# Frequency Response Analysis of Perforated Shells with Uncertain Materials and Damage
## 3 Free Vibration of Reference Configurations  
In this section, we establish the reference configurations on which the numerical experiments below are based and explain the rationale for these experiments. As already given above, the profile functions are \(f_{p}(x_{1})=1\) and \(f_{H}(x_{1})=1+(x_{1}/\pi)^{2}\), for the parabolic and hyperbolic cases, respectively. In Table 1, the first 10 eigenpairs are tabulated for both non-perforated shells with \(t=1/100\). In the two other tables (Tables 2 and 3), the first 10 eigenpairs are listed for perforated configurations, including quasi-random ones. The eigenmodes are characterized using axial (\(K_{x}\)) and angular (\(K_{y}\))  
Figure 6: Hyperbolic shell. Regular \(8\times 8\) pattern. The smaller thickness is chosen to conform with a natural characteristic length scale \(\sqrt[4]{t}\). Figure 7: Representative cells: Layer charts with one hole \(D\). The characteristic length scales are indicated in terms of the dimensionless thickness. In the hyperbolic case for the non-axial layers, the directions depend on the actual geometry, in this case, the decay is in the orthogonal direction to the one indicated by the arrow. wave numbers, where the fractional axial wave numbers refer to symmetric modes that is, the pair \((K_{x},K_{y})=(1/2,4)\) refers to mode \(\varphi(x,y)\sim\cos(x/2)\cos(4y)\) or \(\varphi(x,y)\sim\cos(x/2)\sin(4y)\) due to symmetry. The numbers are universal, that is, no material parameters are included except for the Poisson ratio \(\
u=1/3\). For non-perforated shells, every eigenvalue (disregarding torsional modes) is a double one due to symmetry. Theoretically, the lowest mode can have an order four, but this occurs only at critical values of \(t\) which is not the case here. Even for perforated shells, the eigenvalues are observed to appear in clusters of two. The discrepancies between the eigenvalues in a cluster are a simple measure of homogeneity. In the Halton cases, the discrepancies are larger than in the regular case but still remarkably small. **Remark 1**. : _Even though the regular perforation pattern results in a symmetric configuration, there exist clusters with relatively large discrepancy. If the pattern is a \(n\times n\)-grid, then the modes with angular wave number \(n\), for instance, belong to such a cluster._  
Two further observations: first, the reference eigenvalues cannot be simply scaled to match the perforated ones, and, second, the modes are observed _out of order_. For homogenization, it would be necessary to perform minimization over a set of perforations to get the eigenvalues reasonably approximated, but similar process for the eigenmodes cannot be performed unless the subspaces are the same. At the moment, we are not aware of any efficient way short of computing the whole eigenspaces to accomplish this. Based on these results, we do not attempt to select modes in the frequency response a priori, but compute the results using full systems regardless of the cost. \begin{table}
\begin{tabular}{l c c c c c c c} \hline \hline  & \multicolumn{3}{c}{a Parabolic} & \multicolumn{3}{c}{b Hyperbolic} \\ \hline \(\mathbf{k}\) & \(\mathbf{\lambda}\) & \(\mathbf{K_{x}}\) & \(\mathbf{K_{y}}\) & \(\mathbf{k}\) & \(\mathbf{\lambda}\) & \(\mathbf{K_{x}}\) & \(\mathbf{K_{y}}\) \\ \hline
1 & 0.0325658 & 1/2 & 4 & 1 & 0.129749 & 3/2 & 5 \\
2 & 0.0325658 & 1/2 & 4 & 2 & 0.129749 & 3/2 & 5 \\
4 & 0.0344515 & 1/2 & 3 & 4 & 0.136477 & 2 & 5 \\
5 & 0.0613855 & 1/2 & 5 & 5 & 0.138588 & 2 & 4 \\
6 & 0.0613855 & 1/2 & 5 & 6 & 0.1385888 & 2 & 4 \\
7 & 0.0920303 & 1 & 4 & 7 & 0.142168 & 3/2 & 4 \\
8 & 0.0920303 & 1 & 4 & 8 & 0.142168 & 3/2 & 4 \\
9 & 0.0922696 & 1 & 5 & 9 & 0.156955 & 2 & 6 \\
10 & 0.0922696 & 1 & 5 & 10 & 0.156955 & 2 & 6 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Reference results for the non-perforated test cases. Parameters: \(p=8\), \(t=1/100\); No material scaling.