World Acdemy of Science, Engineering nd Technology Interntionl Journl of Mthemticl, Computtionl, Sttisticl, Nturl nd Physicl Engineering Vol:7, No:, Anlyticl Bsed Trunction Principle of Higher- Order Solution for x / Force Nonliner Oscilltor Md. All Hosen Interntionl Science Index Vol:7, No:, wset.org/publiction/99967 Abstrct In this pper, modified hrmonic blnce method bsed n nlyticl technique hs been developed to determine higher-order pproximte periodic solutions of conservtive nonliner oscilltor for which the elstic force term is proportionl to x /. Usully, set of nonliner lgebric equtions is solved in this method. However, nlyticl solutions of these lgebric equtions re not lwys possible, especilly in the cse of lrge oscilltion. In this rticle, different prmeters of the sme nonliner problems re found, for which the power series produces desired results even for the lrge oscilltion. We find modified hrmonic blnce method works very well for the whole rnge of initil mplitudes, nd the excellent greement of the pproximte frequencies nd periodic solutions with the exct ones hs been demonstrted nd discussed. Besides these, suitble trunction formul is found in which the solution mesures better results thn existing solutions. The method is minly illustrted by the x / force nonliner oscilltor but it is lso useful for mny other nonliner problems. Keywords Approximte solutions, Hrmonic blnce method, Nonliner oscilltor, Perturbtion. M I. INTRODUCTION ANY complex rel world problems in nture re due to nonliner phenomen. Nonliner processes re one of the biggest chllenges nd not esy to control becuse the nonliner chrcteristic of the system bruptly chnges due to some smll chnges of vlid prmeters including time. Thus the issue becomes more complicted nd hence needs ultimte solution. Therefore, the studies of pproximte solutions of nonliner differentil equtions (N.D.Es.) ply crucil role to understnd the internl mechnism of nonliner phenomen. Advnce nonliner techniques re significnt to solve inherent nonliner problems, prticulrly those involving differentil equtions, dynmicl systems nd relted res. In recent yers, both the mthemticins nd physicists hve mde significnt improvement in finding new mthemticl tool would be relted to nonliner differentil equtions nd dynmicl systems whose understnding will rely not only on nlytic techniques but lso on numericl nd symptotic methods. They estblish mny effective nd powerful methods to hndle the N.D.Es. The study of given nonliner problems is of crucil importnce not only in ll res of physics but lso in engineering nd other disciplines, since most phenomen in Md. All Hosen is with the Deprtment of Mthemtics, Rjshhi University of Engineering nd Technology (RUET), Rjshhi-64, Bngldesh (phone: +8877-8887; fx: +88-7-749; e-mil: ll_ruet@yhoo.com). our world re essentil nonliner nd re described by nonliner equtions. It is very difficult to solve nonliner problems nd in generl it is often more difficult to get n nlytic pproximtion thn numericl one for given nonliner problem. There re mny nlyticl pproches to solve nonliner differentil equtions. One of the widely used techniques is perturbtion []-[4], whereby the solution is expnded in powers of smll prmeter. However, for the nonliner conservtive systems, generliztions of some of the stndrd perturbtion techniques overcome this limittion. In prticulr, generliztion of Lindsted-Poincre method nd He s homotopy perturbtion method yield desired results for strongly nonliner oscilltors []-[]. The hrmonic blnce method (HBM) []-[] is nother technique for solving strongly nonliner systems. Usully, set of difficult nonliner lgebric equtions ppers when HBM is formulted. In rticle [], such nonliner lgebric equtions re solved in powers of smll prmeter. Sometimes, higher pproximtions lso fil to mesure the desired