SOLUTION OF DIFFERENTIAL EQUATION FOR THE EULER-BERNOULLI BEAM

Transcription

1 Jourl of Applied Mthemtics d Computtiol Mechics () 57-6 SOUION O DIERENIA EQUAION OR HE EUER-ERNOUI EAM Izbel Zmorsk Istitute of Mthemtics Czestochow Uiversit of echolog Częstochow Pold izbel.zmorsk@im.pcz.pl Abstrct. he pper presets the solutio of fourth order differetil equtio with vrious coefficiets occurrig i the vibrtio problem of the Euler-eroulli bem. he cocerig equtio is writte s first order mtri differetil equtio. o solve the equtio the power series method is proposed. Kewords: Euler-eroulli bem power series method mthemticl modellig Itroductio or certi cses of differetil equtios with vrible coefficiets it is possible to determie their ect solutios [-] usig e.g. homotop lsis [] or the Gree's fuctios method []. However i most cses i order to obti solutio it is ecessr to ppl pproimte methods such s fiite differece method [5] the power series method [6] differetil trsformtio method (DM) - improved lor method [5 7 8] or b the use of the grge multiplier formlism [9]. his pper is cotiutio of cosidertio show t [] reltig to the use of mtri d power series methods for solvig ordir differetil equtios. he work presets s emple for proposed method the solutio to the equtio of motio of the o-uiform bem described ccordig to the Euler-eroulli theor b the equtio of the fourth order. ormultio d solutio of the problem sic cocepts of the procedure At the begiig let us recll the procedure schem for fourth order lier differetil equtio: () ( ) ( ) ( ) ( ) f( ) () ()

2 I. Zmorsk 58 completed b iitil coditios. itroducig fuctios: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) d ( ) ( ) ( ) ( ) s s s f ssumig tht the fuctios pperig i the equtio () re C clss d c be preseted i the form of power series we c rewrite () s first order mtri differetil equtio []: ( ) ( ) ( ) ( ) () I the bove equtio the followig desigtios were chrged: ( ) ( ) ( ) ( ) ( ) [ ] [ ] ( ) ( ) [ ] [ ] f f ( ) ( ) ( ) ( ) ( ) for d boudr coditio is ( ). A solutio of ihomogeeous equtio () i the power series form c be epressed s sum ( ) ( ) ( ) () where E d re determied from the recursive reltios. he first few vlues of those prmeters re: ( ) ( ) ( ) ()

3 em s equtio of motio Solutio of differetil equtio for the Euler-eroulli bem 59 Accordig to the Euler-eroulli theor the motio of the bem legth with vrious cross-sectio re A() d momet of ierti I() (ig. ) is described b the prtil differetil equtio: EI u ( ) A( ) u ρ (5) t where u is the fuctio of deflectio ρ is the mss desit d E is the oug s modulus. Equtio (5) is complemeted b pproprite boudr coditios depedig o the method of fiig the eds of the bem. ig.. A sketch of cosidered bem iωt ( ) ( ) Assumig siusoidl rottio of fuctio u t e the equtio of motio c be rewritte s: d d EI d d ( ) A( ) ρω (6) with turl frequec ω. Net let us ssume the cross-sectio re d the momet of ierti i the form of polomils: A α α I ( ) A I( ) (7) where A A() I I() α is proportiolit fctor of the bem s crosssectio d [ ]. After few trsformtios equtio (6) c be epressed s equtio () where: for ( ) ( ) Ω ( ) ( ) ( ) α ( α ). ( ) ( ) 8 ( α ) ( ) (8)

4 6 I. Zmorsk Prmeter Ω chrcterizes the vibrtio frequec of the bem d is give b formul Ω series: ω ρ A. Note tht fuctios () d () c be preseted s power EI α ( ) ( ) ( ) where (9) Of course we hve to esure the covergece of the series hece: for α > is d for α < is. α α Mtri () occurrig i () is i the form: ( ) () ( ) ( α) ( ) 8( α) Ω d solutio of (6) is give b reltio () for d s i (). I the cse α we ve got vibrtio problem of the Euler-eroulli bem with costt prmeters chrcterizig its phsicl properties. Equtio (5) hs the form: We c ote tht ( ) ( ) ( ) ( ) () Ω () Ω. After coversio ccordig to discussed method mtrices re s follows: O () Ω he solutio of the boudr problem: ( ) ( ) ( ) ( ) ( ) ( ) where: is

5 Solutio of differetil equtio for the Euler-eroulli bem 6 Ω k Ω Ω Ω Ω k k k k ( Ω ) ( Ω ) k k k k ( Ω ) ( Ω ) ( Ω ) E k... Ω E () he geerl solutio of equtio (6) depeds o four costts which re determied from the boudr coditios. or emple the boudr coditios for ctilever bem re ( ) ( ) ( ) ( ) () he coditios t the bem s boudr gives us d t gives so the iitil coditio to the () is [ ] (5) Coclusios I the preset stud the method reducig the fourth order differetil equtio to the form of the first order mtri equtio ws cosidered o the emple of o-uiform bem s vibrtio equtio. Cosidertios show tht it is reltivel simple method of solvig the boudr problem reduced to form which is es for computer implemettio. If ol the fuctios occurrig i the equtios re the pproprite clss d it s possible to epd them i power series the i the itervl of covergece of these series the preseted method c lso be used to solve more comple problems described b differetil equtios. Refereces [] Elishkoff I. ecquet R. Closed-form solutios for turl frequec for ihomogeeous bems with oe slidig support d the other pied Jourl of Soud d Vibrtio 8() [] Che D.-W. Wu J.-S. he ect solutios for the turl frequecies d mode shpes of ouiform bems with multiple sprig - mss sstems Jourl of Soud d Vibrtio 55() 99-. [] Hss H.N. El-wil M.A. A ew techique of usig homotop lsis method for secod order olier differetil equtios Applied Mthemtics d Computtio

6 6 I. Zmorsk [] Kukl S. Zmojsk I. Applictio of the Gree s fuctio method i free vibrtio lsis of o-uiform bems Scietific Reserch of the Istitute of Mthemtics d Computer Sciece 5 () [5] eh -. Jg M-J. Wg C-C. Alzig the free vibrtios of plte usig fiite differece d differetil trsformtio method Applied Mthemtics d Computtio [6] Qisi M.I. A power series solutio for the o-lier vibrtio of bems Jourl of Soud d Vibrtio () [7] Ozgumus O.O. K M.O. lpwise bedig vibrtio lsis of double tpered rottig Euler-eroulli bem b usig the differetil trsform method Meccic [8] Mei C. Applictio of differetil trsformtio techique to free vibrtio lsis of cetrifugll stiffeed bem Computers d Structures [9] Cekus D. ree vibrtio of ctilever tpered imosheko bem Scietific Reserch of the Istitute of Mthemtics d Computer Sciece () -7. [] Kukl S. Zmorsk I. Power series solutio of first order mtri differetil equtios Jourl of Applied Mthemtics d Computtiol Mechics () -8.