Analysis of Stresses and Strains in a Rotating Homogeneous Thermoelastic Circular Disk by using Finite Element Method

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1 Intnatonal Jounal of Comput Applcatons ( Volum 35 No.3, Dcmb 0 Analyss of Stsss an Stans n a Rotatng Homognous Thmolastc Ccula Dsk by usng Fnt lmnt Mtho J. N. Shama Dpatmnt of Mathmatcs Natonal Insttut of Tchnology Hampu H.P 7705 (INDIA Dnka Shama Dpatmnt of Mathmatcs D. B. R. Ambka Natonal Insttut of Tchnology, Jalanha-440 (INDIA Sho Kuma Dpatmnt of Mathmatcs D. B. R. Ambka Natonal Insttut of Tchnology, Jalanha-440 (INDIA ABSTRACT Ths stuy focuss on th fnt lmnt analyss of thmolastc fl n a thn ccula sk subct to a thmal loa an an nta foc u to otaton of th sk. Bas on th two mnsonal thmolastc thos th axsymmtc poblm s fomulat n tms of scon o onay ffntal quaton whch s solv by FM. Futh th tmpatu pofls hav bn mol wth th hlp of hat conucton quaton. Som numcal sults of thmolastc fl a psnt an scuss fo Alumnum (Al ccula sk. Kywos FM; Ccula sk; Axsymmtc; Hat Conucton. INTRODUCTION Rotatng sks a on of th funamntal componnts of many machns an mchansms an a oftn subct to loang o xctaton n th tansvs (out-of-plan cton. Afsa an Go [] analys thmolastc chaactstcs of a thn ccula FGM otatng sk havng a concntc hol an subct to a thmal loa. Yongong t al.. [] an Zhong an Yu [3] stablsh a mchancal mol fo th functonally gant matal (FGM bam wth ctangula coss scton an also scuss th ffcts of th non-homognty paamt on th stbuton of th nomal stss an on th poston of th natual axs fo sval ffnt loang cass. Th fst lna analyss of otatng sks was pfom by Lamb an Southwll [4], thy obtan th soluton fo complt sk of unfom thcknss, accountng fo th ffcts of cntfugal an bnng stsss. Mot [5] has us th Raylgh-Rtz tchnqu to nvstgat th f vbaton chaactstcs of cntally clamp, vaabl thcknss axsymmtc sks wth axsymmtc n-plan stss stbutons. An analytcal soluton n o to stuy fomaton of a otatng sk compos of a lna, lastc, sotopc an homognous matal by applcaton of mchancal an thmal loa hav bn popos by Tmoshnko an Goo [6]. Col an Bnson [7] popos a tchnqu fo tmnng th foc spons to spatally fx loas usng an gn functon appoach that pcts, n avanc, th mos that omnat sk flctons. Bnson [8] scuss th stay flcton of a tansvsly loa, xtmly flxbl spnnng sk usng a hyb of mmban an lat thos whn bnng stffnss of th sk s small. Chn an Bogy [9] obtan th vatvs of th gn valus of a flxbl spnnng sk wth a statonay loa wth spct to ctan paamts n th systm. Futh, n anoth stuy Mot [0] has also bn vlop fnt lmnt mtho (FM pocus fo plats wth sgnfcant mmban stsss. Ngh an Olson [] psnt a FM fomulaton fo analyss of sks th n a boy-fx o a spac-fx co-onat systm. Fnt lmnt Mtho (FM s on of th most succssful an omnant numcal mtho n th last cntuy. It s xtnsvly us n molng an smulaton of ngnng an scnc u to ts vsatlty fo complx gomts of sols an stuctus an ts flxblty fo many non-lna poblms. Th FM s ga as latvly accuat an vsatl numcal tool fo solvng ffntal quatons that mol physcal phnomnon [-3]. Th FM s closly lat to th classcal vaatonal concpt of th Raylgh Rtz mtho [4, 5]. Th mathmatcal thoy of th fnt lmnt has bn vlop an pomot by many sachs (Stang an Fax [6], Babuska t al.. [7] On & Ry [8] an Oua t.al.. [9].Abunant ltatus lat to sk poblms can b foun ov th yas. Th motv bhn th nt sk lat poblms a th nustal applcatons lk ccula saw blas, comput mmoy sks an sc baks. It s notc that many of ths applcatons nvolvs a otatng annula sk subct to statonay loa act tansvsly. In th psnt pap fnt lmnt tchnqu s us to valuat stsss, stans an splacmnt fo vaous cass of tmpatu stbutons fo homognous otatng ccula sk ma up of Alumnum matal.. MATHMATICAL MODL W cons a ccula sk wth a concntc ccula hol as shown n Fg.. Th sk s assum to b otatng wth angula fquncy ω. Th ogn of th pola coonat systm θ s assum to b locat at th cnt of th sk an hol. Fgu. Schmatc agam of a otatng sk wth concntc ccula hol 0

