Numerical Solution of Vibrating Double and Triple-Panel Stepped Thickness Plates

A certain structural optimization of a panel may be achieved by possessing a suitable variation of thickness of plate structure. The importance of types of structures increases when the aero-space and underground structures are considered. Survey of the literature on the flexural vibration of thin plate reveals that the work on this topic has been mainly confined to plates with uniform thicknesses. A relatively few studies have been published on the free vibration of isotropic plates of stepped thicknesses. During the past few decades, many researchers were devoted to mathematical modeling and numerical solution for static elastic multi-structure problems. Since vibration analysis of elastic structures plays important roles in engineering applications, this paper is concerned with Finite Strip-Transition Matrix method (FSTM) for vibration analysis of paneled stepped thickness plate.

Eh ( D) (12 ) ν Regional flexural rigidity of plate Introduction A certain structural optimization of a panel may be achieved by possessing a suitable variation of thickness of plate structure. The importance of types of structures increases when the aero-space and underground structures are considered. Survey of the literature on the flexural vibration of thin plate reveals that the work on this topic has been mainly confined to plates with uniform thicknesses. A relatively few studies have been published on the free vibration of isotropic plates of stepped thicknesses. During the past few decades, many researchers were devoted to mathematical modeling and numerical solution for static elastic multi-structure problems. Since vibration analysis of elastic structures plays important roles in engineering applications, this paper is concerned with Finite Strip-Transition Matrix method (FSTM) for vibration analysis of paneled stepped thickness plate.
Although, Chopra [1] has attempted an exact solution for a simply supported stepped thickness plate with two panels, Warburton [2] pointed out that continuity conditions used by Chopra [1] were incorrect. He presented a modified analytical technique for two paneled stepped plate with different properties of orthotropic. Sakata [3] proposed an approximate formula for estimating the fundamental nature frequency of an isotropic plate with stepped thickness from the natural frequencies of the isotropic plate reduced from the orthotropic one. Recently, Farag [4] applied a closed form solution for vibrating surfaces of partially restrained and clamped double-panel plates via a power matrix exponential method. Xiang and Wang [5] also studied the exact vibration solutions of stepped rectangular plates. Xiang et al. [6,7] extended these studies in cooperation with others to include the case of stepped rectangular mindlin plate and the case of      stepped circular mindlin plate. Gorman and Singha [8] outlined the vibration analysis of stepped cantilever plate using a supper position method. A numerical approach for rectangular stepped plate with sides restrained against rotation has been proposed by Laura and Filipich [9]. They ensured that vibration of plate with stepped thicknesses was studied in a very few number of literature. Filipich et al. [10] used a simple polynomial coordinate function which identically satisfied the restrained boundary conditions of plate with discontinuous thickness. Modal study via a discrete numerical approach, such as finite element method by Mukherjee and Mukhopadhyay [11], or finite strip method by Cheung and Li [12,13] introduces solution accuracy depending upon the suggested element size. Vibrations of plates with variable thickness were studied by Zanzi and Laura [14], Gutierrez et al. [15], Laura and Gutierrez [16] via different methods. Kaabi and Aksu [17,18] applied a modified method to examine the dynamic behavior of rectangular plates with bilinear variation of thickness. Cheung and Li [19] achieved a simple finite strip method to analyze the hunched continuous bridges. Vibration of orthotropic rectangular plate with free edges in the case of discontinuously varying thickness was discussed by Laura et al. [20]. They extended their studies to include two other boundary conditions different from Laura et al. [21]. Farag [22] applied Finite Strip Transition Matrix method to solve the free and forced vibration problems of uniform thickness plate. In extended work, Farag [23] analyzed the stepped thickness plate analytically by means of the matrix exponential method. Recently, a semi and fully discrete finite element methods for investigating vibration analysis of elastic plate-plate structures are proposed by Junjiam et al. [24]. Semie [25] explained the numerical modeling of thin plates using the finite element method. Sanches et al. [26] studied the dynamic stationary response of reinforced plate by the boundary element method.
The present paper offers a modified numerical method relying on reducing the partial differential equation of motion into an ordinary differential one. A convenient basic function is used to obtain the later equation which is solved by Finite Strip Transition Matrix Method (FSTM) inside each panel of plate. Due to the sudden change of thickness between two adjacent panels, continuity conditions must be satisfied to generate the final solution of the entire plate. The known basic function is expressed in a variable parallel to the steps. The reduced differential equation of rectangular stepped plate is still carrying the other unknown variable in the other plate direction. The restrained boundary conditions are included and employed to derive the solution of the particular cases of clamped and simply supported edges.

