Reza Poultangari a a, b, * Mansour Nikkhah-Bahrami

Transcription

1 7 ee d oced Vo Alyss of Sepped Ccul Cyldcl Shells wh Sevel Iemede Suppos Usg Eeded Wve Mehod; Geelzed Appoch Asc A como of vecol fom of wve mehod VWM wh oue epso sees s poposed s ew vehcle fo fee d foced vo lyss of sepped cyldcl shells wh mulple emede flele suppos. The flele suppos c clude spgs wh y popees he possle decos. Bsed o lügge h shell heoy d VWM he efleco popgo d smsso mces fo ccul cyldcl shell e defed. uhemoe coguous veco-m eloshps e eslshed fo fee d foced vo lyss of he ssue cludg y ume of he dscoues he shell hckess o shell seps d emede suppos. Usg hese veco-m elos he equos of moo s well s he sysem couy e well ssfed. Dmeso of hese vecos d mces e compleely depede of he ume of he ppled suppos d geomecl seps he shell. Hece he pese ppoch povdes ecelle compuol dvges d modelg flely comped o he coveol vo lyss mehods vlle he leue. The esuls of he pese sudy e comped wh he esuls vlle he leue s well s he esuls of fe eleme mehod EM d foud ecellece geeme. uhemoe s cse sudy cse cyldcl shell wh hee flele emede suppos d lso hee geomecl seps s cosdeed. The ul fequecy d mode shpes of he ssue e deved d he foced esposes of he shell suec o po lod eco e epoed. Rez Poulg * Msou Nkkhh-Bhm Depme of Mechcl d Aeospce Egeeg Scece d Resech Bch Islmc Azd Uvesy Teh I; poolg@ud.c. School of Mechcl Egeeg College of Egeeg Uvesy of Teh I; * mhm@u.c. hp://d.do.og/.59/ Receved..6 I evsed fom Acceped Avlle ole Keywods Iemede flele suppos; sepped cyldcl shell; eeded vecol-wve mehod; fee d foced vo.

2 8 R. Poulg d M. Nkkhh-Bhm / ee d oced Vo Alyss of Sepped Ccul Cyldcl Shells wh Sevel INTRODUCTION Usg emede suppos fo log ppe ufom o o-ufom shells my duses s commo wy fo poecg he sysem fom he occuece of esoce flue d mpovg he sysem ousess. I hs egd gdy cosdeo fo he suppos ppled clled clsscl suppos s ove smplfed ssumpo g sgfc devos ewee he heoecl d epemel esuls. I s cle h s much s he chcescs of o-gd suppos c ply ge ule o vo of he esposes of he whole vg sysem. I ddo esece of dscouy he shell hckess o shell seps s evle my dusl elemes such s ccumulos flud ks d ppes. Hece followg he mpoce of he suppo chcescs d he oufom hckess of he shell he pccl pplco he eseches hve ee ecouged o eme hese effecs o he vo lyss of shells lke Zhou e l. Qu e l. Che e l. 5 Che e l. d flly Wg e l I hs wy dffee umecl mehods e.g. fe eleme mehod EM lke Slhf d Mohe weghed esdul mehod s Qu e l. geelzed dffeel qudue mehod GDQ s oy e l. 997 d lycl soluos e.g. close fom soluos s hose epeseed y Che e l. wve sed mehod WBM lke Che e l. 5 sfe fuco mehod TM s Zhou e l. 995 d se spce echques SST lke Zhg d Xg 7 e poposed fo lyss of such sysems. Alycl modelg of vos of ufom shells wh eo suppos hs ee ccl ssue he ece lycl sudes pehps due o equeme of ssfyg he couy d he suppo codos smuleously lke Qu Xg e l. Zhg d Xg 6 d oy d m 997. Hece few eseches ee coduced hs e d he poposed mehods e umeed d co sevel lmos. o emple Xg e l. solved he ssue of fee vos of ufom shell wh sevel emede g- o dl gd-suppos. They poposed se-spce echque comed wh dom decomposo mehod ode o ssfy couy s well s suppo codos of he ssue. I hs deco Zhg d Xg 6 wh he sme mehodology poposed Xg e l. solved he polem of fee vos of ufom ope shell wh mulple emede Rg Suppos RS. Alhough he esuls of he mehod poposed Xg e l. e pomsg s lycl mehod he ppled emede RS wee o flele d wee oly lmed he oe deco.e. dl deco. The vo lyss of sepped shells d modelg of he flele suppos e possle usg he umecl mehods e.g. EM whee poposed Qu e l.. Howeve solvg he polem demds sevel degees of feedom fo he ssue d ccodgly eesve memoy esouces d hgh compuol esouces hs mpo evlued Qu e l.. Moeove specl pplcos such s desgg wh opmzo sysems cools vese polems d el me pplcos ccue d fs sysem esposes s well s low sysem memoy e equed smuleously. Thus he meoed pplcos pplyg he coveol d effce umecl mehods lke EM my fl o cpue he shell espose wh esole compuol me d ccucy. I cos wh he umecl mehods h demd hgh memoy d compuol coss he lycl ppoches eques o oly low memoes u lso low compuol coss. Howeve s meoed he vlle lycl mehods e del wh mo lmo o oducg flele emede suppos. Hece s cle h developg ew ppoches whch e cple of povd- Amec Joul of Solds d Sucues

3 R. Poulg d M. Nkkhh-Bhm / ee d oced Vo Alyss of Sepped Ccul Cyldcl Shells wh Sevel 9 g ous fs d ccue esuls fo vo lyss of shells s hghly demded o del wh he desg polems wh less lmos d moe cples. Vecol wve mehod VWM s well-kow mehod whch ws fs oduced y Mce 984 d ppled o fee d foced vo lyss sgh ems. Theefe VWM ws ppled vo lyss of sevel wve gude elemes such s s Hgedo d Ds-Gup 7 sgh-mse 5 d cuved-ems d fmes woee e l. 7 o hee dmesos Me 6 gs Hug e l. ples M e l. 5 d Reo d Mce d memes Bhm e l.. Howeve hs mehod hs o ee eeded fo ohe mechcl elemes such s shell lke geomees. Revew of he leue shows h hee hve ee few emps fo eeso of VWM o moe comple ssues such s esece of mulple geomecl seps o mulple emede suppos cosdeed geomey. Nkkhh-Bhm e l. hve poposed modfed fom of VWM fo fee vo lyss of o-ufom mechcl elemes ems wh y umes of dscoues. They demosed h y dscve comos of VWM mces d he vesos s possle o povde educed ode veco-m eloshp depede of ume of dscouy of he cosdeed ssue. Howeve due o sguly of he efleco d smsso mces he vcy of gd emede suppos he poposed mehod Nkkhh-Bhm e l. co e eeded o he ssues wh emede suppos. To he uhos es kowledge whch s suppoed y ccl leue suvey he VWM hs o ye ee ulzed fo vo lyss of shell elemes o polems wh eo suppos pehs due o lck of eslshme of pope wve vecos d elev mces. The pese sudy ms o eed VWM mehod o he cse of sepped cyldcl shell elemes wh eo flele suppos s es he deco of he wve dspeso. Hece pope wve vecos d elev mces eloshps fo hese kds of wve e e slshed. By oducg cocl eloshp ewee ce wves d he dsplceme feld geel cse sudy cludg possle es he ph of wve moo flele emede suppo ypes d geomecl seps e cosdeed. Bse o ce wve veco moo ech shell segme d commuco ewee he wves ll he shell efces eeded VWM s ulzed fo fee d foced vo lyss of cyldcl shells wh dscoues he hckess he pesece of sevel emede flele suppos. GOVERNING EQUATIONS gue shows cyldcl shell wh he legh hckess h Posso o Youg's modulus E desy ρ d mddle lye dus. The goveg equos of moo sed o lügge heoy fo cyldcl shell s demosed gue c e epessed s follows: u u u whee = e dffeel opeos s show pped A. uhemoe u e he spl dsplcemes of he shell mddle lye he l ccumfeel d dl dec- Amec Joul of Solds d Sucues

