ANALYSIS OF THERMOELASTIC DISC WITH RADIATION CONDITIONS ON THE CURVED SURFACES

Transcription

1 Meils hysics Mechics Receive: Febuy 3 ANAYI OF HERMOEAI DI WIH RADIAION ONDIION ON HE URVED URFAE Rjeesh Ku N.K. b Vio Vghese 3 Depe of Mheics Kuushe Uivesiy Kuushe Hy Ii Depe of Mheics MJ Euciol pus RM Ngpu Uivesiy Ngpu M 4433 Ii 3 Relice Iusies iie Ngpu Mufcuig Divisio Mu Ngpu M 44 4 Ii e-il: jeesh_u@eiffil.co e-il: veeulb@gil.co e-il: vio.vghese@il.co Absc. he picipl i of his ppe is o ivesige he heoelsic pobles i ohoogeeous hic ul isc i which souces e geee ccoig o he lie fucio of he epeue wih copoue effec ue o pil heig bouy coiios of he iio ype. he soluios e bse o heoy of iegl sfoios wih bouy coiios of iio ype o he cuve sufces wih iepee iio coss. he esuls e obie i seies fo i es of Bessel s fucios. oe ueicl esuls fo he epeue chge he isplcee he sess isibuios e show i figues.. Ioucio As esul of he icese usge of iusil cosucio eils he iees i he hel sess pobles hs gow cosiebly ypifie by he ul fis of he exchges be isc oos becuse of is eleey geoey. heefoe ube of heoeicl suies coceig he hve bee epoe so f. Howeve o siplify he lyses los ll he suies wee couce o he ssupio h he uppe lowe sufces of he iscs o cicul e isule o he is issipe wih uifo he sfe coefficies houghou he sufces. Fo exple No e l. ] hs cosiee cicul ple iscusse he sie heoelsic-plsic beig poble ig use of he si icee heoe. Khobge e l. ] hs suie he isibue he supply of hi cicul ple by usig fiie Hel Fouie sfo wih Diichle s ype of bouy coiios. Vghese e l. ] suie heoelsic espose ue o pilly isibue he supply of hollow cylie sucue by usig sfo echique. Nsse 8 9] ivesige pobles ue o he souces i geelie heoelsic boy. Kuli e l. 3] eeie qusi-sic hel sesses i hic ul isc subjece o biy iiil epeue o he uppe fce wih lowe fce eo epeue. Mos of he suies cosiee by vious uhos 3 8 9] hve o cosiee y heoelsic poble fo hic ples wih bouy coiios of iio ype i which souces e geee ccoig o he lie fucio of he epeues which will lso sisfies he ie-epee he coucio equio. he sigificce of foeeioe sfo 5 6 ] ove he pevious use o publishe iegl sfo echiques 3] c be see while obiig he epeue o isplcee of y heigh 3 Isiue of obles of Mechicl Egieeig

2 76 Rjeesh Ku N.K. b Vio Vghese wih hi i bouy coiios of iio ype o he ousie isie sufces wih iepee iio cos. Fo he pevious lieues egig hic isc s cosiee i ws obseve by he uho h o lyicl poceue hs bee esblishe fo hic ul isc cosieig iel he souces geeio wihi he boy specificlly ipce by pil heig.. Foulio of he poble We cosie hic ul isc wih hicess h iel ius exel ius b 3 / occupyig he spce D { x y R : x y b h h} whee x y. e he isc i which iel souces e geee ccoig o lie fucio of he epeue e subjece o pil heig Q / exp ove he uppe sufce h is he Dic el fucio... epeue isibuio. he sie he coucio equio wih iel he geeio is give s follows whee is he iel souce fucio / λ beig he hel couciviy of he eil ρ is he esiy is he cloific cpciy ssue o be cos. Followig 5] we cosie he uegive fucios s he supeposiio of he siple fucio: exp exp o fo he se of beviy we cosie exp 3 b h h. ubsiuig equios 3 i equio yiel 4 whee is he hel iffusiviy of he eil of he isc which is ssue o be cos subjec o he iiil bouy coiios 5

