An Analytical Time Domain Solution for the Forced Vibration Analysis of Thick-Walled Cylinders

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1 1741 An Anlytcl Tme Domn Soluton fo the Foced Vbton Anlyss of Thck-Wlled Cylndes Abstct n ths ppe, we popose tme domn nlytcl soluton fo the foced vbton nlyss of thck-wlled hollow cylndes n pesence of pol othotopy. n ths egd, soluton of the govenng equton s decomposed nto two pts. The ole of the fst one s to stsfy boundy condtons utlzng the method of septon of vbles besdes of Foue sees expnson of the non-homogenous boundy condtons. The second pt hs been lso expessed s the sees of othogonl chctestc functons wth the m of stsfcton of ntl condtons. The poposed nlytcl soluton hs been mplemented to evlute the dynmc esponse of the cylnde n soluton of some smple poblems whch e chosen fom pevous studes. Keywods Foced vbton, Thck-wlled cylndes, Anlytcl tme domn soluton, Pol othotopy Bsh Movhedn Meshkt Botshekn Deptment of Cvl Engneeng, sfhn Unvesty of Technology, sfhn n. b.movhedn@cc.ut.c.; bothsekn.meshkt@gml.com Receved n evsed fom Accepted Avlble onlne NTRODUCTON The ncesng pplcton of thck-wlled cylndes subjected to dynmc nne pessue n dvese felds such s eospce engneeng, cvl engneeng nd submne stuctues hs mde these membes of pmount mpotnce. n ths egds, so mny studes hve been done to compute tme dependent esponses of the both sotopc nd nsotopc cylndes (Hung, 1969; Keles nd Tutuncu, 009; Shke et l. 006; Bb nd Keles, 015; Ghnnd nd Ghoon, 015). n most of these eseches, the tme dependency of the govenng equton hs been elmnted utlzng the Lplce tnsfom (Hung, 1969; Keles nd Tutuncu, 009; Bb nd Keles, 015). Recently, Bb nd Keles (015) poposed n nlytcl soluton fo the nsotopc hollow cylndes unde the ntenl dynmc pessue n Lplce domn. They lso employed the modfed Dubn's numecl nveson to obtn soluton n tme. Snce the tnsfomton of the dynmc esponse fom the Lplce domn to the tme domn poblems s ssocted wth some dffcultes,

2 174 B. Movhedn nd M. Botshekn / An Anlytcl Tme Domn Soluton fo the Foced Vbton Anlyss of Thck-Wlled Cylndes hee the new de s employed fom pevous studes (Shmse nd Booomnd, 011; Movhedn nd Booomnd, 014; Movhedn et l. 013) to popose n nlytcl tme domn soluton fo the govenng dffeentl equton of the mentoned poblem. Ths soluton enbles us to estmte the dynmc esponses of the cylnde,.e. the dl nd hoop stesses o dl dsplcement, wth desble ccucy dectly n tme. The lyout of the ppe s s follows, n the next secton, the model used fo the dynmc nlyss of othotopc hollow cylndes s descbed nd the govenng equtons e deved. n secton 3, the supeposton pncple s employed to expess the soluton n tems of two pts. n secton 4, the poposed soluton s ppled to cses whch wee studed by Bb nd Keles (015) n ode to vldte the study. n ddton, poblem wth non-homogenous ntl condtons s ncluded n ths pt. Fnlly, n secton 5, the summy of the conclusons mde thoughout the ppe e povded. PROBLEM STATEMENT n ths secton, the govenng dffeentl equton of the vbton of thck-wlled hollow cylnde n pesence of pol othotopy s deved. Utlzng the xsymmetc condtons, the dl stn, e, nd tngentl stn, e q e elted to dl dsplcement u, s u u e =, e q =. (1) Consdeng the pol othotopy of the cylnde, the stess-stn elton cn be expessed s és ù éc C ù ée ù 11 1 s = q C1 C e ê ú ê ú ê q ú () whee C 11 nd C e the stffness modules n the dl nd ccumfeentl dectons nd C 1 s the mtel pmete tht ncludes the Posson's effect. Consde n element on the thck wlled hollow cylnde bounded by lnes (, q ) nd ( + d, q + dq). Due to symmety, the dl nd hoop stesses emn constnt long ngul coodnte,.e., [ s q] = 0 nd [ sq q] = 0, nd the she stess component, t q, must be zeo. n ths wy, equlbum equton n the dl decton gves, - ( hs ) ( ) h s s ut (,) + q = h t (3) whee u s the dsplcement component n dl decton tht must be found n pol coodnte nd tme t. Also nd h e the mtel densty nd element s thckness. Fgue 1 shows the geomety of thck wlled hllow cylnde s well s stess component on the specfed element n pol coodnte. Substtutng defntons of (1) nd (), n the bove elton leds to the followng equton, (the element s thckness hs been emoved fom both sdes of (3)), Ltn Amecn Jounl of Solds nd Stuctues 14 (017)

