A comparison study on the harmonic balance method and rational harmonic balance method for the Duffing-harmonic oscillator

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1 Globl Jornl of Pre nd Applied Mthemtics. ISSN - Volme, Nmber (), pp. - Reserch Indi Pblictions A comprison stdy on the hrmonic blnce method nd rtionl hrmonic blnce method for the Dffing-hrmonic oscilltor Md. All Hosen,, M. S. H. Chowdhry *, Mohmmd Yekb Ali, Ahmd Fris Ismil Deprtment of Mthemtics, Rjshhi Uniersity of Engineering nd echnology (RUE), Rjshhi-, Bngldesh. Deprtment of Science in Engineering, Fclty of Engineering, Interntionl Islmic Uniersity Mlysi, Jln Gombk, Kl Lmpr, Mlysi. Deprtment of Mnfctring nd Mteril Engineering, Fclty of Engineering, Interntionl Islmic Uniersity Mlysi, Jln Gombk, Kl Lmpr, Mlysi. Deprtment of Mechnicl Engineering, Fclty of Engineering, Interntionl Islmic Uniersity Mlysi, Jln Gombk, Kl Lmpr, Mlysi. *Corresponding thor s E-mil: szzdbd@iim.ed.my Abstrct he Dffing-hrmonic oscilltor is common model in nonliner sciences nd engineering. In the present pper, the hrmonic blnce method nd rtionl hrmonic blnce method he been introdced to derie the pproximte periods of strongly nonliner Dffing-hrmonic oscilltor. he comprison of two methods is mde to demonstrte tht the rtionl hrmonic blnce method (RHBM) gies lmost similr reslts to next higher-order pproximtion reslts of hrmonic blnce method (HBM). It is highly remrkble tht the soltion procedre in both methods re simple nd tkes less compttionl effort for determining pproximte periods nd shows good greement compred with the exct ones. Keywords:- Dffing-hrmonic oscilltor; Hrmonic blnce method; Nonliner lgebric eqtions; Power series soltion; Rtionl Hrmonic blnce method Mth Sbject Clssifiction Nmber: A, L, N (AMS Mthemtics Sbject Clssifiction )

2 Md. All Hosen et l. Introdction Considerble ttention hs been directed towrds the stdy of nonliner oscilltions which re ppering mthemticlly in the form of nonliner differentil eqtions (NDEs). Obtining exct soltions for NDEs he mny difficlties. It is ery difficlt to sole nonliner problems nd in generl it is often more difficlt to get n nlytic pproximtion thn nmericl one. A few nonliner systems cn be soled explicitly, nd nmericl methods especilly the most poplr Rnge-Ktt forth order method re freqently sed to clclted pproximte soltions. Howeer, nmericl schemes do not lwys gie ccrte reslts especilly the clss of stiff differentil eqtions, chotic differentil eqtion, which present more serios chllenge to nmericl nlysis. In this sittion, mny reserchers he been showed n intensifying interest in the field of nlyticl pproximte techniqes. Poplr method for soling NDEs ssocited with oscilltory systems is Pertrbtion Method [-], which is the most erstile tools ilble in nonliner nlysis of engineering problems nd they re constntly being deeloped nd pplied to eer more complex problems. Howeer, for the strongly nonliner regime pertrbtion method cn not yields desired reslts. king into ccont the boe exchnges, we oght to present new dncement of ll the more effectie methods for finding pproximte nlyticl soltions to complicted nonliner problems in distinctie fields of stdy especilly nonliner oscilltory system hs recently ttrcted the ttention of reserchers, where existing methods he not been fritfl p to now. As reslt, de to conqer these wek-points, in recent yers, nmber of reserchers he deoted their time nd effort to find potent pproches for inestigting of the nonliner phenomen. As the erliest effort, they deeloped lrge riety of pproximte methods commonly sed for nonliner oscilltory systems especilly for strongly nonliner oscilltors inclding He s Homotopy Pertrbtion Method [], Mx-Min Approch Method [], Prmeter Expnsion Method [], Nonliner ime rnsformtion Method [], Algebric Method [], Energy Blnce Method [-], He s Energy Blnce Method [], Rtionl Energy Blnce Method [], He s Freqency-Amplitde Formltion [], Enhnced Cbiction Method [], Reside Hrmonic Blnce Method [], Itertion Method [] nd so on. Howeer, most of these ppers only the first-order pproximtion is considered. In ddition, the forementioned methods lso do not he this bility to gin the soltion in high precision. Frthermore, the soltion procedres re tremendosly difficlt tsk nd cmbersome especilly for obtining higher order pproximtion. In this sittion, we will see tht the Hrmonic Blnce Method [-] nd Rtionl Hrmonic Blnce Method [-] considered in this stdy cn be pplied to strongly nonliner Dffinghrmonic oscilltor. he RHBM discssed by Mickens R.E. nd Semwogerere [], for instnce, hs rrely been pplied to the determintion of periodic soltions of nonliner problems. In fct, to the best of or knowledge, recently Belendez A. et l. [-] s well s Ymgoe S.B. et l. [] sed it to sole simple-term oscilltor eqtion of plsm physics in completely nlytic fshion. Generlly, set of troblesome nonliner lgebric eqtions re fond when HBM nd RHBM is imposed. Sometime nlyticl soltions of these lgebric eqtions fil especilly for

