Nonlinear Electromechanical Stability of a Functionally Graded Circular Plate Integrated With Functionally Graded Piezoelectric Layers

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1653 Nonlina lctocanical Stability of a Functionally Gadd Cicula Plat Intgatd Wit Functionally Gadd Pizolctic Lays Abstact Tis sac dvlos nonlina lctocanical stability of a cicula functionally gadd

1 1653 Nonlina lctocanical Stability of a Functionally Gadd Cicula Plat Intgatd Wit Functionally Gadd Pizolctic Lays Abstact Tis sac dvlos nonlina lctocanical stability of a cicula functionally gadd lat intgatd wit functionally gadd izolctic lays und cossiv adial foc. Gotic nonlinaity is considd in t stain-dislacnt lation using Von-Kaan lation. T stuctu is loadd und canical and lctical loads. Distibution of lctic otntial is considd along t adial and ticknss diction. T to and botto of bot izolctic lays is sot-cicuitd. T ffct of vaious valus of non oognous indx fo bot functionally gadd (FG) and functionally gadd izolctic (FGP) lays can b considd on t sonss of t syst. Futo, a consiv study fo valuation of gotic aats can b fod on t citical loads of t stuctu. Moaad Afi a a Datnt of Solid Mcanic, Faculty of Mcanical ngining, Univsity of Kasan, Kasan, Ian, ail: Afi63@gail.co afi@kasanu.ac.i tt://dx.doi.og/ / Rcivd In Rvisd Fo Acctd Availabl onlin Kywods Stability, functionally gadd izolctic, cicula lat, nonlinaity, functionally gadd atials 1 INTRODUCTION Functionally gadd atials av bn oducd fo usag in nvionnt wit oosit conditions. Oosit conditions ans suc nvionnt tat nds two o o otis fo coving all quints. Fo xal, so sacs av tid to cat a atial tat is alicabl in saccafts. As w know, t tatu of saccaft body at out lays is vy ig wil t tatu at inn lays is not vy ig. Fo ts conditions, t sacs ovid so innovativ atials wit vaiabl otis.

2 1654 M. Afi / Nonlina lctocanical Stability of a Functionally Gadd Cicula Plat Intgatd Wit Functionally Gadd Pizolctic Lays Fo xal a cobination of caic and tal can b usd as functionally gadd atials. T atial otis a cangd gadually fo tal to caic. Tis cang ay b dscibd by a function along t ticknss o ot dinsion of t stuctu. Coosition of functionally gadd atials wit izolctic lnts ooss nw intllignt atials tat can b studid in tis a. T izolctic ffct as bn sntd scintifically by Pi and Jacqus Cui in Pizolctic stuctus a vy alicabl in t industial systs as snso o actuato in vaious gotis suc as lats, cylinds and slls. In od to contol t distibution of t dislacnt o lctic otntial in a izolctic stuctu, functionally gadd izolctic atial (FGPM) can b usd. An invstigation on t litatu can justifis ncssity of tis sac. Wu t al. (00) av usd GDQR (gnalizd diffntial quadatu ul) fo f vibation analysis of solid cicula lats. O zakc t al. (003) av focusd on t buckling load otiization of vaiabl ticknss cicula and annula lats using finit lnt aoac. Ma and Wang (003) av sntd dflction bnding of a functionally gadd cicula lat using t classical nonlina von Kaan lat toy. T lat as bn subjctd to vaious tys of loading suc as canical and tal loadings. As t sults of tat study, nonlina bnding and citical buckling tatu and tal ost-buckling bavio of t FGM lats w discussd. T stability of aatic vibations of cicula lat subjctd to in-lan focs was analyzd using t Liaunov tod by Tylikowski and Fiscut (003). Zou t al. (003) loyd Cbysv Ritz tod in t-dinsional f vibation analysis of cicula and annula lats. Ty usd a lina analysis wit sall stain assution. Basd on t gotic otis of cicula and annula lats, t vibation was dividd into t distinct catgois: axisytic vibation, tosional vibation and cicufntial vibation. Li t al. (004) usd a lag dflction bnding analysis of an axisytic sily suotd cicula lat. T incntal load tcniqu was dvlod fo solving t bnding obl of a tin cicula lat wit lag dflction. Ty av found tat t loyd tcniqu as caability in solution of ngining obls. A cicula lat containing izolctic lays as actuato und static and dynaic canical and lctical loads and using Kicoff lat odl av bn sudid by Skoui t al. (004). xints using a tin cicula aluinu lat stuctu wit distibutd izolctic actuatos w also conductd to vify t analysis and t cout siulations. Kang t al. (005) sntd a closd fo solution fo finding t natual fquncis and od sas of a cicula ototoic lat using Raylig Ritz tod. Ty av found tat t obtaind sults av caability in dsigning of cicula lats suc as wood disk as an ototoic coosit atial. Nosi and Falla (009) studid axisytic and asytic bavio of functionally gadd cicula lats und tansvs canical loading using t fist-od sa dfoation lat toy wit von Kaan non-linaity. By intoducing a stss function and a otntial function, t obl w uncould to fo quations dscibing t intio and dg-zon obls of FG lats. A tubation tcniqu, in conjunction wit Foui sis tod to odl t obl asytis, is usd to obtain t solution fo vaious clad and sily suotd bounday conditions. Cai t al. (009) studid Lag-alitud and gotically Latin Aican Jounal of Solids and Stuctus 1 (015)

