IMPLEMENTING MULTIPLE BOUNDARY CONDITIONS IN THE DQ SOLUTION OF HIGHER-ORDER PDEs: APPLICATION TO...

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, VOL. 39, 1237-1258 (1996)

IMPLEMENTING MULTIPLE BOUNDARY CONDITIONS

APPLICATION TO FREE VIBRATION OF PLATES IN THE DQ SOLUTION OF HIGHER-ORDER PDE’s:

M. MALIK AND C . W. BERT

School of Aerospace and Mechanical Engineering, The University of Oklahoma, 865 Asp Avenue, Room 212, Norman, OK, 73019-0601, U.S.A.

SUMMARY In the context of the free vibration problem of rectangular plates, this paper develops a detailed methodology for implementing multiple boundary conditions in the differential quadrature solutions of higher-order differential equations. It is explained that a certain type of boundary conditions can be built into the differential quadrature weighting coefficients themselves. This simplifies the programming of the differential quadrature solution algorithms. More importantly, however, as shown by the results of fundamental frequencies of a wide spectrum of rectangular plate configurations, the methodology results in strikingly accurate differential quadrature solutions.

KEY WORDS: differential quadrature method; numerical solution methods; higher-order partial differential equations; multiple boundary conditions; free vibration; rectangular plates; thin plate theory

1. INTRODUCTION

The present paper focusses on the differential quadrature method (DQM). The method was proposed by Bellman et aL1s2 in the early seventies and was claimed as a technique capable of providing rapid and accurate solution of initial-value non-linear partial differential equations. This claim has since been corroborated by various studies of the application of the method to general initial and boundary value problems. Consequently, with a development history of only the past two decades, the DQM is now considered as an alternative to other conventional numerical techniques, such as the finite difference and finite element methods.

After the original work of Bellman et a1.,’’2 the DQM developed further through its applica- tions to physical problems. Thus, one finds the work of Mingle3 on non-linear diffusion and Civan and Sliepcevich4- on transport processes, the Thomas-Fermi equation, and integro- differential equations. The DQM was introduced into the general area of structural mechanics by Bert and his associate^.'^^ The very basis of the DQM, as originally conceived in the work of Bellman et al.’” and used in the later works, is in expressing the calculus operator values of the function (i.e. the differential and integral of a variable and the combinations thereof) with respect to a coordinate direction at any discrete point of the solution domain as the weighted sum of the values of the function at all the sampling points chosen in that direction. In this context, a notable contribution is the work of Quan and Changg who showed the DQM to be essentially equivalent to the general collocation method and provided explicit formulae for the polynomial test function

CCC 0029-598 1/96/07 1237-22 0 1996 by John Wiley & Sons Ltd.

Received 17 June 1994 Revised 14 March 1995

1238 M. MALIK AND C. W. BERT

weighting coefficients for first- and second-order derivatives. In a recent work, presumably unaware of the 1989 work of Quan and Changg, Shu and Richards" have presented similar ideas. However, Shu and Richards" provided a recurrence relationship for the polynomial test function weighting coefficients of the general nth order derivatives and demonstrated the application of the DQM to the solution of Navier-Stokes equations for the problem of flow past a circular cylinder.

As with other numerical methods, such as the finite difference and finite element methods, the DQM also transforms the given differential equation(s) into a set of analoguous algebraic equations in terms of the unknown function values at the preselected sampling points in the field domain. Before solving these equations, one invokes the boundary conditions replacing the boundary point equations by the DQ analogue equations of the boundary conditions. This happens to be a rather simple matter with first- or second-order differential equations irre- spective of domain dimensions and whether the boundary conditions are the Dirichlet and/or Neumann type or possibly of the mixed type. In fact, in many of the papers on the DQM, the application problems have been of this t ~ p e . ~ - ~ " - H owever, with the higher-order differential equations, one confronts multiple boundary conditions. In the quadrature solutions, the implementation of such boundary conditions is not a straightforward matter and needs careful consideration.

The quadrature solutions of higher-order differential equations with multiple boundary conditions were introduced through the application of the DQM to structural mechanics p rob le rn~ .~ .~ The governing equation of thin beam or plate flexure is a fourth-order differential equation and has two conditions at each boundary. In their work on the application of DQM to the static analysis of beams and plates, Jang et aL8 proposed the so-called b-technique wherein adjacent to the boundary points of the differential quadrature grid, points are chosen at a small distance 6 z lod5. Then the DQ analogue of the two conditions at a boundary are written for the boundary points and their adjacent b-points.

The b-technique offers an adequate way of applying the double boundary conditions of beam and plate problems and was applied quite successfully to, for example, SS-SS and C-C beams or SS-SS-SS-SS and C-C-C-C In fact, by choosing successive b-points, the technique may be applied to the solution of the governing field equations of order higher than the plate problems. However, an arbitrariness in the choice of the b-value becomes apparent in the actual implementation of the boundary conditions. Thus, while the &value cannot be large (possibly not greater than 0.001) for acceptable solution accuracy, with small b-value, the solution begins to oscillate. The general loss of accuracy with this type of technique for implementing the boundary conditions is actually always shown by some loss of symmetry in the solution (for example in deflection and/or moment values at the grid points) if symmetry conditions are present by virtue of the boundary and loading conditions.

In order to overcome the aforementioned difficulties in the implementation of the boundary conditions, an essentially new concept was suggested by Wang and Bert'* wherein it was proposed that the boundary conditions may be implemented in the weighting coefficient matrices themselves. This was illustrated for the case of a beam simply supported at both ends. Wang and Bert'' pointed out that this technique could be successfully applied only for cases of SS-SS and C-SS (or S S C ) beams and the combinations of these boundary conditions for plates. In that work, the proposed technique of boundary condition implementation was applied to beams and thin isotropic rectangular plates. Later, Bert et al.I3 applied the same technique to the static and free vibration analysis of anisotropic rectangular plates. Extensions in the technique to include free edges in one-dimensional problems of beams and axisymmetric circular plates were further reported by the same group of in~estigat0rs.l~- l 6

IMPLEMENTING MULTIPLE BOUNDARY CONDITIONS 1239

The present work further develops the idea of incorporating the boundary conditions into the weighting coefficient matrices for the differential quadrature solutions of higher-order differential equations. This is done in context of the free vibration problem of thin rectangular plates with various combinations of the boundary conditions. The limitations of the method, as identified in the present paper, are actually different from those which were pointed out by Wang and Bert12. In fact, using the same method, highly accurate solutions can be obtained for plates with two opposite edges clamped. Another type of boundary support of limited practical interest, which has not been considered so far in the DQM literature, is the guided edge. The zero-rotation boundary condition of a guided edge can be incorporated into the weighting coefficient matrices and accurate solutions of the plates with one or more guided edges can be obtained. The boundary conditions (for example, those of a free edge) which cannot be implemented in the weighting coefficient matrices, can of course be handled, if necessary, by the &technique.

