Chapter 3 Vibrations of Completely Free Rectangular Plates
3.3 Convergence Study
,
( yx and the polar coordinate (r,θ) are
2 / 1 2
2 ( ) ]
)
[(x c y b d
r= − + − + − , (3.12)
) (
tan 1
d b y
c x
− +
−
= − −
θ , (3.13)
where b, c, and d are shown in Fig. 3.1.
3.3 Convergence Study
It is one of the typical characteristics of the Ritz method that the obtained frequencies would converge to the exact solutions from the upper bounds if a sufficient number of admissible functions are used. In this section, to verify the accuracy of the solutions and demonstrate the effects of the corner functions on the convergence, convergence studies are presented for completely free square plates (a/b=1.0, as seen in Fig.
3.1) with different notch angles (α=5° or 30°) and notch depths (d/b=0.1,
0.3, or 0.5). The V-notch is located at c/a=0.5. Poison’s ratio υ is taken
equal to 0.3. The numerical results are the nondimensional frequency parameters ωa2 ρh/D for the first five modes. Note that, the first three rigid body modes (zero frequencies) are ignored. The computation was carried out by using FORTRAN programming language with quad precision (34 significant digit accuracy) on a 64-bit computer.Table 3.1 shows the convergence of the frequency parameters for an intact square plate (no V-notch), in which no stress singularities are presented. The frequency data were computed by using polynomial functions with increasing number of terms (I×J) from 3×3 to 10×10.
Note that the frequency parameters for the fourth and fifth modes are exactly identical, which are double roots in an eigenvalue problem. The numerical results are in excellent agreement with those of Leissa (1973), who used beam functions as admissible functions, and those of Filipich and Rosales (2000), who used whole element method. Since the beam functions may not form a complete set of functions, the converged results of Leissa (1973) are larger than the present ones. The present results also show more accuracy than the converged results of Filipich and Rosales (2000). The comparison recognizes the validity of the computation for the part of the polynomial functions.
Table 3.2 shows the convergence of the frequency parameters for a square plate with a very shallow V-notch (d/b=0.03) having large notch angle (α=170°) that causes weak stress singularities at the vertex of the notch. As expected, the admissible functions of polynomials can give
corner functions to the admissible functions only can slightly accelerate the convergence of the numerical solutions for this very shallow, wide angle notch. It demonstrates the validity of the computation for using polynomials and corner functions as admissible functions.
Tables 3.3 to 3.6 show the convergence of the frequency parameters for square plates having a V-notch with various notch angles (α=5° and
30°) and notch depths (d/b=0.1, 0.3 and 0.5). Since the V-notch is much
sharper and deeper than that considered in Table 3.2, the stress singularities at the V-notch would be stronger and the corner functions are expected to show more significant effect on the convergence of the solutions. In these cases under study, the admissible polynomials used alone give solutions with very slow convergence, especially for the case with a sharper (α=5°) or deeper (d/b=0.5) notch. However, supplementing the admissible functions with corner functions significantly accelerates the convergence of the solutions.In the case of Table 3.3, it is found that adding the corner functions into the admissible polynomials may yield ill-conditioned matrices at the number of admissible functions not very large (i.e., 8×8+2×8, 7×7+2×8).
The ill-conditioning is due to numerical roundoff errors. For only using the admissible polynomials, the ill-conditioning also occurs when the number of polynomials (I×J) exceeds 14×14. That is to say, the accurate solutions cannot be obtained for only using the admissible polynomials before the ill-conditioning occurs. However, supplementing the admissible functions with corner functions can give the convergent solutions with high accuracy (4 significant digit convergence) before the
ill-conditioning occurs.
Comparing the results of Table 3.3 with those of Tables 3.4 and 3.5, it is found that the present analysis needs more supplements of corner functions to get the convergent solutions for square plates with a deeper V-notch. Observing the results of Tables 3.4 and 3.6, one can find that more corner functions may not be needed to obtain convergent solutions as α changes from 30° to 5°. Moreover, one may overestimate the numerical solutions of these cases if no supplement of corner functions is involved in the present analysis.
On the basis of the above results, it is recognized that corner functions have significant effects on the convergence of the solutions for square plates with a V-notch. One of the reasons for corner functions having such effects on the convergence is that the corner functions can appropriately describe stress singularity behaviors of moments and transverse shear forces around the vertex of the V-notch. Another is that the corner functions explicitly indicate the existence of the V-notch in the plate under consideration. When the polynomial functions are used along in the Ritz method, the recognition of the existence of the V-notch is only through the integration domain.
Table 3.7 shows the convergence of the frequency parameters for a square plate having a V-notch of α=0° and d/b=0.3, which can be considered as a straight crack. Although a cracked plate and an intact plate have the same integration domain, they have different stiffness. As expected, the solution obtained by using polynomial functions along is
cracked plate. Obviously, it is not suitable to characterize a V-notch in a plate only through the integration domain in the Ritz method as the notch angle or the notch depth becomes smaller. However, the corner functions satisfying the free edge boundary conditions of the V-notch can definitely realize the existence of a V-notch or a crack in the formulation for the Ritz method.