3 B. Movhedn nd M. Botshekn / An Anlytcl Tme Domn Soluton fo the Foced Vbton Anlyss of Thck-Wlled Cylndes ( hs ) ( ) h s s ut (,) + q = h t (4) n whch, c = C11 nd n = C C11 s non-dmensonl pmete tht ndctes degee of nsotopy of mtel. f the vton of ntenl nd extenl pessues e espectvely expessed by functons () stted s follow Pt nd P () t, the govenng boundy condtons t E s = nd = k cn be =- P (), t s = P (). t (5) = = k E Afoementoned condtons cn be expessed n tems of dl dsplcement, u, s follow é æ u 1 u öù é æ u 1 u öù C + P (), t C = - + = PE (), t ê ç ç ú ê ú è ø = è ø = k (6) whee = C11 C1. The genel fom of the ntl dsplcement nd ntl velocty condtons of cn be stsfed by the followng equton é ut (,) ù u(), é ut (,) ù = = u (). (7) t= 0 0 t= 0 0 Fgue 1: The geomety of thck wlled hllow cylnde n pol coodnte. 3 THE SOLUTON METHOD The m hee s to fnd the tme domn nlytcl soluton of the Equton (4) though employng Foue s sees expnson of boundy condtons s well s defnng sutble chctestc poblem to stsfy ntl condtons. n ths egd, we splt the soluton nto two pts s follows: Ltn Amecn Jounl of Solds nd Stuctues 14 (017)

4 1744 B. Movhedn nd M. Botshekn / An Anlytcl Tme Domn Soluton fo the Foced Vbton Anlyss of Thck-Wlled Cylndes ut (,) = u(,) t + u(,) t (8) 1 n the bove elton the ole of ut 1 (,) s to fully stsfy the boundy condtons stted n (6). Afte detemnng ut, 1 (,) obvously, the tsk of stsfyng the ctul ntl condtons emns fo u (,) t whch wll be explned lte. Po to constucton of ut, 1 (,) one should expnd the ght hnd sdes of condtons n (6) n tems of Foue sees n tme;.e. Foue sne sees s: N P () t = A sn w t, A = P ()sn t w tdt, = 1 T 0 T å ò (9) N P () t = B sn w t, B = P ()sn t w tdt, E = 1 T 0 T å ò (10) whee N s the numbe of the bss functons to be used, w = pt nd T s the length of fnte tme ntevl ( t Î [0, T] nsted of t Î [0, )). The mgntude of T my be detemned by nspecton,.e. n successve solutons, one cn enlge T untl the fnl soluton to u conveges to soluton fo smlle tme ntevl, t Î [0, T1], T1 < T. Utlzng the method of septon of vbles leds to expess ut 1 (,) n the followng fom: N u (,) t u snw t = å (11) 1 1, = 1 whee u 1, () s the soluton to the followng odny dffeentl equton whch comes fom substtutng (11) n Equton (4). du 1, du1, w + + ( - n ) u1, = d d c 0. (1) The foementoned equton s known s the Bessel dffeentl equton of ode n. The soluton of whch cn be expessed s the combnton of the Bessel functon of the fst knd, J ( w c), nd the second knd, Y ( w c). n othe wods, the soluton cn be stted s n n u ( ) = cj( wc) + cy( wc) (13) 1, 1, n, n The constnt coeffcents c 1, nd c, n the bove elton e detemned by stsfcton of the dl stess boundy condtons t = nd = k n (6) s é ædu1, 1 u öù 1, C 11 + = -A, ç d ê çè øú = (14) Ltn Amecn Jounl of Solds nd Stuctues 14 (017)

