Journal of Applied and Computational Mechanics, Vol. 3, No. 1, (2017), DOI: /jacm

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1 Jural f Applied ad Cmputatial echaics, Vl., N., (7), -79 DOI:.55/jacm.7. Therm-mechaical liear vibrati aalysis f fluidcveyig structures subjected t differet budary cditis usig Galerki-Newt-Harmic balacig methd. G. Sbamw, B. Y. Ogumla, C. A. Osheku, Departmet f echaical Egieerig, Uiversity f ags, Akka, ags, Nigeria. Cetre fr Space Trasprt ad Prpulsi, Natial Space Research ad Develpmet Agecy, Federal iistry f Sciece ad Techlgy, FCT, Abuja, Nigeria. Received Jauary 9 7; revised Jauary 5 7; accepted fr publicati February 7. Crrespdig authr:. G. Sbamw, mikegbemiiyi@gmail.cm Abstract The develpmet f mathematical mdels fr describig the dyamic behaviurs f fluid cveyig pipes, micr-pipes ad atubes uder the ifluece f sme therm-mechaical parameters results it liear equatis that are very difficult t slve aalytically. I cases where the exact aalytical slutis are preseted either i implicit r explicit frms, high skills ad rigrus mathematical aalyses were emplyed. It is ted that such slutis d t prvide geeral exact slutis. Ievitably, cmparatively simple, flexible yet accurate ad practicable slutis are required fr the aalyses f these structures. Therefre, i this study, apprximate aalytical slutis are prvided t the liear equatis arisig i flw-iduced vibrati f pipes, micr-pipes ad atubes usig Galerki-Newt-Harmic ethd (GNH). The develped apprximate aalytical slutis are shw t be valid fr bth small ad large amplitude scillatis. The accuracies ad explicitess f these slutis were examied i limitig cases t establish the suitability f the methd. Keywrds: Therm-mechaical; N-liear Vibrati; Galerki s methd; Newt-Harmic Balacig Techique; Fluid-cveyig structure.. Itrducti Nliear flw-iduced vibrati f pipes, micr-pipes ad atube has bee the tpic f experimetal, umerical, ad theretical studies as it has attracted a large umber f studies i literatures [-7]. This is because, mdelig the dyamic behaviurs f the structures uder the ifluece f sme therm-fluidic r therm-mechaical parameters fte results i liear equatis ad such are difficult t fid the exact aalytical slutis. I sme cases where decmpsiti prcedures it spatial ad tempral parts are carried ut, the resultig liear equati fr the tempral part cmes i frm f Duffig equati. Applicati f aalytical methds such as Expfucti methd, He s Exp-fucti methd, imprved F-expasi methd, idstedt-picare techiques, qutiet trigmetric fucti expasi methd t the liear equati preset aalytical slutis either i implicit r explicit frm which fte ivlved cmplex mathematical aalysis leadig t aalytic expressi ivlvig a large umber terms. Furthermre, the methds are time-csumig task accmpaied with pssessig high skills i mathematics. Als, they d t prvide geeral aalytical slutis sice the slutis fte cme with cditial statemets (i.e. except i limited circumstaces where exact aalytical slutis are pssible) which make them limited i used as may f the cditis with the exact slutis d t meet up with the practical applicatis sice they give apprximated slutis that hardly prvide a all-ecmpassig uderstadig f the ature f systems i respse t parameters affectig liearity. Als, i practice, aalytical slutis with large umber f terms ad

