Journal of Mecanical Science and Tecnology (9 7~84 Journal of Mecanical Science and Tecnology www.springerlink.com/content/78-494x DOI.7/s6-9-4- Vibration of functionally graded cylindrical sells based on different sear deformation sell teories wit ring support under various boundary conditions M. M. Najafizade * and M. R. Isvandzibaei Department of Mecanical Engineering, Islamic Azad University, Arak Branc, P.O. Box 85/567, Iran (Manuscript Received April 9, 8; Revised October, 8; Accepted Marc 9, 9 -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Abstract In te present work, study of te vibration of tin cylindrical sells wit ring supports made of a functionally gradient material (FGM composed of stainless steel and nickel is presented. Material properties are graded in te tickness direction of te sell according to volume fraction power law distribution. Effects of boundary conditions and ring support on te natural frequencies of te FGM cylindrical sell are studied. Te cylindrical sells ave ring supports wic are arbitrarily placed along te sell and wic imposed a zero lateral deflection. Te study is carried out using different sear deformation sell teories. Te analysis is carried out using Hamilton s principle. Te governing equations of motion of a FGM cylindrical sells are derived based on various sear deformation teories. Results are presented on te frequency caracteristics, influence of ring support position and te influence of boundary conditions. Te present analysis is validated by comparing results wit tose available in te literature. Keywords: Vibration; Functionally gradient materials; Hamilton's principle; Cylindrical sell; Ring support --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------. Introduction Tis paper was recommended for publication in revised form by Associate Editor Eung-Soo Sin * Corresponding autor. Tel.: +9 886465, Fax.: +9 8864578 E-mail address: moammadnajafizade@yaoo.com KSME & Springer 9 Cylindrical sells ave found many applications in industry. Tey are often used as load bearing structures for aircrafts, sips and buildings. Te study of te vibration of cylindrical sells is an important aspect in teir successful application. Te study of te free vibrations of cylindrical sells ave been studied extensively. Among tose wo ave studied te vibrations of cylindrical sells include Arnold and Warburton [], Ludwig and Krieg [], Cung [], Soedel [4], Forsberg [5], Bimaraddi [6], Soldatos and Hajigeoriou [7], Bert and Kumar [8], and Soldatos [9]. Recently, te present autors also presented studies on te influence of boundary conditions on te frequencies of a multi layered cylindrical sell [], and on te free vibrations of rotating cylindrical sells []. In all te above works, tin sell teories based on Love ypotesis were used. Vibration of cylindrical sell wit ring support is considered by Loy and Lam []. Te concept of functionally gradient materials (FGMs was first introduced in 984 by a group of materials scientists in Japan, [], as a means of preparing termal barrier materials. Since ten FGMs ave attracted muc interest as eat-sielding materials. An excellent collection of works on te vibration of cylindrical sells wit termal stresses and deformations can be found in [4, 5], and [6]. Najafizade and Isvandzibaei presented te vibration of functionally graded cylindrical sells based on iger order sear deformation plate teory wit ring support [7]. Vibration study of te FGM sell structures is im-
