A new analytical approach for solving quadratic nonlinear oscillators Alexandria Engineering Journal (2016) xxx, xxx–xxx HO ST E D BY Alexandria University Alexandria Engineering Journal www elsevier[.]
Alexandria Engineering Journal (2016) xxx, xxx–xxx H O S T E D BY Alexandria University Alexandria Engineering Journal www.elsevier.com/locate/aej www.sciencedirect.com ORIGINAL ARTICLE A new analytical approach for solving quadratic nonlinear oscillators Md Mahtab Hossain Mondal, Md Helal Uddin Molla *, Md Abdur Razzak, M.S Alam Department of Mathematics, Rajshahi University of Engineering and Technology (RUET), Kazla, Rajshahi 6204, Bangladesh Received June 2016; revised 27 September 2016; accepted November 2016 KEYWORDS Quadratic nonlinear oscillator; Harmonic balance method; Periodic solution Abstract In this paper, a new analytical approach based on harmonic balance method (HBM) is presented to obtain the approximate periods and the corresponding periodic solutions of quadratic nonlinear oscillators The result obtained in new approach has been compared with that obtained by other existing method The present method gives not only better result than other existing result but also gives very close to the corresponding numerical result (considered to be the exact result) Moreover, the method is simple and straightforward Ó 2016 Faculty of Engineering, Alexandria University Production and hosting by Elsevier B.V This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Introduction The nonlinear problem often arises in exact modeling of phenomena in physical science, mechanical structures, nonlinear circuits, chemical oscillation and other engineering research and study of them is of interest to many researchers For example, the eardrum is the best modeled by quadratic nonlinear oscillator [1] Nowadays, several analytical methods such as homotopy perturbation [2], harmonic balance [3], residue harmonic balance [4], global residue harmonic balance [5], Hamiltonian [6], homotopy analysis [7], max-min [8], coupling of homotopy-variational [9], iterative homotopy harmonic balance method [10], Fourier series solutions with finite harmonic terms [11], and amplitude-frequency formulation [12] have been developed for solving strongly nonlinear oscillators Earlier the classical perturbation methods [13–16] were used to * Corresponding author E-mail address: helal.mathru@yahoo.com (Md Helal Uddin Molla) Peer review under responsibility of Faculty of Engineering, Alexandria University solve weakly nonlinear problems Recently, Hu [17] has used HBM to determine an approximate solution of a quadratic nonlinear oscillator, x ỵ x ỵ ex2 ẳ 0; but the method is not a simple one Hu [17] has obtained two separate harmonic balance solutions respectively for two regions x > and x < The solution is continuous, but the derivative does not exist when it cuts the axis In the present article, a new analytical approximate technique based on HBM is presented to obtain the approximate solution of quadratic nonlinear oscillators Here we obtain one trial solution and the solution is continuous and differentiable everywhere The results are compared with those obtained by Hu [17] (see Appendix A) Formulation and solution method Consider a nonlinear differential equation _ x ỵ x ẳ efx; xị; _ x0ị ¼ a; xð0Þ ¼0 ð1Þ _ is a nonlinear function such that fx; xị _ where fx; xị _ ẳ fðx; xÞ http://dx.doi.org/10.1016/j.aej.2016.11.010 1110-0168 Ĩ 2016 Faculty of Engineering, Alexandria University Production and hosting by Elsevier B.V This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Please cite this article in press as: M Mahtab Hossain Mondal et al., A new analytical approach for solving quadratic nonlinear oscillators, Alexandria Eng J (2016), http://dx.doi.org/10.1016/j.aej.2016.11.010 Md Mahtab Hossain Mondal et al Let us consider, xtị ẳ ac ỵ q cos u ỵ u cos 2u