Natural Frequencies of Isotropic Rectangular Plates in Improved Accuracy

The objectives of this paper are to re-visit one of the most important vibration problems and to present comprehensive lists of accurate natural frequencies for isotropic thin rectangular plates. For this purpose, a simple yet very accurate analytical approach is described to study the free vibration of the plates. In numerical computations, convergence and comparison studies are conducted to collaborate accuracy of the present solution and to demonstrate the improvement of numerical solutions in percentage from previous standard results. Twenty-one tables are then provided to list the lowest six frequency parameters for all possible combinations of three typical edge conditions (clamped, simply supported and free edges). Each table is given in five significant figures for non-Levy type problem and in six significant figures for Levy type problem (i.e., plate with two opposite edges simply supported). These results are presented for five different aspect ratios to follow the same format of the previous standard reference.


Introduction
The free vibration of isotropic thin rectangular plates has been one of the most important problems for a long period of time in mechanical vibration. References up to 1970 on vibrations of general plate shapes are summarized in the famous monograph [1], and the wide coverage for natural frequencies of isotropic rectangular plates was made in 1973 for all possible twenty-one combinations of boundary conditions and five aspect ratios [2]. The numerical results in this reference have been widely accepted as accurate and comprehensive data, where Ritz method with beam functions is used for sixteen combinations not having opposite edges simply supported and the exact solution is employed for five remaining combinations having two opposite edges simply supported. Because of reasonable accuracy, the results are cited as the standard and reliable data in the vibration design book [3].
Besides these classical references, this topic has a long history to date back to an early work by Young in 1950 [4], and since then many papers have appeared including a series of Gorman's work [5][6][7] by using a method of superposition. Those related studies up to the year of 1980 are listed in the author's work [8]. The development in the 1980's is summarized also in review papers [9,10]. More recently, two papers [11,12] presented their approaches to solve this problem.
When one extends the problem to broader sets of the boundary conditions, namely clamped, simply supported and free edges (denoted by capital letters, C, S and F, respectively), there are 3 4 =81 combinations for a rectangular plate fixed in the space. Physically, however, some plates have the identical natural frequencies, for example, in C-S-F-F and C-F-F-S square plates (note that the first letter indicates Edge(1) and the rest in counterclock wise direction). Polya counting theory is applied to solve for the exact number of combinations in Ref. [13], and recently this approach is extended to calculate the number of physically meaningful combinations for generally shaped plates [14].
In this paper, natural frequencies of isotropic rectangular plates are listed in comprehensive way to serve as a new standard data in mechanical design, although the frequencies of the plates with typical edges, such as totally clamped or cantilevered plates, were already solved. For this purpose, Ritz method is employed by using polynomial functions with boundary index, not by the beam functions [2]. With this, better accuracy is realized as resulting in lower values in the upper-bound solutions.

Outline of Analytical Method
A previous solution is used here as in [13] based on the method of Ritz within the classical thin plate theory. This analysis-based solution has a low computational cost and easiness in varying combination in boundary conditions, in contrast to numerical methods such as the finite element method. Figure 1 shows a geometry of rectangular plate and the coordinate system, and the dimension of the plate is given by a×b×h (thickness).
The relation between stress and strain is written in 11 where the elements are given for isotropic material by If one considers the small amplitude (linear) free vibration of a thin plate, the deflection w may be written by where W is the amplitude and ω is a radian frequency of the plate. Then, the maximum strain energy due to the bending is expressed by      (5) where the Dij are the bending stiffnesses and    is a curvature vector The maximum kinetic energy is given by 22  The next step in the Ritz method is to assume that the amplitude as where Amn are unknown coefficients, and () m X  , () n Y  are the functions modified later so that any kinematical boundary conditions are satisfied at the edges.
After substituting Eq.(9) into the energies (5) and (7), the stationary value is obtained by where an integral I is the products The analytical procedure developed thus far is a standard routine of the Ritz method, and the modification is explained next so as to incorporate arbitrary edge conditions into the amplitude W(ξ,η). In the traditional approach, for example, using the beam functions for ) ( m X and ) ( n Y , many different products of regular and hyper trigonometric functions exist for arbitrary conditions and it is difficult to make a unified subroutine to calculate all of the various kinds of integrals.
The present approach introduces a kind of polynomial where B1, B2, B3 and B4 are "boundary indices" [13,15,16] which are added to satisfy the kinematical boundary conditions and are used in such a way as Bi=0 for F (free edge), 1 for S (simply supported edge) and 2 for C (clamped edge). To the C-S-F-F plate, for instance, B1=2, B2=1 and B3=B4=0 are applied. With the boundary indices Bi's and Eqs. (14), the method of Ritz can accommodate arbitrary sets of the edge conditions, and the integrals (12) can be exactly evaluated.

