Effect of Design Parameters and Support Conditions on Natural Frequency of Pipe Excited by a Turbulent Internal Flow

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1 Nube 7 olue 9 July Jounal of Engneeng Effec of Desgn aaees and Suppo Condons on Naual Fequency of pe Eced by a Tubulen Inenal Flow of. D. Najda Nasha Abdulla of. D. Adnan Naj Jael Wjdan Kadhe Saheb College of Engneeng nvesy of Baghdad College of engneeng nvesy of Baghdad College of engneeng nvesy of Baghdad E-al: Najda_abdulla@yahoo.co.uk E-al: adnanaj@yahoo.co E-al: Wjdan98@yahoo.co ABSTACT In hs sudy, he ec of desgn paaees such as ppe daee, ppe wall hckness, ppe aeal and he ec of flud velocy on he naual fequency of flud-sucue neacon n sagh ppe conveyng fully developed ubulen flow wee nvesgae nuecally, analycally and epeenally. Also he ec of suppo condons, sply-sply and claped-claped was nvesgaed. Epeenally, ppe vbaons wee chaacezed by acceleoee ouned on he ppe wall. The naual fequences of vbaon wee analyzed by usng Fas Foue Tansfoe (FFT. Fve es secons of wo dffeen ppe daees of 76. and 5.8 wh wo ppe hcknesses of.7 and. and wo ppe aeals, sanless seel and polyvnyl chlode C n he ange of eynolds nubes fo * o 5* 5 wee suded. Maheacally, he govenng connuy and oenu equaons wee solved nuecally by usng he fne volue o copue he chaacescs of wo densonal ubulen flow. The dynacs of a ppe conveyng flud was descbed by he Tansfe Ma Mehod (TMM whch s povdes a nuecal echnque fo solvng he equaons of ppe vbaons fo sply-sply and claped suppos. The esuls showed ha, he naual fequences ncease wh ppe daee ncease and he naual fequences slghly nceases wh ppe wall hckness ncease. Also, he naual fequences n claped-claped suppoed ppe ae hghe han hose n sply-sply suppoed ppe. KEWODS: pe conveyng flud; Flow-nduced vbaon; Flud sucue neacon. دراسة تا ثيرالمتغيرات التصميمية وشروط الا سناد على التردد الطبيعي لا نبوب يثار داخلي اضط اربي ا.د.نجدت نشا ت عبد الله كلية الهندسة/جامعة بغداد الخلاصة ا.د.عدنان ناجي جميل كلية الهندسة/جامعة بغداد وجدان كاظم صاحب كلية الهندسة/جامعة بغداد بجريان في هذا العمل أجريت د ارسة تجريبية وعددية وتحليلية لتا ثير المتغي ارت التصميمه مثل قطر الا نبوب سمك الجدار ماد ة الانبوب وتا ثير سرعة الماي ع على التردد الطبيعي لا نبوب مستقيم يتقل جريان اضط اربي تام التطور. أيضا تم د ارسة تا ثير نوع المساند (البسيط والمثبت. أنج ازلجانب العملي بقياس إهت ازز الا نبو ب بواسطة جهاز قياس التعجيل الذي ثبت على جدار الا نبوب. وجدت الترددات الطبيعية للا هت ازز با ستعمال محول فورير السريع. إستعملت خمسة مقاطع إختبار مختلفة الاقطار( 76. و 5.8 للصدأ سمك جدار الا نبوب.7 و. و من مادتين مختلفتين هما حديد مقاوم وكلوريد البوليفينل تمت هذه الد ارسة لقيم مختلفه لعدد رينولدز تت اروح بين (* 5-* 5.رياضيا تم حل معادلة الاستم اررية و معادلات الزخم ببناء نموذج عددي با ستخدام طريقة الحج وم المحددة ل حساب خصاي ص الجريان الا ضط اربي 96

