Dynamic response under moving concentrated loads of non uniform rayleigh beam resting on pasternak foundation

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Preston Noel Chandler
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Avilble olie www.pelgireserchlibrry.co Pelgi Reserch ibrry Advces i Applied Sciece Reserch :-8 ISSN: 976-86 ODEN SA: AASRF Dyic respose uder ovig cocered lods o o uior ryleigh be resig o pser oudio P.

vryig velociy wih siply suppored boudry codiio hs bee ivesiged. Alyicl soluio which represes he rsverse displcee respose o he be uder boh cocered orces d sses rvellig o uior velociies ws obied.

1 Avilble olie Pelgi Reserch ibrry Advces i Applied Sciece Reserch :-8 ISSN: ODEN SA: AASRF Dyic respose uder ovig cocered lods o o uior ryleigh be resig o pser oudio P. B. Ojih* M. A. Ibiejugb d B. O. Adejo Depre o Mheicl Scieces Kogi Se iversiy Ayigb Nigeri ABSTRAT I his sudy he dyic respose o o uior Ryleigh be resig o Pser oudio d subjeced o cocered lods rvellig vryig velociy wih siply suppored boudry codiio hs bee ivesiged. Alyicl soluio which represes he rsverse displcee respose o he be uder boh cocered orces d sses rvellig o uior velociies ws obied. To obi he soluio o he ourh order pril diereil equio wih sigulr d vrible coeicies echique bsed o he Geerlized Gleri s Mehod d he sruble s sypoic echique ws eployed. Nuericl resuls i ploed curves re preseed. The resuls show h s he Roory ieri icreses he respose pliudes o he o uior Ryleigh be decreses or boh ovig orce d ovig ss probles. Furherore he resuls show h he respose pliudes o he o uior Ryleigh be decreses wih icrese i he vlues o he sher odulus G or ied vlues o oudio odulus K d Roory ieri. Siilrly s K icreses he G is ore oiceble h h o K. respose pliudes decreses bu he eec o Filly he criicl speed or he ovig ss proble is reched prior o h o he ovig orce or he o uior Ryleigh be proble i he illusrive eple idered. Hece he ovig orce soluio is o se pproiio o he ovig ss proble hereore we co guree sey or desig bsed o he ovig orce soluio sice resoce is reched erlier i he ovig ss proble h i he ovig orce proble. Keywords: ovig ss ovig orce Ryleigh be Pser oudio resoce. INTRODTION The sudy o he behvior o elsic solid bodies bes ples or shell subjeced o ovig lods hs bee he cocer o severl reserchers i pplied heics d egieerig. More speciiclly severl dyicl probles ivolvig he respose o bes o oudio d wihou oudio hve vriously bee cled by Fryb [] d Sdiu d eipholz []. Aog he erlies wor i his re o sudy ws he wor o Soes [] who obied pproie soluio or he respose o be by eglecig he ss o he be. This is becuse he iroducio ieri eec o he ovig ss would e he goverig equio cubersoe o solve s repored i Sisic e l [] recogizig his diiculy pesel [] pplied Ryleigh Riz echiques o reduce he ovig ss proble deied by coiuous diereil equio o pproie syse o discree diereil equios wih lyic coeicies. The syse ws reduced by iie dierece schee or soluio bu o uericl resuls were preseed. Aer his severl reserchers hve pproched his proble by ssuig h he ieri o he ovig lod ws egligible. I c Arye e l [] poied ou i heir sury o wor doe prior o 95 h he udel heicl diiculies ecouered i he proble lie i he c h oe o he coeicies o he lier operor describig he oio is ucio o boh spce d ie. They dded h i is cused by he presece o Dirc-Del ucio s coeicie ecessry or proper descripio o he oio. I is rered his jucure h physiclly his er represes he ierply o he ieril orces due o he discree sses disribued over he srucure durig he oio Fryb []. Arye e l [] lso idered he proble o Pelgi Reserch ibrry