2 Therm-mechaical liear vibrati aalysis f fluid-cveyig structures cditial statemets fr the slutis are t cveiet fr use by desigers ad egieers [8]. Csequetly, recurse has always bee made t umerical methds r apprximate aalytical methds i slvig the prblems [9-]. Hwever, the classical way fr fidig exact aalytical sluti is bviusly still very imprtat sice it serves as a accurate bechmark fr umerical slutis. Als, the experimetal data are useful t access the mathematical mdels, but are ever sufficiet t verify the umerical slutis f the established mathematical mdels. Cmparis betwee the umerical calculatis ad experimetal data fte fail t reveal the cmpesati f mdellig deficiecies thrugh the cmputatial errrs r ucscius apprximatis i establishig applicable umerical schemes. Additially, exact aalytical slutis fr specified prblems are als essetial fr the develpmet f efficiet applied umerical simulati tls. Ievitably, exact aalytical expressis are required t shw the direct relatiship betwee the mdels parameters. Whe such exact aalytical slutis are available, they prvide gd isights it the sigificace f varius system parameters affectig the phemea as it gives ctiuus physical isights tha pure umerical r cmputati methds. Furthermre, mst f the aalytical apprximati ad purely umerical methds that were applied i literatures t liear prblems are cmputatially itesive. Exact aalytical expressi is mre cveiet fr egieerig calculatis cmpare with experimetal r umerical studies ad it serves as a startig pit fr a better uderstadig f the relatiship betwee physical quatities/prperties. It is cveiet fr parametric studies, accutig fr the physics f the prblem ad appears mre appealig tha the umerical slutis. It appears mre appealig tha the umerical sluti as it helps t reduce the cmputati csts, simulatis ad task i the aalysis f real life prblems. Therefre, a exact aalytical sluti is required fr the prblem. Newt-Hamic Balacig techique as prpsed by ai et al. [] is a cmparatively ew techique fr liear cubic-quitic Duffig equati. The simplicity, flexibility i applicati, ad avidace f cmplicated umerical itegrati has give the apprach added advatages ver the previus methds. Cmbiig the Galerki s methd with Newt-Hamic Balacig techique i the aalysis f -liear iitial-budary value prblems, prvides cmplemetary advatages f higher accuracy, reduced cmputati cst ad task as cmpared t the ther methds as fud i literature. Therefre, i this research, aalytical slutis are prvided t the liear partial differetial equatis arisig i flw-iduced vibrati i pipes, micr-pipes ad atubes uder differet budary cditis usig Galerki- Newt-Hamic Balacig ethd. The liear partial differetial equatis were cverted t liear rdiary differetial equatis ad the Newt-Hamic Balacig techique is utilized t prvide exact aalytical slutis t the liear rdiary differetial equatis f vibrati f the structures. The develped aalytical slutis are cmpared with the umerical results ad the results f apprximate aalytical slutis ad gd agreemets reached. The aalytical slutis ca serve as a startig pit fr a better uderstadig f the relatiship betwee the physical quatities f the prblems as it prvides ctiuus physical isights it the prblem tha pure umerical r cmputati methds.. Gverig equatis ad budary cditis Case : Flw-iduced vibrati i pipe Csider a pipe cveyig ht fluid, subjected t stretchig effects ad restig liear ad liear elastic fudati (Pasterak, liear ad liear Wikler fudati) uder exteral applied tesi ad glbal pressure as shw i Fig.. Usig lcal elasticity thery ad Hamilt s priciple, we arrived at the gverig equati f mti as: w w w w EA w EI ( m ) p mf vm f mfv PA T k p x t t x t x () EA w w kw k w N dx x x Fig.. Pipe cveyig ht fluid If the pipe is slightly curved, the the gverig equati becmes Jural f Applied ad Cmputatial echaics, Vl., N., (7), -79

3 . G. Sbamw et. al., Vl., N., 7 w w w w EA w EI ( m ) p mf vm f mfv PA T k p x t t x t x () EA Z w w w Z kw k w N dx x x x x x Where Z (x) is the arbitrary iitial rise fucti. Usig the Galerki s decmpsiti prcedure t separate the spatial ad tempral parts f the lateral displacemet fuctis as w ( x, t ) ( x ) u ( t ) () Where u( t) the geeralized crdiate f the system ad ( x) is a trial/cmparis fucti that will satisfy bth the gemetric ad atural budary cditis. Applyig e-parameter Galerki s sluti give i Eq. () t Eq. (): R x, t x dx (-a) where fr the straight pipe: w w w w EA w R x, t EI ( m ) p mf vm f mf v PA T k p x t t x t x (-b) EA w w kw k w N dx x x Ad fr the slightly curved pipe: w w w w EA w R ( x, t ) EI ( m ) p mf vm f mf v PA T k p x t t x t x (-c) EA Z w w w Z kw k w N dx x x x x x We have the liear vibrati equati f the pipe as: u ( t ) Gu ( t ) ( K C ) u ( t ) Vu ( t ) (5) where ( ) ( ) p f m m x dx d G ( ) ( ) mf u x x dx dx d d K ( ) EI x k ( x ) k p ( x ) dx dx dx EA d C ( ) mf u PA T x dx dx ( ) EA w d ( ) V x N dx k x dx x dx Fr the slightly curved pipe,, G, K ad C are the same but: EA ( ) ( ) Z w w w Z V x N dx k x dx (-a) x x x x x The circular fudametal atural frequecy gives K C (-b) where ( ) EA d C PA T x dx (-c) dx Case : Flw iduced vibrati i fuctially graded micr-pipe Csider the case f flw iduced vibrati i a fuctially graded micrpipe cveyig ht fluid subjected t stretchig effects ad restig liear ad liear elastic fudati uder exteral applied tesi ad glbal Jural f Applied ad Cmputatial echaics, Vl., N., (7), -79

