Research Article Frequency Equations for the In-Plane Vibration of Circular Annular Disks

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1 Advaces i Acoustics ad Vibratio Volume 010, Article ID 50190, 8 pages doi: /010/50190 Research Article Frequecy Equatios for the I-Plae Vibratio of Circular Aular Disks S. Bashmal, R. Bhat, ad S. Rakheja Departmet of Mechaical ad Idustrial Egieerig, Cocordia Uiversity, 1455 De Maisoeuve Bloulevard W., Motreal, QC, Caada H3G 1M8 Correspodece should be addressed to S. Bashmal, bashmal@gmail.com Received 3 October 009; Revised 3 Jue 010; Accepted 5 Jue 010 Academic Editor: Miguel Ayala Botto Copyright 010 S. Bashmal et al. This is a ope access article distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited. This paper deals with the i-plae vibratio of circular aular disks uder combiatios of differet boudary coditios at the ier ad outer edges. The i-plae free vibratio of a elastic ad isotropic disk is studied o the basis of the two-dimesioal liear plae stress theory of elasticity. The exact solutio of the i-plae equatio of equilibrium of aular disk is attaiable, i terms of Bessel fuctios, for uiform boudary coditios. The frequecy equatios for differet modes ca be obtaied from the geeral solutios by applyig the appropriate boudary coditios at the ier ad outer edges. The preseted frequecy equatios provide the frequecy parameters for the required umber of modes for a wide rage of radius ratios ad Poisso s ratios of aular disks uder clamped, free, or flexible boudary coditios. Simplified forms of frequecy equatios are preseted for solid disks ad axisymmetric modes of aular disks. Frequecy parameters are computed ad compared with those available i literature. The frequecy equatios ca be used as a referece to assess the accuracy of approximate methods. 1. Itroductio The out-of-plae vibratio properties of circular disks subjected to a variety of boudary coditios have bee extesively ivestigated e.g., 1 4]. The i-plae vibratio aalyses of circular disk, however, have bee gaiig icreasig attetio oly i the recet years. Much of the iterest could be attributed to importat sigificace of the iplae vibratio i various practical problems such as the vibratio of railway wheels, disk brakes, ad hard disk drives cotributig to oise ad structural vibratio 5 7]. The i-plae vibratio of circular disks was first attempted by Love 8] who formulated the equatios of motio for a thi solid circular disk with free outer edge together with the geeral solutio. The equatios of motio were subsequetly solved by Ooe 9] to obtai the exact frequecy equatios correspodig to differet modes of a solid disk with free outer edge. Hollad 10] evaluated the frequecy parameters ad the correspodig mode shapes for a wide rage of Poisso s ratios ad the vibratio respose to a i-plae force. The i-plae vibratio characteristics of solid disks clamped at the outer edge have bee ivestigated i a few recet studies. Farag ad Pa 11] evaluated the frequecy parameters ad the mode shapes of i-plae vibratio of solid disks clamped at the outer edge usig assumed deflectio modes i terms of trigoometric ad Bessel fuctios. Park 1] studied the exact frequecy equatio for the solid disk clamped at the outer edge. The i-plae vibratio aalyses i the above reported studies were limited to solid disks with either free or clamped outer edge. The i-plae free vibratio of aular disks with