4 R. Poulg d M. Nkkhh-Bhm / ee d oced Vo Alyss of Sepped Ccul Cyldcl Shells wh Sevel os especvely. Usg me hmoc fucos d comple oue epso he ccumfeel deco s o ssfy Eq. he dsplceme felds c e epeseed he especve decos s: gue : Wve s moo homogeeous cyldcl shell wh flele eded suppos. u C λω λω λω e u αce u βce whee C s cos; d e dsplceme os whch e especvely oduced s α=u /u d β=u/u. I Eq. λ s he wve ume s he ccumfeel mode ume o ege cos of he oue ems d ω s ccul fequecy. I fc Eq. cosss of he dsplceme feld egdg o ll possle el d mgy eco esposes of he shell. To o el ps of he clculed esuls hlf-ge epsos of he oue sees s eough fo he ssue.e. = o. To eslsh he pmees of λ d ω Eq. s susued he equo of moo Eq. whch yelds: A B B A C B A A C C E D C 4 whee Λ s dmesoless wve ume s Λ=λ d χ = e he m elemes. The epessos ABCD = e gve Apped B. To cheve he o-vl esuls d hece he o-zeo dsplceme felds he deem of he coeffces m Eq. should e equl o zeo.e. χ =. By defg dmesoless fequecy Ω=ω- ρ/e.5 o fequecy fco s he m sque egevlues [ χ] we c cheve logcl eloshp mog he dmesoless wve umes Λ d fequecy fco Ω wh he d of χ =. These elos e eslshed fo ll kd of veloces of wve s dspeso he shell.e. he phse velocy d he goup velocy of he wves ee e l. 7 d Kczu 6 whee hs ssue wll o e dscussed hee. Mhemclly χ = leds o polyoml of degee fou ems of sque of dmesoless wve ume Λ s follow: A Λ A6 Λ A4 Λ A Λ A 4 Amec Joul of Solds d Sucues

5 R. Poulg d M. Nkkhh-Bhm / ee d oced Vo Alyss of Sepped Ccul Cyldcl Shells wh Sevel whch A =468 e fve coeffces epeseed he pped C. The egh possle oos of Eq. 4 e lyclly deved d dscussed pped D. Accodg o he wve heoy dsplceme felds Eq. c e epessed he fom of ω u U e whee epessos of U c e desced s le comos of egh hmoc fucos he d decos s follows: U U U 4 λ 8 -λ e e e 5 4 λ 8 -λ α e α 5 e e 4 λ 8 -λ β e β e e 5 5 whee epessos d... 8 e he posve d egve gog wve mpludes of kd of ce wves whee fc hey e cg m wves he l deco of he shell. These wve mpludes show he couo of ech coespodg wve he dsplceme felds of he vg shell. These wves c e of kd of pue hmoc o popgg wves o hmocdmpg o sdg wves o pue dmpg o evesce wves wves depedg o he coespodg wve ume vlues see pped D. o he ove composo sed o he cougo of oos of Eq. 4 he wve umes e ged s whee m m m4 m MATRICES O WAVE MOTION. oce d Dsplceme Vecos o fee vo lyss of cyldcl shell v VWM he vecos of posve d egve gog wves goveg he polem e equed o e defed. Thus he followg fom s cosdeed fo wve mpludes: e 5 e 6 e 7 e whee e popgo mces e vecos of wve mpludes d e wves vecos h e ll gog he gh + o lef - decos log wh shell s s depced gue. By defg he vecos of mpludes w d f fo he dsplceme veco w d he foce veco f oh vecos of w d f e cosdeed s follows: w w e f f e whee he ove se of equos U U U U T N M T V e N d T e 7 M e he foce d mome esuls epessos s epeseed he gue. O he ohe hd he vles of T d V e kow s Kchhff Amec Joul of Solds d Sucues

6 R. Poulg d M. Nkkhh-Bhm / ee d oced Vo Alyss of Sepped Ccul Cyldcl Shells wh Sevel pmees whch e oduced he Apped E. I Eq. 7 pme symol ' dces h he em s dvded y ep. gue : oce d mome esuls ccul cyldcl shell. To cosde he mpc of sucul dmpg cyldcl shell he Youg's modulus s doped s E E ˆ γ whee γ s he sucul loss fco d Ê s el coeffce of Youg's modulus eled o he mel used he cylde.. Defo of Mces ψ d φ By defg ce efce mecs s possle o epess he dsplceme d foce vecos y he posve d egve gog wve vecos especvely. Hece y susug Eqs. 5 Eq. 7 yelds followg eloshp: w ψ ψ f 8 whee ψ d e mces of dsplceme vecos d foce vecos especvely. These elos c e we s follows: ψ [ ψ [ 5 5 ψ 6 6 ψ 7 7 ψ ] ] 9 whee vecos of φ d ψ = 8 s elemes of ψ d s follows: ψ D kλ α β β λ D λ β α β λ Dλ k λ α α β λ D α - β λ λ β Amec Joul of Solds d Sucues

7 R. Poulg d M. Nkkhh-Bhm / ee d oced Vo Alyss of Sepped Ccul Cyldcl Shells wh Sevel whch k Eh d D Eh e especvely meme d edg sffess of he shell. The elo ewee popgg wve vecos wh he shell e.g. d o d he gue e mde y he popgo mces. The popgo m elemes he posve d egve deco c e deemed y he followg elo:... whee =. oduced Eq. 6 shows he popgo mces cylde wh he legh of. Defo of Refleco d Tsmsso Mces Tkg o ccou he effecs of flele suppos o he vos of cyldcl shell s ecessy o defe he sffess m. Ths m loco s defed s follows: K K K K K The dgol elemes K e he sffess coeffces of spgs whch e eeded evely y poso o he cylde. The dgol shpe of he m Eq. s ppope fo he le vo lyss of couous sysems such s cyldcl shells... Eded Suppos The oudy codos he gh d lef suppos depced he gue should e ssfed. I ode o ssfy hese oudy codos oe c we: B f Kw A Accodg o gue he vecos of he cde wve mpludes he lef d gh oudes e d especvely. Moeove he efleced wves vecos hese wo oudes e d especvely. Now he efleco m s cosuced y Eqs. 8 d. I hs egd hese equos wee se sed o he cde d efleced wves vecos s follows: R A R B 4 whee R e efleco mces he lef d gh odes of cyldcl shell whch c e defed y he followg A B equos: R K K R K K 5 A A A B B B Amec Joul of Solds d Sucues