In the following, first the governing equations of the reference problem are described. Next some details of basics of the DQM are given. This is followed by a detailed exposition of the technique of implementing the boundary conditions into the weighting coefficient matrices through some examples of combinations of simply supported, clamped, and guide edge boundary conditions. Lastly, the DQM solution results for the fundamental frequencies of thin isotropic rectangular plates are given for a variety of boundary conditions applied along the four edges and compared with the available analytical and approximate solutions.

2. THE REFERENCE PROBLEM

Before going into the details of the differential quadrature solution and implementation of the boundary conditions, the governing equations of the reference problem are recapitulated. On the assumption of a harmonically periodic time response, the analysis of freely vibrating thin rectangular plates of isotropic materials involves essentially the solution of the following eigenvalue differential equation:

w,x,,, + 2PW,xxyy + P W , y y y y - R2W = 0 (1)

where W = W(X, Y ) is the dimensionless mode function corresponding to the dimensionless frequency R; W,xxxx = a4W/aX4, etc.; X = x/a, Y = y /b are dimensionless coor- dinates; a and b are the lengths of the plate edges parallel to the x and y axes, respectively, and L = u/b is the aspect ratio. Further, R = ou2 m, where o is the dimensional circular frequency; D = Eh3/[12(l - v2)] is the flexural rigidity; E, v, and p are respectively, Young’s modulus, Poisson’s ratio, and the density of the plate material; and h is the plate thickness.

From a mathematical standpoint,” the application of the calculus of variations to the energy functional of the classical thin plate theory leads to four possible combinations of essential and natural conditions at an edge of a thin plate. Of these, three combinations correspond to the simply supported (SS), clamped (C), and free (F) boundary conditions. These boundary conditions are commonly realized in engineering practice and are, therefore, conventionally considered in plate analyses. The fourth boundary condition is given by zero rotation (essential condition) and zero effective shear force (natural condition) and has been referred to in the literature as a guided (G) edge condition. In the present work, all four types of boundary conditions are considered. These conditions apply at X = 0 and/or 1, referred to as the X-type edges, and at Y = 0 and/or 1 which are referred to as the Y-type edges.


At a simply supported or a clamped boundary, the transverse deflection of the plate is zero:

w = o (2)

At a simply supported boundary, the condition of zero normal moment can be reduced to

w,xx = 0

at an X-type edge, and

W,YY = 0

(3)

(4)

at a Y-type edge.

given by At a clamped or guided boundary, the condition of zero rotation (normal to the boundary) is

w,x = 0 ( 5 )


w*y = 0 (6)

at a Y-type edge. The condition of zero normal moment at a free boundary is given by

w,,, + vA2 w , y y = 0

A2 w , y y + v w,xx = 0

(7)


(8)

at a Y-type edge. The condition of zero effective shear force at a free or a guided boundary is given by

w,xxx + (2 - V)AZ W , X Y Y = 0 (9)


A2W,YYY + (2 - ~)W.XXY = 0

at a Y-type edge. An additional condition that applies at the corner of two adjacent free edges is given by

3. THE DIFFERENTIAL QUADRATURE ANALOGUES OF THE FUNCTION DERIVATIVES

Consider the unit square domain 0 < X < 1, 0 < Y < 1 of the rectangular plate is Figure 1 in which the number of sampling or grid points shown without any particular spacing are N, and N , in the X- and Y-directions, respectively. In accordance with the differential quadrature method, a given differential equation may be transformed into a set of algebraic equations in terms of the function values Wij at each of the grid points Xi, Yj; i = 1,2,. . . , N,; j = 1,2,. . , , N, . This is done by expressing each of the function derivatives at a grid point as


a linear weighted sum of the values of the function at all the grid points on the particular grid line which passes through that point and is parallel to the co-ordinate direction of the derivative. Thus, an rth order X-partial derivative of W(X, Y) at a sampling point X = Xi on any line parallel to the X-axis may be written as

and, an sth order Y-partial derivative at a sampling point Y = Yj along any line parallel to the Y-axis may be written as

where A!; and B;' are the weighting coefficients. These coefficients are associated with a co- ordinate direction and may be determined by some suitable approximation of the function W(X, Y ) in that direction. The primary requirement for the approximating functions, also referred to as the test functions, is completeness in the sense of the analogous requirement for the interpolation functions in finite element analysis. l8 Of the many possible choices of test function, the most convenient and commonly used choice is the ordered set of polynomials. Thus, one may express

W(X, Y) = @(X)Y(Y) (14)

where @(X) and Y(Y) are the test functions in X- and Y-directions, respectively, such that,

@(X) = x k - l ; k = 1,2,. . . , N , (15)

and

Y(Y) = yl-1; 1 = 1,2,. . . , N , (16)

Substituting equation (14) with equations (15) and (16) into equations (12) and (13), one obtains the following two systems of linear equations:

and

for the coefficients AIL' and B:', respectively. The coefficient matrices in equations (17) and (18) are Vandermonde matrices.2*19 If the

sampling points are unique, a Vandermonde matrix cannot be ~ingular '~ and thus the system of equations (17) and (18) must have unique solutions. These equations may be solved using either linear solvers or Hamming'~'~ analytical method to obtain the weighting coefficients. However, the Vandermonde matrices are known to be inherently ill-conditioned;20 this situation aggra- vates with increasing number of sampling points. Consequently, such solution methods cannot be relied upon for the accuracy of the weighting coefficients which is indeed paramount to the

1242

Y N

3

2

I = 1

M. MALIK AND C. W. BERT

Y

N X 1.1 2 3...

Figure 1. Grid for differential quadrature solution

accuracy of quadrature solutions. The special algorithm of Bjork and Pereyra" is found to yield very accurate weighting coefficients. However, independent of the order of the derivatives and the number and distribution of the grid points, the weighting coefficients are produced most accurately by the explicit formulae of References 9 and 10.

Having the first-order derivative weighting coefficients, all higher-order derivatives may also be obtained using the following recurrence relations:

where [A")] and [B'")] are, respectively, the rth order X-partial derivative and sth order Y-partial derivative weighting coefficient matrices.

The differential quadrature analogue of a mixed derivative, say 8'") W/aX' dY ", follows by the definition of differential operators:

where wk1 = w(xk, Yl). Following the above definitions, the DQ analogue of equation (1) may be obtained as

Note that range of the indices i and k are from 1 to N , and those of the indices j and 1 are from 1 to N,. However, the upper and lower limits of these indices depend on how the boundary conditions are incorporated and are, therefore, not indicated in equation (21).


4. INCORPORATING THE BOUNDARY CONDITIONS IN THE WEIGHTING COEFFICIENT MATRICES

The simplest of the boundary conditions to invoke in equation (21) is the condition of zero displacement ( W ) at a simply supported or clamped edge. This is done by simply ignoring the corresponding grid points in equation (21).

