THEORETICAL STUDY ON PIPE OF TAPERED THICKNESS WITH AN INTERNAL FLOW TO ESTIMATE NATURAL FREQUENCY
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1 Inernaonal Journal of Mechancal Engneerng and Technology (IJMET) Volue 7, Issue, March-Arl 6,., Arcle ID: IJMET_7 Avalable onlne a h:// Journal Iac Facor (6): 9.86 (Calculaed by GISI) ISSN Prn: and ISSN Onlne: IAEME Publcaon THEOETICAL STUDY ON PIPE OF TAPEED THICKNESS WITH AN INTENAL FLOW TO ESTIMATE NATUAL FEQUENCY Nawal H. Al ahey College of Engneerng / Mechancal Dearen, Babylon Unversy / Babl-Iraq ABSTACT Ths research sudy he effec of aered hckness on he free ransverse vbraon of claed free e whch have unfor crcular cross secon conveyng waer by usng aghly z ehod n he wo case, he frs nvolves he e have a consan wall hckness ( ) a claed end equal o ( & ) whle he hckness ( ) a free end changes accordng o he rao ( / =.,.,.6,.8, ). In he second case he hckness a free end ( ) s consan ( & ) whereas he hckness a claed end ( ) changes a rao ( / =.,.,.6,.8, ). The e has a consan nner radus ( ) of ( c or c) and dfferen values of lengh ( & ). Ths sudy shows n he s case he crcal velocy (V c ) of he flud can be decreased a he ncrease n lengh of he e a he sae values of & bu he value of crcal velocy s ncreased wh ncreasng, and he hckness rao ( / ) a he sae lengh of he e. In addon a absence he flow of waer he naural frequency of syse s decreased wh he ncrease n he rao of hckness ( / ) and lengh of e, whereas and are decreased. In he nd case he dynac behavors of he syse a he sae ha n he s case exce ha he naural frequency ncrease wh ncreasng he hckness rao /. A any foraon of he e for unfor secon he naural frequency decreased when he velocy of waer ncreased fro zero o crcal velocy. esuls are coared wh hose avalable n leraure and are found o be n excellen agreeen. Key words: Canlever Pe, Inernal Flow, Taered Thckness Ce hs Arcle: Nawal H. Al ahey, Theorecal Sudy on Pe of Taered Thckness wh an Inernal Flow o Esae Naural Frequency Inernaonal Journal of Mechancal Engneerng and Technology, 7(), 6,.. h:// h:// edor@aee.co
2 Nawal H. Al ahey. INTODUCTION The vbraons resulng fro flud flow causng nosy robles occur n a wde range n ndusral feld fro cvl engneerng, checal rocessng, aerosace and arne srucures. Nabeel and e.al. [], flud flow and he e lne wll be an neracve syse dynacs where couled by he force of flud exered on he e hs force causes he deforaon e hus change he drecon he flow also change flud force. Chol [] nvesgaed he naural frequences of ng syse under effec flud velocy and corols force. I s obaned ha a ceran crcal veloces causng bucklng ye nsably for dfferen boundary condons. Alaa [] suded he effec of he flud flow hrough a e wh resrcon affec he dynac behavor on he vbraon of syse. Wang [] nvesgaed he sac and dynac behavor es conveyng flud aheacally by usng fne dfference ehod. Shnaro [] nvesgaed exerenally he vbraon of hangng ube conveyng flud wh varyng he lengh of he ube. Marjonas [6] nvesgaed flow nduced vbraon n roaon e conveyng flud n hyohess ha he flud s ncoressble and n vscd by usng non lnear equaons of oon whch s derved by fne eleens ehod. Kuer [7] gave analycal roof of sably of e ransed flud n claed nned by a low seed by usng a lug flow odel afer consderaon a ensoned Euler Bernoull bea n arrangeen. Muhsn [8] suded he effec of boundary condons of es on he naural frequency of he syse conveyng flud a dfferen daeer, lengh, e aerals and velocy of flud