Dynamic model of large amplitude vibration of a uniform cantilever beam carrying an intermediate lumped mass and rotary inertia

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1 - 9 Dyamic model of large amplide vibraio of a iform cailever beam carryig a iermediae lmped mass ad roary ieria bsrac I his paper, a mahemaical model of large amplide vibraio of a iform cailever beam arisig i he srcral egieerig is proposed. Two efficie ad easy mahemaical echiqes called variaioal ieraio mehod ad He s variaioal approach are sed o solve he goverig differeial eqaio of moio. To assess he accracy of solios, we compare he resls wih he Rge-Ka h order. excelle agreeme of he approximae freqecies ad periodic solios wih he merical resls ad pblished resls has bee demosraed. The resls show ha boh mehods ca be easily exeded o oher oliear oscillaios ad i ca be prediced ha boh mehods ca be fod widely applicable i egieerig ad physics.. Nikkar *,, S. bagheri, M. Saravi Facly of Civil Egieerig, Uiversiy of Tabriz, Tabriz, Ira Deparme of Mahemaics, Islamic zad Uiversiy, Nor Brach, Nor, Ira Received i 9 Mar I revised form pr * hor ali.ikkar@yahoo.com Keywords Large amplide; Variaioal Ieraio Mehod; He s Variaioal pproach; Noliear oscillaio. INTRODUCTION The Srcral egieerig heory is based po physical laws ad empirical kowledge of he srcral performace of differe ladscapes ad maerials. May egieerig srcres ca be modelled as a sleder, flexible cailever beam carryig a lmped mass wih roary ieria a a iermediae poi alog is spa; hece hey experiece large-amplide vibraio W, ; Herisa ad Marica, ; Cveicai ad Kovacic, 7; Hamda ad Shabaeh, 997; Hamda ad Dado, 997. I geeral, sch problems are o ameable o exac reame ad approximae echiqes ms be resored o. So, may ew echiqes have appeared i he ope lierare sch as: variaioal ieraio mehod He, 7a; Barari e al.,, homoopy perrbaio mehod Torabi ad Yaghoobi, ; Torabi e al., ; Saravi e al.,, homoopy aalysis mehod Liao, ; Kha e al., ad some oher mehods Nikkar e al., ; Ghasempoor e al., ; kbarzade ad Kha, ; liia e al., ; Gaji, ; Torabi e al., ; Sheikholeslami e al.,

2 . Nikkar e al. / Dyamic model of large amplide vibraio of a iform cailever beam carryig a iermediae lmped mass ad roary ieria a; b; Baya e al., ; Rafieipor e al., ; Marica ad Herisa, ; Salehi e al., ; Hamidi e al., The variaioal ieraio mehod VIM ad variaioal approach V which were irodced by He He; 7a;7b are very powerfl mehods i solvig o-liear differeial eqaios ad sdyig oliear vibraio of beams. The Variaioal pproach which is also called He s variaioal approach, as well as he variaioal ieraio mehod, was ilized here o obai he aalyical expressio for he followig model of oliear oscillaios i he srcral egieerig problems: Wih iiial codiios: d d d β d d d d, d This sysem describes he i-modal large-amplide free vibraios of a sleder iexesible cailever beam carryig a iermediae mass wih a roary ieria. The hird ad forh erms i eqaio represe ieria-ype cbic o-lieariy arisig from he iexesibiliy assmpio. The las erm is a saic-ype cbic olieariy associaed wih he poeial eergy sored i bedig. The modal cosas ad β resl from he discreizaio procedre ad hey have specific vales for each mode as described i Hamda ad Dado, 997. DESCRIPTION OF THE VRITIONL ITERTION METHOD To clarify he basic ideas of proposed mehod cosider he followig geeral differeial eqaio, L N g Where, L is a liear operaor, ad N a oliear operaor, g a ihomogeeos or forcig erm. ccordig o he variaioal ieraio mehod, we ca cosrc a correc fcioal as follows: λ L N~ g d Where λ is a geeral Lagrage mliplier, which ca be ideified opimally via he variaioal heory, he sbscrip deoes he h approximaio, ~ is cosidered as a resriced variaio, i.e. δ ~. Lai merica Joral of Solids ad Srcres - 9