2 P. B. Ojih e l Adv. Appl. Sci. Res. :-8 elsic be uder he cio o ovig lods. They ssued he ss o he be o be sller h he ss o he ovig lod d obied pproie soluio o he proble. This is ollowed by he oher eree cse whe he ss o he lod ws sller h he ss o he be. I priculr he dyic respose o siply suppored be rsverse by orce ovig uior speed ws irs sudied by Krylov [5]. He used he ehod o epsio o Eige ucio o obi his resuls. ow [6] lso idered he proble o rsverse oscillios o bes uder he cio o ovig lods or he geerl cse o y rbirrily prescribed lw o oio. He obied his soluio usig Gree s ucios. More recely he proble o he dyic respose o o uior be resig o elsic oudio d uder cocered sses ws cled by Oi [9]. Alysis o his resuls show h he respose pliude o boh ovig orce d ovig ss decrese wih icresig oudio oduli. Oi [8] idered he respose o o uior hi be resig o elsic oudio o severl ovig sses. For he soluio o he proble he used he versile echique o Gleri o reduce he cople goverig ourh order pril diereil equio wih vrible d sigulr coeicies o se o ordiry diereil equios. The se o ordiry diereil equios ws ler sipliied d solved usig odiied sypoic o sruble. Oher sudies o o-uior be iclude Doughls el [] Awodol d Oi [] d Oi d Oole [7]. THE GOERNING EQATION The rsverse displcee o he be whe i is uder he cio o ovig lod is govered by he ourh order pril diereil equio give by: EI R P P O G. where is he spil coordie is he ie is he rsverse displcee is he youg odulus I is he oe o ieri EI is he leurl rigidiy o he srucure while I d re vrible oe o ieri d ss per ui legh o he be respecively. By subsiuig he ovig lod o he or d P g d P d d d covecive ccelerio operor d d io. we hve: c c deied s.. EI R M c c c G g c e dop he eple i Oi S. T. [8] d e I d o be o he or d I I si si Pelgi Reserch ibrry..5.6

3 P. B. Ojih e l Adv. Appl. Sci. Res. :-8 Pelgi Reserch ibrry where is he ss per ui legh o he be. sig equios.5 d.6 i equio. d er soe sipliicio d rerrgee oe obis.7 si 6 si 5 si 9 si si 5 5 c Mg c c c M K G R N where.8 EI N SOTION PROEDRES Equio.7 co be solved by geerlized iie iegrl rsorio becuse he be is o-uior. The pproch ivolves epressig he Dirc del ucio s Fourier ie series d he reducig he ourh order pril diereil equio.7 usig Geerlized Gleri s ehod GGM. The resulig rsored diereil equio is he sipliied usig he odiied sruble s sypoic echique. The geerlized Gleri s ehod is deied by. where is chose such h he desired boudry codiios re sisied. Operio sipliicio By pplyig he geerlized Gleri s ehod. equio.7 c be wrie s. 6 si 5 si 9 si si 5 5 si c Mg ii c i c w w c M K ii G R ii iv N I order o deerie i is required h he epressio o he le hd side o equio. be orhogol o ucio Hece. 6 si 5 si 9 si si 5 5 si d c Mg c c M K G R N ii i ii iv iv w w A rerrgee o equio. d igorig he suio sigs yields

4 P. B. Ojih e l Adv. Appl. Sci. Res. :-8 Pelgi Reserch ibrry. Z g c M K G Z R T N T where T T d d d d '' d while d d d si d si 5 d 5 si 9 d 6 6 iv d 7 5 iv d 8 si 5 iv d 9 si iv d where K G R T T N

5 P. B. Ojih e l Adv. Appl. Sci. Res. :-8 M c [ A B c c[ A B c Mg c [ A B c A d B d A d B d A d B d c I order o solve equio.5 he ucio is chose s he be ucio λ λ λ λ si A B sih h.5.6 So h λ λ λ λ si A B sih h.7 The s d he ode requecies c be deeried by usig he pproprie clssicl boudry codiios. Now subsiuig.6 d.7 io.5 er soe sipliicio d rerrgee oe obis where siϕ A ϕ B sihϕ hϕ.8 λ λ c E E A B c c c B A B g E E N T T S S A Pelgi Reserch ibrry

