On the Missing Modes When Using the Exact Frequency Relationship between Kirchhoff and Mindlin Plates

Transcription

1 On the Missing Modes When Using the Eact Frequenc Relationship between Kirchhoff and Mindlin Plates C.W. Lim 1,*, Z.R. Li 1, Y. Xiang, G.W. Wei 3 and C.M. Wang 4 1 Department of Building and Construction, Cit Universit of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong School of Engineering and Industrial Design & Centre for Construction Technolog and Research, Universit of Western Sdne, Penrith South DC, NSW 1797, Australia 3 Department of Mathematics and Department of Electrical and Computer Engineering, Michigan State Universit, East Lansing, MI 4884, U.S.A. 4 Department of Civil Engineering, National Universit of Singapore, Kent Ridge, Singapore Abstract An eact frequenc relationship eists between Kirchhoff and Mindlin plates of polgonal planform and simpl supported edges. In this paper, we show that the use of this relationship leads to missing vibration modes due to the lack of consideration for the transverse shear modes and coupled bending-shear modes in the relationship. The missing modes appear at relativel high order modes and the were originall discovered b using discrete singular convolution (DSC) method that is capable of accuratel predicting thousands of vibration modes without suffering from numerical instabilit as other methods do. An efficient state-space technique is used to confirm our findings. Using the state-space technique, eact vibration frequencies for transverse shear vibration modes of thick plates can be obtained. Numerical eamples are presented to support our claims. These eact shear frequencies complement the bending frequencies predicted b using the Kirchhoff-Mindlin relationship. It is interesting to note that there are some coupled bending-shear modes that can picked up b the DSC method which provides a complete spectrum of frequencies, even for ver high frequencies. Kewords: discrete singular convolution, high frequenc, Kirchhoff-Mindlin relationship, transverse shear mode, state-space, thick plate, vibration. * Corresponding author: C.W. Lim, Department of Building and Construction, Cit Universit of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong, P.R. China. bccwlim@citu.edu.hk

2 1. Introduction The Kirchhoff (or classical thin) plate theor [1,] is the simplest and most commonl used plate theor for bending, vibration and buckling analses of plates. This theor, however, neglects the effect of transverse shear deformation b the simplified assumption that the normals to the undeformed midplane remain normal after deformation. This theor leads to an overprediction of buckling loads and natural vibration frequencies and an underprediction of deflections when applied to thick plates because of the significant effect of transverse shear deformation in such plates. In 1951, Mindlin [3,4] proposed the first order shear deformation plate theor that relaes the aforementioned normalit assumption. B allowing the straight normals to rotate with respect to the mid-plane of the deformed plate, a constant shear strain is admitted through the plate thickness. This relaation produces two additional degrees of freedom, i.e. the angles of rotation in the two perpendicular directions, in the plate modeling. Owing to the additional degrees of freedom, various analtical solution methodologies applicable to the Kirchhoff plate analsis become invalid and man eact analses available in Kirchhoff plate models cannot be etended to the Mindlin (or thick) plate analsis. In view of the aforementioned problems, Wang and his coauthors [5-9] initiated studies to relate the solutions of Kirchhoff (or thin) plate theor and Mindlin (or thick) plate theor. The derived eact relationships between the two models in bending [5], buckling [6] and vibration [7-9]. Such Kirchhoff-Mindlin relationships ehibit a one-to-one mapping of the eact solutions in both plate models. Consequentl man known eact solutions of thin plates can be easil converted to eact thick plate solutions and these eact relationships can be easil handled b practicing engineers who need not have too much knowledge in the thick plate theor. Nevertheless, there are some inherent pitfalls when using these eact relationships, which is the subject of the present paper. Here, our concern is the missing shear modes in thick plate vibration. As the relationships assume a one-to-one mapping, man phsical modes that eist when the thickness of plate becomes relativel high ma not be captured. The missing modes are due to transverse shear deformation and coupled bending-shear