3 . Nikkar e al. / Dyamic model of large amplide vibraio of a iform cailever beam carryig a iermediae lmped mass ad roary ieria For liear problems, is exac solio ca be obaied by oly oe ieraio sep de o he fac ha he Lagrage mliplier ca be exacly ideified. I his mehod, he problems are iiially approximaed wih possible kows ad i ca be applied i o-liear problems wiho liearizaio or small parameers. The approximae solios obaied by he proposed mehod rapidly coverge o he exac solio. IMPLEMENTTION OF VRITIONL ITERTION METHOD Now, we ca solve he followig eqaio: β 5 wih iiial codiios:, 6 ssme ha he circlar freqecy of he sysem 5 is, we have he followig liearized eqaio: So we ca rewrie Eq. 5 i he form: Where f β 7 f 8 pplyig he proposed mehod, he followig ieraive formla is formed as: λ f d 9 Is saioary codiios ca be obaied as follows: λ' λ, λ" λ,. The Lagrage mliplier, herefore, ca be ideified as; Lai merica Joral of Solids ad Srcres - 9 λ si

4 . Nikkar e al. / Dyamic model of large amplide vibraio of a iform cailever beam carryig a iermediae lmped mass ad roary ieria Lai merica Joral of Solids ad Srcres - 9 Sbsiig he ideified mliplier io Eq. 9 resls i he followig ieraio formla: β d si ssmig is iiial approximae solio has he form: cos d sbsiig Eq. io Eq. 5 leads o he followig residal: ] [ cos ] [ cos cos cos cos cos cos β β β R By he formlaio, we ca obai d R si 5 I he same maer, he res of he compoes of he ieraio formla ca be obaied. I order o esre ha o seclar erms appear i, resoace ms be avoided. To do so, he coefficie of cos i Eq. reqires beig zero, i.e., VIM β 6 DESCRIPTION OF THE HE S VRITIONL PPROCH I 7, he variaioal approach was proposed by He He, 7b. I his mehod, a variaioal priciple for he oliear oscillaios is esablished, he a Hamiloia is cosrced, from which he aglar freqecy ca be obaied by a lacaio mehod. To explai he mehod precisely, cosider he followig geeralized oliear oscillaios wiho forced erms: ',, " f ε 7 where f is a oliear fcio of, ' ad ". For simpliciy, if fcio f depeds o oly, is variaioal fcioal ca be obaied as:

5 . Nikkar e al. / Dyamic model of large amplide vibraio of a iform cailever beam carryig a iermediae lmped mass ad roary ieria where F is he poeial, df / d f. Is Hamiloia reads ' J εf d 8 H ' εf 9 which shold be a ivaria drig he oscillaio. Wih he ivaria, we ca apply he locaio mehod o ideify he freqecy by siable choice of rial fcios He, 6. If, by chace, he exac solio had bee chose as he rial fcio, he i wold be possible o make he residal zero for all vales of by appropriae choice of. From his poi of view, He s variaioal mehod is compleely differe from he classical variaioal mehods, b has lile likeess o Riz mehod. Becase, Riz mehod is oly valid for fixed domai, b i He s variaioal mehod, he iegral bodary is o fixed. So He s fcioal formlaio ca be called as variaioal fcioal wih movable domai. Therefore, He s variaioal approach is qie differe from he classic Riz mehod i calcls of variaio. Followig he approach adoped by he Öziş ad Yildirim, 6 i a previos pblicaio, He s mehod has bee sed o deermie he periodic solios ad he periods of srogly oliear sysems give below i sbseqe hree examples Öziş ad Yildirim, 6. More recely, He s mehod also adoped by D co for self-excied sysems D co, 6a; 6b. 5 IMPLEMENTTION OF HE S VRITIONL PPROCH The goverig eqaio of he sysem is: β,, We ca, ow, easily obai he followig variaioal formlaio: J T β d ssme ha is approximae solio ca be expressed as: cos Where is he freqecy o be deermied ad is he amplide of oscillaio. Sbsiig Eq. io Eq. resls i Lai merica Joral of Solids ad Srcres - 9