6 P. B. Ojih e l Adv. Appl. Sci. Res. :-8 S R G S λ ϕ M Equio.8 is ow he udel equio goverig he proble o he dyic respose o ovig cocered lod o o-uior Ryleigh be resig o Pser elsic oudio. I wh ollows wo cses o equio.8 re discussed. losed or soluio se : The diereil equio describig he respose o o uior Ryleigh be resig o Pser elsic oudio d subjeced o ovig orce y be obied ro equio.8 by seig o i his cse we obi: here siϕ A ϕ B siϕ hϕ.9 ρ E E d g ρ E Solvig equio.9 i cojucio wih iiil codiio he soluio is give by p { ϕ [ h B sih si ] ϕ ϕ ϕ ϕ λ λ λ λ ϕ [ A ϕ siϕ ϕsi } si A B sih h. Thus usig. i. oe obis p { ϕ [ h sih si ] ϕ B ϕ ϕ ϕ λ λ λ λ ϕ [ A ϕ siϕ ϕsi } si A B sih h. Equio. represes he respose o ovig orce or y clssicl Boudry codiios o o uior Ryleigh be o Pser elsic oudio. Pelgi Reserch ibrry 5

7 P. B. Ojih e l Adv. Appl. Sci. Res. :-8 se II: I he ieril er is reied he. This is ered he ovig ss proble. I his cse he soluio o he eire equio.8 is required. As ec soluio o his proble is ipossible odiicio o sruble s echique is eployed. To his ed equio.8 is sipliied d rerrged o e he or q q b qc q g siϕ A sih h ϕ B ϕ ϕ E q. where q q q c c b b b c c b By es o his echique oe sees he odiied requecy correspodig o he requecy o he ovig ss. A equivle ree syse operor deied by odiied requecy he replces equio.. Thus he righhd side o. is se o zero d preer is idered or y rbirry rio deied s Evidely sig. equio. becoes c q b c qc g siϕ A ϕ B sihϕ c hϕ E K [ q ]. [ ]. Reiig ers o oly. he we se i. cse correspodig o he cse whe ieri eec o he ss o he syse is egleced is obied d he soluio o. c be wrie s os D where d re Sice < sypoic soluio o he hoogeous pr o. c be wrie s os.5 where d re slowly ie vryig ucios. Subsiuig equio.5 d is derivives io equio. d eglecig he ers i oe obis Pelgi Reserch ibrry 6

8 P. B. Ojih e l Adv. Appl. Sci. Res. :-8 7 Pelgi Reserch ibrry.6 si si si si b b b b os c os os c os c c K os Reiig ers o oly The vriiol equios re obied by equig he co-eicies o d ers o boh side o he equio 6 o zero Hece oig he ollowig rigooeric Ideiies: si si c c os c c c Si c d eglecig hose ers h do o coribue o he vriiol equio equio.6 reduces o.7. si si os os os The he viriol equios re respecively :.9.8 Ω Solvig equio.8 d.9 respecively oe obis. Z l. u where d re s Thereore whe he eec o he ss o he pricle is idered he irs pproiio o he hoogeous syse is

9 P. B. Ojih e l Adv. Appl. Sci. Res. :-8 where l β os β D is clled he odiied url requecy represeig he requecy o he ree syse due o he presece o he ovig ss. Thus o solve he o hoogeous equio. he diereil operor which cs o he he equivle ree syse operor deied by he odiied requecy β i.e. si ϕ A ϕ B sihϕ d w β ρ d where g ρ E hϕ [ siϕ A ϕ B sihϕ hϕ] is replced by. Solvig equio. i cojucio wih he iiil codiio oe obis epressio or. Thus i view o. p { β [ h B sih si ] ϕ β ϕ β β ϕ ϕ β β β ϕ λ λ λ λ β ϕ [ A β ϕ β β siϕ ϕ si β } si A B sih h. Equio. represes he rsverse displcee respose o ovig orce o he siply suppored o uior Ryleigh be o Pser elsic oudio. ISTRATIE EXAMPES For illusrio o resuls i he oregoig lysis we provide eple o siply suppored uior Ryleigh be. I his cse he uior Ryleigh be hs siple suppors eds X d X. The displcee d he bedig oe vish. Hece osequely or orl odes which iplies I view o. d. λ.. Siilrly λ Thus he ovig orce proble is reduced o o-hoogeeous secod order ordiry diereil equio Pelgi Reserch ibrry 8