In order to understand how the derivative-type boundary conditions could be incorporated in the weighting coefficient matrices, consider the DQ analogue of the derivative a W / a X at the grid points on a line Y = Y j in 0 < X < 1:

or

{W,,>j = CA"'I { W } j

NOW, let a boundary condition on an X-type edge be specified as

W , x = a at X = O (23)

where a is some prescribed constant. The quadrature analogue of the above condition, equation (23), may be written as

N .

(W,x) i j = 1 ak w k j = a, i = I; j = 1,2, . . . , N , k = l

where a k is an array of N, weighting coefficients. These coefficients may be determined simply by the solution of the Vandermonde equations

Now, let the first row of the [A")] matrix in equation (22) be replaced by the ak coefficients, that is,

. . .


Then, the first row of the above matrix equation is actually the quadrature analogue (24) of the boundary condition (23). Thus, the modified weighting coefficient matrix in equation (26) has the boundary condition (23) built in itself. Quite obviously, the [A“)] matrix may also be modified in a similar manner to accommodate a boundary condition of specified gradient W,, at the edge X = 1. Furthermore, if a = 0, equation (23) simply becomes the zero-rotation condition of an X-type clamped or guided edge. In that case, the unique solution of equation (25) will be ak = 0; k = 1,2,. . . , N,. Thus, the zero-rotation condition at X = 0 and/or X = 1 edges are simply incorporated by setting the first and/or last rows of the [A“)] matrix equal to zero.

Next, consider the quadrature rule of the second derivative a2W/aX2 at the grid points on a line Y = Y j written in matrix form as

Now, as another example of the type of boundary condition that can be accommodated in the weighting coefficient matrices, consider

WsXx = fl WVx at X = 0 (28)

which in the quadrature analog form may be written as

( 1 ) N , N .

k = l k = l (W,xx)ij = Aji’wkj = fl(w,x)ij = ( f l & ) w k j , i = 1; J = - . , Ny (29)

Considering the form of equations (27) and (29), it is apparent that if in the [ A ( 2 ) ] matrix, the first row coefficients A:) are replaced by PA:;), that is,

then, the first row of the above matrix equation is actually the quadrature analogue (29) of the boundary condition (28). Thus, the modified weighting coefficient matrix in equation (30) has the boundary condition, equation (28), built in itself.


From the above discussion, it may be seen that, in general, the boundary conditions of the types

arw aqw ar\y

ax. ax. -%F - a and -- --

at an x-edge can be accommodated by simply modifying the appropriate weighting coefficient matrices.

Having modified a weighting coefficient matrix, one can proceed to obtain the higher-order derivatives using the original weighting coefficient matrices as the operator matrices. Thus, for example,

where a matrix with a tilde denotes a weighting coefficient matrix which has the boundary condition(s) built in itself and is to be used later in the quadrature formulation. A matrix denoted with a bar is a derived one; if no conditions are to be imposed on the (r + 1)th derivative, then [A(.+ 1'1 = [A('+') 1. A matrix without any overscript is an original 'as is' weighting coefficient matrix. It is important to realize here that, in general, the [A'"')] matrix is not the same as the [A" + ''1 matrix.

In the present development, the boundary conditions were considered on the X-type edges. The boundary conditions on the Y-type edges can be considered in a similar manner by simply replacing the A-type matrices by the B-type matrices.

A limitation to the above method, that is essentially due to the very basis of the appoximation of derivatives by the differential quadrature method, may be recognized here. Consider, for example, the condition of an X-derivative 8. W/aXr = 0 on Y = 0 boundary. This condition is satisfied only if W(X, 0) = 0, otherwise, one arrives vainly at [A")] = 0.

The quadrature rule for a derivative at a given point involves the function values (as a weighted sum) at the grid points on a line passing through that point and parallel to the co-ordinate direction of the derivative. Accordingly, boundary conditions of the types

cannot be incorporated in the weighting coefficient matrices. In conclusion, the weighting coefficient matrices can be modified to incorporate only

the type of boundary conditions having the derivatives with respect to one spatial co-ordinate only where the spatial co-ordinate direction is along the normal to the respective boundary. It may be added that the coefficient terms in the boundary conditions can be functions of the other co-ordinate direction; for example, one may have a = a(Y) , /3 = jl(Y). However, in that case, the modified weighting coefficients cannot be global, rather, one will have to obtain a set of modified matrices for each grid line. Boundary conditions involving the derivatives with respect to two or more spatial co-ordinates cannot be incorporated into the weighting coefficient matrices. However, such boundary conditions can be invoked by the &technique.

In order to develop a deeper understanding of the methodology of incorporating the boundary conditions in weighting coefficient matrices, consider the following three cases of supports at the two opposite X-type edges of a rectangular plate.


Rectangular plate with both edges X = 0 and X = 1 simply supported

the X-axis. Consider the DQ analogue of the derivative a2 W / a X 2 at the grid points on a line parallel to

Let equation (32) now be written as

0 0 . . . 0 2

A2N, . . .

- . . . - . . .

2 2 2 A N , - l , l A N . - 1,2 ' ' . A N . - 1,N.

0 0 . . . 0 - -

. . .

0

. . .

. . . 2 2

A N , - l , l A N . - 1,2 ' ' . 0 . . . 0

(32)

(33)

then clearly, the boundary condition (3) is satisfied at the two X-type edges X = 0 and X = 1. Let the modified weighting coefficient matrix in equation (33) be

0 . . .

and let the weighting coefficient matrix of the fourth-order derivatives be modified as

Let equation (21) be rewritten replacing the original X-derivative weighting coefficients by the modified coefficients


In equation (36) the zero-moment boundary conditions of the simply supported edge at the two opposite boundaries X = 0 and X = 1 are built in. In addition, if in equation (36) the grid points of the boundaries X = 0 and X = 1 are ignored, then all the requisite boundary conditions of these two opposite edges, W = 0 and W,,, = 0, are automatically satisfied. It may be noted that the Y-derivative weighting coefficients are written in the original form; these may be modified depending on the boundary conditions of the Y-type edges.

Rectangular plate clamped at X = 0 and simply supported at X = 1

be defined to include the boundary condition of the clamped edge, equation (9, at X = 0, Let the DQ analogue of the derivative 8 W /ax at the grid points on a line parallel to the X-axis

where in the right-hand side, the coefficient matrix, [ A ( ’ ) ] , is the modified first-order derivatives weighting coefficient matrix, and is obtained by zeroing the first row of the original matrix [A“)]. Let the second-order weighting coefficient matrix be derived as

and, in order to include the boundary condition (3) at X = 1, a modified second-order weighting coefficient matrix be defined by zeroing the last row of [A(2’ ] , that is,

Once again the modified matrix [ A 4 ] may be obtained using equation (35). Further, in equation (36), if the grid points of the edges X = 0 and X = 1 are ignored, the clamped boundary conditions W = W,x = 0 at X = 0 and the simply supported boundary conditions W = WeXx = 0 at X = 1 are both implemented automatically.


