Free vibration analysis of elastic plate structures by boundary element method

Free vibration analysis of elastic plate structures by boundary element method

MASATAKA TANAKA

Department of Mechanical Engineering, Shinshu University, 500 Wakasato, Nagano 380, Japan

KOUJI YAMAGIWA

Graduate School of Shinshu University

KENICHI MIYAZAKI

Seiko-Epson Co., Ltd

and TAKAHIRO UEDA

Tateishi Electric Co., Ltd

This paper is concerned with an integral equation approach to the free vibration problems of elastic plate structures. In the usual boundary integral equation method, the eigen frequency must be determined by means of the so called direct search of the zero-determinant value of the system matrix. To circumvent these difficulties this paper presents a new integral equation approach and its solution procedure, in which an approximate fundamental solution to the static problem is used for the formulation. The resulting set of integral equations are discretized by means of the boundary-domain element method and reduced to the system of algebraic eigenvalue equations. The potential usefulness of the proposed method is demonstrated through some sample computations.

Key Words: free vibration analysis, plate structure, bending vibration, in-plane vibration, boundary integral equation, boundary-domain element method.

1. INTRODUCTION

The plate structures are widely used in various fields because they have an advantage of larger load-carrying capacity in spite of light weight 1-3. It is one of the most important subjects in engineering to develop an effective solution procedure for the free vibration problems of plate structures. Much knowledge has been already accumulated for both the in-plane and out-of-plane vibration problems of single plates with arbitrary shapes through the application of finite element, finite difference or other analytical methods of solution TM. Recently, the boundary element methods have been successfully applied to these problems 5-1 o. However, there have still been few investigations for the free vibration problem of plate structures composed of elastic plates. In this study, we first discuss the in-plane and out-of-plane vibration problems of a single plate, and then propose a new integral equation formulation and its solution procedure by means of the boundary-domain element method.

As has been well known 5-7, the exact fundamental solutions of the problem under consideration are expressed in terms of the Hankel and Bessel functions. This would often lead to some difficulties in manipulation of the functions and numerical implementation of the

Accepted December 1987. Discussion closes May 1989.

formulation. Much care should be paid to circumvent these difficulties 5 6. In the present method of solution, approximate fundamental solutions of the corresponding static problems are used for the formulation. The resulting boundary integral equations include unknown variables in the inner domain of the plate component, and

Corner Point Ak+l y"

Boundary /-' I Ak-1 Y) Fig. 1. Notation and coordinate system

182 Engineering Analysis, 1988, Vol. 5, No. 4 © 1988 Computational Mechanics Publications

Free vibration analysis of elastic plate structures: M. Tanaka et al.

hence they are solved together with the additional integral equations by means of the boundary-dorrlain element method proposed previously by the senior author ~1. The powerful usefulness of the proposed method is demonstrated through some sample computations. It should be noted that similar ideas to that of the present paper are also successfully applied to the free vibration problem of single plates s.

2. FREE VIBRATION OF SINGLE PLATE

First, we consider the out-of-plane bending vibration of a single plate. We use the Cartesian coordinate system O - xlx2x 3 and the notation as shown in Fig. 1. If the plate vibrates harmonically with the angular frequency o9, the governing equation of the problem can be expressed by

V4 w =1 0)2 phw (1) D

Where V 4 denotes the biharmonic differential operator, p the mass density of the plate material, h the plate thickness, w the lateral deflection of the plate, D the flexural rigidity of the plate. Suppose that the pseudo pressure q defined as

q=0)2phw (2)

is applied to the plate, equation (1) can be formulated into the integral equation by using the fundamental solution of the static bending problem:

, 1 z w * ( x , x ) = ~ r lnr (3)

Now, we consider the weighted residual statement of equation (1) expressed as

f (V4w- q/D)w* df~ = 0 (4)

in which the approximate fundamental solution given by equation (3) is used as the weight function w*. Applying repeatedly the Gaussian divergence theorem, we can finally obtain the boundary integral equations as follows:

