the governing equation for the motion of is [2] where , . The spring force is given by
2.2. TwoDegreeofFreedom
Case 2 2 (TwoMass System with Three Springs). The model of twomass system with three springs is shown in Figure 2. In this system, two equal masses are connected with the fixed supports using spring . The connection between two masses makes a compact item which is a spring with nonlinear properties. The linear coefficient of spring elasticity is and of the cubic nonlinearity is . The system has two degrees of freedom. The generalized coordinates are and .
The mathematical model of the system is [34] where is small nonlinearity . Dividing (2.3) by mass yields
Introducing the new variables
Transforming (2.4) yields
From (2.6), we have
Substituting (2.8) into (2.7) gives
Setting , (2.9) can be written as
Note that the case of corresponds to a hardening spring while indicates a softening one.
Case 3 3 (TwoMass System with a Connection Spring). Similarly, the model of system with one spring is shown in Figure 3. Two masses, and , are connected with a spring in which linear coefficient of rigidity is , and the nonlinear coefficient is . The system has two degrees of freedom.
The generalized coordinates of the system are and . The equation of motion of the system is described by [33]:
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
Solving (2.12) for yields
Substituting (2.14) into (2.13) gives
As mentioned, these models can be transformed to a cubic nonlinear differential equation in general form with different values and . The general form of cubic nonlinear differential is as follows:
3. Basic Idea of PEM
In order to use the PEM, we rewrite the general form of Duffing equation in the following form [7]: where includes the nonlinear term. Expanding the solution , as a coefficient of , and as a coefficient of , the series of can be introduced as follows:
Substituting (3.2)–(3.4) into (3.1) and equating the terms with the identical powers of , we have
Considering the initial conditions and , the solution of (3.5) is . Substituting into (3.6), we obtain
For achieving the secular term, we use Fourier expansion series as follows:
Substituting (3.8) into (3.7) yields
For avoiding secular term, we have
Setting in (3.3) and (3.4), we have:
Substituting (3.11) and (3.12) into (3.10), we will achieve the firstorder approximation frequency (2.10). Note that, from (3.4) and (3.12), we can find that , for all In the following section we will describe the secondorder 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:
Substituting (3.2) and (3.4) into (4.1) and equating the terms with the identical powers of , yields
Considering the initial conditions and , the solution of (4.2) is . Substituting into (4.3), we obtain
It is possible to perform the following Fourier series expansion:
Substituting (4.6) into (4.5) gives
No secular term in requires that
Setting in (3.3) and (3.4) gives
Substituting (4.9) and (4.10) into (4.8), we obtain
From (4.7) and (4.8), then (4.9) can be rewritten in the following form:
The periodic solution of (4.13) can be written [19]
Substituting (4.14) into (4.13) gives
From (4.15), the coefficients (for ) can be written as follows:
Taking into account that , (4.14) yields
To determine the secondorder approximate solution, it is necessary to substitute (4.14) into (4.4). Then secular term is eliminated, and parameter can be calculated. 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 in the following form:
Substituting and (4.8) and (4.18) into (4.4) gives
It is possible to do the following Fourier series expansion:
Substituting (4.20) into (4.19) and collecting, we have
The secular term in the solution for can be eliminated if
Solving (4.22) gives
On the other hand, From (4.16), the following expression for the coefficient is obtained:
Then, we can obtain
From (3.3), (3.4), and (4.8), and taking and considering , we have
Comparing the right hands of (4.25) and (4.26), one can easily obtain the following expression for the secondorder approximate frequency and period
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 and . Substituting and in (4.10), (4.11), (4.26), and (4.27) gives the following results for first and secondorder approximations of the model of nonlinear SDOF Bucking Column system in Case 1:
Also, we can obtain the first and secondorder approximations solutions for Case 2, by substituting and into (4.11), (4.12), (4.27), and (4.28):
Similarly, for and , we obtain the following frequency and period values for Case 3:
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, , of nonlinear differential equation in the cubic form is



Substituting and into (6.1) gives the exact frequency for Case 1:
Substituting and into (6.1), the exact solution of Case 2 is
Using (2.8) and , we can obtain
The first and secondorder analytical approximation for is obtained using (6.4) and therefore, the first and secondorder analytically approximates displacements and obtained using (2.5).
Similarly, substituting and into (6.1), the exact solution of Case 3 is:
After obtaining from (2.14), the first and secondorder analytically approximates displacements and obtained using (2.5).
It should be noted that contains an integral which could only be solved numerically in general. The limitation of amplitude, , in the cubic oscillation equation satisfies ; 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 in the first two cases and for the third case should, respectively, satisfy in (6.4), (6.5), and (6.6) where and .
To illustrate and verify accuracy of this analytical approach, comparisons of analytical and exact results for the practical cases are presented in Tables 1–3 and Figures 4–10. For this reason, we use the following specific parameter and initial values: Case 1: , , , , , , Case 2: , , , , , , and Case 3: , , , , , .
