The He s Amplitude-Frequency Formulation. for Solving Strongly Nonlinear Oscillators. Differential Equations

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1 Adv. Studies heor. Phys., Vol. 5,, no. 8, he He s Amplitude-Frequency Formulation for Solving Strongly Nonlinear Oscillators Differential Equations Jafar Langari Department of Mechanical Engineering Quchan Branch, Islamic Azad University, Quchan, Iran Mehdi Akbarzade Department of Mechanical Engineering Quchan Branch, Islamic Azad University, Quchan, Iran Abstract In this paper, He s amplitude-frequency formulation (HAFF) is used to calculate the periodic solutions of nonlinear strongly oscillators. We find that HAFF works very well for the whole range of initial amplitudes and only one iteration leads to high accuracy of the solutions. Comparison of the result obtained using this method with other methods reveals that this modified method is very effective and convenient. he method is applied to three nonlinear differential equations. Keywords: Nonlinear Oscillators, He s Amplitude-Frequency Formulation, Periodic Solution - Introduction he study of nonlinear problems is of crucial importance in all areas of Physics and Engineering, [-9]. Corresponding author: mehdiakbarzade@yahoo.com

2 3 J. Langari and M. Akbarzade his paper considers the following general nonlinear oscillators in the form: u + f ( u(), t u (), t u () t ) = () Here f is a known mathematical function. u and t are generalized dimensionless displacement and time variables, respectively. If there is no small parameter in the equation, the traditional perturbation methods cannot be applied directly. Recently, considerable attention has been directed towards the analytical solutions for nonlinear equations without possible small parameters. Oscillation systems contain two important physical parameters, i.e. the frequency ω and the amplitude of oscillation, A. So let us consider such initial conditions: According to He s amplitude-frequency formulation [, 7], we choose two trial functions u = A cost and u = Acosωt. Substituting u and u into Eq. (), we obtain, respectively, the following residuals: R = u + f ( u(), t u (), t u () t ) () And R = u + f ( u(), t u (), t u () t ) (3) If, by chance, u or u, is chosen to be the exact solution, then the residual, Eq. () or Eq. (3), is vanishing completely. In order to use He s amplitude-frequency formulation, we set: () R = R cos( t) dt, = π π (5) R = R cos( ωt) dt, = ω Applying He s frequency-amplitude formulation [], we have: ω R ω R (6) R R Where:, ω (7) - Applications In order to assess the advantages and the accuracy of the He s amplitude-frequency formulation (HAFF); we will consider the following examples.

3 He s amplitude-frequency formulation 35.- Example We consider the quintic Duffing oscillator [6]: 5 u + u + εu = (8) With initial condition of: Due to the presence of fifth power nonlinearity, the quintic Duffing equation inherits strong nonlinearity and thus accuracy of approximate analytical methods becomes extremely demanding. Quintic Duffing equation can be found in the modeling of free vibration of restrained uniform beam carrying intermediate lumped mass and undergoing large amplitude of oscillations in the unimodel Duffing type temporal problem [3-5]. According to He s amplitude-frequency formulation [6], we choose two trial functions u = A cost and u = Acosωt where ω is assumed to be the frequency of the nonlinear oscillator Eq (8). Substituting u and u into Eq. (8), we obtain, respectively, the following residuals: 5 5 R = ε A cos ( t) (9) And 5 5 R = Acos( ωt) ω + Acos( ωt) + εa cos ( ωt), () In order to use He s amplitude-frequency formulation, we set: 5 () 5 R = R cos( t) dt ε A, π = = 6 A( 5εA π ω π + π) () π R = R cos( ωt) dt, = = 8 π ω Applying He s frequency-amplitude formulation [], we have: ω R ω R (3) R R Where:, ω = ω () We, therefore, obtain: 5 (5) + ε A 8 he first order approximate solution is obtained, which reads: 5 + ε A 8 (6)