10 P. B. Ojih e l Adv. Appl. Sci. Res. :-8 ρ β E where Kc si. 5 5 E R A 9 A S S 6 Equio. whe solved i cojucio wih he iiil codiios oe obi epressio or Thus ro dd we obi c c si si P β β si E β c β. Equio. represes he rsverse displcee respose o ovig orce o he siply suppored o uior Ryleigh be o Pser elsic oudio. Followig rgues siilr o hose i he previous secio use is de o he odiied Sruble s echique o obi α β β E β s he odiied requecy correspodig o he ree syse due o he presece o he ovig ss hus he ovig ss proble es he or: d g Kc α si d E.5 I view o. he soluio o.5 becoes c c si si g α α si α E α c.6 Equio.6 represes he rsverse displcee respose o ovig ss o he siply suppored o uior Ryleigh be resig o Pser elsic oudio DISSSION OF OSED FORM SOTION The respose pliude o dyicl syses such s his y grow wihou bod. odiio uder which his hppes is ered resoce codiios. I is perie his jucio o esblish codiios uder which resoce occurs. This pheoeo i srucurl d highwy egieerig is o gre cocer o reserchers or i priculr desig egieers becuse or eple i cuses crcs pere deorio d desrucio i srucures. Pelgi Reserch ibrry 9

11 P. B. Ojih e l Adv. Appl. Sci. Res. :-8 Bridges d oher srucures re ow o hve collpsed s resul o resoce occurrig bewee he srucure d soe sigls rversig he. Evidely siply suppored o- uior Ryleigh be resig o Pser elsic oudio d rversed by ovig orce will eperiece resoce whe while he se syse rversed by ovig ss reches he se o resoce wheever Evidely Equios.7 d.9 show h or he se url requecy he criicl speed or he se syse isig o o uior Ryleigh be resig o elsic Pser oudio d rversed by ovig ss is sller h h rversed by ovig orce. Thus resoce is reched erlier i he ovig ss syse h i he ovig orce syse. NMERIA RESTS AND DISSSIONS e shll illusre he lysis proposed i his pper by iderig o uior Ryleigh be o odulus o elsiciy E. 9 N/ he oe o ieri I he be sp legh.9 d he ss per ui legh o he be 758.7g/. he vlue o he oudio odulus is vried bewee / d / he vlues o Roory ieri is vried bewee d.5 he vlues o he sher odulus vried bewee / d 9N/ he resuls re s show o he vrious grphs below or he siply suppored boudry codiio so r idered. Fig : Trsverse displcee o siply suppored o uior Ryleigh be uder he cios o he cocered orces rvellig velociy or vrious vlues o Roory ieri d or ied vlues o oudio odulus d sher odulus G 9 Fro he grphs bove Figures d displys he eec o Roory ieri o he rsverse delecio o he siply suppored o uior Ryleigh be i boh cses o ovig orce d ovig ss respecively. The grphs show h he respose pliude icreses s he vlue o he Roory ieri decreses. Figures d 5 disply he eec o oudio odulus K o he rsverse delecio o siply suppored o uior Ryleigh be i boh cses o ovig orce d ovig ss respecively. The grphs show h icrese i he Roory ieri resuled o decrese i he pliude o vibrio Pelgi Reserch ibrry

12 P. B. Ojih e l Adv. Appl. Sci. Res. :-8 Figures d 6 shows he iluece o sher odulus G o he delecio proile o siply suppored o uior Ryleigh be i boh cses o ovig orce d ovig ss respecively. The grphs show h higher vlues o sher odulus decrese he vibrio o he bes. Fig : Delecio proile o siply suppored o- uior Ryleigh be uder he cios o cocered orces rvellig velociy or vrious vlues o oudio odulus K d ied vlues o Roory ieri.5 d sher odulus G 9 Fig : Respose pliude o siply suppored o uior Ryleigh be uder he cios o cocered orces rvellig velociy or vrious vlues o sher odulus G d or ied vlues o Roory ieri d oudio odulus Pelgi Reserch ibrry