3 M. Afi / Nonlina lctocanical Stability of a Functionally Gadd Cicula Plat Intgatd Wit Functionally Gadd Pizolctic Lays 1655 nonlina vibations of f-dg cicula lats wit gotic ifctions. F vibation analysis of cicula tin lat wit t tys of bounday conditions av bn studid by Yalcin t al. (009). T solution ocdu as bn fod using diffntial tansfo tod. T obtaind sults using tis si-nuical analytical solution tcniqu av bn coad wit sults of Bssl function solution. Axisytic bnding and buckling of fct functionally gadd solid cicula lats av bn studid by Saidi t al. (009) basd on t unconstaind tid-od sa dfoation lat toy (UTST). T obtaind sults av bn coad wit tos sults xtactd using low od sa dfoation tois. Saa and Saidi (009) invstigatd axisytic bnding and sttcing of functionally gadd (FG) cicula lats subjctd to unifo tansvs loading basd on fout-od sa dfoation lat toy (FOST). Gadation of usd atial as bn considd along t ticknss diction basd on a ow law function. Vivio and Vullo (010) intoducd a nw analytical tod fo valuation of lastic stsss and dfoations in t solid and annula cicula lats wit vaiabl ticknss subjctd to tansvs loading. Afi (013) and Afi and Naas (014) sntd nonlina analysis of t functionally gadd izolctic cylind and s, sctivly. T dinsional analysis of a functionally gadd izolctic sll und ulti filds as bn studid by Afi (014). An invstigation on t litatu indicats tat t is no ublisd wok to study t nonlina lctocanical stability of a functionally gadd cicula lat intgatd wit functionally gadd izolctic atials und canical and lctical loads. So usful infoation about lina and nonlina analysis of functionally gadd izolctic atials can b considd in t litatu [Asi t al. 014, Afi and Raii 011, Afi and Raii 01 (a, b, c, d,, f), Raii t al, 011, Kosgofta t al, 009]. FORMULATION Fundantal quations fo canical stability of t functionally gadd lats intgatd wit two functionally gadd izolctic lays at to and botto of lat is dvlod in t snt sction. Classical lat toy (CPT) is usd fo siulation of dfoation coonnts of t lat.,, z is usd fo coonnts of coodinat syst and uw, is usd fo sytic coonnts of lat dfoation. Basd on t abov assutions, t dfoation coonnts can b givn as follows: [Afi and Raii, 01] u (, z ) u0( ) z w (, z ) w 0( ) w 0 ( ) (1) T stain coonnts can b divd fo nonlina stain-dislacnt lation (Von-Kaan) [Afi and Raii, 01] T 1 T u u ( u ) ( u) () Latin Aican Jounal of Solids and Stuctus 1 (015)