differet boudary coditios has also bee addressed i a few studies. The variatios i the i-plae vibratio frequecy parameters of aular disks with free edges were ivestigated as fuctio of the size of the opeig by Ambati et al. 13]. The variatio raged from a solid disk to a thi rig, while the validity of the aalytical results was demostrated usig the experimetal data. Aother ivestigated the free vibratio ad dyamic respose characteristics of a aular disk with clamped ier boudary ad a cocetrated radial force applied at the outer boudary 14]. Irie et al. 15] ivestigated the modal characteristics

2 Advaces i Acoustics ad Vibratio of i-plae vibratio of aular disks usig trasfer matrix formulatio while cosiderig free ad clamped ier ad outer edges. The above reported studies o i-plae vibratio of solid ad aular disks have employed differet methods of aalyses. The fiite-elemet techique has also bee used to examie the validity of aalytical methods e.g., 11, 14]. The exact frequecy equatios of i-plae vibratio, however, have bee limited oly to solid disks. Such aalyses for the aular disks pose more complexities due to presece of differet combiatios of boudary coditios at the ier ad outer edges. This aims at geeralized formulatio for i-plae vibratio aalyses of circular aular disks uder differet combiatios of clamped, free, or flexible boudary coditios at the ier ad outer edges. The equatios of motio are solved for the geeral case of aular disks. The exact frequecy equatios are preseted for differet combiatios of boudary coditios, icludig the flexible boudaries, for various radius ratios, while the solid disk is cosidered as special cases of the geeralized formulatio.. Theory The equatios of the i-plae vibratio of a circular disk are formulated for a aular disk show i Figure 1. The disk is cosidered to be elastic with thickess h, outerradiusb ad ier radius a. The material is assumed to be isotropic with mass desity ρ,youg smoduluse, ad Poisso ratio v. The equatios of dyamic equilibrium i terms of i-plae displacemets alog the radial ad circumferetial directios ca be foud i may reported studies e.g., 11, 16]. These equatios of motio i the polar coordiate system r, θ ca be writte as u r t u θ t C L u r r + 1 r u r r u ] r r C T u r r θ CT 1 1+v u θ r 1 v rθ v C T r 1 v CT u θ r + 1 u θ u ] θ r r r CT 1 1+v u r r 1 v rθ v C T r 1 v u θ θ = 0, C L u θ r θ u r θ = 0, where u r ad u θ are the radial ad circumferetial displacemets, respectively, alog the r ad θ directios, CL = E/ρ1 v adct = E/ρ1 + v. Followig Love s theory 8], the radial ad circumferetial displacemets ca be expressed i terms of the Lamé Potetials φ ad ψ 17], as u r = 1 φ b + 1 ψ, θ u θ = 1 1 φ b θ ψ, 3 where = r/b. 1 b u θ r a θ u r Figure 1: Geometry ad coordiate system used for i-plae vibratio aalysis of a aular disk. Assumig harmoic oscillatios correspodig to a atural frequecy ω, the potetial fuctios φ ad ψ ca be represeted by φ, θ, t = Φ cos θ si ωt, 4 ψ, θ, t = Ψ si θ si ωt, 5 where is the circumferetial wave umber or odal diameter umber. Upo substitutig for u r ad u θ i terms of, Φ,adΨ from to5, i 1, the equatios of motio reduce to the followig ucoupled form: Φ = λ 1Φ, 6 Ψ = λ Ψ, 7 where λ 1 ad λ are odimesioal frequecy parameters defied as λ 1 = ω 1 v ρb E, λ = ω 1+vρb, E = + 1. Equatios 6 ad7 are the parametric Bessel equatios ad their geeral solutios are attaiable i terms of the Bessel fuctios as 18] h z 8 Φ = { A J λ 1 + B Y λ 1 }, 9 Ψ = { C J λ + D Y λ }, 10 where J ad Y are the Bessel fuctios of the first ad secod kid of order, respectively,ada, B, C,adD are the deflectio coefficiets. The radial ad circumferetial displacemets ca the be expressed i terms of the Bessel fuctios by substitutig for