8 4 R. Poulg d M. Nkkhh-Bhm / ee d oced Vo Alyss of Sepped Ccul Cyldcl Shells wh Sevel I s cle h he se of equos of R d A R Eqs. 5 c ssfy he flele oudy B codos fo he oh eds of he cylde... Wve Moo Vcy of Iemede Suppos d Shell Seps I he lycl d sem-lycl soluos povded fo he fee vos of cyldcl shells ely c we fd efeeces o he flele emede suppos he leue. The pese su seco s eded o cosde he effec of eo suppos d shell seps o he vo lyss of he cyldcl shell. Hece gue shows cdece of wve wh e cludg of flele suppo d shell sep. gue : Wves moo he vcy of emede wve es of kd of shell sep d emede suppo sme poso As see gue p of he cde wve veco fe colldg wh he e s sfeed d some of wll e efleced. Thus cosdeg he couy codos d he lce of foces he e ewee he h d he + h segmes of he shell we hve: w w w 6 f - f K m o whch K m s he gdy m of he h emede suppo of he h e h s dgol m cludg of elemes of spg sffess coeffces he elev depede decos.e. d Eq.. Usg Eq. 8 Eq. 6 d ccodg o he equly of geomec d mechcl popees of he cyldcl shell ech segme he followg equly s cocluded: ψ ψ ψ K m 7 Refleco d smsso mces h coec he cde wve veco o he e smed veco d efleced veco e defed s follows: 8 whee s wve smsso m fom lef o gh.e. fom he h o he + h segmes d s he efleco m of he cde wve ewee he wo meoed segmes. Usg Amec Joul of Solds d Sucues

9 R. Poulg d M. Nkkhh-Bhm / ee d oced Vo Alyss of Sepped Ccul Cyldcl Shells wh Sevel 5 he esuls of Eq. 8 d egg Eq. 7 he efleco d smsso mces c e oed s: [ K ] m [ ] K m [ ] 9 whee I s fou y fou dey m. I sme me he efleco d smsso mces he oppose deco.e. gh o lef deco c lso e oed. 4 EXTENDED VECTORIA WAVE METHOD I gue 4 wve moo cyldcl shell wh sevel elsc suppos s show. Accodgly he eloshp ewee he vecos of he oupu wves fom he fs emede suppo.e. d ems of pu wves veco o he suppo.e. d s s follows should e oed h fo homogeeous mels lke he pese ssue we hve : gue 4: Wves moo he vcy of emede es of kd of emede suppos d shell seps. Moeove y cosdeg d d usg Eq. we c we: R A whee s posve gog duco m s defed s follows: I R A R A I sme me he followg elo c e gve ewee d fo he h emede suppo = - s: Amec Joul of Solds d Sucues

10 6 R. Poulg d M. Nkkhh-Bhm / ee d oced Vo Alyss of Sepped Ccul Cyldcl Shells wh Sevel Amec Joul of Solds d Sucues whee he sme s hose gve Eq. 4 o e defed s: I I 4 lly he h segme of he shell he followg elos e eslshed: B R 5 The ove equos coss of fou equos wh fo ukow vecos mely d. Ovously we c solve he ses of Eqs. 5 y ech of hese ukows. o emple Eq. 5 c e solved y s follows: R I B 6 To cheve o-vl soluo fo fee vo lyss of he ssue usg Eq. 6 he deem of he coeffce of should e equl y zeo. I should e oed h Eq. 6 s o uque wy fo clculg he ul fequeces of he shell d we e le o we he sme elos y he ohe wve vecos depced gue 4. o emple ckwd ppoch fo he wves h e movg he vcy of he - h emeded suppo o he ls emede suppo of he shell we hve: 7 uhemoe he h segme o he ls segme of he shell we c we: 8 cosdeg R B d usg Eq. 7 d 8 gves elo ewee d s follows: Γ 9 whee Γ s egve gog duco m whch s defed s: B B R R I Γ I s possle o o geel eloshp ewee d fo he h emede suppo = - s follows: Γ Γ whee Γ o Γ e defed s follows:

11 R. Poulg d M. Nkkhh-Bhm / ee d oced Vo Alyss of Sepped Ccul Cyldcl Shells wh Sevel 7 Amec Joul of Solds d Sucues Γ Γ I Γ Γ Γ I Γ Wve moo he fs shell segme whee depced he gue 4 s cosdeed hee. By cosdeg he cocl eloshps ewee hese wve vecos he followg elos e goveed: A R The ove se of equos coss of fou equos wh fou ukows.e. d. o emple solvg he ove se of equos y gves he followg elo: R Γ I A 4 o o-vl esuls he ove equo he deem of he coeffce m of mus e zeo so we hve: R I A 5 Ivokg Eq. 5 ll of he ul fequeces c e deemed fo sepped shell wh sevel emede suppos wh dvdul y popees fo he ppled suppos. I s eesg o oce h oh deemos epeseed Eq. 6 o 5 povde uque esul fo he fequecy lyss of he shell. To cheve he oos of Eq. 5 Wck-Wllms semmg lgohm c e used ee e l. 7. I ddo y omlzg he wve veco d seg fom Eq. 4 ems of he ul fequeces of he sysem he mode shpes c e oed. 5 ORCED VIBRATION Coceed cug o sesg he fom of g lodg o po lodg e pplcle fo sevel dusl ess e.g. o-dscve ess d vo cool Me 9 Hu e l. 5 Jg e l. Km e l. d Popov 5. Rg eco s le lodg Achech s hmoc lodg y poso log wh he l deco of he shell d dsued o ccle oud he shell. Ideed he mode shpes of he shell c e eced dscly usg he espose of he shell o he g eco. I he ohe hd po lod eco eles us o ece ll possle modes of he cosdeed ssue. I gue 5 sepped ccul cyldcl shell s depced wh wo eded- d sevel emede-suppos d ude g lod eco mddle of y segme of he shell.

