# Regular and Chaotic Vibrations of a Nonlinear
## 2 Model of the System
All analyses have been carried out in a steady state, assuming \(K=100\) and \(N=150\) (where \(KT\) is the time without the  
Figure 3: Flowchart for rotor dynamic response analysis. analysis of the solutions). In calculating the zones, the initial conditions that correspond to the rotor equilibrium conditions were used. Also shown in Figure 4a, the \(k\)-curve calculated from the analysis of a linear system (with no contact of the rotor with the limiter of motion), which is defined as loci of static force values \(q\) for which the maximum displacement \(\rho_{max}=r_{max}/\delta\) during steady-state vibrations is equal to unity (\(\rho_{max}=1\)). Below this curve, the linear analysis suggests no rotor-limiter collisions. However, nonlinear system analysis reveals that in these zones, steady-state vibrations are possible, during which the rotor and the limiter of motion impact the system in a regular or chaotic way. This phenomenon results from the transient phase of motion during which initial sporadic impacts of the rotor against the limiter of motion can change into systematically repeated clashes, resulting in steady-state vibrations of a complex character. These vibrations have been found to only slightly depend on the initial conditions. In the resonance range (for \(\omega\approx 1\)), \(1T\)-periodic vibrations dominate, during which rotor-limiter of motion collisions occur. Figure 4: Influence of the excitation frequency \(\omega\) and static force \(q\) (\(e=0.3\), \(\lambda_{0}=0\)): (**a**) on the character of vibrations, (**b**) on the index \(J\). Figure 4b shows the influence of parameters \(\omega\) and q on the quality index \(J\) (28). The rotor vibration energy increases with the increase of rotational speed (frequency \(\omega\)), with significantly higher vibration levels being noted in the chaotic vibration ranges. Figure 5 shows the areas of periodic, quasi-periodic and chaotic vibrations in the \((\omega,q)\) plane for a motion limiter shifted relative to the rotor by a value of \(\lambda_{0}\) = 0.3. Similarly to Figure 4a, the k curve is also plotted here. Comparing Figures 4a and 5, one can see the influence of the parameter \(\lambda_{0}\) in the higher frequency range. For the \(\lambda_{0}\) = 0 (case of the system: rotor-stator) shown in Figure 4a, chaotic vibrations predominate, whereas for \(\lambda_{0}\) = 0.3 (Figure 5), mostly sub-harmonic oscillations are excited in the system. With the increase of the excitation frequency, the vibration regions appear in the following order: 1T-periodic, 2T-periodic, 3T-periodic and higher order. Chaotic vibrations and quasi-periodic oscillations occur only for high values of \(\omega\) and \(q\). The different character of vibrations for different positions of the limiter of motion is also confirmed by the bifurcation diagrams shown in Figure 6a,b, which correspond to the sections of Figures 4a and 5 for \(q\) = 0.7. The frequency intervals with a set of continuous lines are the regions of sub-harmonic oscillations of type 1:\(n\) (\(nT\)-periodic vibrations), where the number of lines provides information about the order. The mechanism works correctly if the vibration period is equal to the excitation period (single line on the diagram) and, additionally, if no collisions occur. On the bifurcation diagram determined for the rotor-stator system (\(\lambda_{0}\) = 0, Figure 6a), the ranges of chaotic vibrations are present only in the low-frequency range. These are the ranges of 1T-, 2T- and less visible 3T-periodic vibrations, separated from each other by the ranges of chaotic vibrations. In the higher frequency range (for \(\omega\) > 5.8), chaotic or quasi-periodic vibrations are dominant. Analysis of the bifurcation diagram alone does not allow for an unambiguous distinction between these types of vibrations.