M. M. Najafizade and M. R. Isvandzibaei / Journal of Mecanical Science and Tecnology (9 7~84 7 portant. However, study of te vibration of FGM sells wit ring supports is limited. In tis paper a study on te vibration of cylindrical sells wit ring supports made of functionally gradient material (FGM is presented. Te functionally gradient material considered is composed of stainless steel and nickel were te volume fractions follow a power law distribution. Te analysis is carried out using Hamilton s principle. Studies are done for cylindrical sells wit simply supported simply supported SS SS, clamped clamped C-C, free free F F, clamped simply supported C SS, clamped free C-F and free simply supported F SS boundary conditions wit an arbitrary ring support along te axial direction of te cylindrical sell. Results presented include te frequency caracteristics of cylindrical sells wit ring supports, te influence of ring support position and te influence of boundary conditions. Te present analysis is examined by comparing results for functionally graded cylindrical sells witout ring supports wit oters in te literature.. Functionally gradient materials Consider a cylindrical sell of radius R, lengt L and tickness made of FGM. For te cylindrical sell made of FGM te material properties suc as te modulus of elasticity E, Poisson ratio ν and te mass density ρ are assumed to be functions of te volume fraction of te constituent materials wen te coordinate axis across te sell tickness is denoted by z and measured from te sells middle plane and N is te power-law exponent, N. Te functional relationsips between E, ν and ρ wit z for a stainless steel and nickel FGM sell are assumed as [8], z+ E = E( z = ( E E + E z+ ν = ν( z = ( ν ν + ν z+ ρ = ρ( z = ( ρ ρ + ρ N N N ( ( ( From tese equations, wen z = /, E = E, ν = ν and ρ = ρ, and wen z = /, E = E, ν = ν and ρ = ρ. Te material properties vary continuously from material at te inner surface of te cylindrical sell to material at te outer surface of Fig.. Geometry of a cylindrical sell wit ring support. te cylindrical sell. A cylindrical sell composed of functionally gradient material is essentially an inomogeneous sell consisting of a mixture of isotropic materials. For suc a sell, unlike sells composed of fiber-reinforced materials were transverse sear deformation effects can be significant because of te ig elastic modulus to te transverse modulus ratio, if te tickness-to-radius ratio is not more tan.5, classical tin sell teory is valid.. Strains-displacement relationsips Te strain-displacement relationsips for a tin sell are [9], U U A A = + + U α A ( + α A α R R U U A A + + U α A ( + α A α R R U (4 (5 = (6 α α α A( + A( + R U R U = ( + ( α α α α A ( + α A( + A( + α A ( + R R R R α U U = A ( + ( + R α α α A( + A( + α R R α U U = A ( + ( + R α α α A ( + A ( + α R R (7 (8 (9 r r A =, A = ( α α
74 M. M. Najafizade and M. R. Isvandzibaei / Journal of Mecanical Science and Tecnology (9 7~84 Were Fig.. Geometry of a generic sell. were A and A are te fundamental form parameters or Lame parameters, U, U and U are te displacement at any point ( α, α, α, R and R are te radius of curvature related to α, α and α respectively. Te tird-order teory of Reddy [] used in te present study is based on te following displacement field ( α, α α. φ ( α, α α. ψ ( α, α α. β ( α, α ( α, α α. φ ( α, α α. ψ ( α, α α. β ( α, α ( α, α U = u + + + U = u + + + U = u ( Tese equations can be reduced by satisfying te stress-free conditions on te top and bottom faces of te laminates, wic are equivalent to = = at z =± Tus u u U = u( α, α + α. φ( α, α C. α( + φ+ R A α u u U = u( α, α + α. φ( α, α C. α( + φ + R Aα U = u( α, α ( 4 were C =. Substituting Eq. ( into nonlinear strain-displacement relation (4-(9, we can obtain for te tird-order teory of Reddy k k = + α k + α k k k γ γ γ = α + α + γ γ γ ( (4 u u A u ( + + A α AA α R u u A u = ( + + A α AA α R A u A u ( + ( A α A A α A φ φ A k ( + A α AA α φ φ A k = ( + A α AA α A φ A φ k ( ( ( + A α A A α A (5 (6 u φ u A u A u u ( ( + + + ( + φ + k A Rα α Aα Aα α α AA R Aα u φ u A u A u u ( ( + + + ( + φ + k A Rα α Aα Aα α α AA R Aα = C A u φ u A u ( ( ( + ( + + k 4 A Rα A α A A αα A α α A u φ u A u + ( ( + ( + 4 A R α A α A A α α A α α γ u u ( φ + R A α = u u ( φ + γ R A α γ u u ( + φ + R A α = C u u ( + φ + γ R A α γ u u ( + φ + R A α R = C u u ( + φ + R Aα γ R (7, (8 (9 were ( ε, γ are te membrane strains and ( k, k, γ, γ are te bending strains, known as te curvatures. Substituting C = into Eq. ( we get ( α, α α. φ ( α, α ( α, α α. φ ( α, α ( α, α U = u + U = u + U = u ( were Eq. ( is first-order teory of Reddy used in
