Vibration of Plate on Foundation with Four Edges Free by Finite Cosine Integral Transform Method

Transcription

1 854 Vibration of Plate on Foundation with Four Edges Free b Finite Cosine Integral Transform Method Abstract The analtical solutions for the natural frequencies and mode shapes of the rectangular plate on foundation with four edges free is presented b using the finite cosine integral transform method. In the analsis procedure, the classical Kirchhoff rectangular plate is considered and the foundation is modelled as the Winkler elastic foundation. Because onl are the basic dnamic elasticit equations of the thin plate on elastic foundation adopted, it is not need prior to select the deformation function arbitraril. Therefore, the solution developed b present paper is reasonable and theoretical. In order to illuminate the correction of formulations, the numerical results are also presented to comparing with that of the other references. Yang Zhong *, Xue-feng Zhao, Heng Liu Department of Civil Engineering Dalian Universit of Technolog P.R.China Received in 23 Apr 213 In revised form 2 Jun 213 *Author zhong@dlut.edu.cn. Kewords Elastic foundation, Rectangular plate with four edges free, natural frequencies Vibration mode shapes, Finite cosine integral transform 1 INTRODUCTION The vibrations of rectangular thin plates with various boundar conditions are of much importance in all the fields of civil, mechanical, and aerospace engineering. To conduct an accurate free vibration analsis of rectangular plates is necessar for controlling the resonance thus ensuring the safet of plates. Actuall, the vibrations of rectangular thin plates have been extensivel investigated for man ears. The related publications can be counted in thousands, like in Leissa (1993). A literature surve reveals that most previous investigations have dealt with a scheme or technique that is onl suitable for a particular tpe of boundar condition. Due to the mathematical complexit of the situation, it is well known that the analtical solutions are generall available onl for plates that are simpl supported along at least one pair of opposite edges. Leissa gave a surve of research on rectangular plate problems up to 197, like in Warburton (1954) and Leissa (1973). The further overview up to the beginning of this centur is presented in Warburton (1979) and Warburton & Edne (1984). One of the most commonl used methods in free vibration analsis of plates is the Raleigh Ritz energ technique, where appropriate functions associ-

2 Y. Zhong et al. / Vibration of Plate on Foundation with Four Edges Free b Finite Cosine Integral Transform Method 855 ated with various boundar conditions are chosen to describe the lateral deflection of the deformed plates. The chosen functions normall do not satisf both the governing differential equations and boundar conditions. Gorman used the superposition technique to solve approximatel free vibration problems of plates for various geometries and boundar conditions, like in Gorman (198) and Gorman (1982). A set of static beam functions was used to determine the natural frequencies of elasticall restrained plates, like in Bapat et al (1988) and Zhou (1996). Hurlebaus et al. (21) have extended the Fourier series solution to the problems with more complicated boundar conditions than the simpl supported one. The other numerical approaches such as the finite element method as in Yang (1972) and boundar element method as in Zafrang (1995) were usuall adopted b man researchers to analze the plate on elastic foundation. Integral transform is one of the effective approaches to obtain the analtical solutions of some partial differential equations used in elasticit, like in Sneddon (1972). This method has often been utilized to analze some structural engineering problems, like in Sneddon (1981). However, based on the author s knowledge, there are no reports on using the finite integral transform to analze the rectangular plate on elastic foundation, like in Zhong et al (29), Li et al (29), Li et al (211) and Li et al (213). In this paper, the double finite cosine integral transform method is adopted to acquire the theoretical solutions of eigenfrequncies and vibration modes for the rectangular thin plate on foundation with four edges free. In the analsis the elastic foundation was modeled b the Winkler elastic foundation. Because it onl uses the basic dnamic elasticit equations of the thin plate on elastic foundation and there is no need to select the deformation function arbitraril, the developed solution is reasonable. In order to proof the correction of formulations, the numerical results are presented to compare with those from other references. 2 VIBRATION OF PLATE ON FOUNDATION AND INTEGRAL TRANSFORM According to the theor the classical Kirchhoff plate, the governing equation of motion for an unloaded plate on the foundation is 4 w x w w 2 + k 4 D w(x,,t) + ρh D 2 w t 2 = (1) where D = Eh 3 / 12(1 v 2 ) is the flexural rigidit of plate. In which E is Young s moduli. Also is Poisson s ratios. h and ρ are the thickness and the densit of plate. w(x,,t) is the out-ofplane displacement and k is the reaction coefficient of foundation. Assuming a harmonic vibration, one wa write w(x,,t) = W (x, )Sinωt (2) where W (x, ) is the shape function describing the modes of the vibration and ω is the natural circular frequenc of the plate. Substitution of Eg.(2) into Eg.(1) gives Latin American Journal of Solids and Structures 11 (214)