results when >>. In this rticle this limittion is removed. Approximte solutions of the sme equtions re found in which the nonliner lgebric equtions re solved by new prmeter. The higher order pproximtions (minly third pproximtion) hve been obtined for mentioned nonliner oscilltor. However, suitble trunction of these lgebric equtions tkes the solution very close to the previous one but it sves lot of clcultion. This is the min dvntge of the method presented in this rticle. II. THE METHOD Let us consider nonliner differentil eqution x + x ε f x, x ), [ x(), x () ] () ( where f ( x, x ) is nonliner function such tht f ( x, x ) f ( x, x ), nd ε is constnt. A periodic solution of () is tken in the form s x ( ρ cos( t ) + u cos( t ) + v cos( t ) w cos(7 t ) ) () + where, ρ nd ϕ re constnts. If ρ u v nd the initil phse ϕ, solution () redily stisfies the initil conditions [ x(), x () ]. Interntionl Scholrly nd Scientific Reserch & Innovtion 7() 9
World Acdemy of Science, Engineering nd Technology Interntionl Journl of Mthemticl, Computtionl, Sttisticl, Nturl nd Physicl Engineering Vol:7, No:, Substituting () into () nd expnding f ( x, x ) in Fourier series, it is esily tkes to n lgebric identity [ ρ ( ) cos( t) + u( 9 ) cos(t ) + ] () ε[ F (, u, )cos( t) + F (, u, )cos(t ) + ] By compring the coefficients of equl hrmonics of (), the following nonliner lgebric equtions re found where the second term of (8) i.e. x cn be expnded in Fourier series s / / n+ n x b x b cos( t) + b cos( t) + () Herein b, b, re evluted by the reltion ρ( ) ε F, v( ) ε F, u( 9 ) ε F, (4) / 4 b + x cos[(n + ) ϕ] d ϕ n () Interntionl Science Index Vol:7, No:, wset.org/publiction/99967 With help of the first eqution, ϕ is eliminted from ll the rest of (4). Thus (4) tkes the following form ρ ρ + ε F, 8 uρ ε ( ρ F 4 vρ ε ( ρf vf ), 9uF ), Substitution ρ u v, nd simplifiction, second-, third- equtions of () tke the following form u G (, ε,, u, v,, λ ), v G (, ε,, u, v,, λ ),, where G, G, exclude respectively the liner terms of u, v,. Whtever the vlues of nd, there exists prmeter λ (, ε, ) <<, such tht u, v, re expndble in following series () (6) u Uλ + U λ +, v V λ + V λ +, where U, U,, V, V, re constnts. Finlly, substituting the vlues of u, v, from (7) into the first eqution of (), is determined. This completes the determintion of ll relted functions for the proposed periodic solution s given in (). III. EXAMPLE Let us consider the following nonliner oscilltor which ws first studied in detil by Mickens [8] x + x (8) This is conservtive system nd the solution of (8) is periodic. The clssicl hrmonic blnce method [] cnnot be pplied directly. From () the first-order pproximtion solution of (8) is x cos( t) (9) (7) where ϕ t. From () nd using (9) we obtined b, b,, I b, () I b, where I 7 Gmm 6 Gmm () nd so on. Now substituting (9) nd () long with ()-() into (8) nd setting the coefficient of cos ϕ the following lgebric eqution is obtined + I Solving (4) the first-order pproximte frequency is (4) I. 768 () Therefore the first-order pproximtion solution of (8) is (9) i.e. x cos( t) where is given by (). Let us consider second-order pproximtion solution, i.e. x cos( t) + u (cos( t) cos( t)) (6) Substituting (6) long with using ()-() into (8) nd then equting the coefficients of cos( t) nd cos( t), the following equtions re / 4 ( + 44u+ u + u + u ) I ( u) (7) ( 74 + 9u+ 969u + 7u + 7 u ) I / 4 9 u + (8) 87 where I is defined s () nd b, b, re evluted s Interntionl Scholrly nd Scientific Reserch & Innovtion 7() 6