3 Advaces i Acoustics ad Vibratio 3 Φ ad Ψ i ad3. The resultig expressios for the radial ad circumferetial displacemets ca be expressed as: u r = 1 { A X λ 1 + B Z λ 1 b u θ = 1 b where { + C J λ + D Y λ ] C X λ + D Z λ + A J λ 1 + B Y λ 1 ] } } cos θ si ωt, si θ si ωt, X λ i = J λ i = J λ i + λ i J 1 λ i, Z λ i = Y λ i = Y λ i + λ i Y 1 λ i, 11 i = 1, Free ad Clamped Boudary Coditios. Equatios 11 represet the solutios for distributios of the radial ad circumferetial displacemets for the geeral case of a aular disk. The evaluatios of the atural frequecies ad arbitrary deflectio coefficiets A, B, C ad D, however, ecessitate the cosideratio of the i-plae free vibratio respose uder differet combiatios of boudary coditios at the ier ad the outer edges. For the aular disk clamped at the outer edge = 1, the applicatio of boudary coditios u r = 0adu θ = 0 must satisfy the followig for the geeral solutios 11: A X λ 1 + B Z λ 1 + C J λ + D Y λ ] = 0, 13 C X λ + D Z λ + A J λ 1 + B Y λ 1 ] = I a similar maer, the solutio must satisfy the followig for the clamped ier edge = β, where β = a/b is the radius ratio betwee ier ad outer radii of the disk: A X λ1 β + B Z λ1 β + C J λ β + D Y λ β ] = 0, β 15 C X λ β + D Z λ β + A J λ1 β + B Y λ1 β ] = 0. β 16 The coditios ivolvig at the free edges are satisfied whe the radial N r ad circumferetial N rθ i-plae forces at the edge are zero 11], such that N r = Eh ur 1 v b + v u θ θ + v ] u = 0, 17 N rθ = Eh 1 u r 1+vb θ + u θ u ] θ = A direct substitutio of u r ad u θ from 11 i the above equatios would result i secod derivatives of the Bessel fuctios. Alteratively, the above equatio for the boudary coditios may be expressed i terms of Φ ad Ψ through direct substitutio of u r ad u θ from ad3, respectively. The boudary coditios i terms of N r cathusbeobtaied as Φ + + v Ψ Ψ v Φ v Φ + Ψ Rearragig 19 results i + v = 0. Ψ 19 v Φ + 1 v Ψ = 0. 0 The secod order derivative term Φ/ i0 cabe elimiated by addig ad subtractig the term 1// / +λ 1Φ,whichyields v 1 + λ 1 + λ 1 ] v Φ + 1 v Ψ = 0. 1 From 6, it ca be see that the terms withi the first parethesis are idetically equal to zero. Equatio 1 describig the boudary coditio associated with N r ca be further simplified upo substitutios for λ 1 = λ 1 v/, which yields λ Φ + 1 Ψ = 0. Similarly, the boudary coditio equatio associated with N rθ 18 ca be simplified as + 1 Φ + + λ Ψ = 0. 3 Upo substitutios for Φ ad Ψ from 16 i6 ad7, the boudary coditio equatios for the free edges are obtaied, which ivolve oly first derivatives of the Bessel fuctios. For a aular disk with free ier ad outer edges, ad3 represet the coditios at both the ier ad the outer boudaries = 1 ad = β. The equatios for the free edge boudary coditios ca be expressed i the matrix form i the four deflectio coefficiets, as

