Vibration Analysis of Isotropic and Orthotropic Plates with Mixed Boundary Conditions

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1 Tamkang Journal of Science and Engineering, Vol. 6, No. 4, pp. 7-6 (003) 7 Vibration Analsis of Isotropic and Orthotropic Plates with Mied Boundar Conditions Ming-Hung Hsu epartment of Electronic Engineering National Penghu Institute of Technolog Penghu, Taiwan 7, R.O.C. minghung@mail000.com.tw Abstract The vibration response of isotropic and orthotropic plates with mied boundar conditions is numericall modeled using a solution that is based on the differential quadrature method (QM). The QM is applied to each region and with the imposition of appropriate boundar conditions; the problem is transformed into a standard eigenvalue problem. The δ technique is used to treat the various boundar conditions. The results also demonstrate the efficienc of the method in treating this class of engineering problem. Ke Words: ifferential Quadrature Method, Plates, Numerical Methods, Vibration Analsis, Mied Boundar Condition. Introduction A variet of numerical methods are available toda for engineering analsis. Traditionall, there are three numerical methods, which are finite difference method, finite element method and boundar element method. The numerical methods have been used etensivel for solving linear and nonlinear differential equations, and consequentl there are several commerciall developed software packages. The differential quadrature method (QM) is a relativel new method that was introduced b Bellman et al. []. Several researchers have applied the QM to solve a variet of problems in different fields of science and engineering. The QM has wide applications in initial and boundar value problems. Civan et al. [,3] solved multivariable mathematical models using the quadrature method and the cubature method. Han et al. [4] analzed the aismmetric free vibration of moderatel thick annular plates using the differential quadrature method. Chen et al. [5] pointed out that the QM and the CM, due to their global domain propert, are more efficient for nonlinear problems than the traditional numerical techniques such as finite element and finite difference methods. Bert et al. [6,7,8] solved static and free vibration analsis of beams and plates using the QM. Malik et al. [9] has developed a detailed methodolog for implementing multiple boundar conditions in differential quadrature solutions of higher-order differential equations. Rectangular plates have wide applications in civil and mechanical engineering. The dnamic characteristics of rectangular plates are important in engineering designs. Xiang et al. [0] used the Ritz method combined with a variation to solve vibration of rectangular mindlin plates resting on elastic edge supports. Laura et al. [,] calculated the fundamental frequenc coefficient for a rectangular plate with edges elasticall restrained against both translation and rotation using polnomial coordinate functions and the Raleigh-Ritz s method. Grossi et al. [3] analzed the non-uniform plate with the rotational springs at the edges using the Raleigh-Ritz method. Gorman [4] solved the free vibration problem of shear-deformable plates resting on uniform elastic foundations using the modified Superposion-Galerkin method. Omurtag et al. [5] used mied finite element formulation to analze free vibration of orthotropic plates. In this stud, the QM is emploed to solve

2 8 Ming-Hung Hsu the eigenvalue problem of isotropic and orthotropic plates with mied boundar conditions. An overview of the QM to preset the computation of its weighting coefficients and discussion of the selection problem will be offered. The integrit and computational efficienc of the method will be demonstrated in this paper.. The ifferential Quadrature Method The QM uses the basis of the Gauss method in deriving the derivative of a function. It follows that the partial derivative of a function with respect to a space variable can be approimated b a weighted linear combination of function values at some intermediate points in that variet. A differential quadrature approimation at the ith discrete point on a grid at the ais is given b [6] m N f (, ) i ( m) = ij ( j, m A f ) j= for i =,,, N () A differential quadrature approimation at the ith discrete point on a grid at the ais ma be written as N (, ) i ( m) = Bij f (, j ) m f m j= for i =,,, N () ( ) where A n ij and B ( n) ij are the weighting coefficients. The test function can be written as α β (, ) = f for α =,,, N and β =,,, N (3) Substituting Eq. (3) to Eqs. () and (), one obtains N m α α ( m) k Aik = m k = = i for i =,,, N and α =,,, N (4) and N m β β ( m) l Bjk = m k = = j for j =,,, N and β =,,, N (5) The higher-order derivates ma be obtained using following equations ( ) ( ) ( ) A = A A (6) ( 3) ( ) ( ) A = A A ( 4) ( ) ( 3) A = A A M ( m) ( ) ( m ) A = A A ( ) ( ) ( ) B = B B ( 3) ( ) ( ) B = B B ( 4) ( ) ( 3) B = B B M ( m) ( ) ( m ) B = B B ( m) (7) (8) (9) (0) () () (3) where A is the mth order weighting ( m) coefficient matri in the direction and B is the mth order weighting coefficient matri in the direction. The above relation gives the higher order weighting coefficient matri based on the first-order derivative weighting coefficients. The above relations are not restricted to the choice of sampling points. Also calculation of weighting coefficients b these formulae contains a substantial reduction in numerical computations. 3. Choice of the Sampling Points The selection of locations of the sampling points plas a significant role in the accurac of the solution of differential equations. Using equall spaced points can be considered to be a convenient and an eas selection method. A domain is separated into b N, N, points in the, direction. A more accurate solution b choosing a set of unequall spaced sampling points could be obtained. A simple and good choice can be the roots of shifted Chebshev and Legendre points. Bert et al. [6] point out that following nonuniform grid spacing gives better and more reliable result. The inner points are ( i ) π Xi = Cos N 3 for i = 3, 4,, N (4) in the direction,

