A NOTE ON THE DQ ANALYSIS OF ANISOTROPIC PLATES

Transcription

1 A NOE ON HE DQ ANALYSIS OF ANISOROPIC PLAES WEN CHEN *, WEIXING HE **, AND INGXIU ZHONG * * Affiliation: Department of Mechanical Engineering, Shanghai Jiao ong Uniersit, Shanghai 00030, P. R. China ** Affiliation: Department of Electrical Engineering, Jiangsu Uniersit of Science & echnolog, Zhenjiang, Jiangsu 1013, P. R. China. Corresponding author: Wen Chen (PhD), Mail address: P. O. Bo 9601BB, Shanghai Jiao ong Uniersit, Shanghai 00030, P. R. China. Fa: , el: , ctwang@sjtu.edu.cn he total number of pages is 6. 1

2 1. INRODUCION Recentl, Bert, Wang and Striz [1, ] applied the differential quadrature (DQ) and harmonic differential quadrature (HDQ) methods to analze static and dnamic behaiors of anisotropic plates. heir studies showed that the methods were conceptuall simple and computationall efficient in comparison to other numerical techniques. Based on some recent work b the present author [3, ], the purpose of this note is to further simplif the formulation effort and improe computing efficienc in appling the DQ and HDQ methods for these cases.. APPROXIMAE FORMULAS IN MARIX FORM AND REDUCION COMPUAIONS he details about the DQ and HDQ methods see reference [1]. he onl difference between DQ and HDQ methods is to choose different basis functions, namel, the former is based on the polnomials and the latter based on harmonic functions..1 Approimate formulas in matri form Approimate formulas for partial deriaties of function w(,) in two-dimensional domain are gien in matri form b [3] $ w $ w $ w $ w = D $, $, $, $ w = C, 3 wa = B wb = AwC 3 $ w $ w $ w = $ wd, $, $ = B w = wb, (1) where the unknown $w is a n m rectangular matri rather than a ector as in references [1, ], n and m is the number of inner grid points along - and - directions, respectiel. A, B, C and D with subscripts and here stand for the DQ weighting coefficient matrices, modified b the respectie boundar conditions using Wang and Bert s new approach [5], for the 1st, nd, 3rd and th order partial deriaties, respectiel. he superscript means the transpose of the matrices. It is noted that present DQ approimate formulas in matri form can be easil etended to three-dimensional problems. he desired $w in rectangular matri form can be conerted into the conentional ector form b the following Lemma 1.

3 Lemma 1. If A C p m, B C n q and the unknown X C m n, then ec( AXB) = ( A B ) ec( X ) () where ec( ) is the ector-function of a rectangular matri formed b stacking the rows of matri into one long ector, denotes the Kronecker product of matrices. o simplif the presentation, we define ec( AXB) = AXB and ec( X ) = X. Corollar: 1. AX = A I X. XB = I B X 3. AX + XB = A I + I B X ( n ) ( m ) ( ) n m (3) where I n and I m are the unit matri.. Centrosmmetric matrices and computing reduction he weighting coefficient matrices in the DQ and HDQ methods were proen to be either centrosmmetric matri for the deriaties of een order or skew centrosmmetric matri for the deriaties of odd order if a grid spacing is smmetric [3, ]. his is often seen in man situations such as equall spaced grid points or zeros of the Chebshe and the Legendre polnomials. In the following we establish the notations on the centrosmmetric and skew centrosmmetric matrices first [3,, 6]. Definition 1. Let V N N and NV N N denote the set of N N real centrosmmetric and skew centrosmmetric matrices, respectiel, then X=[ ij ] V N N if and onl if N+1-i,N+1-j = ij ; and X=[ ij ] NV N N if and onl if N+1-i,N+1-j = - ij. Lemma. If Q 1, Q V N N ; Q 3, Q NV N N, then Q 1 Q V N N, Q 1 +Q V N N, Q 3 Q V N N 3