3 deformation in thick plate dnamics. If care is not taken while appling the Kirchhoff-Mindlin relationships, important phsical features of the thick plate ma not be realized and thus the results ma be insufficientl interpreted. To analze the missing shear modes, an efficient state-space technique is presented in this paper to derive a sstem of homogenous differential equation for the vibration of a thick plate considering onl the transverse shear deformable modes. This method has been shown to be ver effective in furnishing eact solutions for bending and vibration of plates and clindrical shells [10-1]. It is etended here for the analsis of transverse shear vibration modes of thick plates. Eact shear frequenc solutions are obtained for plates with two opposite sides simpl supported. The eact shear modes complement the bending frequencies predicted from the Kirchhof-Mindlin relationship. However, there are some modes due to the coupling between the bending and the shear dnamics that can not be predicted b the relationship. The can be recovered from the discrete singular convolution (DSC) methods which deliver a complete frequenc spectrum for the vibration analsis of thick Mindlin plates. In addition, the influence of plate boundar conditions on the shear vibration frequencies is eamined. The DSC [13-18] method is another potential numerical approach for plate analses. It is regarded as a novel approach for numerical analsis of singular integrations. The mathematical underpinning of the DSC algorithm is the theor of distributions and wavelet analsis. Man DSC kernels, such as the (regularized) Shannon delta sequence kernel, the (regularized) Dirichlet delta sequence kernel, the (regularized) Lagrange delta sequence kernel and the (regularized) de la Vallée Poussin delta sequence kernel, have been constructed [16]. B appropriatel selecting kernel parameters, the DSC approach ehibits controllable accurac for integration and ecellent fleibilit in handling comple geometries and boundar conditions. In this paper, a complete spectrum of vibration frequencies is resolved using the DSC collocation (DSC-CO) method and the DSC-Ritz method where the latter is an etension of the Ritz procedure using DSC kernels. 3

4 . Modelling and Formulation.1. Problem Definition Consider an isotropic, elastic, rectangular plate of uniform thickness h, length a, width b, Young s modulus E, shear modulus G, Poisson s ratio ν and mass densit ρ, as shown in Fig. 1. The plate is simpl supported on two opposite sides and the other two sides ma assume free, simpl supported or clamped. Here, we intend to determine the vibration frequencies of the plate using three different approaches: (i) the Kirchhoff-Mindlin relationship for bending vibration modes; (ii) the DSC-CO method and the DSC-Ritz method; and (iii) the state-space formulation for shear vibration modes... Frequenc Relationship of Kirchhoff and Mindlin Plates The eact frequencies for the vibration of a simpl supported, rectangular Kirchhoff (or thin) plate are given b [] ω i mπ nπ ( ) ( ) = + / ρh / D a b where i = 1,,, corresponds to the mode sequence number and m, n are the number of half number of half waves. The corresponding frequenc for a Mindlin plate can be deduced via a formal relationship given b [7,8] 6κ G 1 ρh 1 ρh ρh ωi = 1+ ωih ϒ 1+ ωih ϒ 1 D 1 D ω ρh 3κ G ϒ = 1 + κ ( 1 µ ) i (1) () where ω i correspond to the frequenc a Mindlin plate of the same dimensions and material properties, and κ is the shear correction factor.. From Eq. (), accurate frequencies for a simpl supported Mindlin plate can be derived from the Kirchhoff plate solutions given b Eq. (1). These are the fleural vibration modes where a one-to-one mapping between the Kirchhoff plate and the Mindlin plate is possible..3. DSC-Ritz and DSC-CO Formulation Let T denotes a singular kernel and η ( ) be an element of the space of test functions. A singular convolution is defined as 4

