Rational Energy Balance Method to Nonlinear Oscillators with Cubic Term

From the SelectedWorks of Hassan Askari 2013 Rational Energy Balance Method to Nonlinear Oscillators with Cubic Term Hassan Askari Available at: https://works.bepress.com/hassan_askari/4/

Asian-European Journal of Mathematics Vol. 6, No. 2 (2013) 1350019 (8 pages) c World Scientific Publishing Company DOI: 10.1142/S1793557113500198 RATIONAL ENERGY BALANCE METHOD TO NONLINEAR OSCILLATORS WITH CUBIC TERM M. Daeichin School of Mechanical Engineering Sharif University of Technology 11365-9567 Tehran, Iran Meysam.Daeichi@gmail.com M. A. Ahmadpoor Department of Mathematical Sciences Sharif University of Technology 11155-9415 Tehran, Iran M.A.Ahmadpoor1989@gmail.com H. Askari Center of Excellence in Railway Transportation School of Railway Engineering Iran University of Science and Technology Narmak, Tehran 16846, Iran Askari.Iust@gmail.com A. Yildirim Department of Mathematics and Statistics University of South Florida Tampa, FL 33620-5700, USA Ahmetyildirim80@gmail.com CommunicatedbyB.K.Dass Received October 23, 2012 Revised December 12, 2012 Published June 17, 2013 In this paper, a novel approach is proposed for solving the nonlinear problems based on the collocation and energy balance methods (EBMs). Rational approximation is employed as an initial guess and then it is combined with EBM and collocation method for solving nonlinear oscillators with cubic term. Obtained frequency amplitude relationship is compared with exact numerical solution and subsequently, a very excellent accuracy will be revealed. According to the numerical comparisons, this method provides high accuracy with 0.03% relative error for Duffing equation with strong nonlinearity Corresponding author 1350019-1

M. Daeichin et al. in the second-order of approximation. Furthermore, achieved results are compared with other types of modified EBMs and the second-order of harmonic balance method. It is demonstrated that the new proposed method has the highest accuracy in comparison with different approaches such as modified EBMs and the second-order of harmonic balance method. Keywords: Duffing oscillators; rational approximation; energy balance method. 1. Introduction A number of techniques have been already suggested for solving nonlinear equations. Homotopy analysis method (HAM) [18], variational iteration method [17] (VIM), Adomian decomposition method (ADM) [3] and perturbation methods [22] are the prominent and the potent approaches that have been employed for solving several nonlinear problems [1, 2, 4, 7, 19, 24, 32, 33]. In addition, Hamiltonian approach (HA) [16], energy balance method (EBM) [12], frequency amplitude formulation (FAF) [14], max-min approach (MMA) [15], variational approach (VA) [13] can be mentioned as novel ways that have been utilized for evaluating lots of sophisticated nonlinear equations [5, 8, 11, 25 31, 34]. HA, EBM, FAF, MMA and VA are so straightforward in comparison with HAM, VIM, ADM and perturbation methods and they have been modified by several researchers for extracting analytical solutions of the nonlinear problems. For instance, Younesian et al. [31] have amended EBM using Petrov Galerkin method. Moreover, HA has been modified by Yildirim et al. [27]. In addition, Durmaz and Kaya [10] have modified EBM using Petrov Galerkin approach and collocation method [9]. In this paper, EBM is modified using rational approximation. Rational approximation have been originally proposed by Mickens [20] for analyzing of nonlinear problems and then, this novel idea has been employed by several researchers for solving diverse forms of nonlinear equations [6, 21]. In the present study, the aforementioned idea is used and expanded with combining EBM. It is proved, this new modification of EBM has the highest accuracy in comparison with other kinds of its modifications. Furthermore, achieved result of developed method is numerically compared with second-order of harmonic balance method [23]. 2. Classical Energy Balance Method In this part, EBM is succinctly illustrated. First, we consider Eq. (2.1) as a generalized nonlinear oscillator and next according to the He s procedure, variational and Hamiltonian functions are constructed in Eqs. (2.3) and (2.4), respectively. with the following initial conditions: ẍ + f(x) = 0 (2.1) x(0) = A, ẋ(0) = 0 (2.2) 1350019-2