ỵ v cos 3u ỵ ị 2ị be a solution of (1), where a; c; q are constants, u ¼ x t and x ¼ 2p is a frequency of nonlinear oscillation, here T is a perT iod If q ¼ c u v and the initial phase u0 ¼ 0; solution Eq (2) readily satisfies the initial conditions _ x0ị ẳ a; x0ị ẳ 0: Substituting Eq (2) in Eq (1) and expand_ in a Fourier series, it turns to an algebraic identity ing fðx; xÞ a qð1 c u_ ị cos u ỵ u1 4u_ ị cos 2u ẳ eẵF1 a; c; u; ị cos u ỵ F2 a; c; u; Þ cos 2u : ð3Þ 4au2 ve q 3u þ ae 6cu þ 4u2 vq q2 ¼0 13ị 9auv2 e q8v ỵ ae16cv ỵ 9uv uqịị ẳ 0: 14ị Neglecting the higher order terms more than two such as u2 v and uv2 from Eqs (13) and (14) and also dividing Eqs (13) and (14) by q we obtain as follows 15ị 3u ỵ ae 6cu ỵ 4u2 vq q2 ẳ 8v ỵ ae16cv ỵ 9uv uqị ẳ 0: 16ị Equating the coefficients of equal harmonics of Eq (3), the following nonlinear algebraic equations are found: Substituting q ¼ c u v in Eqs (15) and (16) then we obtain qð1 c u_ ị ẳ eF1 ; u1 4u_ ị ẳ eF2 ; ae1 ỵ c2 ị 6u 2aec ỵ u ỵ 5cu ỵ 7u2 ỵ v2 ị ẳ 17ị aue1 ỵ cị 8v aeu2 ỵ 16cv ỵ 10uvị ẳ 0: 18ị v1 9u_ ị ẳ eF3 ; ð4Þ with the help of second equation, u_ is eliminated from all the rest of Eq (4) Thus Eq (4) takes the following form qu_ ẳ q ỵ eF1 qc; 3qu ẳ qeF2 4ueF1 ỵ 4quc; 8qv ẳ qeF3 9veF2 ỵ 9qvc; 5ị using q ẳ c u v and simplifying, second, thirdequations of Eq (5) takes the following nonlinear algebraic equations G1 ða; e; c; u; v; ị ẳ 0; G2 a; e; c; u; v; ị ẳ 0; : Here the coefficient of u of Eq (17) is and ae 1=2; e > On the other hand, the coefficient of v of Eq (18) is and v fully depends on u Therefore, Eqs (17) and (18) can be solved in power series by choosing a small parameter k ¼ ae=6 Thus we obtain u ẳ 1 ỵ cị2 k 21 ỵ 5cịk2 ỵ 31 ỵ 18c ỵ 31c2 ịk3 19ị 17 3c ỵ 489c2 ỵ 395c3 k4 6ị These types of algebraic equations have been solved by the power series method introducing a small parameter (see [18,19] for details) which provides desired results v ẳ 1 ỵ cị k2 ỵ ỵ 7cịk3 13 ỵ 242c ỵ 635c2 ịk4 : 4 20ị Now Eq (9) can be solved for c by substituting the values of u and v from Eqs (19) and (20) But we use another equation to find the value of c When u ! p, x (presented in Example Consider the quadratic nonlinear equation in the following form _ x ỵ x ỵ ex2 ẳ 0; x0ị ẳ a; x0ị ẳ 0: 7ị The third-order approximate solution is chosen in the following form Table Comparison of approximate periods with the corresponding exact period and Hu [17] for e ¼ a Exact Hu [17] Er(%) Present study Er(%) where 0.10 6.3116 q ¼ c u v and u ¼ x t: 0.20 6.4114 0.30 6.6294 0.40 7.1246 0.45 7.7065 0.46 7.9052 0.47 8.1672 0.48 8.5452 0.49 9.2080 6.3112 0.006 6.4095 0.029 6.6226 0.103 7.0962 0.399 7.6277 1.023 7.8014 1.313 8.0233 1.762 8.3278 2.544 8.8118 4.303 6.3116 0.000 6.4114 0.000 6.6290 0.006 7.1206 0.056 7.6939 0.163 7.8884 0.213 8.1466 0.252 8.5250 0.236 9.2202 0.132 x ẳ ac ỵ q cos u ỵ u cos 2u þ v cos 3uÞ ð8Þ Substituting Eq (8) into Eq (7) and expanding in a Fourier series and equating the constant terms and the coefficients of cos u, cos 2u and cos 3u respectively, we obtained the following equations as c ỵ ae 2c2 ỵ u2 ỵ v2 ỵ q2 ẳ 9ị auve ỵ q1 ỵ 2ace ỵ aueị qx2 ẳ 10ị u ỵ ae 2cu ỵ vq ỵ q2 4ux2 ẳ 11ị v ỵ ae2cv ỵ uqị 9vx2 ẳ 0: 12ị By elimination of x2 from Eqs (10)–(12), we obtained the following equations as Where Er(%) denotes the absolute percentage error Please cite this article in press as: M Mahtab Hossain Mondal et al., A new analytical approach for solving quadratic nonlinear oscillators, Alexandria Eng J (2016), http://dx.doi.org/10.1016/j.aej.2016.11.010 0.6 0.6 0.4 0.4 0.2 0.2 -0.2 x x Solving quadratic nonlinear oscillators t 10 -0.2 t 10 -0.6 -0.6 -0.8 -0.8 -1 -1 Figure Represents a comparison of the obtained from present analytical approximate solution (denoting by circles) with numerical ones (denoting by solid line) and also with known results [17] (denoting by dash lines) for e ¼ and the initial amplitude a ¼ 0:45 Figure Represents a comparison of the obtained from present analytical approximate solution (denoting by circles) with numerical ones (denoting by solid line) and also with known results [17] (denoting by dash lines) for e ¼ and the initial amplitude a ¼ 0:49 1.2 0.6 0.8 0.4 0.4 x 0.2 x -0.4 -0.4 -0.2 0 t 10 -0.4 -0.4 -0.8 -0.6 -1.2 -0.8 -1.6 -1 -2 Figure Represents a comparison of the obtained from present analytical approximate solution (denoting by circles) with numerical ones (denoting by solid line) and also with known results [17] (denoting by dash lines) for e ¼ and the initial amplitude a ¼ 0:47 t 10 Figure Represents a comparison of the obtained from present analytical approximate solution (denoting by circles) with numerical ones (denoting by solid line) for e ¼ 0:5 and the initial amplitude a ¼ 0:96 0.6 0.4 dx/dt 0.6 0.4 x 0.2 0 -0.2 0.2 t 10 -1 -0.8 -0.6 -0.4 x -0.2 0.2 0.4 0.6 -0.2 -0.4 -0.4 -0.6 -0.8 -0.6 -1 Figure Represents a comparison of the obtained from present analytical approximate solution (denoting by circles) with numerical ones (denoting by solid line) and also with known results [17] (denoting by dash lines) for e ¼ and the initial amplitude a ¼ 0:48 Eq (8)) becomes að1 2c 2uÞ, which is equal to b (see [17] for details) The value of b is obtained from the algebraic equation Figure Phase portrait: Represents a comparison of the present analytical approximate solution (denoting by circles) with numerical ones (denoting by solid line) for initial conditions are _ x0ị ẳ 0:47 and x0ị ẳ a2 ea3 b2 eb3 ỵ ẳ : 3 ð21Þ The above equation has three solution, but e b Therefore, Please cite this article in press as: M Mahtab Hossain Mondal et al., A new analytical approach for solving quadratic nonlinear oscillators, Alexandria Eng J (2016), http://dx.doi.org/10.1016/j.aej.2016.11.010 Md Mahtab Hossain Mondal et al 0.6 0.6 0.4 dx/dt 0.4 dx/dt 0.2 0.2 -1 -0.8 -0.6 -0.4 x -0.2 0.2 0.4 0.6 -1 -0.2 -0.8 -0.6 -0.4 x -0.2 0.2 0.4 0.6 -0.2 -0.4 -0.4 -0.6 -0.6 Figure Phase portrait: Represents a comparison of the present analytical approximate solution (denoting by circles) with numerical ones (denoting by solid line) for initial conditions are _ x0ị ẳ 0:48 and x0ị ẳ Figure Phase portrait: represents a comparison of the Hu [17] approximate solution (denoting by circles) with numerical ones (denoting by solid line) for initial conditions are x0ị ẳ 0:48 and _ x0ị ẳ 0.5 0.7 0.3 0.5 dx/dt dx/dt 0.7 0.3 0.1 -1 -0.8 -0.6 -0.4 x -0.2 -0.1 0.2 0.4 0.1 0.6 -1 -0.3 -0.8 -0.6 -0.4 Figure Phase portrait: Represents a comparison of the present analytical approximate solution (denoting by circles) with numerical ones (denoting by solid line) for initial conditions are _ x0ị ẳ 0:49 and x0ị ẳ 22ị and then ð23Þ Substituting the value of u from Eq (19) in Eq (23), we can determine the value of c But it has no analytical solution We use an iteration formula to solve it Eq (23) can be written as a1 2c 21 ỵ cị2 Qị ẳ b 24ị where Q ẳ k 21 ỵ 5cịk2 þ 3ð1 þ 18c þ 31c2 Þk3 17 3c ỵ 489c2 ỵ 395c3 k4 choosing Q constant, we can easily solve Eq (24) as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a ỵ 2aQ ỵ a2 2a2 Q 2abQ c¼ : 2aQ 0.4 0.6 -0.5 -0.7 að1 2c 2uị ẳ b: 0.2 -0.3 -0.5 pp b ẳ ỵ 2ae 3 4ae 4a2 e2 =ð4eÞ; x -0.2 -0.1 ð25Þ ð26Þ Eq (24) has two solutions Here ỵ is used before the surd since jcj < b: -0.7 Figure 10 Phase portrait: represents a comparison of the Hu [17] approximate solution (denoting by circles) with numerical ones (denoting by solid line) for initial conditions are x0ị ẳ 0:49 and _ x0ị ẳ Results and discussion Based on the harmonic balance method, a straightforward analytical approximate technique has been presented to determine the approximate solution of quadratic nonlinear oscillators The method is in agreement with the corresponding numerical solutions and gives similar results to those