Harmonic Balance Solution of Mulholland Equation

Transcription

1 Hrmoni Bne Souion of Muhon Equion M. Sifur Rhmn M. A Hosen n M. Shmsu Am Absr oowing new hrmoni bne meho pproime souions of vn er Po s equion hve been eermine ner he imi ye. The meho is eenbe o higher orer noniner iffereni sysem hving imi ye. In his pper seon pproime souion of Muhon equion hir orer iffereni equion is foun. The souion shows goo greemen wih he numeri souion. Ine Terms Hrmoni bne meho imi ye Noniner osiion Perioi souion. I. INTRODUCTION Sef-eie sysems SES hve ong hisory in he fie of mehnis [-]. A sef-eie osior is sysem whih hs some soure of energy upon whih i n rw. One of is mos prominen feures is he eisene of sbe imi ye in phse spe emerging from bne beween energy gin n issipion. The imi ye opoogy is inepenen by he inii oniions. Reeny SESs hve been propose s funmen oos for he onro n reuion of friion [-]. The possibe infuene of sef-eiion ynmis on friion fore is bse on he ie h when imi ye is esbishe hen imie hnge of eern oniions nno esroy i n sysem persiss on is friioness osiing moion. In SES he mping is funion of posiion n he mos gener piure n be esribe by ienr s iffereni equion f g. Equions of his in rise irey in vrious mehni ppiions. One of he suie equions wihin his ss is he vn er Po s equion whih posses n unique imi ye. Mny nyi pprohes hve been eveope for pproiming perioi souions of Eq.. The mos wiey use mehos re he perurbion mehos whereby he souion is epne in power series of sm Mnusrip reeive Augus ; revise Deember.. A. Auhor is wih he Deprmen of Mhemis Rshhi Universiyof Engineering n Tehnoogy RUET Rshhi Bngesh orresponing uhor o provie phone: 8877; e-mi: msr_mh_8@yhoo.om. S. B. Auhor is wih he Deprmen of Mhemis Rshhi Universiyof Engineering n Tehnoogy RUET Rshhi Bngesh orresponing uhor o provie phone: ; e- mi: _rue@yhoo.om. T. C. Auhor is wih he Deprmen of Mhemis Rshhi Universiyof Engineering n Tehnoogy RUET Rshhi Bngesh orresponing uhor o provie phone: 8878; e- mi: msm@yhoo.om. prmeer. The P meho [] KBM meho [] n mui-ime epnsion meho [7] re imporn mong hem. The hrmoni bne HB meho [8-7] is noher ehnique for eermining perioi souions of noniner iffereni equions by using he rune ourier series. Sine he erivion of higher pproimion is ompie he firs n seon pproime souions re usuy eermine. The vnge of HB meho is h he souion gives esire resu hough nonineriies beome signifin. Reeny Shmsu [8] hs presene new ehnique for obining higher pproimion of some srongy noniner iffereni sysems. His [8] ehnique is esier hn he eising HB meho n he souions over he gener inii vue probem i.e. [ α β ]. Usuy i is require o sove se of noniner gebri or gebri-rnsenen equions oring o HB meho. The numeri souion of hose equions gives eeen resus; bu mny uhors see [-7] for eis moify he HB souion o eermine unnown oeffiiens in nyi pproh. Some uhors eene he perurbion mehos o e noniner osiions esribe by hir orer iffereni equions. oieb [-] use he HBM o invesige imi yes of boh seon- n hir orer noniner probems. The im of he ries [-] ws quiive ype suies; bu he uhors i no ery eermine he pproime souions of hose probems. The im of he presen rie is o fin he pproime souions of he hir orer noniner probems espeiy Muhon equion bse on he new HBM presene [8-]. The souion shows goo greemen wih numeri souion hough he noniner erm beomes signifin. II. THE METHOD e us onsier noniner iffereni equion f where is onsn. In gener Eq. hs mpe souion; bu in some of he ses i hs perioi souion e.g. ner imi ye of Muhon equion. A perioi souion of Eq. is hosen in he form os os sin where n re onsns. In gener he unnown funions n re eermine ogeher wih n he inii phse.

