Fractional Order Thermoelastic Deflection in a Thin Circular Plate

Transcription

1 Aville ://vuedu/ Al Al M ISSN: Vol Iue Decee Alicion nd Alied Meic: An Inenionl Jounl AAM Fcionl Ode eoelic Deflecion in in Cicul Ple J J ii S D We C Deuk 3 nd J Ve 4 Deen of Meic D Aedk College Deekooi Ngu-44 M Indi iijie@gilco Deen of Meic Lxinyn Iniue of ecnology Ngu-4433 M Indi dwe@ediffilco 3 Deenof Meic RM Ngu Univeiy Ngu-4433 M Indi kcdeuk@ediffilco 4 Deen of Alied Meic Pilli HOC College of ngg & ec Ryni Univeiy of Mui M Indi jyoive@gilco Ac Received: Augu 6; Acceed: Feuy 4 7 In i wok qui-ic uncouled eoy of eoeliciy ed on ie fcionl e conducion equion i ued o odel in cicul le woe lowe ufce i inined zeo eeue wee e ue ufce i inuled e edge of e cicul le i fixed nd cled Inegl nfo ecnique i ued o deive e nlyicl oluion in e yicl doin e nueicl eul fo eeue diiuion nd el deflecion e coued nd eeened giclly fo Coe eil eywod: ui-ic; eoeliciy; fcionl; inegl nfo; el deflecion; olic; e conducion MSC No: 355 6A33 4A38 74F5 898

2 AAM: Inen J Vol Iue Decee Inoducion Duing e econd lf of e wenie cenuy conidele oun of eec in fcionl clculu w ulied in engineeing lieue Indeed ecen dvnce of fcionl clculu e doined y oden exle of licion in diffeenil nd inegl equion yic ignl oceing fluid ecnic vicoeliciy eicl iology nd elecoceiy ee i no dou fcionl clculu ecoe n exciing new eicl eod of oluion of divee ole in eic cience nd engineeing I i genelly known inege-ode deivive nd inegl ve cle yicl nd geoeic ineeion Howeve in ce of fcionl-ode inegion nd diffeeniion i i no o Since e ence of e ide of diffeeniion nd inegion of iy no necey inege ode ee w no ny ccele geoeic nd yicl ineeion of ee oeion fo oe n 3 ye In Podluny i i own geoeic ineeion of fcionl inegion i Sdow on e wll nd i Pyicl ineeion i Sdow of e e clicl eoy of eoeliciy oued uc inee in ecen ie due o i nueou licion in engineeing diciline uc nucle eco deign ig enegy icle cceleo geoel engineeing dvnced icf ucue deign ec e e conducion of clicl couled eoy of eoeliciy i olic in nue nd ence edic infinie eed of ogion of e wve Clely i condic e yicl oevion Hence evel non-clicl eoie uc Lod-Suln 967 eoy Geen Lindy eoy 97 ve een ooed in wic e Fouie lw nd e olic e conducion equion e elced y oe coliced equion wic e yeolic in nue edicing finie wve ogion Geen nd Ngdi 993 develoed e eoy of eoeliciy wiou enegy diiion Cndeki 986 gve eview of eoeliciy wi econd ound ii e l udied viou ole in cylindicl doin in e conex of genelized eoelic eoie Recenly ii e l 6 udied dynic ole in fcionl ode eoeliciy wi finie wve eed In e l decde udy on ui-ic eoeliciy incooing e ie fcionl deivive gined oenu Povenko udied viou ole on qui ic fcionl ode eoeliciy Boley nd Weine 96 udied e ole of el deflecion of n xiyeic eed cicul le in e ce of fixed nd ily uoed edge Roy couduy 973 dicued e nol deflecion of in cled cicul le due o ye eing of concenic cicul egion of e ue fce i ifie e iedeenden e conducion equion Deuk nd ogde 5 deeined quiic el deflecion in in cicul le due o illy diiued nd xiyeic e uly on e oue cuved ufce wi e ue nd lowe fce zeo eeue Deuk e l 9 udied qui-ic el deflecion ole of in cled cicul le due o e geneion Deuk e l 4 dicued e el ee in ily uoed le wi el ending oen wi e ouce I i een e lieue deling wi ole of qui-ic uncouled fcionl ode eoeliciy i liied o infinie doin nd o f no one udied ole on el deflecion in e conex of fcionl ode eoeliciy Hence in e een udy n effo een de o develo eicl odel o udy el deflecion in e conex of fcionl ode eoeliciy fo finie in cicul le unde conn eeue dii-