(a) diagram
(b) diagram
(a) diagram
(b) diagram
(a) diagram
(b) diagram
(a) diagram
(b) diagram
Figures 4–6, which are correspond to Case 1, indicate the comparison of this analytical method for different parameter with initial values , , , and , and , and , and , , , , and which are in an excellent agreement with exact solutions.
Figures 7 and 8 represent the and diagrams which obtained analytically and exactly solving of Case 2 with different parameter and initial values with , and , , , , , . Also, the corresponding diagrams (, ) 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: , , , , , and , , , , , .
According to these tables and figures, the difference between analytical and exact solutions is negligible. In other words, the firstorder approximate results of PEM are accurate, but we significantly improve the percentage error from lowerorder to secondorder 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 secondorder approximate frequencies and periods for Single and TwoDegreesOfFreedom (SDOF and TDOF) systems. Excellent agreements between approximate frequencies and the exact one have been demonstrated and discussed, and the discrepancy of the secondorder approximate frequency 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: where is a constant. Imposing initial conditions , yields
Equating (A.1) and (A.2) yields or equivalently
Integrating (A.4), the period of oscillation is
Substituting into (A.5) and integrating where
The exact frequency is also a function of and can be obtained from the period of the oscillation as
References
 R. E. Mickens, Oscillations in Planar Dynamic Systems, Scientific, Singapore, 1966.
 A. H. Nayfeh and D. T. Mook, Nonlinear Oscillations, WileyInterscience, New York, NY, USA, 1979. View at: MathSciNet
 R. P. Agarwal, S. R. Grace, and D. O'Regan, Oscillation Theory for Second Order Dynamic Equations, vol. 5 of Series in Mathematical Analysis and Applications, Taylor & Francis, London, UK, 2003. View at: MathSciNet
 J.H. He, NonPerturbative Methods for Strongly Nonlinear Problems, Dissertation. deVerlag im Internet GmbH, Berlin, Germany, 2006.
 A. H. Nayfeh, Perturbation Methods, John Wiley & Sons, New York, NY, USA, 1973. View at: Zentralblatt MATH  MathSciNet
 J.H. He, “Homotopy perturbation method for bifurcation on nonlinear problems,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 6, p. 207, 2005. View at: Google Scholar
 J.H. He, “An elementary introduction to recently developed asymptotic methods and nanomechanics in textile engineering,” International Journal of Modern Physics B, vol. 22, no. 21, pp. 3487–3578, 2008. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 A. R. Ghotbi, A. Barari, and D. D. Ganji, “Solving ratiodependent predatorprey system with constant effort harvesting using homotopy perturbation method,” Mathematical Problems in Engineering, vol. 2008, Article ID 945420, 8 pages, 2008. View at: Google Scholar  MathSciNet
 S. S. Ganji, D. D. Ganji, S. Karimpour, and H. Babazadeh, “Applications of He's homotopy perturbation method to obtain secondorder approximations of the coupled twodegreeoffreedom systems,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 10, no. 3, pp. 305–314, 2009. View at: Google Scholar
 S. T. MohyudDin and M. A. Noor, “Homotopy perturbation method for solving fourthorder boundary value problems,” Mathematical Problems in Engineering, vol. 2007, Article ID 98602, 15 pages, 2007. View at: Google Scholar  Zentralblatt MATH  MathSciNet
 L.N. Zhang and J.H. He, “Homotopy perturbation method for the solution of the electrostatic potential differential equation,” Mathematical Problems in Engineering, vol. 2006, Article ID 83878, 6 pages, 2006. View at: Google Scholar  MathSciNet
 I. Khatami, M. H. Pashai, and N. Tolou, “Comparative vibration analysis of a parametrically nonlinear excited oscillator using HPM and numerical method,” Mathematical Problems in Engineering, vol. 2008, Article ID 956170, 11 pages, 2008. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 Y.X. Wang, H.Y. Si, and L.F. Mo, “Homotopy perturbation method for solving reactiondiffusion equations,” Mathematical Problems in Engineering, vol. 2008, Article ID 795838, 5 pages, 2008. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 J.H. He, “Modified LindstedtPoincaré methods for some strongly nonlinear oscillations—II: a new transformation,” International Journal of Nonlinear Mechanics, vol. 37, no. 2, pp. 315–320, 2002. View at: Publisher Site  Google Scholar  MathSciNet
 D.H. Shou and J.H. He, “Application of parameterexpanding method to strongly nonlinear oscillators,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 8, no. 1, pp. 121–124, 2007. View at: Google Scholar
