Journal of pplied and Computational Mechanics, Vol, No 1, (016), 5-1 DOI: 10055/jacm016169 Periodic Solutions of the Duffing Harmonic Oscillator by He's Energy Balance Method M El-Naggar 1, G M Ismail 1 Department of Mathematics, Faculty of Science, Benha University Benha, 1518, Egypt, Department of Mathematics, Faculty of Science, Sohag University Sohag, 85, Egypt, gamalm010@yahoocom Received June 016; revised ugust 5 016; accepted for publication ugust 8 016 Corresponding author: G M Ismail, gamalm010@yahoocom bstract Duffing harmonic oscillator is a common model for nonlinear phenomena in science and engineering This paper presents He s Energy Balance Method () for solving nonlinear differential equations Two strong nonlinear cases have been studied analytically The analytical results of the are compared with the solutions obtained by applying He s Frequency mplitude Formulation (FF) and numerical solutions using Runge-Kutta method The results show that this method is potentially presented to solve high nonlinear oscillator equations Keywords: Energy balance method, Frequency amplitude formulation, Duffing harmonic oscillator, Periodic solutions 1 Introduction The study of given nonlinear problems is of crucial importance not only in all areas of physics but also in engineering and other disciplines, since most phenomena in our world are essentially nonlinear and are described by nonlinear equations It is very difficult to solve nonlinear problems, and in general it is often more difficult to extract an analytic approximation from a numerical one for a given nonlinear problem There are many analytical approaches to solve nonlinear differential equations One of the widely used techniques is perturbation [1-], whereby the solution is expanded in powers of a small parameter However, for the nonlinear conservative systems, the generalizations of some of the standard perturbation techniques overcome this limitation Several approaches have been proposed for dealing with the Duffing harmonic oscillator; for example, the harmonic balance method [5-9], the energy balance method [10-1], the Hamiltonian approach [1], the homotopy perturbation method [1-16], the parameter expansion method [17, 18], the frequency amplitude formulation [19, 0], the variational iteration method [1-] and other new methods [-0] In this paper, He's energy balance method is employed to solve the Duffing harmonic nonlinear oscillator problem and to compare the results with the numerical solution Basic Idea of He's Energy Balance Method Let us consider a general nonlinear oscillator in the following formula u f ( u ) 0, (1)
6 M El-Naggar, G M Ismail Vol, No 1, 016 in which u and t are the generalized dimensionless displacement and time variables, respectively Its variational principle can be easily obtained as t 1 ( ) ( ) J u u F u dt 0 () where F ( u ) f ( u ) du () Its Hamiltonian, therefore, can be written as or 1 ( ) ( ), () H u F u F 1 R ( t ) u F ( u ) F ( ) 0 (5) The Oscillation system contains two important physical parameters, i e, the frequency and the amplitude of oscillation Therefore, let us consider such initial conditions as: ssume that its initial approximate guess can be expressed as u (0), u (0) 0 (6) u cos( t ) Substituting Eq (7), into u term of Eq (5), yields (7) 1 R ( t ) sin t F ( cos t ) F ( ) 0 (8) Since Eq (7) is only an approximation to the exact solution, Eq (8) cannot be made zero everywhere Collocation at gives F ( ) F (9) Its period can be determined using the relation T as T F F (10) pplications In order to assess advantages and the accuracy of He's energy balance method, we should consider the following two examples 1 Example 1 s for the first example, let us consider the following nonlinear oscillator [1] which is an example of conservative nonlinear oscillatory systems having a rational form for the restoring force as u u u 0, (11) 1u with the initial conditions Journal of pplied and Computational Mechanics, Vol, No 1, (016), 5-1
