A boundary element solution to the vibration problem of plates

A boundary element solution to the vibration problem of plates

Journal of Sound and Vibration (1990) 141(2), 313-322 A BOUNDARY ELEMENT SOLUTION TO THE VIBRATION PROBLEM OF PLATES J. T. KATSIKADELIS Departmen...

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Journal

of Sound and

Vibration (1990) 141(2), 313-322

A BOUNDARY

ELEMENT SOLUTION TO THE VIBRATION PROBLEM OF PLATES J. T. KATSIKADELIS

Department of Civil Engineering, National Technical University, Zografou Campus, GR- 15173 Athens, Greece (Received 3 March 1989, and in final form

16 November 1989)

A boundary element method is presented for the dynamic analysis of thin elastic plates of arbitrary shape. In addition to the boundary supports the plate may be also supported on point or line supports in the interior. Both free and forced vibrations are considered. The case of support excitation is also taken into account. The method utilizes the fundamental solution of the static problem to establish the integral representation for the deflection. The domain integrals involving the inertia forces are evaluated by employing an efficient Gauss integration technique over domains of arbitrary shape. This procedure yields a mass matrix and the equation of motion is derived with respect to the Gauss integration nodal points. Numerical results are presented to illustrate the method and demonstrate its efficiency.

1. INTRODUCTION It is evident from Leissa’s monograph [l] that the methods of solution of the vibration problems of plates prior to 1969 were highly dependent on the shape of the plate. The method of separation of variables, the Rayleigh-Ritz method and Galerkin method were the most widely used. However, realistic problems of dynamic analysis of thin elastic plates can be solved only numerically. The most popular method of solution of plate vibration problems today is the finite element method (FEM). Since that time there have been too many papers to list here. More information about the application of the FEM to plate vibrations can be found in text books on finite elements such as the one by Zienkiewicz [2]. It should be noted here that all large commercial finite element programs have the capability of solving plate vibration problems. Recently, the Boundary Element Method (BEM) [3] appeared as a possible alternative to the domain-type numerical solution methods such as FEM or FDM. The BEM is a numerical method ideally suited for elastodynamic problems [4]. Basically, there are two approaches for treating plate vibration problems by using BEM: (a) the conventional BEM in which the dynamic fundamental solution is used (Dynamic BEM); (b) the BEM in which the static fundamental solution is used (Static BEM), i.e., the fundamental solution of the biharmonic equation, which also requires domain discretization. Dynamic BEM has been used by many investigators to treat free [5-121 as well as forced vibrations [13-141. Static BEM with finite element domain discretization has been employed by Bizine [ 151 and Tanaka et al. [ 161. A variation of Static BEM has been presented by Katsikadelis and Sapountzakis [ 171 and Katsikadelis et al. [ 181. In the last approach the Green function of the corresponding boundary value problem is first established numerically by using BEM. Subsequently, it is employed to obtain an integral representation of the deflection which is free from boundary terms. 313 0022-460X/90/170313+

10 %03.00/O

@ 1990

Academic

Press

Limited

314

J. T. KATSIKADELIS

The major advantage of Static BEM is that it yields a typical linear algebra eigenvalue problem which can considerably reduce the required computation. On the other hand, Dynamic BEM requires searching for zeros of the determinant of the matrix the elements of which are complicated complex transcendental functions (complex Hankel functions) involving the frequency within the argument. A disadvantage of the static formulation is that it requires domain discretization in addition to the boundary one. This fact may diminish the merits of the method if the domain discretization is performed using the finite element technique. The trend in research today is to utilize the static method for plate vibration problems despite its shortcomings. Thus, efforts are made to alleviate the method from the domain discretization either by using the dual reciprocity method to relate the domain inertia forces to the boundary [19] or by inventing techniques to evaluate the domain integrals without the need for finite element domain discretization. The purpose of the present paper is to present an efficient variation of Static BEM. In the procedure the domain integrals are evaluated by using Gauss integration over domains of arbitrary shape. This reduces the work required for the preparation of data considerably, while the method retains most of the advantages of a BEM solution. Numerical examples are presented to illustrate the efficiency of the method and demonstrate its advantages.

