# Regular and Chaotic Vibrations of a Nonlinear
## 2 Model of the System
With the increase of the distance between the limiter and the rotor (parameter \(\lambda_{0}\)), the ranges of subsequent \(nT\)-periodic vibrations become more distinct, and they shift towards lower frequencies (\(\lambda_{0}\) = 0.3, Figure 6b). The subsequent regions of sub-harmonic vibrations of increasing order are separated by narrow zones of quasi-periodic and chaotic oscillations. Figure 5: Influence of the excitation frequency \(\omega\) and static force \(q\) on the character of vibrations (\(e\) = 0.3, \(\lambda_{0}\) = 0.3). Bifurcation diagrams are often used to analyse rotor vibrations because they provide a lot of interesting information regarding the character of vibrations [10; 11; 15]. It should be pointed out that in the parameter range for which quasi-periodic or chaotic vibrations take place, the solutions of Equations (20)-(25) are sensitive to even very small changes of parameters; the chaotic solutions change easily into quasi-periodic solutions or sub-harmonic solutions of a high order. This is confirmed by the stroboscopic portraits (Figure 7) obtained respectively for chaotic, sub-harmonic and quasi-periodic oscillations. In the case of \(nT\)-periodic vibrations, the stroboscopic portrait is a set of n points (Figure 7b); for quasi-periodic vibrations, this portrait is a closed curve (Figure 7a), and for chaotic vibrations, it is a fractal (Figure 7c,d). The shape of the fractal in Figure 7c suggests that the chaotic vibrations are probably the result of successive bifurcations of 4T-periodic vibrations. However, whether the vibrations are quasi-periodic or chaotic, in both cases, there always appear to be harmful impacts between the rotor and limiter. Figure 8a,b illustrate the influence of the parameters \(\omega\), \(e\) and \(\lambda_{0}\) on the character of vibrations, respectively, for \(q\) = 0.4, \(\lambda_{0}\) = 0.3 and \(q\) = 0.4, \(e\) = 0.5. The linear analysis of the steady-state vibrations predicts the occurrence of internal impacts only in the area above curve \(k\) (Figure 8a) or between curves \(k_{1}\) and \(k_{2}\) (Figure 8b). In the resonant regions, mostly \(T\)-periodic oscillations are set in with the full rubbing between the rotor and the limiter of motion. In this region, the rotor axis rotates in the same direction as the rotor revolutions (forward whirl). Outside the resonance region, all types of vibrations take place with partial rubbing between the rotor and limiter. Figure 8b is determined for large values of parameters \(q\) and \(e\) (i.e., small values of clearance \(\delta\)). For this reason, in the case of the rotor-stator system (\(\lambda_{0}\) = 0), collisions occur practically in the entire frequency range, which results in periodic vibrations for lower frequencies and chaotic vibrations for higher frequencies. For smaller values of parameters \(q\) and \(e\), vibrations without collisions are possible. For higher values of \(|\,\lambda_{0}\,|\), quasi-periodic oscillations occur, and by further increasing the value of \(|\,\lambda_{0}\,|\) there appear zones of \(T\)-periodic vibrations with no internal impacts. For example, for a relatively high excitation amplitude \(e=0.5\) and for the static load \(q=0.4\), in the region lying to the right of the curve \(k_{2}\) in Figure 8b, there are practically no steady-state impacts between the rotor and the limiter of motion when \(|\,\lambda_{0}\,|>0.4\) and \(3<\omega<6\). This may suggest that in order to protect the stator, one can use a limiter of motion close to the shaft supports. Figure 8: \(Cont.\)  
Figure 9 shows the bifurcation diagram obtained by the stroboscopic method, which corresponds to the section of Figure 8b. This graph illustrates the effect of the limiter of motion position on the type of excited vibrations for \(\omega=8\). For \(|\lambda_{0}\,|<0.25\), chaotic oscillations predominate (as might be suggested by the more irregular envelope of the strobe points), whereas for \(0.25<|\lambda_{0}\,|<0.8\), there occur mostly quasi-periodic vibrations.