4 1656 M. Afi / Nonlina lctocanical Stability of a Functionally Gadd Cicula Plat Intgatd Wit Functionally Gadd Pizolctic Lays Using abov lation, t in-lan coonnts of stain can b obtaind as follows: u0 w 0 1 w z ( ) z u0 z w 0 z u w w 0( ) w 0( ) z 0 z (3) T silifid notation can b sown as follows: 0 1 w ( ) w, 0 u u0 1 w 0, (4) Aft dtination of stain coonnts, t constitutiv quations can b saatly divd fo bot functionally gadd and functionally gadd izolctic lays. Ts quations fo functionally gadd lay a: [Afi and Raii, 01; Gau and Rana, 014] { } { ( )} z z 1 1 { ( )} z z 1 (5) And fo functionally gadd izolctic lays a: [Afi and Raii, 01] C C z z C C zz (6) In od to colt t constitutiv lations, in tis st, t lctic otntial distibution ust b dfind. A t dinsional distibution of lctic otntial can b sntd as ultilication of two functions on toug adial diction () and anot toug ticknss diction f ( z ). (, z ) ( ) f ( z ) (7) w, f ( z ) ust satisfy lctic otntial bounday conditions along t ticknss (tansvs) diction. T sot-cicuitd bounday conditions is considd fo bot to and botto izolctic lays. By loying a scond od aoxiation fo lctic otntial distibution along t z diction, w ll av: [baii and Rastgo, 008] z (, z ) f ( ){1 ( ) } (8) Latin Aican Jounal of Solids and Stuctus 1 (015)

5 M. Afi / Nonlina lctocanical Stability of a Functionally Gadd Cicula Plat Intgatd Wit Functionally Gadd Pizolctic Lays 1657 Using t abov quation, t lctic fild coonnts can b divd as follows: ( z, ) (, z ) f k, x k, z x ( z, ) f, ( x, y, z ) z f, z z k (9) By substitution of lctic fild coonnts fo q. (9) into constitutiv quations of izolctic lays (q. (6)), w will av: C ( z ) C ( z ) f, z f, z C ( z ) C ( z ) f, z f, z (10) In od to attain t final govning quations, t sultant of foc and onts unit widt ust b valuatd. N M ij ij ( ) ( ) ( ) ( ) ijdz, ij, zdz ij (11) As ntiond abov, t intgal ust b valuatd along t ticknss diction. Sinc t lat is includd two diffnt atials, abov intgal ust b dcoosd into two intgal ( ) ij ( ) ij ij ij N dz dz dz ( ) ij ( ) ij ij ij M zdz zdz zdz (1) Substituting t stss lations in ts of stain coonnts and lctic otntial snts t sultant of foc and onts as follows: 1 N { } { } A1 A A3 A4 A5, A N { } { } A3 A4 A7 A8 A9, A M { } { } A A11 A4 A1 A13, A M { } { } A4 A15 A8 A16 A17, A (13) w, ntiond cofficints, A ay b considd in Andix A. i i Latin Aican Jounal of Solids and Stuctus 1 (015)