Equation of Motion
The isotropic rectangular plates with elastically restrained boundary conditions in the two directions ζ and η of plates are considered here. The plate thickness shown in figure 1 is a uniform in ζ -direction, while it changes suddenly in η -direction at a step i S Referring to the domain of the isotropic region i R , the regional dimensionless partial differential equation of motion of plate vibration is: The different symbols are denoted in the nomenclatures part where the suffix i means that the mentioned magnitudes are calculated inside the region R i .

Method of Solution
The partial differential equation (1) figure 1. The basic function of strip [23], is applied to reduce the partial differential equation of motion into an ordinary differential one. The reduced differential equation is solved numerically by the FSTM as an initial value problem under the proposed initial boundary conditions at =

Reduction technique
The regional displacement of the stepped plate is: ζ is known regional shape function satisfying the plate boundary conditions [23] at = ( 0, 1) ζ and m i (V ( )) η is unknown regional longitudinal function to be determined at = ( 0, 1) η .
Eq. (2) is applied to reduce the partial differentia Eq. (1) Inside the Region = i R ; i 1,2,3 into [4]: Where Consequently, the equation of motion for any panel becomes:   The ordinary differential equation (4) is transformed within a region i R to a linear system of differential equations expressed as [22]: The general solution of the ordinary differential equation is: Where i δ is the width of strip j κ bounded by two nodal lines − j j 1 K ,K inside region i R .
The regional Transition Matrix K i [Y ] is calculated according to the Runge Kutta method such as: Where I is the unit matrix?

Continuity conditions
The Magnitudes of displacement W slope shear v Q must satisfy the continuity conditions at the sudden step S i so that: The symbols ℜ i i , L denote positions immediately after and before the sudden step S i respectively.
is step thickness ratio at step

Results and Discussions
Different cases of plates composed of different panels with unequal thicknesses and panel widths are investigated by the present technique. This study takes in account the variation of the aspect ratios β and various magnitudes of panel thickness ratio  Table 1 with those obtained by Xiang et al. [5] using an exact solution and using closed form solution [4].
The comparisons are available only for the case of double panel square plate with panel width ratio + = The natural frequencey parameters mn λ in the first six modes are obtained in Table 2 for the cases of 3-panels square full clamped plates CCCC with panel width ratio . The results show that the natural frequency parametrer inreases by decreasing 2 α . Table 3 shows the natural frequencey parameters mn λ for the cases of 3-panels square full plates CE R CE R with two opposite edges clamped and other edges ellastically restrained against rotation with restrained coefficient varying from 0 to ∞ . The results are obtained for cases of panel width ratio This case is carried out for 3-panel square plate, SE R SE R , simply supported at two opposite edges and partially restrained against rotation at the other edges ( Similarly, the case of square isotropic stepped plate CSCS with two opposite edges clamped and other edges simply supported is carried out as shown in figures 3-a,b,c. The frequency parameters mn λ are recorded for the case of panel width ratio + =    and aspect ratio β of the intire plate varying from 1 to 2 is investigated as shown in figures 4-a,b,c. The resuls show that the parameter mn λ increases by increasing the aspect ratio β for all recorded modes.

Conclusion
The finite strip transition matrix method FSTM described here involvoes a numerical solution of stepped paneled plate with classical and restrained boundary conditions. This method is a combination between the strip and transition matrix method to solve the vibration problem of stepped plates as an initial value proplem. Transition matrix method is a semi analytical method relying on estmiating the numerical solution of the intial value problem by means of Range Kutta method. The plate domain is divided into paneled regoins consisting of strips bounded by nodal lines. Each strip is governed by the transition matrix formula which transite from one strip to another via nodal vectors until the final edge is reached. Several cases of double and triple panel plates are investigated for the variation of thickness ratio, aspect ratio, panel width ratio and boundary conditions. To show the accuracy of the present method, the results have been summerized and compared with those obtained by other methods. A good exteremely agreement of results is found for all compared cases.