obtained by Hu [17] But it has already been mentioned that (in Section 2), he solved the equation by two steps This is the disadvantage of Hu’s method First, we have calculated the approximate periods of Eq (7) obtained in this paper for several values of amplitude, a and e ¼ and the results have been presented in Table We have also included the corresponding exact period and other existing results [17] Finally, we have determined the approximate solution of Eq (7) by using present method for e ¼ 1; a ¼ 0:45; e ¼ 1; a ¼ 0:47; e ¼ 1; a ¼ 0:48; e ¼ 1; a ¼ 0:49; e ¼ 0:5; a ¼ 0:96 and other existing solutions obtained in [17] and those results are compared with the corresponding numerical solution obtained by fourthorder Runge-Kutta method All the results have been shown in Figs 1–5 It is noted that we have compared only our results with numerical solution in Fig Then the results have been compared by phase plane (Figs 6–10) Earlier Hu [17] Please cite this article in press as: M Mahtab Hossain Mondal et al., A new analytical approach for solving quadratic nonlinear oscillators, Alexandria Eng J (2016), http://dx.doi.org/10.1016/j.aej.2016.11.010 Solving quadratic nonlinear oscillators obtained only the first approximation by harmonic balance method In our solution we have used third harmonic So, we have obtained a solution by Hu’s technique containing third harmonic (please see Appendix A) to properly compare our solution to that of Hu From these figures, we observe that the present method gives better result than other existing results [17] Furthermore, we see that the present technique is very close to the numerical result Also by the present method we can solve another quadratic nonlinear oscillator _ x ỵ x ỵ ex_ ẳ 0; x0ị ¼ a; xð0Þ ¼ but by the Hu [17] method it cannot be solved Here in [17] although we can find a but b is not found by that method Conclusion In the present work, an analytical approximate technique based on the harmonic balance method has been presented to obtain approximate solution of quadratic nonlinear conservative oscillators The method is straightforward and the determination of the solution is quite easy for various quadratic nonlinear conservative oscillators On the other hand, the existing method is used for certain cases Substituting Eq (34) into Eq (33) and expanding in a Fourier series and equating the coefficient of cos u and cos 3u respectively, we obtained the following equations as 2b2 e bp 16 bpu 32 2 1 ỵ b eu b eu bpx2 ỵ bpux2 ẳ 15 35 4 ð35Þ 2b2 e 16 bpu 352 2 b eu ỵ ỵ b eu bpux2 ¼ 0: 15 21 315 Now, by eliminating x2 from Eqs (35) and (36), we obtained the following equation as 47 608 2 2 b ep b3 epu þ b2 p2 u þ b epu b p u 30 35 315 16 3 b epu ẳ 0: uẳ 7ae 394ae ỵ 35pị and ỵ 2ae In [17], Hu presented only first approximate solution of the form but here we use (for convergent) pp ỵ 2ae 3 4ae 4a2 e2 b¼ : 4e We can find the second approximate solution as x ẳ a1 uịCosẵu ỵ uCosẵ3uị: 28ị Substituting Eq (28) into Eq (27) and expanding in a Fourier series and equating the coefficient of cos u and cos 3u respectively, we obtained the following equations as 2a2 e ap 16 apu 32 2 1 ỵ a eu ỵ a eu apx2 ỵ apux2 ẳ 0: 15 35 4 ð29Þ 2a2 e 16 apu 352 2 ỵ a eu ỵ a eu apux2 ¼ 0: 15 21 315 ð30Þ Now, by eliminating x2 from Eqs (29) and (30), we obtained the following equation as 47 608 a ep ỵ a3 epu ỵ a2 p2 u a epu 30 35 315 16 a2 p2 u2 ỵ a3 epu3 ẳ 0: ð31Þ Neglecting the higher order terms of u more than one of Eq (31) and we obtain as uẳ 7ae : 394ae ỵ 35pị 32ị If we replace e by e in Eq (27) we obtain the following form _ x ỵ x ex2 ẳ 0; x0ị ẳ b; x0ị ẳ0 33ị we can find second approximation solution in the form x ẳ b1 uịCosẵu ỵ uCosẵ3uị: 34ị 38ị pp 3 4ae 4a2 e2 4e bẳ 27ị 37ị Neglecting the higher order terms of u more than one of Eq (37) and we obtain as Appendix A _ x ỵ x ỵ ex2 ẳ 0; x0ị ẳ a; x0ị ẳ 0: 36ị By solving