4 Therm-mechaical liear vibrati aalysis f fluid-cveyig structures pressure. Applyig strai gradiet ad cupled stress theries fllwed by Hamilt s priciple [8], we arrived at the gverig equati f mti fr the fuctially graded micrpipe give as: w 8 w w w w GA eq l l ( ) 5 EI GAeq l l l 5 m p mf vmf x x t t x t EA w EA w w mf v PA T k p k w k w N dx x x x If the pipe is slightly curved, the the gverig equati fr the FG micrpipe becmes: w 8 w w w w GI eq l l ( ) 5 EI eq GAeq l l l 5 mp mf vmf x x t t x t EA eq w EA eq Z w w w Z mf v PA T k p k w k w N dx x x x x x x where r ri EI E ( r ) z da E ( r ) r si ( )( rdrd ) eq A r ri GI G ( r ) z da EG ( r ) r si ( )( rdrd ) eq A r ri EA E ( r ) da E ( r )( rdrd ) eq A r ri GA G ( r ) da G ( r )( rdrd ) eq A ( ) r r i r r E r E E r ri r ri G( ) r r i r r r G G r ri r ri ( ) r r i r r r r ri r ri m A p r r i r r ( r ) r ri r ri i which l, l ad l are the idepedet legth scale parameters embedded i the cstitutive equatis f the higher rder stresses. If we fllw the same prcedure f applyig the Galerki s decmpsiti prcedure t separate the spatial ad tempral parts f the lateral displacemet ad the apply the e-parameter Galerki s sluti, we arrived at the same liear vibrati equati fr the micrpipe as: u ( t ) Gu ( t ) ( K C ) u ( t ) Vu ( t ) (9) But at this time, we have (7) (8) ( ) ( ) p f m m x dx d G ( ) ( ) mf u x x dx dx 8 d d K EI GAeq l l l ( ) ( ) ( ) 5 x GA eq l l 5 k x k p x dx dx x dx EA d C ( ) mf u PA T x dx dx EA d d V ( x ) N dx k dx x dx dx Jural f Applied ad Cmputatial echaics, Vl., N., (7), -79

5 . G. Sbamw et. al., Vl., N., 7 Fr the slightly curved FG micrpipe,, G, K ad C are the same but EA Z w w w Z d V ( x ) N dx k dx x x x x x dx The circular fudametal atural frequecy gives: where Case : Flw-iduced vibrati i atube K C () ( ) EA d C PA T x dx dx Csider a sigle-walled carb atube cveyig ht fluid, subjected t stretchig effects ad restig liear ad liear elastic fudati uder exteral applied tesi ad glbal pressure. Fllwig the Erige s lcal elasticity thery [-] ad Hamilt s priciple, we arrived at the gverig equati f mti fr the sigle-walled carb atube (SWCBT) as: w w w w EA w EA ( ) w w EI mp mf vmf mfv PA T k p dx x t t x t x x x w w w w w w ( mp mf ) k k w w vm f x t x t x x x x t kw k w ea EA w EA mf v w w PA T k p dx x x x If the atube is slightly curved, the the gverig equati fr the atube becmes: w w w w EA w EI ( m ) p mf vm f mfv PA T k p x t t x t x EA Z w w w Z N dx k w k w x x x x x e a w w w ( m p mf ) k k w w w w w vm f x t x t x x x x t EA w EA Z w w w Z mf v PA T k p N dx x x x x x x Fr atube cveyig fluid, the radius f the tube is assumed t be the characteristics legth scale, Kudse umber is larger tha -. Therefre, the assumpti f -slip budary cditis des t hld ad mdified mdel shuld be used. Therefre, we have: U avg, slip v K VCF ak K () U avg, slip v K where K is the Kudse umber, σ v is tagetial mmet accmmdati cefficiet which is csidered t be.7 fr mst practical purpses [7]. Therefre, a k a ta ak a b K U a K U VCF U ad Eq. () culd be writte as v avg, slip k avg, slip avg, slip v K B () () () (5) () Jural f Applied ad Cmputatial echaics, Vl., N., (7), -79