2 of. D. Najda Nasha Abdulla of. D. Adnan Naj Jael Wjdan Kadhe Saheb Effec of Desgn aaees and Suppo Condons on Naual Fequency of pe Eced by a Tubulen Inenal Flow ثناي ي الابعاد. قد تم بناء نموذج رياضي اعتمد على استخدام طريقة المصفوفات الانتقالية لحل معادلات إهت ازز الا نبوب لتوضيح تا ثير الاهت ازز لا نبوب يحمل ماي ع.اظهرت النتاي ج با ن للترددات الطبيعية تزيد بزيادة قطر الا نبوب والترددات الطبيعية تزداد بنسبه بسيطة بزيادة سمك جدار الا نبوب. بينت النتاي ج ايضا الترددات الطبيعية في المسند المثبت أعلى من تلك للمسند البسيط. كلمات البحث: أنبوب ينقل ماي ع الا هت ازز المستحث بواسطة الجريان تداخل الهيكل و الماي ع. -INTODCTION png syses play a vey poan ole n vaous ndusal applcaons. They ae used n any engneeng applcaons fo conveyng gases and fluds ove a wde ange of epeaues and pessues. These applcaons nclude hydaulcs, flud ansfe, coolng wae and fuel supply. Flow-nduced vbaon of a ppe conveyng flud s a consequence of flud flow hough he ppes, and s wdely ecognzed as a ajo concen n he desgn of any ndusal applcaons. Flow nduced vbaon s caused n sucues by focng due o e vaan pessue acng on he suface of he sucue. Many eseaches have been caed ou sudes on he vbaon of a ppe conveyng flud. The naual fequences and ccal veloces of lanaed ccula cylndcal shells wh fed-fed ends conveyng flud was suded by [Chang and Chou 995], dynac chaacescs equaons wee obaned unde he assupon of haonc oon, and he naual fequences coespondng o each flow velocy wee found. [See Seo e al. 5] suded he foced vbaon esponse of a ppe conveyng haoncally pulsang flud by usng he fne eleen. The dapng and sffness aces vaed wh e, and he pedced he seady sae esponse of he ppe whou usng he e daa of he esponse. [ousf e al. ] also nvesgaed ppes conveyng pulsang flowng flud. Bolon s was eployed o spl he boundaes fo he sable egons. The esulng coupled-odnay dffeenal equaons wee decoupled by neglecng he ec of he ass ao e and solved analycally. [Zhang and Chen ] nvesgaed nonlnea vbaon of ppes conveyng flud n he supeccal ege. They focused on he nonlnea vbaon aound each bfucaed equlbu. [Wang e al. ] eaned he dynacs of sply suppoed flud-conveyng ppes wh geoec pefecons by consdeng he negal paal dffeenal equaon of oon. [-Mn e al. ] solved analycally he lnea dynacs of flud-sucue neacon n a ppelne conveyng flud by usng he eleen- Galekn o evaluae he naual fequency of ppe conveyng flud a dffeen bounday condons. The ec of he nduced vbaon of a sply suppoed ppe conveyng flud wh a escon was heoecally and epeenally nvesgaed by [Mahd ]. A ansfe a was pleened o descbe he dynac esponse of a ppe conveyng flud and a nuecal echnque was used fo solvng wo-densonal ncopessble seady vscous flow fo he ange of eynolds nube(5<e<. [Mousa ] nvesgaed he fee vbaon behavo of sepped ohoopc cylndcal shells, by usng he cobnaon of Flügge s shell heoy, he ansfe a appoach and he obeg negaon. The sudy of flow nduced vbaon daws on hee dscplnes: ( flud dynacs, ( echancal vbaon, and ( sucual echancs. An poan feaue s he equeen o deal wh neacons beween flud oon and ovng sucues. Ths objecve can be obaned by beakng he poble no wo secons: flud and sucue. The pessue dop along he ppe and velocy wll obaned fo he flud odel and wll be poed o a sucual odel. The second pa deals wh sucue vbaon due o flowng flud by usng a ansfe a (TMM. -FID DNAMIC MODE Fo he wo densonal sulaon he govenng equaons fo asyec and ncopessble flud wee obaned by wng 97