12 8 R. Poulg d M. Nkkhh-Bhm / ee d oced Vo Alyss of Sepped Ccul Cyldcl Shells wh Sevel Amec Joul of Solds d Sucues gue 5: Wves moo he vcy of lodg poso. As geel cse s possle o choose y y fuco cludg coceo po lodg he ccumfeel deco usg oue epsos. Effecs of he g hmoc lodg o sucul esposes of he shell c e oed usg he wve moo lyss wh he shell he vcy of he lodg poso. Couy d foce lcg ewee h d + h segmes of he shell ecly he lodg poso e ssfed y he followg elos: e e - e e f f w w 6 whee dces d + he ove equo e efeed o he lef d gh sdes of he lodg posos especvely. uhemoe s he g lodg veco cosss of foce d mome lodg he elev decos s follows: T f f T f 7 whee f d T e dsued foces d oque he eled decos especvely. Susug Eq. 8 Eq. 6 gves: 8 o he veco-m fom c e we s: 9 whee d e he oppose gog wve veco mpludes oh sdes of lodg poso gue 5. Solvg he ove equo yelds: Π Π Π Π Π 4 om he wve moo lyss he oh sdes of he lodg poso we c fd: 4

13 R. Poulg d M. Nkkhh-Bhm / ee d oced Vo Alyss of Sepped Ccul Cyldcl Shells wh Sevel 9 whee mces of d e defed s: 4 By elmg d fom Eqs. 4 d 4 we c we: I Π Π 4 Accodgly fo ce lod veco wh y fucos of d c e clculed fom Eq. 4. Usg oue epso y fucos of c e povded fo lod veco. Accodgly we c use of he followg elo: T f T f f e 44 whee s veco of oue pmees d c e oed s follows: f T f f T e d 45 o modelg he po lod eco y poso o he shell e.g. = d gue 5 he po lod c e epessed ems of Dc del fuco: f T f T f f e 46 whee d T e mpludes of dsued foces d oque he elev decos especvely. Hece he oue coeffce pmees he pesece of po lodg c e oed s follows: T f T f f e f T f f T e d 47 I he ove equo f d T e he po foces d coceed oques especvely. Hece fo po lodg supeposg of ccumfeel mode umes fom zeo o fy s equed. Howeve s pcclly mpossle o dop ll of he mode umes d hece oly fe umes of o uced c e used fo supeposo depedg o he equed egee- Amec Joul of Solds d Sucues

14 4 R. Poulg d M. Nkkhh-Bhm / ee d oced Vo Alyss of Sepped Ccul Cyldcl Shells wh Sevel g ccucy. Usg Eqs. 7 8 d d wve mpludes oed fom Eq. 4.e. d fo y dsce he lef o gh sdes of lodg poso he mpludes of he dsplceme felds o decos e oed fom he followg elos: U U U U U U [] [ β β [ β β [] [α α α [α α α α ] α ] β β ] β β ] [ β [α α 5 [α α 5 5 [ β α α ] α α ] β β β ] β β β ] Accodgly usg Eqs. 4 d 48 ll of dsplceme esposes fo he h d + h segmes of he shell c e oed fo y posos o h d + h segmes of he shell. I sme me d y usg Eqs. d sml eloshps fo he ohe segmes of he shell c e fouded esly. 6 NUMERICA RESUTS I hs seco fee d foced vos of sepped ccul cyldcl shells ffeced y he suppos chcescs wll e emed usg he defed veco-m elos of VWM he pevous secos. I some possle cses he esuls of VWM wll e comped wh he umecl esuls of EM d he esuls vlle he leue. Hee he followg commo geomecl d mechcl popees e doped fo he cses sudes: he shell me dus = m desy ρ=78 kg/m Youg modulus E= GP d Posso s o =.. Moeove ll esuls d chs e deemed wh he help of self-we pogm MATAB. All of he oed esuls e clculed wh PC wh dul coe CPU of ype of Iel Peum G.GHz d.9gb usele m. Howeve s he code ws o opmzed fo pllel compug pcclly oly oe CPU of he compue ws ulzed clculos. To eme he effecs of he suppo chcescs o ul fequeces d sucul esposes of he ssue ce ypes of flele suppos e ulzed. The flele suppos e E I o E V whch e defed s follows: E I E II E III : Idces h he suppo s gd he ll decos ecep oe flele deco.e. o o deco especvely. E IV E V : Idces h he suppo s fee he ll decos ecep oe flele deco.e. o deco especvely. Assumg vey lge sffess fo he spgs he suppos e.g. of he shell wh flele suppos he VWM wh flele suppos educes o VWM fo shell wh gd clsscl oudy codos. o emple fo he oudy codos of She Dphgm SD ll he dl d ccumfeel sffess should e cosdeed s K= K d he ool d l sffess should e equl o zeo.e. K= d K= d so o. Amec Joul of Solds d Sucues

15 R. Poulg d M. Nkkhh-Bhm / ee d oced Vo Alyss of Sepped Ccul Cyldcl Shells wh Sevel 4 6. ee Vo Polem 6.. equecy Alyss Tles d e eguled o compe he esuls of fequecy lyss of sepped shell wh elsc eded suppos usg he poposed mehod d hose epoed Qu e l. d Che e l. 5. Qu e l. hve solved he ssue of fee d foced vo lyss sepped shell wh flele eded suppos usg umecl ppoch modfed vol pcple d lessques weghed esdul mehod med y dom decomposo mehod DDM. I sme me Che e l. 5 hve ulzed lügge h shell heoy d he wve sed mehod o WBM d hey hve lyzed he sme ssue doped y Qu e l. ude po lod eco. Hece y cosdeg he pesece of oly oe sep he shell geomey d eglecg he eo suppos.e. K m Eq. 9 s hose Qu e l. d Che e l. 5 compso ewee he esuls of VWM d he esuls epoed y Qu e l. d Che e l. 5 s pefomed. The esuls of hs compso e summzed Tle. The symol of m Tle deoes he l mode ume. As c e see Tle hee s good geeme ewee he esuls of VWM d he esuls vlle leue fo he cse of sepped shell wh flele suppos. Tle shows he fequecy lyss fee vos of hee sepped cyldcl shell wh dffee l leghs d SD-SD eded suppos oed y usg VWM d he esuls oed sed o DDM y Qu e l. d WBM y Che e l. 5 especvely. By eglecg ll flele emede suppos mely K m Eq. 9 he pese wok s comple wh hose epoed y Qu e l. d Che e l.. Despe of dffeeces shell heoy used y Qu e l. le dces good pomy ewee ll esuls oed y he poposed mehod d hose gve he leue. To eme he ccucy of he VWM he pesece of emede suppos he esuls of pese mehod e comped wh he esuls of Xg e l.. I he sudy of Xg e l. peseed ec soluo se-spce echque SST o solve he polem of vos of ufom ccul cyldcl shell sed o he Goldeveze Novozhlov shell heoy y usg dffee clssc eded suppos e.g. clmped C SD d fee suppos d dffee umes of emede gd suppos. I ode o possle compso wh he sudy of Xg e l. he hckess of he shell he pese sudy should e cosdeed s ufom. Hece he efce mces of ψ d e equed o e equl fo ech segme.e. ψ ψ... ψ d... Eq. 9. I fc Eqs. d ll e depede fom he shell leghs ech segme d he leghs of he shell segmes oly ppe he popgo m elos. Compso of he fequecy fco oed fo ufom shell y VWM d SST fo dvese ccumfeel d l mode umes.e. = d m= 4 s mde les d 4 he pesece of wo d hee umes of emede RS especvely. Despe he dffeeces he ppled heoy hs sudy d hose ulzed he efeece Xg e l. vey good pomy of he gve esuls wh oh mehodologes c e see especlly he cse of he shells. Amec Joul of Solds d Sucues

