Research Article An Exact Series Solution for the Vibration of Mindlin Rectangular Plates with Elastically Restrained Edges

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1 Shock and Vibration Volume 04, Article ID 8670, pages Research Article An Exact Series Solution for the Vibration of Mindlin Rectangular Plates with Elastically Restrained Edges Xue Kai, Wang Jiufa, Li Qiuhong, Wang Weiyuan, and Wang Ping College of Mechanical and Electrical Engineering, Harbin Engineering University, Harbin 5000, China Correspondence should be addressed to Wang Jiufa; Received 4 March 03; Accepted 9 September 03; Published 5 February 04 Academic Editor: Nuno Maia Copyright 04 Xue Kai et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. An analysis method is proposed for the vibration analysis of the Mindlin rectangular plates with general elastically restrained edges, in which the vibration displacements and the cross-sectional rotations of the mid-plane are expressed as the linear combination of a double Fourier ine series and four one-dimensional Fourier series. The use of these supplementary functions is to solve the possible discontinuities with first derivatives at each edge. So this method can be applied to get the exact solution for vibration of plates with general elastic boundary conditions. The matrix eigenvalue equation which is equivalent to governing differential equations of the plate can be derived through ug the boundary conditions and the governing equations based on Mindlin plate theory. The natural frequencies can be got through solving the matrix equation. Finally the numerical results are presented to validate the accuracy of the method.. Introduction Rectangular plates are important structural elements and the analysis of the vibration is very important for the design of plate-type structures in aerospace, electronic, mechanical, marine, nuclear, and structural engineering. Thus, many researchershavedonemuchworkandgotalotofresults. However, this research is based on the classical Kirchhoff hypothesis. This theory neglects the effect of shear deformation and rotary inertia which result in the over-estimation of vibration frequencies. This deviation will increase with increag plate thickness. To improve the results, the Mindlin first-order plate can be employed. So the vibration of Mindlin rectangular plates has begun to gain attention. The effective methods used to analyze the vibration of Mindlin rectangular plates include Rayleigh-Ritz method [ 4] and some numerical methods, such as differential quadrature method (DQM) [5, 6] and discrete gular convolution method (DSC) [7, 8]. Several analysis methods were also proposed by some researchers. Hashemi et al. derived the exact close form characteristic equations and their associated eigenfunctions for the thick rectangular plates with two opposite sides simply supported [9, 0]. Gorman used the superposition method to obtain a solution for the Mindlin plates [, ]. Most existing studies are limited to the classical homogeneous boundary conditions. Recently, Li proposed a Fourier series method for the vibration analysis of arbitrarily supported beam [3]. The flexural displacement of the beam is sought as the linear combination of a Fourier series and an auxiliary polynomial function. Subsequently, this method is extended to the flexural and in-plane vibration of rectangular plates under general boundary conditions. The flexural and in-plane displacement of the plate is sought as the linear combination of a double Fourier ine series and auxiliary series functions [4 6]. It has been shown that this solution methodworksverywellforvariousedgesupports. In this paper, an improved Fourier series method is employed to analyze the free vibration of Mindlin rectangular plates with general elastic boundary supports, in which the vibration displacements and the cross-sectional rotations of the mid-plane are expressed as the linear combination of a double Fourier ine series and four one-dimensional Fourier series. The possible discontinuities problems, which maybe encountered in the displacement and rotations partial differentials along the edges, can be solved by these