3 Vibration Analsis of Isotropic and Orthotropic Plates with Mied Boundar Conditions 9 ( i ) π Yi = Cos N 3 for i = 3, 4,, N (5) in the direction, and boundar points are X (6) X X = δ (7) N = δ (8) X = (9) N in the direction. Y (0) Y Y = δ () = N δ () Y = (3) N in the direction. Where, δ is small distance in the direction, δ is small distance in the direction, X = /a, Y = /b, a is the length of the plate in the direction and b is the length of the plate in the direction. 4. Vibration Analsis of the Isotropic Plates The analsis of freel vibrating thin rectangular plates of isotropic materials involves the essential solution of the following eigenvalue differential equation [7]: Y λ + λ =Ω 4 4 where λ = a/b, Ω = ωa ( ρ h/), W (4) = Eh 3 /[( v )] is the fleible rigidit, W is the non-dimensional transverse deflection of the plate, where E is Young s modulus, v is Poisson s ratio, ρ is the densit of the plate material respectivel, and h is the plate thickness. For a simpl supported or a clamped boundar, the transverse deflection of the plate is zero: W (5) For a simpl supported boundar, the condition of zero normal moment can be reduced to (6) at the direction edge, and (7) at the direction edge. The condition of zero normal moment at a free boundar is given b + νλ (8) at the direction edge, and λ + ν (9) at the direction edge. The condition of zero effective shear force at a free or a guided boundar is given b 3 + ( ν) λ (30) at the direction edge, and 3 3 λ + 3 ( ν) (3) at the direction edge. It is assumed that the stiffness of the elastic restraint for the rotational restraint k φ can be epressed as K φ = k φ a/ (3) at the direction, and K φ = k φ b/ (33) at the direction. The constraint boundar is given b W = K φ at the direction edge, and = K φ W at the direction edge. The QM formula of the eigenvalue differential equation is given b N N N ( 4) ( ) ( ) Aik Wkj + λ Aik Bjl Wkl k= k= l= N 4 (4) ij ij ij l= (34) (35) + λ B W Ω W (36) Combination of the governing equation with mied boundar conditions will be substituted into the following sstem of linear equations Elu Eru Wb 0 Eld E = (37) rd Wi Ω Wi

4 0 Ming-Hung Hsu where the subscript b and i refer to the locations at the boundar and the interior regions, respectivel. The vector {W b } and {W i } are the normal deflection vectors corrsponding to the boundar and interior points. Substituting Eq. (37) into a general eigenvalue form, one obtains [E]{ W i } = Ω {W i } (38) where [ ] E = E E E E rd ld lu ru (39) B solving the eigenvalue problem of Eq. (38), the frequenc parameters will be got. 5. Vibration Analsis of the Orthotropic Plates The analsis of freel vibrating thin rectangular plates of orthotropic materials involves the essential solution of the following eigenvalue differential equation [7]: + Y λ 4 W λ =Ω where λ = a/b, ωa ρh/ ( ) 3 /[ v W = E h ( v )], 3 /[ v = E h ( v )], = ν, 3 66 G h / =, (40) Ω= + 66, E is Young s modulus in the direction, E is Young s modulus in the direction, G is shear modulus in the - plane, v is Poisson s ratio for transverse strain in the direction when stressed in the direction, v is Poisson s ratio for transverse strain in the direction when stressed in the direction, ρ is the densit of the plate material, and h is the plate thickness. For a simpl supported or a clamped boundar, the transverse deflection of the plate is zero: W (4) For a simpl supported boundar, the condition of zero normal moment can be reduced to + λ (4) at the direction edge, and + λ (43) at the direction edge. It is assumed that the stiffness of the elastic restraint for the rotational restraint k φ, can be epressed as K φ = k φ a/ (44) in the direction, and K φ = k φ b/ in the direction. The constraint boundar conditions are considered: W + λ = K φ (46) at the direction edge, and W + = K φ λ (47) at the direction edge. Eq. (40) can be written in terms of Q approimation as follows: N N N ( 4) ( ) ( ) Aik Wkj + λ Aik Bjl Wkl + 66 k= k= l= N 4 (4) + λ Bij Wij Ω Wij (48) + l= 66 Combination of the governing equation with boundar conditions will be substituted into the following sstem of linear equations Flu Fru Wb 0 Fld F = (49) rd Wi Ω Wi where the subscript b and i refer to the locations at the boundar and the interior regions, respectivel. The vector {W b } and {W i } are the normal deflection vectors in corresponding to the boundar and interior points. Substituting Eq. (49) into a general eigenvalue form, one obtains [F]{W i } = Ω {W i } (50) where [ F ] = Frd Fld Flu F ru (5) B solving the eigenvalue problem of Eq. (50), the frequenc parameters will be obtained. 6. Results and iscussion Table shows the frequenc parameters of