2 Aoring o he KBM meho n pproime souion is hosen in powers of sm prmeer nmey os u u where boh n re ime epenen funions sisfying wo firs orer iffereni equions A ω B n he unnown funion B B A A re eermine sube o he oniion h u u eue he firs hrmonis. The onsn ω is he unperurbe frequeny of he osiion for Muhon equion ω. In gener KBM meho is use o isuss rnsien. However he meho is use o invesige perioi souion in whih vnishes n beomes onsn see [8]. Cery he pproime souion Eq. is hosen in form of he KBM meho; bu he eerminion of he phse n unnown funions re ifferen from he KBM meho. Now subsiuing Eq. ino Eq. n epning he funion f in ourier series we obin [ ] sin sin os os sin 7 sin os 7 os By ompring he oeffiiens of equ hrmoni we obin 7 7 When ω we use new prmeer << wih Ο n sove he equions of Eq. in powers of s Differeniing Eq. wie wih respe o n subsiuing we obin he inii oniions equions. sin os os os sin sin sin os os 7 Thus from Eqs. -7 we mesure he vues of n. Empe Consier he Muhon equion. 8 e us onsier rune form of Eq. s. sin os os rom Eq. 8 n Eq. we esiy obin sin os 7 os [ sin os os sin HOH ] sin where HOH sns for he higher orer hrmonis. Compring he oeffiiens of equ hrmonis we obin

3 Herein four unnown quniies wi be ue from four noniner equions of Eq.. Aoring o he KBM meho wo noniner gebrirnsenen equions re usuy sove o ue n for he sme inii oniions whever he orer of he pproime souion is. Bu we hve o sove four equions for he hrmoni bne souion oring o he propose meho. I is no iffiu o sove numeriy he sysem esribe by Eq.. Bu we sove he si noniner gebri-rnsenen sysem wih ess effor oring o he prinipe of [8]. Aoring o [8] we sh be be o fin n pproime souion of equion in he form of Eq. s Ο Ο..7.8 Ο Ο. rom Eq. we obin n hen. iny subsiuing he vues of n ino Eq. 7 We obin he vues of n whih represens he inii vues of n for he sey-se souion. Resus n Disussion: In orer o es he ury of n pproime souion some uhors [8] ompre nyi souions o hose obine by he numeri ehniques. We hve ompre suh n pproime souion of he Muhon equion Eq. 8 o he numeri souion for ifferen vues of. irs of we po in ig. he seon pproime souion of Eq. 8 for. wih inii oniions [ ] in whih he unnown oeffiiens re ue by he Eq. n subsiuing hese vues in Eq. 7 we obin n. Then orresponing numeri souion hs been ompue by Runge-Ku fourh-orer meho. In ig. b he seon pproime perurbion souion see Appeni A n he orresponing numeri souion hve been poe for he sme vues of wih he inii oniions [.8. 7 ]. Compring he figures i is er h he hrmoni bse souion of Eq. 8 shows beer oiniene wih he numeri souion hn he perurbion souion originy presene by Muhon []. In ig. we hve poe he seon pproime souion n he numeri souion when.. In his se we hve ue he inii oniions [ ]. The figure inies h he hrmoni bse souion gin shows goo oiniene wih he numeri souion. In ig. b we hve so ompre he perurbion souion o he numeri souion when.. in his se wih inii oniions re [ ]. In his figure he perurbion souion hs grey evie from he numeri souion. Thus inresing wih he vues of he perurbion souion oses i suibiiy whie he hrmoni bse souion shows goo greemen wih he numeri souion. Appeni A A seon pproime souion perurbion of Eq. 8 is [] α os ω α α os ω α 8 sin ω Ο α 8 sin ω α os ω A. where α.8 α. 8 n ω.8. Ο Anowegemen The uhors re grefu o Mr. Shw Hossin Assisn Professor Deprmen of Engish RU for his hep in nguge.