5 + Ft () = ( T* η)() t = Tt ( ) η( d ) (3) where t indicates the time variable and a dumm variable. Depending on the form of the kernel T, the singular convolution is the ke issue for a wide range of problems in science and engineering, e.g., Hilbert transform, Abel transform, and Radon transform. However singular kernels cannot be directl applied in computers because the are tempered distributions and do not have a value anwhere. Hence, the singular convolution in Eq. (3) is of little direct numerical merit. In order to avoid the difficult of using singular epressions directl in computer, we need to construct sequences of approimations ( T α ) to the distribution T lim Tα ( ) T( ), (4) α α 0 where α 0 is a generalized limit. Obviousl, for the singular kernels of the delta tpe, T ( ) = δ( ), each element in the sequence, T ( ), is a delta sequence kernel. With a sufficientl smooth approimation, it is useful to consider a discrete singular convolution (DSC) F () t = T ( t )( f ) α k α where Fα () t is the approimation of Ft () and { k } is an appropriate set of discrete points on which the DSC in Eq. (5) is well defined. Note that, the original test function η ( ) is replaced b f. ( ) k α k (5) As the Fourier transform of the delta distribution is unit in the Fourier domain, the distribution can be regarded as a universal reproducing kernel [13] f ( ) = δ ( ) f ( ) d (6) As a consequence, delta sequence kernels are approimate reproducing kernels or bandlimited reproducing kernels that provide a good approimation to the universal reproducing kernel in certain frequenc bands. There are man delta sequence kernels arising in the theor of partial differential equations, Fourier transforms, signal processing and wavelet analsis, with completel 5

6 different mathematical properties. For the purpose of numerical computations, the delta sequence kernels of both (i) positive tpe and (ii) Dirichlet tpe are of particular importance and the have ver distinct mathematical and numerical properties. For simplicit, we focus on two tpical kernels of the Dirichlet tpe, Shannon s delta sequence kernel sin( α) δα( ) = (7) π and a simplified de la Vallée Poussin delta kernel 1 cos( α) cos( α) δα() = πα (8) to realize the proposed DSC method. According to the theor of distributions, the smoothness, regularit and localization of a tempered distribution can be improved b a function of the Schwartz class. It is suggested in [19] that a delta regularizer Rσ () is used in regularizing a delta kernel. A good eample is the Gaussian R ( ) = e σ σ Therefore, Shannon s delta sequence kernel in Eq. (7) and de la Vallée Poussin delta kernel in Eq. (8) can be modified in their regularized form as (9) and δ σα, ( ) sin( α) e π = σ (10) δ ( ) 1 cos( α) cos( α) πα = e σ (11) σα, These regularized kernels converge etremel fast when used for approimating functions and their derivatives. For practical computational purposes, the can be truncated for a finite domain. For different implementations of the DSC algorithm, the two popular methods are the 6

7 Ritz method and the collocation formulation. The DSC-collocation is a local method, in which the derivative of a function at a particular point in the coordinate domain is computed b as a few neighborhood grid points. While the Ritz method is classicall a global method, it requires the full set of grid points in a computational domain to compute a derivative. However, the DSC-Ritz can be formulated as a local method because the DSC kernels have time-frequenc localization. The Ritz approach to the Mindlin plate vibration problem is based on the energ principle. B assuming a set of admissible trial functions with independent amplitude coefficients, a closer upper bound for the frequenc could be achieved b minimizing the energ functional with respect to the coefficients. In the DSC-Ritz method, a new set of trial functions is emploed, which is able to approimate the deflection of the whole domain and at the same time satisfies the prescribed boundar conditions. These trial functions are formed from the product of sets of two-dimensional localized DSC kernels and basic functions which associate the piecewise boundar geometric epressions. The new shape α k functions ϕ ( α = w, θ, θ ) can be epressed as α α k k l ϕ (, ξ η) = f (, ξ η) ϕ (1) where b ϕ b α is the basic function. The two-dimensional DSC kernel M N N fk = k= 1 i= 1 j= 1 M fk (, ξη) are given k = 1 (, ξη) δσij(, ξη) (13) where N is the number of grid points adopted in both ξ- direction and η- direction, and M = N, k = ( i -1) N + j (14) Note that δ σij in Eq. (13) is a DSC delta kernel and the two-dimensional forms for the aforementioned two DSC kernels in Eqs. (10-11) can be constructed b tensor products as δ σij (, ξ η) = for Shannon s kernel, and sin[( π/ )( ξ ξi)]sin[( π/ )( η ηj)] e ( π / )( ξ ξ )( η η ) i j i i [( ξ ξ ) / σ ] [( η η ) / σ ] e (15) 7