6 Therm-mechaical liear vibrati aalysis f fluid-cveyig structures 5 w w w w EA w EI ( m m ) m,, VCF U m VCF U PA T k x t t x t x p f f avg slip f avg slip p EA Z w w w Z N dx k w k w x x x x x (7) w w w w w w ( mp m ), f k k w w mf VCF Uavg slip x t x t x x x x t ea EA w EA Z, w w w Z mf VCF Uavg slip PA T k p N dx x x x x x x Agai, fllwig the same prcedural methd, the Galerki s decmpsiti prcedure ad the apply the eparameter Galerki s sluti, we arrived at the same liear vibrati equati fr the atube as: u ( t ) Gu ( t ) ( K C ) u ( t ) Vu ( t ) (8) Here, we have ( ) ( ) ( ) ( ) ( ) ( ) ( ) d d m p m f x x ea x m f u x ea dx dx dx d d d G ( x ) m ( ) ( ) ( ) f v ea x ea dx dx dx dx d d d d K ( x ) EI k ( x ) k p k ea ea k p dx dx dx dx dx EA d d C ( ) ( ) m f u PA T x e a dx dx dx EA d d N ( ) ( ) ( ) dx k x k ea x x dx dx x V ( x ) dx EA d k ( ea) ( x ) ( ea) N dx x x dx V Fr the slightly curved atube,, G, K ad C are the same but: EA Z w w w Z d N ( ) ( ) ( ) dx k x k ea x x x x x x dx x ( x ) dx (9-a) EA Z w w w Z k ( ea) ( x ) ( ea) N dx x x x x x x ad the circular fudametal atural frequecy gives: K C (9-b) where: EA ( ) d d C ( ) PA T x e a dx (9-c) dx dx. The iitial ad budary cditis The structures (pipe, micr-pipe ad atube) may be subjected t ay f the fllwig budary cditis. i. Clamped-Clamped (dubly clamped) Where the trial/cmparis fucti are give as: csh cs ( x ) csh x cs x sih x si x sih si () r Jural f Applied ad Cmputatial echaics, Vl., N., (7), -79

7 . G. Sbamw et. al., Vl., N., 7 sih si ( x ) csh x cs x sih x si x csh cs where are the rts f the equati cs csh. The iitial ad the budary cditis are: w (, x ) A w (, x ) (-a) w (, t ) w '(, t ) w (, t ) w '(, t ) (-b) The applicatis f space fucti as give abve fr clamped-clamped will ivlve lg calculatis ad expressis i fidig, G, K, C, ad V, alteratively, a plymial fucti f the frm Eq. () ca be applied fr this type f supprt system: ( x ) a a X a X a X a X () where X x. Applyig the budary cditis ( x ) X X X a () Orthgal fucti shuld satisfy the equati a ( X ) ( X ) dx (5) Substitute Eq. () it Eq. (5), we have a 7 () 5 a 7a 5a 5a a Fr a =, arrived at a 5. fr the first mde. () ii. Clamped-Simple supprted The trial/cmparis fucti is give as: csh cs ( x ) csh x cs x sih x si x sih si where are the rts f the equati, ta tah. The iitial ad the budary cditis are: w (, x ) A w (, x ) (8-a) w (, t ) w '(, t ) w (, t ) w ''(, t ) (8-b) Alteratively, a plymial fucti f the frm Eq. (9) ca be applied fr this type f supprt system: 5 ( x ) X X X a (9) usig rthgal fuctis, a.5 fr the first mde. (7) iii. Simple-Simple supprted ( x ) si x, si () The iitial ad the budary cditis are: w (, x ) A w (, x ) (-a) w (, t ) w ''(, t ) w (, t ) w ''(, t ) (-b) Alteratively, a plymial fucti f the frm Eq. () ca be applied fr this type f supprt system. ( x ) X X X a () O usig rthgal fuctis, a. fr the first mde iv. Clamp-Free (catilever) csh cs ( x ) csh x cs x sih x si x sih si () Jural f Applied ad Cmputatial echaics, Vl., N., (7), -79

8 r Therm-mechaical liear vibrati aalysis f fluid-cveyig structures sih si ( x ) csh x cs x sih x si x csh cs cs csh. The iitial ad the budary cditis are: are the rts f the equati w (, x ) A w (, x ) (5-a) w (, t ) w '(, t ) w ''(, t ) w '''(, t ) (5-b) Alteratively, a plymial fucti f the frm Eq. () ca be applied fr this type f supprt system: x X X X a Als, with the aid f rthgal fuctis, a.5 fr the first mde 7 () ( ) (). Nliear atural frequecy by Newt-balacig methd If we substitute t, the liear equati (8) t be slved ca be writte as ( K C ) V u ( ) u ( ) u ( ) (7) Sice fr the udamped clamped-clamped, clamped-simple ad simple-simple supprted structures, G =. Applyig Newt s prcedure, the displacemet ad the squared agular frequecy ca be expressed as u u u (8) (9) Substitutig Eq. (8) ad (9) results it Eq. (7) gives ( ) K C V u u u u u u () O liearizig Eq. () with respect t the crrecti terms u ad, we have ( K C ) V u u u u () ( K C ) V u u u I rder t fid the peridic sluti f Eq. (8), assume a iitial apprximati t u u ( ) () The abve Eq. () is a peridic fucti f f perid. O substitutig ( ) it Eq. () gives: ( K C ) V Acs u Acs A cs ( K C ) V Acs A cs Acs () Eq.() ca be expressed as: Acs u ( t ) f Acs f Acs u () x where u is a peridic fucti f f perid. We have the fllwig Furier series expasi: i i f Acs a cs i (5) b f x Acs bics i (-a) i where the Furier cefficiets are: ( K C ) A VA a VA ( K C ) VA a b b VA (-b). First-rder aalytical apprximati Fr the first-rder aalytical apprximati, we get: u (7) Jural f Applied ad Cmputatial echaics, Vl., N., (7), -79