16 4 R. Poulg d M. Nkkhh-Bhm / ee d oced Vo Alyss of Sepped Ccul Cyldcl Shells wh Sevel E I -E I E I -E II E II -E II Che e l. 5 Pese Dff. % Che e l. 5 Pese Dff. % Qu e l. Pese Dff. % Tle : Nul fequecy fco Ω d E I -E I E II -E II o E III -E III eded suppos fo shell wh oe sep /=.5 d h/=. h/h=; elsc suppo sffess: K = 5 7 N m - d K =.5 8 N m - d m=. Amec Joul of Solds d Sucues

17 R. Poulg d M. Nkkhh-Bhm / ee d oced Vo Alyss of Sepped Ccul Cyldcl Shells wh Sevel 4 Che e l. 5 m= m= m= Pese Dff. % Che e l. 5 Pese Dff. % Che e l. 5 Pese Dff. % m=4 m=5 m= Tle : Compso of fequecy fcos fo hee-sepped cyldcl shell wh SD SD eded suppos /= /= /= 4/=.5 d h/=. h/h= h/h= h4/h=4. Amec Joul of Solds d Sucues

18 44 R. Poulg d M. Nkkhh-Bhm / ee d oced Vo Alyss of Sepped Ccul Cyldcl Shells wh Sevel h m Xg e l. = = = Pese Dff.% Xg e l. Pese SD-SD eded suppos: Dff.% Xg e l. Pese Dff.% C-C eded suppos: eded suppos: Tle : Compso of fequecy fco Ω fo shell wh wo emede g suppos d dffee hckesses d eded suppos codos. I hs deco fudmel fequecy fcos e epeseed le 5 gve y VWM f d SST ppled Xg e l. fo dffee ufom shell leghs d hckesses d he Amec Joul of Solds d Sucues

19 R. Poulg d M. Nkkhh-Bhm / ee d oced Vo Alyss of Sepped Ccul Cyldcl Shells wh Sevel 45 pesece of dffee umes of emede RS. Ag good pomy c e see he oed esuls y he wo mehods ppled. h m Xg e l. = = = Pese Dff. % Xg e l. Pese Dff. % SD-SD eded suppos: Xg e l. Pese Dff. % C-C eded suppos: eded suppos: Tle 4: Compso of fequecy fco Ω fo shell wh hee emede g suppos d dffee hckesses d eded suppos codos Amec Joul of Solds d Sucues

20 46 R. Poulg d M. Nkkhh-Bhm / ee d oced Vo Alyss of Sepped Ccul Cyldcl Shells wh Sevel h Xg e l..974 =7.669 = = =.59 = =.659 = = SD-SD C-C - Pese Xg e l. Pese Two umes of RS /= /=/ =7. =.594 = =.545 Thee umes of RS /= /= / =/ = = = = Xg e l =5.567 =.989 =.555 =.656 = =.988 =4.566 = Pese Tle 5: Compso of fudmel fequecy fco he pesece of vous umes of emede RS. f gues 6 show he effec of he wve es loco o he ul fequeces of ccul cyldcl shells. Hee he emed es e of ype of shell sep d emede dl suppos RS. I ode o mke f compso ll codos e cosdeed he sme s hose gve Che e l. 5. Hece he geomecl d mechcl cosdeos he pesece d he sece of he emede suppo e s follows: he hckess o h/=. he sep hckess o h/h=.5.5 d 4 he legh o dus o /=5 d d flly sep loco d emede suppo loco s /=.5. gues 6 show h he oed esuls he sece of he emede suppo hve good geeme wh hose epoed he leue Che e l. 5. Ge mpc of he esece of eo suppo o cesg of oh em mode- d fudmel-fequecy fcos s clely vsle he ll cses cosdeed fo gues 6. I s vey logcl pheome whee hppeed due o he cese of he whole sysem gdy ffeced y he ppled eo suppo. Hece depedg o he hckess o h/h he gees vo of ll fequeces ppomely occus fo he emede ype of RS ppled evl of.4</<.6. As c e see hese fgues he mmum fudmel fequecy fco fo ll of he shell legh os.e. /=5 d he pesece of emede RS hppeed hckess o of h/h=.5 d he poso o of /=.4. Howeve he cse of mmum em mode fequecy fco depedg o he shell legh o hppes hckess o of h/h=.5 fo /= o hckess o of h/h=.5 fo /=5. Amec Joul of Solds d Sucues

21 R. Poulg d M. Nkkhh-Bhm / ee d oced Vo Alyss of Sepped Ccul Cyldcl Shells wh Sevel 47 equecy Pmee Ω udmel equecy Ω f Wve Be oco / Wve Be oco /.4.7 equecy Pmee Ω udmel equecy Ω f Wve Be oco / Wve Be oco / c d gue 6: Bem mode- c d fudmel-fequecy pmees d vesus he loco of he wve e / fo SD SD shell wh d whou emede suppo h/h= h/h=.5 Δ h/h=4 d h/h=.5 cludg geomecl chcescs of: /=5 d cd /=. 6.. Mode Shpes Usg Eq. 6 o 4 s possle o o ll mode shpes of he cyldcl shell. I hs seco dl mode shpes fo em modes.e. =m= he l decos of shell wh dffee umes of emede RS d SD-SD eded suppos e ploed gues 7 o 9 By hese cosdeos we e le o cg ssfco of he suppo codos d couy smuleously. Two cses fo emo he mode shpes e cosdeed hee. I he fs cse ufom shell d he e cse sepped shell wh dffee umes of emede g suppos e cosdeed hee. I he cse of ufom shell he effecs of shell hckess o mode shpes coespodg wh em mode fequeces e depced gues 7 fo wo shell leghs.e. /= d 5 especvely. I hs fgue he omlzed dl dsplceme e ploed fo h/=..5 d. he pesece of hee umes of emede RS. Due o he geomey d suppo codo chose fo he ssue symmecl mode shpes e epeced. As c e see hese fgues he slope of he Amec Joul of Solds d Sucues