5 Vibration Analsis of Isotropic and Orthotropic Plates with Mied Boundar Conditions the isotropic plates that are supported as all of edges are simpl supported with torsion spring constraint as shown in Figure. In the models of QM, the data were used as follows: 6 sampling points and δ, δ 5. Table shows that the magnitude of the Ω increases when K φ and K φ increases. Then difference between Ω solved using the QM and Ω obtained in reference [4] are from 0.70% to 0.358%. The results are found to be in good agreement with results in the literature. Table shows the frequenc parameters of the isotropic plate that is supported as all of edges are simpl supported ecept one edge is simpl supported with torsion spring as shown in Figure. It is observed that with an increase in K φ, Ω are increased. Table 3 shows the frequenc parameters of the isotropic plates that are supported as all of edges are simpl supported ecept two edges are simpl supported with torsion spring as shown in Figure 3. The results show that Ω increases when K φ, K φ increase. Ω is sensitive to the torsion spring stiffness. Table. The frequenc parameters of the isotropic plates that are supported as all of edges are simpl supported ecept one edge is simpl supported with torsion spring for different K φ. K φ Ω (QM) Table 3. The frequenc parameters of the isotropic plates that are supported as all of edges are simpl supported ecept two edges are simpl support with torsion spring for different K φ, K φ. K φ, K φ ( QM ) Ω Simpl supported Table. The frequenc parameters of the isotropic plates that are supported as all of edges are simpl supported with torsion spring for different K φ, K φ. K φ, K φ Ω (QM) Simpl supported Simpl supported Ω (Reference [4]) ifference(%) Simpl supported with torsion spring Simpl supported with torsion spring Simpl supported with torsion spring Simpl supported with torsion spring Simpl supported with torsion spring Figure. The plate is supported as all of edges are simpl supported with torsion spring constraint Figure. The plate is supported as all of edges are simpl supported ecept one edge is simpl supported with torsion spring. Table 4 shows the frequenc parameters of the isotropic plates that are supported as all of edges are simpl supported ecept three edges are simpl supported with torsion spring as shown in Figure 4. The calculated results displa the higher K φ, K φ introduce the higher Ω. The frequenc parameters of the orthotropic plate that is supported, as all of edges are simpl supported is presented in Table 5. In order to justif the model, the orthotropic plates results are compared to the analtical epressions given b Leissa [7]. For the orthotropic plates with simpl supported, the analtical epression for the natural angular frequenc given b Reference [7] is,

6 Ming-Hung Hsu Table 4. The frequenc parameters of the isotropic plates that are supported as all of edges are simpl supported ecept three edges are simpl supported with torsion spring for different K φ, K φ. K φ, Kφ Ω ( QM ) Table 5. The frequenc parameters of the orthotropic plate that are supported as all of edges are simpl support. m,n Ω (QM) Ω (Eq. 5) ifference 0 ( 0 ), , , , , , , , , Simpl supported Simpl supported Simpl supported with torsion i Figure 3. The plate is supported as all of edges are simpl supported ecept two edge is simpl supported with torsion spring. π ωmn = m + ( + 66 ) m n λ + n λ a ρ (5) In order to compare the results of QM, a well-known frequenc parameter definition is used, ρh Ω=ωmna (53) + 66 Simpl supported with torsion In the model, the material properties of the square orthotropic plate are /( + 66 ) =.543 and Simpl supported with torsion spring Simpl supported Simpl supported with torsion spring Simpl supported with torsion spring Figure 4. The plate is supported as all of edges are simpl supported ecept three edge is simpl supported with torsion spring. /( + 66 ) = In the models, the data where used as follows: 6 sampling points and δ, δ 5. The difference between Ω 0 from QM and Ω from equations (5) and (53) is from 0.% to 0.%. Table 6 shows the frequenc parameters of the orthotropic plates that are supported, as all of edges are simpl supported with torsion spring for different K φ,k φ as shown in Figure. The results show that Ω increases when K t increases.