4 4 Advaces i Acoustics ad Vibratio = 0 Table 1: Frequecy equatios for the solid disks correspodig to free ad clamped edge coditios. Boudary coditios at = 1 Clamped Free Radial J 1 λ 1 = 0 λ 1 J 0 λ 1 = 1 vj 1 λ 1 Circumferetial J 1 λ = 0 λ J 0 λ = J 1 λ = 1 X λ 1 X λ = X J λ 1 J 1 λ J 1 λ 1 ] + X 1 λ 1 J 1 λ ] = λ λ /J 1 λ J 1 λ 1 X >1 λ QJ λ ]X λ 1 QJ λ 1 ] = Q 1 where Q = λ / 1 λ J λ 1 X λ 1 λ Y λ 1 Z λ 1 X λ J λ Z λ Y λ J λ 1 X λ 1 Y λ 1 Z λ 1 X λ λ J λ Z λ λ Y λ β λ J λ1 β 1 β X λ1 β β λ Y λ1 β 1 β Z λ1 β β X λ β β J λ β β Z λ β β Y λ β β J λ1 β β X λ1 β β Y λ1 β β Z λ1 β β X λ β β λ J λ β β Z λ β β λ Y λ β A B C ={0}. D 4 The determiat of the above matrix yields the frequecy equatio for the aular disk with free ier ad outer edge coditios. For the clamped ier ad outer edges, the equatios for the boudary coditios ca be obtaied directly from 13 to 16, such that X λ 1 Z λ 1 J λ Y λ J λ 1 Y λ 1 X λ Z λ A X λ1 β Z λ1 β β J λ β β Y λ β B ={0}. β J λ1 β β Y λ1 β X λ β Z λ β C D 5 I the above equatios, 4 ad5, the top two rows describe the boudary coditio at the outer edge, while the bottom two rows are associated with those at the ier edge. The equatios for the boudary coditios ivolvig combiatios of free ad clamped edges ca thus be directly obtaied from the above two equatios. For the free ier edge ad clamped outer edge, deoted as freeclamped coditio, the matrix equatio comprises the tip two rows of the matrix i 5 ad the lower two rows from 4. For the clamped ier edge ad free outer edge, deoted as Clamped-Free coditio, the matrix equatio is formulated i the similar maer usig the lower ad upper two rows from 5ad4, respectively. The i-plae vibratio aalysis of a solid disk ca be show as a special case of the above geeralized formulatios. Upo elimiatig the coefficiets associated with Bessel fuctio of the secod kid, Equatios 4ad5reduce to those reported by Ooe 9] for free solid disk ad by Park 1] for the clamped solid disk. The frequecy equatio correspodig to differet values of for the solid disks ivolvig the two boudary coditios are summarized i Table 1,whereX λ 1 is the derivative of the Bessel fuctio J evaluated at the outer edge = 1. For aular disks, simplified frequecy equatios ca be obtaied for the axisymmetric modes. These equatios where expressed i Table for the four combiatios of boudary coditios... Flexible Boudary Coditios. I the above aalysis, the boudary coditios cosidered are either clamped or free. However, Flexible boudary coditios may be cosidered more represetative of may practical situatios. The proposed formulatios ca be further employed to the i-plae vibratio of solid as well as aular disks with flexible boudary coditios. Artificial sprigs may be applied to describe the flexible boudary coditios at the ier or the outer edge of a aular disk. A umber of studies o the aalysis of out-of-plae vibratio characteristics of circular plates ad cylidrical shells have employed uiformly distributed artificial sprigs aroud the edge to represet a flexible boudary coditios or a flexible joit 19 ]. The effects of flexible boudary coditios o the iplae free vibratio of circular disks have bee cosidered i a recet by the authors 3] usig the Rayleigh- Ritz approach. Artificial sprigs, distributed alog the radial ad circumferetial directios at the free outer ad/or ier edges, were cosidered to simulate for flexible boudary coditios. The exact solutio of the frequecy equatios for the disk with flexible supports ca be attaied from 11 together with the cosideratio of the flexible boudary coditios. The coditios ivolvig flexible edge supports at the ier ad outer edges are satisfied whe the radial N r ad circumferetial N rθ i-plae forces at the edges