2 Shock and Vibration x Figure:AMindlinplatewithgeneralelasticboundarysupport. supplementary functions. Then, an exact solution for Mindlin rectangular plates with arbitrary elastically restrained edges can be obtained. Finally several numerical examples and the comparisons with those results reported in the literature are presentedtovalidatetheaccuracyofthepresentapproach.. Mathematical Modeling and Solution Methodology Consider a rectangular Mindlin plate elastically restrained along all edges, as shown in Figure. Theboundaryconditions are physically realized in terms of three kinds of restraining springs (translational, rotational, and torsional springs) attached to each edge. Different boundary conditions can be directly obtained by changing the stiffness of springs. The governing differential equations for free vibration of a Mindlinplatearegivenby kgh ( w x w y ψ x x ψ y w y ) ρh t =0, D( ψ x x D( ψ y y μ ψ x y μ ψ y x y ) kgh( w x ψ x) ρh ψ x t =0, μ ψ y x μ ψ x x y ) kgh( w y ψ y) ρh ψ y t =0, where w is the transverse displacement, ψ x and ψ y are the slope due to bending alone in the respective planes, k is the shear correction factor to account for the fact, G = E/(μ) is the shear modulus, μ is the Poisson s ratio, D=Eh 3 /(( μ )) is the flexural rigidity, ρ is the mass density, and h is the thickness of the plate. In terms of transverse displacements and slope, the bending and twisting moments and the transverse shearing forces in plates can be expressed as M x =D( ψ x x μ ψ y y ), M y =D( ψ y y μ ψ x x ), y () M xy = μ D( ψ x y ψ y x ), Q y =kgh( w y ψ y), Q x =kgh( w x ψ x). There are three forces along every edge and they are the bending moment, the twisting moment, and the shearing forces. Three kinds of restraining springs (rotational, torsional, and translational springs) along every edge can be corresponding to these three forces. The boundary conditions for an elastically restrained rectangular plate are as follows: k x0 w= Q x, K x0 ψ x = M x, K yx0 ψ y = M xy, at x=0; k xa w=q x, K xa ψ x =M x, K yxa ψ y =M xy, at x=a; k y0 w= Q y, K y0 ψ y = M y, K xy0 ψ x = M xy, at y=0; k yb w=q y, K yb ψ y =M y, K xyb ψ x =M xy, at y=b, where k x0 and k xa (k y0 and k yb ) are the translation spring constants, K x0 and K xa (K y0 and K yb )aretherotational spring constants, and K yx0 and K yxa (K xy0 and K xyb )arethe torsional spring constants at x=0and x=a(y =0and y = b), respectively. All classical homogeneous boundary conditions can be easily derived by simply setting each of the spring constants to be infinite or zero. According to the Mindlin plate theory, the transverse displacement of the plate median surface and the rotations of the cross-section, respectively, along the x direction and the y direction are utilized. In this study, these quantities are expressed in form of improved Fourier series expansions [6]: w(x,y) = A mn (λ m x) (λ n y) (ξ lb (y) d lm (λ mx) ξ la (x) f () (3) ln (λ ny)),