4 36 J. Langari and M. Akbarzade In order to compare homotopy perturbation method result we write homotopy perturbation method solution, by J. H. He s result [6]: 5 (7) + ε A 8.- Example We consider the cubic-quintic nonlinear oscillator. A cubic-quintic Duffing oscillator of a conservative autonomous system can be described by the following second-order differential equation with cubic-quintic nonlinearities []: 3 5 u + u + εu + εu = (8) With initial condition of: Cubic-quintic Duffing equation can be found in the nonlinear dynamics of a slender elastica [8], the generalized Pochhammer-Chree (PC) equations and the compound Korteweg-de Vries (KdV) equation. We choose two trial functions u = A cost and u = A cosωt where ω is assumed to be the frequency of the nonlinear oscillator Eq. (8). Substituting u and u into Eq. (8), we obtain, respectively, the following residuals: R = εa cos ( t) + εa cos ( t) (9) R = Acos( ωt) ω + Acos( ωt) + εa cos ( ωt) + εa cos ( ωt), () In order to use He s amplitude-frequency formulation, we set: () R = R cos( t) dt εa π εa π, π = + = π 6 3 A( ωπ+ π + 8εA π+ 5εA π) () π R = R cos( ωt) dt, = = 8 π ω And by same manipulation as Example, the frequencyamplitude formulation reads: 3 5 (3) + εa + εa 8 In order to compare with homotopy perturbation method we write homotopy perturbation method solution, by J. H. He s result [6]: 3 5 () + εa + εa 8

5 He s amplitude-frequency formulation Example 3 Considering the following nonlinear oscillator governed by [6]: u + u( + u ) = (5) With initial condition of: Similarly, we choose two trial functions u = Acost and u = Acosωt. Substituting u and u into Eq. (5), we obtain, respectively, the following residuals: R = Acos t + ( + A sin t) Acost (6) R = A cos( ωt) ω + ( + A ω sin ωt) Acosωt, (7) Similarly, the frequency-amplitude formulation reads: (8) A In order to compare with artificial parameter-decomposition method we write artificial parameter-decomposition method, by J.I. Ramos [7]: (9) A 3- Conclusions he He s amplitude-frequency formulation (HAFF) suggested in this paper is an efficient tool for calculating periodic solutions to nonlinear oscillatory systems. All of the examples show that the results of the present method are in excellent agreement with those obtained by other methods. his method introduces a capable tool to solve this kind of non-linear problems. he basic idea described in this paper is expected to be further employed to solve other similar strongly nonlinear oscillators. References [] A. Fereidoon, Y. Rostamiyan, M. Akbarzade, Davood Domiri Ganji, Application of He s homotopy perturbation method to nonlinear shock damper dynamics, Archive of Applied Mechanics 8 (6) ()6-69.

6 38 J. Langari and M. Akbarzade [] D. D. Ganji, N. Ranjbar Malidarreh, M. Akbarzade, Comparison of Energy Balance Period for Arising Nonlinear Oscillator Equations (He s energy balance period for nonlinear oscillators with and without discontinuities), Acta Applicandae Mathematicae: An International Survey Journal on Applying Mathematics and Mathematical Applications 8 (9) [3] Hui-Li Zhang, Application of He s amplitude-frequency formulation to a nonlinear oscillator with discontinuity, Computers and Mathematics with Applications 58 (9) [] H. Hu, Solution of a mixed parity nonlinear oscillator: Harmonic balance, Journal of Sound and Vibration, 99 (7) [5] H. Babazadeh, D. D. Ganji, and M. Akbarzade, HE S ENERGY BALANCE MEHOD O EVALUE HE EFFEC OF AMPLIUDE ON HE NAURAL FREQUENCY IN NONLINEAR VIBRAION SYSEMS, Progress In Electromagnetics Research M, (8), Vol., 3 5. [6] Ji-Huan, He, Non Perturbative Methods for Strongly Nonlinear Problems, first edition, Donghua University Publication, (6). [7] J.I. Ramos, Solution An artificial parameter-decomposition method for nonlinear oscillators: Applications to oscillators with odd nonlinearities, Journal of Sound and Vibration, 37 (7) [8] M.N. Hamdan, N.H. Shabaneh, On the large amplitude free vibrations of a restrained uniform beam carrying an intermediate lumped mass, Journal of Sound and Vibration. 99 (997) [9] M. Akbarzade, D. D. Ganji, and H. Pashaei, ANALYSIS OF NONLINEAR OSCILLAORS WIH u n FORCE BY HE S ENERGY BALANCE MEHOD, Progress In Electromagnetics Research C, (8): Vol. 3, [] M. Akbarzade, J. Langari, D. D. Ganji, A Coupled Homotopy Variational Method and Variational Formulation Applied to Nonlinear Oscillators With and Without Discontinuities, Journal of Vibration and Acoustics AUGUS, Vol. 33 / 5-. Received: February,

Application of He s Amplitude - Frequency. Formulation for Periodic Solution. of Nonlinear Oscillators

Adv. heor. Appl. Mech., Vol.,, no. 7, - 8 Application of He s Amplitude - Frequency Formulation for Periodic Solution of Nonlinear Oscillators Jafar Langari* Islamic Azad University, Quchan Branch, Sama