Rectangular plate simply supported at X = 0 and guided at X = 1

Let the DQ analogue of the derivative 8 W / 8 X at the grid points on a line parallel to the X-axis be defined to include the zero-rotation boundary conditions of the guided edge, equation (5), at X=l

. . .

. . .

. . . A'" A(l) * ( I )

N , - l . l N , - 1 . 2 * . ' N , - 1,N

0 0 . . . 0

where on the right-hand side, the coefficient matrix, [x")], is the modified first-order derivative weighting coefficient matrix, and is obtained by zeroing the last row of the original matrix [ A " ) ] .

Let the second-order weighting coefficient matrix be derived as

[A'2'] = [A'"] [ p ] (41) and, in order to include the boundary condition (3) at X = 0, a modified second-order weighting coefficient matrix is defined by zeroing the first row of the [A")] matrix, that is,

0 0 . . . 0

. . .

. . .

The modified matrix [A"'4)] may again be obtained using equation (35). Further, in equation (36), if the grid points of the edge X = 0 are ignored, both of the simply supported boundary conditions W = W , x x = 0 at X = 0 invoked. Also, the guided edge condition Wsx = 0 at X = 1 is built into the weighting coefficients. However, the zero-effective-shear-force condition at X = 1 has to be invoked by writing the DQ analogue of this condition:

c 2:) w k j + (2 - v)12 c $' w k , = 0; i = N, (43) k k I

where the Y-derivative weighting coefficients, written in the original form, may be modified depending on the boundary conditions of the Y-type edges.

5. COMPUTATIONAL ASPECTS

A factor decisive to the accuracy of the differential quadrature solutions is the choice of the sampling or grid points. A natural choice of the grid points is of the equally spaced points in each


co-ordinate direction. However, with the exception of a few cases of two opposite edges simply supported, equally spaced grid points do not generally present a good choice with respect to solution accuracy vis-a-vis the number of sampling points. A better choice for the positions of the grid points between the first and the last points at two opposite edges is that corresponding to the zeros of orthogonal polynomials. For example, the zeros of Chebyshev polynomials have been used in lubrication" and anisotropic plate13 problems, and zeros of Legendre polynomials have been used the in plate/elastic-half space interaction problem.22 Another choice which is found to be even better than the Chebyshev and Legendre points is the set of points used by Shu and Richards." These are given by the equations

( i - 1)n . , i = l , 2 ,..., N, xi=- l-cos- 1 2 '[ N,-1

and

i = 1,2¶, . . , N y (45)

in the X- and Y-directions, respectively. The grid point co-ordinate equations, equations (44) and (49, do not include the &points.

These points may be included wherever needed; the co-ordinates of these points adjacent to the opposite edges would be simply 6 and 1 - 6. Thus, in the X-direction

- i = 3,4,. . . , (N, - 2) (i - l)n xi=- 1 -cos- 2 " N , - 3 1 '

and similarly in the Y-direction

(i - 1)n . j = 3,4,. . . , ( N , - 2) 1 Yi = - 1 - cos- 2 '[ N , - 3

(47)

From the discussion in the previous section, it may be concluded that for the rectangular plate configurations formed from the combination of only two types of support, namely simple support and clamped support, at their four edges, the DQ analogue equation (21) may be written in terms of modified weighting coefficients of both X- and Y-derivatives. Thus,

with i = 2,3, . . . , ( N , - 1) and j = 2,3,. . . , (Ny - 1). It may be noted that in equation (48), the grid points of the four edges, X = 0, X = 1, Y = 0 and Y = 1 are ignored and all the boundary conditions of the four edges are actually built into the equation.

The assembly of equation (48) for all values of the indices i and j results in the following eigenvalue equation of size (N, - 2) x ( N y - 2)

(49) [S] {W} - Q2[I] { W } = 0

where the eigenvalue matrix [S] is comprised of the modified weighting coefficients.


As was pointed out earlier, the boundary conditions of a free edge cannot be incorporated into the weighting coefficient matrices. In this case the DQ analogue equation (21) may be written by ignoring i and/or j indices at the grid points of the free edge line and of the corresponding points of an inner grid line parallel and next to the free edge. The inner grid lines correspond to i = 2 and i = N, - 1 for the boundary conditions on the X-type edges, and to j = 2 and j = N, - 1 for the boundary conditions on the Y-type edges. As will be seen later, these inner grid lines may be the one from the usual grid or from the grid with adjacent &lines. It may be recalled that the sampling point co-ordinates are obtained from equations (44) and (45) for a usual grid and from equations (46) and (47) for the grid with Mines.

Considering an X-type edge, the DQ analogue of the free edge boundary conditions are

and

where in one of these two equations i = 1 and in the other i = 2 at an X = 0 edge or i = N , and i = N, - 1 at an X = 1 edge. Note, however, that in both equations (50) and (51), the X- derivative weighting coefficients need to be of the boundary point itself. This is taken care of by the index m which will have the values m = 1 and m = N , at the X = 0 and X = 1 edges, respectively.

The DQ analogue of the boundary conditions of a Y-type free edge are

and

wherein one of these two equations j = 1 and in the other j = 2 at a Y = 0 edge or j = N , and j = N, - 1 at a Y = 1 edge. In both, equations (52) and (53), n = 1 and n = N, at the Y = 0 and Y = 1 edges, respectively.

In addition to the above conditions, at the corner of two perpendicular free edges

with appropriate values of the indices i and j at the given corner. It may be noted that equations (50) through (54) are written in terms of the original weighting coefficients; of course these coefficients would be of the modified form if opposite to the free edge, there is an SS-, C- or G-edge.

It may be recalled that in the case of a guided edge, it becomes possible to incorporate one of the boundary conditions (viz. zero rotation) in the weighting coefficient matrices and one needs to have the DQ analogue of the other boundary condition (i.e. zero effective shear force) corresponding to the grid points of the guided edge. Thus, it may seen that in either case, if the plate has one or more free and/or guided edges, the assembly of the DQ analogue of the differential equation


and the requisite DQ analogues of the boundary conditions at the respective grid points leads to algebraic equations of the following form:

Here, in equation (55), the subscripts b and d are used to indicate the grid points used for writing the DQ analogue for the boundary conditions and the governing differential equation, respectively. Obviously, in order to transform equation (55) into the eigenvalue equation of the form of equation (49), the deflection components of the grid points corresponding to the former grid points, i.e. the column { wb}, need to be eliminated. This situation is identical to that in finite element analysis involving elements with internal nodes (not connected to the boundary nodes) wherein the corresponding degrees of freedom are eliminated by the static condensation tech- n i q ~ e . " ~ ~ ~ The same technique is used in the present work to eliminate the { w b ) vector in equation (55) and reduce it to the same form as equation (49).