3 A comprison stdy on the hrmonic blnce method lrge mplitde. In present stdy, this limittion is remoed. Approximte soltions of the sme eqtions re fond in which the nonliner lgebric eqtions re soled by new prmeter. Using RHBM, the second-order pproximte period deried here is more ccrte nd closer to the next higher-order thn obtined by HBM. Considering the interesting property tht the proposed techniqe not only proides ccrte reslts bt lso it is more conenient nd effectie for soling more complex nonliner problems.. Appliction.. Soltion pproches of HBM to the Dffing-hrmonic oscilltor Let s consider the Dffing-hrmonic oscilltor nd initil conditions x x ( x ), x( ), x (). () Eq. () cn be rewrite nother form s x x x x () Let s consider two-term soltion, i.e., x ( cos( t) cos( t)) for the Eq. (). Sbstitting this soltion long with into Eq. (), it redces to ( ) cos( t) cos( t) ( ) cos( t) ( ) cos( t)] HOH, () where HOH stnds for higher-order hrmonics terms. Now compring the coefficients of eql hrmonic trems, the following reltions re: ( ) () From the first eqtion of Eq. (), it cn esily written s ( ) ( ) () Applying Eq. () into second eqtion of Eq. () with simplifiction, we get the following nonliner lgebric eqtion of : For the Dffing-hrmonic oscilltor, the power series soltion of presented in Eq. () is inlid. Herein is sbstitted by into Eq. () nd then eqting the coefficients of,, yields ()

4 Md. All Hosen et l,, It cn cler be seen tht the coefficients of,,, respectiely in the three eqtions of Eq. () re the sme constnt i.e.. herefore, by choosing, in the three eqtions of Eq. (), the eqtions of,, cn be written s: ( ( ( ), ),, ). he power series soltion of Eq. () in terms of is, Now sbstitting the le of, () (). where,, re clclted by, () Eq. () into Eq. (), the second-order pproximte period of Dffing-hrmonic oscilltor is... () In sme mnipltion discssed boe, the method cn be sed to determine higherorder pproximtions. In this stdy, third-order pproximte soltion is x( t) cos( t) (cos( t) cos( t) (cos( t) cos( t)). () Sbstitting Eq. () into the Eq. () nd eqting the coefficients of cos( t ), cos( t ) nd cos( t ) eql to zero the following eqtions re yields