{" cw+ Jr (V*w-M*T. + T*M,-w*V.)dF

+ Z [[M*,w- w*M.3

= 0) 2 jf~ phw*w c df~ (5a)

T*M.- if* V.) dF

+

=0)~ I- ph~*w~df~ (5b) d,

which c is a constant depending only on the geometrical property of the boundary and equal to 1/2 if the boundary is smooth. The asterisked functions in equations (5) are the set of fundamental solutions corresponding to

equation (3), and those denoted by the tilde their normal derivatives. The deflection ~ multiplied by ~'* and also M*t denotes the relative displacement with respect to the observation point on the boundary, so that the corresponding boundary integrals can be evaluated in the sense of Cauchy's principal value integral as shown in Ref. 1 5. The variables T., M,, V. and M.t stand for the slope, bending moment, shearing force and twisting moment on the boundary, respectively. The quantity denoted by the symbol [1" -l] means the discontinuity jump at a corner point of the plate, and I]-M.t]] implies the concentrated force acting there. The notation ~ in front of the discontinuities indicates the summation over the whole corner points except for the one coinciding with the observation point. As has been shown in Refs 8 and 1 2, the discontinuity []-M,,]] can be approximated by the slope. 7". before and after the corner point under consideration by using the following relation:

~T. M. , = - D(1 - v) ~ - (6)

where v denotes Poisson's ratio and O( )lOt the derivative along tangential direction. Through such a procedure we can eliminate the corner variable JIM.t]] in the numerical implementation of the formulation. Other details with respect to the variables and the implementation of equations (5) can be found in the literature 12-14.

Since the approximate fundamental solution is used for the formulation, the r.h.s, of the boundary integral equations (5) includes the unknown variable w~ in the inner domain of the plate. Therefore, an additional equation is needed to solve the problem without any iterative solution procedure. For this purpose, we use the integral equation expressed by

w~+ fr (V*w-M*T,+ T*M,-w*V.)dF

+Z JIM*w-

= 09 2 [ phw*wc df~ Jn

(7)

The boundary integral equation (5) and the additional integral equation (7) are discretized by means of the boundary-domain element method 11 as in the previous paper for elastic buckling analysis of plate structures x4. The boundary-domain element method proposed in Ref. 11 can use any higher-order finite elements as the domain elements to discretize the domain integrals included in the integral equations (5) and (7). However, it should be noted that the basic idea of the boundary-domain element method for the free vibration problem of single plates was originally proposed by Bezine 8 and the constant domain element was successfully applied to numerical computation of some sample problems. If we employ the same solution scheme as Bezine's and take into account the homogeneous boundary conditions, we can finally obtain the discretized set of equations in the following matrix form:

[ GE] { XB} = (D2[JE] {Wc} (8)

{we) + [Go]{x.] = 0)2[JJ{w } (9)

Engineerin9 Analysis, 1988, Vol. 5, No. 4 183


where {X~} denotes the column vector of nodal unknowns on the boundary and {w~} the column vector of nodal unknowns regarding the deflection w~ of the inner domain of the plate. Elimination of {XB} from the above two sets of equations can lead to the algebraic set of eigenvalue equations expressed as

jointed edge of plate components, we can finally obtain the global set of equations, which are solved under the appropriate boundary conditions prescribed on the periphery of the structure.

We first express the discretized sets of equations with respect to the integral equations (5) and (11) as follows:

([JE] -- [G~][G~] - x[SE]){We} = 5 {We} (10)

from which the eigen frequency and the eigen mode can be computed by using the subroutine library available at most computing centres.