The eigenvalues of the [S] matrix are obtained by the technique of inverse iteration with shifting.23 The mode shapes given by the vector { w d } , are obtained simultaneously. Also, with the shifting technique the lowest and higher order eigenvalues may be obtained with equal ease.

6. RESULTS AND DISCUSSION

The numerical results being presented in this section serve to demonstrate simultaneously the efficacy of the technique of implementing the boundary conditions and further the versatility of the differential quadrature analysis for a wide spectrum of rectangular plate configurations. Conse- quently, the results contained herein are for the fundamental mode frequencies only but with a large number of boundary condition combinations at the four edges of the rectangular plates.

As mentioned earlier, the boundary conditions being considered are of simply supported (SS), clamped (C), free (F), and guided (G) edges. The number of possible plate configurations with SS, C, and F edges are twenty one. The published work on the free vibration frequencies of rectangular plates is enormous; see for example, References 24-27. Of these, the work of LeissaZ6 is the most complete in that it presents the frequency data of all twenty-one plate configurations for the first nine modes and for a wide range of aspect ratios. In this work, the frequencies of the six plate configurations with a pair of opposite edges simply supported are exact since in these cases, equation (1) may be solved analytically. For the remaining fifteen cases, frequencies were obtained by the Rayleigh-Ritz method with thirty-six terms (six each in the x- and y-directions) of beam characteristic functions. Later works, such as those using orthogonal polynomials in the Rayleigh-Ritz method2'# and others based on some variations of Rayleigh-Ritz and Kantorovich methods29. 30 have claimed more accurate results. However, none of these works are as complete as that of Leissa26 and with the exception of plates with two or more adjacent edges free, the differences in their data from those of Leissa26 are no more than marginal from the engineering standpoint. For these reasons, the data of Leissa26 are employed for evaluating the accuracy of the results of the quadrature solutions.

The numerical results are presented in Tables I-IV. The rectangular plate configurations in Tables I, II(a) and II(b) are those that are formed with possible combinations of simply supported, clamped, and free edge conditions. Table 111 addresses some observations on the convergence and accuracy of the differential quadrature solutions uis-a-ois the number of grid points. The results in Table IV are for plates with one or more guided edges. In Tables I, II(a), II(b), and IV, the differential quadrature results are produced with N, = N, = 15 using either equations (44) and


Table 1. Dimensionless fundamental frequencies of rectangular plates having one pair of opposite edges simply supported

1 = afb

Plate type 215 213 1 312 512 ~~~ ~~

1. SS-SS-SS-SS* 114874 14.25610 19-73921 32’07621 71.55463 2. ss-c-ss-ss 11.75022 15.57828 23.64632 42.52780 103.92265

(1 1.7502) (15.5783) (23.6463) (42.5278) (103.9227) 3. SS-SS-SS-F 10.12607 10-67117 11.68454 13-71109 18-80092

(10-1259) (10.6712) (11.6845) (137111) (18.8009) 4. SSC-ss-c 12.13469 17-37300 28.95085 5634807 14548394

(12.1347) (17.3730) (28.9509) (56.3481) (145.4839) 5. SS-C-SS-F 10.18892 10.97524 12.68736 1682247 3062766

(10.1888) (109752) (12.6874) (168225) (30.6277) 6. SS-F-SS-F 9.76013 9.69832 9.63138 9.55818 9-48414

(9.7600) (9-6983) (9.6314) (95582) (9.4841)

* DQM values of the SS-SS-SS-SS plate match exactly with the values from the theoretical formula: Oil = xz(l + AZ). Frequencies of other cases compared with the data of LeissaZ6 (the values in parentheses). In all cases in this table, N, = N, = 15 sampling points are used in equations (44) and (45)