World Acdemy of Science, Engineering nd Technology Interntionl Journl of Mthemticl, Computtionl, Sttisticl, Nturl nd Physicl Engineering Vol:7, No:, Interntionl Science Index Vol:7, No:, wset.org/publiction/99967 / 4 ( + 44u+ u + u + u ) I b (9) / 4 ( 74 + 9u+ 969u + 7u + 7 u ) I b () 87 nd so on. After simplifiction, (7) tkes the form ( + 44u + u + u ( u) 4 + u ) I () By elimintion of from (8) with the help of () nd simplifiction, the following nonliner lgebric eqution of u is obtined s 4 99u 7 u 48u 8u u λ + + + + 87 87 where λ () 8 The power series solution of () in terms of λ is 4 6 99λ 7λ 99λ 87λ u λ + + + () 7 7 Now substituting the vlue of u from () into (), the second-order pproximte frequency is / 4 ( + 44u+ u + u + u ) I.694 / ( u) (4) Thus the second-order pproximtion solution of (8) is x cos( t) + u(cos( t) cos( t)) where u nd re respectively given by () nd (4). It is observed tht solution (6) mesures better result when (7)-(8) is truncted s / ( + 44u+ u / ) I ( u) () And / ( 74 + 9u+ 969 u / ) I 9 u + (6) 87 Seeing tht ()-(6), it is cler tht the higher order terms of u (more thn second) re ignored; but hlf of the second order terms re considered. Now from () we cn esily obtin + + ( 44u u / ) I (7) ( u) Substituting () into (6) nd then simplifying we get 79u 6u u λ + + 44 (8) where λ is defined s (). The power series solution of (8) is 79 6 4 46 4 u λ + λ λ λ + λ + (9) 44 968 Substituting the vlue of u from (9) into (7) the secondorder frequency in trunction form is ( + 44u+ u / ) I.694 / trun / ( u) () In similr wy, the method cn be used to determine higher order pproximtions. In this rticle, third-order pproximte solution of (8) is x cos( t) + u(cos( t) cos( t)) + v(cos( t) cos( t)) () Substituting () together with () nd () into (8) nd equting the coefficients of cos ϕ, cosϕ nd cos ϕ the relted functions re obtined from the following equtions ( + 44u+ u + u / + v+ 8uv+ 7 v ) I ( uv) () / (74 9u969u 7u + v+ 6 uv+ ) I 9 u 87 () (4 647u 9u 99u + 8768v + uv + ) I v + 4 (4) where I is defined s (). Using () we cn esily written s / ( + 44u+ u + u + v+ 8uv+ 7 v ) I () ( uv) Interntionl Scholrly nd Scientific Reserch & Innovtion 7() 6
World Acdemy of Science, Engineering nd Technology Interntionl Journl of Mthemticl, Computtionl, Sttisticl, Nturl nd Physicl Engineering Vol:7, No:, By eliminting from () nd (4) with the help of () nd simplifiction, the following nonliner lgebric equtions of u nd v re obtined 9 u (74 9u 969u 7u / + v + 6uv / + ) I (44) 87 4 99u 7 u 7u 8v uv 74 u v u λ + + + + + + + 74 (6) 4 8u 6u 79u u 9487 uv 6 u v v μ + + + + 87 87 87 87 (7) v + (4 647u 9u 99u + uv / + ) I 4 / + 8768v Here, using (4) it cn be written s (4) Interntionl Science Index Vol:7, No:, wset.org/publiction/99967 where λ is defined s () nd μ. 7 The lgebric reltion between λ nd μ is 9λ μ (8) 46 v Therefore, (7) tkes the form 9λ 8 6 79 9487 6 46 87 87 87 87 4 u u u u uv u v + + + + (9) The power series solutions of (6) nd (9) re obtined in terms of λ u λ +.7λ + 6.67776λ 4 6.674λ 7.77λ +, v.686λ +.488λ.99λ 4 9.464λ +.78876λ +. (4) (4) Now substituting the vlues of u nd v from (4)-(4) into (), the third-order pproximte frequency is / ( + 44u+ u + u + v 8uv 7 v ) I.7779 + + (4) / ( uv) Therefore third-order pproximtion periodic solution of (8) is defined s () where u, v nd re respectively given by (4)-(4). The third pproximte solution () mesures lmost similr result when ()-(4) re truncted s ( + 44u + u + u / + v + 8uv / ) I ( u v) (4) / ( + 44u+ u + u / + v+ 8 uv/ ) I (46) ( u) By eliminting from (44)-(4) with the help of (46) nd simplifiction, the following nonliner lgebric equtions of u nd v re obtined 4 99u 4 u 7u 8v 9uv u v u λ + + + + + + + (47) 4 8u 6u 89u 6 u 88 uv 99 u v v μ + + + + 87 87 87 87 (48) where λ nd μ re defined s () nd (7). In similr wy, the power series solutions of (47)-(48) in terms of λ re u λ +.7λ + 6.869λ.849. + 4 λ λ v.686λ +.488λ.86λ.9 + 8.4497 + 4 λ λ (49) () Substituting the vlues of u nd v from (49)-() into (46), nd simplifying we get the third-order pproximte frequency in truncted form is ( + 44u+ u / 8 / ).7776 / + u + v+ uv I trun / ( u) () Therefore third-order pproximtion periodic solution of (8) is defined s () where u, v nd re respectively given by (49)-(). Interntionl Scholrly nd Scientific Reserch & Innovtion 7() 6