9 8. G. Sbamw et. al., Vl., N., 7 (8) ad therefre u ( ) u( ) Acs (9) O substitutig Eq. (5), (), (7) ad (8) it Eq. (), expadig the resultig expressi i a trigmetric series ad the settig the cefficiet f cs t zer results i the first-rder aalytical apprximati. The crrespdig apprximate aalytical peridic slutiu ( ) ca be expressed as: i which u ( ) Acs (5) K C VA (5) where agular frequecy is the first-rder aalytical apprximati. We ca write Eq. (5) directly i terms f the primary parameters i the gverig equatis as: K C VA u( ) Acs (). Secd-rder aalytical apprximati Fr the secd-rder aalytical apprximati, we set u( ) c cs cs (5) By substitutig Eq. (5), () ad (5) it Eq. (), fllw by a trigmetric series expasi ad the settig c ad the cefficiets f cs ad cs t zer result i a set f simultaeus equati i terms f Slvig Eqs. (5) ad (55) simultaeusly, we have a bc bc A A c (5) a b c b c c (55) V A a Ab a A A b b 8a K C V 8 A (5) Aa A V c A b b 8a K C V 8 A (57) The crrespdig apprximate aalytical peridic sluti u ( ) is give as u ( ) u u A cs t c cs t cs t (58) Recall frm Eq. (9) that. Therefre, the secd-rder aalytical apprximati frequecy is give as: Frm Eqs. (5) ad (5), we have (59) K C V K C V 8 9A 9A () K C V 8 A We ca re-write Eq. (58) directly i terms f the primary parameters i the gverig equatis by substitutig Eqs. (57) ad (59) it Eq. (58) t have Jural f Applied ad Cmputatial echaics, Vl., N., (7), -79

10 Therm-mechaical liear vibrati aalysis f fluid-cveyig structures 9 K C V K C V 8 9A 9A ( ) u Acs t K C V 8 A K C V K C V 8 9A 9A cs t () K C V 8 A A V K C V 8 A K C V K C V 8 9A 9A cs t K C V 8 A. Third-rder aalytical apprximati Fr the third-rder aalytical apprximati, the relevat expressis must be exteded. Gig back t Eqs.(8), (9) ad (), replacigu ( ), u ( ), ad by u ( ), u ( ), ad, respectively, i the equatis, we have: 9 Acs c cs cs u ( K C ) V Acs c cs cs Acs c cs cs () ( K C ) V Acs c cs cs Acs c cs cs As befre, Eq. () ca be expressed as: Acs c 9cs cs Acs c 9cs cs f Acs c cs cs f x A cs c cs cs u Fr the third-rder aalytical apprximati, we set: u ( ) c cs cs c cs cs 5 () Substitute Eq. () it Eq. (), emplyig a trigmetric series expasi ad the settig the cefficiets fcs, cs ad cs5 t zer yields a set f simultaeus equati i terms f c, c ad as give i Eq. (5-7): K C A c A c A c V A c Vc A c Vc c K C A c V A c Vc A c Vc Vc A c V A c Vc c A c K C 9 c A c V A c Vc Vc c A c V A c Vc 9V c K C c 9 9 A c V A c V A c V c K C c 9 8 c () (5) () Jural f Applied ad Cmputatial echaics, Vl., N., (7), -79

11 7. G. Sbamw et. al., Vl., N., 7 A c Vc A c Vc A c V V c c Vc K C A c V A c Vc c 5 (7) Slvig Eq. (5-7), we have where c c Vc A c V A c Vc A c V A c Vc K 9 C Vc c c A c V Vc A c V A c Vc 9V c K C 9 A c Vc A c Vc 5 K C A c V A c Vc A c Vc 9 A c V A c V A c V c K C 7 9 Vc K C A c V A c Vc 8 5 A c V A c Vc K C A c Vc 9 A c A c 8c A c (8) (9) (7) It shuld be ted that: u ( ) u ( ) u ( ) (7) Substitutig Eq. (58) ad Eq. (), we have: u ( ) A c c cs t c c c cs t c cs 5 t (7) Where the third-rder atural frequecy is give as: K C V K C V 8 9A 9A K C V 8 A (7) Jural f Applied ad Cmputatial echaics, Vl., N., (7), -79

