## 5 Optimization of the Three-Layer One-Segment Tube
### Maximization of the Distance between Natural Frequencies and a Selected Frequency  
The main optimization task in the proposed approach is the maximization of the distance between an arbitrarily selected excitation frequency \(p\) and the neighboring natural frequencies (see Equation (8)). The considered values of \(p\) are 30, 40, 50, 60, 70, and 80 Hz. The minimal difference between any natural frequency and any of the arbitrarily selected excitation frequencies is significantly higher than the targeted 10% of the excitation frequency. Actually, in every case, it reaches 30% of the excitation frequency \(p\), whereas the minimum difference \(|p-f_{i}|\) for every \(i\) should be higher than 10% of \(p\) to avoid the resonance phenomenon. The results obtained from the ANN\({}_{1235}\) network are presented in a graphical form in Figure 6. Successive rows in Figure 6 present the results of \(f_{1}\) maximization (the bottom row) and the maximization of the distance between arbitrarily selected excitation frequencies (30-80 Hz) and natural frequencies calculated for MODEL1. Figure 5: Optimization of \(f_{1}\), MODEL1 with three layers of laminate. The green boxes represent the distance in the frequency space around a selected excitation frequency (30-80 Hz) that is free from natural frequencies and obtained when a particular set of lamination angles is applied. The blue vertical lines represent the natural frequencies calculated for the same lamination angles. It is clearly visible that the ANN\({}_{1235}\) network gives the expected results. Table 5 shows a kind of lookup table that enables the quick selection of lamination angles that lead to resonance avoidance for any excitation force frequency up to about 72 Hz. For example, for the excitation frequency \(p=45\) Hz, the suggested lamination angle set \([-64.5/61/-29.4]\) leads to a distance from \(p=45\) Hz to the closest natural frequency \(f_{i}\) of at least 17.4 Hz (or, expressed as a percentage, 32.1%). No calculations are necessary for resonance avoidance once Table 5 is ready. Both problems, namely, the maximization of \(f_{1}\) and the maximization of frequency gaps (eight optimization cases in total), were successfully solved using the ANN\({}_{1235}\) network. The network ANN\({}_{1235}\) was trained and tested using only 3413 FE calls during pattern set generation and 400 FE calls during the verification of obtained results. If it is assumed that the number of FE calls necessary to perform the same tasks using the classical optimization approach exceeds 70,000 (eight optimization cases, each needing about 9000 FE calls), then, in the GA+FEM approach, the number of FE calls reaches 4,000,000. The number of FE calls in the application of the proposed GA+ANN procedures decreases by at least one order of magnitude compared with the classical optimization approach. Figure 6: Optimization of the distance between natural frequencies and selected excitation frequencies with the ANN\({}_{1235}\) network and angle increment of \(`0\lx@arcdegree\). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Excitation** & \multicolumn{2}{c}{**Suggested**} & \multicolumn{2}{c}{**Minimal Distance**} \\
**Frequency** & **Lamination** & \multicolumn{2}{c}{\(|\boldsymbol{p}-\boldsymbol{f}_{i}|\)} \\
**Range [Hz]** & **Angles** & **[Hz]** & **[\%]** \\ \hline
0.0–27.2 & \([51.5/-6.3/77.5]\) & \(10.5\) & \(38.5\) \\
27.2–38.4 & \([90/-81.4/-90]\) & \(14.6\) & \(38.2\) \\
38.4–42.6 & \([35.9/40.5/31.9]\) & \(14.1\) & \(33.2\) \\
42.6–54.2 & \([-64.5/61/-29.4]\) & \(17.4\) & \(32.1\) \\
54.2–65.7 & \([81.9/14.4/-54.8]\) & \(17.0\) & \(25.9\) \\
65.7–71.8 & \([62.8/73.6/-71.4]\) & \(20.0\) & \(27.8\) \\ \(>\)71.8 & \([63.7/-55.3/-76.7]\) & — & — \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Suggested lamination angles for different excitation force frequencies. The results of \(f_{1}\) maximization show the stabilization of the fundamental natural frequency value; further increasing the number of layers does not result in an associated increase in \(f_{1}\).