HARMONIC BALANCE SOLUTION OF COUPLED NONLINEAR NON-CONSERVATIVE DIFFERENTIAL EQUATION

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1 GNIT J. nglesh Mth. So. ISSN - HRMONIC LNCE SOLUTION OF COUPLED NONLINER NON-CONSERVTIVE DIFFERENTIL EQUTION M. Sifur Rhmn*, M. Mjeur Rhmn M. Sjeur Rhmn n M. Shmsul lm Dertment of Mthemtis Rjshhi University of Engineering n Tehnology RUET, Rjshhi, nglesh *msr_mth_@yhoo.om Reeive.. ete.. STRCT moifie hrmoni lne metho is emloye to etermine the seon roximte solutions to oule nonliner ifferentil eution ner the limit yle. The solution shows goo greement with the numeril solution. Keywors: Hrmoni lne metho, Coule vn er Pol eution, Nonliner osilltor. Introution The system of two eutions freuently rises in nonliner osilltions, nonliner ynmis n mthemtil hysis et. Rn n Holmes [] first roose the system of two oule vn er Pol eutions with liner iffusive ouling. They investigte the roerties of ertin erioi motion of two ientil vn er Pol osilltors with wek nonliner ouling. Reently, Neem et. l. [] hs foun roximte first integrl for system of two oule vn er Pol osilltors with liner iffusive ouling. There re mny nlytil rohes for roximting erioi solutions of the nonliner systems. The most wiely use methos re the erturtion methos, in whih the solution is exne in ower series of smll rmeter. The LP metho [], KM metho [-] n multi-time exnsion metho [-] re imortnt mong them. Usully, lower orer e.g., first or seon roximte solution is etermine y the erturtion methos ue to voi lgeri omlexities. To tkle similr nonliner rolems, there re more imortnt roximtion tehniues. One of them is the itertion tehniue see [-]. The hrmoni lne H metho [-] is nother tehniue for etermining erioi solutions of nonliner ifferentil eutions y using the trunte Fourier series. Sine the erivtion of higher roximtion is omlite, the first n seon roximte solutions re usully lulte. The vntge of H metho is tht the solution gives esire result though nonlinerities eome signifint. The im of this rtile is to fin seon roximtion to oule vn er Pol s eution following moifie hrmoni lne tehniue.

2 Rhmn et l.. The solution metho Let us onsier nonliner oule ifferentil eution u u ε f U, u ε u u, ε << where n x x u, U, x x α, β. erioi solution of E. is hosen in the form x os x os os os sin sin os os sin L sin L where, n re onstnts. In generl the unknown funtions, j, j, j, j, j,, L re etermine together with n the initil hse, n. Now sustituting E. into E. n exning the funtion f U, u in Fourier series, we otin os os sin L ε[ F,,,, L os F,,,, Los L G,, f g,,,,,,,,,, ε os ε ε sin ε ε os ε ε sin ε n Los Lsin G Lsin g, f,,,,,,, os L sin L os L,,, sin L, Lsin L Lsin L Lsin L os os sin L ε[ F,,,, L os F,,,, Los L ]

3 Hrmoni lne Solution G,, f g,,,,,,,,, ε sin ε, ε os ε ε sin ε Los Lsin G Lsin g ε os ε, f,,,,, os L,,, sin L,,, Lsin L Lsin L os L sin L Lsin L y omring the oeffiients of eul hrmoni from E. n E., we otin εf ε, ε F ε ε G ε, ε G ε n ε f ε, ε f ε ε g ε, ε g ε Utilizing the first eution of E., we eliminte tkes the following form G ε ε F ] from ll the rest. Thus E. ε ε F εf ε }/ { { ε ε F εg ε }/ gin utilizing the first eution of E., we eliminte E. tkes the following form g ε f ε { ε f ε ε f ε }/ { from ll the rest. Thus ε f ε ε g ε }/ We use new rmeter ε, << n ε, << with ε Ο n solve the thir-, fourth-et. eutions of E. n E. in owers of n resetively s

