Free vibration of a magneto-electro-elastic toroidal shell

Transcription

1 icccb Nottingha Unisity Pss Pocdings of th Intnational onfnc on opting in iil and ilding ngining W izani (dito) F ibation of a agnto-lcto-lastic tooidal shll. Rdkop patnt of Mchanical ngining, Unisity of Ottaa, Ottaa, anada X.. Wang patnt of iil ngining, Unisity of hanto, hanto, PR hina Abstact In cnt yas, significant pogss has bn ad in th dlopnt and analysis of sat atials and stcts. Most analysis ok on stcts has dalt ith latily sipl gotis, naly flat plats and cylindical shlls. Fo ths gotis, any stdis ha bn condctd daling ith sch coplications as fnctional gading of atials, otion contol, and th ffct of th intaction of agntic, lctic, thal and lastic poptis. In th ok of this stdy, shlls of tooidal goty a considd. pcifically, an analysis of th f ibation of a thick-alld tooidal shll is condctd, taking into accont th intaction of agntic, lctical, and lastic poptis. A si-analytical soltion is psntd, basd on th lina thoy of lasticity, ith slts coptd sing th diffntial adat thod (QM). Rslts obtaind a copad ith slts gin in th litat fo latd gotis. h pap conclds ith an alation of th significanc of th lcto-agntic ffct in th ang of th gotic paats considd. Kyods: agnto-lcto-lastic atial, tooidal shll, ibations, diffntial adat Intodction Rcnt ok on agnto-lcto-lastic atials has dalt ainly ith flat plat and cylindical shll gotis. oltions ha bn psntd, fo xapl, fo a solid agnto-lcto-lastic cylind by chanan (), and a lti-lay thick-alld cylindical shll by Annigi t al. (6). oth of ths stdis ad s of th finit lnt thod. Rlatily f stdis to dat ha dalt ith toi o tooidal shlls. Aong thos copltd a th stdis by zo and Wang (), ho considd th ibation contol of a thin tooidal shll panl, and Rdkop and Y (), ho considd th static analysis of a pizolctic axisytic tooidal ing. In this stdy, th ali ok on th ibation of agnto-lcto-lastic plats and shlls is xtndd to co thick-alld tooidal shlls. h appoach takn aks s of th diffntial adat thod, and is alidatd by copaing slts ith thos fo to pios stdis gin in th litat. N slts a dtind fo thick-alld lastic tooidal shlls, ith and ithot agnto-lcto ffcts. onclsions a dan abot th significanc of th intacti ffct. Goty and atial h tooidal shll has a bnd adis of R, and is bondd by inn and ot cicfntial sfacs of coss-sctional adii i and o spctily (Fig. ). h shll is coplt in th idional

2 and cicfntial dictions, and φ. y assing th od shap in th cicfntial diction, th athatical soltion is dcd to th to dinsions of th coss-sction, and. h diffnt atial idalizations a considd in this stdy, isotopic lastic, othotopic lastic, and agnto-lcto-lastic. Fo th thid of ths idalizations, an othotopic lationship is assd btn th stsss and stains, and th atial constants a assd to confo to th folation discssd by Abodi () and chanan (). Fo th aios idalizations, a singl lay of hoognizd atial is assd fo th nti shll. Fig. Vi shoing fist adant of a thick-alld tooidal shll hoy h coplt st of ations fo th ibation of a agnto-lcto-lastic cylind has bn psntd by chanan () and Annigi t al. (6). h ations consid th-dinsional otion, and a in ts of lastic displacnts and agntic and lctic potntials. hy a xtndd h to co a thick-alld agnto-lcto-lastic tooidal shll. h lina stain-displacnt lations in tooidal coodinats (Rdkop 99) a gin by h,, and a th chanical displacnts in th adial, idional, and cicfntial dictions,, and φ (Fig. ), and Rcos. h coponnts of th agntic fild cto i a latd to th agntic potntial ψ thogh In a siila ay, th coponnts of th lctic fild cto i a latd to th lctic potntial Φ thogh cos sin sin cos 6 ψ ψ ψ