by usng bea heory. Ivan [9] nvesgaed he flow nduced vbraon a unfor and aered hckness n dfferen boundary condons (claed claed & ned ned ) by usng fne eleens ehod. A. Marzan and e.al. [] used Wnkler ye elasc foundaon o sudy s effec on he sably e flud conveyng flud a ransverse oon o deerne he fluer velocy. Al [] suded he dynac anners of a e ranssson flud a lanar flow akng no consderaon general boundary condons as colan aeral wh lnear and roaonal srngs. Shankarachar and e.al. [] nvesgaed he dynac behavor of e conveyng flud he frequency equaons s derved for classcal boundary condons where he frequency of syse decreased wh ncreasng he velocy of flow. In hs aer, can be obaned he frequency by usng aroxae for whch reresened by aylegh z ehod of canlever e wh an nernal flow whch have aered hckness n he wo cases whch have dfferen rao beween hckness a claed and free end, esaed he naural frequency of vbraons a dfferen values of nner radus, he hckness a claed and free end, dfferen values of velocy flow of waer and dfferen values of lengh.. THEOETICAL ANALYSIS Fgures () show he unfor cross secon of claed free e a aered hckness of lengh L, nner radus, he hckness a claed end, and a free end can be derved h:// 6 edor@aee.co
3 Theorecal Sudy on Pe of Taered Thckness wh An Inernal Flow To Esae Naural Frequency Fg (-a) Canlever e of aered Thckness / Fro Fg.(-a) :- ( x ) / (L-x) = ( ) /L Fro fg (-b):-( x ) / x = ( ) /L Afer slfy above relaons yelds: Fg (-b) Canlever e of aered Thckness / (-a) (-b) x = (-x/l )+ (x/l) () In aered hckness of e a lengh of ar of e (x), A(x) = π ( where xo = ( + x ), herefore (x) = ρ * A(x), and xo xo ), herefore I x) 6 ) = π x, I(x) = π/( ( x x x x. Now he rocedure of ayegh-z o s aled derve he naural frequency for ransverse oon of aered cross secon of canlever e. Le us use he sle wo er aroxaon Benoray []. Y r c y x) c y ( ) () ( x x x c Y r c () L L By usng above equaons he values of j and k j can be esaed Benoraya [] : j j L L ( x) y y dx () '' '' ( x) y y j j k E I (6) Afer negraon equaon () accordng o e where he e s ey fro flud can be yelded:- = π L [ / + /6], = π L [ / + /7] = π L [ /6 + /8], =. (-a) h:// 7 edor@aee.co
4 Nawal H. Al ahey h:// 8 edor@aee.co f (x) = f *A f (x) herefore f (x) = π f Now afer usng equaon () and negraon yelds: { f = f * L/, f = f *L/6 = f, = f * L/7 } (-b) Now he eloyen sueroson beween equaons (-a) and equaons (-b) wll be obaned: f, f, f, f (7) Afer negraon of equaon () he followng relaons whch reresen he sffness of e as follows:- L E k (8) L E k (9) k k (9-a) L E k () Now can be wre he oher relaons of ass and sffness n he arx for as follow : k k n n k k n n c c = () or n general arx noaon as : c M K n () The evaluaon of hs deernan rovdes an esaon of he wo fundaenal naural frequences ω and ω for he e carryng flud whch s no oved. In order o colee he naural frequency of e when he flud oves a any velocy,
5 Theorecal Sudy on Pe of Taered Thckness wh An Inernal Flow To Esae Naural Frequency frsly he crcal velocy of flow should be deerned for unfor canlever e fro he flowng equaon Ivan [9], V c =.87 L E I / () f A f Thus he naural frequency (ω) of a e a any velocy of flud can be found fro he followng equaon: n V V f c Blvens [] () Thus V f s reresened he velocy of he flow.. ESULTS AND DISCCUSION The consrucon of es effec on he raccal alcaon subsequenly effec on qualy erforance. Table () shows coarson of he naural frequency of he frs ode for ransverse free