6 . Nikkar e al. / Dyamic model of large amplide vibraio of a iform cailever beam carryig a iermediae lmped mass ad roary ieria 5 J π β Makig J saioary wih respec o, accordig o He s mehod He, 7b, we obai: π J 8 β π β So he freqecy ca be approximaed as: β V 5 ad zero-order approximae solio from Eq. is: β cos 6 6 RESULTS ND DISSCUSIONS mahemaical model describig he process of large amplide vibraio of a iform cailever beam arisig i he srcral egieerig is proposed. Two differe aalyical mehods are applied o solve he dyamic model of he large amplide o-liear oscillaio eqaio. I order o verify he precisio of he mehods, crre resls were compared wih Rge-Ka h order mehod i Tables ad. I is observable ha or resls are i excelle agreeme wih he resls provided by Rge-Ka h order. The behavior of, obaied by VIM ad V a π ad β.5 is show i Figs. ad. The compariso of he displaceme verss ime for resls obaied from VIM, V ad Rge-Ka h order has bee depiced i Figs. ad for π, β.5 ad 5. Table Compariso bewee VIM & V wih ime marchig solio for he moio eqaio 5, whe.5s, π ad β.5 VIM V RKF ERRORVIM ERRORV Lai merica Joral of Solids ad Srcres - 9

7 6. Nikkar e al. / Dyamic model of large amplide vibraio of a iform cailever beam carryig a iermediae lmped mass ad roary ieria Table Compariso bewee VIM & V wih ime marchig solio for he moio eqaio 5, whe.5s, β VIM V RKF ERRORVIM ERRORV Figre VIM deflecio a π ad β.5 Figre V deflecio a π ad β.5 Lai merica Joral of Solids ad Srcres - 9

8 . Nikkar e al. / Dyamic model of large amplide vibraio of a iform cailever beam carryig a iermediae lmped mass ad roary ieria 7 Figre Compariso bewee VIM ad V periodic solios verss ime wih Rge-Ka h order for β. ad 5. Figre 5 Compariso bewee VIM ad V periodic solios verss ime wih Rge-Ka h order for β. ad 5. 7 CONCLUSIONS I his sdy, a mahemaical model describig he process of large amplide vibraio of a iform cailever beam arisig i he srcral egieerig is proposed. Variaioal approach ad efficie approximae mehod VIM is employed o derive he oliear vibraio of a iform cailever beam. The freqecy of boh mehods is exacly he same. Comparig wih merical Lai merica Joral of Solids ad Srcres - 9

9 8. Nikkar e al. / Dyamic model of large amplide vibraio of a iform cailever beam carryig a iermediae lmped mass ad roary ieria Resls, i is show ha he approximae aalyical solios are i very good agreeme wih he correspodig solios. Refereces kbarzade, M., Kha, Y.,. Dyamic model of large amplide o-liear oscillaios arisig i he srcral egieerig: alyical solios. Mahemaical ad Comper Modellig, 55:8 89. liia, M., Gaji, D. D., Gorji-Badpy, M.,. Nmerical sdy of mixed covecio i a iclied wo sided lid drive caviy filled wih aoflid sig wo-phase mixre model. Ieraioal Commicaios i Hea ad Mass Trasfer, 8: 8-5. Barari,., Kaliji, H. D., Ghadimi, M., Domairry, G.,. No-liear vibraio of Eler-Berolli beams. Lai merica Joral of Solids ad Srcres 8:9-8. Baya, Mahmod., Pakar, Ima., Baya, Mahdi.,. alyical sdy o he vibraio freqecies of apered beams. Lai merica Joral of Solids ad Srcres 8:9-6. Cveicai, L., Kovacic, I., 7. Paramerically excied vibraios of a oscillaor wih srog cbic egaive olieariy. Joral of Sod ad Vibraio, :. D co. M. 6a. Self-exied sysems: alyical deermiaio of limi cycles. Chaos, Solios ad Fracals, :79-7. D co. M. 6b. Deermiaio of limi cycles for a modified va der Pol oscillaor. Mechaics Research Commicaios :9-98. Gaji, D. D.. semi-alyical echiqe for o-liear selig paricle eqaio of Moio. Joral of Hydro-evirome Research 6: -7. Ghasempor, S., Vahidi, J., Nikkar., Mighai, M.,. alyical pproach o Some Highly Noliear Eqaios by Meas of he RVIM. Research Joral of pplied Scieces, Egieerig ad Techology, 5: 9-5. Hamda, M. N., Shabaeh, N. H., 997. O he large amplide free vibraios of a resraied iform bea carryig a iermediae lmped mass. Joral of Sod ad Vibraio, 99:7 76. Hamda, M. N., Dado, M.H.F., 997. Large amplide free vibraios of a iform cailever beam carryig a iermediae lmped mass ad roary ieria. Joral of Sod ad Vibraio, 6:5 68. Hamidi, S.M., Rosamiya, Y., Gaji, D.D., Fereidoo,.,. ovel ad developed approximaio for moio of a spherical solid paricle i plae coqee flid flow. dvaced Powder Techology, hp://dx.doi.org/.6/j.ap..7. He, J. H. 6. Some asympoic mehods for srogly oliear eqaios. Ieraioal Joral of moder Physics B, :-99. He, J.H., 7a. Variaioal ieraio mehod - some rece resls ad ew ierpreaios. J Comp ppl Mah, : 7. He, J. H. 7b. Variaioal approach for oliear oscillaors. Chaos, Solios ad Fracals, : 9. He, J.H.,. sympoic mehods for soliary solios ad compacios. bsrac ad applied aalysis, doi:.55//9679 Herisa, N., Marica, V.,. Explici aalyical approximaio o large-amplide o-liear oscillaios of a iform cailever beam carryig a iermediae lmped mass ad roary ieria. Meccaica, 5: Kha, Y., Taghipor, R., Fallahia, M., Nikkar,.,. ew approach o modified reglarized log wave eqaio. Neral Compig ad pplicaios, doi:.7/s Liao, S. J.,. Beyod perrbaio: irodcio o homoopy aalysis mehod. Chapma & Hall/CRC Press, Boca Rao. Lai merica Joral of Solids ad Srcres - 9