Convergence and Comparison of the Solution
In numerical studies, the material constants are assumed for isotropic materials, and Young's modulus E and Poisson's ratio ν are included in the frequency parameters Ω in Eqs. (8). Poisson's ratio still affects values of the frequency parameters, and a constant of ν=0.3 is used throughout in the paper, except for the comparison in Table 2 with results obtained by another author who used ν=0.333 [5] [7]. For thin plates, a value of plate thickness does not affect the frequency parameters, unlike in the analysis of shallow shells (panels) [15,16]. Table 1 presents convergence study of frequency parameters with increase of series terms in Eq. (9). Since the amplitude function satisfies kinematical conditions exactly, this Ritz method yields upper-bound solutions, and therefore all the frequency parameters converge from above. For uniform edge conditions, such as C-C-C-C, S-S-S-S and F-F-F-F plates, the frequency parameters converge within five significant figures even for small number of terms as in 8×8 terms. When boundary conditions are mixed in one plate, however, the converge speed deteriorates as seen C-F-F-F plate. The worst result with respect to the convergence is given for C-C-F-F plate, where numerical instability occurs for the 14×14 solution, but these are still exceptions among twenty-one sets of edge conditions. For these two cases, the results in the following tables are obtained by using 12×12 solutions. Table 2 is a comparison study with values of Gorman for C-C-C-C [6], C-F-F-F (ν=0.333) [5] and F-F-F-F plates (ν=0.333) [7]. The exact values can be obtained for S-S-S-S plate. In the all results presented in the table, good agreement is obtained with the present results. A method of Gorman, known as superposition method, is well known to yield accurate numerical results, although the formulation is a little cumbersome. Based on the convergence and comparison studies in these two tables, the accuracy of the present solutions is well established.

Comprehensive Results for Non-Levy Type Problem
Tables 3 presents lists of frequency parameters in five significant figures of totally clamped rectangular plates (C-C-C-C plate) for the lowest six modes. Aspect ratio is varied from a/b=0.4, 2/3, 1(square), 1.5 and 2.5. This table uses the same format as in Ref. [2] and these results are presented by using the 12×12 solutions, together with the frequency parameters obtained by Leissa [2]. The difference is also calculated for each pair by using  Table 3 Frequency parameters Ω=ωa (ρh/D ) 1/2 for C-C-C-C plates. Table 5 Frequency parameters Ω=ωa (ρh/D ) 1/2 for C-C-C-F plates.  Table 4 Frequency parameters Ω=ωa (ρh/D ) 1/2 for C-C-C-S plates. Table 6 Frequency parameters Ω=ωa (ρh/D ) 1/2 for C-C-S-S plates.  Table 7 Frequency parameters Ω=ωa (ρh/D ) 1/2 for C-C-S-F plates. Table 9 Frequency parameters Ω=ωa (ρh/D ) 1/2 for C-S-C-F plates.  Table 8 Frequency parameters Ω=ωa (ρh/D ) 1/2 for C-C-F-F plates.  Table 11 Frequency parameters Ω=ωa (ρh/D ) 1/2 for CSFF plates.  Table 12 Frequency parameters Ω=ωa (ρh/D ) 1/2 for C-F-C-F plates.  Table 15 Frequency parameters Ω=ωa (ρh/D ) 1/2 for S-S-F-F plates.  Table 19 Frequency parameters Ω=ωa (ρh/D ) 1/2 for S-C-S-C plates.  Table 20 Frequency parameters Ω=ωa (ρh/D ) 1/2 for S-C-S-S plates. As seen in the table, all the differences are between -0.10% -0.02%, and all the present values are very slightly lower than those in Ref. [2], and due to the fact that all the present solution satisfies exactly the kinematical condition at the edges, it is observed that the present results are more accurate and closer to the exact values (assuming that they exist). The mode numbers (i.e., number of half-waves in the mode shape) are listed for each frequency in Ref. [2].
Tables 4 to 17 are in the same format as in Table 3, and the frequency parameters are tabulated for fourteen sets of edge conditions, where the exact solutions are not available due to the condition that two opposite edges are not simply supported (non-Levy type problem).
In all pairs of the present and reference values (i.e., 450 pairs= (5 aspect ratios)×(6 modes)×(15 tables) ), all the differences are negative (the present values are lower) and the differences are all within one percent.