2. STATEMENT OF THE PROBLEM AND FORMULATION OF THE BOUNDARY EQUATIONS When a homogeneous, isotropic, thin and linear elastic plate is subjected to a transverse loading f(P, r), under the assumption of small deflections, its flexural vibrations are governed by the differential equation DV4W+pti=f

in R,

(1)

where R is the two-dimensional plane region occupied by the plate bounded by the boundary C = lJ’,Iy Ci (see Figure l), W = W( P, T), P: (x, y) E R and T 3 0 is the time; D = Eh3/ 12( 1 - v’) is the flexural rigidity of the plate, E, v being the elastic constants of the material, and h the thickness of the plate; p is the surface mass density of the plate; f = f (P, T) is the transverse loading; and V4 = d/ax4 + 2a4/ax2 ay2 + a4/ay4 is the biharmonic operator.

YA

Figure

1. Two-dimensional

region

R occupied

by the plate.

PLATE

Moreover, the deflection boundary C [20]:

BOUNDARY

W must

ELEMENT

satisfy

“,(p)w+%(P)~=%(P,

SOLUTION

the following

T),

315

boundary

conditions

on the

P,(p)aWlan+fb(p)MW pet,

= P3(P, T),

720.

@a, b)

Here the functions (Yk,& are specified on the boundary. The conventional boundary conditions, i.e., clamped, simply supported, free or guided edge, are obtained by specifying appropriately these functions. Equations (2a, b) are the most general linear boundary conditions and they also include support excitations (LYE,& # 0). Finally, MW and VW denote the bending moment M,, and the effective shear force V,, along the boundary, and they are given as w+ (v - 1) dZ W/d],

MW = -D[VZ

VW = -D[(a/an)V' The deflection

W(P, 7) is also subjected W(P,O)=W(P),

where

c(P)

W - (v - l)(a/as)(a’

to the initial

functions

13a, b)

conditions

W(P,O)=ct(P),

and h(P, 0) are prescribed

W/an at)].

PER,

of the spatial

(3)

co-ordinate

I?

For a plate with corners which are not prevented (free or elastically restrained) boundary conditions (2a, b) must be supplemented by the corner condition

the

(5)

c,kW+CZk[ITW~=C3k(7).

where the constants elk, c2k and the function c3k( T) are specified on the corner k. I[TWj is the fictitious corner force due to the jump of discontinuity of the twisting moment TW at the corner. For any two functions w and v which are four times continuously differentiable in R and three times continuously differentiable on C the following reciprocal identity is valid [21]: vGV2w

(vV4w-wV4v)da= By applying solution,

relation

(6) to the function

o = (1/87~D)r’ln

-w$V’c-~V*w+d”v:v an

W satisfying

equation

r=lP-QI,

p,

r,

of the biharmonic equation DV4v = 6( P - Q), the following the deflection W(P, T) is obtained:

(6)

(1) and to the fundamental

QER integral

(A,L?+A,X+A,@+A,!P)

-&-

ds. an

representation

for

ds.

IC Here the kernels

Ai( r), i = 1,2,3,4,

are given by

A,(r) = -cos A,(r) = -f(2r and the following

notation

n = w,

A,(r) = In r+ 1,

q/r,

ln r+r)

cos

Q,

A,(r) = ar2 In r,

(8a, b) (8c, d)

has been used: X = a W/an,

@=V’W,

V = (a/an)V’W.

(9)

316

J. T. KATSIKADELIS

Notice that for the line integral r = IP - 41, whereas for the domain integrals r = IP - Q(, P, Q E R, q E C; cp is the angle between the direction of r and the normal II to the boundary at point q. Application of the operator V2 = a2/ax2+a2/ay2 to equation (7) results in the integral representation of the Laplacian as

V’W(p,

T)&

pA2tid+-

IC

(A,@+A,Y)

ds.

(10)

Equation (7) involves five unknown quantities: i.e., the deflection W inside the domain quantities R = n(s, r), X = X(s, r), @ = @(s, T) and Y = Y (s, 7). Four additional equations are established by using the boundary equation method [20]. According to this method the boundary conditions (2a, b), by virtue of equations (3a, b), the notation (9), and the use of intrinsic co-ordinates can be written as R and the boundary

(11, 12) Moreover, upon letting point P +p E C in equations boundary integral equations are derived: &2=-

1 D

A4f dir-; II

,oA,+dda-II

R

1 CY@=D

A,f do-; II

R

(7) and (10) the following two

ds,

(A,L2+A,X+A,@+A,?P)

(13)

I C

R

II

pA,+ddaR

(A,O+A,‘P)

ds.

(14)

I c

Here (Yis the angle between the tangents at point p (see Figure 1). 3. NUMERICAL SOLUTION The system of equations (7), (ll), (12), (13) and (14) can be solved numerically. The differential equations are treated by using the finite difference method and the boundary integrals by using the boundary element method (see Figure 2). The domain integrals are evaluated by using Gauss integration over domains of arbitrary shape by developing a technique which is to be referred to here as the finite sector method

Figure

2. Discretization

of the boundary.