5 B. Movhedn nd M. Botshekn / An Anlytcl Tme Domn Soluton fo the Foced Vbton Anlyss of Thck-Wlled Cylndes 1745 é ædu1, 1 u öù 1, C 11 + = B, ç d ê çè øú = k (15) Fnlly, fte dong some smplfctons, ut 1 (,) s theefoe wtten s whee Y ( w c) j + J ( w c) j u (,) t = sn t 1 N c n 1, n, å w (16) = 1 C11 j3, + j4, j = ck( n- 1) BJ ( w c) + c( n-1) AJ ( wk c) 1, n n - k w [ B J ( w c) + AJ ( wk c)] n-1 n-1 (17) j =-cn ( - 1)[ kby ( w c) + AY ( w k c)], n n + k w [ B Y ( w c) + A J ( wk c)] n-1 n-1 (18) 3 n n n n j3, = c( n-1) [ J( wkcy ) ( wc) - J( wcy ) ( wkc)] (19) j =+ w { c( n -1)[- J ( w k c) Y ( w c) + kj ( w c) Y ( w k c) 4, n n-1 n n-1 - kj ( w k c) Y ( w c) + J ( w c) Y ( w k c)] n-1 n n-1 n + kw [ J ( w k c) Y ( w c) -J ( w k c) Y ( w k c)]} n-1 n-1 n-1 n-1 (0) At ths pont, the second pt of elton (8) must be detemned. n ths egd, the method of septon of vbles s ppled by substtutng n (4) whch yelds: u t (,) = u() Tt () (1) () () + - du du n u() d d 1 dtt () = = - b, u () ctt () dt () whee - b s the septon constnt. The solutons of the two septed odny dffeentl equtons n () fo u () nd Tt () e espectvely expessed s: nd u () = cj ( b ) + cy ( b ) (3) 3 n 4 n Tt () = sncbt + bcoscbt (4) Ltn Amecn Jounl of Solds nd Stuctues 14 (017)

6 1746 B. Movhedn nd M. Botshekn / An Anlytcl Tme Domn Soluton fo the Foced Vbton Anlyss of Thck-Wlled Cylndes n ode to detemne the unknown coeffcents of c 3, c 4 nd b, chctestc poblem must be fomed by substtutng (1) n the homogenous fom of stess boundy condtons,.e., Pt () = P() t= 0 n (6), whch esults n followng system of lgebc equtons, E AC=0 b (5) n the bove elton, A b s mtx dependng on b, nd C s 1 vecto contnng the coeffcents of c 3 nd c. 4 The components of A b e: é bjn- 1( b) + (1- n) Jn( b) byn- 1( b) + (1-n) Yn( b) ù A b =. (6) kbjn- 1( kb) + (1- n) Jn( kb) kby n-1( kb) + (1-n ) Yn( kb) ê ú k k To hve non-tvl soluton to (5), the detemnnt of A s set zeo. A b = 0. (7) The bove mentoned ssue non-stndd egenvlue poblem whch should be solved fo b. Moeove, by substtutng the oots of (7),.e. b, j = 1,,... n (5), the coespondng null spce j Cj, j = 1,,... wth components c 3,j nd c 4,j, hve been computed. n ths wy, the second pt of the ut (,) s theefoe wtten s M u (,) t = å u () éj sncbj t + bj cos cbj tù ê ú, (8), j j = 1 n whch u, j( ) = c3, jjn( bj) + c4, jyn( bj) nd ( b j, j) denote set of new unknown coeffcents to be detemned by stsfyng the ntl dsplcement nd velocty condtons n (7). Utlzng the othogonlty of the set u, j() wth espect to weght functon w () =, (See Hldebnd (1976), fo moe detls on the popetes of Stum-Louvlle poblems), these coeffcents hve been detemned by nsetng (8) nd (11) n (7) s follows j = k ò é N ù u, j() u0() u1, () w - å d ê = 1 ú, k ò cbj éu, j() ù d (9) Ltn Amecn Jounl of Solds nd Stuctues 14 (017)