3 Shock and Vibration 3 ψ x (x, y) = B mn (λ m x) (λ n y) ψ y (x, y) = (ξ lb (y) d lm (λ mx) ξ la (x) C mn (λ m x) (λ n y) (ξ lb (y) f d 3 lm (λ mx) ξ la (x) f 3 ln (λ ny)), ln (λ ny)), where A mn, d lm, f ln, B mn, d lm, f ln, C mn, d 3 lm,andf3 ln are the expansion coefficients, λ m =mπ/a, λ n =nπ/b, a and b are the length and width, respectively, and ξ a (x) = a πx π a a 3πx π a, ξ a (x) = a πx π a a 3πx π a, ξ b (y) = b πy π b b (5) π b, ξ b (y) = b πy π b b π b. Theoretically, there is an infinite number of these supplementary functions. However, one needs to ensure that the selected functions will not nullify any of the boundary conditions. It is easy to verify that ξa(0) = ξa(a) = ξa (a) = 0, ξa (0) =, ξa(0) = ξa(a) = ξa (0) = 0, ξa (a) =, similar conditions exist for the supplementary function in ydirection. Though these conditions are not necessary, they can simplify the subsequent mathematical expressions and the corresponding solution procedures. One will notice from (4) that beside the standard double Fourier series, four gle Fourier series are also included. The potential discontinuity associated with the x-derivative and y-derivative of the original function along the four edges can be transferred onto these auxiliary series functions. Then, the Fourier series would be smooth enough in the whole solving domain. Therefore, not only is this Fourier series representation of solution applicable to any boundary conditions but also the convergence of the series expansion canbeimproved. Substituting (4) into the boundary conditions, for example, k x0 w= Q x,atx=0;onecanhave k x0 ( A mn λ n yξ b (y) ξ b (y) d m ) d m (4) = kgh( f n λ ny B mn λ n y ξ b (y) d m ξ b (y) d m ). (6) In order to derive the constraint equations for the unknown coefficients, all the e terms and the auxiliary series functions will be expanded into Fourier ine series. The related formulas are provided in Appendix A. Then,by equating the coefficients for the like terms on both sides, one can obtain the following equations: k x0 ( A mn β n d m β n d m ) = kgh(f n B mn β n β n d m ). d m Similarly, the substitution of (4) into the remaining boundary conditions will lead to eleven equations that can be obtained from (3): K x0 ( B mn β n d m β n d m ) = D(f n μ( ( λ q )C mq τ q η n d 3 m η n d 3 m )), K yx0 ( C mn β n d 3 m β n d 3 m ) = μ D(f3 n η n k xa ( ( ) m A mn ( λ q )B mq τ q d m η n d m ), (β ln =kgh(f n ( ) m B mn ( ) m d lm )) (β ln (7) ( ) m d lm )),

4 4 Shock and Vibration K xa ( ( ) m B mn = D(f n μ( K yxa ( ( ) m C mn (β ln ( ) m d lm )) ( λ q ) ( ) m C mq τ q (η ln = μ D(f3 n (β ln (η ln ( ) m d 3 lm ))), ( ) m d 3 lm )) ( λ q ) ( ) m B mq τ q ( ) m d lm )), k y0 ( A mn α m f n α mf n ) = kgh(d m C mn α m f 3 n α mf 3 n ), K y0 ( C mn α m f 3 n α mf 3 n ) K yb ( ( ) n C mn =D(d 3 m μ( K xyb ( (α lm p=0 (γ lm ( ) n B mn = μ D(d m ( ) n f 3 ln )) ( λ p ) ( ) n B mp ε p ( ) n f ln ))), (α lm p=0 (γ lm ( ) n f ln )) ( λ p ) ( ) n C pn ε p ( ) n f 3 ln )). (8) When all the series expansions are truncated to m=m and n=nin numerical calculations, the twelve equations can be rewritten in a matrix form as where HB = QA, (9) = D(d 3 m μ( p=0 γ m ( λ p )B mp ε p f n γ mf n )), K xy0 ( B mn α m f n α mf n ) p=0 = μ D(d m k yb ( γ m ( ) n A mn ( λ p )C pn ε p f 3 n γ mf 3 n ), (α lm =kgh(d m ( ) n C mn ( ) n f ln )) (α lm ( ) n f 3 ln )), H, H, H, H, H, H, H = [.. d.,. ] [ H, H, H, ] Q, Q, Q,3 Q, Q, Q,3 Q = [..,. ] [ Q, Q, Q,3 ] A = A 00,A 0,...,A MN,B 00,B 0,...,B MN, C 00,C 0,...,C MN } T, B = d 0,d,...,d M,f 0,f,...,f N,d 0,d,...,d M, f 0,f,...,f N,d3 0, d 3,...,d3 M,f3 0,f3,...,f3 N }T. (0) The elements of the matrices H and Q are defined in Appendix B. By substituting (4) into the governing differential equation (), as mentioned earlier, all the e terms and the auxiliary series functions are expanded into Fourier ine series.

