results for plates with rectangular voids. The calculated results are coincident with the experiment results in that paper, which verifies the effectiveness of the Extended Dirac Functions Method In fact, plates with cylindrical voids are more common than plates with rectangular voids(e. g, concrete slabs in buildings and bridges). However, cylindrical voids are much more complicated to account for with the extended dirac function than rectangular voids, and the consequent mathematical derivation becomes more tedious. Even so, the wide usage of these plates makes the solution necessary. Up to now, few or no papers on the vibration of plate with cylindrical voids is reported. Of course, the composite model derived at the end of this paper is also a simple method to get the natural frequencies of cylindrical voided plate. But it is not so powerful not only because that the influence resulted from voids' locations and radii can not be considered but also because it is only suitable for much more densely located and small radii voids The objective of this paper is to supply a simplified method for studying the dynamic response of plates with cylindrical voids. The discontinuous variations in the rigidity and mass density of the rectangular plates resulted from cylindrical voids are expressed as a function of the extended Dirac function, which is a different concept from Takabatake 's in that the modified variables are rigidity and thickness. It will be shown that the natural frequencies of such plates can be analytically solved for voids of arbitrary positions and radii. With the resulting simpler formula, one can conveniently calculate the natural frequencies of plates with cylindrical voids. We will also derive that the natural frequencies of plates with voids depend on the locations and radii of the voids, and Poisson s ratio as well as the plate dimensions, which is different from the solid plate. Accordingly, a very simple formula is supplied in this paper to obtain the natural frequencie of plates with cylindrical voids, which will be very convenient to use in engineering The present paper also provides an optimization tool to get the natural frequencies required by means of changing voids locations and radii, e. g, the maximum, the minimum natural frequencies or the lightest plate having the same natural frequencies as the solid plate It is worth mentioning that although the numerical results are obtained only for simply supported plate the method is applicable to other boundary conditions, when appropriate functions Fmn are selected in advance. For example, Fmn for clamped plate have been supplied by Wu and Zhang(1991,1989) EXTENDED DIRAC FUNCTION METHOD Governing equation of Bending Motion As shown in Figure 1, a rectangular plate of width a, length b, and thickness h has arbitrarily positioned cylindrical voids. The cylindrical voids are parallel to the y-direction, go through the whole plate, and are disposed symmetrically with respect to the middle plane of the plate. The coordinate origin is selected at a comer of the plate. The plate material is assumed to be isotropically elastic
Kirchhoff assumptions are assumed to be suitable in the paper. That is to say, the thickness of the plate is small with respect to its length and width and deflection in the middle plane is also small. The governing equation of the bending motion of plates with cylindrical voids can be written at2 ax =9(x,y,) Where u is the Poisson,'s ratio of the plate material, h is the plate thickness, q(x, y t)is the extemal load applied in the plate transversely. D, p are the modified flexural rigidity and mass density due to cylindrical voids, respectively, Do, Po the initial flexural rigidity and mass density of the plate without voids P=po5(x), D=Don(x), Do =Eh (2a,b,c) 12(1-x2) where, 5=1-2a(x(x-x) (x)(x-x) Where the function a, (x)=2 r2-(x-x)?/h is used to describe the characteristic of cylindrical voids. N is the total number of the cylindrical voids; x, is the location of the /"void, and r its radius. A(x-xi is extended Dirac function defined jrx∈(x; (4) Free Transverse Vibration Problem Let q(x,y, r)=0, and after separating time and space variables (i.e. w(r, y, t)=Wx, y)(t)),E 7()+a27(t)=0 (5a) a2+ADomi a2「,ai a2w D-2+uD O2 ow phaw=0 aa Where W(x y)is the plate modal shape, and T'(t) is the time- function multiplier. The extended Dirac function yields the following results
A(x-x)f(x)dr=2 fEs)ds -1)"f(()d2 With the help of the Galerkin method, the modal shape becomes Fx)=∑WmFm(xy Where Fmn(x, y) are the displacement functions satisfying the boundary conditions of the plate for the modes m and n, and Wmm are the weight factors. Finally, the natural frequencies of the plate with cylindrical voids are: Yph The natural frequencies coefficients Amn are related to Gmnmt, Hmmm through 入mH a Fmn+un ay2 lx a*F. a- Fmr +2(1-A)7 Fmndxdy x H Solution for Simply Supported Plates In the case of simply supported plates, Fmn x, v)=sin(mm/a)sin(n y/b). Using the orthogonality lation of sine functions, Gmm and Hmnmn beco me +兀5b|5m (12)
H 4h Where (-)y*+2) ot h (k-1)!(k+1)! (-1) k(k +m万) (15) Where dimensionless variables F=r /a,h=h/a, b=b/a. In order to satisfy the hin plate assumption, let h <0.2 in the following calculation In the particular case of a plate without void (i.e,r=0), Eqs( 8)"(15)yield the well-known natural frequencies of a rectangular plate(e. g, Cao, 1989) As follows amn=m+ xk (16) The ratios an=@mn a0mn of the natural frequencies of plate with and without voids can expressed such as 1-Ti (11) T2 @ll 5n +(4+2) 6r r b 兀n h Numerical Results and Concluding Remarks As an example, a number of first natural frequencies on for simply supported plates with voids of equal radius F= r and different radii are calculated, in which Poissons ratio, is 0. 17; b=1. Note that F <h/2. Then
1 1-T1 61 Q011 ere T11 r--(μ x石分02)2(D) (21) (k-1)!(k+1)! ∑万-2系第F2 (-1){万) (22) k!(k+1)! In the particular case r =r (i.e, F=r), we assume that the distance is the same between every two adjacent voids. Tul and T2l can be simplified as 1l1 h31-+2 Scos 2 x,(-1)万)2 (k-1)(k+1)! (23a) Ni r cos] xi(-1)t r) N (k+1)! In fact, the summation terms in Equations(23a)and(23b)converge so quickly that only the first three series are accurate enough for the solution Thus(23a)and (23b)can be rewritten as 万31-(+2)义)c0s2 Tu=ot Fo N-05+?(m (24a) 6 2(1+ 12 The general equations will now be applied in particular cases to estimate the effects of void radii and distributions on the natural frequencies of rectangular plates. The parameter h will be confined to values smaller than 0.2 to remain the assumption of thin plates. The Poisson s ration u will be set equal to 0.17. Figure 2 examines the effects of number of voids on the first natural frequency. It shows the ratio 8n of natural frequencies for square plate with N=3, 4, 5, 6 voids equally spaced (i. e x =i/(N+1)). Figure 2 tells us that the greater the void radius, the greater the natural frequencies of the plate, especially with the thicker of the plate. But the plate thickness cannot be too large to match the thin plate assumption. Eqs(8),(12)and (13)show that Zorn are independent of the material coefficients but Amn are dependent on Poissons ratio. The natural frequency coefficients of the solid plate are only a function of its sizes but those of the plate with voids are not