4 Rhmn et l. j j n j j j, j, j, j, j, j, j, j, j, j, j, j, LL LL LL LL j,,l j,,l Now sustituting the vlues of,, L,, L n,, L,, Lfrom E. n E. resetively into the first eution of E. n E. resetively, we etermine n, n then n. To etermine stey stte solution we strt from x n x. Thus we otin sin sin os L sin sin os L where n. Finlly, sustituting the vlues of,,, L n,,, Linto seon,, eution of E., E. n E., we solve them for,, n. Then sustituting the vlues of,, n into the eution, x x os os os os sin sin L L We otin the vlue of n, whih reresents the initil vlue of x n x for the stey-stte solution.. Exmle Consier oule vn er Pol eution u u ε U u ε u u, ε << x x where u, U x x n α, β Let us onsier erioi solution of the form

5 Hrmoni lne Solution sin os sin os os sin os sin os os x x Sustituting E. into E., we get sin os os [ sin os ε sin sin sin os os ] os HOH HOH ] os os os sin sin sin [ ε n [ sin os sin os os ε sin sin sin os

6 Rhmn et l. os ] os HOH HOH ] os os os sin sin sin [ ε where HOH stns for the higher orer hrmonis. Comring the oeffiients of eul hrmonis, we otin ε ε ε ε ε ε ε ε ε ε n ε ε ε ε ε ε

7 Hrmoni lne Solution ε ε ε ε ε ε ε ε ε With the hel of the first eution of E. n E., we eliminte n resetively from the seon, thir, fifth n sixth eution of E. n E., we otin ε ε ε ε ε ε ε ε

8 Rhmn et l. ε ε n ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε To etermine stey stte solution we strt from x n x. Thus we otin

9 Hrmoni lne Solution sin sin sin sin os os sin sin os os Herein fourteen unknown untities,,,,,,,,,,,, n will e lulte from fourteen nonliner eutions esrie y E., E., E. n E.. Certinly E. n E. reresents set of nonliner ε ε lgeri eutions with smll rmeter n. ε ε ε oring to [], we shll e le to fin n roximte solution of E. n E. in the form n,,,,,,,,,,,,,,,,,,,,,,,, L L L L L L L L Sustituting E. n E. into E. n E. resetively n euting the oeffiients of n on oth sies, we will hve system of liner eutions of, L,,, L,,, L,,, L,,,,,,,,,,,,,,,,,,,,,,, L, L,,,L. Solving these eutions, we otin,, L L L L

10 Rhmn et l. n L L L L Finlly, sustituting the vlues of,,, n,,, into first eution of E., E. n E., we solve them for,, n. Then sustituting the vlues of,, n into the eution x os x os os os sin sin os os sin sin We otin the vlue of x n x, whih reresents the initil vlue of x n x for the stey-stte solution.. Results n Disussion In orer to test the ury of n roximte solution, some uthors [,, ] omre nlytil solutions to those otine y numeril tehniues. We hve omre suh n roximte solution of E. to numeril solution for ε. n ε.. First of ll we lot in Fig. n Fig., the seon roximte solution for ε., with initil onitions x., x, x., x, where unknown L

11 Hrmoni lne Solution oeffiients,,,,,,, ; mlitues, n initil hses, re lulte y the first eutions of E. n E. together with Es., - n then sustituting these vlues of,,,,,,,,,,, in E., we otin x n x. Then orresoning numeril solution hs een omute y Runge-Kutt fourth-orer metho. From the figure it is ler tht the nlytil solution shows goo oiniene with the numeril solution. In Fig., we hve shown the orresoning hse ifferene etween two osilltors. In Fig. n Fig., the seon roximte solution of E. for ε. with the initil onitions x., x, x., x,,,,,,,, where unknown oeffiients re ; mlitues, n initil hses, re lulte y the first eutions of E. n E. together with eutions, - n then sustituting these vlues of,,,,,,,,,,, in E., we otin x n x. Then orresoning numeril solution hs een omute y Runge-Kutt fourth-orer metho. From the figure it is ler tht the nlytil solution shows goo oiniene with the numeril solution. In Fig., we hve shown the orresoning hse ifferene etween two osilltors.... x Fig. t... x Fig. t Fig. Fig. Fig. shows the hrmoni lne solution of x of E. whih is enote y.. n the orresoning numeril solution is enote y. Here ε.,.,.,.,.,.,.,.,.,.. n.,.,.,.,.,.,. In Fig., we oserve the hrmoni lne solution of x of E. whih is enote y. n the orresoning numeril solution is enote y. Here ε.,.,.,.,.,.,.,

