Analytical Solution of Nonlinear Cubic-Quintic. Duffing Oscillator Using Global Error. Minimization Method

Transcription

1 Adv. Studies Theor. Phys., Vol. 6,, o., Aalytical Solutio of Noliear Cubic-Quitic Duffig Oscillator Usig Global Error Miimizatio Method A. Kargar ad M. Akbarzade Departmet of Mechaical Egieerig Qucha Brach, Islamic Azad Uiversity, Qucha, Ira Abstract A modified variatioal approach called Global Error Miimizatio (GEM) method is developed for obtaiig a approximate closed-form aalytical solutio for oliear oscillator differetial equatios. The proposed method coverts the oliear differetial equatio to a equivalet miimizatio problem. A trial solutio is selected with ukow parameters. Next, the GEM method is used to solve the miimizatio problem ad to obtai the ukow parameters. Keywords: Noliear Oscillators; Aalytical approximate solutios; Global Error Miimizatio method. Correspodig author: ali_kargar_57@yahoo.com

2 468 A. Kargar ad M. Akbarzade - Itroductio There are several methods used to fid approximate solutios to oliear problems such as homotopy perturbatio method [], eergy balace method [3] ad Amplitude-Frequecy Formulatio [4] were used to hadle strogly oliear systems. Our cocer i this paper is the derivatio of a approximate aalytical solutio for a oliear oscillatory differetial equatio. To do this we modify the variatioal approach proposed by He [5] ad develop a method called GEM (Global Error Miimizatio) []. - Basic idea I this sectio the Global Error Miimizatio (GEM) method is itroduced ad developed. The method is systematically described ad will result i a approximate aalytic solutio for the strogly oliear oscillator ODEs. Cosider a geeral secod-order oliear oscillator differetial equatio: u + F( u, u, u) =, u() = A, u () = () With iitial coditios: u() = A, u () = B () Defiitio: Cosider the oliear system (); we defie the followig fuctioal for the oscillator equatio, called the global error fuctioal: Eu (, u, u) = T u + Fu (, u, u) dt (3) ( ) π T =, ω is the primary atural frequecy where E is a cotiuous fuctioal. ω The solutio of Eq. () ca be expressed i the form of Fourier series []: (4) ut () = a + a cos( ωt) + bsi( ωt) = ( ) Where a, a, b are costats. These ukow costats could ot be determied for the case of ifiite Fourier series. However, we ca approximate Eq. (4) by a fiite series: m (5) ut () = a + a cos( ωt) + bsi( ωt) = ( ) I this paper, a atural ad efficiet method will be developed for determiig these ukows. The oliear problem () is first coverted to the miimizatio problem

3 Aalytical solutio 469 (3). We directly substitute the trial solutio (5) i the miimizatio problem. The solutios of the miimizatio problem are the ukow costats of Eq. (5). 3- Applicatio A cubic-quitic Duffig oscillator of a coservative autoomous system ca be described by the followig secod-order differetial equatio with cubic-quitic oliearities [8]: u + u+ ε u + u = u = A u = (6) 3 5 ε, (), () We begi the procedure with the simplest trial solutio: u ( t) = Acos( ω t), u () = A, u () = (7) Next, we covert Eq. (6) to the miimizatio problem (3): T 3 5 ( ) π Eu (, u, u) = u + u+ εu + εu dt, T= ω (8) By replacig u () t = A cos( ωt ) i Eq. (8) ad performig the itegratio we get: A 8 4 E = ( 945A ε π + 88A επ + A ε π 9 ω + 4A επ+ 9π 384ωπ + 9ωπ 4 4 ) (9) E The solutio could be foud through the coditio = : ω ω = A ε + 36A ε A ε + 8A ε ε + 8A ε + 64A ε + 536A ε () If we add oe more term to the trail fuctio (Eq. (7)) we ca easily obtai the secod order approximatio. I order to compare with Modified Lidsted-Poicare solutio: Double series Expasio, we write J. H. He s result [6]:

4 47 A. Kargar ad M. Akbarzade ω = + 3 ε A + 5 ε A () Table Compariso of the GEM method with Modified Lidsted-Poicare solutio [6] ( ε = ε = ). A GEM method Modified Lidsted- Poicare I caseε =, ε = ε, Eq. (6) turs to the well-kow Duffig equatio ad its oliear agular frequecy ca be obtaied fromeq. (), which reads: 4 () ω = A ε+ 3 64A ε + 536A ε+ 4 The exact frequecy of the periodic motio of the Duffig equatio is give by [6]: π π + εa dx ωexact = msi x ε A Where m = For compariso, the exact frequecy obtaied by ( + ε A ) itegratig Eq. (3) ad the approximate frequecy computed by Eq. () are listed i Table. (3) Table Compariso of the GEM method with exact solutio. ε A GEM method Exact solutio

5 Aalytical solutio Coclusios The GEM method was successfully applied to oliear Cubic-Quitic Duffig Oscillator. The method is useful to obtai aalytical solutio for all oscillators ad vibratio problems, such as i the fields of civil structures, fluid mechaics, electromagetics ad waves, etc. This paper shows oe step i the attempt to develop a ew oliear aalytical techique i absece of small parameters. Refereces [] Fereidoo A., Rostamiya Y., Akbarzade M., Domiri Gaji Davood, Applicatio of He s homotopy perturbatio method to oliear shock damper dyamics, Archive of Applied Mechaics, 8 (6), () [] Farzaeh V., Akbarzadeh Tootoochi Ali, Global Error Miimizatio method for solvig strogly oliear oscillator differetial equatios, Computers ad Mathematics with Applicatios, 59 () [3] Gaji D. D., Rajbar Malidarreh N.., Akbarzade M., Compariso of Eergy Balace Period for Arisig Noliear Oscillator Equatios (He s eergy balace period for oliear oscillators with ad without discotiuities), Acta Applicadae Mathematicae, 8 (9), [4] He J. H., Some asymptotic methods for strogly oliear equatios, It. J. Mod. Phys. B, (6) [5] He J. H., Variatioal approach for oliear oscillators, Chaos Solitos ad Fractals, 34 (7) [6] Ji-Hua, He, No Perturbative Methods for Strogly Noliear Problems, dissertatio.de -Verlag im iteret GmbH, Berli, (6). Received: December,