Rational Energy Balance Method variational and Hamiltonian functions can be constructed as: T [ ] 4 1 J(x) = 2 ẋ2 + F (x) dt, (2.3) where F (x) = f(x)dx and T = 2π ω : 0 H(x) = 1 2 ẋ2 + F (x) (2.4) and based on He s method, the residual function is defined as R(t) =H(x) H(x(0)). (2.5) In the classical procedure of EBM, A cos ωt is assumed as an initial guess for Eq. (2.1) and it is inserted in Eq. (2.5). In addition, ωt = π 4 is used as location point and subsequently the following algebraic equation is obtained: 1 R(t, ω) = (ωt π 4 ) 2 A2 ω 2 sin 2 ωt + F (A cos ωt) F (A) =0. (2.6) 3. Rational Energy Balance Method In this section, first, rational approximation is briefly introduced and then this approximation is combined with classical EBM. As illustrated in Sec. 1 of the paper, the following function is exerted as an approximation for the analyzing of the nonlinear problems by Mickens [20]. in which x N M = N n=0 [A n cos(2n +1)θ + B n sin(2n +1)θ] 1+ M m=1 [C m cos(2m)θ + D m sin(2m)θ], (3.1) θ = ωt, (3.2) where ω is the frequency of the oscillation, {A n,b n,c m,d m } are a set of constants and M and N are integers [20]. As defined by Mickens, the first approximation of Eq. (3.1) reaches the following function: And the second approximations can be defined as [20]: x 0 0(t) =A 0 cos θ. (3.3) x 1 0 (t) =A 0 cos θ + A 1 cos 3θ, (3.4) x 0 1 (t) = A 0 cos θ (3.5) 1+C 1 cos 2θ Eq. (3.4) has been already used by several researchers for modifying diverse kinds of analytical techniques [9, 10, 27]. In this study, Eq. (3.5) is exerted as a second approximation and it is already combined with EBM. By substituting Eq. (3.5) 1350019-3

M. Daeichin et al. into Eq. (2.5), Eq. (3.6) is obtained as an original residual function for Eq. (2.1). R(t) = 1 ( A0 ω sin(ωt)(1 + C 1 cos(2ωt) 2A 0 C 1 ω cos(ωt)sin(2ωt) 2 (1 + C 1 cos(2ωt)) 2 ( ) A0 cos ωt + F F (A). (3.6) 1+C 1 cos 2ωt For finding A 0,C 1,ω, two location points are employed based on the collocation method. According to the collocation method, Eqs. (3.7) (3.8) are defined as residual functions that must be solved. 1 R(t, ω) = (ωt π 6 ) 2 And + F 1 R(t, ω) = (ωt π 3 ) 2 + F ( A0 ω sin(ωt)(1 + C 1 cos(2ωt) 2A 0 C 1 ω cos(ωt)sin(2ωt) ( A0 cos ωt 1+C 1 cos 2ωt ) 2 ) 2 (1 + C 1 cos(2ωt)) 2 ) F (A) =0. (3.7) ( A0 ω sin(ωt)(1 + C 1 cos(2ωt) 2A 0 C 1 ω cos(ωt)sin(2ωt) ( A0 cos ωt 1+C 1 cos 2ωt ) 2 (1 + C 1 cos(2ωt)) 2 ) F (A) =0. (3.8) Furthermore, based on the initial conditions, the third equation for the finding the unknown parameters of Eq. (3.6) is: A 0 1+C 1 = A. (3.9) By simultaneously solving Eqs. (3.7) (3.9), the unknown parameters of Eq. (3.6) is easily obtained. 4. Applications In this part, rational EBM is utilized for extracting analytical solution of the nonlinear oscillator with cubic terms, namely the duffing equation: ẍ + x + εx 3 =0. (4.1) Based on Eq. (2.3), the variational function can be defined as: T J(x) = ( 12ẋ2 + x 2 + 14 ) εx4 dt. (4.2) 0 1350019-4