7 B. Movhedn nd M. Botshekn / An Anlytcl Tme Domn Soluton fo the Foced Vbton Anlyss of Thck-Wlled Cylndes 1747 b j = k ò k ò u () u () d, j 0 éu, j() ù d. (30) n ths wy, the dl dsplcement of thck-wlled cylnde cn now be wtten s: N c Yn( w / c) j1, + Jn( w / c) j, ut (,) = å sn( wt) + C j + j M å j = 1 = , 4, [ c3, jjn( bj) + c4, jyn( bj)] éj sn( cbjt) bj cos( cbjt) ù +. (31) 4 RESULTS AND DSCUSSONS The poposed method hs been used fo both homogenous nd nonhomogeneous ntl dsplcement condtons. n the both cses the specfctons of othotopc hollow cylnde e tken fom Bb nd Keles (015), s C 11 = 1, = 1, = 1 nd k =. Moeove, the nlytcl soluton n (31) hs been computed usng the fst 100 tems of the sees of ut 1 (,) wth T = 80 fo the Foue sn sees expnson of Pt () n (9), nd the fst 100 tems of the sees of u (,) t,.e. M = 100. n Tble 1, the fst 15 sets of egenvlues, b j, nd to of the components of the elted egenvecto, C j, hve been povded fo thee types of mtel wth dffeent degees of nsotopy. n = 0.5, α = 3 n =, α = n = 1, α = j β j c 3,j / c 4,j β j c 3,j / c 4,j β j c 3,j / c 4,j Tble 1: Results of the fst 15 sets of the non-stndd egenvlue poblem n (5) fo thee nsotopy's types of the mtel. Ltn Amecn Jounl of Solds nd Stuctues 14 (017)

8 1748 B. Movhedn nd M. Botshekn / An Anlytcl Tme Domn Soluton fo the Foced Vbton Anlyss of Thck-Wlled Cylndes n the cse of homogenous ntl condtons, thee dffeent dynmc nne pessue functons wee consdeed s Pt () = 1, Pt () = 1- exp(0.8) t nd Pt () = 1- cos(0.8) t to vldte esults wth those stted n Bb nd Keles (015) (The vtons of oute pessue ws not consdeed n Pt= ). n ths egds, the vton of dl dsplcement, the mentoned efeence,.e. () 0 E ut (,), nd hoop stess, s q(,) t t = e llustted n Fgues to 4. As cn be seen, the esults of n Bb nd Keles (015) nd pesented method follow sml tends n evluton of both u nd s q. () (b) Fgue : Vton of () ut (, ) nd (b) q(,) t s due to Pt () = 1 fo thee dffeent degees of nsotopy. () (b) Fgue 3: Vton of () ut (, ) nd (b) q(,) t fo thee dffeent degees of nsotopy. s due to Pt () = 1- exp(0.8) t Ltn Amecn Jounl of Solds nd Stuctues 14 (017)

9 B. Movhedn nd M. Botshekn / An Anlytcl Tme Domn Soluton fo the Foced Vbton Anlyss of Thck-Wlled Cylndes 1749 () Fgue 4: Vton of () ( t, ) (b) u nd (b) s (,) t due to Pt () = 1- cos(0.8) t q fo thee dffeent degees of nsotopy. As mentoned pevously, the pesented nlytcl soluton s ble to pedct dynmc esponse of the hollow cylnde even n pesence of nonhomogeneous ntl nd extenl pessue boundy condtons. n ths egds, the second smple poblem hs been chosen to nvestgte the foced vbton of the othotopc cylnde due to followng vtons of the nne nd oute pessues, P () t = 1- cos(0.8), t P () t = 1 (3) The ntl dsplcement nd ntl velocty condtons e lso consdeed s follows: 0 0 Fgue 5 depcts the vtons of (,) n = 0.5 nd 3 E u () = ( -)( - k), u () = 0 (33) = t two tme steps s t nd q (,) t s wthn the thckness of the cylnde wth t =. t = nd 5 n ode to povde desgn cte fom the stndpont of ftgue of othotopc cylndes, the vtons of s q(,) t nd sq ( k, t ) e llustted n Fgue 6 fo thee dffeent degees of nsotopy. As cn be seen, ncesng degees of nsotopy, n, wll esult n n ncese n the fequency of cylnde's esponse Ltn Amecn Jounl of Solds nd Stuctues 14 (017)