7 Vibration Analsis of Isotropic and Orthotropic Plates with Mied Boundar Conditions 3 Table 6. The frequenc parameters of the orthotropic plates that are supported as all of edges are simpl supported with torsion spring for different K φ,k φ. K φ,k φ Ω (QM) Table 7. The frequenc parameters of the orthotropic plates that are supported as all of edges are simpl supported ecept one edge is simpl supported with torsion spring for different K φ. K φ Ω (QM) Table 8. The frequenc parameters of the orthotropic plate that are supported as all of edges are simpl supported ecept two edges are simpl supported with torsion spring for different K φ, K φ. K, K φ φ Ω ( QM ) Table 7 shows the frequenc parameters of the orthotropic plates that are supported as all of edges are simpl supported ecept one edge are simpl supported with torsion spring as shown in Figure. The results show that K φ increases when Ω increases. Table 8 shows the frequenc parameters of The orthotropic plate that are supported, as all of edges are simpl supported ecept two edges are simpl supported with torsion spring as shown in Figure 3. The results show that Ω increases with increases in K φ and K φ. Table 9 presents the frequenc parameters ofthe orthotropic plate that are supported as all of edges are simpl supported ecept three edges are simpl supported with torsion spring as shown in Figure 4. The results show that when K φ,k φ increases, Ω increases. The frequenc parameters of the orthotropic plates that are supported, as all of edges are simpl supported with different δ, δ is appeared in Table 0. It ma be saw from Table 0, the difference of Ω solved using the QM with δ, δ = 0 6, and Ω calculated using Eq. (5) is 0.%. Table 9. The frequenc parameters of the orthotropic plate that are supported as all of edges are simpl supported ecept three edges are simpl supported with torsion spring for different K φ, K φ. K φ,k φ Ω (QM) Table presents the frequenc parameters of the orthotropic plates with different K φ. The geometr of a rectangular orthotropic plate divided into four subdomains with mied boundar conditions is showed in Figure 5. The data used in the analsis are as follows: a/b =.7, v.3, E = N/m, E = N/m, ρh/ρ s h =., ρh/ρ s h =.5, ρh/ρ s3 h =.0, ρh/ρ s4 h.8, and G = N m, ρ s, ρ s, ρ s3 and ρ s4 are the densit of the subdomains S, S, S3 and S4. ue to the multiple support discontinuities along the edge, the dnamic characteristics of a rectangular plate with mied boundar conditions are much more comple than the rectangular plates with support continuities along the edges. The results indicate that the frequenc parameters of the orthotropic plates decrease as K φ increases. The results indicate that the eigenvalue of the orthotropic plates decreases as K φ increases. Table shows the frequenc parameters of the plates with different ρ s h. The numerical results indicate that ρ s h is a significant influence on the eigenvalue of the orthotropic plates. It is clear that the frequenc parameters of the orthotropic plates decrease as ρ s h increases. Similar results for eigenvalue of the orthotropic plate with different ρ s h, ρ s3 h and ρ s4 h are listed in Tables 3, 4 and Concluding Remarks This paper presents, to the author s knowledges, the first known vibration analsis of rectangular isotropic and rectangular orthotropic plates with edges elasticall restrained against rotational displacements using the QM. The present method is, however, a ver general approimation technique that is able to provide vibration solutions for plates with an combination of boundar conditions and elasticall restrained edges. It is concluded that the demonstrated accurac and simplicit of the QM make it a good candidate for modeling more complicated cases of vibration of isotropic plates and orthotropic plates with mied boundar conditions.