PLATE

BOUNDARY

ELEMENT

SOLUTION

317

[22]. According to this method the two-dimensional domain R is divided into a finite number of sectors by straight lines emanating from a point inside the region (common vertex of sectors) and reaching the boundary (see Fig 3). For domains with complex geometry more than one vertex may be used. Subsequently, each sector is mapped onto a triangle on which a ready-to-use Gauss-Radau integration scheme is employed. Thus, an integral over the domain R may be approximated as

II

s g(Q) duo = C k=l

R

II

Rt

g(Q)IUQ)I

da, = % : k=l

C:g(Q,k)l.Wjk)l,

(15)

,=I

where S is the number of sectors, WIis the number of Gauss-Radau points in the kth sector, Cj” and 0,” (j = 1,2,. . . , m) are the weight factors and the Gauss-Radau points in the kth sector, and J( Qf) are the values of the Jacobian of the transformation which transforms the kth sector onto the triangle Rr. The transformation that maps the sector onto a triangle has been given by Katsikadelis [22] as

where X = f(j)is the equation of the sector base in local co-ordinates (see Figure 4). When this technique is used with BEM, the sector base consists of a group of consecutive boundary elements and it is convenient to approximate the function Z = f(l)by an interpolating polynomial rather than to use the analytic expression for the curve. Thus,

Figure

3. Two-dimensional

domain

L

>

divided

7

into four sectors.



I

Global axes Figure

4. Mapping

of a sector onto a triangle.

318

J. T. KATSIKADELIS

if a polynomial approximation is used, the sector base is chosen so that it can be represented as n=f(jq = (Yg+(Y*p+(Y*JZ+** -+a$. (17) n) are computed from the coordinates of the nodal and/or TheC0effiCientsoi(i=1,2,..., the extreme points of the boundary elements. Thus, by using constant boundary elements to approximate the unknown boundary quantities, unevenly spaced finite difference schemes to approximate the derivatives and a collocation technique at the Gauss points in the interior of the plate the following system of simultaneous differential equations is established:

(184

W=B5+C$i’+[A5,

AS2 AS3 AJ[fi

(lab)

X Q, ‘I’]‘,

where n=[n,

L& ’ **

LIJT,

X=[X,

a=[@

CD2 ***

@J,

W=[?P,

x2 Y2

*** a**

XJ’, ?PJT

(1%

are the values of the unknown boundary quantities at the nodal points of the N boundary elements, W=[W, w, *** W,]’ (20) are the values of the deflection W at the M Gauss integration points, A, (i = 1,2,3,4,5; j = 1,2,3,4) and Ci (i = 3,4,5) are constant matrices while all other quantities depend on time. Solving equations (18a) for the boundary quantities 0, X, Cp and w, and substituting the results into equations (18b) gives the following system of ordinary differential equations: MW+W=F, (21) Equation (21) is the equation of motion of the plate with respect to the Gauss integration nodal points. M is an M x M generalized mass matrix and F is an M x 1 force vector. When forced vibrations are considered (Ff 0) equation (21) can be solved numerically by using time step integration. For free vibrations the eigenfrequencies and mode shapes are established by solving the corresponding linear algebra eigenvalue problem. 4.

EXAMPLES

Four numerical examples are presented in this section to demonstrate the effectiveness of the method. The results have been obtained on a CDC CYBER 171-8 computer using a program written on the basis of the numerical procedure developed in the previous section. 4.1.

FREE

VIBRATIONS

4.1.1. Example 1 A circular simply supported plate with radius a and a Poisson ratio of v = 0.30 is considered. In Table 1 the first six eigenfrequencies are presented as obtained by the

PLATE BOUNDARY

ELEMENT SOLUTION

TABLE

Eigenfrequency

parameter A = am

319

1

of a simply supported circular plate of radius a

A”

Present method

Niwa et al. [7]

Providakis and Beskos [14]

Exact

Al

2.221 3.726 5.050 5.437 6.284 6.926

2.2 3.8 5.1 5.5 6.4 7.0

2.22 3.73 5.06 5.45 6.32 6.96

2.222 3.728 5.061 5.452 6.321 6.963

A2

A3 A, As Ah

present

method,

dynamic

fundamental

with the exact

BEM

by the indirect ones

solution which

[14].

were

[7] The

and by the BEM results

computed

with the frequency

of these three methods

by numerical

evaluation

domain

are compared

of the frequency

equation:

4.1.2.