M. M. Najafizade and M. R. Isvandzibaei / Journal of Mecanical Science and Tecnology (9 7~84 75 te present study wic is based on te following displacement field. Substituting Eq. ( into nonlinear strain-displacement relation (4-(9, we can obtain for te first-order teory of Reddy k = + α k k γ = γ Were u u A u ( + + A α AA α R u u A u = ( + + A α AA α R A u A u ( + ( A α A A α A φ φ A k ( + A α AA α φ φ A k = ( + A α AA α A φ A φ k ( ( ( + A α A A α A γ u u ( φ + R A α = u u ( φ + γ R A α ( ( ( (4 (5 were ( ε, γ are te membrane strains and k is te bending strain, known as te curvatures. 4. Stress-strain relationsips Consider a cylindrical sell wit internal ring support as sown in Fig.. R is te radius, L is te lengt, is te tickness and a, is te position of te ring support along te axial direction of te cylindrical sell. Te reference surface is cosen to be te middle surface of te cylindrical sell were an ortogonal coordinate system x, θ, z is fixed. Te deformations of te sell wit reference to tis coordinate system are denoted by U, U and U in te x, θ and z directions, respectively. For a tin cylindrical sell, te stress-strain relationsip are defined as, σ Q Q σ Q Q σ = Q44 σ Q 55 σ Q 66 (6 For an isotropic cylindrical sell te reduced stiffness Q (i, j=, and 6 is defined as ij E Q = Q = ν (7 ν E Q = ν (8 E Q44 = Q55 = Q66 = ( + ν (9 were E is Young's modulus and ν is Poisson's ratio. Defining { Aij, Bij, Dij, Eij, Fij, Gij, Hij} / 4 5 6 Qij {,,,,,, } = / α α α α α α dα ( were Q ij are functions of z for functionally gradient materials. Here A ij denote te extensional stiffness, D ij te bending stiffness, B ij te bending-extensional coupling stiffness and Eij, Fij, Gij, H ij are te extensional, bending, coupling, and igerorder stiffness. 5. Te stress resultants For a tin cylindrical sell te force and moment results are defined as N σ M σ N d M d N σ M σ = σ α, = σ α α P σ P σ P d d P σ P σ = σ α α, = α α ( (
76 M. M. Najafizade and M. R. Isvandzibaei / Journal of Mecanical Science and Tecnology (9 7~84 Q σ R σ Q = dα, = α dα σ R σ ( 6. Te equations of motion for vibration of a generic sell Te equations of motion for vibration of a generic sell can be derived by using Hamilton's principle wic is described by δ ( Π Kdt =, Π = U V (4 t t were K, Π, U and V are te total kinetic, potential, strain and loading energies, t and t are arbitrary time. Te kinetic, strain and loading energies of a cylindrical sell can be written as K = ρ( U& + U& + U& dv (5 αα α U = ( σ + σ + σ + σ + σ d αα α (6 V = ( qδ U + qδu + qδu AAd αdα (7 αα Te infinitesimal volume is given by dv = A A dα dα dα (8 wit te use of Eqs. (-(9 and substituting into Eq. (4, we get te equations of motions a generic sell for te tird-order teory of Reddy ( NA A ( N A Q + N AA α α Aα R PCA PC A PCA ( + ( α R R α α R A CR CP AA + AA R R u&& u&& Cu&& = ( ui && + && φi + [ C( + && o φ + + ] I R A α R C && φ C u&& u + I4 ( + && && φ+ I6 (9 R R R A α ( N A A ( NA Q N + + AA α α Aα R PCA PC A PCA + ( + ( α R R α α R A CR CP AA AA + R C && φ = ( ui && + && o φi+ I4 R R u&& u Cu + c ( + && && && φ + + I R Aα R C ( u && + && + u (4 R R A φ I6 α PCA A AA CP A ( + + ( ( / N α R α A α A A ( P AC / A P C A + N + ( R A α α α ( PC PC A ( PC ( α α α A α α α PC A ( QA ( CR A ( + α A α α α PCA ( QA ( CR A ( + α R α α CP A PCA A ( ( α α α R A A A ( PC α A α u u = { ui && o + C ( + ( I α A α A && φ && φ u&& + C ( + ( I4 CI6(( ( α A α A Rα A && φ u&& A u&& + ( + α A A α Aα α u&& && φ u&& A u&& + ( ( + ( + R A A A A α α α α α ( M A ( CP A A A + + M CP α α α α ( M A ( P C A + + CR AA AAQ Aα Aα CP AA ui φ I CuI ( C φ R && && && && + = + + + u&& C u&& u&& u C I + C ( + && && φ + I ] 4 R A α R Aα ( M A ( C AP A A + + M CP α α α α 6 } (4 (4