6 1658 M. Afi / Nonlina lctocanical Stability of a Functionally Gadd Cicula Plat Intgatd Wit Functionally Gadd Pizolctic Lays Dfind coonnts of t sultant of foc and onts ust satisfy quilibiu quations as follows: N N N 0 M M M w 1 w F, c ( ) 0 (14) Substitution of sultant of foc and onts in quilibiu quations yilds: 1 {,, } {,, } A1, A, A3, A4, A5, A6, { } { } { A1 A3} { A A4} { A3 A7} { A4 A8} (1 ) (1 ), { A5 A9} { A6 A10} 0 3 {,, } {,, } A, A11, A4, A1, A13, A14, 1 1 (15) 3 { } { } { A A4} { A11 A15} { A4 A8} { A1 A16} (1 ) (1 ), w 1 w { A13 A17} { A14 A18 } F, c ( ) 0 lctic dislacnt quations along t t dictions a [Gau and Rana, 014]: D ( z ) ( z ), f z, z f D ( z ) ( z ) f f z z z z, zz, z (16) Discag quation tat ilis divgnc of lctic dislacnt vaniss toug t izolctic sction yilds tid quation as follows: [baii and Rastgo, 008] D D Dz (. D) dz ( ) dz z (17) Substitution of lctic dislacnt quations into discag quation yilds:, A19 {, } A0 {, } A1 {, } A {, } A3 {, } A4 {, } A A A A , 7 (18) In tis st, w can collct t ssntial quations fo valuation of t sults of t obl. Ts quations includ two canical quations and on lctical quation. Latin Aican Jounal of Solids and Stuctus 1 (015)

7 M. Afi / Nonlina lctocanical Stability of a Functionally Gadd Cicula Plat Intgatd Wit Functionally Gadd Pizolctic Lays {,, } {,, } A1, A, A3, A4, A5, A6, { } { } { A1 A3} { A A4} { A3 A7} { A4 A8} (1 ) (1 ), { A5 A9} { A6 A10} 0 3 {,, } {,, } A, A11, A4, A1, A13, A14, { } { } { A A4} { A11 A15} { A4 A8} { A1 A16} (1 ) (1 ), w 1 w { A13 A17} { A14 A18 } F, c ( ) 0 (19), A19 {, } A0 {, } A1 {, } A {, } A3 {, } A4 {, } A A A A , 7 By considing t dislacnt filds and lctic otntial as follows: n n n u un sin( )sin( ), w w n sin( )sin( ), n sin( )sin( ) (0) a b a b a b W can valuat t sults of t obl in ts of diffnt gotical and atial otis. Bfo final valuation of t sults, t distibution of atial otis ust b dfind. 3 RSULTS AND DISCUSSION Bfo solution of t obl, it is aoiat to dfin t atial otis fo t FG and FGP lays. Fo FG lay, it is assud tat t botto of t lat is stl and to of tat is caic. Tfo t distibution of t atial otis fo FG lay is (baii and Rastgo 008): 1 z n ( z ) ( c )( ) - z (1) w, ( z ), ( z ) c, is ticknss of lastic solid sction of t lat and n is t non-oognous indx of caic-tal sction of t lat. Figu 1 sow t distibution of odulus of lasticity along t ticknss diction of t lat in ts of diffnt valus of non oognous indx. Latin Aican Jounal of Solids and Stuctus 1 (015)

8 1660 M. Afi / Nonlina lctocanical Stability of a Functionally Gadd Cicula Plat Intgatd Wit Functionally Gadd Pizolctic Lays T distibution of t canical and lctical otis fo t two FGP lays can b suosd as a ow function along t ticknss diction as follows (Kosgofta t al 009; Afi and Raii, 010): z n ( z ) ( ) z i () ( w, i snts t valu of t all canical and lctical coonnts at ticknss of t izolctic sction. Ot nuical aats a considd as: z and is c 3.810, 10, 7.610, , 0.16 a a a VN VN N, N Figu 1: Distibution of vaiabl odulus of lasticity of FGM along t ticknss diction. Aft dfining t atial otis, buckling load of FG cicula lat intgatd wit izolctic lays can b valuatd. As a fist cas study, t ffct of ticknss of sat lays can b invstigatd on t buckling loads. Fo tis study, t valus of a considd. Latin Aican Jounal of Solids and Stuctus 1 (015)