13 P. B. Ojih e l Adv. Appl. Sci. Res. :-8 Fig : Respose pliude o siply suppored o uior Ryleigh be uder he cio o cocered ss rvellig velociy or vrious vlues o Roory ieri d or ied vlues o sher odulus G 9 d oudio odulus Fig 5: Respose pliude o siply suppored o uior Ryleigh be uder he cio o cocered ss rvellig velociy or vrious vlues o oudio odulus K d or ied vlues o sher odulus G 9 Roory ieri Pelgi Reserch ibrry

14 P. B. Ojih e l Adv. Appl. Sci. Res. :-8 Fig 6: Respose pliude o siply suppored o uior Ryleigh be uder he cio o cocered orces rvellig velociy or vrious vlues o sher odulus G d or ied vlues o oudio odulus K d Roory ieri.5 Fig 5.7: opriso o he displcee respose o ovig orce d ovig ss cses o o uior siply suppored Ryleigh be or ied vlues o R.5 K d G9 Pelgi Reserch ibrry

15 P. B. Ojih e l Adv. Appl. Sci. Res. :-8 Tble : Resuls or vrious vlues o roory ieri R O wih ied vlues o sher odulus G O 9 d oudio odulus K O or boh cses o ovig orce d ovig ss MOING FORE MOING MASS Tsec R R.5 R.5 R.5 R R.5 R.5 R.5..7E-.E- 7.8E E-5.7E-.E- 5.77E-5.E E-.96E-.E-.7E- 9.8E-.96E-.7E-.7E-..E-.E- 6.E- 5.8E-.E-.E- 5.8E-.E-. 6.8E-.87E-.E- 9.E- 6.8E-.87E- 9.E- 6.7E-.5.E-.6E-.7E-.5E-.E-.6E-.5E- 7.55E E-.8E-.9E E-.9E- 7.E-.7.6E-.58E-.9E-.8E-.6E-.58E-.8E- 6.E-.8.E- 5.5E-.5E-.E-.E- 5.5E-.E- 5.9E-.9.6E- 5.96E-.E-.E-.6E- 5.96E-.E- 6.E E-.78E- 8.8E E- 8.8E- 6.7E-..55E- 5.E-.E- 6.79E-.55E- 5.E- 6.79E- 6.69E-..7E-.5E- 6.E- 6.E-.7E-.5E- 6.E- 5.E-..5E-.7E-.66E- 6.E-.5E-.7E- 6.E-.7E-..69E-.8E-.67E-5 5.E-.69E-.8E- 5.E-.87E E- -5.8E- -.6E-.8E- 7.E- -5.8E-.8E- -.E E- -.9E- -.6E- -.5E- -.8E- -.9E- -.5E- -.5E-.7 -.E- -.58E- -.5E E- -.E- -.58E E- -.9E E- -.59E- -6.6E- -.E- -.8E- -.59E- -.E- -.E E- -5.6E- -.5E- -.E- -.76E- -5.6E- -.E- -5.E-. -.6E- -5.5E- -.6E- -.E- -.6E- -5.5E- -.E- -7.9E E- -5.9E- -.E- -.E- -.58E- -5.9E- -.E- -8.E-. -.7E- -.E- -.E- -.E- -.7E- -.E- -.E E-. -.7E- -.68E- -.55E E- -.7E- -.68E E- -6.5E E- -.E- -.E- -.E- -.55E- -.E- -.E E E- -.6E- -.8E- -.E- -.6E- -.6E- -.E- -5.5E E- -.8E- -.9E- -.E- -.85E- -.8E- -.E E E- -6.9E- -.E E- -.E E E- -.6E- -.E E- -.68E- -.6E E- -.85E-.9-9.E- -.5E- 9.E E- -9.E- -.5E- -.59E- -.6E-. -.8E- -9.7E-.7E-.E- -.8E- -9.7E-.E-.8E-. 7.5E- -.7E-.5E- 5.57E- 7.5E- -.7E- 5.57E-.67E-..6E-.6E-.E- 7.E-.6E-.6E- 7.E-.96E-..7E-.E- 6.9E- 7.76E-.7E-.E- 7.76E-.E-..E-.E-.E- 7.8E-.E-.E- 7.8E-.7E-.5.9E-.5E-.5E- 8.E-.9E-.5E- 8.E- 5.96E-.6.76E-.5E-.7E- 9.8E-.76E-.5E- 9.8E- 7.8E-.7.8E- 5.6E-.E-.E-.8E- 5.6E-.E- 8.E-.8.65E- 6.8E-.5E-.E-.65E- 6.8E-.E- 7.76E E-.9E-.8E E-.8E- 6.9E-..8E- 5.9E-.9E-.E-.8E- 5.9E-.E-.9E- Pelgi Reserch ibrry

