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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

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

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,, dF- u'~,,,p,, dF = po 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 unk