8 δ σij (, ξ η) = cos[( π/ )( ξ ξi)] cos[( π/ )( ξ ξi)] e ( π/ )( ξ ξi ) cos[( π/ )( η ηj)] cos[( π/ )( η ηj)] e ( π/ )( η ηj ) i [( ξ ξ ) / σ ] j [( η η ) / σ ] for de la Vallée Poussin kernel. In all the above kernels, ξ i and η j are grid point along the ξ- ais and η- ais, respectivel. The parameters σ and are chosen as (16) σ = r ; 3 =, where is the grid spacing, and r is an adjustable parameter determining the radius of influence. Details on how to accommodate the geometric boundar conditions are given b Lim et al. [18, 0, 1]. For the DSC-CO method, the solution of the Mindlin plate vibration problem is obtained b directl solving the discretized partial differential equations. In the DSC-CO, the approimate solution is sought from a finite set of N DSC kernel functions. Numericall to solve the plate vibration governing equation, it is necessar to give a matri approimation to the differential operator so that the action of the operator can be realized. The DSC approimation to the n th order derivative of a function ϕ ( α = w, θ, θ ) can be rewritten as n kl B n α k+ M ϕ n α n q= q = c k kl, Bϕ q (17) l= k M where c, are a set of DSC weights and can be calculated through the DSC kernel, and q is the direction of differentiation ( q = ξη, ), and n ( = 0,1,, ) is the order of differentiation. The regularized Shannon kernel is used in the present work. B substituting Eq. (17) into the governing differential equations, a sstem of linear algebraic equations for the governing equations can be obtained. Before calculating the eigenvalues, appropriate boundar conditions are to be implemented. The reader is referred to Refs. [15] and [16] for an elaboration about the DSC-CO method and its applications. α 8

9 .4. Eact Solution for Shear Vibration of Thick Plates Considering onl the magnitude of vibration where the time dependent function sinω t has been simplified, the governing differential equations for a thick plate based on the Mindlin plate theor can be epressed as [4] w w κ Gh + θ + + θ + ρhω w = 0 (18) 3 (1 ) D w h D θ θ ν θ θ ρ ν κ Gh + θ + ω θ = 0 1 (19) 3 (1 ) D w h D θ θ ν θ θ ρ ν κ Gh + θ + ω θ = 0 1 where θ (, ), (, ) (0) θ are rotations in the -ais and -ais, w (, ) is the transverse displacement, D Eh 3 1( 1 ν ) frequenc, and = is the fleural rigidit, ω is the angular κ is the shear correction factor. If the plate vibrates in a shear mode, i.e. w(, ) = 0, the governing differential equations reduce to 3 (1 ) D h D θ θ ν θ θ ρ ν κ Ghθ + ω θ = 0 1 (1) 3 (1 ) D h D θ θ ν θ θ ρ ν κ Ghθ + ω θ = 0 1 () For a simpl supported rectangular plate of side length a and width b, the rotations in Eqs. (1) and () can be epressed as nπ mπ θ (, ) = Acos sin (3) a b nπ mπ θ (, ) = Bsin cos (4) a b where n and m are the number of halfwaves of the vibration mode along the and directions, and A and B are unknown coefficients to be determined, respectivel. B 9

10 substituting Eqs. (3) and (4) into Eqs. (1) and (), the non-trivial frequenc ω can be obtained eplicitl and epressed in terms of the non-dimensional frequenc parameter λ ( = ( ωb / π ) ρh / D) as follows b m π h n π h λ = ν κ + + π h b a 6(1 ) 1 where a/ b is the aspect ratio and h/ b is the thickness ratio of the plate, respectivel. (5) For a plate with simpl supported edges opposite to each other, the state-space technique must be adopted to obtain the eact solutions. Assuming that the two simpl supported edges are parallel to the -ais, the rotations in Eqs. (1) and () can be epressed as []: mπ θ (, ) = φ ( ) sin (6) b mπ θ (, ) = φ ( ) cos (7) b where φ () and φ ( ) are unknown functions to be determined. Equations. (6) and (7) satisf the simpl supported boundar conditions on edges at = 0 and = b. B substituting Eqs. (6) and (7) into Eqs. (1) and (), the following differential equation sstem can be derived ( ψ )' = Hψ (8) where ψ = [ φ ' ' T ( φ ) φ ( φ ) ], the prime represents the derivative with respect to and H is a 4 4 matri with the following non-zero elements: H = H 1 (9) 1 34 = H 1 D(1 ν )( mπ / b) / + κ Gh ρhω /1 = (30) D ( mπ / b)(1 +ν ) H 4 = (31) ( mπ / b)(1 + ν ) H 4 = 1 ν (3) 10