12 Rhmn et l..,.,.,. n.,.,.,.,.,.,.... x Fig. t Fig. Fig. shows the freueny ifferene etween x n. n.. x, when ε.,.,... x Fig. t... x t Fig. Fig. Fig. In Fig. we oserve the hrmoni lne solution of x of E. whih is enote y.. n the orresoning numeril solution is enote y. Here ε.,.,.,.,.,.,.,.,.,.,. n.,.,.,.,.,.,. In Fig. we oserve the hrmoni lne solution of x of E. whih is enote y.. n the orresoning numeril solution is enote y. Here ε.,.,.,.,.,.,.,.,.,.,. n.,.,.,.,

13 Hrmoni lne Solution.,.,.... x Fig. t Fig. Fig. shows the freueny ifferene etween x n x, when ε.,.,. n.. REFERENCES [] Rn, R. H. n Holmes, P. J., ifurtion of erioi motions in two wekly oule vn er Pol osilltors, Int. J. Nonliner Meh., Vol.. -,. [] Neem, I. n Mhome, F. M., roximte first integrls for system of two oule vn er Pol osilltors with liner iffusive ouling, Mthemtil n omuttionl lition, Vol., No.. -,. [] Mrion, J.., Clssil Dynmis of Prtiles n System Sn Diego, C: Hrourt re Jovnovih,. [] Krylov, N.N. n N.N. ogoliuov, Introution to Nonliner Mehnis, Prineton University Press, New Jersey,. [] ogoliuov, N. N. n Yu.. Mitroolskii, symtoti Methos in the Theory of Nonliner Osilltions, Gorn n reh, New York,. [] Nyfeh,.H., Perturtion Methos, J. Wiley, New York,. [] Nyfeh,. H., Introution to Perturtion Tehniue, John Wylie,. [] Lu He-Xing n Li Wei-Dong n itertion metho for single-egree-of-freeom nonliner ynmil eution vne in Virtion Engineering, Vol.,. [] H. Hu n J. H. Tng, lssil itertion roeure vli for ertin strongly nonliner osilltors, J. Soun Vi., Vol.,. -,. [] Mikens, R.E., Osilltion in Plnr Dynmi Systems, Worl Sientifi, Singore,. [] West, J. C., nlytil Tehniues for Nonliner Control Systems, English Univ. Press, Lonon,. [] Mikens, R. E., Comments on the metho of hrmoni lne, J. Soun Vi., Vol.,. -,.

14 Rhmn et l. [] Mikens, R. E., generliztion of the metho of hrmoni lne, J. Soun Vi., Vol.,. -,. [] Lim, C. W. n. S. Wu, new nlytil roh to the Duffing-hrmoni osilltor, Physis Letters, Vol.,. -,. [] Lim, C. W. n. S. Wu, urte roximte nlytil solutions to nonliner osillting systems with non-rtionl restoring fore, vne in Virtion Engineering, Vol.,. -,. [] Wu,.S., W.P. Sun n C.W. Lim, n nlytil roximte tehniue for lss of strongly nonliner osilltors Int. J. Nonliner Meh., Vol.,. -,. [] He, J. H., Moifie Linste-Poinre methos for some strongly nonliner osilltions-i: exnsion of onstnt, Int. J. Nonliner Meh., Vol.,. -,. [] Lim, C. W. n S. K. Li, urte higher-orer nlytil roximte solutions to nononservtive nonliner osilltors n lition to vn er Pol me osilltors, Int. Journl of Mehnil Sienes, Vol.,. -,. [] Ymgoue, S.. n T. C. Kofne, On the nlytil roximtion of me osilltions of utonomous single egree of freeom osilltors, Int. J. Nonliner Meh., Vol.. -,. [] M. Shmsul lm, M. E. Hue n M.. Hossin new nlytil tehniue to fin erioi solutions of nonliner systems, Int. J. Nonliner Meh., Vol.. -,.