22 48 R. Poulg d M. Nkkhh-Bhm / ee d oced Vo Alyss of Sepped Ccul Cyldcl Shells wh Sevel dl dsplceme felds.e. du/d ll suppo posos deceses he hcke d shoe cyldcl shells. Also he shoe cyldes hve hghe sesvy o he shell hckess.. h/=. h/=.5 h/=.. h/=. h/=.5 h/=. Nomlzed U Nomlzed U m 4 5 m gue 7: Mode shpes of homogeeous SD-SD shell wh hee emede g suppos fo m== em mode d dffee codos s: h/=..5. /= /=5. The mode shpes of he shell wh dffee umes of emede RS coespodg wh em mode fequeces e ploed gues 8 fo wo shell leghs.e. /= d 5 especvely. Hece hee cses e cosdeed hese fgues. Cse -: oe emede RS / =.5 Cse -: wo umes of emede RS / = / =.5 d Cse -: hee umes of emede RS / = / = / =.5. I hese fgues ssfyg of couy d ll he suppo codos ppled o such usymmecl codos c e see ovously he ll gve esuls. I gues 9 o 9f dl mode shpes e depced he l deco of sepped shell wh dffee comos of emede g suppos. Hece hee cses of shell hckess d seppes e cosdeed s: Cse -: ufom Shell Cse -: hee seppes he shell; /= / =/=.5 d h/=. h/h= h/h= d h4/h=4 d flly Cse -: wh sme codo y cse - ecep hckesses of h/h= h/h=5 d h4/h=7. I gues 9 9c d 9e ol shell legh s /=5 d shell legh h cosdeed gues 9 9d d 9f s /=5. Nume of he eo suppos used hese fgues s especvely s oe eo suppo /=.5 gues 9 d 9 wo eo g suppos /=/=.5 gues 9c d 9d d hee eo suppos /= /= /=.5 gues 9e d 9f. These fgues dce h couy d suppo codos of he ssue e compleely ssfed y he poposed mehod smuleously. Howeve y chgg he shell hckess fom cse - o cse - dl defomos omlzed modes e gded d mosly eded o he he sdes of he shell. 6. oced Vo I hs seco sepped ccul cyldcl shell wh SD-SD eded suppos d hee emede suppos of dffee ypes s cosdeed. Ohe cosdeos e: egh o /=5 hckess os h/=. d h/h= h/h= h4/h=. I ll of he followg fgues he mplude of he po lod eco s cosdeed N. The po lodg e ppled o he shell d Amec Joul of Solds d Sucues

23 R. Poulg d M. Nkkhh-Bhm / ee d oced Vo Alyss of Sepped Ccul Cyldcl Shells wh Sevel 49 /=/8 mposed hes segmes of he shell. I ll gphs of fequecy esposes fequecy ges e cosdeed fom o 5 Hz d he seps e Hz. Cse Cse Cse. Cse Cse Cse.8.8 Nomlzed U.6 Nomlzed U m m gue 8: Mode shpes coespodg wh em mode of he shell wh h/=.5 d SD-SD eded suppos hee cses of umes of ppled emede RS d wo shell leghs of /= /= Vefco of VWM Resuls To cheve ccue esuls wh low clculo me mmum umes of uco of ccumfeel mode ume whch ws desced Sec. 6 s equed o e deemed. gues depcs he dl d ccumfeel dsplceme esposes due o eco of po lod f N ecly he mddle of he fs segmes of he shell =/ gue 5 he dl deco. The esuls of hese fgues e ploed fo some dffee uco of. I hese fgues good covege esuls fo ll ucos.e. = d 4 s clely vsle. Howeve he dffeeces ewee esuls of = d 4 e eglgle d hece = s chose fo evlug d plog he equecy Respose ucos R. Usg VWM d wh he o-opmzed self-wg code he me elpsed me fo clculo whe = ws ou.9 secods. Due o lck of pope compso esuls he leue o evlue he equed VWM soluo me d s umecl pefomce we hve ulzed he fe eleme mehod povded y ANSYS fo vldo d evluo of he esuls. The foced vo lyss gve y VWM fo clsscl eded d emede suppos.e. SD d RS s cosdeed. The ulzg ANSYS eced shell wh po lod he mddle of he fs segme s he hes segme of he sepped shell s modeled usg he fou ode elemes of shell 6. Amog sevel composos of mpped meshg hee dffee meshg syles e chose fo evluo of EM covegece. These cses e: Cse -: 6 Cse -:8 d Cse -: 4 odes he d decos especvely. The esuls of ANSYS fo dffee gd szes of cses - o - e ploed gues c d d. As c e see fom gues c d d cses - d - e covege d ppomely c cove ll possle peks coespod wh ll he ul fequeces. I ovell compso ewee he esuls of VWM d he esuls of ANSYS s EM he esuls of oh mehods e good geeme he cse - fo oh of he dl d l dsplceme esposes. These compsos ewee VWM d EM e mde he g- Amec Joul of Solds d Sucues

24 5 R. Poulg d M. Nkkhh-Bhm / ee d oced Vo Alyss of Sepped Ccul Cyldcl Shells wh Sevel ufes e d f. The compuol me fo clculos of cse - ws ou mues usg wo CPU coes whee s ppomely 59 mes gee h hose gve y VWM ulzg oly oe CPU coe..8.8 Cse Cse Cse Nomlzed U.6.4. Cse Cse Cse Nomlzed U m m.8 Cse Cse Cse.8 Cse Cse Cse Nomlzed U.6.4 Nomlzed U m m c d.8 Cse Cse Cse.8 Cse Cse Cse Nomlzed U.6.4 Nomlzed U m m e f gue 9: Mode shpes of shell coespodg wh em mode =m= d shell leghs of ce /= d df /=5: Oe /=.5 cd wo /= /=.5 d ef hee /= /= /=.5 emede suppos hee dffee cses of sepped shells. Amec Joul of Solds d Sucues

25 R. Poulg d M. Nkkhh-Bhm / ee d oced Vo Alyss of Sepped Ccul Cyldcl Shells wh Sevel Po odg d lely Iemede Suppos Sepped shell espose o he dl d l po lod ecos /=.5 s evlued gues he pesece of hee flele emede suppos of kd of E IV llused gue 5. Hece he gues dffee sffess vlues K= 9 d N/m e cosdeed fo he ppled emede suppos d dl d l esposes o he ecos e ploed. Oe of hese vlues.e. N/m e coespods o he dl gd suppo o RS especvely d hece s possle o ce he R chges le wh chges of he suppo chcescs. om hese fgues depeds of dl o l ecos he followg esuls c e cocluded: s he sffess of he emede suppos decesed l mplude U s moe sesve o he dl eco d vce ves dl mpludes U s moe ffeced y he l eco. om hese fgues educo of oh dl d l mpludes cocluded. Howeve he oed esuls fo mos of peks show fesml chges he pek posos. gues -d depc h some of he peks oh dl d l decos e vshed o educed sogly he cse of K= 7 N/m. I s pomsg oucome s dces h he eel eco c e mposed sfely ce lmed ge of fequecy chges whou y esoce polem y pope seg of he suppo sffess. Howeve should e oed h he polem of sly he shell vos wh he flele suppos could e mpo ssue h s o cosdeed he pese sudy. I gues he effec of he poso of E IV flele suppos o he sucul espose of he shell s depced whe K= 8 N/m. Hee ode o eme he effec of he suppo posos fou dffee cses e cosdeed. I he cses emed ll of he eo suppos e RS ecep: Cse -4: whou flele suppo Cse -4: he lef sde eo suppo s flele Cse -4: he mddle eo suppo s flele d Cse 4-4: he gh sde eo suppo s flele. gues d lluse dl d l esposes of he shell ude dl eco especvely. I sme me he dl d l esposes of he shell ude l eco e llused gues c d d especvely. I ovell cocluso fom hese fgues he followg esuls c e cocluded: he effec of he suppo chcescs o he sucul dl d l esposes of he shell s decly eled o he dsce of he suppo poso o he eco poso. I s eesg h he l shell esposes e moe sesve o he chcescs of emoe suppos he pesece of he dl ecos. Vce ves he pesece of l ecos he dlly shell espose s moe sesve o he chcescs of he emoe suppos fom he lodg loco.e. he suppos h e posoed f fom he lodg loco. Ths cocluso s mpo fo o-desucve ess NDT puposes d ful deeco of he ppled suppos. 6.. Dsued Dmpg Effecs The effec of he sucul dmpg he fom of dsued dmpg ype o he dl d l dsplceme espose s depced gues d. om hese fgues s cle h he mpludes of peks deceses popool o he mgude of he sucul loss fco γ. Howeve he coespodg fequeces of he peks do o show y sgfc depedecy o γ d lmos he peks e depede of γ. Howeve elsewhee ecep pek posos he shell mpludes p- Amec Joul of Solds d Sucues