5 Advaces i Acoustics ad Vibratio 5 Table : Frequecy equatios of axisymmetric modes for aular disks. Boudary coditios radial Circumferetial ier outer Clamped Clamped J 1 λ 1 Y 1 λ 1 β J 1 λ 1 βy 1 λ 1 = 0 J 1 λ Y 1 λ β J 1 λ βy 1 λ = 0 Free Clamped Free Free Free Clamped λ 1 /1 vj 0 λ 1 +J 1 λ 1 ] λ 1 /1 vy 0 λ 1 β+1/βy 1 λ 1 β] λ 1 /1 vy 0 λ 1 +Y 1 λ 1 λ 1 /1 vj 0 λ 1 β+ 1/βJ 1 λ 1 β] = 0 λ 1 /1 vj 0 λ 1 J 1 λ 1 ]Y 1 λ 1 β J 1 λ 1 βλ 1 /1 vy 0 λ 1 Y 1 λ 1 ] = 0 J 1 λ 1 λ 1 /1 vy 0 λ 1 β+ 1/βY 1 λ 1 β] + λ 1 /1 vj 0 λ 1 β+ 1/βJ 1 λ 1 β]y 1 λ 1 = 0 J 1 λ +λ J 0 λ ] /βy 1 λ β+λ Y 0 λ β] /βj 1 λ β+λ J 0 λ β] Y 1 λ +λ Y 0 λ ] = 0 J 1 λ λ J 0 λ ]Y 1 λ β J 1 λ βy 1 λ λ Y 0 λ ] = 0 J 1 λ /βy 1 λ β+λ Y 0 λ β] /βj 1 λ β+ λ J 0 λ β]y 1 λ = 0 Table 3: Exact frequecy parameters of i-plae vibratio of a solid disk with free edge v = 0.3. Mode = 1 = = 3 = Table 4: Exact frequecy parameters of i-plae vibratio of a solid disk with clamped edge v = Mode = 1 = = 3 = are equal to the respective radial ad circumferetial sprig forces, such that N r = Eh ur 1 v b + v u θ θ + v ] u r = K r u r, θ, N rθ = Eh 1 u r 1+vb θ + u θ u ] 6 θ = K θ u θ, θ, where K r ad K θ are the radial ad circumferetial stiffess coefficiets, respectively. Itroducig the odimesioal stiffess parameters, K r = K r b1 v /Eh ad K θ = K θ b1 + v/eh,6cabewritteas 1 1 λ + K r Φ K r Ψ = 0, K θ Φ λ K θ Ψ = 0. The applicatio of the above coditios yields the matrix equatios for the disk with flexible supports at the ier ad outer edges, similar to 4. The frequecy parameters are subsequetly obtaied through solutio of the matrix equatios. The above boudary equatios reduce to those i ad3 for the free edge coditios by lettig K r = 0 ad K θ = 0. Furthermore, the clamped edge coditio ca be represeted by cosiderig ifiite values of K r ad K θ. Equatios 7 further show that the flexible edge support coditios ivolve combiatios of the free ad clamped edge coditios. 3. Results The frequecy parameters λ 1 derivedfordifferet combiatios of boudary coditios are compared with those reported i differet studies to demostrate the validity of the proposed formulatio. For this purpose, the frequecy parameters of a solid disk with free ad clamped outer edge are iitially evaluated ad compared with those reported by Hollad 10] ad Park1], respectively. The results preseteditables3 ad 4 for the free ad clamped outer edge coditios, respectively, were foud to be idetical to those reported i 10, 1] for solid disks with clamped edge. The exact frequecy parameters for the aular disks were subsequetly obtaied uder differet combiatios of boudary coditios at the ier ad outer edges. The edge coditios are preseted for the ier followed by that of the outer edge. For istace, a Free-Clamped coditio refers to free ier edge ad clamped outer edge. The solutios obtaied for coditios ivolvig free ad clamped edges Free-Free, Free-Clamped, Clamped-Clamped, ad Clamped-Free are compared with those reported