2 Intnatonal Jounal of Comput Applcatons ( Volum 35 No.3, Dcmb 0 3. BOUNDARY CONDITIONS Th sk cons n th psnt stuy s subct to a tmpatu gant fl. Th nn sufac of th sk s assum to b fx to a shaft so that sothmal contons can b pval on t. Th out sufac of th sk s f fom any mchancal loa an mantan at unfom tmpatu gant. Thus, th bounay contons of th poblm a gvn by ( a, u 0, T 0 b, 0, T ( ( 0 wh u an not splacmnt an stss along th aal cton. 4. FORMULATION OF TH PROBLM Whn a matal s subct to a tmpatu gant fl, t xpncs a stss asng fom an ncompatbl gn-stan. gnstans (Dhalwal an Sngh [0] a non-lastc stans o f xpanson stans that vlop n a boy u to vaous asons, such as phas tansfomaton, pcptaton, tmpatu chang, tc. n th psnt stuy, th gnstan s assocat wth th thmal xpanson of th sk. Snc th matal of th sk s sotopc, th thmal gnstan at a pont s th sam n all ctons whch can b gvn by ( T( ( Wh T ( s th chang n tmpatu at any stanc. Th total stan s th sum of th lastc stan an th gn stan. Thus, th componnts of th total stan a gvn by, Wh an (3 componnt of th total stan an an a th aal an ccumfntal a th aal an ccumfntal componnts of th lastc stan. Th lastc stans a lat to stsss by Hook s law. Thus ; (4 Wh an a th aal an ccumfntal stss componnts, spctvly. Th two mnsonal qulbum quaton n pola coonats th nta foc u to otaton of th sk s gvn by 0 0 wh s angula otaton of th sk an s bng th angula fquncy of th vbaton mos n th sk. Bcaus of symmty,, vanshs an a npnnt of. Thus, th scon q. (5 s ntcally satsf an th fst qulbum quaton s uc to (5 ( 0 (6 Now, substtut F F F F F F nto qs. (6 an (4 gvs ; Stan-splacmnt latons fo th axsymmtc poblm a u u, (8 Fom q. (8, t s sn that two stan componnts a lat by (. By makng us of q. (7 nto ths laton, w gt F F F (3 0 (9 Th hat conucton quaton fo a ynamc coupl thmolastc sol s gvn by [Dhalwal an Sngh [0]] K (7 T T ( 0 T C t Wh K s th thmal conuctvty, T (0 C -Spcfc hat at constant stan an 0 bng unfom fnc tmpatu Th quatons (9 an (0 consttut th mathmatcal mol consstng of scon o ffntal quatons whch povs us th functon F an th componnts of stss. 5. TMPRATUR FILD W shall cons followng th cass of thmal vaatons n th sk: Cas I: Dsk havng unfom tmpatu stbuton. In ths cas, w hav T( T0, 0 ( Cas II: Dsk at stay stat tmpatu stbuton. t In ths cas 0, so that th hat conucton quaton (0 taks th fom T 0 Upon solvng ths quaton wth th hlp of thmal contons (4, w obtan bt0 T( bt0 log( / a, ( Cas III: Non-hat conuctng sk. In ths cas thmal conuctvty K 0, so that q. (0, las to th tmpatu laton gvn by