8 4 Ming-Hung Hsu Table 0. The frequenc parameters of the orthotropic plates that are supported as all of edges are simpl supported for different δ, δ. δ, δ Ω (QM) Ω (Eq. 53) ifference (%) Table. The frequenc parameters of the orthotropic plates with different K φ K φ (K φ a / ) ρ s h/ρh ρ s h/ρh ρ s3 h/ρh ρ s4 h/ρh Ω (QM) Table. The frequenc parameters of the orthotropic plates with different ρ s h K φ (K φ a/ ) ρ s h/ρh ρ s h/ρh ρ s3 h/ρh ρ s4 h/ρh Free edge Ω (QM) S4 S3 Table 3. The frequenc of the orthotropic plates with different ρ s h Free edge S S Free edge K φ (K φ a/ ) ρ s h/ρh ρ s h/ρh Free edge Simpl supported with torsion spring ρ s3 h/ρh ρ s4 h/ρh Figure 5. The orthotropic plate with mied boundar conditions Ω (QM)

9 Vibration Analsis of Isotropic and Orthotropic Plates with Mied Boundar Conditions 5 Table4. The frequenc parameters of the orthotropicplates with different ρ s3 h K φ (K φ a/ ) ρ s h/ρh ρ s h/ρh ρ s3 h/ρh ρ s4 h/ρh Ω (QM) Table 5. The frequenc parameters of the plates with different ρ s4 h K φ (K φ a/ ) ρ s h/ρh ρ s h/ρh ρ s3 h/ρh ρ s4 h/ρh Ω (QM) References [] Bellman, R. E., Kashef, B. G. and Casti, J., ifferential Quadrature: a Technique for Rapid Solution of Nonlinear Partial ifferential Equations, Journal of Computational Phsics, Vol. 0, pp (97). [] Civan, F. and Sliepcevich, C. M., Application of ifferential Quadrature to Transport Processes, Journal of Mathematical Analsis and Applications, Vol. 93, pp. 06- (983). [3] Civan, F., Solving Multivariable Mathematical Models b the Quadrature and Cubature Methods, Numerical Methods for Partial ifferential Equations, Vol. 0, pp (994). [4] Han, J. B. and Liew, K. M., Aismmetric Free Vibration of Thick Annular Plates, International Journal of Mechanical Science, Vol. 4, pp (999). [5] Chen, W. and Zhong, T., The Stud on the Nonlinear Computations of the Q and C Methods, Numerical Methods for Partial ifferential Equations, Vol. 3, pp (997). [6] Bert, C. W., Wang, X. and Striz, A. G., Convergence of the Q Method in the Analsis of Anisotropic Plates, Journal of Sound and Vibration, Vol. 70, pp (994). [7] Bert, C. W., Wang, X. and Striz, A. G., ifferential Quadrature for Static and Free Vibration Analsis of Anisotropic Plates, International Journal of Solids and Structures, Vol. 30, pp (993). [8] Bert, C. W., Wang, X. and Striz, A. G., Static and Free Vibration Analsis of Beams and Plates b ifferential Quadrature Method, Acta Mechanica, Vol. 0, pp. -4 (994). [9] Malik, M. and Bert, C. W., Implementing Multiple Boundar Conditions in the Q Solution of Higher-order PE s: Application to Free Vibration of Plates, International Journal for Numerical Methods in Engineering, Vol. 39, pp (996). [0] Xiang, Y., Liew, K. M. and Kitipornchai, S., Vibration Analsis of Rectangular Mindlin Plates Resting on Elastic Edge Supports, Journal of Sound and Vibration, Vol. 04, pp. -6 (997). [] Laura, P. A. A. and Grossi, R. O., Transverse Vibration of a Rectangular Plate Elasticall Restrained Against Rotation Along Three Edges and Free on the Fourth Edge, Journal of Sound and Vibration, Vol. 59, pp (978). [] Laura, P. A. A. and Gross, R. O., Transverse Vibrations of Rectangular Plates with Edges Elasticall Rrestrained Against Translation and Rotation, Journal of Sound and Vibration, Vol. 75, pp (98). [3] Grossi, R. O. and Nallim, L. G., A Note on the Strain Energ Stored in Rotational Springs at the Plate Edges of Non-uniform Thickness, Journal of Sound and Vibration, Vol. 06, pp (997).

10 6 Ming-Hung Hsu [4] Gorman,. J., Accurate Free Vibration Analsis of Shear-deformable Plates with Torsional Elastic Edge Support, Journal of Sound and Vibration, Vol. 03, pp (997). [5] Omurtag, M. H. and Kadioglu, F., Free Vibration Analsis of Orthotropic Plates Resting on Pasternak Foundation b Mied Finite Element Formulation, Computers and Structures, Vol. 67, pp (998). [6] Malik, M., and Bert, C. W., Implementing Multiple Boundar Conditions in the Q Solution of Hogher-order PE s: Application to Free Vibration of Plates, International Journal for Numerical Methods in Engineering, Vol. 39, pp (996). [7] Leissa, A. W., Vibration of Plate, NASA SP-60, U.S.A. (969). Manuscript Received: Ma 6, 003 Revision Received: Jul. 5, 003 Accepted: Oct. 3, 003