4 Referenes [] Nyfeh AH Moo DT. Noniner osiions. John Wiey n Sons; 7. [] Den Hrog JP. Mehni vibrions. h e. New Yor: Dover pub.; 8. [] Bogrz R Ryze B. Dry friion sef-eie vibrions nysis n eperimen. Eng Trns 7; : 7-. [] D Auno M. A simpe moe for ow friion sysems. In: Bhushn B eior. No ASI Series Vo. ;. p. -. [] Mrion J. B. Cssi Dynmis of Pries n Sysem Sn Diego CA: Hrour Bre Jovnovih 7. [] Kryov N.N. n N.N. Bogoiubov Inrouion o Noniner Mehnis Prineon Universiy Press New Jersey 7. [7] Bogoiubov N. N. n Yu. A. Miroposii Asympoi Mehos in he Theory of Noniner Osiions orn n Breh New Yor. [8] Nyfeh A.H. Perurbion Mehos J. Wiey New Yor 7. [] Miens R.E. Osiion in Pnr Dynmi Sysems Wor Sienifi Singpore. [] Wes J. C. Anyi Tehniques for Noniner Conro Sysems Engish Univ. Press onon. [] Miens R. E. Commens on he meho of hrmoni bne J. Soun Vib. Vo. pp [] Miens R. E. A generizion of he meho of hrmoni bne J. Soun Vib. Vo. pp [] im C. W. n B. S. Wu A new nyi pproh o he Duffinghrmoni osior Physis eers A Vo. pp. -7. [] Wu B.S. W.P. Sun n C.W. im An nyi pproime ehnique for ss of srongy noniner osiors In. J. Noniner Meh. Vo. pp [] He J. H. Moifie inse-poinre mehos for some srongy noniner osiions-i: epnsion of onsn In. J. Noniner Meh. Vo. 7 pp. -. [] im C. W. n S. K. i Aure higher-orer nyi pproime souions o non-onservive noniner osiors n ppiion o vn er Po mpe osiors In. Journ of Mehni Sienes Vo. 8 pp. 8-. [7] Ymgoue S. B. n T. C. Kofne On he nyi pproimion of mpe osiions of uonomous singe egree of freeom osiors In. J. Noniner Meh. Vo. pp. 8-. [8] M. Shmsu Am M. E. Hque n M. B. Hossin A new nyi ehnique o fin perioi souions of noniner sysems In. J. Noniner Meh. Vo. pp [] M. Sifur Rhmn M. E. Hque n S. S. Shn Hrmoni bne souion of noniner iffereni equion non-onservive Journ of Avnes in Vibrion Engineering Vo. pp. -. [] R. J. Muhon Non-iner osiions of hir-orer iffereni equion In. J. Noniner Meh. Vo. pp [] H. P. W. oieb Hrmoni bne pproh o perioi souion of noniner er equion J. Soun Vib. Vo. 7 pp [] H. P. W. oieb Hrmoni bne pproh o imi ye for noniner er equion J. Soun Vib. Vo. 7 pp ig. ig. : Hrmoni bne souion of Eq. 8 is enoe by ο n orresponing numeri souions is enoe by. Here n re ue by Eq. wih inii oniions [ ] when.. The vues of unnowns re ig. b ig. b: Perurbion souion of Eq. 8 see [] wih inii oniions [.8. 7 ] when. is enoe by - - n orresponing numeri souions is enoe by. The vues of unnowns re α.8 α. 8 ω..

5 ig. ig. : Hrmoni bne souion of Eq. 8 is enoe by ο n orresponing numeri souions is enoe by. Here n re ue by Eq. wih inii oniions ] when [ The vues of unnowns re ig. b ig. b: Perurbion souion of Eq. 8 see [] wih inii oniions [ ] when. is enoe by - - n orresponing numeri souions is enoe by. The vues of unnowns re α.8 α. 8 ω. 8.