6 6 Advaces i Acoustics ad Vibratio Table 5: Frequecy parameters of i-plae vibratio of a aular disk with Free-Free coditios. Radius ratio β Referece 15] = 1 = = 3 = 4 preset Referece Preset Referece Preset Referece 15] 15] 15] Preset Table 6: Frequecy parameters of i-plae vibratio of a aular disk with Free-Clamped coditios. Radius ratio β Referece 15] = 1 = = 3 = 4 Preset Referece Preset Referece Preset Referece 15] 15] 15] Preset Table 7: Frequecy parameters of i-plae vibratio of a aular disk with Clamped-Clamped coditios. Radius ratio β Referece 15] = 1 = = 3 = 4 preset Referece preset Referece preset Referece 15] 15] 15] preset Table 8: Frequecy parameters of i-plae vibratio of a aular disk with Clamped-Free coditios. Radius ratio β Referece 15] = 1 = = 3 = 4 preset Referece preset Referece preset Referece 15] 15] 15] preset Table 9: Frequecy parameters of i-plae vibratio of a aular disk with flexible boudary coditios. Boudary coditios Elastic-Free Elastic-Clamped Elastic-Elastic Free-Elastic Clamped-Elastic = 1 = = 3 = 4 Referece preset Referece preset Referece preset Referece 19] 19] 19] 19] preset

7 Advaces i Acoustics ad Vibratio 7 by Irie et al. 15] i Tables 5, 7, 8, 9, respectively.the simulatios results were obtaied for two differet values of the radial ratios β = 0. ad 0.4, ad v = 0.3. The results show excellet agreemets of the values obtaied i the preset with those reported i 15], irrespective of the boudary coditio ad radius ratio cosidered. The exact frequecy parameters of the aular disk are further ivestigated for boudary coditios ivolvig differet combiatios of free, clamped, ad elastic edges. The solutios correspodig to selected modes are obtaied for Elastic-Free, Elastic-Clamped, Elastic-Elastic, Free- Elastic, ad Clamped-Elastic coditios are preseted i Table 9. The results were attaied for β = 0. ad v = 0.3. The odimesioal radial ad circumferetial stiffess parameters were chose as K r = 1adK θ = 1. The results are also compared with those obtaied usig the Raleigh-Ritz methods, as reported i 3]. The comparisos reveal very good agreemets betwee the aalytical ad the reported results irrespective of the boudary coditio cosidered. The results suggest that the proposed frequecy equatios could serve as the referece for approximate methods o i-plae vibratio characteristics of the aular disks with differet combiatios of edge coditios. 4. Coclusios The characteristics of i-plae vibratio for circular disks are ivestigated uder differet combiatios of boudary coditios. The goverig equatios are solved to obtai the exact frequecy equatio of solid ad aular disks. Frequecy equatios are preseted for differet combiatios of boudary coditios, icludig flexible boudaries, at the ier ad outer edges. The odimesioal frequecy parameters obtaied by the preset approach compare very well with those available i literature, irrespective of the boudary coditio ad radius ratio cosidered. The exact frequecy parameters ca serve as a referece to assess the accuracy of approximate methods. The preseted frequecy equatios ca be umerically evaluated to obtai the iplae modal characteristics of circular disk for a wide rage of costraits coditios ad geometric parameters. Nomeclature A, B, C, D :Deflectiocoefficiets of the exact solutio a: Ier radius of the aular disk b: Outer radius of the aular disk CL: E/ρ1 v CT: E/ρ1 + v E: Youg s modulus of disk h: Thickess of the aular disk J : Bessel fuctio of the first kid of order K r, K θ : Radial ad circumferetial stiffess K r, K θ : coefficiets Nodimesioal radial ad circumferetial stiffess parameters : Nodal diameter umber N r, N rθ : Radial ad circumferetial i-plae forces r: Radial coordiate u r, u θ, u z : Radial, circumferetial, ad ormal displacemets of the disk Y : Bessel fuctio of the secod kid of order z: Normal coordiate β: Radiusratio θ: Circumferetial coordiate λ: Nodimesioal