Example 2

A square plate with side length a subjected studied.

The computed

eigenfrequencies

eigenfrequencies

are in good agreement

to mixed

and mode

boundary

conditions

shapes are presented

has been

in Figure 5. The

with those given by Leissa [l].

(b)

(d)

Figure 5. Mode shapes and eigenfreqencies of a square plate subjected to mixed boundary conditions (SS-F-SS-C). The contour lines are drawn Aw = 0.02 a part. (a) Mode 1, w, = 12.79 (12.69); (b) mode 2, o2 = 32.98 (33.06); (c) mode 3, wg = 43.04 (41.70); (d) mode 4, wq = 63.40 (63.01). The values in the parentheses are taken from reference [l].

320

J. T. KATSIKADELIS

4.1.3. Example 3 A rectangular plate with a point and a line support has been studied to demonstrate the capability of the presented method to treat problems having internal supports. The results are presented in Table 2. The comparison with analytic results is possible only for the cases in which the internal support is symmetric because for the antisymmetric modes the corresponding eigenfrequencies are identical with those of the plate without internal supports. In Table 2 the eigenfrequencies corresponding to the same mode shape are denoted by the same superscript.

TABLE

Eigenfrequencies 0 = wa2m

b/2

b/2

K

K

a/2

I

i_-_________ i a/2 a/2 K SC H

I-----_---_

a/2

U

0.320E +02(‘) 0*377E+O2 0449E + 02@’ 0.613E+02’3’ 0*757E+02 0~877E +02(4) Oe982E + 02’5’ Oe108E + 03@’ 0*112E+03 O.l30E+03”’

0.320E + 02”’ 0*423E+O2 0.613E + 02’3’ 0.680E + 02 0.982E + 02”’ 0.108E + 03@’ 0*112E+03 0*119E+03 O.l30E+03”’ O.l42E+03

0.2

0.4

0.6

a/2

e = 0.2 b

e=O*O

0.0

2

of a rectangular simply supportedplate with internal supports (a/b = 0.75)

0*237E+02 0*520E+02 0.576E+02 0.8548+02 0*861E+02 0*989E+02 O.l14E+03 0*120E+03 0*131E+03 O.l45E+03

0.8 Time, /

I.0

a/2

n

Analytic O.l54E+02 0*321E+02”’ 0.450E + 02’2’ 0.598E+02 0.617E+02’3’ 0.894E + 02’4’ 0.944E + 02 0.987E + 02”’ 0.111E+03’6’ 0.128E+03”’

I.2

1.4

R m.n R 1.1 R 1.2 R 2.1

n

1.3 R 2.2 R 2.3 R 3.1 R I,4 R 3.2 R 2,4

I

Figure 6. Time history of the response ratio R(t) = W(P, T)/ W,, of a suddenly uniformly loaded simply supported square plate. -, Exact; 00, BEM.

PLATE

BOUNDARY

ELEMENT

SOLUTION

321

Figure 7. Maximum response D = max R(t) of a square simply supported plate subjected to a rectangular impulse. T, is the fundamental period. Key as Figure 6. 4.2.

FORCED

4.2.1.

Example

VIBRATIONS

4

A simply supported square plate with side a subjected to a suddenly applied uniformly distributed load q of (i) infinite duration and (ii) of small duration (impulsive load) has been studied. The results are presented in Figure 6 and Figure 7. In both cases the numerical results obtained are in excellent agreement with those obtained by an exact series solution. 5. CONCLUDING

REMARKS

A variation of the Static BEM for the dynamic analysis of plates has been described. The main conclusions drawn from this investigation are the following: (a) both free and forced vibrations can be treated; (b) the method is well-suited for computer aided analysis; (c) plates of arbitrary shape subjected to any type of boundary conditions and excitation force can be analyzed, and support excitation also can be considered; (d) the method is effective and the obtained results are accurate; (e) the method has all the advantages of the static BEM for plate dynamic analysis as it circumvents the computational difficulties arising from the use of a complex dynamic fundamental solution; (f) the use of Gauss integration over domains of arbitrary shape increases the efficiency of the Static BEM since it alleviates it from domain finite element discretization. ACKNOWLEDGMENTS

The author wishes to thank his doctoral students, Diplom Civil Engineers Mr C. Kandilas and Mr E. Sapountzakis, for their assistance in obtaining the numerical results. REFERENCES 1. A. W. LEISSA 1969 Vibration of Plutes (NASA SP-160). Washington, DC.: U.S. Government Printing Office. 2. 0. C. ZIENKIEWKZ 1977 The Finite Element Method New York: McGraw-Hill. 3. C. A. BREBBIA 1978 The Boundary Element Method for Engineers London: Pentech Press. 4. D. E. BESKOS 1987 Applied Mechanics Reviews 40,1-23. Boundary element methods in dynamic analysis. 5. J. VIVOLI 1972 Acustica 26, 562-567. Vibrations des plaques et potentials des couches. 6. J. VIVOLI and P. FILIPPI 1974 Journal of the Acoustical Society of America 55, 562-567.