3 Jounal of Engneeng olue 9 July Nube 7 98 boh he connuy and he oenu equaons n cylndcal coodnaes, hen dscadng he devaes wh espec o he cylndcal coodnae. The anspo equaons wh k-ε odel have he followng fo: [Wang and Zhang 5] Connuy equaon: ( ( ( Whee s aal velocy, and s noal velocy. Moenu equaon n aal decon: ( ( ( Moenu equaon n adal decon: ( ( ( Equaon fo knec enegy of ubulence: ( ( k k k k k k ( ( σ σ ε k ( Equaon fo dsspaon ae of knec enegy of ubulence: ( ( ε σ ε σ ε ε ε ε ( ( ( ε ε C C k k (5 The dsspaon ae, ε of he enegy s wen as: k / ε (6 Whee k s he knec enegy of he flow and s he nvolved lengh scale. Ths s hen elaed o he ubulen vscosy μ based on he andl ng lengh odel, [Tennekes, and uley 97], ε k C (7 Whee C s an epcal consan and s he densy of he flud. And s he ecve vscosy defned as: ubulen olecula ecve (8 The poducon of knec enegy of ubulence k s gven by : k (9 Based on an eensve eanaon of a wde ange of ubulen flows, he consan paaees used n he equaons wll ake he followng values: C μ.9; C.; C.9; σ k. and σ ε. The fne volue s pobably he os popula used fo nuecal dscezaon n CFD. The govenng equaons

4 of. D. Najda Nasha Abdulla of. D. Adnan Naj Jael Wjdan Kadhe Saheb Effec of Desgn aaees and Suppo Condons on Naual Fequency of pe Eced by a Tubulen Inenal Flow n he dffeenal fos ae negaed ove each conol volue. The esulng negal consevaon laws ae eacly sasfed fo each conol volue and fo he ene doan, whch s a dsnc advanage of he fne volue. Each negal e s hen conveed no a dscee fo, hus yeldng dscezed equaons a he cenods, o nodal pons, of he conol volues. In he case of calculaon of velocy coponens, dffeen conol volues fo he ones used fo he calculaon of ohe vaables (e.g. pessue, ubulence knec enegy and s dsspaon ae wll be used. Fo eaple, he velocy coponen n he -decon,, s calculaed a he faces ha ae noal o he decon, and he conol volue fo s dsplaced one half conol volue fo he an conol volue n -decon fgue ( and he conol volue fo dsplaced one half conol volue fo he an one n adal decon fgue (. The dffuson and advecon volue negals can be conveed no suface negals ove he suface S of he conol volue usng he Gauss Dvegence Theoe, hus unds s φ Γφ nds s Sφd v ( Whee n s he coponen of he ouwad noal suface veco. Ths equaon conans hee es whch need o be dscesed: a dffuson e, a convecon e, and a souce e. The oenu equaon also conans a pessue e whch does no sasfy a anspo equaon. In he followng, a descpon of he dscesaon of hese fou es n -D s gven. Dffuson e Dffuson es ae usually dscesed usng cenal dffeencng, as follows φ ΓφAe ΓφAw φ ( E ( W Γ nds φ φ φ φ ( e ( w s δ δ ΓφAn ΓφAs ( φn φ ( φ φs ( δ n ( δ s De( φe φ Dw( φ φw Dn( φn φ Ds( φ φs ( De Γφ Ae Dw e n whch (δ w, (δ, ec. and whee δ s he dsance beween he especve node cene and, and A s he suface aea of he especve cell face. Convecon e The convecon e n equaon ( s negaed as he su of flues ove he fou faces suoundng he conol volue, hus u s nds Souce e u φe u φw u Γφ φn A w u φs ( The souce e s dscesed as follows, S d s φ Sφ ( φ Whee S s he aveage value of he souce S Ф houghou he conol volue. When φ S Ф s a funcon of Ф, S s decoposed no a soluon-ndependen pa, b, and a soluondependen pa, hus S,S ( φ b Sφ - TANSFE MATIX METHOD Ths analycal and he nuecal analyss of he pesen wok ncludes he analyss of sucual dynacs of he ppe by Tansfe Ma Mehod (TMM o fnd he naual fequency, ode shapes and esponse of he syse. 99