3 9 C Deuk e l uion y qui ic oc Coe eil i coen fo nueicl uoe nd e eul fo eeue nd el deflecion e dicued nd illued giclly Foulion of e ole Conide in cicul le of ickne occuying ce D defined y z woe lowe ufce i inined zeo eeue wee e ue ufce i inuled e conn e flux i lied on e fixed cicul oundy nd eicl odel i eed conideing non-locl Cuo ye ie fcionl e conducion equion of ode fo in cicul le e definiion of Cuo ye fcionl deivive i given y Podluny 999 f n n n d f d n d n n Fo finding e Llce nfo e Cuo deivive equie knowledge of e iniil vlue of e funcion nd i inege deivive of e ode k n f L f n * k k f f k n n i e Llce n- wee e eik denoe e Llce nfo wi eec o ie fo ee e eeue of e le z i ifying ie fcionl ode diffeenil equion z z 3 wi oundy condiion 4 z 5 z z 6 nd unde zeo iniil condiion

4 AAM: Inen J Vol Iue Decee 7 9 el deflecion 7 8 e diffeenil equion ifying e deflecion funcion 9 4 M D i given Deuk e l 9 wee M i e el oen of e le defined M z z dz D i e flexul igidiy of e le denoed 3 D nd e e coefficien of e line el exnion e Young odulu nd Poion io of e le eil eecively nd Since e edge of e cicul le i fixed nd cled ie uil-in edge 3 quion 3 o 3 coniue e eicl foulion of e ole 3 Soluion o oin e exeion fo eeue funcion z nfo nd i invee nfo ove e vile nd Weine 96 z ; we fi define e finie Fouie in e nge z defined in Boley

5 9 C Deuk e l z z dz 4 z wee z z 5 nd z in z e e oiive oo of e ncendenl equion co king e inegl nfo of quion 3 8 nd wi e id of nfo quion 4 one oin z Secondly we define e finie Hnkel nfo nd i invee nfo ove e vile e nge in d wee J J nd e e oiive oo of e ncendenl equion

6 AAM: Inen J Vol Iue Decee 7 93 J Now we ly e inegl nfo o quion 6-9 nd wi e id of nfo quion one oin 3 4 king e Llce nfo of quion nd lying iniil condiion 3-4 we ge e oluion follow * * 5 On enging e e in quion 5 we ge ] [ * 6 On lying invee Llce nfo o quion 6 we ge 7 wee L ] [ 8 Hee eeen e Mig-Leffle funcion e euling doule nfo of eeue i inveed ucceively y en of e inveion foule 5 nd We oin e exeion of eeue z

7 94 C Deuk e l z z co 9 3 Deeinion of el deflecion Uing quion 9 in quion one oin M in co 3 Aue e oluion of quion 9 ifie e condiion 3 C J J 3 wee e e oiive oo of e ncendenl equion J I cn e eily own 3 Now Hence e oluion 3 ifie e condiion 3 Now Uing e following well known eul C J 33 4 C J J 34

8 AAM: Inen J Vol Iue Decee 7 95 J J 35 in quion 34 one oin J C Alo in co M 37 On ilifying ove equion we ge co in M 38 Suiuing quion 36 nd 38 ino quion 9 one oin co in 4 D J C 39 co in J D C 4 Suiuing quion 4 ino quion 3 one oin

9 96 C Deuk e l D J J co in J 4 4 Nueicl clculion 4 Dienion Fo e ke of convenience we cooe: diu of in cicul le ickne of in cicul le 4 Meil oeie nd e nueicl clculion een cied ou fo Coe Pue in cicul le wi e eil oeie : 6 el diffuiviy 34 el conduciviy k 386 W / k 3 deniy 8954kg / ecific e c 383J / kg Poion io 35 coefficien of line el exnion Lé conn Roo of ncendenl equion 65 nd e e oo of ncendenl equion J We e fo convenience / D X 647 nd e g e loed fo fcionl ode ee 5 5 deicing wek nol nd ong conduciviy nd fixed ie 5 Figue nd deic e diiuion of eeue el deflecion long e dil diecion fo viou vlue of fcionl ode ee

10 el Deflecion eeue AAM: Inen J Vol Iue Decee 7 97 e nueicl clculion een cied ou in Ml 3 oging envionen e Mig-Leffle funcion ued in e e wee evlued following Podluny α=5 α= α=5 α= Figue eeue Diiuion Funcion Figue el Deflecion Funcion X α=5 α= α=5 α= Figue eeen e eeue diiuion long e dil diecion Fo e ce 55 e vlue of e eeue ow n incee wi eec o diu Fo e ce e vlue of e eeue iniilly incee u o 7 nd en decee in e nge 7 Fo e ce deicing clicl eoeliciy e en of g i coleely diffeen coed o fcionl eoeliciy 55 I ould e noed fo nge e g ow wek conduciviy Fo e g decie nol conduciviy nd fo e g deic ong conduciviy e ce coincide wi Geen nd Ngdi eoy Figue eeen el deflecion long e dil diecion I cn e oeved fo e ce 5 5 e deflecion incee wi incee in diu nd in x-

