Free Vibration Analysis Of Rectangular Plates Using Higher Order Finite Layer Method

In this paper a higher order finite layer formulation based on the auxiliary nodal surface (ANS) technique for a free vibration analysis of rectangular plates is presented. The free vibration analysis has been performed using the generalized jacobi iteration method, thus finding the natural frequencies and modes of vibration. In this study, authors consider the effects of different types of material and dimensions of the plate on its free vibration analysis. Many examples have been studied to show the good performance of the higher order finite layer with one ANS for free vibration analysis of plate.

Due to scientific and technological development of engineering, rectangular plates have been widely used in different engineering fields. The rectangular plates are commonly used as structural components in aerospace, mechanical, nuclear, marine and structural engineering [1].
A study on the free vibration analysis of plates should be made first before the forced vibration analysis in order to investigate vibrational characteristics of plate (fundamental natural frequency of plate) and to prevent the resonance from occurring. The vibration behavior of structure defined as special spectra consist of an infinite number of frequencies and modes which can be found by knowing geometrical shape, mass distribution, stiffness and boundary conditions of the plate [2].
For years, there has been a large amount of research work on the free vibration analysis of plate structure. In 1970, Sinivas [3] developed a three-dimensional linear, small deformation theory of elasticity solution by the direct method for the free vibration of simply supported thick rectangular plates. The free vibration of isotropic plates with various types of boundary conditions, for three different aspect ratios was investigated by Cheung and Chakrabarti [4] in 1972 by using lower order finite layer technique. The resultant frequencies for the smallest thickness/span ratio are close to that existing for thin plates, while frequencies for higher thickness/span ratios tend to be of lower value. The closed-form solutions are obtained in 1999 by Meunier and Shenoi [5] for finding the natural frequencies of sandwich plate panels by using higherorder shear deformation theory. Petrolito and Golley [6] in 2000 develop a finite strip-element for the vibration analysis of thick plates. The method uses a combined polynomial and trigonometric interpolation scheme that enables all boundary conditions to be correctly treated. The global equations are derived in the usual manner of the finite element method, and natural frequencies of vibration can be found by solving a linear eigenvalue problem. In 2003, Sheikh et al. [7] analyzed the free vibration for both the thin and thick plates. The solution depends on Reissner-Mindlin theory by adopting a new type of triangular element with three nodes at corners, three others at mid of the three sides and one internal node at the center of element.
In the present study, a higher order finite layer with a second order polynomial has been used for the free vibration analysis of plates. The higher order layer produced by introducing an auxiliary nodal surface at mid the distance between the upper and lower surfaces of the lower order layer.

:
The plate was divided into a number of horizontal layers in the direction of the thickness of a layer. These layers may be of lower or higher order and for finite layer formulation the layers may be imagined as nodes as shown in (Fig. 1). By selecting functions satisfying the boundary conditions in two directions, the philosophy of the finite strip method can be extended to layered systems. The resulting method is called the finite layer method (FLM), and this method is useful for layered materials, rectangular in plan form [8].
The general form of the displacement function of a layer element is given as a product of polynomials in the thickness direction and continuously differentiable smooth series in the other two directions.
With the stipulation that such series (called basic functions) should satisfy the boundary conditions at the edges of the layer thus a three-dimensional problem is reduced to one-Dimensional problem.
For a layer element the generalized displacement functions can be constructed as [9, 10]: For a simply supported case owing to the othogonality of trigonometric series, the terms of the series can be decoupled and may be solved separately.

:
As it has been mentioned, the displacement function of any layer has the form as in equation (6) Many computer programs in FORTRAN language have been developed to solve numerical problems in free vibration analysis for rectangular plates. The original programs are those provided in 1991 by Majeed [10] who used it in static analysis for rectangular plates. These programs are suitable for dynamic analysis of rectangular plates which have different boundary conditions. : In order to demonstrate the capability and efficiency of the formulation presented and the reliability of the higher order finite layer with one auxiliary nodal surface in dealing with free vibration analysis of plates, typical examples have been studied.
: A thin plate simply-supported at the four edges is analyzed by the higher order finite layer taking different aspect ratio (a / b). Natural frequencies for the lowest six modes of vibration obtained and compared with the results reported by Leissa [11] and presented in Table- Fig. 2), (Fig. 3) and (Fig. 4) show the comparison between the values resulted from present method and those found by Lee [12].

:
Results are presented in terms of the natural frequency ω (rad/sec). The plate has the same material and geometrical properties in the previous example except the thickness will be variable here.
In Figs. (5) and (6) the relation between the lowest natural frequency and the thickness/span ratio (h/a) are illustrated for two plates having different boundary conditions, the results are compared with the available data computed for the same plates by Ref. (13).
In order to compute natural frequencies of a simply supported orthotropic square plate. Elastic constants as that given by Srinvivas et al 1970 [3]  There were many advantages from using finite layer method specially for analyzing the plates which compound from different layers. The higher order finite layer method is used successfully to study the vibration characteristics by finding the natural frequencies of a rectangular plate subjected to different edge conditions. Computed frequencies were found to agree very well with the corresponding results available in the literature.
It is clear from the results that the natural frequency of the plate will be increased by increasing the aspect ratio (a/b) and the plate thickness.
3. Srinivas S., Jaga Rao C.V., and Rao A.K. "An exact analysis for vibration of simply-supported homogeneous and laminated thick rectangular plates", .
4. Cheung Y.K., and Chakrabarti S. "Free vibration of thick layered rectangular plates by a finite layer method", .
7. Sheikh A.H., Dey P. and Sengupta D. "Vibration of thick and thin plates using a new triangular element", . 12. Lee S.J. "Free vibration analysis of plates by using a four-node finite element formulated with assumed natural transverse shear strain", .
13. Qian L.F., Batra R.C. and Chen L.M. "Free and forced vibrations of thick rectangular plates using higher order shear and normal deformable plate theory and meshless Petrov-Galerkin (MLPG) method", .
14. Kant T. and multiplayer plates based on higher order refined theories", Swaminathan K. "Free vibration of isotropic, orthotropic and .