vbraons of e coarson of he naural frequency of he frs ode n he dfferen values of velocy of flow fro V f = o V f = V c where hose are coared beween he aylegh-z ehod n he resen work and he fne eleen ehod (FEM) n Ivan [9] a claed-free boundary for unfor e, D=., =., L=. The resuls based on he an roeres of aeral E=7 Ga, ρ=8 kg/. Fgures ( &) show ha he frs ode of vbraon of aered hckness n absence flow (V f =) as a funcon of he rao ( / ) oban for he aylegh z ehod for varaon values of nner radus ( ), he lengh of e (L) and hckness a claed end ( ). I s clearly seen ha he naural frequency ncreased wh he ncreased n hckness ( ) and he nner radus ( ), Ths anners llusraed he sran energy of srucure ncreased wh ncrease n he hckness and he radus herefore ha s caused ncreased he sffness of syse. In he sae fgures he naural frequency decreased wh ncrease n he raon of hckness ( / ) and he lengh (L) ha s causes ncreasng n he ass for he e and he waer whch caused an ncrease n he knec energy of he srucure. In he fgures ( o 9) he naural frequency as a funcon wh he velocy of flow (V f ) of waer for of e a dfferen values of ( / ),, &L where he naural frequency decreased wh ncreased n he velocy of flow. Fgures ( & ) show ha he frs ode of vbraon of aered hckness n absence flow (V f =) as a funcon of he rao ( / ) for varaon values of nner radus ( ), he lengh of e (L) and hckness a free end ( ). Can be seen ha he naural frequency ncreased wh he ncrease n hckness ( ), he nner radus ( ) and rao of hckness ( / ).Ths behavor llusraed he sran energy of srucure ncreased wh ncrease n he hckness and he radus. In he sae fgures he naural frequency decreased wh ncreased n he lengh of e as lke n he above case. Fgures ( o 9) show he naural frequency also decrease wh ncreased he velocy of flow of all dfferen srucures of e because of velocy of flow ose ressure on he wall and caused deforaon of he e herefore caused decrease n he elascy of e and he naural frequency of he syse. h:// 9 edor@aee.co
6 Naural frequency wn (rad/sec) Nawal H. Al ahey Table Naural frequency (rad/sec) of ransverse vbraons of e n dfferen value of velocy of flow Velocy V f (/s)..m. F.E.M. Dfference δ % % % % % % % V c =9.87 δ= [(-z ehod FEM ehod)/ -z ehod] *% 8 s. ode, =, Vf= L=, =. L=, =. L=, =. L=, = Thckness rao (/) Fgure Naural frequency for s ode as a funcon of hckness rao (/) n dfferen values of radus & lengh, absence flow and =. h:// edor@aee.co
7 Naural frequency wn (rad/sec) Naural frequency wn (rad/sec) Theorecal Sudy on Pe of Taered Thckness wh An Inernal Flow To Esae Naural Frequency 7 s. ode, =, Vf= L=, =. L=, =. L=, =. L=, = Thckness rao (/) Fgure Naural frequency for s ode as a funcon of hckness rao ( / ) n dfferen values of radus & lengh, absence flow and =. 8 6 L=, =., = /=. /=. /=.6 /=.8 /= Velocy of flow Vf (/sec) Fgure Naural frequency for s ode as a funcon of velocy of flow V f n dfferen values of hckness rao ( / ) a one eer lengh, radus =. and =. h:// edor@aee.co
8 Naural frequency wn (rad/sec) Naural frequency wn (rad/sec) Nawal H. Al ahey 8 6 L=, =., = /=. /=. /=.6 /=.8 /= Velocy of flwo Vf (/sec) Fgure Naural frequency for s ode as a funcon of velocy of flow V f n dfferen values of hckness rao ( / ) a one eer lengh, radus =. and =. L=, =., = /=. /=. /=.6 /.8 /= Velocy of flow Vf (/sec) Fgure 6 Naural frequency for s ode as a funcon of velocy of flow V f n dfferen values of hckness rao ( / ) a wo eer lengh, radus =. and =. h:// edor@aee.co