3 Intnatonal Jounal of Comput Applcatons ( Volum 35 No.3, Dcmb 0 T0 T ( (3 C( v Upon substtutng th valus of an fom qs. (3 nto q. (3 an aangng th tms, w gt T0 F T( F (4 C T0 F F F C (5 Ths tmpatu stbutons gvn by qs.(, ( an (4 hols n th oman a b. 6. FINIT LMNT FORMULATION Substtutng th valus of fom quatons (, ( an (4 n q. (9 an followng a stana fnt lmnt sctzaton appoach, th oman of th sk s v aally nto N numb of lmnts of qual sz an abov q. (9 can b tansfom to th followng systm of smultanous algbac quatons: K F L ;, : wh K L (6,,..., f ( ( N ( ( ( f ( ( 3 ; (Cas-I bt0 f ( ( 3 ( v 3 C T 0 f ( C T0 ; ; (Cas-II ; (Cas-III (7 Th symbol us n th abov quaton ncats th lmnt numb whch s us to sctz th oman of th sk. Onc th valu of F calculat, vaous componnts of stss, stan an splacmnt can b asly calculat by th followng latons:, F F u F F F F F F 7. RSULT AND DISCUSSION 3 In ths scton, som numcal sults of thmolastc fl.. ffnt componnts of stss, stan an splacmnt a psnt fo Al sk. Th mchancal an thmal popts of th sk a sam as that of Afsa an Go []. Th Posson s ato s takn as 0.3 whch s constant thoughout th matal. Th accuacy of th fnt lmnt sults s vf by vayng th lmnt sz us to sctz th sk an t s foun that all th sults convg vy wll whn th lmnt sz s mm fo a sk of a = 5 mm an b = 50 mm. To xamn th ffct of tmpatu stbuton pofls on th componnts of stss, stan an splacmnt, th ffnt cass a cons. Futh th angula sp ω= a/s an th ato of th out aus to th nn aus b / a = 0. Fg. llustats th vaaton of aal stss vsus ( a /(b a fo vaous cass of tmpatu stbuton. It s obsv that th aal stss s maxmum n magntu fo Cas-II fo th low ang of ( a /(b a valu. But as th ato ( a /(b a ncass th aal stss bhavo also gt chang to Cas-III > Cas-II > Cas-I. Th vaaton of ccumfntal stss wth ( a / (b a ato fo ffnt cass s shown n Fg. 3. It s obsv fom th fgu that ccumfntal stss s postv fo th whol ang of ( a/ (b a ato fo cas I an ncass slghtly n th low ang of ( a/ (b a ato.. fom an thn ncass gaually fom Fo cas II, ccumfntal stss n ngatv n low ang of ( a / (b a ato.. fom 0-0.3appox., an changs ts sgn as w mov towas th hgh valus of ( a /(b a ato. In cas III, t s obsv that th ccumfntal stss s ngatv fo all low valus of ( a /(b a ato an s almost constant fo ( a / (b a ato. A p s obsv fo ccumfntal stss fo ths cas whn th valu of ( a /(b a ato achs about 0.7, an aft that a sun ncas n th valu of ccumfntal stss s notc towas th postv n. Th tn can b wttn to show th vaaton of ccumfntal stss wth th ( a /(b a ato fo th low ang s Cas III >Cas II > Cas I (n magntu, but th tn changs fo th hgh ang as Cas III> Cas I> Cas II. Fg. 4 shows th vaaton of aal stan along th ( a /(b a ato fo th ffnt cass. It s obsv that aal stan s almost constant fo Cas I. An fo Cas II an Cas III th bhavo of aal stan s almost smla fo low ang of ( a /(b a ato as n both cass aal stan s stctly changs ts sgn fom postv to ngatv valus. Fg. 5 shows th vaaton of ccumfntal stan fo vaous cass of tmpatu stbutons. It s obsv that ccumfntal stan ncass lnaly up to ( a /(b a =0. thn mans constant an postv fo Cas I fo whol ang of ( a /(b a ato. In cas II, ccumfntal stan shows a fluctuatng

4 Intnatonal Jounal of Comput Applcatons ( Volum 35 No.3, Dcmb 0 bhavo acoss th whol th ang of ( a /(b a ato. In Cas III, a qut ffnt bhavo s obsv as ccumfntal stan s ngatv up to ( a / (b a =0.8 an aft that t sunly ncass an bcom postv. Th vaaton of splacmnt wth ( a / (b a ato s pot n Fg. 6 fo vaous cass of tmpatu stbutons. A postv splacmnt wth a stay bhavo s obsv fo Cas I an th Cas II shows a smla bhavo wth oppost sgn. Dsplacmnt shows a fluctuatng bhavo n low ang of ( a /(b a ato (fom 0-0.6, thn ncass shaply n th hgh ang of ( a /(b a ato. Fg. 4: Raal Stan vsus ( a /( b a fo vaous cass of tmpatu stbuton. Fg. : Raal Stss vsus ( a /( b a fo vaous cass of tmpatu stbuton. Fg. 5: Ccumfntal Stan vsus ( a /( b a fo vaous cass of tmpatu stbuton. Fg. 3: Ccumfntal Stss vsus ( a /( b a fo vaous cass of tmpatu stbuton. Fg. 6: Dsplacmnt vsus ( a /( b a fo vaous cass of tmpatu stbuton. 3