3 Assing a copltly copld atial (chanan ), th coplt st of constitti lations is gin in atix fo as μ μ μ ε ε ε h,, φφ, φ, φ, 6 a th chanical stss coponnts,,, φ a th coponnts of th lctic displacnt, and,, φ a th coponnts of th agntic indction. h poptis ij, ij, ij, ε ij, ij, μ ij in th constitti atix a assd to b knon constants. h ations of otion (Rdkop 99, chanan ) a h (Rcos)/, cos/, sin/, (Rcos)/, (Rcos)/, and ρass dnsity. obining th constitti lations, stain-displacnt lations, th agntic and lctic fild lations, and th ations of otions yilds fi goning ations in ts of th chanical displacnt coponnts,, and, and th potntial fnctions ψ and Φ. h goning ations a spplntd by bonday conditions hich i, fo th tooidal shll pobl, that th coponnts,, φ,, and b zo on th sfacs i and o. iffntial adat soltion A si-analytical soltion is soght fo th ibation pobl that satisfis th id syty conditions of a tooidal shll that is coplt in both th idional and cicfntial dictions (Rdkop ). h fi basic nknons a takn as Φ Φ Φ ρ ρ ρ t t t

4 ) (,,, U (, )cos sinωt Φ(,,, Φ(, )sin sinωt ) (,,, V (, )cos sinωt ψ (,,, ψ (, )sin sinωt (,,, W (, )sin sinωt h is th cicfntial a nb, ω th cicla fncy, and t th ti. bstittion into th goning ations lads, fo ach choic of, to a to-dinsional athatical pobl in and to b sold nically. In th psnt stdy, th pobl is sold sing th QM (h ). In th application of th thod to this pobl, a sh of sapling points is fist dfind in th coss-sctional plan. iatis in th goning ations and bonday conditions a placd by sis inoling th als of th basic fnctions at th sapling points. nfocnt of th goning ations and bonday conditions at th sapling points lads to a st of siltanos ations in ts of th als of th basic nknons at th sapling points, and th nknon fncy ω. hoghot in this stdy, th slts a gin in ts of a non-dinsional fncy, Ωω o ρ/.. Validation opaisons a psntd fo to spaat pobls to indicat th alidity of th psnt appoach. Fo th fist pobl, slts fo a on-dinsional QM soltion fo th natal fncis of a agnto-lcto-lastic solid cylind a copad ith slts gin ali by chanan (). Fo th scond pobl, slts fo a to-dinsional QM soltion fo th natal fncis of an isotopic tooidal shll a copad ith slts gin ali by chanan and Li (). abl. opaison of fncis Ω fo agnto-lcto-lastic* solid cylind ith slts by chanan, (abl ), nd cicfntial haonic, k fs to axial od of ibation k... ch. QM ch. QM ch. QM Mod *.6,.,,.,, 66.98,, -/, ε.6, ε.6,.797,.768,, μ -8.9, μ 9.976,.,.9, ρ. Fo th fist alidation pobl, th QM as -pogad fo a solid cylind. Folloing chanan (), th shap of th od of ibation is assd in th cicfntial and axial dictions, yilding a on-dinsional athatical pobl in th adial diction. In abl, QM slts a copad ith th slts of chanan () (abl ). h copaison is fo th nd haonic in th cicfntial diction, and fo th diffnt ods in th axial diction (haing k als of.,., and.). o sts of als a gin fo th QM, fo and adial sapling points. It is sn fo abl that th is a apid congnc in th QM soltion, and that clos agnt ith th ali slts is obtaind (axi diffnc lss than %).