26 5 R. Poulg d M. Nkkhh-Bhm / ee d oced Vo Alyss of Sepped Ccul Cyldcl Shells wh Sevel pomely e depede of he sucul loss fcos d hey e o ude sgfc fluece of γ foced vo of he shell. = = =4 4 5 = = =4 R og U R og U equecy Hz equecy Hz 4 Cse Cse Cse Cse Cse Cse 5.5 R og U 4 5 R og U equecy Hz equecy Hz c d 4 EM VWM Dff. % EM VWM Dff. % R og U equecy Hz R og U equecy Hz e f gue : Covegecy emo of sucul dl d l esposes of sepped shell wh hee emede g suppos /= /= /=.5 d shell hckess h/=. d h/h= h/h= h4/h= d shell legh of /=5 usg VWM d EM cd d compg hem ef. Amec Joul of Solds d Sucues

27 R. Poulg d M. Nkkhh-Bhm / ee d oced Vo Alyss of Sepped Ccul Cyldcl Shells wh Sevel 5 R og U RS E I K = E I K = 9 R og U RS E I K = E I K = equecy Hz equecy Hz 4 5 RS E I K = E I K = RS E I K = E I K = 9 R og U R og U equecy Hz equecy Hz c d gue : Sucul esposes of sepped shell wh hee flele emede suppos d SD-SD eded suppos fo dl d l dsplceme mpludes fo dl eco d c dl d dl dsplceme mpludes fo l eco he po lod poso. 7 CONCUSION The m oecve of hs ppe ws poposg eesle mehod fo lyss of fee d foce vo of shells wh y umes of seps d flele emede suppos. To model he ssue d sed o lügge h shell heoy sepped ccul cyldcl shell ws cosdeed wh wo eded- d my emede-suppos whch ll of hem e flele wh depede chcescs. The suppo codos s well s he couy of he sepped shell wee ssfed ccuely y oducg wve vecos gog he posve-egve deco log wh he shell s d deemg elev efleco d popgo mces. Bsed o VWM coguous d eesle m-kd elos wee poposed fo fee d foced vo lyss of he ssue whee coed of 4 vecos d 4 4 mces depede of he ume of seps d emede suppos ulzed. o fee vo lyss of he shell he poposed mehod ws successfully employed o o he especve ul fequeces of he sysem. uhemoe he shell sucul esposes wee vesged ude po lod ecos. The oed esuls wee successfully vefed gs hose vlle he leue d hose oed y EM. The esuls of he pese sudy fee d foced vos of he shell c e summzed s follows: Amec Joul of Solds d Sucues

28 54 R. Poulg d M. Nkkhh-Bhm / ee d oced Vo Alyss of Sepped Ccul Cyldcl Shells wh Sevel. Sucul esposes of vg shell e hghly ffeced y he popees of s emede suppos d gog hs mpo fdg y usg smplfyg ssumpos clsscl suppos c led o sgfc eos pedco of vol ehvo of he ssue.. Accodg o he oed esuls he poposed mehod s lycl mehod povdes ccue esuls oh fee d foced vos of he shell whou y lmo popees d umes of he seps o he flele emede suppos whch wee ppled. c. I ovell compso he poposed mehod shows ge umecl pefomce vesus EM especlly he cse of po lod ecos of he ssue. I hs cse lhough ge umes of uced modes s eeded o e supeposed he equed compuol me y VWM ws ou 59 mes less h h gve y EM ANSYS. d. The oed esuls show h he l dl shell esposes e moe sesve o he chcescs of emoe suppos he pesece of dl l ecos. Ths could e mpo cocluso fo ful deeco of he ppled suppos. Cse 4 Cse 4 Cse 4 Cse Cse 4 Cse 4 Cse 4 Cse 4 4 R og U R og U equecy Hz equecy Hz 5 R og U Cse 4 Cse 4 Cse 4 Cse 4 4 R og U Cse 4 Cse 4 Cse 4 Cse equecy Hz equecy Hz c d gue : Sucul esposes of sepped shell wh hee flele emede suppos d SD-SD eded suppos fo dl d l dsplceme mpludes fo dl eco d c dl d d l dsplceme mpludes fo l eco he po lod posos. Amec Joul of Solds d Sucues

29 R. Poulg d M. Nkkhh-Bhm / ee d oced Vo Alyss of Sepped Ccul Cyldcl Shells wh Sevel 55 γ= γ=.5 γ=. 4 5 γ= γ=.5 γ=. R og U R og U equecy Hz equecy Hz gue : Sucul esposes of sepped shell wh hee flele emede suppos he dl d l decos of he shell d fo dffee sucul dmpg cosdeos. I should e oed hs ed h he poposed mehod s fully cple fo pplyg o ccue desg of sevel flele suppos wh y popees fo ce sepped cyldcl shell log wh he shell s. We eleve h he poposed mehod c e eeded d ckles complced ssues such s o-homogeeous shells d shells mde of compose lyes o fucolly gded mels. The pese mehod lso seems cple of eg ppled s poweful ss fo cool of he shell vo polems d pedcg d vefyg he NDT esuls whch c e suec of fuue sudes. Refeeces Achech J.D.. Recpocy elsodymcs Cmdge Uvesy. Bhm A. Ilkh M.R. Nkkhh-Bhm M.. Wve popgo echque fo fee vo lyss of ul ccul d secol memes. Joul of Vo d Cool : 7. Che M. Xe K. Xu K. Yu P. Y Y. 5. Wve sed mehod fo fee d foced vo lyss of cyldcl shells wh dscouy hckess ASME Joul of Vo d Acouscs 7: Che Y. J G. u Z.. ee vo lyss of ccul cyldcl shell wh oufom elsc oudy codos Ieol Joul of Mechcl Sceces 74: -. Hgedo P. Ds-Gup A. 7. Vos d wves couous mechcl sysems Joh Wley & Sos d Egld. Hu S.D. H. Tzou H.S. 5. Pecso mcoscopc cuos of polc cyldcl shell eflecos Joul of Vo Acous 7: -. Hug D. Tg. Co R.. ee vo lyss of pl og gs y wve popgo. Joul of Soud d Vo : Jg J. Yue H.H. Deg Z.Q. Tzou H.S. Cyldcl shell cool wh cee- d coe-plced phooscve skew-qud cuo sysems. Joul of Vo d Acouscs 4:45-5. Kczu D.G. 6. Epesso fo dec evluo of wve ume cyldcl shell vo sudes usg he lügge equo of moo Joul of he Acouscl Socey of Amec 9: Km H.S. Soh J.W. Jeo J. Cho S.-B.. Reduco of he dg soud of sumeged fe cyldcl shell sucue y cve vo cool Sesos :-47. ee S.-K. Mse B.R. Be M.J. 7. Wve popgo efleco d smsso cuved Bems Joul of Soud d Vo 9: Amec Joul of Solds d Sucues