3 Alysis of heoelsic isc wih iio coiios o he cuve sufces h h b h h Q h b 4 h b 3 / exp whee is he Dic Del fucio hvig b ; is cos; Q is he he flux wih cos segh; is he hel couciviy coefficie of he eil is he efeece epeue... hel isplcees hel sess. he Nvie s equios i he bsece of boy foces fo xisyeic wo-iesiol heoelsic poble c be expesse s 7]: u e e u u 8 whee u u e he isplcee copoes i he il xil iecios especively he ilio e s u u u e. he isplcee fucios i he cyliicl cooie syse e epesee by he ooie s heoelsic isplcee poeil ove s fucio s 4] u 9 u i which ooie s heoelsic poeil us sisfy he equio he ove s fucio us sisfy he equio whee. he copoe of he sesses e epesee by he use of he poeil ove s fucio s

4 whee υ e he she oulus oisso s io especively. he bouy coiios o he cio fee sufce fucios e b 7 Equios o 7 cosiue he heicl foulio of he poble. 3. oluio of he poble 3.. oluio of he he coucio poble. I oe o solve equio 4 ue he bouy coiio 6 we fisly iouce he iegl sfo 6] of oe ove he vible. e be he pee of he sfo he he iegl sfo is ivesio heoe e wie s b p g g / p p g g 8 whee g p is he sfo of g wih espec o ucleus p. Applyig he sfo efie by equio 8 o he equios ig io ccou equio 6 we obi / exp Q exp whee is he sfoe fucio of. he eigevlues e he posiive oos of he chceisic equio Y b J b Y J. 78 Rjeesh Ku N.K. b Vio Vghese

5 Alysis of heoelsic isc wih iio coiios o he cuve sufces he eel fucio c be efie s 79 J Y Y b] Y J J b] wih J i J ij Y i Y iy fo i b b] i which J Y e Bessel fucios of fis seco i of oe p especively. We iouce he ohe iegl sfo se i 5] h espos o he bouy coiios give i equio 7 s h f f f f h 3 Applyig he sfo efie by equio 3 i equios 9 usig equio we obi h Q / exp exp 6 whee is he sfoe fucio of. he sybol es fucio i he sfoe oi he ucleus is give by he ohogol fucios i he ievl h h s Q cos W si whee Q 3 4 cos h W cos h 3 4 si h h sih h Q W ] Q W ]. h he eigevlues e he posiive oos of he chceisic equio cos h si h]cos h si h] cos h si h]cos h si h]

6 Usig Eq. 6 equio 4 yiel H 7 whee exp 3 h Q H. he geel soluio of equio 7 is fucio H exp exp. 8 Usig 8 i equio 5 yiel he vlue of ubsiuig he esulig vlue of i equio 8 yiel exp exp H H. 9 Applyig ivesio heoes of he sfo ules efie by equios 8 o he equio 9 yiel ] exp exp 3 he ccoplishig ivesio heoes of he sfo ules efie by equios 3 o equio 3 he epeue is obie s: ] exp exp 3 whee H. ig io ccou he fis equio of equio 3 he epeue isibuio is filly epesee by ] exp exp exp 3 he fucio give i equio 3 epeses he epeue evey is ll pois of hic ul isc of fiie heigh whe hee e coiios of iio ype wih pil heig o he uppe sufce. 8 Rjeesh Ku N.K. b Vio Vghese

7 3.. oluio of he hel sess poble. Refeig o he fuel equio is soluio 3 fo he he coucio poble he soluio fo he isplcee fucio e epesee by he ooie s heoelsic isplcee poeil govee by equio e epesee by ] exp exp exp. 33 iilly he soluio fo ove s fucio e ssue so s o sisfy he govee coiio of equio s ] exp exp exp ] cosh sih. 34 I his e wo isplcee fucios i he cyliicl cooie syse e fully foule. Now i oe o obi he isplcee copoes we subsiue he vlues of heoelsic isplcee poeil ove s fucio i equios 9 oe obis ] exp exp u exp sih cosh 35 ] exp exp u ] cosh sih si 4 cos si W Q exp. 36 hus ig use of he wo isplcee copoes he ilio is esblishe s ] exp exp e cosh 4 sih cosh 3 exp ] sih cosh. 37 he he sess copoes c be evlue by subsiuig he vlues of heoelsic isplcee poeil fo equio 33 ove s fucio fo equio 34 i equios oe obis 8 Alysis of heoelsic isc wih iio coiios o he cuve sufces

