## 4 Results and Discussion for Geiger Domes
### Unstable Regions
#### 4.3.2 Domes with a Closed Upper Section (Type A)  
The regular type A dome was characterized by 18 infinitesimal mechanisms, while the modified type B dome had 8. This means that 18 and 11 main unstable regions were determined, respectively. However, in this type of dome, there are three additional frequencies depending on the initial prestress. The influences of the initial prestress level on the resonance frequencies are shown in Figures 9 and 10 and Table 6.  
The behavior of type A domes is quite different from that of type B domes. This is due to the minimum prestress level, which is much greater. There is also no similarity  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & \multicolumn{4}{c}{**Regular**} & \multicolumn{4}{c}{**Modified**} \\ \cline{2-9}  & \multicolumn{4}{c}{**Resonance frequency \(\eta\) [Hz]**} & \\ \cline{2-9}  & \(\eta\)**19** & \(\eta\)**20** & \(\eta\)**21** & \(\eta\)**22** & \(\eta\)**9** & \(\eta\)**10** & \(\eta\)**11** & \(\eta\)**12** \\ \hline \(S_{min}\) & 31.4 & 31.6 & 34.8 & 88.6 & 30.7 & 31.0 & 35.2 & 85.4 \\ \(S_{max}\) & 36.7 & 36.9 & 41.2 & 88.8 & 33.4 & 33.6 & 38.6 & 85.5 \\ Relative increase & 16.9\% & 16.8\% & 18.4\% & 0.2\% & 8.8\% & 8.4\% & 9.7\% & 0.1\% \\ \hline \multicolumn{9}{c}{**Natural frequency \(f\) [Hz]**} \\ \cline{2-9}  & \(f\)19 & \(f\)20 & \(f\)21 & \(f\)22 & \(f\)9 & \(f\)10 & \(f\)11 & \(f\)12 \\ \cline{2-9} \(S_{min}\) & 15.7 & 15.7 & 17.3 & 44.3 & 15  
between the behaviors of regular and modified type A domes. In the case of regular domes, the areas of unstable regions do not depend on the frequencies (Figure 13), e.g., for \(S_{min}=24\) kN, the areas corresponding to the first \(\eta 1\) and eighteenth \(\eta 18\) resonance frequencies are comparable, \(A_{\eta 1}=0.54\ j^{2}\) and \(A_{\eta 18}=0.62\ j^{2}\), whereas for the second resonance frequency \(\eta 2\), this is \(A_{\eta 2}=0.15\ j^{2}\). There is no rule except one that applies to all four domes: as the initial prestress level \(S\) increases, the resonant frequencies also increase, and the range of the unstable regions decreases.  
In turn, modified type A domes are not sensitive to the pulsatility index \(v\) (Figure 14). The boundaries of the instabilities practically coincide, as in the cases of type B domes for \(S\geq 35\) kN.  
Analyzing the results obtained for the additional three resonant frequencies (Table 6: \(\eta 19\), \(\eta 20\), \(\eta 21\) for a regular dome and \(\eta 9\), \(\eta 10\), \(\eta 11\) for a modified dome), it can be seen that these frequencies do not depend on the pulsatility index \(v\) (\(\eta(v=0\div 0.75)=\text{const.}\)). The boundaries of instabilities coincide; however, these frequencies depend on the initial prestress level \(S\). The regular domes are more sensitive to the initial prestress. The relative increase vs. \(S_{min}\) is approximately 16.8-18.4%, while for the modified domes, it is approximately 8.4-9.7%. It should be noted that the behavior of the resonance frequencies is similar to the behavior of the natural frequencies.  
In turn, as in the case of type B domes, the next resonance frequencies, i.e., \(\eta>\eta 21\) and \(\eta>\eta 11\), for regular and modified domes, respectively, are independent of the pulsatility index \(v\) and of the initial prestress level \(S\) (Table 6). The relative increase is less than 1% for all frequencies. These resonance frequencies are twice as high as the natural frequencies.