11 3 D( mπ / b) + κ Gh ρh ω /1 H 43 = (33) [ D(1 ν ) / ] A general solution of Eq. (8) can be obtained as H ψ = e c (34) where c is a 4 1 constant vector that can be determined b the boundar conditions of the two edges parallel to the -ais and i H e is the general matri solution of Eq. (8) []. The two edges parallel to the -ais ma have the following prescribed boundar conditions when the shear vibration of the plate is considered M 0, M = 0, if the edge is free (35,36) = = M 0, θ = 0 if the edge is simpl supported (37,38) θ 0, θ = 0 if the edge is clamped (39,40) = where M and M are bending moment and twisting moment in the plate, respectivel, and are defined b M θ = D θ + ν (41) M 1 ν θ = D θ + (4) In view of Eq. (34), a homogeneous sstem of equations can be derived b implementing the boundar conditions of the plate along the two edges parallel to the -ais [Eqs. (35)-(40)] and is given b Kc = 0 (43) where K is a 4 4 matri. The vibration frequenc ω ma be determined when the determinant of K in Eq. (6) is equal to zero. 11

12 3. Numerical Eamples In this section, a few numerical eperiments are designed to illustrate the missing modes when using the Kirchhoff-Mindlin relationship for determining shear related vibration modes. 3.1 Simpl supported (SSSS) thick plate Table 1 shows the dimensionless frequencies obtained using the Kirchhoff-Mindlin relationship, eact shear mode solutions from Eq. (5), DSC-Ritz method and DSC-CO method for a plate with a thickness ratio h/ a = 0.1. Prior to mode-113, the Kirchhoff-Mindlin relationship provides eact mapping with respect to the solutions of DSC-Ritz and DSC-CO indicating that there eist no transverse shear modes in this range. At mode 113 and beond, the mismatch begins to appear. As shown in Table 1, modes-113 to 115 are pure transverse shear modes corresponding to (0,1), (1,0) and (1,1) for the half-wave numbers denoted b ( mn, ). The number of missing modes increases for higher frequencies because the shear effect becomes more and more significant when the wavelength becomes shorter. Although the mismatch of frequencies onl appears from mode-113, it is epected that mismatches occur at much lower modes if the plate is thicker. To prove this prediction, two more eamples for the SSSS plate are presented in Tables and 3, in which the thickness ratios are ha= 0. and 0.5 respectivel. Indeed, the transverse shear modes start to appear as low as at mode-7 and mode-18, respectivel. It is epected that the transverse shear modes begin to dominate the lower order modes when thickness of the plate is further increased. It is interesting to note that there eists another class of coupled bending-shear vibration modes involving coupled terms of w θ or w θ besides the pure bending and pure transverse shear modes as indicated in Table 1. These modes cannot be predicted b the Kirchhoff-Mindlin relationship because the θ or θ effects are not considered nor the be predicted b the eact shear mode solutions from Eq. (5) because the fleural effect is not considered. It is also 1