9 M. Afi / Nonlina lctocanical Stability of a Functionally Gadd Cicula Plat Intgatd Wit Functionally Gadd Pizolctic Lays 1661 Sown in figu is t distibution of buckling load fo diffnt valus of atio of ticknss in ts of non oognous indx. T obtaind sults indicat tat wit incasing t atio of ticknss, t buckling load of lat dcas. Sown in figu 3 is t distibution of buc- kling load fo diffnt valus of out adius of lat ( b ) in ts of non oognous a indx. T obtaind sults indicat tat wit incasing t valus of out adius of lat ( b ), t buckling load dosn t obying a unifo bavio. Fo slctd valus of out adius ( 1.5,,3 ), it is obsvd tat fo incasing t valu of fo 1.5 till, t buckling load incass and tn wit incasing t fo till 3, t buckling load dcass. a Figu : Buckling load of a FGP cicula lat fo of diffnt valus of izolctic ticknss ( ) in ts of non oognous indx. Latin Aican Jounal of Solids and Stuctus 1 (015)

10 166 M. Afi / Nonlina lctocanical Stability of a Functionally Gadd Cicula Plat Intgatd Wit Functionally Gadd Pizolctic Lays Figu 3: Buckling load of a FGP cicula lat fo diffnt valus of out adius ( b ) in ts of non oognous indx. a Figu 4: Buckling load of a FGP cicula lat fo diffnt valus of odulus lasticity of caic ( c ) in ts of non oognous indx. Latin Aican Jounal of Solids and Stuctus 1 (015)

11 M. Afi / Nonlina lctocanical Stability of a Functionally Gadd Cicula Plat Intgatd Wit Functionally Gadd Pizolctic Lays 1663 As anot sults of tis sac, t ffct of stiffnss of atial can b considd on t sults of t obl. Tis invstigation can b fod by loying a dinsionlss aat suc as ( c ). Tis study can b fod fo t valus of ( c 1.9,1.45,1 ) and in ts of vaious non oognous indx. T obtaind sults in tis figu indicat tat t bavio of buckling load vsus incasing o dcasing t atio of odulus of lasticity ( c ) is not unifo. Wit incasing t c fo 1 till 1.45, t buckling load incass and tn wit dcasing t c fo 1.45 till 1.95, t buckling load dcass. 4 CONCLUSION lctocanical stability of a functionally gadd cicula lat intgatd wit two functionally gadd izolctic lays und adial cossiv load as bn studid in tis a. All canical and lctical otis can b vaid along t ticknss diction. T ffct of diffnt gotical and atial aats as bn considd on t buckling load of t a. T obtaind sults in tis a can dict ngins fo oduction of lctocanical stuctus in tcnical alication. Tis analysis sows tat loting a functionally gadd atial offs vaious otions fo otiizd dsign. valuation of stability and buckling load is iotant fo alication of t izolctic stuctus in diffnt conditions. T snt sults is alicabl fo ngin in fabication of izolctic stuctus as lctocanical lnts (snso o actuato). So iotant sults a xssd as follows: 1. Invstigation on t ffct of atio of ticknss on t buckling load of cicula lat indi- cats tat wit incasing t atio of ticknss, t buckling load of lat unifoly dcas.. Invstigation on t ffct of out adius of lat ( b ) indicats tat t buckling load a fo incasing t valu of fo 1.5 till incass and tn wit incasing t fo till 3, dcass. T sa conclusion ay b obsvd in studying t ffct of t atio of odulus of lasticity ( c ).Wit incasing t c fo 1 till 1.45, t buckling load incass and tn wit dcasing t c fo 1.45 till 1.95, t buckling load dcass Acknowldgnts T autos would lik to gatfully acknowldg t financial suot by Univsity of Kasan. Gant Nub: /0. Latin Aican Jounal of Solids and Stuctus 1 (015)