30 56 R. Poulg d M. Nkkhh-Bhm / ee d oced Vo Alyss of Sepped Ccul Cyldcl Shells wh Sevel oy C.T. m K.Y Vo of cyldcl shells wh g suppo Ieol Joul of Impc Egeeg M Y. Zhg Y. Keedy D. 5. A symplecc lycl wve sed mehod fo he wve popgo d sedy se foced vo of ecgul h ples Joul of Soud d Vo 9:96 4. Mce B.R Wve efleco d smsso ems Joul of soud d vo 97:7-46. Me C. 9. Hyd wve/mode cve cool of edg vos ems sed o he dvced Tmosheko heoy Joul of Soud d Vo :9 8. Me C. 6. Epemel vldo of wve vo lyss of comple vos wo-soy mellc spce fme sed o he Tmosheko edg heoy. ASME Joul of Vo d Acouscs 8:. Me C. Mse B.R. 5. Wve efleco d smsso Tmosheko ems d wve lyss of Tmosheko em sucues. Joul of Vo d Acouscs 7:8-94. Nkkhh-Bhm M. Khoshy-A M. Rsekh-Sleh N.. Modfed wve ppoch fo clculo of ul fequeces d mode shpes y o-ufom ems Sce Ic 8B: Popov A.V. 5. A cousc-emsso echque of foced vos used fo flw deeco h-wlled pessue vessels. Russ Joul of Nodesucve Tesg 4: Tsled fom Defekoskopy 4:7. Qu M.S Rece esech dvces I he dymc ehvo of shells P : Homogeeous Shells. ASME Appled Mechcs Revews 55: Qu Y. Che Y. og X. Hu H. Meg G.. ee d foced vo lyss of ufom d sepped ccul cyldcl shells usg dom decomposo mehod Appled Acouscs 74: Reo J.M. Mce B.R.. Clculg he foced espose of wo-dmesol homogeeous med usg he wve d fe eleme mehod Joul of soud d vo : Slhf R. Mohe M.. e eleme fo cyldcl h shells ude hmoc foces e Elemes Alyss d Desg 5: 8 9. T.oy C. Y.m K. Shu C Alyss of cyldcl shells usg geelzed dffeel qudue Shock d Vo 4: Wg C. M. Swddwuhpog S. T J Rz mehod fo vo lyss of cyldcl shells wh g sffees Joul of Egeeg Mechcs :4 4. Xg Y. M Y.. Kpoch S.. Ec soluos fo vo of cyldcl shells wh emede g suppos Ieol Joul of Mechcl Sceces 44: Zhg. Xg Y. 6. Vo of ope ccul cyldcl shells wh emede g suppos Ieol Joul of Solds d Sucues 4: Zhg. Xg Y. 7. Ec soluo fo vo of sepped ccul cyldcl shells Joul of soud d vo 99: Zhou H. W. v B. W... ee vo of cyldcl shells wh elsc-suppo oudy codos Appled Acouscs 7: Zhou J. Yg B Dsued sfe fuco mehod fo lyss of cyldcl shells AIAA Joul : APPENDIX A Dffeel opeos of he equo of moo: c A. Amec Joul of Solds d Sucues

31 R. Poulg d M. Nkkhh-Bhm / ee d oced Vo Alyss of Sepped Ccul Cyldcl Shells wh Sevel 57 Amec Joul of Solds d Sucues c c whee he ove equos c=e/ρ-.5 s he speed of dspeso of logudl wves d =h/.5 s dmesoless coeffce. APPENDIX B. COEICIENTS O EEMENTS O THE MATRIX [ ] The coeffces of Eq. e s follows: A B B D A C A C A C E B. APPENDIX C. COEICIENTS O THE POYNOMIA EQUATION Coeffces A =468 peseed Eq. 4 e gve s ellow 4 8 A Ω Ω A Ω Ω A Ω Ω Ω Ω Ω Ω Ω Ω Ω A Ω Ω Ω A C. APPENDIX D. THE WAVE NUMBER AND REQUENCY ANAYSIS The oos of he Eq. 4 epese he ehvo of he elsodymc wve popgo he shell o he fequecy chges. I hs egd fou ou of egh possle oos of wve umes oed e

32 58 R. Poulg d M. Nkkhh-Bhm / ee d oced Vo Alyss of Sepped Ccul Cyldcl Shells wh Sevel eled o posve gog wves d cos fou ohe emg oos e ssoced wh egve gog wves. om hese oos fou dffee ypes of wves c e deeced depedg o he como of he el d mgy vlues of oos s Zhou w l. d Kczu 6: Type : Λ ζ Popgg wves Type : Λ ζ evesce wves Type : Λ ζ ζ 4 Sdg wves Type 4: Λ ζ ζ 4 Sdg wves Whee ζ =... 4 e el vlues. All egh dmesoless wve ume e defed s he oos of he Eq. 4 depedely s follows: Λ 5 Λ 7 G G 6 6 G9 G G 4 G G9 G G 4 G 9 9 Λ 6 Λ 7 G 6 G 6 G9 G G 4 G 9 G9 G G 4 G 9 D. whee he coeffces G = e povded he follows. Geelly he sece of dmpg vo of he cyldcl shells ll he fou ypes of wves.e. popgg decyg d sdg ype d 4 c e ceed dffee fequecy ges. The pmees used he Eqs. D. e defed s: G A4 A A6 A A G / G A 9A A A 7 A A 7 A A 7A A A / / G 4 G G 4 G G G 5 G 4 G 6 G G 7 A6 / 4A8 / G8 G5 / A8 G / 9 G6 / G 4 A8 G 8A / A8 G A6 / A8 G 4A4 A6 / A / G A4 / A8 G 4 A6 / A8 G 5 / A8 G 6 A6 / 4A8 G 7 4A4 / A8 / G A / G A G G G G G G G G G G G5 G G G G 8 D. APPENDIX E oce d momeum esuls: N M T N M M M V U U D U k U U U D U U N k U U U U M D U U M D U U U M D E. Amec Joul of Solds d Sucues