13 observed that a complete spectrum of accurate frequenc solutions can be furnished b both the DSC-Ritz and DSC-CO methods. The latter method ehibits better accurac when compared to the eact solutions of the Kirchhoff-Mindlin relationship and the eact shear mode solutions. Therefore, DSC-Ritz and DSC-CO are potential methods for analzing high frequencies in regions where no eact solution eists or other numerical methods, such as the finite element method, encounter failure. For more information on the high frequenc analsis, please refer to Lim et al. [18]. 3. Other thick plates with two opposite sides simple supported (SFSF and SCSC) To further illustrate the potential of the DSC-Ritz algorithm, two more eamples for thick plates are presented. These are plates with two opposite sides simpl supported, while the other two sides ma be free (named SFSF plate), or clamped (named SCSC plate). These results are presented in Tables 4 and 5, respectivel. Here, the eact Kirchhoff-Mindlin frequenc relationships are not available. As there is no eplicit analtical solution such as that in Eq. (5) available, the state space technique presented in Sec..4 needs to be emploed to obtain numerical shear mode solutions b solving Eq. (43). Matching of the shears modes from the state space technique with the complete frequenc spectrum as presented in Tables 4 and 5 ma create confusion. However, bearing in mind that the Ritz method alwas overestimates the vibration frequencies, the matching has been done such that the DSC-Ritz solutions are alwas a little higher that the state space solutions. The pure transverse shear modes start to appear at mode-38 and mode-30, respectivel, for the SFSF and SCSC plates. These eamples, again, clearl demonstrate that pure shear vibration modes do eist and an eact relationships linking the Kirchhoff and Mindlin plates, if available in the future, must be implemented in care. To help understand the phsical nature of pure shear vibration modes, two eamples are presented in Figs. and 3 for a square simpl supported plate with ab= 1, hb= 0.1, n = 3, m = (for Fig. ) and m = 3 (for Fig. 3), respectivel. As observed, these vibration modes onl involve deformation in planes parallel to the mid-plane of the plate caused b shear deformation or rotation of the normals originall perpendicular to the 13

14 mid-plane. It is also obvious that such modes vibrate at a lower frequenc if the plate is thicker. 4. Conclusions This paper addresses the missing vibration modes of thick Mindlin plates when using a Kirchhoff-Mindlin frequenc relationship. The relationship, valid for simpl supported and polgonal plates, gives a one-to-one mapping of vibration frequencies between a Kirchhoff plate and a Mindlin plate. It has been served as a useful and convenient tool for analzing complicated thick plates based on the simple thin plate theor. However, the limitation of such a relationship has not been detected previousl. As the Kirchhoff plate accounts onl for the bending effect, the transverse shear dnamics is neglected and it plas an important role in the frequenc spectrum of Mindlin plates. The pure shear modes are predicted b a state-space technique and are found to complement the bending modes predicted b the relationship. The state-space technique is capable of providing eact frequenc parameters for the Mindlin plates. A complete and accurate vibration spectrum of the thick Mindlin plates, including the modes due to bending, shear, and their coupling, are obtained b using discrete singular convolution (DSC) methods, with which the missing modes were original discovered. Numerical eamples are designed to support the present claim. Further findings include the fact that the pure transverse shear modes and mode coupling between shear and bending enter into the lower order mode ranges as the thickness of the Mindlin plate increases. Acknowledgement This research is supported in part b a Young/Junior Scholars (YSS) Funding of the Cit Universit of Hong Kong. The authors wish to thank Dr Yibao Zhao for providing the DSC-CO results. 14