10 . Nikkar e al. / Dyamic model of large amplide vibraio of a iform cailever beam carryig a iermediae lmped mass ad roary ieria 9 Marica, V., Herisa, N., Opimal homoopy perrbaio mehod for srogly oliear differeial eqaios. Noliear Sciece Leers, : 7 8. Nikkar,., Mighai, Z., Saghebia, S.M., Nojabaei, S.B., Daie, M.,. Developme ad Validaio of a alyical Mehod o he Solio of Modellig he Pollio of a Sysem of Lakes. Research Joral of pplied Scieces, Egieerig ad Techology, 5:96-. Öziş, T., Yildirim., 6. Deermiaio of freqecy-amplide relaio for Dffig-Harmoic oscillaor by he eergy balace mehod. Special Isse of Compers ad Mahemaics wih pplicaios, 57:8-87. Rafieipor, H., Lofavar,., Masoori, M.H.,. New alyical pproach o Noliear Behavior Sdy of symmerically LCBs o Noliear Elasic Fodaio der Seady xial ad Thermal Loadig. Lai merica Joral of Solids ad Srcres, 95:5-55. Salehi, P., Yaghoobi, H., Torabi, M.,. pplicaio of he Differeial Trasformaio Mehod ad Variaioal Ieraio Mehod o Large Deformaio of Cailever Beams der Poi Load. Joral of Mechaical Sciece ad Techology, 6 9: Saravi, M., Hermaa, M., Ebarahimi khah, H.,. The compariso of homoopy perrbaio mehod wih fiie differece mehod for deermiaio of maximm beam deflecio. Joral of Theoreical ad pplied Physics, 7:8, doi:.86/ Sheikholeslami, M., Soleimai, S., Gorji-Badpy, M., Gaji, D. D., Seyyedi, S. M., a. Naral covecio of aoflids i a eclosre bewee a circlar ad a sisoidal cylider i he presece of mageic field. Ieraioal Commicaios i Hea ad Mass Trasfer, 99: 5-. Sheikholeslami, M., Gorji-Badpy, M., Gaji, D. D., b. Mageic field effecs o aral covecio arod a horizoal circlar cylider iside a sqare eclosre filled wih aoflid. Ieraioal Commicaios i Hea ad Mass Trasfer, 97: Torabi, M., Yaghoobi, H.,. Novel Solio for cceleraio Moio of a Verically Fallig Spherical Paricle by HPM-Pade pproxima. dvaced Powder Techology, : Torabi, M., Yaghoobi, H., Saedodi, S.,. ssessme of Homoopy Perrbaio Mehod i Noliear Covecive-Radiaive No-Forier Codcio Hea Trasfer Eqaio wih Variable Coefficie. Thermal Sciece, 5:6-7. Torabi, M., Yaghoobi, H., ziz,.,. alyical Solio for Covecive-Radiaive Coiosly Movig Fi wih Temperare Depede Thermal Codciviy. Ieraioal Joral of Thermophysics, :9-9. W, B.S., Lim, C.W., Ma Y.F.,. alyical approximaio o large-amplide oscillaio of a o-liear coservaive sysem. Ieraioal Joral Noliear Mechaics, 8:7. Lai merica Joral of Solids ad Srcres - 9