5 Nube 7 olue 9 July Jounal of Engneeng To descbe he suaon a each node, fou quanes us be defned; he deflecon (, he slope (Ф, he oen (M, and he shea foces (. Also, wo flud physcal popees wll be added, whch ae velocy ( and pessue ( coespond o he copessve and Cools foces due o flow nduced vbaon. These s quanes can be aanged n a veco Z{, Ф, M,,, }, whch descbes he sae of he syse a node, and s called he sae veco, [Fancs 98]. W M M (7 W epesens he copessve and cenfugal foce plus he cooles foce, Fo sple bea heoy, he deflecon and slope of ppe eleen ( and of canleve of fleual sffness (EI subjeced o bendng oen and shea foce appled a s fee end gven by [Beads996] as follows: The flud flow and ppe sucue ae a dynac neacve syse, and coupled by he foces eeed on he sucue by he flud. These dynac foces ae of wo ypes. M 5W χ (8 EI EI ( GA 8EI p -Copessve foces, whch esul due o he ec of flud pessue and he change n oenu of he flud. These foces noally lead o bucklng nsably. - Cooles foce, whch esuls due o he flud oaon, he oaon has he ec of couplng he e and space. The copessve and cenfugal foce plus he Cooles foce can be wen as: [Mahd ] y W(, f. Ap.( (5 y f. ( - Feld a In ode o deene he ansfe a of any ppe eleens abay oenaed n space, a poon of ppe us be consdeed n (- y plane. Consde he ppe poon only beween ( and (- n he (- y plane as shown n fgue (. The equlbu of assless ppe of lengh ( as follows: -The sus of vecal foces on he ppe ae equal o zeo,.e. W (6 - The sus of oens abou pon (- ae equal o zeo oo, o: M W Φ (9 EI EI 8EI Whee χ s he nuecal faco by whch he aveage shea sess us be ulpled n ode o allow fo s dsbuon ove he ansvese secon. G, s he odulus of gdy wh espec o he followng equaon. [Beads996]. E G ( υ Whee ( υ osson ao. ( Subsung equaon (6 and equaon (7 no equaon (8 and equaon (9 gves: - - Φ W 8(EI M (EI - χ ( GA p 6(EI ( W Φ Φ - M - - ( (EI (EI 8(EI - on a The pon a fo a pacula node wh concenaed ass can be obaned fo he foce equaon of ass ( : 9

6 of. D. Najda Nasha Abdulla Effec of Desgn aaees and Suppo Condons on Naual of. D. Adnan Naj Jael Fequency of pe Eced by a Tubulen Inenal Flow Wjdan Kadhe Saheb 9 ( F p f y p f ω ( ( Whee: p f ω ( : s he nea foce noduced due o vbang ass fo haonc oon a he fequency (ω. Slaly as n feld a, he above equaon can be wen n densonless fos as: ( Φ Φ (5 M M (6 p f ( Ω (7 Whee, ( Ω f p EI ω (8 Also, (9 ( Fo he above equaons, he pon a can be wen as follows: M Φ M Φ Ω ( O, [ ] [ ][ ] Z. Z ( - Bounday condons The bounday condons n he ansfe a, gve he descpon of he sae veco paaees a he suppoed ends of he ppe. I eans he sae vecos a saon ( o[ ] Z, and saon (n o[ ] n Z ae epesen he bounday condon of he ppe. -Fo penned- penned suppos, he deflecon and he oen ae equal o zeo n penned suppoed ppe ends, and he ohe paaees ddn equal o zeo, as shown below: [ ] Φ Z o [ ] Φ Z n ( -Fo claped- claped suppos case, he deflecon and he slope ae equal o zeo, and he ohe paaees have a value ohe han zeo, as shown below: [ ] M Z o [ ] M Z n ( - ANATICA SOTIONTECHNIE Equaon of oon s paal dffeenal equaon wh espec o X and τ fo ppe vbaon, ( ( y y y pa y EI p f f p f (5 I can be solved by usng he followng assupon,