World Acdemy of Science, Engineering nd Technology Interntionl Journl of Mthemticl, Computtionl, Sttisticl, Nturl nd Physicl Engineering Vol:7, No:, Interntionl Science Index Vol:7, No:, wset.org/publiction/99967 IV. RESULTS AND DISCUSSIONS We illustrte the ccurcy of modified hrmonic blnce method by compring the pproximte solutions previously obtined with the exct frequency ex. In prticulr, we will consider the solution of (8) using the clssicl hrmonic blnce method []. This method is procedure for the determining nlyticl pproximtions to the periodic solutions of the differentil equtions using truncted Fourier series representtion []. Like, the homotopy purterbtion method, the hrmonic blnce method cn be pplied to nonliner oscilltory problems where liner terms does not exist, the nonliner terms re not smll, nd there is no perturbtion prmeter. For this nonliner problem, the exct vlue [4] of the frequency is.74 ex ( ) () The first, second nd third-order pproximte frequencies nd their Reltives Errors ( R. E.) or percentge errors obtined in this rticle by pplying modified hrmonic blnce technique to mentioned conservtive nonliner oscilltor re the following.768 ( ).9% ().694 trun ( ).% (4).7779 trun ( ).% () where R.E. were clculted by using the following eqution i ( ) e ( ) R. E. i,, (6) ( ) e In [] pproximtely solved (8) using clssicl hrmonic blnce method. They determined the following results for the first nd second-order pproximte ngulr frequencies in orders.49 ( ).% (7).64 ( ).7% (8) Also in [] pproximtely solved (8) using nother improved hrmonic blnce method tht incorportes slient fethers of both Newton s method nd the hrmonic blnce method. They clculted the following results for the first, second nd third-order pproximte ngulr frequencies in orders.768 ( ).6% (9).69 ( ).% (6).778 ( ).% (6) In [] lso pproximtely solved (8) using modified He s homotopy perturbtion method. They chieved the following results for the first, second nd third-order pproximte ngulr frequencies re s follows.768 ( ).6% (6).686 ( ).7% (6) /.79 ( ).4% (64) Compring ll the pproximte frequencies the ccurcy of the result obtined in this pper is better thn those obtined previously by [] nd [] nd is lmost similr those obtined by []. It hs been mentioned tht the procedure of [] nd [] is lborious especilly for obtining the higher pproximtions. V. CONCLUSION Bsed on modified hrmonic blnce method, n nlyticl technique hs been presented to determine pproximte periodic solutions of conservtive nonliner oscilltor. In compred with the previously published methods, determintion of solutions is strightforwrd nd simple. The pproximte ngulr frequency presented in this rticle using third-order pproximte principle with reltive error is.%. The dvntges of this method include its nlyticl simplicity nd computtionl efficiency, nd the bility to objectively better greement in third-order pproximte solution. To sum up, we cn sy tht the method presented in this rticle for solving nonliner oscilltor cn be considered s n efficient lterntive of the previously proposed methods. ACKNOWLEDGMENT The uthor is grteful to the Rjshhi University of Engineering nd Technology for which provided vrious supports. Interntionl Scholrly nd Scientific Reserch & Innovtion 7() 6