Periodic Solutions of the Duffing Harmonic Oscillator by He's Energy Balance Method 7 du u (0), (0) 0 (1) dt For this problem, u 1 1 f ( u ) u and F ( u ) u ln(1 u ) 1u Its variational formulation can be easily established as t 1 u 1 J ( u ) u ln 1 u dt 0 (1) Its Hamiltonian, therefore, can be written as follows: 1 u 1 1 R ( t ) u ln 1 u ln 1 0 (1) We use the trial function (7), as used in the method of energy balance, to determine the angular frequency u cos t (15) Substituting Eq (15) into Eq (1), yields 1 cos t 1 1 R ( t ) sin t ln 1 cos t ln 1 0 (16) If we collocate at t we obtain 1 1 1 ln 1 ln 1 0 16 (17) or ln(1 ) ln(1 ), T (18) In order to compare He's energy balance's result with He's frequency amplitude, we write Fan's result [1] 1 (19) 1 Substituting Eq (18) into Eq (15), we can obtain the approximate solution as ln(1 ) ln(1 ) u a cos t (0) Table 1: Comparison of He's Energy balance solution with He's frequency amplitude formulation FF [1] 001 1 1 01 1000 1000 0 10005 1000 0 10066 1006 06 1089 108 08 10766 10750 1 1151 1195 5 618 597 10 86610 866101 100 86605 86605 Journal of pplied and Computational Mechanics, Vol, No 1, (016), 5-1
8 M El-Naggar, G M Ismail Vol, No 1, 016 FF FF 1 10 FF FF 100 1000 Fig 1 Comparison of analytical solutions with the numerical solution The above results are in good agreement with the results obtained by He's frequency amplitude formulation [1] and the numerical integration results obtained by using the Runge-Kutta method as illustrated in Table 1 and Fig 1 Example This is an example of a conservative nonlinear oscillator system having an irrational elastic item [1]: u u u 0, 1u (1) with the initial conditions where du u (0), (0) 0, () dt u f ( u ) u 1 u Its variational formulation is t 1 u 1 J ( u ) u ln 1 u dt 0 Journal of pplied and Computational Mechanics, Vol, No 1, (016), 5-1 () By a similar manipulation, as illustrated in previous example, we obtain 1 cos t 1 1 R ( t ) sin t ln 1 cos t ln 1 0 ()
From Eq () and with t we have Periodic Solutions of the Duffing Harmonic Oscillator by He's Energy Balance Method ln(1 ) ln(1 ), T (5) In order to compare with He's frequency amplitude, we write Fan's result [1] as Then, we can obtain the following approximate periodic solution 1 1 (6) 1 ln(1 ) ln(1 ) u cos t (7) Table Comparison of He's Energy balance solution with He's frequency amplitude formulation FF [1] 001 1119 1119 01 11158 11158 0 1089 1088 0 17595 17581 06 17 169 08 1955 198 1 1551 1557 5 10588 105 10 100681 100656 100 100007 100007 1000 1 1 q 9 FF FF 1 10 FF FF 100 1000 Fig Comparison of analytical solutions with the numerical solution Table and Fig Show an excellent agreement between the results obtained from the energy balance method, He's frequency amplitude formulation [1] and numerical results using Runge-Kutta method Journal of pplied and Computational Mechanics, Vol, No 1, (016), 5-1
0 M El-Naggar, G M Ismail Vol, No 1, 016 Conclusion n analytical method called energy balance method has been successfully used to found approximate periods for strongly nonlinear Duffing harmonic oscillator The approximate periods for such nonlinear problems show a good agreement with the numerical solutions In comparison with the previously published methods, the determination of solutions is straightforward and simple To sum up, we can say that the energy balance method applied in this paper to determine approximate periods for a Diffing harmonic oscillator can be considered as an efficient alternative of the previously proposed methods References [1] He, J H, Variational iteration method: a kind of nonlinear analytical technique: some examples, International Journal of Non-Linear Mechanics, Vol, No, pp 699-708, 1999 [] He, J H, Variational