7 Nube 7 olue 9 July Jounal of Engneeng ( X τ φ( X ep( Ωτ, (6 Whee Ω s he densonless fequency defned n equaon (8. e, ( X C j ep( j X φ (7 J Whee C j ae consans whch can be found by usng bounday condon. Cobnng equaons (6 and (7 ( X, ep( Ωτ C j ep( j X τ (8 J Whee j ae he oos of polynoal equaon The esulng chaacesc fo equaon of oon s, ( γ β Ω j j j (9 To solve he poble of fee vbaon fo a gven sucue fs s bounday condons us be known. The wo bounday condons ae claped- claped, and pnnedpnned, o descbe he classcal bounday condons pedance values wee aken o be zeo o nfny values. - Claped- claped condon: The claped- claped bounday condons ay be wen as:- X a ( In ode o evaluae he naual fequency fo he syse unde consdeaon, equaon (9 can be subsued no he bounday condons equaon (. Ths yelds he followng a equaon, ep ep ( ep( ep( ep( ( ep( ep( ep( -nned- pnned condon: C C C C ( The nned- pnned bounday condon ay be wen as:- a ( In Sla pocess, he a equaons fo pnned-pnned condon can be obaned as: ep ( ep( ep( ep( ( ep( ep( ep( ep X C C C C ( Fo a gven flow velocy, seveal densonless fequences ae chosen and nseed no he chaacesc equaon. A MATHAB poga wll be used o deene he egenvalues and consequenly o copue he deenan. The poga wll choose he fequency ha coesponds o he deenan closes o zeo. New se of fequences ae chosen nea hs pon and he pocess s epeaed unl convegence wll occu. 5-EXEIMENTA WOK Wae was used as a wokng flud n all ess. Epeens wee conduced n a wae flow loop consuced fo ha pupose and s shown scheacally n fgue (. In hs sudy, fve dffeen es secons of C ppe and dawn / sanless seel ppe wh lengh of 6 wee used. The es secon consss of nechangeable ppe daees of 76., 5.8, and 8. wh. and.7 hckness. The suay of echancal popees of he ppe 9