5 A comprison stdy on the hrmonic blnce method ) ( () Another form of the first eqtion of Eq. (), ) ) ( ( () By omitting from second nd third eqtions of Eq. () with the help of Eq. () nd simplifiction, the following nonliner lgebric eqtion of nd re:., () In the cse for Dffing-hrmonic oscilltor, the power series soltions of nd presented in Eq. () re inlid. Herein is sbstitted by nd is sbstitted by into Eq. () nd then eqting the coefficients of,, yields

6 Md. All Hosen et l,, (), (), () In Eqs. ()-() the eqtions of,,, cn be written in nother form s ) (, () ) (, () ), ( () ), ( () where is defined in Eq. () nd. he lgebric reltion between nd is () Now soling Eq. () nd Eq. () nd then Eq. () nd Eq. () simltneosly in

7 A comprison stdy on the hrmonic blnce method terms of : () Sbstitting the les of nd where,, nd,, re clclted by Eq. () into Eq. (), the third-order pproximte period of the Dffing-hrmonic oscilltor is:.. ().. Soltion pproches of RHBM to the Dffing-hrmonic oscilltor A cos( t) Let s consider two-term soltion, i.e., x ( t) nd sbstitting into cos( t) Eq. (), it redce to ( ) cos( t) A [ ( A A ) cos( t) ( A A A A A ) cos ] HOH, where HOH represents the higher-order hrmonic terms. Now from Eq. (), we compring the coefficients of eql hrmonic terms, the following eqtions re: ( ) A ( ( A ( A A A A A A ) cos( ) A A t) ) A ) After pplying initil conditions Eqs. ()-() tke the following form () () ()

8 Md. All Hosen et l ) ( ) ( () ) ( ) ( () Eq. () cn be written nother form s ) )]( ( [ () By elimintion of from the Eq. () with the help of Eq. () nd simplifiction, the following nonliner lgebric eqtion of is: () Here mentioned tht, the Dffing-hrmonic oscilltor, the power series soltion of presented in Eq. () is inlid. Herein is sbstitted by into Eq. () nd then eqting the coefficients of,, yields, ()

9 A comprison stdy on the hrmonic blnce method,. From Eq. () it cn be written s ( () ), where. he power series soltion of Eq. () in terms of is () Sbstitting Eq. () into Eqs. ()-(), it tkes the form of first order lgebric eqtions of nd nd the les re:. nd. () Now sbstitting the le of where,, re clclted by Eq. () nd Eq. () into Eq. (), the second-order pproximte period of Dffinghrmonic oscilltor is... () () (). Reslts nd discssions We otline the exctness of estimted pproximte periods obtined by HBM nd RHBM. Compring ll the periods with existing periods preiosly obtined by seerl thors nd the exct period ex. For Dffing-hrmonic oscilltor, the exct period is ex..

10 Md. All Hosen et l which is stted in Belendez A. et l. []. he second- nd third-order pproximte periods obtined by HBM re the following:..,... nd in RHBM.. In Ref. [] pproximtely soled () sing Homotopy Pertrbtion method in two different cses. hey clclted the following pproximte period of oscilltion in orders.., b... In Ref. [] pproximtely soled () sing energy blnce method tht incorportes slient fethers of freqency-mplitde reltion. hey clclted the following pproximte period of oscilltion s:... Also in Ref. [] pproximtely soled () sing clssicl hrmonic blnce method. hey determined the following pproximte period of oscilltion.. Compring ll the pproximte periods with corresponding exct period, it cn be seen tht the pproximte periods obtined by this stdy show n excellent greement nd is better thn those obtined preiosly by Belendez A. et l. [], Ozis. et l. [] nd Lim C.W. et l. []. It hs been mentioned tht, sing RHBM, the second-order pproximte period is lmost similr to third-order pproximte period obtined by HBM. Interesting isse is here tht, most of the existing methods inclding Belendez A. et l. [], Hiling W. et l. [], Ozis. et l. [], Znig A.E. et l. [], H H. [], Mickens R.E. [], Lim C.W. et l. [] nd H H. et l. [] he considered only first-order pproximtion which leds low ccrcy. Moreoer, the soltion procedres re cmbersome especilly for obtining higher pproximtions. he dntges of this method inclde its simplicity nd compttionl efficiency, nd the bility to objectiely find better greement thn seerl existing reslts.. Conclsion Using HBM nd RHBM to obtined pproximte periods for strongly nonliner Dffing-hrmonic oscilltor. he soltion procedre of the introdced methods re ery simple, esy nd strightforwrd. In Dffing-hrmonic oscilltor, the pproximte periods obtined sing introdced methods show mch better greement with the corresponding exct period thn the periods of the other existing techniqes. High ccrcy of the pproximte periods obtined from Dffing-hrmonic oscilltor reels the erstility of the introdced methods in soling highly nonliner clss of problems. o entirety p, it cn sy tht the introdced methods is better nd efficient lterntie thn the existing.