frequecy parameters v: Poisso s ratio : Nodimesioal radial coordiate of the disk ρ: Mass desity of the disk φ, ψ: LaméPotetials Φ, Ψ: Radial variatios of LaméPotetials ω: Radia atural frequecy. Refereces 1] A. Leissa, Vibratio of Plates, Nasa SP-160, ] J. Rao, Dyamics of Plates, Narosa Publishig House, ] Ö. Civalek, Discrete sigular covolutio method ad applicatios to free vibratio aalysis of circular ad aular plates, Structural Egieerig ad Mechaics, vol. 9, o., pp , ] Ö. Civalek ad H. Ersoy, Free vibratio ad bedig aalysis of circular Midli plates usig sigular covolutio method, Commuicatios i Numerical Methods i Egieerig, vol. 5, o. 8, pp , ] K. I. Tzou, J. A. Wickert, ad A. Akay, I-plae vibratio modes of arbitrarily thick disks, Joural of Vibratio ad Acoustics, vol. 10, o., pp , ] H. Lee ad R. Sigh, Self ad mutual radiatio from flexural ad radial modes of a thick aular disk, Joural of Soud ad Vibratio, vol. 86, o. 4-5, pp , ] D. J. Thompso ad C. J.C. Joes, A review of the modellig of wheel/rail oise geeratio, Joural of Soud ad Vibratio, vol. 31, o. 3, pp , ] A. Love, A Treatise o the Mathematical Theory of Elasticity, Dover Publicatios, New York, NY, USA, ] M. Ooe, Cotour vibratios of isotropic circular plates, Joural of the Acoustical Society of America, vol. 8, pp , ] R. Hollad, Numerical studies of elastic-disk cotour modes lackig axial symmetry, Joural of Acoustical Society of America, vol. 40, pp , ] N. H. Farag ad J. Pa, Modal characteristics of i-plae vibratio of circular plates clamped at the outer edge, Joural of the Acoustical Society of America, vol. 113, o. 4, pp , ] C. I. Park, Frequecy equatio for the i-plae vibratio of a clamped circular plate, Joural of Soud ad Vibratio, vol. 313, o. 1-, pp , ] G.Ambati,J.F.W.Bell,adJ.C.K.Sharp, I-plaevibratios of aular rigs, Joural of Soud ad Vibratio, vol. 47, o. 3, pp , ] V. Sriivasa ad V. Ramamurti, Dyamic respose of a aular disk to a movig cocetrated, i-plae edge load, Joural of Soud ad Vibratio, vol. 7, o., pp. 51 6, 1980.

8 8 Advaces i Acoustics ad Vibratio 15] T. Irie, G. Yamada, ad Y. Muramoto, Natural frequecies of i-plae vibratio of aular plates, Joural of Soud ad Vibratio, vol. 97, o. 1, pp , ] J.-S. Che ad J.-L. Jhu, O the i-plae vibratio ad stability of a spiig aular disk, Joural of Soud ad Vibratio, vol. 195, o. 4, pp , ] A. C. Erige ad E. S. Suhubi, Elastodyamics, vol., Academic Press, New York, NY, USA, ] D. G. Zill ad M. R. Culle, Advaced Egieerig Mathematics, Joes ad Bartlett Publishers, ] P. A. A. Laura, L. E. Luisoi, ad J. J. Lopez, A ote o free adforcedvibratiosofcircularplates:theeffect of support flexibility, Joural of Soud ad Vibratio, vol. 47, o., pp , ] K. S. Rao ad C. L. Amba-Rao, Lateral vibratio ad stability relatioship of elastically restraied circular plates, AIAA Joural, vol. 10, o. 1, pp , ] P. A. A. Laura, J. C. Paloto, ad R. D. Satos, A ote o the vibratio ad stability of a circular plate elastically restraied agaist rotatio, Joural of Soud ad Vibratio, vol. 41, o., pp , ] C. S. Kim ad S. M. Dickiso, The flexural vibratio of thi isotropic ad polar orthotropic aular ad circular plates with elastically restraied peripheries, Joural of Soud ad Vibratio, vol. 143, o. 1, pp , ] S. Bashmal, R. Bhat, ad S. Rakheja, I-plae free vibratio of circular aular disks, Joural of Soud ad Vibratio, vol. 3, o. 1-, pp. 16 6, 009.

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