12 1664 M. Afi / Nonlina lctocanical Stability of a Functionally Gadd Cicula Plat Intgatd Wit Functionally Gadd Pizolctic Lays Rfncs Afi, M., Raii, G.H. (010), To lastic analysis of a functionally gadd cylind und intnal ssu using fist od sa dfoation toy, Sci. Rs. ssays, 5(1), Afi, M., Raii, G.H. and Kosgofta, M.J. (011), Otiizd dsign of a cylind und canical, agntic and tal loads as a snso o actuato using a functionally gadd izoagntic atial, Int. J. Pys. Sci., 6(7): Afi, M. and Raii, G.H. (011a), Non lina analysis of a functionally gadd squa lat wit two sat lays as snso and actuato und noal ssu, Sat. Stuct. Syst., 8(5): Afi, M. and Raii, G.H. (01b), Studying t nonlina bavio of t functionally gadd annula lats wit izolctic lays as a snso and actuato und noal ssu, Sat. Stuct. Syst., 9(): Afi, M. and Raii, G.H. (01c), T-dinsional ulti-fild quations of a functionally gadd izolctic tick sll wit vaiabl ticknss, cuvatu and abitay nonoognity, Acta. Mc., 3(3): Afi, M., Raii, G.H. and Kosgofta, M.J. (01d), lcto lastic analysis of a ssuizd tickwalld functionally gadd izolctic cylind using t fist od sa dfoation toy and ngy tod, Mcanika., 18(3): Afi, M., Raii, G.H. and Kosgofta, M.J. (01), xact solution of a tick walld functionally gadd izolctic cylind und canical, tal and lctical loads in t agntic fild, Sat. Stuct. Syst., 9(5): Afi, M., Raii, G.H. (01f), T ffct of nonoognity and nd suots on t to lastic bavio of a clad-clad FG cylind und canical and tal loads, Int. J. Ps. Vs. Piing : Afi, M., Raii, G.H. (01g), Consiv tolastic analysis of a functionally gadd cylind wit diffnt bounday conditions und intnal ssu using fist od sa dfoation toy, Mcanika, 18(1), Afi, M. (013), Nonlina tolastic analysis of tick-walld functionally gadd izolctic cylind, Acta. Mc. 4, Moaad Afi, (014), A colt st of quations fo izo-agnto-lastic analysis of a functionally gadd tick sll of volution, Latin Aican Jounal of Solids and Stuctus 11 (11): Asi, S.R, Faajou, A., Bogi, M., Hassani, A.H., (014), Tal ffcts on t stability of cicula ga sts via nonlocal continuu canics. Latin Aican Jounal of Solids and Stuctus, 14(4): Afi, M., Naas, N., (014), Nonlina lcto to lastic analysis of a tick sical functionally gadd izolctic sll, Coos Stuct, 118, Cai, C. Touzé, C. Toas, O. (009) Non-lina vibations of ifct f-dg cicula lats and slls, uo. J. Mc. A/Solids. 8: baii, F. and Rastgo, A. (008), An analytical study on t f vibation of sat cicula tin FGM lat basd on classical lat toy, Tin. Wall. Stuct. 46 (1): Gau, A. M., Rana, D. S. (014), Sa Wav Poagation in Pizolctic-Pizolctic Coosit layd stuctu, Lal. A. J. Solids. Stu. 11 (13): Kosgofta, M.J. Gobanou Aani, A. and Afi, M. (009), Tolastic analysis of a tick walld cylind ad of functionally gadd izolctic atial, Sat. Mat. Stuct., 18 (11). Kang, W. L, N-H. Pang, S. Cung, W. Y. (005), Aoxiat closd fo solutions fo f vibation of ola ototoic cicula lats, Al. Acoust. 66: Li, Q.S. Liu, J. Xiao, H.B. (004) A nw aoac fo bnding analysis of tin cicula lats wit lag dflction, Int. J. Mc. Sci. 46: Latin Aican Jounal of Solids and Stuctus 1 (015)