Comprehensive Results for Levy Type Problem
It is widely known that the exact solution is available for vibration of isotropic rectangular plates when a pair of opposite edges are simply supported. The solution is already presented in Ref. [2]. This means that it is not necessary to use approximate method to calculate natural frequencies. In this paper, however, six tables are prepared in the same format to demonstrate that this Ritz solution can provide frequency parameters in the accuracy with same degree. The present results are given with six significant figures, and compared to the exact values in Ref. [2].
Tables 18 to 23 list up the frequency parameters for rectangular plates with S-S-S-S, S-C-S-C, S-C-S-S, S-C-S-F, S-S-S-S-F and S-F-S-F, respectively. Among 180 pairs of the present and exact frequencies (i.e., 180 pairs= (5 aspect ratios)×(6 modes)×(6 tables) ), almost all pairs show the exact match with six significant figures, only with four exceptions showing very slight difference over 0.01 percent.
Table 24 provides classification of physically meaningful sets of edge conditions to give the identical natural frequencies. For example in this table, " Table 4" gives the frequency parameters of C-C-C-S plate and the results are identical with S-C-C-C, C-S-C-C and C-C-S-C.  Table 3 C1 C-C-C-C Table 4 C2 C-C-C-S S-C-C-C, C-S-C-C, C-C-S-C Table 5 C3 C-C-C-F F-C-C-C, C-F-C-C, C-C-F-C Table 6 C4 C-C-S-S S-S-C-C, S-C-C-S, C-S-S-C Table 7 C5 C-C-S-F F-S-C-C, F-C-C-S ,S-F-C-C, S-C-C-F, C-F-S-C, C-S-F-C, C-C-F-S Table 8 C6 C-C-F-F F-F-C-C, F-C-C-F, C-F-F-C Table 9 C7 C-S-C-F F-C-S-C, S-C-F-C, C-F-C-S Table 10 C8  Table 12 C10 C-F-C-F F-C-F-C Table 13 C11 Table 14 C12 Table 15 C13 Table 16 C14 Table 17 C15 F-F-F-F Table 18 A1 S-S-S-S Table 19 A2 S-C-S-C C-S-C-S Table 20 A3 S-C-S-S S-S-S-C, S-S-C-S, C-S-S-S Table 21 A4 S-C-S-F F-S-C-S, S-F-S-C, C-S-F-S Table 22 A5

Table 23 A6 S-F-S-F F-S-F-S
(1) Plates not having two opposite edges simply supported (2) Plates having two opposite edges simply supported Table 24 Correspondance of the present tables with those in Ref. [2].  Figure 2 presents the distribution of lowest six frequency parameters of square plates for twenty-one sets of boundary conditions. The boundary conditions are aligned in the increasing order of the fundamental frequencies, and the fundamental frequencies may appear almost linearly increasing. Among them, SFFF and FFFF plates have rigid body motions and only the selfequilibrium modes (i.e, modes with some nodal lines) exist. Also, SSSS, CCSS and CCCC plates may appear to have only four distinct modes because two modes include a pair of identical frequencies, and FFFF plate does five distinct modes due to the same reason.

Conclusions
The aim of this study was to establish a new standard for natural frequencies of isotropic rectangular plates and to summarize the lists of frequencies for all possible combinations of three classical boundary conditions (i.e., free, simply supported and clamped edges). After the accuracy of the Ritz solution was established for non-Levy type problems (i.e., plates not having two opposite sides simply supported), these frequency parameters were given in five significant figures in the lists and were compared to the other list [2]. The differences were all less than one-percent. For Levy type (i.e., plates with two opposite sides simply supported) problems where the exact solutions are available, two sets of frequency parameters were in good agreement showing the differences lower than 0.001 percent.
The present results for most commonly needed information in mechanical vibration will be valuable standard for comparison with numerical methods in future studies and design data.