Next, we shall show briefly a similar solution procedure for the in-plane free vibration problem of elastic plates. By assuming that the plate is in the plane stress state, and applying the approximate fundamental solution of the static problem, that is, Kelvin's solution to the integral equation formulation, we can obtain the boundary integral equation as follows X°:

ck~u,,, + p*,,,u,, d F - u'~,,,p,, dF = p o 2 u~mu m df~ dr dr

(11)

Also in this case, the r.h.s, of the boundary integral equation includes the unknown variable u,, in the inner domain of the plate. Hence, we use an additional integral equation expressed

Uk+ I

(12)

Discretization of equations (11) and (12) through the boundary-domain element method and taking account of the homogeneous boundary conditions can lead to

[HE]{XE] = coZ[RE]{Uc} (13)

{u~} + [H~] {XE} = ~02[RE] {Uc} (14)

Eliminating the nodal unknows {XE}, we can obtain

~ 1 ([R~] - [Hc][HE]- [Re]){ud = ~ - {u~} (15)

[S~] {~,, } - JR.] {£°} = o) 2 [AB] { ~;i.}

From equations (7) and (12) we can obtain

(16)

u ~ 2 F A ] ~ u ~ {~}+[S ,]{r . } - [R ,]{~°}= t. ,j~w~, (17)

In equations (16) and (17) we have introduced the following notation:

{u}~=kut u2 w l

{"c}T=["~c .~A { f } r = t p t h p2 h V.J

(18)

where the symbol [ J denotes a row vector. In order to transform the variables in the local

coordinate system to those in the global coordinate system, we use the following relationships:

{M.} = [ ]{M.# (19)

where the superimposed bar denotes the variables in the global coordinate system, and the transformation matrix [A] can be given by

[11 ml nl [A]= 12 m2 n2 (20)

13 m3 n3 0 0 0

In equation (20) 1~, m~ and n~ (i = 1, 2, 3) are the directional cosines of the axes x'~, x~ and x; with respect to the global coordinate axis x~, respectively.

If equations (16) and (17) are expressed in the global coordinate system by using equation (19), it follows that

{S"] {t} -- [R"] {'M.} -- 2 [A. ]{ .J (21)

{~;} + [St]{t} - [/~] {~io} -co- 2[At]{w~},o (22)

where

This algebraic set of eigenvalue equations can be easily solved as in the out-of-plane free vibration problem.

[g . ] = [ S . ] [ A ]

[S,] =[Sl][A ]

= [R.] [A]

[R,] = [ g , ] [ A ] (23)

3. FREE VIBRATION P R O B L E M OF PLATE STRUCTURES

The out-of-plane and in-plane vibrations are combined in the free vibration problem of assembled plate structures. To analyse this problem we introduce a local coordinate system to each plate component of the structure. Then, the discretized sets of equations derived in the previous chapter can hold for each plate component. These equations expressed in the local coordinate system are assembled into the global coordinate system which is used in common for the whole structure. By taking account of the equilibrium and compatibility conditions on the

Equations (21) and (22) of every plate component are assembled into the global set of the whole structure. Then, we consider the compatibility and equilibrium conditions on the jointed edge of plate components. Suppose that L plate components are jointed together, these conditions can be expressed as follows:

(1) Compatibility of displacements {u} and slope T,;

(24)

fllT~=fl2T~ . . . . . flLT~ (25)

184 Engineering Analysis, 1988, Vol. 5, No. 4


(2) Equilibrium of forces {f} and moment M,;

{fl} + {f2} + . . . + {fL} = 0 (26)

fllM~ +f12M2 + . . . +ilL M L = 0 (27)

It is assumed in equations (26) and (27) that no external forces and moments are applied at the jointed edge. In addition, the coefficient fli is defined such that #i= 1 when the plate component has the same tangential direction as that of the reference plate component and otherwise #i= - 1 .

If the homogeneous boundary conditions expressed as

w = 0, T, = 0 (clamped)

w = 0, M, = 0 (simply supported)

M, = 0, V, = 0 (free) (28)

are taken into account, the assembled set of equations for the whole structure can be obtained as follows:

[H.]{X} =o~Z[G.]{u} (29)

Fig. 2. Element division of square plate

{u} (30)

where {u}r=[uc w d and {X} denotes the nodal unknowns on the boundary and the jointed edges of the structure. Elimination of {X} from equations (29) and (30) can finally lead to the following set of algebraic eigenvalue equations:

([Gc] - [Hc] [HB] -~ [G.]){u} = ~ {u} (31)

from which we can easily compute the eigen frequency o~ and the corresponding eigen mode.