11 98 C Deuk e l iu 3nd en gdully decee o zeo I i zeo e oue cicul edge wic coincide wi e oundy condiion ioed on e in cicul cylinde Wen e fcionl ode ee deciing wek conduciviy i deflecion i ig wee fo ce nol conduciviy nd ong conduciviy i deflecion i le Hence one cn y el conduciviy of el decee i deflecion incee 5 5 I i noed fo e g cnging vlue of fcionl ode ee e eed of wve ogion i ffeced Hence i cn e n ion fco fo deigning new eil licle o el life iuion 5 Concluion e eoy of eoeliciy ed on ie fcionl e wve equion ooed y Povenko [8] i ued o odel finie cylinde e ce nd coeond o wek nd ong conduciviy eecively wile coeond o nol conduciviy We eic ouelve o e qui-ic uncouled eoy neglecing e inei e in e equion of oion nd e couling e e qui-ic een of eoelic ole i oile if e elxion ie of ecnicl ocillion i conidely le n e elxion ie of e e conducion oce e oivion eind e conideion of e fcionl eoy i i edic eded eone o yicl iuli een in nue In el life iuion e ole deling wi finie doin e ion u unfounely due o e colexiy involved in odeling finie doin e lieue i liied o ole on infinie doin Hence i ole w develoed fo finie cylinde We cn uize in in cicul le e eeue nd deflecion e inveely ooionl o e el conduciviy of el In e ce e ie fcionl e conducion equion ineole e ndd olic e conducion equion nd e yeolic wve equion Likewie e eoeliciy ineole e clicl eoy of el ee wiou enegy diiion inoduced y Geen nd Ngdi 993 nd diing e ogion of econd ound ee Cndeki 986 RFRNCS Boley B A nd Weine J H 96 eoy of el See Wiley; New Yok USA Cndeki DS 986 eoeliciy wi Second ound: A eview Al Mec Rev Vol Deuk C nd ogde N L 5 An invee qui-ic el deflecion ole fo in cled cicul le J e See Vol Deuk C We SD nd ulkni VS 9 ui-ic el deflecion of in cled cicul le due o e geneion J e See Vol

12 AAM: Inen J Vol Iue Decee 7 99 Deuk C ndi MV nd u R 4 el ee in ily uoed le wi el ending oen wi e ouce Meil Pyic nd Mecnic Vol Geen A nd Lindy A 97 eoeliciy J liciy Vol -7 Geen A nd Ngdi PM 993 eoeliciy wiou enegy diiion J liciy Vol Lod H nd Suln Y A 967 Genelized Dynicl eoy of eoeliciy Jounl of e Mecnic nd Pyic of olid Vol Podluny I Geoeic nd yicl ineeion of fcionl inegion nd fcionl diffeeniion Fcionl Clculu nd Alied Anlyi 5:4 Podluny I 999 Fcionl Diffeenil quion An Inoducion o Fcionl Deivive Fcionl Diffeenil quion Soe Meod of ei Soluion nd Soe of ei Alicion Acdeic Pe Sn Diego-Boon-New Yok-London-okyo-oono Povenko Y 5 Fcionl e conducion equion nd ocied el ee J e See Vol 8 83 Povenko Y 9 eoeliciy wic ue fcionl e conducion equion J M Sci Vol Povenko Y 9 eoy of eoeliciy ed on e ce-ie-fcionl e conducion equion Py Sc Vol 36-6 DOI: 88/3-8949/9/36/47 Povenko Y Signling ole fo ie-fcionl diffuion-wve equion in lflne in e ce of ngul yey Nonline DynVol Povenko Y Fcionl Cneo-ye equion nd genelized eoeliciy J e See Vol Povenko Y eoie of el ee ed on ce-ie-fcionl eleg equion Co M Al Vol Roy Coudui S 973 A Noe On ui-sic el Deflecion of in Cled Cicul Ple due o R-ye Heing of Concenic Cicul Region of e Ue Fce Jounl of e Fnklin Iniue Vol ii J J ed G D nd Deuk C 5 Genelized eoelic diffuion ole in ick cicul le wi xiyeic e uly Ac Mec Vol 6-34 DOI: 7/ ii J J ed G D nd Deuk C 5 wo dienionl genelized eoelic diffuion in lf ce unde xiyeic diiuion Ac Mec Vol 6 No DOI: 7/ ii J J ed G D nd Deuk C 6 A Bief Noe on Genelized eoelic Reone in Hlf Sce due o Peiodiclly Vying He Souce unde Axiyeic Diiuion Inenionl Jounl of eodynic Vol 9-6 DOI: 554/ijo ii J J ed G D nd Deuk C 6 Dynic ole of fcionl ode eoeliciy fo ick cicul le wi finie wve eed J e See Vol 39 No -3 DOI: 8/