16 P. B. Ojih e l Adv. Appl. Sci. Res. :-8 Tble : Resuls or vrious vlues o oudio odulus K O wih ied vlues o sher odulus G O 9 d roory ieri R O or boh cses o ovig orce d ovig ss MOING FORE MOING MASS Tsec K K K K K K K K..E-.E-.6E-.56E-.6E-.6E-.56E-.8E-..9E-.E-.68E-.85E-.9E-.6E-.E-.67E-..6E-.E- 7.E- 5.9E- 8.E- 7.96E- 5.8E-.55E-..E-.9E-.9E- 9.5E-.8E-.7E- 9.89E- 7.9E-.5.E-.7E-.E- 7.76E-.E-.8E-.E- 6.8E E-.E-.9E-.57E-.98E-.5E- 9.5E E- 6.57E-.E- 9.7E- 5.9E-.87E- 9.68E- 9.7E E- 7.E-.E-.5E- 6.8E- 5.6E- 8.E- 8.57E E- 9.6E- 7.6E- 5.6E- 8.8E- 9.E E- 9.E- 6.7E-.657.E- 6.99E-. 9.5E-.8E-.7E- 7.8E- 5.59E-.E-.5E- 8.6E E- 9.E- 5.6E-.88E-.6E- 8.6E-.E-..57E- 8.58E-.E- 5.9E-.8E-.E-.5E-.8E-..E E- -.6E-.E- -.E- -6.7E- 5.87E-5.E E- -.E- -.E- 9.58E-5 -.E- -.7E- -.7E- -.E E- -.76E- -.E- -.55E- -.5E- -.7E- -.7E- -.5E E- -.9E- -.57E- -.E- -.8E- -.E- -.59E- -5.E-.8-8.E- -.7E- -5.E- -6.8E- -.6E- -.8E- -.98E- -.9E E- -.57E- -9.9E- -7.7E- -.E- -.8E- -7.5E E E- -.9E- -9.6E E- -.9E E E- -.E- -.5E- -.E- -.6E- -.6E- -.E- -8.9E E- -.8E- -.E- -9.7E- -.97E E- -9.7E-. -.7E- -.79E- -.7E- -.E E E- -8.E E E E- -9.8E- -.7E E- -.E E E- -.5E- -.E- -.79E E E E- -6.8E- -.8E- -6.5E- -.6E- -.E- -8.5E- -7.5E E- -5.E- -.6E- -7.E- -.6E- -.7E- -8.7E- -.8E E- -.6E- -6.7E- -.9E- -.6E- -.6E- -6.6E- -.E E E- -.8E- -.E- -.5E- -.5E- -.E-. -.E- 7.E-.99E-.E E-.E-.E-.E-. -.E-.77E-.9E-.E-.9E-.8E-.7E- 8.5E E-.E-.E-.7E-.E-.76E- 5.E-.66E-..E- 5.8E-.8E- 6.85E-.E-.8E-.9E-.56E-. 5.6E- 5.E- 7.89E- 6.59E- 5.88E-.6E- 5.59E- 7.67E E-.96E-.E-.E- 6.8E-.E- 8.6E- 7.8E E E- 6.99E-.E-.5E- 8.E E-.5E-.8E E-.E-.E E-.5E- 9.6E- 5.E-.86E-.7E- 8.8E E- 9.8E-.5E-.E-.E- 9.8E- 9.7E-. 9.E-.5E- 8.8E- 8.58E-.77E-.9E- 7.E- 6.8E- Pelgi Reserch ibrry 5