3 856 Y. Zhong et al. / Vibration of Plate on Foundation with Four Edges Free b Finite Cosine Integral Transform Method 4 W x W W + λw = 2 (3) 4 where λ = k ρhω 2 D In order to solve the partial differential equation (3), the double finite cosine integral transform approach [11] is exploited. If f (x, ) is a function of the two independent variables x and, defined on the square < x < a, < < b, the definition of double finite cosine integral transform is presented b the equation a b f (m,n) = f (x, )cosα m dxd (4) The inversion formula can be derived as f (x, ) = 1 ab f (,) + 2 ab + 4 ab f (m,)cosα m x + 2 ab f (m,n)cosα m f (,n)cosβ n (5) where α m = mπ / a and β n = nπ / b. a and b are the length and the width of the plate respectivel. The double integral transform of the first partial derivative term appeared in Eg.(3) ma readil be written as b 3 W ( 1) m x 3 a b 4 W x cos α 4 m dxd = α 4 m W (m,n) + x=a 3 W 2 α x 3 m x= ( 1) m W cosβ x x=a x n d x= (6) and the third term can be written as Latin American Journal of Solids and Structures 11 (214)

4 Y. Zhong et al. / Vibration of Plate on Foundation with Four Edges Free b Finite Cosine Integral Transform Method 857 a 3 W ( 1) n 3 a b 4 W cos α 4 m dxd = β 4 n W (m,n) + =b 3 W 3 = 2 W β n ( 1) n W cosα =b m x dx = (7) The second term is split into two parts. The first part considers the partial derivative with respect to first a b 4 a W cos α 2 m dxd = ( 1) n W cosα =b m xdx + = b 3 W ( 1) m 3 W cosβ x 2 x 2 n d + α 2 m β 2 n W (m,n) x=a x= (8) while the second part considers the partial derivative with respect to x first a b 4 b W cos α 2 m dxd = ( 1) m W cosβ x x=a n d + x= 3 W ( 1) n 3 W =b = cosα x dx + α 2 m m β 2 n W (m,n) a (9) Substitution of equations (6-9) into equation (3) leads to a (α 2 m +νβ 2 n ) ( 1) n [(α m 4 + 2α m 2 β n 2 + β n 4 ) + λ]w (m,n) = +(α m 2 +νβ n 2 ) b (β 2 n +να 2 m ) ( 1) m x +(β n 2 +να m 2 ) a b + 2 W 2 =b + 2 W 2 ( 1) n W cosα =b m xdx = + 2 W 2 x=a x + 2 W 2 ( 1) m W cosβ x x=a x n d x= = x= cosα m x dx \ cosβ n d (1) Latin American Journal of Solids and Structures 11 (214)

5 858 Y. Zhong et al. / Vibration of Plate on Foundation with Four Edges Free b Finite Cosine Integral Transform Method The boundar conditions of a free plate are [15] Q x = D x + 2 W 2 = at x = and x = a (11) M x = D( 2 W + v 2 W 2 ) = at x = and x = a (12) M x = D(1 v) 2 W x = at x = and x = a (13) Q = D + 2 W 2 = at = and = b (14) M = D( 2 W 2 + v 2 W ) = at = and = b (15) M x = D(1 v) 2 W x = at = and = b (16) Of cause, there is another simplified expression of the boundar conditions for a free plate. Substituting the boundar conditions that are described b equation (11) and equation(14) into equation(1), one can obtain +(α m 2 +νβ n 2 ) +(β n 2 +να m 2 ) [(α m 4 + 2α m 2 β n 2 + β n 4 ) + λ]w (m,n) = a b ( 1) n W cosα =b m xdx = ( 1) m W cosβ x x=a x n d x= (17) Because the right-hand side of the equation(17)is definite integral, it is the constant. Let a W = ( 1) n W b cosα =b m xdx ; = ( 1) m W cosβ = x x=a x n d x= Therefore, equation (17) can be written as Latin American Journal of Solids and Structures 11 (214)