Table II(a). Dimensionless fundamental frequencies of rectangular plates with various simply supported, clamped, and free edge combinations

~~~ ~

1 = a/b

Plate type 215 213 1 312 512

7. c-c-c-c 2364164

0.027% 8. C4-C-SS 23.44302

- 0.013%

(23.648)

(23.440)

9. C-C-C-F 22.55146

0.113% 10. C-SS-C-F 22.52314

0.093% 11. C-F-C-F 22.32661

0.087% 12. C-F-SS-F 15.32956

0.341%

(22.577)

(22.544)

(22.346)

(15.382)

27.00800 (27.010)

0.007% 25.86438

(25.861)

22,93102 (23.015)

0.365% 22.78101 (22.855)

0.324% 22.23380

(22.314) 0.359%

15.24598 (1 5.340)

0.513%

- 0.013%

35.98926 (35992)

0.008 % 31.82119

(3 1.829) 0.025%

23.99523 (24.020)

0.103% 23.43890 (23460)

0.090% 2224306

(22.272) 0.130%

15.171 89 (1 5.285)

0.740%

60.76799 (60.772)

0.007 yo 48.16001

(48.167) 0.0 1 5 yo

26.56965 (26-73 1)

0.604 % 24.62921

(24.775) 0.588%

22.05834 (22.215)

0705% 15,09643

0.792% (15.217)

147-76023

0 Yo

(107.07)

(14740)

107.04246

0.026% 37.54703

(37.656) 0.289%

28.43849 (28.564)

0.439% 21.96590

(22.1 30) 0.742%

15.00560 (1 5.128)

0.809% ~~~~~

The values in parentheses are the frequency data of Leissa.z6 Percentage values indicate the relative difference with respect to Leissa’s data. In all cases in this table, N, = Ny = 15 sampling points are used. Grid point co-ordinates for Case 7-9 are obtained from equations (44) and (45). For the remaining cases, equations (46) and (47) are used 6 = lo-’


Table II(b). Dimensionless fundamental frequencies of rectangular plates with various simply supported, clamped, and free edge combinations

I = a/b

Plate type 215 213 1 312 512

13. C-C-SS-SS 16.84752

0.009 Yo 14. C-C-SS-F 15.71685

(16.849)

(1 5.696) - 0.133%

15. C-SS-SS-F 15.67146 (1 5.649) - 0.144%

16. SS-SS-F-F 1.32062 (1.3201)

17. SS-F-F-F 2.73216 (2.6922)

18. F-F-F-F 3.43389 (3.4629) 04338%

- 0.039%

- 1.484%

19.95123

0.004% 16-25229

(16287) 0.213%

1603425 (16067)

0.204% 2-23054

(2-2339) 0.150% 4.59624 (4.481) 2.572% 9-34307 (8.9459)

- 4.440%

(19.952) 27.05413

(27.056) Om7 Yo

17.77422 (1 7-61 5) - 0.904%

16.99260 (16865) - 0.761 %

335619 (3-3687) 0.371 % 691513

(6.6480) - 4.018%

1391187 (13.489) - 3.135%

44.89027 (44.893)

0.006 % 20-92208 (21.035)

0537% 18.43325

0.576% 501860

(5.0263) 0.153%

10.45755 (9.849 8)

21-02150 (20.128) - 4.439%

(18.540)

- 6.170%

105.29701 (105.31)

0.0 12 Yo 33.53876

(33578) 0.1 17%

23.01571 (23.067)

0.222% 8-25438

(8.2506) 0.046%

1481530 (14.939)

21.46173 (21.643)

0.838%

0.828%

The values in parentheses are the frequency data of Leissa.26 Percentage values indicate the relative difference with respect to Leissa’s data. In all cases in this table, N , = N , = 15 sampling points are used. Grid point co-ordinates for Case 13 are obtained from equations (44) and (45). For the remaining cases, equations (46) and (47) are used 6 =

(45) or equations (46) and (47). This is done more for uniformity rather than mere accuracy, since in many cases, good accuracy could be obtained with fewer grid points, and in a few cases, one might consider more grid points for improved accuracy. In all the tables, the frequency data are obtained for the aspect ratios I = a/b = 2/5,2/3, 1,312, and 512; this is the same range as used in the work of Leissa.26 Furthermore, in all cases involving free and/or guided edges, the Poisson’s ratio was taken as 0-3.

First, consider Table I in which each of the six rectangular plates has one pair of the opposite edges simply supported. As mentioned earlier, the data for these configurations given by LeissaZ6 are exact frequency values. It may seen from Table I that the DQ solution values are almost exact. Also, it may be remarked that although the given DQ results in Table I are obtained with fifteen grid points in each co-ordinate direction; results with an absolute maximum error less than 1 per cent may be obtained with N, = N y = 7 for plates having more than two edges simply supported and with N, = N , = 9 for plates having just two (opposite) edges simply supported. It should also be noted that in the three plate configurations with free edges (Cases 3,5, and 6), no 6-type grid lines were needed and the grid point co-ordinates were obtained from equations (44) and (45).

Next, in Tables II(a) and II(b), twelve plate configurations with various other combinations of simply supported, clamped, and free edge conditions are considered. These tables provide a comparison with Leissa’s data in terms of relative percentage difference:

Leissa’s value - DQM solution value Leissa’s value

relative percentage difference = 100 x

N

ul P

Tab

le 11

1. C

onve

rgen

ce o

f diff

eren

tial q

udra

ture

sol

utio

ns

1 =

afb

Leiss

a*

7 9 11

13

15

Leiss

a*

7 9 11

13

15

22.5

44

24.1

43

- 7.

09%

23

.41 1

22.6

25

22.58

3

22.5

23

- 3.

85%

- 0

.36%

- 0.

17%

0.09

%

2.69

22

2.81

42

2.74

83

2.737

8

2.73

40

2.73

22

- 4.5

3%

- 2.

08%

- 1.

70%

- 1.

55%

- 1.

48%

C-S

S-C

-F

Plat

es

22.8

55

2346

0 24

.775

28

.616

24

.798

25

.352

2483

7 16

.668

23

.718

22.8

60

24.3

87

24.4

95

22-8

14

23-6

33

24.5

76

22.7

81

23.4

39

24.6

29

- 2

52%

- 5

.71%

- 2.

33%

- 8.

67%

28

.9%

4.

27%

- 0.

02%

- 3.

95%

1.

13%

018%

-0

.74%

0.

80%

032%

0.

09%

0.

59%

SS-F

-F-F

Pl

ates

4.

481

6.64

80

9.84

98

4.63

61

6.89

72

10.5

35

4.58

19

6853

4 10

422

4588

1 6.

8840

10

438

4.59

36

6.90

36

10.4

50

- 3

.46%

- 3.

75%

- 6.

96%

-225

%

-3.0

9%

-5.8

1%

- 2

.39%

- 3.

55%

- 5.

97%

-2.5

1%

- 3

84%

- 6.

09%

4-

5962

6-

9151

10

.458

- 2

.57%

- 4.

02%

- 6.

17%

28-5

64

29-1

17

28.0

88

28.3

63

28.4

13

28.4

38

- 1.

94%

1.67

%

0.70

%

0.53

%

044%

14.9

39

14.5

73

2.45

%

15.0

22

- 0.

55%

14

-816

0.

82%

14

.817

O

-82%

14

-815

0.

84%

1.32

01

1.33

56

1.17

%

1.32

31

1.32

19

1.32

13

- 0.0

9%

1.32

06

- 0-

04%

3.46

29

3.25

33

6-05

%

3442

7 0.

58%

3-

4352

0.

80%

3.

4344

0.

82%

34

339

0.84

%

- 0.

23%

- 0.

14%

SS-S

S-F-

F Pl

ates

2.

2339

33

687

5.02

63

2.22

26

3-31

74

5014

9 0.

51%

1.

52%

0.

23%

2-

2224

3.

3318

5W

O1

0.52

%

1.10

%

052%

2.

2265

3.

3456

5.

0103

0.

33%

0.

69%

0.

32%

2.2

283

3-35

24

5015

0 0.

25%

04

8%

023%

2.

2305

3.

3562

5.

0186

0.

15%

03

7%

0-15

%

F-F-

F-F

Plat

es

8.94

59

13.4

89

2012

8 9.

0188

14

.234

20

292

9.32

77

13.7

80

20.9

87

9.32

15

13-8

17

2097

4

9.33

36

13.8

70

2099

8

9.34

31

1391

2 21

.022

- 0

.81%

- 5

.53%

- O

-82%

- 4

.27%

- 2

.16%

- 4

'27%

- 4.

20%

- 2

.43%

- 4

.20%

- 4.

33%

- 28

2%

- 4.3

2%

- 4.

44%

-

3.14

%

- 4.

44%

8.25

06

8.35

20

8.27

08

8.26

04

- 0

.12%

- 1

.23%

- 0

'24%

3 8.

2576

5

- 0

.09%

R r z tl

8.25

44

0

005%

4 21

.643

m

20

333

1

21.5

17

21-4

72

21.4

66

21-4

62

m

x

6.05

%

058%

0.79

%

082%

084%

* Leis

sa's

data

z6 in

thi

s lin

e; D

QM

dat

a an

d th

eir p

erce

ntag

e diff

eren

ce w

ith r

espe

ct to

Leis

sa's

data

giv

en in

oth

er li

nes.

In a

ll ca

ses,

the

sam

plin

g po

ints

are

ob

tain

ed fr

om e

quat

ions

(46)

and

(47)

with

d =

lo-'