Rational Energy Balance Method And the Hamiltonian can be shown by: H(x) = 1 2 ẋ2 + x 2 + 1 4 εx4 = 1 2 A2 + 1 4 εa4. (4.3) According to the He s classical EBM [12], the first approximate frequency-amplitude relationship of Eq. (4.1) is equal to ω = 1+ 3 4 εa2. (4.4) By employing Eq. (3.5) as a second approximation and based on Eqs. (3.7) and (3.8), a coupled algebraic equation is obtained as: 2ω 2 (A 0 (2 + C 1 ) 6A 0 C 1 ) 2 +6A 2 0 (2 + C 1) 2 +9εA 4 0 (2A2 + εa 4 )(2 + C 1 ) 4 =0, (4.5) 6ω 2 (A 0 (2 C 1 ) 2A 0 C 1 ) 2 +2A 2 0 (2 C 1) 2 + εa 4 0 (2A2 + εa 4 )(2 C 1 ) 4 =0. (4.6) By simultaneously solving Eqs. (3.9), (4.5) (4.6), natural frequencies and other unknown parameters can be readily achieved. 5. Discussion and Numerical Results For evaluating the correctness of the modified version of EBM by rational approximation, it is numerically compared with exact solutions. For a large range of amplitude, obtained natural frequencies of Eq. (4.1) by means of REBM are tabulated in Table 1. In accordance with this table, it can be concluded, REBM gives very Table 1. Comparison of obtained results of different kinds of analytical approaches for Eq. (4.1). εa 2 ω ex ω FEBM c ω FEBM GP [10] ω SEBM c [9] ω SEBM GP ω SHBM [23] ω REBM (Relative (Relative (Relative (Relative (Relative (Relative error %) error %) error %) error %) error %) error %) 0.5 1.1708 1.1726 1.1619 1.1702 1.1702 1.1707 1.1707 0.16% 0.76% 0.05% 0.05% 0.001% 0.001% 1 1.3178 1.3229 1.3038 1.3161 1.3163 1.3178 1.3178 0.39% 1.06% 0.13% 0.11% 0.0043% 0.0043% 5 2.1504 2.1795 2.1213 2.1406 2.1426 2.1509 2.1507 1.35% 1.35% 0.46% 0.36% 0.0262% 0.015% 10 2.8666 2.9155 2.8284 2.8500 2.8535 2.8678 2.8672 1.70% 1.33% 0.58% 0.45% 0.0419% 0.0217% 100 8.5336 8.7178 8.4261 8.4700 8.4842 8.5390 8.5360 2.16% 1.25% 0.75% 0.57% 0.0643% 0.0287% 1000 26.8107 27.4044 26.4764 26.6055 26.6519 26.8289 26.8187 2.21% 1.24% 0.77% 0.59% 0.0681% 0.03011% 5000 59.9157 61.2454 59.1692 59.4559 59.5599 59.9563 59.9337 2.22% 1.24% 0.77% 0.59% 0.06783% 0.0301% 1350019-5