4 he following lemma 3 is presented on the Kronecker product of the centrosmmetric and skew centrosmmetric matrices. he proofs are straightforward and thus omitted for the sake of breit. Lemma 3. If A 1, A V N N and B 1, B NV N N, then A 1 A V N N, B 1 B V N N, A 1 B 1 NV N N. he centrosmmetric and skew centrosmmetric matrices can be factorized into two smaller sub matrices in the calculation of their determinant, inerse and eigenmodes. herefore, the respectie computing effort and storage requirements can be reduced b 75 percent and 50 percent, respectiel. he details on centrosmmetric and skew centrosmmetric matrices can be found in references [3,, 6]. 3. ON ANISOROPIC PLAES he equation goerning the behaiors of mid-plane smmetric laminated plates is gien b [1] ( ) D w + D w + D + D w + D w + D w 11, 16, 1 66, 6,, = q + ρhω w N w, N w, N W,, () where D are the plate stiffness, h is the total plate thickness, ρ is the densit, w is the model ij deflections, q is the pressure onl for deflection analsis, ω is the natural frequenc onl for free ibration analsis, N and N are uniform compression in-plane loads in the - and - directions for buckling analsis. In terms of the present DQ and HDQ approimate formulas (1) with N = N = N and N = 0, we hae ( ) 3 D D w$ + D βc wa $ + β D + D B wb $ + D β A wc $ ( ) + D β wd $ = qa + ϖ w$ Na B w$ + wb $, (5)

5 where β=a/b denotes the aspect ratio, ϖ = ρha ω. he relatie boundar conditions hae been taken into account in the formulation of weighting coefficient matrices, no additional equations are required. Appling Lemma 1 and relatie corollaries ields 3 [ D 11( D I) + D 16β( C A) + β ( D 1 + D 66 )( B B ) + D 6β ( A C ) + D β ( I D )] w = qa + ϖ w Na ( B I + I B ) w. (6) he aboe formulation equation is equialent to equation (13) in references [1, ]. he present procedures obiousl simplif formulation effort and are much easier for programming, and the resulting formulation has an eplicit matri form. It can be concluded that the approimate formulas in matri form in the DQ and HDQ methods are much simpler, more compact and conenient to formulate partial differential operators in two-dimensional domain than the conentional ones in polnomial form presented b Cian and Sliepceich [7]. For problems with smmetric boundar conditions such as the simpl supported or clamped anisotropic plates as discussed in references [1, ], it is straightforward that A, A, C and C are skew centrosmmetric matri and B, B, D and D are centrosmmetric matri when the uniform grid points or the zeros of the Chebshe polnomials are used. According to Lemmas and 3, the resulting coefficient matri in the formulation equation (6) is a centrosmmetric matri. herefore, the reduction algorithm based on the factorization properties of centrosmmetric matri [3,, 6] is applicable for the cases, namel, the computing effort and storage requirements are reduced to 75 percent and 50 percent as much as that in references [1, ]. Bert et al. [1, ] pointed out that the DQ and HDQ methods were er competitie technique to analze static and dnamic behaiors of the anisotropic plates. he present work makes the methods more computationall efficient and easier to be used for these problems. 5

6 REFERENCES 1) C. W. Bert, X. Wang and A. G. Striz 1993 International Journal of Solids and Structures 30, Differential quadrature for static and free ibration analsis of anisotropic plates. ) C. W. Bert, X. Wang and A. G. Striz 199 Journal of Sound and Vibration 170, Conergence of the DQ method in the analsis of anisotropic plates. 3) Wen Chen, Yongi Yu and Xinwei Wang 1996 Numerical Methods for Partial Differential Equations 1, Reducing the computational effort of the differential quadrature method. ) Wen Chen, Xinwei Wang and ingiu Zhong 1996 Communications in Numerical Methods in Engineering 1, he structure of weighting coefficient matrices of harmonic differential quadrature and its applications. 5) X. Wang and C. W. Bert 1993 Journal of Vibration and Sound 16, A new approach in appling differential quadrature to static and free ibrational analsis of beams and plates. 6) L. Datta and S. D. Morgera 1989 Circuits Sstems Signal Process 8, On the reducibilit of centrosmmetric matrices applications in engineering problems. 7) F. Cian and C. M. Sliepceich 198 Journal of Mathematical Analsis and Applications 101, 3-3. Differential quadrature for multi-dimensional problems. 6