8 of. D. Najda Nasha Abdulla of. D. Adnan Naj Jael Wjdan Kadhe Saheb Effec of Desgn aaees and Suppo Condons on Naual Fequency of pe Eced by a Tubulen Inenal Flow daees, of each es secon ae shown n Table (. The acceleoee ype 68 Buel & Kjae s used o easue he acceleaon of he esed odel. B&K chage aplfe ype 65 s used fo sgnal aplfcaon pupose and powe supply fo ansduces. The dgal osclloscope whch uses IGO DSD wh bul-n FFT analyze s used o dsplay he esponse of waves eac fo he acceleoee, due o vbaon of he sucues. A ubne flow ee (XIAN wae ee wh upsea saghenes s used o easue he wae flow ae wh ange.-. /s. 6-ESTS AND DISCSSIONS The naual fequences of he ppe conveyng flud ae vey poan fo he nepeaon of ppe conveyng flud esponse daa. A self ecaon analyss was pefoed fo dffeen cases. Fgues (5 and (6 show he fs hee naual fequences of wo ppes wh dffeen daees and specfcaons as enoned above fo sply-sply and claped-claped suppos condons especvely. Tables ( and ( show he copason of naual fequences values of ppe suppoed wh sply-sply and claped-claped suppos condons especvely. I was obseved ha, he fs naual fequency was slghly affeced by he daee sze fo boh ypes of seleced suppos, whle he second and hd naual fequences wee sgnfcanly affeced by he ppe daee sze. I was noed ha he ppe wh daee of 76. had hghe naual fequences values han he ppe wh daee of 5.8 fo boh of suppos. In geneal, he naual fequency depends decly on sffness and oal ass of ppe conveyng flud. The sffness self depends on oen of nea. So he nceasng n daee sze wll cause nceasng n oen of nea, heefoe nceasng n sffness, yelds nceasng n naual fequency. Fgues (7 and (8 pesen he nfluence of ppe daee on he fs hee naual fequency fo sply-sply and clapedclaped especvely. Effec of he desgn paaees on naual fequences ncease wh nceasng ode of naual fequency, eans ha he ec s sall n he fs fequency and hs ec wll ncease n he second naual fequency and oe ncease n he hd fequency. Two hcknesses,.7 and. fo C ppe wh daee of 5.8 wee seleced o nvesgae he ec of ppe hckness on he naual fequency of he ppe. Fgues (9 and ( show he fs hee naual fequency of wo ppes wh dffeen wall hcknesses fo sply-sply and claped-claped condons especvely. The naual fequency wee calculaed nuecally by neacon beween flud and sucue. The esuls showed ha, he hckness of ppe had lle ec on he naual fequences values fo boh suppos, as lsed n ables ( and (5. Because he change n ppe hckness wll cause vey lle ec on he oen of nea and hs wll gve vey lle ec on he sffness and heefoe he naual fequency. Fgue ( and ( llusaes he fs hee naual fequences fo 5.8 daee of C ppe fo sply-sply and clapedclaped condons especvely. Two ppe aeals wee seleced, C and sanless seel ppe wh 8. daee wh hckness of.5. o sudy he ec of ppe aeal on he naual fequences, Fgues ( and ( show he fs hee naual fequences of wo ppes dffe n aeal fo sply-sply and clapedclaped condons especvely. The naual fequences wee calculaed nuecally by neacon beween flud and sucue. Fgues (5 and (6 show he ec of ppe aeal on he fs hee naual fequency fo sply-sply and claped-claped condons especvely. The fequency of he syse s anly depends on he sucual sffness of he ppe. As he ppe aeal 9

9 Nube 7 olue 9 July Jounal of Engneeng sffens was nceased, he naual fequency was nceased due o he dec elaonshp beween he. Fgues (7 and (8 show he epeenal FFT specu of sply- sply of sanless seel ppe aeal and clapedclaed of C ppe aeal especvely. These fgues show ha, he naual fequences values of sanless seel ppe s hghe han hose of C ppe fo boh suppos wh dffeence aos as lsed n ables (6 and (7 fo sply-sply and claped- claped suppos especvely. Ths s because of he sanless seel ppe s uch sffe han C due o s physcal popees. A deenaon of naual fequency of he ppe wh he sae aeal and densons as oulned n he epeenal seup descpons wee caed ou usng ANSS, fo a sply-sply and claped-claped bounday condons, and he coespondng ode shapes fo hose sulaons ae shown n fgue (9 fo C ppe wh flud velocy of /s, 76. daee and hckness of CONCSIONS Fo he esuls obaned, he followng conclusons can be obseved:- -Geneally, he flud flow velocy educes he naual fequences of he ppe conveyng flud, fo he paccal ange of flud veloces (.7-5. /s, decease s vey lle. -The Foue ansfo povdes a fequency doan epesenaon of he sgnal and he esuls show ha he fs naual fequences was consdeed o be as a donan fequences. -The naual fequences ncease as ppe daees ncease and he naual fequences slghly nceases as ppe wall hckness nceases. -The naual fequences values of seel ppe ae hghe han C ppe. Also, he values of naual fequences n claped-claped suppoed ppe ae hghe han hose n splysply suppoed ppe. Table ( Mechancal popees of he ppes. No Oue daee ( Thckness ( Inne daee ( Modulus of elascy (N/ * -9 Densy (Kg/ Maeal C C C C Sanless seel 9