12 Therm-mechaical liear vibrati aalysis f fluid-cveyig structures 7 Als, Eq. (9) ca be writte directly i terms f the primary parameters ad related parameter t the primary parameters i the gverig equatis as: A V 78 A K C V A K C V K C V u ( ) 8 9A 9A K C V cs 8 A t A V K C V 8 A K C V K C V 8 9A 9A K C V cs 8 A t K C V K C V 8 9A 9A K C V cs 5 8 A t Fllwig the prcedure give abve, we write a geeralized expressi fr the kth-rder f aalytical apprximati as: k k j u ( ) d cs j cs j (75) j k k k (7) (7) Als, it ca easily be see that as the liear term, V teds t zer, the frequecy rati f the liear frequecy t the liear frequecy, teds t : b lim (77) V b Where ( K C ) b (78-a) It shuld be pited ut that whe the liear term, V is set t zer, we recvered the liear atural frequecy. Als, as the amplitude, A teds t zer, the frequecy rati f the liear frequecy t the liear frequecy, teds t. b Jural f Applied ad Cmputatial echaics, Vl., N., (7), -79

13 7. G. Sbamw et. al., Vl., N., 7 Furthermre, it ca easily be shw that lim A b Ad fr very large values f the amplitude A, we have (78-b) lim u ( t ) A (79) t lim (8) A b Where, K, C ad V are respective itegral values f the respective space fuctis f the respective budary cditis ad structure csidered. O a geeral te, fllwig ai et al. [7], it ca easily be shw that the exact atural frequecy f the fluid-cveyig structures is give as: Fr the geeral case i this wrk, dt, exact si t (8-a) VA K C VA (8-b) ( K C ) VA Where whe the liear term, V is set t zer, we recvered the liear atural frequecy. Althugh, it is very difficult t geerate a geeral exact sluti f Eq. (78). Hwever, usig series itegrati methd, we develped a aalytical sluti fr the liear atural frequecy as: exact N N (8)... 8 We als develped a apprximated aalytical sluti: apprx. 5 (8) It ca easily be see that as the liear term teds t zer, the frequecy rati f the apprximated liear frequecy t the exact liear frequecy, teds t. exact lim (8) V exact Als, it culd be bserved that as the amplitude A teds t zer, the frequecy rati f the apprximated liear frequecy t the exact liear frequecy, teds t. exact lim A exact (85). Results ad Discussi The first five rmalized mde shapes f the beams fr clamped-clamped, simple-simple, clamped-simple ad clamped-free supprts are shw i Fig. -5. Als, the figures shw the deflectis f the beams alg the beams spa at five differet buckled ad mde shapes. Figs. -9 shw the cmparis f hyperblic-trigmetric ad the plymial fuctis fr the rmalized mde shapes f the beams fr clamped-clamped, simple-simple, clamped-simple ad clamped-free supprts. The figures depict the validity f the develped plymial fuctis i this wrk as there are very gd agreemets betwee the hyperblic-trigmetric ad the develped plymial fuctis. Jural f Applied ad Cmputatial echaics, Vl., N., (7), -79

14 Therm-mechaical liear vibrati aalysis f fluid-cveyig structures 7 de shape fucti st mde shape d mde shape rd mde shape th mde shape 5 th mde shape de shape fucti st mde shape d mde shape rd mde shape th mde shape 5 th mde shape Dimesiless beam leght Fig.. The first five rmalized mde shaped f the beams uder clamped-clamped supprts Dimesiless beam leght Fig.. The first five rmalized mde shaped f the beams uder simple-simple supprts st mde shape d mde shape rd mde shape th mde shape st mde shape d mde shape rd mde shape th mde shape de shape fucti 5 th mde shape de shape fucti 5 th mde shape Dimesiless beam leght Fig.. The first five rmalized mde shaped f the beams uder clamped-simple supprts Dimesiless beam leght Fig. 5. The first five rmalized mde shaped f the beams uder clamped-free (catilever) supprts.8 Hyperblic-Trigmetric fucti Plymial fucti Hyperblic-Trigmetric fucti Plymial fucti. de shape fucti...8. de shape fucti Dimesiless beam leght Fig.. Nrmalized mde shaped f the structures uder clamped-clamped supprts fr Hyperblic-Trigmetric ad Plymial fuctis Dimesiless beam leght Fig. 7. Nrmalized mde shaped f the structures uder simple-simple supprts fr Hyperblic-Trigmetric ad Plymial fuctis Jural f Applied ad Cmputatial echaics, Vl., N., (7), -79