6 Y. Zhong et al. / Vibration of Plate on Foundation with Four Edges Free b Finite Cosine Integral Transform Method 859 W (m,n) = I (α 2 +νβ 2 m m n ) + (β 2 n +να 2 m ) (α 4 m + 2α 2 m β 2 n + β 4 n ) + λ (18) Substitution of equation (18) into equation (5) gives W (x, ) = (C m + D m J )cosα m x + (C n I + D n )cosβ n +2 C mn ) cosα m (19) I where C mn = m (α 2 m +νβ 2 n ) (α 4 m + 2α 2 m β 2 n + β 4 n ) + λ ; D = (β 2 n +να 2 m ) mn (α 4 m + 2α 2 m β 2 n + β 4 n ) + λ It is clear that the equation(19)can meet the boundar conditions described b equations (11),(13),(14) and (16). From the remaining boundar conditions presented b equations (12) and (15), one can obtain α 2 m (C m + D m J ) + v (β 2 n C n I + D on )cosβ n +2 (α 2 m + vβ 2 n )( C mn )cosβ n = (2) ( 1) m α 2 m (C m + D m J ) + v β 2 n (C n I + D n )cosβ n +2 ( 1) m (α 2 m + vβ 2 n )( C mn )cosβ n = (21) β 2 n (C n I + D n ) + v α 2 m (C m + D m J )cosα m x +2 (β 2 n + vα 2 m )( C mn )cosα m x = (22) ( 1) n β 2 n (C n I + D n ) + v α 2 m (C m + D m J )cosα m x +2 ( 1) n (β 2 n + vα 2 m )( C mn )cosα m x = (23) Latin American Journal of Solids and Structures 11 (214)

7 86 Y. Zhong et al. / Vibration of Plate on Foundation with Four Edges Free b Finite Cosine Integral Transform Method Equations(2)and equation(21)make α 2 m (C m + D m J ) + 2 (α 2 m + vβ 2 n )(C mn ) cosβ n = (24),3,5,3,5 Similarl, equations (23) and (24) make β 2 n (C n I + D n ) + 2 (β 2 n + vα 2 m )(C mn ) cosα m x = (25),3,5,3,5 Each coefficient of the cosα m x and cosβ n has to be vanish. What follows is a sstem of homogeneous algebraic equations α 2 m (C m + D m J ) =,3,5 (α 2 m + vβ 2 n )(C mn ) =,3,5 β 2 n (C n I + D n ) =,3,5 (β 2 n + vα 2 m )(C mn ) =,3,5 (26) Eqs. (26) forms the sstem of four groups of an infinitel large number of linear equations in terms of the unknowns,, I and J. Non-trivial solution of those equations requires the coefficient matrix to vanish to an desired degree of accurac. Non-trivial solution of Eqs. (26) requires the coefficient matrix to vanish. From this determinant the eigenfrequencies of the plate are calculated. The associated vibration modes are given b equation(19)after inserting the eigenfrequencies. The infinite series that occur in the corresponding equations (see Eqs. (19) and (26)) have been evaluated without an truncation using MATLAB [15] ; this has been done b specifing the upper limit of the summation index as infinit. The evaluation is exact since corresponding closed-form equivalents are automaticall substituted in MATLAB. 3 NUMERICAL RESULTS The first page includes the title of the paper, the names of the authors and their institutions, and an abstract. The following vertical spacing must be used: The previousl derived expressions are applied to a rectangular thin plate on the foundation with four edges free. The geometrical and Latin American Journal of Solids and Structures 11 (214)

8 Y. Zhong et al. / Vibration of Plate on Foundation with Four Edges Free b Finite Cosine Integral Transform Method 861 material properties are given as a = b = 4. m, ν =.15, E = MPa, h =.2 m, k = N/m 3 and ρ = 175 kg/m 3 In order to make the comparison with other method, the eigenfrequencies and the vibration modes are computed b the superposition method [7] and b the present approach. The calculation results are shown in Table 1. It is obvious that the results b two different methods are in excellent agreements. This also validates the present approach is correct. Fig. 1-4 illuminates the corresponding vibration modes respectivel. Table 1 the nature frequencies of a plate Frequencies (Hz} Superposition [7] Present paper Figure 1 the 11 order vibration mode Figure 2 the 12 order vibration mode Latin American Journal of Solids and Structures 11 (214)