5 Shock and Vibration 5 Then, by equating the coefficients for the like terms on both sides, one can obtain the following equations: A mn ( λ m ) (d lm ( λ m )β ln f ln φ lm)a mn ( λ n ) (d lm φ ln ( λ n )f ln α lm) B mn ( λ m )ε mp ( p=0 d lm ( λ m)ε mp β ln f C mn ( λ n )τ nq p=0 ln γ lm) f 3 (d 3 lm η ln ln ( λ n)τ nq α lm ) ρω kg (A mn β n d m f n α m β n d m f n α m) =0, D(B mn ( λ m ) μ (B mn ( λ n ) (β ln d lm ( λ m )φ lmf ln ) (φ ln d lm α lmf ln ( λ n ))) D(C mn ( λ n ) μ μ (C mn ( λ m ) ( p=0 (φ ln d 3 lm α lmf 3 ln ( λ n )) (β ln d 3 lm ( λ m )φ lmf 3 ln )) B mn λ m λ n ε mp τ nq ( p=0 d lm η ln ( λ m )ε mp f kgh( A mn ( λ n )τ nq f ln γ lm ( λ n )τ nq ))) (d lm η ln ln α lm ( λ n )τ nq ) C mn ρhω (C mn (d 3 lm β ln f 3 ln α lm)) (d 3 lm β ln f 3 ln α lm)) = 0. () μ p=0 ( C mn λ m λ n ε mp τ nq p=0 ( p=0 d 3 lm η ln ( λ m )ε mp f 3 kgh( A mn ( λ m )ε mp ( p=0 B mn ρhω (B mn ln γ lm ( λ n )τ nq ))) β ln d lm ( λ m)ε mp γ ln f (β ln d lm α lmf ln )) (β ln d lm α lmf ln )) = 0, ln ) Writing in matrix form, we have CA DB ρhω kg (EA FB) = 0. () Substituting (9),thefinalsystemequationscanbeobtained as where (K ρhω M) A = 0, (3) kg C, C, C,3 C = [ C, C, C,3 ], [ C 3, C 3, C 3,3 ] D, D, D, D = [ D, D, D, ], [ D 3, D 3, D 3, ] E, E, E,3 E = [ E, E, E,3 ], [ E 3, E 3, E 3,3 ]

6 6 Shock and Vibration Table : The first seven frequency parameters Ω =(ωb /π )(ρh/d) / for F-F-F-F Mindlin rectangular plates. M=N Table : The first seven frequency parameters Ω =(ωa )(ρh/d) / for S-F-S-F Mindlin rectangular plates. a/b h/a Method Present [9] Present [9] Present [9] Present [9] Present [9] Present [9] F, F, F, F = [ F, F, F, ], [ F 3, F 3, F 3, ] (4) where K = C DH Q and M = E FH Q.Theelement of the matrices C, D, E,andF are defined in Appendix B.The natural frequencies and eigenvectors can be obtained through solving (3).Then,thephysicalmodeshapesalsocanbegot ug (4)and(9). 3. Result and Discussion Several examples involving various boundary conditions will bediscussedinthissection.toavoidanycomparisonofthe roundoff results, which might be unrealistic, the nondimensional frequency is used. For the analysis, Poisson s ratio μ= 0.3 and shear correction factor k = 5/6 are used. First, convergence studies are carried out. Table gives the frequencies calculated by ug different number of terms in the series expansion for F-F-F-F Mindlin rectangular plates (for a/b = and h/a = ).Itshowsthattheresultsarevery accurate when M and N are small numbers. When M and N are larger than 0, results are almost invariant. So this method has a very good convergence characteristic. In order to evaluate the accuracy of the present method, the comparisons with those results reported in the literature arecarriedout.firstofall,consideringaplatewithtwoopposite simple edges and the other two edges free (S-F-S-F). A simple edge can be obtained by setting the translational and torsional spring constants to be infinite (here the infinite is represented by a very large number, D 0 7,andD is the flexural rigidity) and the rotational spring constants to be zero. A free edge can be obtained when all the spring constants are zero. In Table, the first seven nondimensional frequency parameters, Ω = (ωa )(ρh/d) /,aregivenwith different aspect ratios and thickness ratios. At the same time, a comparison of the exact solution in [9] isalsopresented. Theresultsshowthatthecalculatedfrequenciesareexcellent. As mentioned earlier, the