15 References 1. Timoshenko SP, Woinowsk-Krieger S. Theor of plates and shells, nd ed., McGraw-Hill: New York, Leissa AW. Vibration of plates. NASA SP-160, Scientific and Technical Information Office, NASA: Washington DC, Reissner E. The effct of transverse shears deformation on the bending of elastic plate. Journal of Applied Mechanics 1945; 1: Mindlin RD. Influence of rotar inertia and shear in fleural motion of isotropic, elastic plates. Journal of Applied Mechanics 1951; 18: Wang CM, Redd JN and Lee KH. Shear Deformable Beams and Plates: Relationships with Classical Solutions, Elsevier: Singapore, Redd JN, Wang CM. Relationship between classical and shear deformation theories on aismmetric circular plates. AIAA Journal 1997; 35: Wang CM. Natural frequencies formula for simpl supported Mindlin plates. Journal of Vibration and Acoustics 1994; 116: Liew KM, Wang CM, Xiang Y, Kitipornchai S. Vibration of Mindlin Plates: Programming the p-version Ritz Method. Elsevier: Oford, Wang CM. Vibration frequencies of simpl supported polgonal sandwich plates via Kirchhoff solutions. Journal of Vibration and Vibration 1996; 190: Xiang Y, Wang CM. Eact buckling and vibration solutions for stepped rectangular plates. Journal of Sound and Vibration 00; 50: Xiang Y, Wang CM, Kitipornchai S. Eact buckling solutions for rectangular plates under intermediate and end uniaial loads. Journal of Engineering Mechanics, ASCE, 003, 19: Xiang Y, Ma YF, Kitipornchai S, Lim CW, Lau CWH. Eact solutions for vibration of clindrical shells with intermediate ring supports. International Journal of Mechanical Sciences 00, 44: Wei GW. Discrete singular convolution for the solution of the Fokker-Planck equations. Journal of Chemical Phsics 1999; 110: Wei GW. Wavelets generated b the discrete singular convolution kernels. Journal of Phsics A 000; 33: Wei GW. Vibration analsis b discrete singular convolution. Journal of Sound and Vibration 001; 44: Wei GW, Zhao YB, Xiang Y. Discrete singular convolution and its application to the analsis of plates with internal supports. I Theor and algorithm. International Journal for Numerical Methods in Engineering 00;55: Xiang Y, Zhao YB, Wei GW. Discrete singular convolution and its application to the analsis of plates with internal supports. II Comple supports. International Journal for Numerical Methods in Engineering 00;55: Lim CW, Li ZR and Wei GW, DSC-Ritz method for high-mode frequenc analsis of thick shallow shells, I. J. Num. Meth. Eng., in press, Wei GW, Zhang DS, Kouri DJ, Hoffman DK. Lagrange distributed approimating functionals. Phsical Review Letters 1997; 79: Liew KM, Lim CW. A Ritz vibration analsis of doubl-curved rectangular shallow shells using a refined first-order theor. Computer Methods in Applied Mechanics and Engineering 1995; 17:

16 1. Liew KM, Lim CW. Vibration studies on moderatel thick doubl-curved elliptic shallow shells. Acta Mechanica 1996; 116: Xiang, Y, Wei, GW. Eact solutions for buckling and vibration of stepped rectangular Mindlin plates. International Journal of Solids and Structures 004; 41:

17 Table 1. Comparison of frequenc parameter λ ωa ρh/ D / π = for an SSSS thick square Mindlin plate with ν = 0.3 and h/ a = 0.1. Values in parenthesis indicate the mode sequence number corresponding to Kirchhoff-Mindlin relationship [8]. Kirchhoff-Mindlin Eact solutions for DSC-Ritz Method Mode DSC-CO Relationship pure transverse number de la Vallée Shannon Method (mode number) Shear modes (m,n) Poussin (0,1) (1,0) (1,1) (113) (114) (0,) (,0) (1,) (,1) (115) (116) (117) (,) (0,3) (3,0) (118) (119) (1,3) (3,1) (10) (11) (1) (13) (14) (15) (,3) (3,) (0,4) (4,0) (16) (17) (1,4) (4,1) (3,3) (18) (19) (130) (131) (,4) (4,) (13) (133) (134) (135) (0,5) (5,0) (3,4) (4,3) (1,5)

18 (5,1) (136) (137) (,5) (5,) (138) (139) (140) (141) (4,4) (14) (143) (144) (3,5) (5,3) (145) (146) (0,6) (6,0) (1,6) (6,1) (147) (148) (149) (150) (,6) (6,) (151) (15) (5,4) (4,5) (153) (154) (01) (8,9) (0,14) (341) (386) (11,15) (6,19) (11,18) Coupled bending-transverse shear ( w θ or w θ ) vibration modes 18