9 Naural frequency wn (rad/sec) Naural frequency wn (rad/sec) Theorecal Sudy on Pe of Taered Thckness wh An Inernal Flow To Esae Naural Frequency 6 L=, =., = /=. /=. /=.6 /=.8 /= 6 7 Velocy of flow Vf (/sec) Fgure 7 Naural frequency for s ode as a funcon of velocy of flow V f n dfferen values of hckness rao ( / ) a wo eer lengh, radus =. and =. L=, =., = /=. /=. /=.6 /=.8 /= Velocy of flow Vf (/sec) Fgure 8 Naural frequency for s ode as a funcon of velocy of flow V f n dfferen values of hckness rao ( / ) a wo eer lengh, radus =. and =. h:// edor@aee.co
10 Naural frequency wn (rad/sec) Naural frequency wn (rad/sec) Nawal H. Al ahey 7 6 L=, =., = /=. /=. /=.6 /=.8 /= Velocy of flow Vf (/sec) Fgure 9 Naural frequency for s ode as a funcon of velocy of flow V f n dfferen values of hckness rao ( / ) a wo eer lengh, radus =. and =. 8 s. ode, =, Vf= L=, =. L=, =. L=, =. L=,= Thckness rao (/) Fgure Naural frequency for s ode as a funcon of hckness rao ( / ) n dfferen values of radus & lengh, absence flow and =. h:// edor@aee.co
11 Naural frequency wn (rad/sec) Naural frequency wn (rad/sec) Theorecal Sudy on Pe of Taered Thckness wh An Inernal Flow To Esae Naural Frequency 7 s. ode, =, Vf= L=, =. L=, =. L=, =. L=, = Thckness rao (/) Fgure Naural frequency for s ode as a funcon of hckness rao ( / ) n dfferen values of radus & lengh, absence flow and =. L=, =., = /=. 8 /=. /=.6 /=.8 /= Velocy of flow Vf (/sec) Fgure Naural frequency for s ode as a funcon of velocy of flow V f n dfferen values of hckness rao ( / ) a one eer lengh, radus =. and =. h:// edor@aee.co
12 Naural frequency wn (rad/sec) Naural frequency wn (rad/sec) Nawal H. Al ahey 8 6 L=, =., = /=. /=. /=.6 /=.8 /= Velocy of floww Vf (rad/sec) Fgure Naural frequency for s ode as a funcon of velocy of flow V f n dfferen values of hckness rao ( / ) a one eer lengh, radus =. and =. 8 6 L=, =., = /=. /=. /=.6 /=.8 /= Velocy of flow Vf (/sec) Fgure Naural frequency for s ode as a funcon of velocy of flow V f n dfferen values of hckness rao ( / ) a one eer lengh, radus =. and =. h:// 6 edor@aee.co
13 Naural frequency wn (rad/sec) Naural frequency wn (rad/sec) Theorecal Sudy on Pe of Taered Thckness wh An Inernal Flow To Esae Naural Frequency 7 L=, =., = /=. /=. /=.6 /=.8 /= Velocy of flow Vf (/sec) Fgure Naural frequency for s ode as a funcon of velocy of flow V f n dfferen values of hckness rao ( / ) a one eer lengh, radus =. and =. L=, =., = /=. /=. /=.6 /=.8 /= Velocy of floww Vf(/sec) Fgure 6 Naural frequency for s ode as a funcon of velocy of flow V f n dfferen values of hckness rao ( / ) a wo eer lengh, radus =. and =. h:// 7 edor@aee.co
14 Naural frequencu wn (rad/sec) Naural fequency wn (rad/sec) Nawal H. Al ahey L=, =., = /=. /=. /=.6 /=.8 /= 6 7 Velocy of floww Vf (/sec) Fgure 7 Naural frequency for s ode as a funcon of velocy of flow V f n dfferen values of hckness rao ( / ) a wo eer lengh, radus =. and =. L=, =., = /=. /=. /=.6 /=.8 /= 6 7 Velocy of flow Vf (/sec) Fgure 8 Naural frequency for s ode as a funcon of velocy of flow V f n dfferen values of hckness rao ( / ) a wo eer lengh, radus =. and =. h:// 8 edor@aee.co