series expansion will have to be truncated in numerical calculations. In this example and all thesubsequent calculations, thesetting M=N=is used. Specially, in order to compare with the results in [9], shear correction factor k = is used. Three more classical cases (C-F-F-F, C-F-S-F, and C- S-S-F) were considered, and the corresponding frequency parameters are listed in Tables 3, 4, and 5. A clamped edge can be viewed as all the spring constants set to be infinite. The frequency parameters solved by Ritz method and DSC method also are given as a comparison. A good agreement is also observed among these solutions. This method not only can solve the Mindlin rectangular plates with classical boundary conditions but also can solve plates with the elastical supports. Xiang et al. [3]andGorman []andzhou[4] also researched Mindlin plates with elastically restrained edges, but they both used two kinds of springs (rotational and translational springs) to realize the elastical

7 Shock and Vibration 7 Table 3: The first seven frequency parameters Ω =(ωb /π )(ρh/d) / for C-F-F-F Mindlin rectangular plates. a/b h/b Method Present [] Present [] Present [] Present [] Present [] Present [] Table 4: The first seven frequency parameters Ω =(ωb /π )(ρh/d) / for C-F-S-F Mindlin rectangular plates. a/b h/b Method Present [8] Present [8] Present [7] Present [7] Present [7] Present [7] Table 5: The first seven frequency parameters Ω =(ωb /π )(ρh/d) / for C-S-S-F Mindlin rectangular plates. a/b h/b Method Present [8] Present [8] Present [8] Present [8] Present [8] Present [8] supports along every edge. As mentioned earlier, three kinds of springs along every edge are needed to truly realize the general elastic supports, including all classical homogeneous boundary conditions. Table 6 gives the first seven frequency parameters for Mindlin rectangular plates (for a/b = and h/a = 0.) with two opposite free edges, the other two edges only elastically restrained against translation, and Table 7 gives the first seven frequency parameters for the plates with twooppositefreeedges,theothertwoedgessymmetrically elastically restrained. Table 8 lists the frequency parameters when the translational, rotational, and torsional spring constants are all varying at x = a. For simplicity, all the restraining springs are assumed to have the same stiffness in the three examples. Table 8 showsthattheresultsare invariant when K is larger than 0 7, so it is appropriate to set infinite to be D 0 7 in this paper.

8 8 Shock and Vibration Table 6: The first seven frequency parameters Ω =(ωb /π )(ρh/d) / for Mindlin rectangular plates with two opposite free edges, the other two edges only elastically restrained against translation k x0 = k xa =K D. K Method Present [4] Present [4] Present [4] Present [4] Table 7: The first seven frequency parameters Ω =(ωb /π )(ρh/d) / for Mindlin rectangular plates with two opposite free edges, the other two edges symmetrically elastically restrained k x0 = k xa = K Dand K x0 = K xa = K D. K K Method Present [4] Present [4] Present [4] Table 8: The first seven frequency parameters Ω =(ωb /π )(ρh/d) / for C-C-F-C Mindlin rectangular plates with translational, rotational, and torsional restraints at x=a, k xa = K xa = K yxa = K D. K Conclusions An improved Fourier series method is proposed to analyze the free vibration of Mindlin rectangular plates with general elastic boundary supports. The