19 Table. Comparison of frequenc parameter λ ωa ρh/ D / π = for an SSSS thick square Mindlin plate with ν = 0.3 and h/ a = 0.. Values in parenthesis indicate the mode sequence number corresponding to Kirchhoff-Mindlin relationship [8]. Mode number Kirchhoff-Mindlin Relationship (mode number) Eact solutions for pure transverse shear modes (m,n) DSC-Ritz Method Shannon De la Vallée Poussin (1) () (3) (4) (5) (6) (7) (8) (9) (10) (11) (1) (13) (14) (15) (16) (17) (18) (19) (0) (1) () (3) (4) (5) (6) (0,1) (1,0) (7) (8) (1,1) (0,) (,0) (9) (30) (1,) (,1) (,) (31) (3) (33) (0,3) (3,0) (1,3) (3,1) (34) (35) (36) (37) (,3) (3,)

20 (38) (39) (0,4) (4,0) (1,4) (4,1) (40) (41) (3,3) (,4) (4,) (4) (43) (44) (45) (46) (47) (0,5) (3,4) (4,3) (5,0) (1,5) (5,1) (48) (49) (50) (51) (5) (,5) (5,) (4,4) (53) (54) (3,5) (5,3) (55) (56) (0,6) (6,0) (57) (58) (59) (60) Coupled bending-transverse shear ( w θ or w θ ) vibration modes 0

21 Table 3. Comparison of frequenc parameter λ ωa ρh/ D / π = for an SSSS thick square Mindlin plate with ν = 0.3 and h/ a = 0.5. Values in parenthesis indicate the mode sequence number corresponding to Kirchhoff-Mindlin relationship [8]. Mode number Kirchhoff-Mindlin Relationship (mode number) Eact solutions for pure transverse shear modes (m,n) DSC-Ritz Method Shannon de la Vallée Poussin (1) () (3) (4) (5) (6) (7) (8) (9) (10) (11) (1) (13) (14) (15) (16) (17) (0,1) (1,0) (1,1) (18) (19) (0,) (,0) (1,) (,1) (0) (1) () (,) (0,3) (3,0) (3) (4) (1,3) (3,1) (5) (6) (7) (8) (,3) (3,) (9) (30) (0,4) (4,0) (1,4) (4,1) (3,3)

22 (31) (3) (33) (,4) (4,) (34) (35) (36) (37) (0,5) (3,4) (4,3) (5,0) (1,5) (5,1) (38) (39) (40) (41) (,5) (5,) (4,4) (4) (43) (44) (45) (3,5) (5,3) (46) (47) (0,6) (6,0) (1,6) (6,1) (48) (49) (50) (51) (5) (6,) (,6) (4,5) (5,4) Coupled bending-transverse shear ( w θ or w θ ) vibration modes

23 Table 4. Comparison of frequenc parameter λ ωa ρh/ D / π = for an SFSF thick square Mindlin plate with ν = 0.3 and h/ a = 0.. Mode number Pure transverse shear modes via state space method (m,n) DSC-Ritz Method Shannon de la Vallée Poussin (1,1) (1,) (,1) (1,3) (1,4) (,) (1,5) (3,1) (3,) (,3)

24 (1,6) (,4) (3,3) (4,1) (4,) (,5) (,6) (1,7) (1,8) (3,4) (4,3) (5,1) (,7) (5,) (,8) (3,5) (4,4) (3,6) (1,9) (5,3) (1,10) (6,1) (6,) (3,7)

25 Table 5. Comparison of frequenc parameter λ ωa ρh/ D / π = for an SCSC thick square Mindlin plate with ν = 0.3 and h/ a = 0.. Mode number Pure transverse shear modes via state space method (m,n) DSC-Ritz Method Shannon de la Vallée Poussin (1,1) (1,) (,1) (1,3) (,) (1,4) (3,1) (,3) (,4) (3,) (1,5) (1,6)

26 (4,1) (3,3) (,5) (4,) (,6) (3,4) (1,7) (5,1) (4,3) (1,8) (5,) (3,5) (,7) (3,6) (,8) (4,4) (5,3) (6,1) (1,9)

27 z b a h Fig. 1. Geometr of a thick plate. 7

28 Fig.. Pure shear vibration mode for a square simpl supported plate with ab= 1, hb= 0.1, n = 3, m = and λ = Fig. 3. Pure shear vibration mode for a square simpl supported plate with ab= 1, hb= 0.1, n = 3, m = 3 and λ =