It may be seen from Tables II(a) and II(b) that except for the last two cases of SS-F-F-F and F-F-F-F plates, the fundamental frequencies of Cases 7-16 obtained by the differential quadrature solution are in close agreement with Leissa’s data; in fact, in most of these cases the relative difference is well below 1 per cent. It may be noted that the plate configurations of Cases 7-15 are the ones in which the boundary conditions of at least one pair of opposite edges could be built into the pertinent weighting coefficient matrices. However, the differential quadrature frequencies for SS-F-F-F and F-F-F-F plates seem to be in error when compared with Leissa’s corresponding data. In about one-half of the cases of Tables II(a) and II(b), 6-type grid lines adjacent to the boundary edges were needed. Such cases are indicated in the footnotes of these two tables.

Table 111 provides some information on the dependence of the accuracy and convergence of the DQ solutions on the number of sampling points. The table includes four cases of Tables II(a) and II(b) which were considered inferior with respect to either the convergence rate or the accuracy. In each case, five values of sampling points, namely, N,, N , = 7,9,11, 13 and 15 were taken and sampling points were obtained from equations (46) and (47) with 6 = lo-’. Of these four cases, the convergence trend of the DQ solution in the cases of C - S X - F and SSSS-F-F is obvious. The SSF-F-F and F-F-F-F plates do not exhibit any convergence trend in fact the DQM solution for these cases is quite erratic. Even more erratic behaviour was observed with C-SS-F-F, C-C-F-F, and C-F-F-F plates. These are the only three configurations of Leissa’s work, which for obvious reasons, were left out in the reported results of this paper. It may be noted that of these five cases, the boundary conditions of F-F-F-F plate cannot be built into the DQ analogues and in the remaining cases, the boundary conditions are only partially incorporated into the DQ analogues.

The fundamental frequencies of the some seventeen plate configurations with one or more guided edges obtained by the DQM are given in Table IV. The guided edge boundary condition is of limited practical interest and, possibly for this reason, has remained ignored in the plate literature. The guided edge is a mathematically plausible boundary condition. Also, from a mathematical standpoint, the axes of symmetry in symmetric modes of vibrations are equivalent to guided edges. The equivalence has actually been used in plate vibration analyses; see for example the recent work of Gorman3’ on the free vibration analysis of orthotropic rectangular plates. Recently, Bert and Malik3’ identified and analysed additional configurations of plates having at least one edge guided. It has also been found that among these additional configurations, similar to the well-known case of one pair of opposite edges simply supported, equation (1) may be solved analytically and frequency equations in explicit or transcendental form be determined for two other cases in which one pair of opposite edges in such that (a) both the edges are guided and (b) one edge is simply supported and the other (opposite) edge is guided. This is possible basically for the same reason as for the plates having a pair of opposite edges simply supported: the modes of G-G and SS-G beams are described by trigonometric functions cos (m - 1)aX and sin (2m - l)nX/2, respectively. The frequencies generated from the solution of the characteristic equations would of course be the exact values. Bert and Malik32 have obtained comprehensive data of the free vibration frequencies on various modes for such cases.

The plate configurations in Table IV are of the three aforementioned types of which exact frequency data have been obtained by the solution of the characteristic equations.32 It may be recalled that the guided edge is of the type in which one boundary condition can be built into the weighting coefficient matrices. Thus, all the plate configurations in Table IV are those in which the boundary conditions of at least one pair of the opposite edges could be built into the pertinent weighting coefficient matrices. This feature is indeed reflected in the DQ solution data of Table IV; all these fundamental frequency values match to the number of the digits given in the table with the analytical solution values reported in Reference 32. Thus, giving the exact values would have been a mere repetition of the data given in Table IV.


Table IV. Dimensionless fundamental frequencies of rectangular plates with one or more guided edges

3, = a/b

Plate type 215 213 1 312 512

19. SS-SS-SS-G 20. ss-c-SSrG 21. SS-G-SS-F 22. SS-G-SS-G 23. C-SS-C4 24. C-SS-SS-G 25. SS-SS-G-G 26. C - S S G 4 27. SS-SS-G-F 28. SS-C-G-F 29. SS-G-G-F 30. SS-F-G-F 31. C-G-SS-G 32. C-G-CG 33. SSrG4-G 34. c-G-G4 35. SS-G-F-G

10.26439 10.34455 940977 9.86960

22.59267 15,71498 2.86219 5.81936 2.75921 2.89012 2,44001 2.4 1 748

15.41 82 1 22.37329

2,46740 5.59332 3.008 15

10.96623 11-35736 9.77630 9.86960

22-99377 16.25 197 3.56402 6.26090 324533 3.80890 2.42458 2.39476

1 5.41 821 22.373 29

2.46740 5.59332 6-09382

12.33701 13.68577 9,73624 9.86960

23.8 1563 17.33175 4.93480 7.23771 4.03369 5.70387 2.40785 2.37812

1 5.4 1821 22-37329

2.46740 5,59332

15.41 821

~~

15.42126 19.74590 9.68035 9.86960

25.82839 19.88128 8.01905 9.77231 540480

10.05836 2.38954 2.36651

1 5.4 1 821 22.37329 2.46740 559332

15.41821

25.29086 41.18466

9.59073 9,86960

33.341 14 28,69490 17.88866 18.96041 840570

24-09540 2.37 104 2-35881

15.41 82 1 22.37329

2.46740 5.59332

15.41821

In all cases, the sampling points are obtained from equations (44) and (45) with N, = N, = 15

7. CONCLUDING REMARKS

This paper has attempted to address in a generalized manner the important aspect of implementing multiple boundary conditions in the DQ solution of higher-order differential equations. The context is the free vibration analysis of a thin rectangular plate with a variety of boundary conditions in homogeneous form at the four edges of the plate. The solution domain is square (in normalized co-ordinates) with its sides parallel to the co-ordinate axes. The governing differential equation of the problem is fourth-order partial differential equation with sets of two boundary conditions at each edge. In the past, the boundary conditions of this problem have been handled with some limited success; however, the implementation of the plate boundary conditions in the DQ solution remains an intriguing matter. The motivation to the present work came from some ideas, also presented in some earlier works, of incorporating the boundary conditions into the weighting coefficients. The present paper develops these ideas, establishing the technique of incorporating the boundary conditions on a sound basis and identifying its limitations. It has been shown that boundary conditions involving individual normal derivatives of any order may be built into the differential quadrature weighting coefficients. Consequently such boundary conditions are built into the DQ analogue of the governing differential equation itself. Thus, one does away with invoking these conditions during the solution stage of the system equations that are formed by the assembly of the DQ analogue equations.