M. Daeichin et al. outstanding results in the second approximation for the Duffing equation. Moreover, obtained natural frequencies of diverse kinds of EBMs and harmonic balance approach have been tabulated in the mentioned table. As seen from Table 1, REBM has the highest accuracy among all of the represented methods in Table 1. According to Table 1, REBM reaches to very exceptional accuracy for very strong nonlinearity in comparison with the other methods. For example, when εa 2 = 5000, relative errors of SEBM C, SEBM GP, SHBM and REBM are 0.77%, 0.59%, 0.06783% and 0.0301%, correspondingly. It is apparent from the abovementioned conclusion that REBM are so potent in comparison with other forms of modified EBM. Besides, it can be concluded that REBM is more effective than SHBM for solving the Duffing equation. 6. Conclusion In this study, a novel technique was explained based on the collocation method, EBM and rational approximation. Initially, classical EBM was illustrated and then rational approximation was briefly explicated. In continuation, by combination of the EBM and the rational approximation, a novel modified version of the EBM was proposed and subsequently it was applied to the well-known Duffing equation. Furthermore, to examine accuracy of the proposed method, natural frequencies were tabulated and compared with exact numerical solutions. From the tabulated frequencies, it can be concluded that the modified method is so potent for solving nonlinear problems. In addition, the modified method was compared with diverse types of EBM and second-order of the harmonic balance method. According to Table 1 of this paper, accuracy of these techniques can be sorted as: SEBM C < SEBM GP < SHBM < REBM. Future research can examine accuracy of REBM to the other kinds of nonlinear oscillators. Furthermore, it can be a very interesting idea to find a way for employing rational elliptic function [35] and combining it with classical EBM for solving nonlinear problems. References 1. T. A. Abassy, M. A. El-Tawil and H. Zoheiryb, Modified variational iteration method for Boussinesq equation, Comput. Math. Appl. 54 (2007) 955 965. 2. M. A. Abdou and A. Soliman, New applications of variational iteration method, Physica D: Nonlinear Phenomena 211 (2005) 18. 3. G. Adomian, A review of the decomposition method in applied mathematics, J. Math. Anal. Appl. 135 (1988) 501 544. 4. M.R.Alam,Y.LiuandD.K.P.Yue,Braggresonanceofwavesinatwo-layerfluid propagating over bottom ripples. Part I. Perturbation analysis, J. Fluid Mech. 624 (2009) 191 224. 5. H. Askari, Z. Saadatnia, D. Younesian, A. Yildirim and M. KalamiYazdi, Approximate periodic solutions for the Helmholtz Duffing equation, Comput. Math. Appl. 62 (2011) 3894 3901. 1350019-6