approach for nonlinear oscillators, Chaos Solitons and Fractals, Vol, No 5, pp 10-19, 007 [] He, J H, Variational iteration method - some recent results and new interpretations, Journal of Computational and pplied Mathematics, Vol 07, No 1, pp -17, 007 [] He, J H, Wu, X H, Construction of solitary solution and compaction-like solution by variational iteration method, Chaos Solitons and Fractals, Vol 9, No 1, pp 108-11, 006 [5] Mickens, R E, Mathematical and numerical study of the Duffing-harmonic oscillator, Journal of Sound and Vibration, Vol, No, pp 56-567, 000 [6] Hu, H, Tang, J H, Solution of a Duffing-harmonic oscillator by the method of harmonic balance, Journal of Sound and Vibration, Vol 9, No, pp 67-69, 006 [7] Lim, C W, Wu, B S, new analytical approach to the Duffing-harmonic oscillator, Physics Letters, Vol 11, No -5, pp 65-7, 00 [8] Guo, Z, Leung, Y T, Yang, H X, Iterative homotopy harmonic balancing approach for conservative oscillator with strong odd-nonlinearity, pplied Mathematical Modelling, Vol 5, No, pp 1717-178, 011 [9] Leung, Y T, Guo, Z, Residue harmonic balance approach to limit cycles of non-linear jerk equations, International Journal of Non-Linear Mechanics, Vol 6, No 6, pp 898-906, 011 [10] Ozis, T, Yildirim,, Determination of the frequency-amplitude relation for a Duffing harmonic oscillator by the energy balance method, Computers and Mathematics with pplications, Vol 5, No 7-8, pp 118-1187 [11] Ganji, D D, Esmaeilpour, M, Soleimani, M, pproximate solutions to Van der Pol damped nonlinear oscillators by means of He's energy balance method, International Journal of Computer Mathematics, Vol 87, No 9, pp 01-0, 010 [1] Yazdi, M K, Khan, Y, Madani, M, skari, H, Saadatnia, Z, Yildirim,, nalytical solutions for autonomous conservative nonlinear oscillator, International Journal Nonlinear Sciences and Numerical Simulation, Vol 11, No 11, pp 979-98, 010 [1] Yildirim,, Saadatinia, Z, skari, H, Khan, Y, Yazdi, M K, Higher order approximate periodic solutions for nonlinear oscillators with the Hamiltonian approach, pplied Mathematics Letters, Vol, No 1, pp 0-051, 011 [1] Khan, Y, Wu, Q, Homotopy perturbation transform method for nonlinear equations using He's polynomials, Computers and Mathematics with pplications, Vol 61, No 8, pp 196-1967, 011 [15] Belendez,, Gimeno, E, lvarez, M L, Mendez, D I, Hernandez,, pplication of a modified rational harmonic balance method for a class of strongly nonlinear oscillators, Physics Letters, Vol 7, No 9, pp 607-605, 008 [16] Belendez,, Mendez, D I, Fernandez, E, Marini, S, Pascual, I, n explicit approximate solution to the Duffing-harmonic oscillator by a cubication method, Physics Letters, Vol 7, No, pp 805-809, 009 [17] Sedighi, H M, Shirazi, K H, Vibrations of micro-beams actuated by an electric field via Parameter Expansion Method, cta stronautica, Vol 85, pp 19-01 [18] Sedighi, H M, Shirazi, K H, Zare, J, Novel equivalent function for deadzone nonlinearity: applied to analytical solution of beam vibration using He s Parameter Expanding Method, Latin merican Journal of Solids and Structures, Vol 9, pp -51, 01 [19] He, J H, Solution of nonlinear equations by an ancient Chinese algorithm, pplied Mathematics and Computation, Vol 151, No 1, pp 9-97, 00 [0] El-Naggar, M, Ismail, G M, pplications of He s amplitude-frequency formulation to the free vibration of strongly nonlinear oscillators, pplied Mathematical Sciences, Vol 6, No, pp 071-079, 01 [1] He, J H, Variational iteration method a kind of non-linear analytical technique: some examples, International Journal of Non-Linear Mechanics, Vol, No, pp 699-708, 1999 [] He, J H, Variational approach for nonlinear oscillators, Chaos Solitons and Fractals, Vol, No 5, pp 10-19, 007 Journal of pplied and Computational Mechanics, Vol, No 1, (016), 5-1
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