4. NUMERICAL COMPUTATIONS AND DISCUSSION

A new computer program has been developed on the basis of theoretical considerations mentioned before. We will show here the numerical results obtained for some sample problems. It is noted that only constant boundary elements and also constant domain elements have been used for the present computations.

The first example is the out-of-plane bending vibration of a square plate. The boundary conditions are assumed such that the whole boundary is simply supported or clamped. The element division is illustrated in Fig. 2 where the whole boundary is equally divided and also the inner domain into triangular domain elements of equal size. To check the numerical accuracy computation is performed under the following conditions:

(1) The number of boundary elements is changed as 12, 16, 20, 32, and 48, while the number of domain elements is kept constant as 72 triangular elements.

(2) The number of triangular domain elements is changed as 8, 32 and 72, while the number of boundary elements is kept constant as 48 on the whole boundary.

The results obtained are summarized in Figs 3 and 4 for the clamped boundary conditions. The error in these

i_ o i,. t_ lid

Fig. 3. e r r o r

40.0

5 0 . 0

2 0 . 0

I0.0

I I I

\\\\\\\N\\

lNNNNNX\/

• M o d e I _

• M o d e 2

I - I I -

0 2 0 4 0 6 0 8 0

Number of Internot Cetts Influence of boundary element mesh on numerical

figures are calculated such that

Numerical solution- Analytical solution Error(~o) =

Analytical solution

xl00

In Tables 1 and 2 comparison is made between the numerical results obtained and the analytical ones 1'2. It can be seen that better accuracy can be achieved for lower eigen modes.

Next, we compute the in-plane free vibration problem of a rectangular plate as shown in Fig. 5. The dement division used is illustrated in Fig. 6. In Table 3 the numerical results obtained are compared with the analytical ones 2.

Engineering Analysis, 1988, Vol. 5, No. 4 185

Free vibration analysis oj elastic plate structures: M. Tanaka et al.

-- 0 .0 ~t ¢. o I . I _ UJ

- 10 .0

-20 .0

-:50.0

Fig. 4. error

~ ~ r - ' - - - - - - 7 f

/I , ~o.o , I I Mode2 / /

/I •

0 10 20 :50 40 50

Number of Boundary ELements

Influence of domain element mesh on numerical

Table 1. Eigen frequencies of bending vibration (clamped plate)

Clamped

Mode ~o Exact Error (%)

1 1.231 1.228 0.24 2 2.565 2.505 2.40 3 3.828 3.694 3.63

Table 2. Eioen frequencies of bending vibration (simply supported plate)

Supported

Mode w Exact Error (%)

1 0.676 0.674 0.30 2 1.717 1.685 1.90 3 2.778 2.695 3.08

Table 3. Eigen frequencies of in-plane vibration

Mode o~ Exact Error (%)

1 43.1 43.0 0.40 2 133.2 128.9 3.40 3 222.3 214.8 3.50

I "

Fig. 6. Element division of rectangular plate

v / f f " . . , I ,

/ ( ' / J ',I" , , /

J / J / /

J /

J Z _

J J

J A D

C

Fig. 7. Assembled plate structure and element division

r I 1

I I I

I

I I

Fig. 5. plate

x2

c) © i i [ e ~ e

0 0 0 0 0

Uni-directional free vibration

z y - -

Xl

0

of rectangular

Now, we proceed to computation of the free vibration problem of assembled plate structures. Take for example the plate structure composed of three equal rectangular plate components as shown in Fig. 7. The aspect ratio of the smaller to the larger edge is assumed to be 0.5. It is also assumed that the boundary ABCD is clamped and the other boundary is free. We use in this case the rectangular domain element, and divide the whole boundary and each jointed edge into 30 and 6 constant boundary elements of an equal size, respectively, as shown in Fig. 7.

In Figs 8 to 10are shown the eigen modes of the first three eigen frequencies (~o 1 =0.409, ~o 2 =0.493 and (03 = 1.139).