15 Naural frequency wn (rad/sec) Theorecal Sudy on Pe of Taered Thckness wh An Inernal Flow To Esae Naural Frequency 7 L=, =., = 6 /=. /=. /=.6 /=.8 /= Velocy of flow Vf (/sec) Fgure 9 Naural frequency for s ode as a funcon of velocy of flow V f n dfferen values of hckness rao ( / ) a w eer lengh, radus =. and =.. CONCLUSION The flowng conclusons can be deduced fro he resuls of he resen sudy, he naural frequency of es conveyng flow of flud a hckness rao / decrease wh ncreased he rao of hckness agans ha es whch hckness rao / where he naural frequency ncreased wh ncreasng he rao of hckness. In he oher hand he ncreasng of nner radus of he syse wll rse he naural frequency bu he ncreasng he lengh of he e caused reduced he naural frequency also ncreasng he velocy of flow caused decreasng he frequency of he syse. LIST OF SYMBOLS A Cross secon area a of e claed end ( ). A Cross secon area of e a free end ( ) A(x) Cross secon area of e a ar of lengh (x) ( ) A f Cross secon area of flud ( ) c & c Consans E Modulus of elascy (N/ ) L Lengh of he e () I Second oen of area ( ) I(x) Second oen of area a ar of lengh(x) ( ) f Mass of flud er un lengh (kg/) (x) Mass of e er ar of lengh x (kg/) Thckness of e a claed end () Thckness of e a free end () x Thckness of e a any ar of lengh of e Inner radus of e (). o Ouer radus of e a claed end () h:// 9 edor@aee.co
16 Nawal H. Al ahey o xo V f V c x Y r Ouer radus of e a free end () Ouer radus of e a ar of lengh x Velocy of flud (/sec) Crcal velocy of flud flows n he e (/sec). Lengh of ar of e (). Dslaceen (alude of e () ρ Mass densy of e aeral (kg/ ) ρ f Mass densy of flud n he e (waer) (kg/ ) ω Naural frequency of e a velocy of flow V f (rad/sec) ω n Fundaenal naural frequency of e n absence of flow (rad/sec) EFENCESES [] Nabeel K. Abd Al-Sahb a, Adnan N. Jaeel b, Osaah F. Abdulaeef a*, Invesgaon no he Vbraon Characerscs and Sably of a Welded Pe Conveyng Flud, J. (JJMIE), (),. [] Chol H. & Song H.,Ou of lane vbraons of angled es conveyng flud, Journal of he Korea Socey, (), 99, 6-6. [] Alaa A.M.H., The effec of nduced vbraon on a e wh a resrcon conveyng flud, Ph.D. heses, Unversy of echnology,. [] Wang & Bloo, Sably ssues of concenrc es conanng seady and ulsale flows, J.F. and srucure,. [] Shnaro & Masak, Sably and bfurcaon's of ube conveyng flow, graduae school of scence, unversy of Tokyo, Jaan,. [6] Marjonas Bogdevčus, Nonlnear dynac analyss of roang e conveyng flud by he fne eleens ehod, J. Transor, 8, (),, -8. [7] G.L. Kuer & eal, On sably of a claed-nned e conveyng flud, Faculy of cvl engneerng and geoscences, delf unversy of echnology, delf, The Neherlands HEON,9(), (). [8] Mohsn J. Jwege & Zahd I. Mohaed, Vbraon characerscs of dfferen cross secon es wh dfferen end condons, Journal of Eng, & Tech., 8(8),, 6-6. [9] Ivan Gran, "Flow nduced vbraons n es, a fne eleen aroach" Cleveland sae unversy, May,. [] A. Marzan a, M. Mazzo a, E. Vola a, P. Vor b & I. Elshakoff b*,"fem Forulaon for Dynac Insably of Flud-Conveyng Pe on Non unfor Elasc Foundaon", J. of Mechancs Based Desgn of Srucures and Machnes, Vol., Issue,. 8-9,. [] Al H. AI-Hll & Thaer J. Nayesh, "Free vbraon characerscs of elascally suored e conveyng flud", Journal (NUCEJ), Vol. 6, No.,. 9-9,. [] Shankarachar M Suar, M. adhakrshna, P. aesh Babu," Flud Induced Png Vbraon wh Elascally esraned Dfferen End Suors", Journal (IJEET), Vol., Issue,. [] Ercan Serf Kaya, Takuro Kaayaa and Toshaka Yaao, Sesc Characerscs of he Folded Canlever Shear Srucure. Inernaonal Journal of Cvl Engneerng and Technology, (),, [] Benoraya, Benaroy "Mechancal Vbraon", Prence Hll, Inc., U.S.A., 998 [] Blevnes, "Flow nduced vbraon" Kreger ublshng coany, Malabar Florda, nd edon,. h:// edor@aee.co
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