17 P. B. Ojih e l Adv. Appl. Sci. Res. :-8 Tble : Resuls or vrious vlues o sher odulus G O wih ied vlues o oudio odulus K O d roory ieri R O.5 or boh cses o ovig orce d ovig ss MOING FORE MOING MASS Tsec G G 9 G 9 G 9 G G 9 G 9 G 9..5E-.6E-.5E-.88E-.E-.E-.E-.98E-..69E-.68E-.6E-.E-.E-.E-.8E-.5E-. 7.7E- 7.E- 6.9E-.E-.E-.E- 9.7E- 5.98E-..E-.9E-.9E-.66E-.E-.9E-.E- 8.8E-.5.E-.E-.E-.E-.76E-.7E-.E- 8.8E-.6.6E-.E-.E-.97E- 5.8E- 5.6E-.8E- 7.5E-.7.E-.E- 9.6E- 6.9E- 6.79E- 6.57E-.9E- 7.6E-.8.E-.E- 9.5E- 6.7E- 7.6E- 7.E-.9E- 9.7E-.9.E-.E-.E-.9E E-.E E-.E-.E E- 8.57E-..9E-.7E-.9E-.68E- 5.5E-.8E-.5E- 5.6E-. 9.6E- 9.E- 6.89E-.5E-.E-.87E- 9.8E-.66E-..5E-.E-.5E-.E-.5E- 8.58E-.85E-.8E-. -.E- -.6E-.88E-.8E- -8.E E- -.E-.E-.5 -.E- -.E-.E-5-8.E-6 -.E- -.E- -5.8E-.7E-.6 -.E- -.E- -.7E- -.6E- -.8E- -.76E- -7.8E- -.9E E- -.57E- -.59E- -.7E- -.6E- -.9E- -.E- -5.6E E- -5.E- -6.9E- -.E- -.E- -.7E E E E- -9.9E- -.E- -.6E- -.6E- -.57E- -.97E- -5.E E- -.E- -.E- -.9E- -.6E- -.6E- -6.E-. -.5E- -.5E- -.E- -5.E- -.8E- -.E- -.5E- -8.7E-. -.8E- -.E- -9.5E- -6.E- -.6E- -.8E E-. -.E- -.7E- -9.8E- -6.7E- -.8E- -.79E- -.9E E E E- -.E- -.9E- -5.9E E- -.6E- -7.5E-.5 -.E- -.6E- -.E- -.E- -6.6E E- -6.E E- -.8E- -.8E- -.67E- -6.E- -6.8E- -.8E- -6.E E- -.6E- -6.8E- -.E- -5.6E- -5.E- -.E E E- -6.7E- -.6E- -.E- -.9E- -.6E- -.6E- -.85E E- -.5E- -.5E- -.E- -.E E- -.8E-..E-.99E- 8.85E E-6.88E- 7.E-.5E-.67E-..78E-.9E-.E-.7E-.5E-.77E-.E-.5E-..59E-.E-.67E-.E-.5.E-.E-.5E-. 5.E-.8E- 6.99E-.6E- 5.6E- 5.8E- 9.5E-.76E-. 7.5E- 7.89E-.9E-.6E E-.9E- 6.76E-.5.8E-.E-.E-.9E- 6.6E-.96E-.E- 9.E-.6.5E-.59.E- 5.E- 5.8E-.8E-.E- 9.5E-.7.58E-.5E- 9.E- 6.5E-.8E-.6E-.88E- 8.E-.8.E-.5E- 9.5E- 6.8E-.6E-.E-.95E- 7.E-.9.E- 9.8E-.E-.9E-.7E-.7E-.9E- 8.E E- 8.8E-.E-.E-.9E-.5E-.6E- 9.E- Pelgi Reserch ibrry 6