M. M. Najafizade and M. R. Isvandzibaei / Journal of Mecanical Science and Tecnology (9 7~84 77 ( M A ( P C A + + CR AA AAQ Aα Aα CP AA [ u && I && φ I Cu && I ( C && φ R + = + + u&& C u&& u&& u + C I + C ( + && && φ + I. 4 6 R A α R Aα ] (4 wit te use of Eqs.(-(5 and substituting into Eq. (4, we get te equations of motion a generic sell for te first-order teory of Reddy ( N A A ( N A Q + N AA α α A α R = [ ui && +φ&& o I] (44 ( NA A ( N A Q N + + AA α α Aα R = ui && +φ&& o I (45 AA AA ( QA ( QA N + N = I u R R α α && o (46 ( MA A ( MA + M + AAQ α α Aα ] = ui && + && φi (47 ( MA A ( MA + M + AAQ α α A α [ u I φ I ] = && + && (48. For Eqs. (9-(48 are defined as Ii = ρα dα (49 i 7. Te equations of motion for vibration of a cylindrical sell Te curvilinear coordinates and fundamental form parameters for a cylindrical sell are R = a, R =, A = a, A =, α = α, α = θ, α = x (5 Substituting relationsip (5 into Eqs. (9-(4 te equations of motions for vibration of cylindrical sell wit te tird-order teory of Reddy are converted to a N N Iu ( I CI φ u CI && + = && + && (5 N P + C + Q CR + CP C C C C = ( I + I + I 6 u&& + ( I CI + I4 I6 && φ a a a a ( C C I I 6 u && (5 a a P C P P Q Ca + N C a a R Q R C P + Ca ++ C a u C u && φ = CI I + ( CI 4+ CI6 + a C C && C u u ( I4 I6 φ && && + I 6 + CI6 a a a C u&& + I6 u&& I (5 a M P M P a + Ca + C CR a+ aq = Iu&& + CIu&& + ( I+ CI 4 CI6 && φ ( CI 4 CI6 u && + (54 M P M P C a + Ca CR a C + aq + CR = ( ICI I4 u&& a C u + ( I + CI 4 && && φ I4 (55 a Substituting relationsip (5 into Eqs. (44-(48 te equations of motions for vibration of cylindrical sell wit te first-order teory of Reddy are converted to N N a + = Iu&& + I && φ (56 N + Q = Iu&& + Iφ&& (57 Q Q N a = u&& I (58 M M a + aq= Iu&& I && φ (59 M M a + aq= Iu&& I && φ (6 8. Analysis Te displacement fields for a cylindrical sell wit
78 M. M. Najafizade and M. R. Isvandzibaei / Journal of Mecanical Science and Tecnology (9 7~84 an arbitrary number of ring support and te displacement fields wic satisfy tese boundary conditions can be written as φ ( x u = A cos( nθ cos( ωt u = Bφ( x sin( nθcos( ωt P ξ i u = Cφ ( x Π ( x ai cos( nθ cos( ωt (6 i= φ( x φ = D cos( nθ cos( ωt φ = Eφ( x sin( nθcos( ωt were, A, B, C, D and E are te constants denoting te amplitudes of te vibrations in te x, θ and z directions, φ and φ are te displacement fields for iger order deformation teories for a cylindrical sell, φ ( x is te axial function tat satisfies te geometric boundary conditions, a i is te position of te it ring support, P denotes te number of ring supports, ξ i is a parameter aving a value of wen a ring support exists and wen tere is no ring support, n denotes te number of circumferential waves in te mode sape and ω is te natural angular frequency of te vibration. For Te displacement fields defined in Eq. (6 only te transverse displacement is restrained on a ring support. Te axial function φ ( x is cosen as te beam function as []: λmx λmx φ( x = γcos( + γcos( L L λmx λmx ζm( γsin( + γ4sin( L L (6 were γ i ( i =,...,4 are some constants wit value or cosen according to te boundary condition. λ m, are te roots of some transcendental equations and ζ m are some parameters dependent on λ m. Te γ i ( i =,...,4, te transcendental equations and te parameters ζ m for te six boundary conditions are considered []. Te geometric boundary conditions for clamped, free and simply supported boundary conditions can be expressed matematically in terms of φ ( x as: Clamped boundary