15 7. G. Sbamw et. al., Vl., N., 7.8 Hyperblic-Trigmetric fucti Plymial fucti.8 Hyperblic-Trigmetric fucti Plymial fucti.... de shape fucti..8. de shape fucti Dimesiless beam leght Fig. 8. Nrmalized mde shaped f the structures uder clamped-free supprts fr Hyperblic-Trigmetric ad Plymial fuctis Dimesiless beam leght Fig. 9. Nrmalized mde shaped f the structures uder clamped-simple supprts fr Hyperblic- Trigmetric ad Plymial fuctis Fig. illustrates the effects f budary cditis the liear amplitude-frequecy respse curves f the atube. Als, the figure shws the variati f frequecy rati f the atube with the dimesiless maximum amplitude f the structure uder differet budary cditis. Frm, the result, it shw that frequecy rati is highest i the beam which is clamped-free (catilever) supprted beam ad lwest with clamped-clamped beam. The lwest frequecy rati f the clamped-clamped beam is due t high stiffess f the beam with this type f budary cditis i cmparis with ther types f budary cditis. The fudametal liear vibrati frequecy is the lwest rt f the resultig characteristics equati. It ca be see frm the figure, i ctrast t liear systems, the liear frequecy is a fucti f amplitude s that the larger the amplitude, the mre pruced the discrepacy betwee the liear ad the liear frequecies becmes Clamped-Simple supprted Clamped-Clamped supprted Simple-Simple supprted Clamped-Free supprted.5. u = m/s u = m/s u = 5 m/s.5.5 Frequecy rati..5. Frequecy rati Dimesiless maximum amplitude Fig.. Effects f budary cditis the liear amplitude-frequecy respse curves f the atube Dimesiless maximum amplitude Fig.. Effects f fluid-flw velcity the liear amplitude-frequecy respse curves f CNT Fig. shws effects f fluid-flw velcity the liear amplitude-frequecy respse curves f pipe. It is bserved that with icrease f the legth r by extesi, the aspect rati f the pipe, the liear vibrati frequecies f the structure decreases while liear vibrati frequecies icreases with the icrease i the fluidflw velcity. 8 Deflecti, w(x,t) (x - m) Time (sec) Fig.. idpit deflecti time histry fr the liear aalysis f SWCBT whe K=. ad U= m/s Jural f Applied ad Cmputatial echaics, Vl., N., (7), -79

16 Therm-mechaical liear vibrati aalysis f fluid-cveyig structures 75 Fig. illustrates the midpit deflecti time histry fr the liear aalysis f SWCBT whe K =. ad U= m/s while Fig. presets the midpit deflecti time histry fr the liear aalysis f SWCBT whe K =. ad U= 5 m/s. Als, Fig. depicts the midpit deflecti time histry fr the liear aalysis f SWCBT whe K =. ad U= 5 m/s. Cmparig Fig. ad,the stretchig effects the dyamic behaviurs f the SWCNT is depicted. It is depicted that icrease i the slip parameter leads t decrease i the frequecy f vibrati ad by extesi, decrease i the critical velcity. 8 Deflecti, w(x,t) (x - m) Time (sec) Fig.. idpit deflecti time histry fr the liear aalysis f SWCBT whe K=. ad U= 5 m/s 8 Deflecti, w(x,t) (x - m) Time (sec) Fig.. idpit deflecti time histry fr the liear aalysis f SWCBT whe K=. ad U= 5 m/s Fig. 5 shws the cmparis f the liear vibrati with liear vibrati f the SWCNT. It culd be see i the figure that the discrepacy betwee the liear ad liear amplitudes icreases with icremet f the maximum vibrati. The liear effects shw the stretchig effects. As stretchig effect icreases, the stiffess f the system icreases which csequetly, icreases i the atural frequecy ad the critical fluid velcity. 8 iear vibrati N-liear vibrati Deflecti, w(x,t) (x - m) Time (sec) Fig. 5. Cmparis f midpit deflecti time histry fr the liear ad liear aalysis f CBT whe K=. ad U= 5 m/s Jural f Applied ad Cmputatial echaics, Vl., N., (7), -79