10 1750 B. Movhedn nd M. Botshekn / An Anlytcl Tme Domn Soluton fo the Foced Vbton Anlyss of Thck-Wlled Cylndes () (b) (c) (d) Fgue 5: Vton of () s (,), (b) s (,5), (c) s q (,) nd (d) q (,5) s wthn the cylnde s thckness. () (b) Fgue 6: Vton of () q(,) t s, (b) s ( k, t ) due to Pt () = 1- cos(0.8) t nd P () t = 1 E fo thee dffeent degees of nsotopy. q Ltn Amecn Jounl of Solds nd Stuctues 14 (017)

11 B. Movhedn nd M. Botshekn / An Anlytcl Tme Domn Soluton fo the Foced Vbton Anlyss of Thck-Wlled Cylndes CONCLUSONS n the pesent study, sem nlytcl tme domn soluton hs been poposed fo the govenng equton to the vbton of thck-wlled hollow cylnde n the pesence of pol othotopy. The effects of dffeent mtel popetes nd ntenl pessue vtons on the dynmc esponses of hollow cylnde hve been nvestgted. The suffcent ccucy of the pesented method hs been lso llustted n compson of the obtned esults wth those epoted n Bb nd Keles (015). Fnlly, the supeotes of the pesented soluton cn be lsted s follows: Employng the nlytcl soluton, the dynmc esponse of the cylnde cn be evluted dectly n tme wth no need to use ny tnsfomton such s nvese Lplce tnsfom. The poposed scheme cn be used to evlute dynmc esponse of pol othotopc cylndes n pesence of exteo pessue o non-homogenous ntl condtons, whch my be useful fo desgnng puposes. The pesented method cn be extended to evlute tnsent esponse of the ppe conveyng flud due to ntenl nd extenl tempetue vtons. Refeences Bb, S., Keles,., (015). A novel ppoch to foced vbton behvo of thck-wlled cylndes. ntentonl Jounl of Pessue Vessels nd Ppng 137: 7. Ghnnd, M., Ghoon, H., (015). Elstc nlyss of pessuzed thck FGM cylndes wth exponentl vton of mtel popetes usng TSDT, Ltn Amecn jounl of solds nd stuctues, 1(6): Hldebnd, F.B., (1976). Advnced clculus fo pplctons nd edton, Pentce-Hll nc. (New Jesey). Hung, C.L., (1969). On foced vbton of nsotopc cylndes. Appled Scentfc Resech 0(1):1-15. Keles,., Tutuncu, N., (009). Effect of nsotopy on xsymmetc dynmc esponse of thck-wlled cylndes. ntentonl Jounl of Pessue Vessels nd Ppng 86: Movhedn, B., Booomnd, B., (014). The soluton of dect nd nvese tnsent het conducton poblems wth lyeed mtels usng exponentl bss functons. ntentonl Jounl of Theml Scence 77: Movhedn, B., Booomnd, B., Soght, S., (013). A Tefftz method n spce nd tme usng exponentl bss functons: Applcton to dect nd nvese het conducton poblems. Engneeng Anlyss wth Boundy. Element methods 37: Shke, M., Akhlgh, M., Hosen, S.M., (006). Vbton nd dl wve popgton velocty n functonlly gded thck hollow cylnde. Composte Stuctues 76: Shmse, B., Booomnd, B., (011). Exponentl bss functons n soluton of lmnted stuctues. Composte Stuctues 93: Ltn Amecn Jounl of Solds nd Stuctues 14 (017)