condition φ( = φ ( L = (6 Free boundary condition φ ( = φ ( L = (64 Simply support boundary condition φ( = φ ( L = (65 Substituting Eq. (6 into Eqs. (5-(55 and substituting Eq. (6 into Eqs. (56-(6, for te tird-order teory and te first-order teory can be expressed in matrix form as u A u B = φ D φ [ ] ω [ ] C u M C E (66 Te eigenvalue equations are solved by imposing te non-trivial solution condition and equating te determinant of te caracteristic matrix det ( Cij Mij ω = to zero. Expanding tis determinant, a polynomial in even powers of ω is obtained βω o + βω + βω + βω + βω + β = o (67 8 6 4 4 5 were β i( i =,,,,4,5 are some constants. Eq. (67 is solved and five positive and five negative roots are obtained. Te five positive roots obtained are te natural angular frequencies of te cylindrical sell FGM wit ring support based tird-order teory and first-order teory. Te smallest of te five roots is te natural angular frequency studied in te present study. 9. Results and discussion To validate te present analysis, results for simply supported-simply support and clamped-clamped FG cylindrical sells are compared wit Loy and Reddy [] and wit Najafizade and Isvandzibaei [7]. Te comparisons sow tat te present results agree well wit tose in te literature. Te functional graded cylindrical sell is composed of nickel on its inner surface and stainless steel on its outer surface. Te material properties for stainless steel and nickel, calculated at T = K, are presented in Table.
M. M. Najafizade and M. R. Isvandzibaei / Journal of Mecanical Science and Tecnology (9 7~84 79 Table. Comparison of natural frequency (Hz for a FG cylindrical sell wit SS-SS boundary condition. L =. cm, R = 5.8 cm, =.5 cm, E =.7788* N m, ν =.7756, ρ = 866 kg m n m Reddy [] Najafizade & Isvandzibaei [7] tird order first order 5.7 4.6 4. 4.9 564. 565. 564.6 564. 894. 89. 89.5 89.8 4 46.9 47. 47.7 47.6 5 6.9 5.8 5.5 5. 6 4799.6 4789.8 4795.6 4794. 4. 95. 95.9 94.4 45. 45. 48.7 46.4 66. 664. 667.5 665.6 4 94.6 9.7 9.8 9. 5 78.8 58.7 57.6 54.6 6 4.9 9. 89.5 88.7 Table. Comparison of natural frequencies (Hz for FG cylindrical sell wit clamped-clamped (C-C boundary condition. ( m=, L/ R=, / R=. m n Ref. [] tird order first order.4.4.7.9.7.5.7.7.78 4.89.8.85 5.6.4.48.847.844.89.4.7.5.58.5.6 4...9 5.45.48.4 Table. Properties of materials. Coefficients Stainless steel E (Nm ν ρ ( kgm E (Nm ν Nickel ( kgm P.4 9.6 866.95 9. 89 P P.79-4 -. -4 -.794-4 P -6.54-7.797-7 -.998-9 P.7788.7756 866.598. 89 ρ were, Po, P, P, P and P are te coefficients of temperature T( K expressed in Kelvin and are unique to te constituent materials. Te material properties P of FGMs are a function of te material properties and volume fractions of te constituent material. From te comparisons presented in Tables -, it can be seen tat te present analysis is accurate as te results obtained wit te present analysis agreed well wit tose in te literature. In tis paper, studies are presented for a functional graded cylindrical sell wic is supported by a ring arbitrarily placed along te axial direction of te sell. Tis is
8 M. M. Najafizade and M. R. Isvandzibaei / Journal of Mecanical Science and Tecnology (9 7~84 7 6 5 4 5 5 n N=.5 N=.7 N= N= N=5 N=5.7.6.5.4.....4.6.8. n SS-SS C-C F-F Fig.. Variation of te natural frequencies (Hz against circumferential wave number n for a FG cylindrical sell wit a ring support wit te different volume fraction N under SS-SS boundary condition (m=, /R=., L/R=, =...8.6.4. 