Eigenfrequencies of thin plates and layer potentials.

J. T. KATSIKADELIS

322

7. Y. NIWA, S. KOBAYASHI and M. KITAHARA 1981 Theoretical and Applied Mechanics 29, 287-307. Eigenfrequency analysis of a plate by the integral equation method. 8. Y. NIWA, S. KOBAYASHI and M. KITAHARA 1982 in Developments in Boundary Element Methods-2 (editors P. K. Banerjee and R. P. Shaw), 143-176. Determination of eigenvalues by boundary element method. London: Applied Science. 9. G. K. K. WONG and J. R. HUTCHINSON 1981 in Boundary Element Methods (editor C. A. Brebbia), 272-289. An improved boundary element method for plate vibrations. Berlin: SpringerVerlag. 10. M. KITAHARA 1985 Boundary Integral Equation Methods in Eigenvalue Problems of EIastcF dynamics and Thin Plates. Amsterdam: Elsevier. 11. R. HEUER and H. IRSCHIK 1987 Acta Mechanica 66, 9-20. A boundary element method for eigenvalue problems of polygonal membranes and plates. 12. H. IRSCHIK, R. HEUER and F. ZIEGLER 1987 in Boundary Elements ZX, Vol. 2 (editors C. A. Brebbia, W. L. Wendland and G. Kuhn), 35-45. BEM using Green’s functions of rectangular domains: static and dynamic problems of bending of plates. Berlin: Springer-Verlag. 13. G. BBZINE and D. GAMBY 1982 Journal de Mkanique 7’heorique et Applique’e 1, 451-466. Etude de mouvements transitoires de flexion d’une plaque par la mCthode des equations inttgrales de front&e. 14. C. P. PROVIDAKIS and D. E. BESKOS 1988 in Boundary Elements X, Vol. 4: Geomechanics, Wave Propagation and Vibrations (editor C. A. Brebbia), 403-413. Dynamic analysis of plates. Berlin: Springer Verlag 15. G. BBZINE 1980 Mechanics Research Communications 7, 141-150. A mixed boundary integral finite element approach to plate vibration problems. 16. M. TANAKA, K. YAMAGIWA, K. MIYAZAKI and T. UEDA 1987 Proceedings of the 1st Japan-China Symposium on Boundary Element Methods (editors M. Tanaka and Du Quinghua), 375-384. Integral equation approach to free vibration problems of assembled plate structures. Oxford: Pergamon Press. 17. J. T. KATSIKADELIS and E. J. SAPOUNTZAKIS 1987 in Boundary Elements ZX, Vol. 2: Stress Analysis Applications 51-67. Numerical evaluation of the green function for the biharmonic equation using BEM with application to static and dynamic analysis of plates. Berlin: SpringerVerlag. 18. J. T. KATSIKADELIS, E. J. SAPOUNTZAKIS and E. G. ZORBA 1988 in Boundary Elements X, Vol. 4: Geomechanics, Wave Propagation and Vibrations, 431-444. A BEM approach to static and dynamic analysis of plates with internal supports. Berlin: Springer-Verlag. 19. D. NARDINI and C. A. BREBBIA 1982 Proceedings of the Fourth International Conference on Boundary Element Methods, Southampton University, September 1982. A new approach to free vibration analysis using boundary elements. Berlin: Springer-Verlag. 20. J. T. KATSIKADELIS and A. E. ARMENAKAS 1989 American Society of Mechanical Engineers, Journal of Applied Mechanics 56, 364-374. A new boundary equation solution to the plate problem. 21. J. T. KATSIKADELIS and A. E. ARMENAKAS 1984 American Society of Mechanical Engineers, Journal of Applied Mechanics 51, 574-580. Analysis of clamped plates on elastic foundation by the boundary integral equation method. 22. J. T. KATSIKADELIS 1990 (to be published). A Gaussian quadrature technique for regions of arbitrary

shape.