10 of. D. Najda Nasha Abdulla of. D. Adnan Naj Jael Wjdan Kadhe Saheb Effec of Desgn aaees and Suppo Condons on Naual Fequency of pe Eced by a Tubulen Inenal Flow Table ( Copason of s hee naual fequences of sply-sply suppoed ppe wh dffeen daees fo dffeen s. pe daees Fequency Hz TMM Epeenal ANSS Analycal Fn Fn Fn Fn Fn Fn Table ( Copason of s hee naual fequences of claped-claped suppoed ppe wh dffeen daees fo dffeen s. pe daees Fequency Hz TMM Epeenal ANSS Analycal Fn Fn Fn Fn Fn Fn Table ( Copason of s hee naual fequences of sply-sply suppoed ppe wh dffeen hckness fo dffeen s. pe daees Fequency Hz TMM Epeenal ANSS Analycal.7. Fn Fn Fn Fn Fn Fn

11 Nube 7 olue 9 July Jounal of Engneeng Table (5 Copason of s hee naual fequences of claped-claped suppoed ppe wh dffeen hckness fo dffeen s. pe daees Fequency Hz TMM Epeenal ANSS Analycal.7. Fn Fn Fn Fn Fn Fn Table (6 Copason of s hee naual fequences of sply-sply suppoed ppe wh dffeen aeals fo dffeen s. pe daees Fequency Hz TMM Epeenal ANSS Analycal C SS Fn Fn Fn Fn Fn Fn Table (7 Copason of s hee naual fequences of claped-claped suppoed ppe wh dffeen aeals fo dffeen s. pe daees Fequency Hz TMM Epeenal ANSS Analycal C SS Fn Fn Fn Fn Fn Fn

12 of. D. Najda Nasha Abdulla of. D. Adnan Naj Jael Wjdan Kadhe Saheb Effec of Desgn aaees and Suppo Condons on Naual Fequency of pe Eced by a Tubulen Inenal Flow Fgue ( Conol volue fo -velocy. Fgue ( Conol volue fo -velocy. Fgue ( End foces and oens fo assless ppe. 97

13 Nube 7 olue 9 July Jounal of Engneeng Fgue ( Scheac daga of flow loop. Fgue (5 The fs hee naual fequences fo sply-sply suppoed ppe fo dffeen daees. Fgue (6 The fs hee naual fequences fo claped-claped suppoed ppe fo dffeen daees. Fgue (7 The fs hee naual fequences fo sply suppo o dffeen ppe daees. Fgue (8 The fs hee naual fequences fo claped suppo fo dffeen ppe daees. 98

14 of. D. Najda Nasha Abdulla of. D. Adnan Naj Jael Wjdan Kadhe Saheb Effec of Desgn aaees and Suppo Condons on Naual Fequency of pe Eced by a Tubulen Inenal Flow Fgue (9 The fs hee naual fequences fo sply-sply suppoed ppe fo dffeen wall hcknesses. Fgue ( The fs hee naual fequences fo claped-claped suppoed ppe fo dffeen wall hcknesses. Fgue ( The fs hee naual fequences fo sply suppo fo dffeen ppe wall hcknesses. Fgue ( The fs hee naual fequences fo claped suppo fo dffeen ppe wall hcknesses. Fgue ( The fs hee naual fequences fo sply-sply suppoed ppe fo dffeen ppe aeals. Fgue ( The fs hee naual fequences fo claped-claped suppoed ppe fo dffeen ppe aeals. 99

15 Nube 7 olue 9 July Jounal of Engneeng Fgue (5 The fs hee naual fequences fo sply suppo fo dffeen odulus of elascy. Fgue (6 The fs hee naual fequences fo claped suppo fo dffeen odulus of elascy. Fgue (7 FFT Specu of ppe velocy of 8. daee wh.5 hckness, sanless seel, sply-sply suppoed ppe. Fgue (8 FFT Specu of ppe velocy of 8. daee wh.5 hckness, sanless seel, claped-claped suppoed ppe. s ode 7. Hz nd ode 8. Hz d ode 65.6 Hz Fgue (9 Mode shapes and ppe defoaon fo claped-claped suppoed C ppe wh d