5 5 n C-C SS-SS F-F C-SS C-F F-SS Fig. 4. Variation of te natural frequencies (Hz wit te circumferential wave number n for a functionally graded cylindrical sell wit a ring support. (m =, / R=., L / R=, a / L=.. carried out by setting ξ i = in Eq. (6. Altogeter six boundary conditions, simply supported-simply supported SS-SS, clamped-clamped C-C, free-free F- F, clamped-simply supported C-SS, clamped-free C-F and free-simply supported F-SS boundary conditions, are considered in te study. Fig. sows te variation of te natural frequency (Hz wit te position of te ring support for a FG cylindrical sell. Te position of te ring support as a significant influence on te natural frequency and its influence varied wit te volume fraction N. Fig. 4 sows te variation of te natural frequency wit te circumferential wave number n for a functional graded cylindrical sell wit a ring support at a =.L. Te frequencies for te six boundary conditions increased wit te circumferential wave number. Tis increase in frequencies is most significant Fig. 5. Variation of te natural frequencies (Hz versus te position of te ring support for SS-SS, C-C and F-F boundary conditions (m =, n =, /R=., L/R=..7.6.5.4....5.5 n C-SS C-F F-SS Fig. 6. Variation of te natural frequencies (Hz versus te position of te ring support for C-SS, C-F and F-SS boundary conditions (m =, n =, /R=., L/R=. wen n increased from to and for n greater tan te frequencies increase gradually wit te circumferential wave number. Tis frequency beavior indicates tat te lowest frequency for a functional graded cylindrical sell wit a ring support occurs at n =. Figs. 5 and 6 sow te variation of te natural frequencies wit position of te ring support. Te position of te ring support as a significant influence on te natural frequencies and tis influence varied wit te boundary conditions. For a functionally graded cylindrical sell wit ring support wit same endconditions applied in bot edges, suc SS-SS, C-C and F-F boundary conditions, te natural frequencies are te greatest wen te ring support is in te middle of te functionally graded cylindrical sell. Te natural frequencies decreased as te ring support moved away from center towards eiter end of te sell. Tus te natural frequencies curve is symmetrical about te center of te sell, see Fig. 5. Tis symme-
M. M. Najafizade and M. R. Isvandzibaei / Journal of Mecanical Science and Tecnology (9 7~84 8 try of te frequency curve is as expected since te end conditions are symmetrical about te ring support. For a functionally graded cylindrical sell wit ring support wit different end-conditions, suc as C-SS, C-F and F-SS boundary conditions, te natural frequencies curve are not symmetrical about te center of te sell, see Fig. 6. Figs. 7 and sow te variation of te natural frequencies FGM sell wit position of te ring support at different L/R ratios for te six boundary conditions. From te figures, te influence of te ring support position on te natural frequencies is generally significant at large L/R ratio. It can be seen tat boundary conditions ave some effects on tis influence. For example in Fig. 7 te natural frequencies difference between = and.5 at L/R= is 6.% and L/R= is 54.6% wile in Fig. 8 te natural frequencies difference between = and.5 at L/R= is 5.5% and L/R= is 5.%..8.6.4..5.5 L/R= L/R=6 L/R= Fig. 7. Variation of te natural frequencies (Hz wit te position of te ring support at different L/R ratios for SS-SS boundary conditions. (m=, n=, /R=...8.6.4..5.5 L/R= L/R=6 L/R= Fig. 9. Variation of te natural frequencies (Hz wit te position of te ring support at different L/R ratios for F-F boundary conditions. (m=, n=, /R=...8.6.4..5.5 L/R= L/R=6 L/R= Fig.. Variation of te natural frequencies (Hz wit te position of te ring support at different L/R ratios for C-SS boundary conditions. (m=, n=, /R=...8.6.4. L/R= L/R=6 L/R=.8.6.4. L/R= L/R=6 L/R=.5.5.5.5 Fig. 8. Variation of te natural frequencies (Hz wit te position of te ring support at different L/R ratios for C-C boundary conditions. (m=, n=, /R=. Fig.. Variation of te natural frequencies (Hz wit te position of te ring support at different L/R ratios for C-F boundary conditions. (m=, n=, /R=.