5 Intnatonal Jounal of Comput Applcatons ( Volum 35 No.3, Dcmb 0 8. CONCLUSION An analytcal soluton fo thmolastcty qulbum quatons of a thn axsymmtc otatng sk ma of an sotopc matal s psnt. Th vaaton of ffnt componnts of stss, stan an splacmnt n aal cton s masu by applyng thmal loa wth th hlp of fnt lmnt mtho. It s foun that th thmolastc fl n sk s sgnfcantly nflunc by th tmpatu stbuton pofl. Thus, th thmolastc fl n a sk can b contoll an optmz by contollng ths paamts. Fnally th mol s hlpful n sgnng ccula cutt o gnng sk. 9. ACKNOWLDGMNTS Th autho Dnka Shama thankfully acknowlgs th fnancal assstanc pov by MHRD, GOI, Nw Dlh. 0. RFRNCS [] A.M. Afsa, J. Go, Fnt lmnt analyss of thmolastc fl n a otatng FGM ccula sk, Appl. Math. Mo. 34, (00. [] L. Yongong, Z. Hongca, Z. Nan, D. Yao, Stss analyss of functonally gant bam usng ffctv pncpal axs, Int. J. Mch. Mat. Ds., (005. [3] Z. Zhong, T. Yu, Analytcal soluton of a cantlv functonally ga bam, Compos. Sc. Tchnol. 67, (007. [4] H. Lamb, R.V. Southwll, Th vbatons of a spnnng sk, 9 Pocngs of th Royal Socty (Lonon Ss A, 99, [5] C.D. Mot J, F vbaton of ntally stss ccula sks, J. ngng. Inu. 87, (965. [6] S.P. Tmoshnko, J.N. Goo, Thoy of lastcty, Tata McGaw-Hll, 3 ton, Nw Yok (970. [7] K.A. Col, R.C. Bnson, A fast gn functon appoach fo computng spnnng sk flctons, J. Appl. Mch. 55, (988. [8] R.C. Bnson, Obsvatons on th stay-stat soluton of an xtmly flxbl spnnng sk wth tansvs loa, J. Appl. Mch. 50, (983. [9] J.S. Chn & D.B. Bogy, ffcts of loa paamts on th natual fquncs an stablty of a flxbl spnnng sk wth a statonay loa systm. J. Appl. Mch. 59, (99. [0] C.D. Mot J, Stablty contol analyss of otatng plats by fnt lmnt: mphass on slots an hols, J. Dyn. Sys. Mas. Contol 94, (97. [] G.L. Ngh, M.D. Olson, Fnt lmnt analyss of otatng sks, J. Sou. Vb. 77, 678 (98. [] J.N. Ry, An Intoucton to th Fnt lmnt Mtho, Tata McGaw-Hll, 3 ton, 005. [3] D.V. Hutton, Funamntal of Fnt lmnt Analyss, Tata McGaw-Hll, st ton, 004. [4] T. J. Chung, Fnt lmnt analyss n flu ynamcs, McGaw Hll, Nw Yok, 978. [5] M.J. Tun, R.W. Clough, H. C. Matn & L.P. Topp, Stffnss an flcton analyss of complx stuctus, J. Ao. Sc. 3, (956. [6] G. Stang, G.J. Fx, An Analyss of fnt lmnt mthos, Pntc-Hall Inc., nglwoo Clffs, Nw Jsy, 973. [7] I. Babuska, A.K. Azz, Lctus on th mathmatcal founatons of th fnt lmnt mtho, mathmatcal founatons of th fnt lmnt mtho wth applcaton to patal ffntaton quatons. A. K. Azz. Acamc Pss, 97. [8] J.T. On, J.N. Ry, Intoucton to mathmatcal thoy of fnt lmnts, John Wly & Sons, Nw Yok, 976. [9] A. Oua, A. Fahm, M. M. Toufgh, Nonlna analyss of tansnt spag by th coupl Fnt lmnt Mtho, Int. J. Mch., 5, 35-39, (0. [0] R.S. Dhalwal, A. Sngh, Dynamc Coupl Thmolastcty, Hnustan Publshng Copoaton Nw Dlh - (Ina. 4

Load Equations. So let s look at a single machine connected to an infinite bus, as illustrated in Fig. 1 below.

Load Equations. So let s look at a single machine connected to an infinite bus, as illustrated in Fig. 1 below. oa Euatons Thoughout all of chapt 4, ou focus s on th machn tslf, thfo w wll only pfom a y smpl tatmnt of th ntwok n o to s a complt mol. W o that h, but alz that w wll tun to ths ssu n Chapt 9. So lt