13 M. Afi / Nonlina lctocanical Stability of a Functionally Gadd Cicula Plat Intgatd Wit Functionally Gadd Pizolctic Lays 1665 Ma, L.S. Wang, T.J. (003). Nonlina bnding and ost-buckling of a functionally gadd cicula lat und canical and tal loadings, Int. J. Solids. Stuct. 40: Nosi, A. Falla, F. (009) Non-lina analysis of functionally gadd cicula lats und asytic tansvs loading, Int. J. Non-Lina Mc. 44: O zakc, M. Tays, N. Kolcu, i, F. (003), Buckling analysis and sa otiization of lastic vaiabl ticknss cicula and annula lats II. Sa otiization, ng. Stuct. 5: Raii, G.H. Afi, M. and Kosgofta, M.J. (011), Alication and analysis of functionally gadd izolctical otating cylind as canical snso subjctd to ssu and tal loads, Al. Mat. Mc.-ng., 3 (8): Saa, S. Saidi, A.R. (009) Axisytic bnding analysis of tick functionally gadd cicula lats using fout-od sa dfoation toy, uo. J. Mc. A/Solids. 8: Saidi, A.R. Rasouli, A. Saa, S. (009) Axisytic bnding and buckling analysis of tick functionally gadd cicula lats using unconstaind tid-od sa dfoation lat toy, Coos. Stuct. 89: Skoui,. M. Hu, Y. R. Dung Ngo, A. (004) Modling of a cicula lat wit izolctic actuatos, Mcatonics. 14: Tylikowski, A. Fiscut, K. (003) Stability and stabilization of cicula lat aatic vibations, Int. J. Solids. Stuct. 40: Vivio, F. Vullo, V. (010). Closd fo solutions of axisytic bnding of cicula lats aving non-lina vaiabl ticknss, Int. J. Mc. Sci. 5: Wu, T.Y. Wang, Y.Y. Liu, G.R. (00). F vibation analysis of cicula lats using gnalizd diffntial quadatu ul, Cout. Mtods. Al. Mc. ngg. 191: Yalcin, H. S. Aikoglu, A. Ozkol, I. (009) F vibation analysis of cicula lats by diffntial tansfoation tod, Al. Mat. Cout. 1: Zou, D. Au, F.T.K. Cung, Y.K. Lo, S.H. (003) T-dinsional vibation analysis of cicula and annula lats via t Cbysv Ritz tod, Int. J. Solids. Stuct. 40: Andix A 1 3 ( z ) dz, z ( z ) dz, z ( z ) dz A C ( z ) dz, A zc ( z ) dz, A C ( z ) dz, A zc ( z ) dz 5 6 z, z 7 8 A f ( z ) dz, A f ( z ) dz, A C ( z ) dz, A zc ( z ) dz 9 10 z, z 11 1 A f ( z ) dz, A f ( z ) dz,, A z C ( z ) dz, A z C ( z ) dz 13 ( ), 14 z, z ( ), A z f z dz A z f z dz A z, z ( ), ( ), 3 4 z, z z C ( z ) dz, A z C ( z ) dz A f ( z ) dz, A f ( z ) dz, A ( z ) dz, A z ( z ) dz A z dz A z z dz A 5 z ( ), 6 z ( ), 7 zz ( ), zz A z dz A z dz A z f dz f ( z ) dz, A f ( z ) dz Latin Aican Jounal of Solids and Stuctus 1 (015)

Loss factor for a clamped edge circular plate subjected to an eccentric loading

Loss factor for a clamped edge circular plate subjected to an eccentric loading ndian ounal of Engining & Matials Scincs Vol., Apil 4, pp. 79-84 Loss facto fo a clapd dg cicula plat subjctd to an ccntic loading M K Gupta a & S P Niga b a Mchanical Engining Dpatnt, National nstitut