5. C O N C L U S I O N S

In this paper, the boundary-domain element method has been presented for free vibration analysis of the elastic


r

/ J

/ /

/


Fig. 10. 3rd eigen mode (0)3 = 1.139)

Fig. 8. 1st eigen mode (0) 1 =0.409)

J

I r f /

/ ~ f f

f f

/ f I

L U "

Fig. 9. 2nd eioen mode (o92 = 0.493)

plate structures. A new compute r p rogram has been developed on the basis of the theory, and some sample problems were computed by means of the computer program, whereby the potential usefulness of the proposed method was demonstra ted. The most

impor tant point is that use of the boundary -domain element method can yield the set of algebraic eigenvalue equations. This implies that the present method is a powerful alternative to the usual integral equat ion method using the exact fundamental solution in which a transcendental equation must be solved to determine the eigenvalues 5,6.

Finally, the authors are grateful to Dr G. Bezine for his useful comments and suggestions on the numerical implementat ion of the corner variable [[M,t]] th rough private communicat ions .

R E F E R E N C E S

1 Timoshenko, S. P. and Woinowsky-Krieger, S. Theory of Plates and Shells, 2nd edn., McGraw-Hill, 1959

2 Szilard, R. Theory and Analysis of Plates, Prentice-Hall, Englewood Cliffs, 1974

3 Donnell, L. H. Beams, Plates and Shells, McGraw-Hill, 1976 4 Zienkiewicz, O. C. The Finite Element Method in Engineering

Science, McGraw-Hill, 1971 5 Niwa, Y., Kobayashi, S. and Kitahara, M. Determination of

Eigenvalues by Boundary Element Methods, Chapter 6, Developments in Boundary Element Methods, (Eds P.K. Banerjee and R. P. Shaw), Appl. Sci. Publishers, London, 1982, 2, 143-176

6 Kitahara, M. Boundary Integral Equation Methods to Eigenvalue Problems of Elastodynamics and Thin Plates, Elsevier, 1985

7 Gospodinov, G. and Ljutskanov, D. The Boundary Element Method Applied to Plates, Appl. Math. Modelling, 1982, 6, 237- 244

8 Bezine, G. A Mixed Boundary Integral-Finite Element Approach to Plate Vibration Problems, Mech. Res. Comm., 1980, 7, 141-150

9 Tsuta, T., Yamaji, S. and Fujita, H. A Boundary Element Method of Elasto-Plastic 3-D Plate Bending and Applications, Proc. 1st Japan Natl. Symp. on BEM, JASCOME, Tokyo, in Japanese, 1984, 301-306

10 Nardini, D. and Brebbia, C. A. Boundary Integral Formulation

Engineering Analysis, 1988, Vol. 5, No. 4 187

Free vibration analysis q f elastic plate structures: M. Tanaka et al.

of Mass Matrices for Dynamic Analysis, Chapter 7, Topics in Boundary Element Research, (Ed. C.A. Brebbia), Springer- Verlag, 1985, 2, 191 207

11 Tanaka, M. and Tanaka, K. On a New Boundary Element Solution Scheme for Elastoplasticity, lngenieur-Archiv, 1982, 50, 289 295

12 Tanaka, M. and Miyazaki, K. A Direct Boundary Element Method for Elastic Bending Analysis of Plates, Trans. Japan Soc. Mech. Engrs., Ser. A, in Japanese, 1985, 51, 163(~1641

13 Tanaka, M. and Miyazaki, K. Elastic Stress Analysis of Assembled Plate Structures by Boundary Element Method,

Boundary Elements VII, (Eds C.A. Brebbia and G. MaierJ, Springer-Verlag and Computational Mechanics Publications, 1985, 4/3-4/16

14 Tanaka, M. and Miyazaki, K. Boundary Element Method for Solving Elastic Buckling Problems of Assembled Plate Structures, Boundary Elements VIII, (Eds M. Tanaka and C. A. Brebbia), Springer-Verlag and Computational Mechanics Publications, 1987 547 560

15 Stern, M. A General Boundary Integral Formulation for the Numerical Solution of Plate Bending Problems, Int. J. Solids and Structures, 1979, 15, 769 782


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Free vibration analysis of elastic plate structures by boundary element method