## 4 Results and Discussion for Geiger Domes
### Unstable Regions
#### 4.3.1 Domes with an Open Upper Section (Type B)  
The regular type B dome was characterized by 31 infinitesimal mechanisms, while the modified type B dome had 21. Therefore, 31 and 21 main unstable regions were determined, respectively. For regular domes, the main unstable regions corresponding to the following resonance frequencies are shown: \(\eta 1\), \(\eta 10\), \(\eta 20\), and \(\eta 31\) (Figure 8), while for the modified dome, they are \(\eta 1\), \(\eta 7\), \(\eta 14\), and \(\eta 21\) (Figure 9). Three cases of initial prestress, i.e., \(S_{min}=2\) kN, \(S=15\) kN, and \(S=35\) kN, are presented. Comparing the Ince-Strutt maps with the stable and unstable regions for both domes, similarities are apparent. First, the area of the unstable regions is larger at higher frequencies. Second, at \(S_{min}=2\) kN (Figures 8a and 9a), the unstable regions are larger than those at \(S=15\) kN (Figures 8b and 9b). In turn, at \(S=35\) kN (Figures 8c and 9c), the boundaries of the unstable regions practically coincide. This means that the resonance frequencies \(\eta\) do not depend on the pulsatility index \(v\), and the risk of the excitation of unstable motion decreases. \begin{table}
\begin{tabular}{c c c c c} \hline  & \multicolumn{2}{c}{**Regular Geiger Domes**} & \multicolumn{2}{c}{**Modified Geiger Domes**} \\ \cline{2-5}  & **Type A** & **Type B** & **Type A** & **Type B** \\ \hline
**3** & 0.9213 & 0.9213 & 0.8010 & 0.8010 \\
**4** & 2.0061 & 2.0061 & 2.0061 & 2.0061 \\
**C1** & 1.7391 & 1.7391 & 1.7391 & 1.7391 \\
**C2** & 0.8696 & 0.8696 & 0.7560 & 0.7560 \\
**C3** & - & 0.3623 & - & 0.2167 \\
**C4** & - & 0.5072 & - & 0.3034 \\
**C5** & - & - & 0.2356 & 0.2356 \\
**C6** & - & - & 0.2270 & 0.2270 \\ \hline \end{tabular}
\end{table} 
Table 4: _Cont._Differences are also apparent. In the case of a regular dome (Figure 8), the resonance frequencies \(\eta\) are higher, and the unstable regions for the same level of initial prestress \(S\) are larger than those of a modified dome (Figure 9). To compare the behavior of the domes, the influence of the initial prestress \(S\) on the area of unstable regions \(A_{\eta}(S)\) is shown in Figure 6. As can be seen, at low values of initial prestress (\(S\in<2\) kN; \(20\) kN \(>\)), the area of the unstable regions of a regular dome (Figure 10a) is greater than that of a modified dome (Figure 10b). For example, for the lowest prestress level \(S_{min}=2\) kN, the areas of the unstable region corresponding to the first and last resonance frequencies are larger by 14% and 47%, respectively. However, further compression significantly narrows the unstable areas, and the areas are similar for both domes. Figure 8: Limits of the four main selected unstable regions of the regular type B dome: (**a**) \(S_{min}=2\) kN; (**b**) \(S=15\) kN; (**c**) \(S=35\) kN. Figure 9: Limits of the four main selected unstable regions of the modified type B dome: (**a**) \(S_{min}=2\) kN; (**b**) \(S=15\) kN; (**c**) \(S=35\) kN. The influence of the initial prestress level \(S\) on the distribution and range of unstable regions was measured by the nondimensional parameter \(\lambda\) (15) (Figure 11). In the case of the regular dome (Figure 11a), as the initial prestress level increases, the changes in the range of areas are greater. For example, at \(S=15\) kN, the areas are smaller by approximately 53% than the areas at the minimum prestress level, while in the case of the modified dome (Figure 11b), they are about 45% smaller. Parameter \(\lambda\) decreases exponentially as the initial prestress level increases for both domes, and at the maximum level, \(\lambda=0.02\) and \(\lambda=0.04\) (the unstable regions decrease by 98% and 96%). As can be seen, the changes in the range of unstable regions corresponding to almost all resonance frequencies are comparable between the domes, except for the first region (this region corresponds to the first resonance frequency \(\eta 1\)). As was stated earlier, the unstable regions are the greatest at the lowest prestress level \(S_{min}=2\) kN.