World Acdemy of Science, Engineering nd Technology Interntionl Journl of Mthemticl, Computtionl, Sttisticl, Nturl nd Physicl Engineering Vol:7, No:, Interntionl Science Index Vol:7, No:, wset.org/publiction/99967 REFERENCES [] Mrion, J. B., Clssicl Dynmics of Prticles nd System (Sn Diego, CA: Hrcourt Brce Jovnovich), 97. [] Krylov, N.N. nd N.N. Bogoliubov, Introduction to Nonliner Mechnics, Princeton University Press, New Jersey, 947. [] Bogoliubov, N. N. nd Yu. A. Mitropolskii, Asymptotic Methods in the Theory of Nonliner Oscilltions, Gordn nd Brech, New York, 96. [4] Nyfeh, A.H., Perturbtion Methods, J. Wiley, New York, 97. [] Awrejcewicz, J., I.V. Andrinov nd L.I. Mnevitch, Asymptotic Approches in Nonliner Dynmics (New trends nd pplictions), Springer, Berlin, Heidelberg, pp. 4-9, 998. [6] Cheung, Y.K., S.H. Mook nd S.L. Lu, A modified Lindstedt- Poincre method for certin strongly nonliner oscilltors, Int. J. Nonliner Mech., Vol. 6, pp. 67-78, 99. [7] He, J.-H., Modified Lindstedt-Poincre methods for some strongly nonliner oscilltions-i: expnsion of constnt, Int. J. Nonliner Mech., Vol. 7, pp. 9-4,. [8] Ozis, T. nd A. Yildirim, Determintion of periodic solution for / u force by He s modified Lindstedt-Poincre method, J. Sound Vib., Vol., pp. 4-49, 7. [9] Belendez, A., T. Belendez, A. Hernndez, S. Gllego, M. Ortuno nd C. Neipp, Comments on investigtion of the properties of the period for the nonliner oscilltor x + ( + x ) x, J. Sound Vib., Vol., pp. 9-9, 7. [] Belendez, A., C. Pscul, S. Gllego, M. Ortuno nd C. Neipp, Appliction of modified He s homotopy perturbtion method to obtin higher-order pproximtions of n x force nonliner oscilltor, Physics Letters A, Vol. 7, pp. 4-46, 7. [] Belendez Appliction of He s homotopy perturbtion method to the Duffing-hrmonic oscilltor, Int. J. Non-liner Sci. nd Numer Simultion, Vol. 8, pp. 79-88, 7. [] Belendez, A., A. Hernndz, A. Mrquez, T Belendez nd C. Neipp, Appliction of He s homotopy perturbtion method to nonliner pendulum, Europen J. of Physics, Vol. 8, pp. 9-4, 7. [] Hu, H., Solution of qudrtic nonliner oscilltor by the method of hrmonic blnce, J. Sound Vib., Vol. 9, pp. 46-468, 6. [4] Mickens, R.E., Oscilltion in Plnr Dynmic Systems, World Scientific, Singpore, 996. [] West, J. C., Anlyticl Techniques for Nonliner Control Systems, English Univ. Press, London, 96. [6] Mickens, R. E., Comments on the method of hrmonic blnce, J. Sound Vib., Vol. 94, pp. 46-46, 984. [7] Mickens, R. E., A generliztion of the method of hrmonic blnce, J. Sound Vib., Vol., pp. -8, 986. 4 / [8] Mickens, R. E., Oscilltions in n x potentil, J. Sound Vib., Vol. 46 (), pp. 7-78,. [9] Lim, C. W. nd B. S. Wu, A new nlyticl pproch to the Duffinghrmonic oscilltor, Physics Letters A, Vol., pp. 6-7,. [] Lim, C. W., S. K. Li nd B. S. Wu, Accurte higher-order nlyticl pproximte solutions to lrge-mplitude oscillting systems with generl non-rtionl restoring force, J. Nonliner Dynmics, Vol. 4, pp. 67-8,. [] Wu, B.S., W.P. Sun nd C.W. Lim, An nlyticl pproximte technique for clss of strongly nonliner oscilltors Int. J. Nonliner Mech., Vol. 4, pp. 766-774, 6. [] Belendez, A., A. Hernndz, A. Mrquez, T Belendez nd C. Neipp, Anlyticl pproximtions for the period of nonliner pendulum, Europen J. of Physics, Vol. 7, pp. 9-, 6. [] Shmsul Alm, M., M. E. Hque nd M. B. Hossin A new nlyticl technique to find periodic solutions of nonliner systems, Int. J. Nonliner Mech., Vol. 4 pp. -4, 7. [4] H.P.W. Gottlieb, Frequencies of oscilltors with frctionl-power nonlinerities, J. Sound Vib., Vol. 6, pp. 7-66,. [] Mickens, R. E., Truly nonliner oscilltions, World Scientific Publishing Co. Pte. Ltd., Singpore,. Md. All Hosen ws born in Sirjgnj, Bngldesh, in 98. He received the bchelor degree in Mthemtics nd mster degree in Applied Mthemtics from University of Rjshhi (RU) in 4 nd, respectively. His min reserch interests re in Nonliner Differentil Equtions nd Nonliner Dynmicl Systems. Currently, he is servicing s n Assistnt Professor in the Deprtment of Mthemtics, Rjshhi University of Engineering nd Technology, Rjshi-64, Bngldesh. Prof. Hosen is life time member in Bngldesh Mthemticl Society. Interntionl Scholrly nd Scientific Reserch & Innovtion 7() 64