The procedures for implementation of the boundary conditions and formulation of the quadrature analog equations were detailed in reference to the free vibration analysis of thin isotropic rectangular plates. The quadrature-solution results of a large number of rectangular plates having various combinations of the boundary conditions were compared with the available exact and approximate results. The quadrature-solution results were found to be of very high accuracy; however, the accuracy of the solutions depends on how accurately the boundary conditions could be invoked. To be specific, such cases are ones having at least a pair of opposite edges that are either simply supported, clamped, or guided. On the other hand, for plates having two or more adjacent edges free, the quadrature method could not yield results of any dependable accuracy; the matter of proper implementation of such boundary conditions in DQ solutions is indeed open to further investigation.

It may be pointed out that the boundary conditions considered in the present work are of homogeneous form. In so far as the weighting coefficients may be determined for a given normal derivative equal to some specified non-zero value, the weighting coefficient matrices can be modified to incorporate non-homogeneous boundary conditions as well.

Although this work has been presented in the context of plate vibration problems, it is believed that the ideas can be implemented for the DQ solutions of other engineering and physical science field problems.

ACKNOWLEDGEMENTS

The authors are grateful to the referees whose thorough reviews and precise comments were very helpful in the revision of the original manuscript of the present paper.

REFERENCES

1. R. E. Bellman and J. Casti, ‘Differential quadrature and long term integration’, J . Math. Anal. Appl., 34, 235-238

2. R. Bellman, B. G. Kashef and J. Casti, ‘Differential quadrature: a technique for the rapid solution of non-linear partial

3. J. 0. Mingle, ‘The method of differential quadrature for transient non-linear diffusion’, J . Math. Anal. Appl., 60,

(1 97 1).

differential equations’, J. Comp. Phys., 10,4&52 (1972).

559-569 (1977). 4. F. Civan and C. M. Sliepcevich, ‘Application of differential quadrature to transport processes’, J. Math. Anal. Appl., 93.711-724 119831

5. F. Civan and’C. M. Sliepcevich, ‘On the solution of the Thomas-Fermi equation by differential quadrature’, J . Comp.

6. F. Civan and C. M. Sliepcevich, ‘Solving integro-differential equations by quadrature method‘, in A. Haji-Sheikh and T. Huang (eds.), Integral Methods in Science and Engineering, Hemisphere Publishing Co., Washington DC, 1986, pp. lob1 13.

7. C. W. Bert, S. K. Jang and A. G. Striz, ‘Two new approximate methods for analyzing free vibration of structural components’, AIAA J., 26, 612418 (1988).

8. S. K. Jan& C. W. Bert and A. G. Striz, ‘Application of differential quadrature to static analysis of structural components’, Int. j . numer. methods eng., 28, 561-577 (1989).

9. J. R. Quan and C. T. Chang, ‘New insights in solving distributed system equations by the quadrature method-I. analysis’, Comp. Chem. Eng., 13, 779-788 (1989).

10. C. Shu and B. E. Richards, ‘Application of generalized differential quadrature to solve two-dimensional incompressible Navier-Stokes equations’, Int. j . numer. methodsjuids, 15, 791-798 (1992).

11. M. Malik and C. W. Bert, ‘Differential quadrature solution for steady state incompressible and compressible lubrication problems’, ASME J. Trib., 116, 296-302 (1994).

12. X. Wang and C. W. Bert,‘A new approach in applying differential quadrature to static and free vibrational analyses of beams and plates’, J. Sound Vib., 162, 566-572 (1993).

13. C. W. Bert, X. Wang and A. G. Striz, ‘Differential quadrature for static and free vibration analysis of anisotropic plates’, Int. J . Solids Struct., 30, 1737-1744 (1993).

14. X. Wang, C. W. Bert and A. G. Striz, ‘Differential quadrature analysis of deflection, buckling, and free vibration of beams and rectangular plates’, Contput. Struct., 48, 47-79 (1993).

Phys., 56,343-348 (1984).

1258 M. MALIK A N D C. W. BERT

15. X. Wang, C. W. Bert and A. G. Striz, ‘Free vibration analysis of annular plates by the DQ method‘, J. Sound Vib., 164,

16. C. W. Bert, X. Wang and A. G. Striz, ‘Static and free vibrational analysis of beams and plates by differential

17. I. H. Shames and C. L. Dym, Energy and Finite Element Methods in Structural Mechanics, Taylor & Francis, Bristol,

18. K. H. Huebner, The Finite Element Methodfor Engineers, John Wiley & Sons, New York, 1975. 19. R. W. Hamming, Numerical Methodsfor Scienrisr and Engineers, Dover Publications Inc., New York, 1973. 20. W. H. Press, B. P. Flannery, S. A. Teukolsky and W. T. Vetterling, Numerical Recipes in C, Cambridge University

21. A, Bjork and V. Pereyra, ‘Solution of Vandermonde system of equations’, Math. Comp., 24, 893-903 (1970). 2 2 M. Ma& C. W. Bert and A. R. Kukreti, ‘Differential quadrature solution of uniformly loaded circular plate resting on

elastic half-space’, Proc. 1st Int. Con$ on Contact Mechanics. Computational Mechanics Publications, Southampton,

173-175 (1993).

quadrature method’, Acta Mech., 102, 11-24 (1994).

Pennsylvania, 1985.

yress, Cambridge, 1988.

(1993), pp. 385-396. 23. K-J. Bathe, Finite Element Procedures in Engineering Analysis, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1982. 24. D. Young, ‘Vibration of rectangular plates by the Ritz method: ASME J. Appl. Mech., 17, 448453 (1950). 25, G. B. Warburton, ‘The vibration of rectangular plates’, Proc. Inst. Mech. Eng., 168, 371-384 (1954). 26. A. W. Leissa, ‘The free vibration of rectangular plates’, J. Sound Vib., 31, 257-293 (1973). 27. R. B. Bhat, ‘Natural frequencies of rectangular plates using characteristic orthogonal polynomials in Rayleigh-Ritz

method‘, J. Sound Vib., 102, 493499 (1985). 28. S. M. Dickinson and A. Di Blasio, ‘On the use of orthogonal polynomials in the Rayleigh-Ritz method for the study of

the flexural vibration and buckling of isotropic and orthotropic rectangular plates’, J . Sound Vib., 108,5142 (1986). 29. R. B. Bhat, P. A. A. Laura, R. G. Gutierrez, V. H. Cortinez and H. C. Sanzi, ‘Numerical experiments on the

determination of natural frequencies of transverse vibrations of rectangular plates of non-uniform thickness’, J. Sound Vib., 138, 205-219 (1990).

30. R. B. Bhat, J. Singh and B. Mundkur, ‘Plate characteristic functions and natural frequencies of vibration of plates by iterative reduction of partial differential equation’, ASME J . Vib. Acoust., 115, 177-181 (1993).

31. D. J. Gorman, ‘Accurate free vibration analysis of the completely free orthotropic rectangular plate by the method of superposition’, J . Sound Vib., 165, 409-420 (1993).

32. C. W. Bert and M. Malik, ‘Frequency equations and modes of free vibrations of rectangular plates with various edge conditions’, Proc. Inst. Mech. Eng., J . Mech. Eng. Sci., 208, 307-319 (1994).

IMPLEMENTING MULTIPLE BOUNDARY CONDITIONS IN THE DQ SOLUTION OF HIGHER-ORDER PDEs: APPLICATION TO...

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Transcript of IMPLEMENTING MULTIPLE BOUNDARY CONDITIONS IN THE DQ SOLUTION OF HIGHER-ORDER PDEs: APPLICATION TO...