8 M. M. Najafizade and M. R. Isvandzibaei / Journal of Mecanical Science and Tecnology (9 7~84..8.6.4. L/R= L/R=6 L/R=.5.5 /R=. /R=. /R=.5.5.5.5.5 Fig.. Variation of te natural frequencies (Hz wit te position of te ring support at different L/R ratios for F-SS boundary conditions. (m=, n=, /R=. Fig. 5. Variation of te natural frequencies (Hz wit te position of te ring support at different /R ratios for F-F boundary conditions. (m=, n=, L/R=.5 /R=. /R=. /R=.5.5 /R=. /R=..5.5 /R=.5.5.5.5.5 Fig.. Variation of te natural frequencies (Hz wit te position of te ring support at different /R ratios for SS-SS boundary conditions. (m=, n=, L/R= Fig. 6. Variation of te natural frequencies (Hz wit te position of te ring support at different /R ratios for C-SS boundary conditions. (m=, n=, L/R=. /R=. /R=..5 /R=..5 /R=. /R=.5 /R=.5.5.5.5.5.5.5 Fig. 4. Variation of te natural frequencies (Hz wit te position of te ring support at different /R ratios for C-C boundary conditions. (m=, n=, L/R= Fig. 7. Variation of te natural frequencies (Hz wit te position of te ring support at different /R ratios for C-F boundary conditions. (m=, n=, L/R=
M. M. Najafizade and M. R. Isvandzibaei / Journal of Mecanical Science and Tecnology (9 7~84 8.5.5.5.5 /R=. /R=. /R=.5 Fig. 8. Variation of te natural frequencies (Hz wit te position of te ring support at different /R ratios for F-SS boundary conditions. (m=, n=, L/R= Figs. and 8 sow te variation of te natural frequencies FG sell wit position of te ring support at different /R ratios for te six boundary conditions. From te figures it is apparent tat te frequencies are iger at larger /R ratios. Te influence of te ring support position is significant at small /R ratios. Te frequencies are also iger at large /R ratios.. Conclusions A study on te vibration of functionally graded (FG cylindrical sell wit a ring support arbitrarily placed along te sell composed of stainless steel and nickel as been presented. Material properties are graded in te tickness direction of te sell according to volume fraction power law distribution. Te study is carried out using different sear deformation sell teories wit Hamilton s principle. Studies are carried out for cylindrical sells wit simply supported simply supported SS SS, clamped clamped C-C, free free F F, clamped simply supported C SS, clamped free C-F and free simply supported F SS boundary conditions wit an arbitrarily ring support along te axial direction of te cylindrical sell. Studies were made on te frequency caracteristics, te influence of ring support position and te influence of boundary conditions. Te study sowed tat a ring support as significant influence on te frequencies and te extent of tis influence depends on te position of te ring support and te boundary conditions of te functionally graded cylindrical sell. Te study sows tat te frequency caracteristics of te functionally graded cylindrical sells are similar to omogeneous isotropic cylindrical sells. However, tis is because te functionally graded cylindrical sells exibit interesting frequency caracteristics wen te constituent volume fractions are varied. Tis is done by varying te power law exponent N. Te study sowed tat for a functionally graded cylindrical sell wit ring support wit same endconditions applied in bot edges, suc SS-SS, C-C and F-F boundary conditions, te natural frequencies are te greatest wen te ring support is in te middle of te functionally graded cylindrical sell and natural frequencies decreased as te ring support moved away from center towards eiter end of te sell, Tus te natural frequencies curve is symmetrical about te center of te sell, Tis symmetry of te frequency curve is as expected since te end conditions are symmetrical about te ring support, but for a functionally graded cylindrical sell wit ring support wit different end-conditions, suc as C-SS, C-F and F-SS boundary conditions, te natural frequencies curve is not symmetrical about te center of te sell. Te present analysis is validated by comparing