Eqs (29) and (32), we can determine the approximate frequency x ¼ x1 Also by solving Eqs (35) and (38), we can determine the approximate frequency x ¼ x2 Finally we have determined the corresponding approximate period T1 and T2 as well as x and x_ of oscillation for phase plane (Figs and 10) for the value of a ¼ 0:48 and a ¼ 0:49 References [1] R Porwal, N.S Vyas, Damped quadratic and mixed-parity oscillator response using Krylov-Bogoliubov method and energy balance, J Sound Vib 309 (2008) 877–886 [2] J.H He, Homotopy perturbation technique, Comput Met Appl Mech Eng 178 (1999) 257–262 [3] R.E Mickens, Oscillations in Planar Dynamics Systems, World Sci, Singapore, 1996 [4] Z Guo, X Ma, Residue harmonic balance solution procedure to nonlinear delay differential systems, J Appl Math Comp 237 (2014) 20–30 [5] P Ju, X Xue, Global residue harmonic balance method for large-amplitude oscillations of a nonlinear system, J Appl Math Model 39 (2015) 449–454 [6] J.H He, Hamiltonian approach to nonlinear oscillators, Phys Lett A 374 (2010) 2312–2314 [7] S.J Liao, A.T Cheung, Application of homotopy analysis method in nonlinear oscillations, ASME J Appl Mech 65 (1998) 914–922 [8] J.H He, Max-min approach to nonlinear oscillators, Int J Nonlinear Sci Numer Simul (2008) 207–210 [9] M Akbarzede, J Langari, D.D Ganji, A coupled homotopyvariational method and variational formulation applied to nonlinear oscillators with and without discontinuities, ASME J Vib Acoust 133 (2011) 044501 Please cite this article in press as: M Mahtab Hossain Mondal et al., A new analytical approach for solving quadratic nonlinear oscillators, Alexandria Eng J (2016), http://dx.doi.org/10.1016/j.aej.2016.11.010 [10] Z Guo, A.Y.T Leung, The iterative homotopy harmonic balance method for conservative Helmholtz-Duffing oscillators, J Appl Math Comp 215 (2010) 3163–3169 [11] A.C.J Luo, B Yu, Complex period-1 motions in a periodically forced, quadratic nonlinear oscillator, J Vib Cont 21 (5) (2015) 896–906 [12] A.M El-Naggar, G.M Ismail, Applications of He’s amplitudefrequency formulation to the free vibration of strongly nonlinear oscillators, J Appl Math Sci (2012) 2071–2079 [13] N.N Bogoliubov, Yu.A Mitropolskii, Asymptotic Methods in the Theory of Nonlinear Oscillations, Gordan and Breach, New York, 1961 [14] A.H Nayfeh, D.T Mook, Nonlinear Oscillations, John Wiley and Sons, New York, 1979 Md Mahtab Hossain Mondal et al [15] N.N Krylov, N.N Bogoliubov, Introduction to Nonlinear Mechanics, Princeton University Press, New Jersey, 1947 [16] A.H Nayfeh, Perturbation Methods, John Wiley and Sons, New York, 1973 [17] H Hu, Solution of a quadratic nonlinear oscillator by the method of harmonic balance, J Sound Vib 293 (2006) 462–468 [18] M Shamsul Alam, M.E Haque, M.B Hossian, A new analytical technique to find periodic solutions of nonlinear systems, Int J Nonlinear Mech 42 (2007) 1035–1045 [19] M.A Hossain, M.S Rahman, M.S Alam, M.R Amin, An analytical technique for solving a class of strongly nonlinear conservative systems, J Appl Math Comp 218 (2012) 5474– 5486 Please cite this article in press as: M Mahtab Hossain Mondal et al., A new analytical approach for solving quadratic nonlinear oscillators, Alexandria Eng J (2016), http://dx.doi.org/10.1016/j.aej.2016.11.010 ... Hossain Mondal et al., A new analytical approach for solving quadratic nonlinear oscillators, Alexandria Eng J (2016), http://dx.doi.org/10.1016/j.aej.2016.11.010 Solving quadratic nonlinear oscillators. .. The above equation has three solution, but e b Therefore, Please cite this article in press as: M Mahtab Hossain Mondal et al., A new analytical approach for solving quadratic nonlinear oscillators, ... press as: M Mahtab Hossain Mondal et al., A new analytical approach for solving quadratic nonlinear oscillators, Alexandria Eng J (2016), http://dx.doi.org/10.1016/j.aej.2016.11.010 [10] Z Guo, A. Y.T