9 862 Y. Zhong et al. / Vibration of Plate on Foundation with Four Edges Free b Finite Cosine Integral Transform Method Figure 3 the 22 order vibration mode Figure 4 the 44 order vibration mode 4 CONCLUSIONS The present paper shows that the eigenfrequencies and the vibration modes of a classical Kirchhoff rectangular plate on foundation with four edges free can be calculated b the integral transform method. The approach is based on the finite cosine integral transform for the classical thin plate equation. One of the advantages of the method is that the results converge quickl and can be calculated with desired accurac. The other one is that after finishing the analtical derivation for the frequenc determinant the calculation of eigenfrequencies and the vibration modes for given material data and geometr becomes straightforward. The present method is convenient for sensitivit studies. As the analtical solutions for the computation of eigenfrequencies and the vibration modes of a thin plate on the foundation with four edges free are of paramount im- Latin American Journal of Solids and Structures 11 (214)

10 Y. Zhong et al. / Vibration of Plate on Foundation with Four Edges Free b Finite Cosine Integral Transform Method 863 portance for man applications such as in the design of building foundations and the rigid pavements of highwa and airport, the present approach of analsis provides an efficient procedure for accurate results which should be of academic and practical importance. References Bapat, A. V., Venkatramani, N., Suranaraan, S., (1988), Simulation of classical edge conditions b finite elastic restraints in the vibration analsis of plates, Journal of Sound and Vibration 12: Gorman, D.J., (198), A comprehensive stud of the free vibration of rectangular plates resting on smmetricall distributed uniform elastic edge supports, Journal of Applied Mechanics 56: Gorman, D.J., (1982), Free Vibration analsis of rectangular plates Elsevier North Holland, Inc. Hurlebaus, S. L., Gaul, J., Wang, T. S., (21), An exact series solution for calculating the natural frequencies of orthotropic plates with completel free boundar, Journal of Sound and Vibration 244: Leissa, A.W., (1993), Vibration of Plates, Acoustical Societ of America. Leissa, A.W., (1973), The free vibrations of rectangular plates, Journal of Sound and Vibration 31: Li, R., Zhong, Y., Tian, B., Liu, Y.M., (29), On the finite integral transform method for exact bending solutions of full clamped orthotropic rectangular thin plates. Applied Mathematics Letters 22: Li, R., Zhong, Y., Tian, B., (211), On new smplectic superposition method for exact bending solutions of rectangular cantilever thin plates. Mechanics Research Communications 38: Li, R., Zhong, Y., Li, M.L., (213), Analtic bending solutions of free rectangular thin plates resting on elastic foundations b a new smplectic superposition method. Proceedings of the Roal Societ A 46: Sneddon, Ian. H., (1972), The use of integral transforms McGraw-Hill, Inc. Sneddon, Ian. H., (1981), The application of integral transform in elasticit McGraw-Hill, Inc. Warburton, G.B., (1954), The vibrations of rectangular plates, Proceeding of the Institute of Mechanical Engineers, Series A 168: Warburton, G.B., (1979), Response using the Raleigh Ritz method, Journal of Earthquake Engineering and Structural Dnamics 7: Warburton, G.B., Edne, S.L., (1984), Vibrations of rectangular plates with elasticall restrained edges, Journal of Sound and Vibration 95: Yang, T.Y., (1972), A finite element analsis of plate on two parameters foundation model, Computer and Structure 2: Zafrang, A.E., (1995), A new fundamental solution for boundar element analsis of thick plate on Winkle foundation, Int. J. Num. Eng. 38: Zhong, Y., Li, R., Liu, Y.M., Tian, B., (29), On new smplectic approach for exact bending solutions of moderatel thick rectangular plates with two opposite edges simpl supported. International Journal of Solids and Structures 46: Zhou, D., (1996), Natural frequencies of rectangular plates using a set of static beam functions in the Raleigh Ritz method, Journal of Sound and Vibration 189: Latin American Journal of Solids and Structures 11 (214)