18 P. B. Ojih e l Adv. Appl. Sci. Res. :-8 Tble : copris o he displcee respose o ovig orce d ovig ss o o uior siply suppored Ryleigh be or ied vlues o Roory ieri.5 K d G 9. Tsec MOING FORE MOING MASS.6E-.6E-..68E-.9E-. 7.E- 8.E-..9E-.8E-..E-.E-.5.E-.57E-.6.E- 5.9E-.7.E- 6.8E-.8.E- 7.6E-.9.8E- 6.7E-..7E- 5.59E-. 9.E-.88E-..E-.8E-. -.6E- -.E-. -.E- -.E-.5 -.E- -.5E E- -.8E-.7-5.E- -.6E E- -.E E E- -.6E-. -.E- -.97E-. -.7E E- -.7E-. -.6E- -.E E- -.6E E- -.6E E- -.6E E- -.E-.9.99E- -.77E-..9E-.9E-..E-.E-..8E-.E E- 5.88E-..E- 6.8E E-.6.5E E- 5.E E-.E E-.77E-..6E-.6E- ONSION The proble o vibrios o o uior Ryleigh be resig o elsic Pser oudio d rsverse by cocered sses rvellig velociy hs bee ivesiged. Illusrive eple ivolvig siply suppored boudry codiio ws preseed. The soluios hihero obied re lyzed d resoce codiios or he vrious probles re esblished. Resuls show h: Resoce is reched erlier i syse rversed by ovig ss h i h uder he cio o ovig orce. As he sher odulus G Roory ieri d oudio odulus K icreses he pliude o o uior Ryleigh be uder he cio o ovig lods ovig velociy decreses. b he he vlues o he sher odulus G d Roory ieri re ied he displcee o o uior Ryleigh be resig o elsic Pser oudio d rversed by sses rvellig wih velociy. c For ied vlue o il orce sher odulus d oudio odulus he respose pliude or he ovig ss proble is greer h h o he ovig orce proble or he illusred ed codiio idered. d I hs bee esblished h he ovig orce soluio is o upper boud or ccure soluio o he ovig ss i uior Ryleigh bes uder ccelerig lods. Hece he o- relibiliy o ovig orce soluio s se pproiio o he ovig ss proble is coired. Pelgi Reserch ibrry 7

19 P. B. Ojih e l Adv. Appl. Sci. Res. :-8 e I he illusred eples or he se url requecy he criicl velociy or ovig ss proble is sller h h o he ovig orce proble. Hece resoce is reched erlier i he ovig ss proble. REFERENES [] Awodol T.O. d Oi S.T : Jourl o he Nigeri ssociio o Mheicl Physics vol 7 pp 9-6 [] Ayre R. S; Jcobse. S; Hsu R. S 95: Trsverse vibrio o oe d o wo-sp bes uder he cio o ovig ss lod. Proceedigs o he irs.s. Niol ogress o pplied echics Illiois Isiue o echology hicgo Illiois pp [] Dougls R hrles d Frederic Y.M : Q.JI Mech. Appl. Mh. 55 pp 9-8. [] Fryb 5: ibrios o solid srucures uder ovig lods. Jourls o Nigeri Associio o Mheicl Physics vol. 9. [5] Krylov A.N 95: Mheicl collecio o ppers o he Acdey o scieces vol. 6 Peersburg. [6] ow A.N 95: Phil. Mg. Ser. 7 vol. 9 No. 7 pp [7] Oi S.T d Ooloe 5: ABAS- Jourl o he Mheicl ssociio o Nigeri vol No. A pp [8] Oi S.T 996: Abcus Jourl o heicl ssociio o Nigeri vol. o pp 7-88 [9] Oi S.T : Jourl o he Nigeri Sociey o Egieers 5 pp [] Pesel E 95: Igeieur Archive. ol. 9 pp [] Sdiu S d eipholz H.H.E 987: Igeieur Archive ol. 57 pp -. [] Sisic M.M; Euler J.A; Mogoery S.T 97: Igeieur Archive vol. 9 pp 95. [] Soes G 89: Discussio o diereil equio relig o he breig o rilwy bridges Trslios o he bridge philosophicl sociey 85 pp Pelgi Reserch ibrry 8

A new approach to Kudryashov s method for solving some nonlinear physical models

A new approach to Kudryashov s method for solving some nonlinear physical models Ieriol Jourl of Physicl Scieces Vol. 7() pp. 860-866 0 My 0 Avilble olie hp://www.cdeicourls.org/ijps DOI: 0.897/IJPS.07 ISS 99-90 0 Acdeic Jourls Full Legh Reserch Pper A ew pproch o Kudryshov s ehod