11 A comprison stdy on the hrmonic blnce method. Acknowledgement he thors wold like to cknowledge the finncil spports receied from the Interntionl Islmic Uniersity Mlysi, Ministry of higher edction Mlysi throgh the reserch grnt FRGS---.. References: [] Nyfeh A.H., Pertrbtion Methods, J. Wiley, New York,. [] Elms N. nd Boyci H., A new pertrbtion techniqe in soltion of nonliner differentil eqtion by sing rible trnsformtion, Appl. Mth. & comp., ; : -. [] Azd A.K., Hosen M.A. nd Rhmn M.S., A pertrbtion techniqe to compte initil mplitde nd phse for the Krylo-Bogolibo-Mitropolskii method, mkng J. Mth., ; (): -. [] Belendez A., Hernndez A., Belendez., Fernndez E., Alrez M.L. nd Neipp C., Appliction of He s homotopy pertrbtion meyhod to the Dffinghrmonic oscilltor, Int. J. Nonliner Sci. Nmer. Siml., ; (): -. [] Azmi R., Gnji D.D., Bbzdeh H., Dodi A.G. nd Gnji S.S., He s Mx-Min method for the reltiistic oscilltor nd high order Dffing eqtion, Int. J. Mod. Phys. B, ; : -. [] Azimi M. nd Azimi A., Appliction of prmeter expnsion method nd ritionl itertion method to strongly nonliner oscilltor, rends in Applied Sciences Reserch, ; (): -. [] Hiling W. nd Kwok-wi C., Anlyticl soltions of generlized Dffinghrmonic oscilltor by nonliner time trnsformtion method, Phys. Lett. A, ; (-): -. [] Akbri M.R., Gnji D.D., Mjidin A. nd Ahmdi A.R., Soling nonliner differentil eqtions of Vnderpol, Ryleigh nd dffing by AGM, Front. Mech. Eng., ; (): -. [] Ozis. nd Yıldırım A., Determintion of the freqency-mplitde reltion for Dffing-hrmonic oscilltor by the energy blnce method, Compters nd Mthemtics with Applictions, ; : -. [] Askri H., Sdtni Z., Esmilzdeh E. nd Yonesin D., Mlti-freqency excittion of stiffened tringlr pltes for lrge mplitde oscilltions, J. Sond Vib., ; : -. [] Yonesin D., Askri H., Sdtni Z. nd Yzdi K.M., Anlyticl pproximte soltions for the generlized nonliner oscilltor, Applicble Anlysis, ; (): -. [] Drmz S., Demirbg S.A. nd Ky M.O., High order He s energy blnce method bsed on colloction method, Int. J. Nonliner Sci. Nmer. Siml., ; : -. [] Deichin M., Ahmdpoor M.A., Askri H. nd Yildirim A., Rtionl energy blnce method to nonliner oscilltors with cbic term, Asin-Eropen j. mth., ; ():. [] Yonesin D., Askri H., Sdtni Z. nd Yzdi K.M., Freqency nlysis of

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