5 Fo th scond alidation pobl, folloing chanan and Li (), th shap of th od of ibation is assd in th cicfntial diction, yilding a to-dinsional athatical pobl in th adial and idional dictions. In abl, slts fo th QM a copad ith slts in abl 6 of chanan and Li (). h copaison is fo th nd haonic in th cicfntial diction, and fo th diffnt thicknss als (inn adis of.,., and.7, ot adis of.). o sts of als a gin fo th QM, fo shs of x6 and x6 (sapling points in th adial and idional dictions). It can b sn fo abl that th is apid congnc in th QM soltion, and that clos agnt ith th ali slts is obtaind (axi diffnc lss than %). abl. opaison of fncis Ω fo isotopic lastic* tooidal shll ith slts by chanan & Li, (abl 6),, o., R. i...7 &L QM &L QM &L QM Mod x6 x6 x6 x6 x6 x *.,.,,,, 66, ij, ε ij, ij, μ ij, ij, ρ. 6 Rslts N slts a psntd in this sction fo to pobls. h fist pobl concns th ibation of a thick-alld othotopic lastic tooidal shll. In abl, non-dinsional fncis a psntd fo th nd haonic fo nin gotic cass. h lastic poptis ij, as gin in th tabl, cospond to th als sd by chanan (). In copaing th slts fo th tooidal adis cas R., ith cosponding QM slts fo isotopic tooidal shlls in abl, it is sn that th changs in atial poptis ij lad to only ino changs in th fndantal fncis, i.. th ibation slts a ostly dpndnt on th goty in this ang of th paats. abl. Rslts fo fncis Ω fo othotopic lastic* tooidal shll,, o. R... i i i Mod *.6,.,,.,, 66.98, ij, ε ij, ij, μ ij, ij, ρ. h scond pobl concns th ibation of a thick-alld agnt-lcto-lastic tooidal shll. In abl, non-dinsional fncis a psntd fo th sa nin gotic cass considd in th fist pobl (abl ). h atial poptis gin in th tabl no also incld intacti ffcts. In copaing th slts of abl ith th cosponding als in abl, it can b sn

6 that, fo th ang of gotic paats considd, latily ino changs in fncis a podcd by th considation of th intacti ffcts. abl. Rslts fo fncis Ω fo agnto-lcto-lastic* tooidal shll,, o R... i i i Mod *.6,.,,.,, 66.98,, -/, ε.6, ε.6,.797,.768,, μ -8.9, μ 9.976,.,.9, ρ. 7 onclsions A si-analytical soltion has bn psntd to th ibation pobl of a agnto-lcto-lastic thick-alld tooidal shll sing th diffntial adat thod. A copaison of als obtaind sing th cnt appoach indicats good agnt ith slts obtaind by oth thods. N slts obtaind fo ply lastic shlls and fo agnto-lcto-lastic shlls sho ino diffncs in th coptd natal fncis in th ang of th gotic paats considd in th stdy. Acknoldgnt h fist atho gatflly acknoldgs th sppot of th Natal cincs and ngining Rsach oncil of anada. Rfncs AOUI, J.,. Micochanical analysis of flly copld lcto-agnto-tho-lastic ltiphas coposits. at Matials and tcts,, ANNIGRI, A.R., GANAN, N., and WARNAMANI,., 6. F ibations of siply sppotd layd and ltiphas agnto-lcto-lastic cylindical shlls. at Matials and tcts,, UANAN, G.R.,. F ibation of an infinit agnto-lcto-lastic cylind. Jonal of ond and Vibation, 68, -6. UANAN, G.R., LIU, Y.J.,. An analysis of th f ibation of thick-alld isotopic tooidal shlls. Jonal of ond and Vibation, 68, -6. RKOP,. 99. A displacnt soltion in tooidal lasticity. Intnational Jonal of Pss Vssls and Piping, (), RKOP,., YU, X.,. tatic bhaio of a pizolctic tooidal ing,. In: nd Intnational onfnc on Adancs in tctal ngining and Mchanics (AM ),, san, Koa. RKOP,.. Vibation of ltilay fnctionally gadd thick shlls of oltion,. Accptd fo: th Pan- Aican ongss of Applid Mchanics (PAAM XI), Janay, Igaz Falls, azil. U,.,. iffntial Qadat and Its Application in ngining. lin: ping-vlag. ZOU,.., WANG,.W.,. Vibation contol of tooidal shlls ith paalll and diagonal pizolctic actatos, Jonal of Pss Vssl chnology,, 7-76.