general boundary conditions are physically realized with the uniform distribution of springs on each boundary edge, and different boundary conditions can be directly obtained by changing the stiffness ofthesprings.thevibrationdisplacementsandthecrosssectional rotations of the mid-plane are sought as the linear combination of a double Fourier ine series and four gle auxiliary series functions, respectively. The use of these supplementary functions is to solve the discontinuity problems which were encountered in the displacement and rotations partial differentials along the edges. The unknown expansion coefficients can be solved through ug the boundary conditions and the governing equations. In this method, analytical solution is derived for the vibrations of Mindlin rectangular plates with general elastic boundary support. Finally, the numerical results and the comparisons with those reported in the literature are presented to validate the accuracy of the method. Appendices A. Supplementary Series We have ξ a (x) = a πx π a a 3πx π a = α m λ m x, 4a, m = 0, 3π α m = a ( 4m )π 6a (9 4m )π,,

9 Shock and Vibration 9 ξ a (x) = a πx π a a 3πx π a = α m λ m x, a, m = 0, 3π α m = a( )m ( 4m )π 6a( )m (9 4m )π,, ξ b (y) = b πy π b b π b = β n λ n y, 4b, n = 0, 3π β n = b ( 4n )π 6b (9 4n )π,, ξ b (y) = b πy π b b π b = β n λ n y, b, n = 0, 3π β n = b( )n ( 4n )π 6b( )n (9 4n )π,, ξ a (x) = πx 4 a 3 3πx 4 a = γ m λ m x,, m = 0, π γ m = ( ) m ( 4m )π 9( )m (9 4m )π,, ξ a (x) = πx 4 a 3 3πx 4 a = γ m λ m x, 0, m = 0, γ m = ( 4m )π 9 (9 4m )π,, ξ b (y) = πy 4 b 3 4 b = η n λ n y,, n = 0, π η n = ( ) n ( 4n )π 9( )n (9 4n )π,, ξ b (y) = πy 4 b 3 4 b = η n λ n y, 0, n = 0, η n = ( 4n )π 9 (9 4n )π,, ξ a (x) = π πx 8a a 9π 3πx 8a a = φ m λ m x,, m = 0, a φ m = ( 4m )a 7 (9 4m )a,, ξ a (x) = π πx 8a a 9π 3πx 8a a = φ m λ m x,, m = 0, a φ m = ( ) m ( 4m )a 7( )m (9 4m )a,, ξ b (y) = π πy 8b b 9π 8b b = φ n λ n y,, n = 0, b φ n = ( 4n )b 7 (9 4n )b,, ξ b (y) = π πy 8b b 9π 8b b = φ n λ n y,, n = 0, b φ n = ( ) n ( 4n )b 7( )n (9 4n )b,, λ m x= p =0, ε mp = m =0 p=0 ε mp λ p x p=0, ε mp =0,, λ n y= q =0, ε nq = n =0, ( ) p, pπ m=p, 0, p (( ) mp ) m =p, (m p )π, ε nq λ q y,, ε nq =0,, ( ) q qπ n=q, 0, q (( ) nq ) n =q, (n q )π. (A.)

10 0 Shock and Vibration B. Definitions of Matrices We have C, } s,t = (λ m λ n )δ st, C, } s,t = λ m δ st ε mp, p=0 C,3 } s,t = λ n δ stε nq, D, } s,m =( λ m β n φ n )δ mm, D, } s,m =( λ m β n φ n )δ mm, D,3 } s,n =( λ n α m φ m )δ nn, D,4 } s,n =( λ n α m φ m )δ nn, D,5 } s,m = λ mβ n δ mm ε mp, p=0 D,6 } s,m = λ mβ n δ mm ε mp, p=0 D,7 } s,n =γ mδ nn, D,8 } s,n =γ mδ nn, D,9 } s,m =η nδ mm, D,0 } s,m =η nδ mm, D, } s,n = λ nα m δ nn τ nq, D, } s,n = λ nα m δ nn τ nq, E, } s,t =δ st, E, } s,t =0, E,3 } s,t =0, F, } s,m =β nδ mm, F, } s,m =β nδ mm, F,3 } s,n =α mδ nn, F,4 } s,n =α mδ nn, F,5 } s,m =0, F,6} s,m =0, F,7 } s,n =0, F,8} s,n =0, F,9 } s,m =0, F,0} s,m =0, F, } s,n =0, F,} s,n =0, H, } n,m = k x0 β n, H, } n,m = k x0 β n, H,3 } n,n = kchδ nn, H,4 } n,n =0, H,5 } n,m = kchβ n, H,6 } n,m = kchβ n, H,7 } n,n =0, H,8} n,n =0, H,9 } n,m =0, H,0 } n,m =0, H, } n,n =0, H,} n,n =0, Q, } s,t =k x0 δ mm, Q, } s,t =kghδ mm, Q,3 } s,t =0, (B.) where m = 0,,...,M, m = 0,,...,M, n = 0,,...,N,,,...,N, s = m(n)n, andt=m (N)n. Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper. Acknowledgments The authors acknowledge the support of this work by the National Natural Science Foundation of China (no ) and the Fundamental Research Funds for the Central Universities (HEUCF070). The authors would like to express their cere thanks to the reviewers for the constructive and positive comments. References [] K. M. Liew, Y. Xiang, and S. Kitipornchai, Transverse vibration of thick rectangular plates I. Comprehensive sets of boundary conditions, Computers and Structures, vol. 49, no., pp. 9, 993. [] K. M. Liew, K. C. Hung, and M. K. Lim, Vibration of mindlin plates ug boundary characteristic orthogonal polynomials, Journal of Sound and Vibration,vol.8,no.,pp.77 90,995. [3] Y.Xiang,K.M.Liew,andS.Kitipornchai, Vibrationanalysis of rectangular mindlin plates resting on elastic edge supports, Journal of Sound and Vibration,vol.04,no.,pp. 6,997. [4] D. Zhou, Vibrations of Mindlin rectangular plates with elastically restrained edges ug static Timoshenko beam functions with the Rayleigh-Ritz method, International Journal of Solids and Structures,vol.38,no.3-33,pp ,00. [5] F.-L. Liu and K. M. Liew, Analysis of vibrating thick rectangular plates with mixed boundary constraints ug differential quadrature element method, JournalofSoundandVibration, vol. 5, no. 5, pp , 999. [6] P. Malekzadeh and S. A. Shahpari, Free vibration analysis of variable thickness thin and moderately thick plates with elastically restrained edges by DQM, Thin-Walled Structures, vol. 43, no. 7, pp , 005. [7] Y.Hou,G.W.Wei,andY.Xiang, DSC-Ritzmethodforthefree vibration analysis of Mindlin plates, International Journal for Numerical Methods in Engineering, vol.6,no.,pp.6 88, 005.

11 Shock and Vibration [8] Y.Xiang,S.K.Lai,andL.Zhou, DSC-elementmethodforfree vibrationanalysisofrectangularmindlinplates, International Journal of Mechanical Sciences,vol.5,no.4,pp ,00. [9] S. H. Hashemi and M. Arsanjani, Exact characteristic equations for some of classical boundary conditions of vibrating moderately thick rectangular plates, International Journal of Solids and Structures,vol.4,no.3-4,pp ,005. [0] S. H. Hashemi, K. Khorshidi, and H. Rokni Damavandi Taher, Exact acoustical analysis of vibrating rectangular plates with two opposite edges simply supported via Mindlin plate theory, JournalofSoundandVibration,vol.3,no.4-5,pp , 009. [] D. J. Gorman, Free vibration analysis of Mindlin plates with uniform elastic edge support by the superposition method, Journal of Sound and Vibration,vol.07,no.3,pp ,997. [] F. Ohya, M. Ueda, T. Uchiyama, and M. Kikuchi, Free vibration analysis by the superposition method of rectangular Mindlin plates with internal columns resting on uniform elastic edge supports, Journal of Sound and Vibration, vol.89,no.-,pp. 4, 006. [3] W. L. Li, Free vibrations of beams with general boundary conditions, Journal of Sound and Vibration, vol.37,no.4,pp , 000. [4] W. L. Li, Vibration analysis of rectangular plates with general elastic boundary supports, Journal of Sound and Vibration, vol. 73, no. 3, pp , 004. [5] J. Du, W. L. Li, G. Jin, T. Yang, and Z. Liu, An analytical method for the in-plane vibration analysis of rectangular plates with elastically restrained edges, Journal of Sound and Vibration,vol. 306, no. 3 5, pp , 007. [6] W.L.Li,X.Zhang,J.Du,andZ.Liu, Anexactseriessolution for the transverse vibration of rectangular plates with general elastic boundary supports, Journal of Sound and Vibration, vol. 3, no. -, pp , 009.

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