16 of. D. Najda Nasha Abdulla of. D. Adnan Naj Jael Wjdan Kadhe Saheb Effec of Desgn aaees and Suppo Condons on Naual Fequency of pe Eced by a Tubulen Inenal Flow EFEENCES Beads C. E., Sucual baon: Analyss and Dapng, New ok: Halsed ess, 996. Blevns. D., Flow Induced baons. an Nosand enhold d, New ok, 977. Chang J.S. and Chou W.J., Naual Fequences and Ccal eloces of Fed- Fed anaed Ccula Cylndcal Shells Conveyng Flud, Jounal of Copues and Sucues, ol.57, No.5, pp.99-99, 995. Fancs, Mechancal baon, Theoy and Applcaon, d Edon, Addson-Wesley, SA, 98. Mahd A.A., The Effec of Induced baon on a pe wh a escon Conveyng Flud, h.d., Thess, unvesy of echnology,. Mousa A., Eac Soluons fo he baon of Ccufeenally Sepped Ohoopc Ccula Cylndcal Shells, Copes endus Mecanque 9, pp.78 78,. adousss M.., Flud-Sucue Ineacons: Slende Sucues and Aal Flow. Acadec ess, Aseda, he Nehelands, 998. See Seo., Jeong W.B., Jeong S.H., oo W.S., and Jeong O.K., Fequency esponse Analyss of Cylndcal Shells Conveyng Flud sng Fne Eleen Mehod, Jounal of Mechancal Scence and Technology, ol. 9, No., pp. 65-6, 5. Tennekes H., and J. uley, A Fs Couse n Tubulence. MIT ess, Cabdge, 97. Wang., Da H.., an., Dynacs of Sply Suppoed Flud-Conveyng pes wh Geoec Ipefecons, Jounal of Fluds and Sucues 9,. 97 6,. Wang X., and Zhang N., Nuecal Analyss of Hea Tansfe n ulsang Tubulen Flow n a pe, Inenaonal Jounal of Hea and Mass Tansfe 8 ( ousf A. E., Jweeg M. J., and Isal M.. Closed-Fo Soluon fo Evaluang he ncpal Insably egons fo Consevave pes Conveyng ulsang Flowng Flud, Tansacons of he ASME Novebe pp.-7,. an-e Zhang and -un Chen, Inenal esonance of pes Conveyng Flud n he Supeccal ege, Nonlnea Dyn., 67, pp.55 5,. -Mn H., B.,. And ue Z., Naual fequency Analyss of flud Conveyng pelne wh Dffeen Bounday Condons, Nuclea Eng. And Desgn ol., 6-67,. NOMENCATE Sybols Descpon ns C, C, C Consans n he sandad k-e odel - d pe daee E Modulus of elascy N/ Fn,Fn,Fn The fs hee naual fequences Hz F Feld a - G Modulus of gdy N/ I Second oen of aea k Tubulen knec enegy /s engh of he ppe M Bendng oen N. f Mass of flud pe un lengh kg/ p Mass of ppe pe un lengh kg/ 9

17 Nube 7 olue 9 July Jounal of Engneeng Mean pessue N/ on a - k oducon of ubulen knec enegy /s Shea foce N adal coodnae -, Mean velocy n aal and adal decon /s W Cools and copessve foces N X ongudnal coodnae - Tansvese dsplaceen Z Sae veco - Geek Sybols ν osson ao - χ Nuecal faco - ω Ccula naual fequency ad/sec φ Slope o Dynac vscosy of he flud kg/.s Densy kg/ ε Tubulen enegy dsspaon /s Densy kg/ Ω Densonless ccula naual fequency - j oo of polynoal equaon - Supescps ef - gh - 9