Although the main idea of the paper is an introduction of the Polya counting theory to an engineering counting problem that may be encountered in applied mechanics, this author equally appreciates the interest shown by Professor Bert in the proposed Ritz method to calculate natural frequencies of anisotropic plates with arbitrary boundary conditions. Professor Bert raised two constructive comments that are answered in order.
The first comment is that the present author took up only three classical boundary conditions (i.e., free, simply supported, and clamped edges) in numerical examples and did not consider the fourth boundary condition of a guided or sliding edge with zero effective shear force and bending moment. The function (20) in the x and y-direction (1) is not applicable in its direct form to the fourth boundary condition but it is widely accepted that the fourth condition is not as important as the first three ones. It may be possible to apply the present function to the fourth boundary condition by adding a constant term to give a constant displacement caused by the guided or sliding edge and also constraining the slope at the edge.
The second comment, which is more important, is on convergence rates of the present solution applied to anisotropic plates. Before commenting on that, I have to make it clear that the caption in Table 2 of 1 was erroneous. The convergence result in the table was for a specially orthotropic square plate. (i.e., anisotropic plate with a fiber orientation angle θ=0 deg), not for skew orthotropic square plates (θ=30 deg). This was obvious that the converged values in Table 2 are in exact agreement with those of specially orthotropic plates in Table 5 of 1.
| Convergence of frequency parameters Ω of diagonally orthotropic square plates (θ=45 deg, , and |
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| Convergence of frequency parameters Ω of diagonally orthotropic square plates (θ=45 deg, , and |
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Convergence of the fundamental frequency parameter by Ashton’s Ritz method. Whiteney’s Fourier, Bert’s DQM, and the present Ritz method. (Data are replotted from Fig. 4 in 2.)
Convergence of the fundamental frequency parameter by Ashton’s Ritz method. Whiteney’s Fourier, Bert’s DQM, and the present Ritz method. (Data are replotted from Fig. 4 in 2.)
The present author has an opinion that use of the Ritz method with (modified) polynomial functions yields very accurate upper bounds with advantages in applying to arbitrary boundary conditions and in computation time, when it is used with the following points in mind (3).
• The first few terms of the polynomial (say, ten) give rapid convergence of the solution, but the use of higher order polynomials (say, 20 or more terms) tends to make the eigenvalue equation numerically unstable, unless it is somehow modified.
• The plate region considered should have a regular plan, such as rectangular and elliptical plates. For plates of irregular geometry, e.g., with cutouts or L-shaped plates, the solution accuracy deteriorates.
In summary, the Ritz method with modified polynomials is a valuable and recommendable approach. The only problem is that because it is very easy to use and guarantees good accuracy, one cannot escape this easiness and does not create new methodology.