17 7. G. Sbamw et. al., Vl., N., Deflecti, w(x,t) (x - m) - - Deflecti, w(x,t) (x - m) Time (sec) Time (sec) Fig.. Effects f slip parameter the deflecti f the liear vibrati (a) K=. (b) K =.5 Effects f slip parameter, Kudse umber the vibrati f the atube is shw i Fig.. It is depicted that icrease i the slip parameter leads t decrease i the frequecy f vibrati ad by extesi, decrease i the critical velcity. Dimesiless Frequecy,(Imagiary, mde ) 8 - e a/ =. e a/ =.5 e a/ = Dimesiless flw velcity (a) Dimesiless Frequecy(Imagiary, mde ad mde ) 8 - (e a/) mde =. (e a/) mde =.5 (e a/) mde =. (e a/) mde =. (e a/) mde =.5 (e a/) mde = Dimesiless flw velcity (b) Dimesiless Frequecy (Real, mde ) e a/ =. e a/ =.5 e a/ = Dimesiless flw velcity Jural f Applied ad Cmputatial echaics, Vl., N., (7), -79 Dimesiless Frequecy (Real, mde ad mde ) 5 (e a/) mde =. (e a/) mde =.5 (e a/) mde =. (e a/) mde =. (e a/) mde =.5 (e a/) mde = Dimesiless flw velcity (c) (d) Fig. 7. a-d Effects f lcal parameter ad fluid flw velcity the atural frequecy f the liear vibrati I rder t ivestigate the vibratial ad stability behavirs f the structure, effects f fluid flw velcity the atural frequecies are carried ut. As the flw velcity is icreased frm zer, the atube becmes mre flexible ad the fudametal frequecies decreases as shw i Fig. 7 a-d. The atural frequecy becmes zer whe the flw velcity f the cveyed fluid exceeds a certai value which crrespds t the critical flw velcity. Als, the Figures depict the critical speeds crrespdig t the divergece cditi fr differet values f the system s parameters. It shuld be pited ut that as the ttal mass f the system which cmprised f the masses f fluid ad CNT icreases, the frequecy f vibrati decreases. Whe the system becmes stiffer (the stiffess f the system due t bedig ad shears rigidities whse values deped the secd mmet f iertia ad the crss-sectial areas f the CNC ad the fluid), the atube vibrates at higher frequecies. Fr a CNT with certai characteristics, the term, m fu is cstat alg the atube s legth ad the system s flexibility decreases as the flw velcity becmes greater. The real ad imagiary parts f the eigevalues related t the tw lwest mdes with differet

18 Therm-mechaical liear vibrati aalysis f fluid-cveyig structures 77 atube parameters are pltted fr a rage f dimesiless velcity, -. It is fud that the frequecy f the first mde becmes zer i viciity f dimesiless velcity.8 while fr the secd mde, i the viciity f.. Whe the flw velcity reaches the critical velcity, bth the real ad imagiary parts f the flexural frequecy are equal t zer fr the first mde, which crrespds t a pit f buded eutral stability. Whe the velcity f the fluid is greater tha the critical velcity, the vibrati frequecy is zer. Althugh, the effect f the lcal parameter flexural frequecy is t very sigificat, it is csidered i this study because the small scale ca affect the critical flw velcity, as shw i Fig. 7. It culd be iferred that by icreasig the lcal parameter, the atural frequecy ad the critical velcity decreased. Fig. 8a-b shws the effects f temperature chage the frequecies f the CNT. Frm the figure, as the temperature chage icreases, the atural frequecies ad the critical flw velcity f the structure icrease. 5 Chage i Temp. = K Chage i Temp. = 5K Chage i Temp. = 5K 5 Chage i Temp. = K Chage i Temp. = 5K Chage i Temp. = 55K Dimesiless Frequecy Dimesiless Frequecy Dimesiless flw velcity Dimesiless flw velcity (a) (b) Fig. 8. a-b Effects f temperature ad fluid flw velcity the atural frequecy f the liear vibrati at lw/rm temperature.8.. u t. -. A = u Fig. 9. Phase-space curve f u t versus u Fig. 9 shws a phase-space/plae curve f u t versus u which shws the behaviur f the scillatr. The phase plts shw the behaviur f the scillatr whe the amplitude is varied. It is peridic with ceter (, ) where stability cditis ca be bserved. This situati is cmm i ufrced, udamped cubic Duffig scillatrs. 7. Cclusi I this paper, uified aalytical slutis are develped fr the liear equatis arisig i flw-iduced vibratis thrugh pipes, micr-pipes ad atubes usig Galerki-Newt-Balacig ethd (GNB). The results shwed that the alterati f liear flw-iduced frequecy frm liear frequecy is tied t icreasig amplitude f scillatis, flw velcity, ad thermal parameters. It ca be ccluded that as a startig pit fr uderstadig the relatiship betwee the physical quatities i these prblems, the develped aalytical slutis ca prvide ctiuus physical isights it the prblems tha pure umerical r cmputatial methds ad ca als serve as a methdt verify the umerical r apprximate aalytical slutis. Jural f Applied ad Cmputatial echaics, Vl., N., (7), -79

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