Rational Energy Balance Method 6. A. Belendez, E. Gimeno, T. Belendez and A. Hernandez, Rational harmonic balance based method for conservative nonlinear oscillators: Application to the Duffing equation, Mech. Res. Comm. 36 (2009) 728 734. 7. L. Cveticanin, M. KalamiYazdi, H. Askari and Z. Saadatnia, Vibration of a two-mass system with non-integer order nonlinear connection, Mech. Res. Comm. 43 (2012) 22 28. 8. L. Cveticanin, M. K. Yazdi and H. Askari, Analytical approximations to the solutions for a generalized oscillator with strong nonlinear terms, J. Engrg. Math. 77 (2012) 211 223. 9. S. Durmaz, S. A. Demirbag and M. O. Kaya, High order He s energy balance method based on collocation method, Int. J. Nonlinear Sci. Numer. Simul. 11 (2010) 1 5. 10. S. Durmaz and M. O. Kaya, High order energy balance method to nonlinear oscillators, J. Appl. Math. 2012 (2012), Article ID:518684, 7 pp. 11. J. Fan, He s frequency-amplitude formulation for the Duffing harmonic oscillator, Comput. Math. Appl. 58 (2009) 2473 2476. 12. J. H. He, Preliminary report on the energy balance for nonlinear oscillations, Mech. Res. Comm. 29 (2002) 107 111. 13. J. H. He, Variational approach for nonlinear oscillators, Chaos Solitons Fractals 34 (2007) 1430 1439. 14. J. H. He, An improved amplitude-frequency formulation for nonlinear oscillators, Int. J. Nonlinear Sci. Numer. Simul. 9(2) (2008) 211 212. 15. J. H. He, Max-Min approach to nonlinear oscillators, Int. J. Nonlinear Sci. Numer. Simul. 9(2) (2008) 207 210. 16. J. H. He, Hamiltonian approach to nonlinear oscillators, Phys. Lett. A 374 (2010) 2312 2314. 17. J. H. He, G. C. Wu and F. Austin, The variational iteration method which should be followed, Nonlinear Sci. Lett. A 1 (2010) 130. 18. S. J. Liao, An approximate solution technique not depending on small parameters: A special example, Int. J. Non-Linear Mech. 30 (1995) 371 380. 19. S. J. Liao, On the homotopy analysis methods for nonlinear problems, Appl. Math. Comput. 147 (2004) 499 513. 20. R. E. Micknes, A generalization of the method of harmonic balance, J. Sound Vibration 111(3) (1986) 515 518. 21. R. E. Micknes and D. Semwogerere, Fourier analysis of rational harmonic balance approximation for periodic solutions, J. Sound Vibration 195 (1996) 417 429. 22. A. H. Nayfeh and D. T. Mook, Nonlinear Oscillations (Wiley, New York, 1979). 23. B. S. Wu, W. P. Sun and C. W. Lim, An analytical approximate technique for a class of strongly non-linear oscillators, Internat. J. Non-Linear Mech. 41 (2006) 766 774. 24. A. Yildirim, Variational iteration method for inverse problem of diffusion equation, Int. J. Numer. Methods Biomed. Engrg. 26 (2010) 1713 1720. 25. A. Yildirim, H. Askari, Z. Saadatnia, M. KalamiYazdi and Y. Khan, Analysis of nonlinear oscillations of a punctual charge in the electric field of a charged ring via a Hamiltonian approach and the energy balance method, Comput. Math. Appl. 62 (2011) 486 490. 26. A. Yildirim, H. Askari, M. K. Yazdi and Y. Khan, A relationship between three analytical approaches for the nonlinear problems, Appl. Math. Lett. 77 (2012) 1729 1733. 27. A. Yildirim, Z. Saadatnia, H. Askari, Y. Khan and M. KalamiYazdi, Higher order approximate periodic solutions for nonlinear oscillators with the Hamiltonian approach, Appl. Math. Lett. 24 (2011) 2042 2051. 1350019-7

M. Daeichin et al. 28. D. Younesian, H. Askari, Z. Saadatnia and M. KalamiYazdi, Frequency analysis of strongly nonlinear generalized Duffing oscillators using He s frequency-amplitude formulation and He s energy balance method, Comput. Math. Appl. 59 (2010) 3222 3228. 29. D. Younesian, H. Askari, Z. Saadatnia and M. KalamiYazdi, Free vibration analysis of strongly nonlinear generalized Duffing oscillators using He s variational approach and homotopy perturbation method, Nonlinear Sci. Lett. A 2 (2011) 11 16. 30. D. Younesian, H. Askari, Z. Saadatnia and M. KalamiYazdi, Analytical approximate solutions for the generalized nonlinear oscillator, Appl. Anal. 91 (2012) 965 977. 31. D. Younesian, H. Askari, Z. Saadatnia and A. Yildirim, Periodic solutions for the generalized nonlinear oscillators containing fraction order elastic force, Int. J. Nonlinear Sci. Numer. Simul. 11(12) (2010) 1027 1032. 32. D. Younesian, H. Askari, Z. Saadatnia and A. Yildirim, Analytical solution for nonlinear wave propagation in shallow media using the variational iteration method, Waves Random Complex Media 22 (2012) 133 142. 33. D. Younesian, Z. Saadatnia and H. Askari, Analytical solutions for free oscillations of beams on nonlinear elastic foundations using the variational iteration method, J. Theoret. Appl. Mech. 50 (2012) 639 652. 34. D. Q. Zeng, Nonlinear oscillator with discontinuity by the max-min approach, Chaos Solitons Fractals 42 (2009) 2885 2889. 35. A. E. Zuniga, C. A. Rodriguez and O. M. Romero, On the solution of strong nonlinear oscillators by applying a rational elliptic balance method, Comput. Math. Appl. 60 (2010) 1409 1420. 1350019-8