results wit tose available in te literature. References [] R. N. Arnold and G. B. Warburton, Flexural vibrations of te walls of tin cylindrical sells, Proceedings of te Royal Society of London, 97 (948 8-56. [] A. Ludwig and R. Krieg, An analysis quasi-exact metod for calculating eigen vibrations of tin circular sells, J. Sound Vibration, 74 (98 55-74. [] H. Cung, Free vibration analysis of circular cylindrical sells, J. Sound Vibration, 74 (98-59. [4] W. Soedel, A new frequency formula for closed circular cylindrical sells for a large variety of boundary conditions, J. Sound Vibration, 7 (98 9-7. [5] K. Forsberg, Influence of boundary conditions on modal caracteristics of cylindrical sells, AIAA J, (964 8-89. [6] A. Bimaraddi, A iger order teory for free vibration analysis of circular cylindrical sells, Int, J. Solids Structures, (984 6-6. [7] K. P. Soldatos, A comparison of some sell teories used for te dynamic analysis of cross-ply laminated circular cylindrical panels, J. Sound Vibration, 97 (984 5-9. [8] C. W. Bert, M.Kumar, vibration of cylindrical sell of biomodulus composite materials, J. Sound vibra-
84 M. M. Najafizade and M. R. Isvandzibaei / Journal of Mecanical Science and Tecnology (9 7~84 tion, 8 (98 7-. [9] K. P. Soldatos, A comparison of some sell teories used for te dynamic analysis of cross-ply laminated circular cylindrical panels, J. Sound Vibration, 97 (984 5-9. [] K. L. Lam and C. T. Loy, Effects of boundary conditions on frequencies caracteristics for a multi-layered cylindrical sell, J. Sound Vibration, 88 (995 6-84. [] K. Y. Lam and C. T. Loy, Analysis of rotating laminated cylindrical sells using different tin sell teories, J. Sound Vibration, 86 (995-5. [] C. T. Loy and K. Y. Lam, Vibration of cylindrical sells wit ring support, I.Joumal of Impact Engineering, 5 (996 455-47. [] M. Koizumi, Te concept of FGM Ceramic Transactions, Functionally Gradient Materials, Japan, (99. [4] Anon, FGM components PM meets te callenge, Metal Powder Report, 5 (996 8-. [5] X. D. Zang, D. Q. Liu and C. C. Ge, Termal stress analysis of axial symmetry functionally gradient materials under steady temperature field, Journal of Functional Materials, 5 (994 45-465. [6] R. C. Weterold, S. Seelman and J. Z. Wang, Use of functionally graded materials to eliminate or control termal deformation, Composites Science and Tecnology, 56 (996 99-4. [7] M. M. Najafizade and M. R. Isvandzibaei, Vibration of functionally graded cylindrical sells based on iger order sear deformation plate teory wit ring support, Acta Mecanica, 9 (7 75-9. [8] Y. Obata and N. Noda, Steady termal stresses in a ollow circular cylinder and a ollow spere of a functionally gradient material, Journal of Termal stresses, 7 (994 47-487. [9] M. M. Najafizade and B. Hedayati, Refined Teory for Termoelastic Stability of Functionally Graded Circular Plates, Journal of Termal Stresses, 7 (4 857-88. [] W. Soedel, Vibration of sells and plates, Marcel Dekker, INC, New York, USA, (98. [] C. T. Loy, K. Y. Lam and J. N., Reddy, Vibration of functionally graded cylindrical sells, Int. J. Mecanical Science, 4 (999 9-4. [] S. C. Pradan, C. T. Loy, K. Y. Lam, J. N. Reddy, Vibration caracteristics of functionally graded cylindrical sells under various boundary conditions, Applied Acoustics, 6 ( -9. M. M. Najafizade received is BS degree in 995 from Azad University (Arak and te Ms Degree in 997 from Azad University (Arak, and is P.D. degree in from Science and Researc Branc Islamic Azad University (Teran, Iran, all in mecanical Engineering. He is member of faculty in Islamic Azad University (Arak since 998. He teaces courses in te areas of dynamics, teory of plates and sells and finite element metod. He as publised more tan articles in journals and conference proceeding. Moammad Reza Isvandzibaei received is Ms Degree from Azad University (Arak, and now e is te student of P.D. in university of Pune, (India all in mecanical Engineering. He is member of faculty in Islamic Azad University (Andimesk.