 N. H. Sweilam and R. F. AlBar, “Implementation of the parameterexpansion method for the coupled van der pol oscillators,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 10, no. 2, pp. 259–264, 2009. View at: Google Scholar
 L.N. Zhang and L. Xu, “Determination of the limit cycle by He's parameterexpansion for oscillators in a ${u}^{3}/(1+{u}^{2})$ potential,” Zeitschrift für Naturforschung A, vol. 62, no. 78, pp. 396–398, 2007. View at: Google Scholar
 L. Xu, “Application of He's parameterexpansion method to an oscillation of a mass attached to a stretched elastic wire,” Physics Letters A, vol. 368, no. 34, pp. 259–262, 2007. View at: Publisher Site  Google Scholar
 J.H. He, “A review on some new recently developed nonlinear analytical techniques,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 1, no. 1, pp. 51–70, 2000. View at: Google Scholar  Zentralblatt MATH  MathSciNet
 A. Marathe and A. Chatterjee, “Wave attenuation in nonlinear periodic structures using harmonic balance and multiple scales,” Journal of Sound and Vibration, vol. 289, no. 45, pp. 871–888, 2006. View at: Publisher Site  Google Scholar
 H. P. W. Gottlieb, “Harmonic balance approach to limit cycles for nonlinear jerk equations,” Journal of Sound and Vibration, vol. 297, no. 12, pp. 243–250, 2006. View at: Publisher Site  Google Scholar  MathSciNet
 J.H. He, “Linearized perturbation technique and its applications to strongly nonlinear oscillators,” Computers & Mathematics with Applications, vol. 45, no. 1–3, pp. 1–8, 2003. View at: Google Scholar  Zentralblatt MATH  MathSciNet
 J.H. He, “Preliminary report on the energy balance for nonlinear oscillations,” Mechanics Research Communications, vol. 29, no. 23, pp. 107–111, 2002. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 S. S. Ganji, D. D. Ganji, Z. Z. Ganji, and S. Karimpour, “Periodic solution for strongly nonlinear vibration systems by He's energy balance method,” Acta Applicandae Mathematicae, vol. 106, no. 1, pp. 79–92, 2009. View at: Publisher Site  Google Scholar
 S. S. Ganji, D. D. Ganji, and S. Karimpour, “He's energy balance and He's variational methods for nonlinear oscillations in engineering,” International Journal of Modern Physics B, vol. 23, no. 3, pp. 461–471, 2009. View at: Google Scholar
 M. Rafei, D. D. Ganji, H. Daniali, and H. Pashaei, “The variational iteration method for nonlinear oscillators with discontinuities,” Journal of Sound and Vibration, vol. 305, no. 45, pp. 614–620, 2007. View at: Publisher Site  Google Scholar  MathSciNet
 J.H. He and X.H. Wu, “Construction of solitary solution and compactonlike solution by variational iteration method,” Chaos, Solitons & Fractals, vol. 29, no. 1, pp. 108–113, 2006. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 J.H. He, “Variational approach for nonlinear oscillators,” Chaos, Solitons & Fractals, vol. 34, no. 5, pp. 1430–1439, 2007. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 L. Xu, “Variational approach to solitons of nonlinear dispersive $K(m,n)$ equations,” Chaos, Solitons & Fractals, vol. 37, no. 1, pp. 137–143, 2008. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 J.H. He, “Iteration perturbation method for strongly nonlinear oscillations,” Journal of Vibration and Control, vol. 7, no. 5, pp. 631–642, 2001. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 M. A. Noor and S. T. MohyudDin, “Variational homotopy perturbation method for solving higher dimensional initial boundary value problems,” Mathematical Problems in Engineering, vol. 2008, Article ID 696734, 11 pages, 2008. View at: Google Scholar  Zentralblatt MATH  MathSciNet
 D. D. Ganji, M. Rafei, A. Sadighi, and Z. Z. Ganji, “A comparative comparison of He's method with perturbation and numerical methods for nonlinear vibrations equations,” International Journal of Nonlinear Dynamics in Engineering and Sciences, vol. 1, no. 1, pp. 1–20, 2009. View at: Google Scholar
 L. Cveticanin, “Vibrations of a coupled twodegreeoffreedom system,” Journal of Sound and Vibration, vol. 247, no. 2, pp. 279–292, 2001. View at: Publisher Site  Google Scholar
 L. Cveticanin, “The motion of a twomass system with nonlinear connection,” Journal of Sound and Vibration, vol. 252, no. 2, pp. 361–369, 2002. View at: Publisher Site  Google Scholar
 R. Vito, “On the stability of vibrations of particle in a plane constrained by identifical nonlinear springs,” International Journal of Nonlinear Mechanics, vol. 9, no. 84, p. 325, 1974. View at: Publisher Site  Google Scholar
 E. A. Jackson, Perspectives of Nonlinear Dynamics. Vol. 1, Cambridge University Press, Cambridge, UK, 1989. View at: MathSciNet
Copyright
Copyright © 2009 S. S. Ganji et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.