8 ] exp exp cosh ] { ]} ] sih cosh exp 38 ] exp exp cosh ] { ]} ] sih cosh exp 39 ] exp exp cosh ] { } ] sih 3 exp 4 ] exp exp sih ] cos si { W Q } ]] cosh sih exp pecil cse ueicl clculios eig 4 /. 43 ubsiuig he vlue of equio 43 i equios 3-4 we obie he expessios fo he epeue sesses especively fo ou ueicl iscussio. 8 Rjeesh Ku N.K. b Vio Vghese

9 Alysis of heoelsic isc wih iio coiios o he cuve sufces he ueicl copuios hve bee cie ou fo Aluiu el wih pee =.65 c b=3. c h=. c; oulus of elsiciy E = N/c ; she oulus =.7 6 N/c ; oisso io =.8; hel expsio coefficie = c/c- º; hel iffusiviy =.86 c /sec; hel couciviy =.48 cl sec - /c º wih which e he posiive oos of he sceel equio J Y b J b Y which e he posiive oos of he sceel equio 3 cos h si h] cos h 4 si h] 4 cos h si h] cos h 3 si h]. I he foegoig lysis e pefoe by seig he iio coefficies coss i.86 i 3 i i 4 so s o obi cosieble heicl sipliciies. I oe o exie he ifluece of pil heig o he uppe sufce of hic isc we pefoe he ueicl clculio fo ie = ueicl viios i il iecio o he uppe sufce = whee pil he supply is pplie e epice i he followig figues wih he help of copue pog. he eive ueicl esuls fo he equios 3-4 hs bee illuse gphiclly efe Figs. -5 fo he hic isc wih iel he souce pil heig o is fl sufce =. Figue shows he viio of epeue chge log he il of he hic ul isc he hee sufce =. I is evie h epeue fucio ise gully iceses wih ie. 83 Fig.. epeue isibuio. As show i Fig. he viio of hel sess i he il iecio he ipoi of he isc. Fo he figue he locio of pois of iiu sess occus he e pois hough-he-il iecio while he hel sess espose e xiu he ieio

10 84 Rjeesh Ku N.K. b Vio Vghese so h oue eges es o exp oe h he ie sufce leig ie p beig ue esile sess. Fig.. Ril sess isibuio. Figue 3 shows he geil sess isibuio followig ecesig e log he il iecio ue o copessive sesses wihi coceic egio. Fig. 3. geil sess isibuio. Figue 4 shows he viio of he xil sess isibuio which is siil i ue bu copively lge i giue o il sess copoe. Figue 5 shows he viio of he sheig sess log he il iecio of he hic ul isc he hee sufce =. his she sess follows siusoil ue wih high ces oughs log he il iecio.

11 Alysis of heoelsic isc wih iio coiios o he cuve sufces 85 Fig. 4. Axil sess isibuio. Fig. 5. he sess isibuio. 5. oclusios I his suy we hve ee heoelsic poble of hic ul isc i which souces e geee ccoig o he lie fucio of he epeue. We successfully esblishe obie he epeue isibuio isplcees sess fucios wih iiol pil heig Q / exp vilble he ege h of he isc. he i oe o exie he vliiy of bouy vlue poble we lye s picul cse wih heicl oel fo ueicl clculios wee cie ou. We y coclue h he syse of equios popose i his suy c be pe o esig of useful sucues o chies i egieeig pplicios i he eeiio of heoelsic behviou wih iio.

12 86 Rjeesh Ku N.K. b Vio Vghese Refeeces ] M. Ishih Y. igw R. Kwu N. No // Joul of hel esses ] N.. Khobge K.. Deshuh // Joul of hel esses ] V.. Kuli K.. Deshuh // h ] A.E.H. ove A eise o he Mheicl heoy of Elsiciy Dove publicios Ic New Yo ] E. Mchi A. Fsulo // Ai ell Rele Accei elle sciee i oio ] E. Mchi. Zgblich // oc. Eibugh Mh. oc ] N. No R. Hesi Y. igw hel esses ylo Fcis New Yo 3. 8] M.E. Nsse // J. he. esses ] M.E. Nsse // J. he. esses ] V. Vghese N.W. Khobge // Ieiol Joul of Applie Mheics & isics 4 M