## 5 Results and Discussion
It should be noted that in this case the third critical frequency shifts quite strongly to the right, in the direction of increasing natural frequencies, from 1.76 to 2.14, which also imposes certain restrictions on the choice of operating speed range in the absence of magnetic field influence [29]. The numerical values of the amplitudes for this case are presented in Table 3.  
When the rotor cavity is half and two-thirds full, i.e., at \(\gamma=1.67\) and \(\gamma=1.25\), the character of changes in the amplitude and frequency of natural oscillations of the system is similar to the case when the rotor cavity is one-third filled. In this case, the system oscillates with smaller amplitudes by almost an order of magnitude lower than when the rotor cavity is one-third full (see Figures 8 and 9). In the case of half filling, the second and third critical frequencies are still greater than the first critical frequency [51]. Accordingly, a stronger shift in critical frequencies is observed compared to those listed above, for example, the second critical frequency appears already at \(\tau=0.61\), while the third already at \(\tau=2.61\). When the cavity is filled by two-thirds, the amplitudes of the first and second critical frequencies are commensurate with each other and with the above case. The second critical frequency shifts in the direction of decreasing angular velocities and appears at \(\tau=0.59\), whereas the third critical frequency shifts to the left in the direction of its increase and appears at \(\tau=3.02\). With an increase in the parameter \(b_{m}\), the oscillation amplitudes at the second and third critical speeds of the rotor and foundation, as before, are damped until their full damping. The numerical values of the amplitudes for this case are presented in Tables 4 and 5. \begin{table}
\begin{tabular}{l l l l} \hline \hline \multirow{2}{*}{\(B\), \(\gamma=2.6\)} & **First Critical Speed** & \multicolumn{2}{l}{**Second Critical**} \\  & **Amplitude Value** & **Speed** & **Amplitude Value** \\ \hline \(b_{m}=0\) & & 0.188180917, \(\tau=0.64\) & 38.0983794, \(\tau=2.14\) \\ \(b_{m}=0.5\) & & 0.188128512, \(\tau=0.64\) & 35.05368501, \(\tau=2.14\) \\ \(b_{m}=5\) & 0.114550189, \(\tau=0.16\) & 0.185743424, \(\tau=0.64\) & 4.620198303, \(\tau=2.14\) \\ \(b_{m}=25\) & 0.114543039, \(\tau=0.16\) & 0.148131171, \(\tau=0.64\) & 0.442357104, \(\tau=2.14\) \\ \(b_{m}=50\) & 0.113852547, \(\tau=0.16\) & 0.100918717, \(\tau=0.64\) & 0.180448011, \(\tau=2.15\) \\ \(b_{m}=100\) & 0.113189756, \(\tau=0.16\) & 0.06088884, \(\tau=0.64\) & 0.093096022, \(\tau=2.16\) \\ \(b_{m}=500\) & 0.109089699, \(\tau=0.16\) & – & – \\ \(b_{m}=5000\) & 0.124710542, \(\tau=0.16\) & – & – \\ \hline \multirow{2}{*}{\(D\), \(\gamma=2.6\)} & **First Critical Speed** & \multicolumn{2}{l}{**Second Critical**} \\  & **Amplitude Value** & **Suped** & **Third Critical Speed** \\ \hline \(b_{m}=0\) & 0.074299992, \(\tau=0.16\) & 0.130906778, \(\tau=0.64\) & 37.42981166, \(\tau=2.14\) \\ \(b_{m}=0.5\) & 0.074296638, \(\tau=0.16\) & 0.130872119, \(\tau=0.64\) & 34.43775859, \(\tau=2.14\) \\ \(b_{m}=5\) & 0.074266809, \(\tau=0.16\) & 0.129169584, \(\tau=0.64\) & 4.551320535, \(\tau=2.14\) \\ \(b_{m}=25\) & 0.074142119, \(\tau=0.16\) & 0.101854993, \(\tau=0.64\) & 0.41732892, \(\tau=2.14\) \\ \(b_{m}=50\) & 0.074004213, \(\tau=0.16\) & 0.067028512, \(\tau=0.64\) & 0.1481861679, \(\tau=2.14\) \\ \(b_{m}=100\) & 0.073787317, \(\tau=0.16\) & 0.036021818, \(\tau=0.64\) & 0.053425977, \(\tau=2.14\) \\ \(b_{m}=500\) & 0.074595252, \(\tau=0.16\) & 0.01092513, \(\tau=0.66\) & 0.007393946, \(\tau=2.14\) \\ \(b_{m}=5000\) & 0.140446071, \(\tau=0.16\) & – & – \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Critical Speeds. Figure 6: Amplitude–frequency characteristic of natural oscillations of the rotor for \(s=1\) and \(\gamma=2.6\); (a) general view; (b) enlarged view. Figure 7: Amplitude–frequency characteristic of natural oscillations of the foundation for \(s=1\) and \(\gamma=2.6\); (**a**) general view; (**b**) enlarged view.