# Regular and Chaotic Vibrations of a Nonlinear
## 2 Model of the System
The Cartesian components of the impact force are calculated from the following equations:  
\[F_{x}=N\cos\phi-T\sin\phi \tag{14}\]  
\[F_{y}=N\sin\phi+T\cos\phi \tag{15}\]  
where the angle \(\phi\) defines the location of the contact point between the rotor and the limiter of motion and is defined as follows: equations:  
\[\sin\phi  
solving the differential Equations (20)-(25), Runge-Kutta algorithms with variable integration steps were used: 4 th order one and a more accurate 7 th order one. The equations have been integrated over a given time interval (0, \(NT\)), where \(T\) is the period of the exciting force. The study of vibration type has been carried out in a steady state, analysing the solutions in the interval (\(KT\), \(NT\)) for this purpose. The vibration analysis was primarily based on the results of the spectral analysis obtained using the FFT algorithm. In order to improve the assessment of the studied courses, the periodicity conditions of the solutions have been checked in parallel. Additionally, it has been verified whether there are collisions present in the rotor-stator system by noting their average number during the excitation period. The algorithm presented in Figure 3 has been used, among others, to make bifurcation diagrams by finding the components of the state vector at prescribed time instants (the stroboscopic method). In the case of models described by many parameters, it is more useful to determine the zones of the parameters in which the system displays the same qualitative behaviour. It is a more involved problem since it calls for the determination of the vibration type (sub-harmonic, quasi-periodic or chaotic) and the mode of vibration (forward or backward whirl). The algorithm presented in Figure 3 has been used to determine the areas of various types of vibrations by changing the values of two selected system parameters during the calculations. The number of impacts of the rotating element with the limiter during the forcing period and whether there is full or partial rubbing between the rotor and the limiter during the impact are also of importance. Additionally, an index \(J=E/m_{1}\omega_{0}{}^{2}\) related to the kinetic energy of rotor vibrations has been determined. With the introduction of dimensionless quantities, this index takes the following form:  
\[J=\frac{1}{2(N-K)T}\int\limits_{KT}^{NT}\left[u_{1}^{\prime 2}+u_{2}^{ \prime 2}+\rho_{1}(u_{3}^{\prime 2}+u_{4}^{\prime 2})\right]d\tau \tag{28}\]  
Selected results of the qualitative analysis are discussed below, where the analysis is limited to the influence of the parameters \(\omega\), \(e\), \(q\) and \(\lambda_{0}\). A small difference between the rigidities of the rotor supports has been assumed (\(c_{11}=0.475\), \(c_{12}=0.525\), \(l_{1}=l_{2}=l\)), which results in couplings (\(\lambda_{1}=-0.05\)) which have an important effect on the mode shapes. Thanks to the assumed asymmetry, spatial vibrations are excited in the analysed system. The following set values of the parameters have been used in the numerical calculations: \(\rho_{0}=0.01\), \(\rho_{1}=0.05\), \(\mu_{2}=2\), \(\gamma_{0}=1000\), \(\gamma_{2}=100\), \(\kappa_{1}=0.9975\), \(C_{0}=0.001\), \(C_{1}=0.1\), \(C_{2}=0.01\), \(\Delta=1\), \(\lambda_{1}=-0.05\),\(f_{S}=0.1\). Additionally, the values \(R/\delta=25\), \(V_{S_{min}}/\Omega\delta=0.1\) have been used in modelling the friction force. Figure 4 shows the influence of the excitation frequency (parameter \(\omega\)) and static force (parameter \(q\)) on the vibration type (Figure 4a) and index \(J\) (Figure 4b) for the case of the rotor-stator system \(\lambda_{0}=0\) and \(e=0.3\). In order to determine the vibration type, the spectra of the state vector coordinates have been analysed. In the case of periodic system responses, the conditions of periodicity of solutions have been additionally examined, which allows us to determine the order of subharmonic vibrations (i.e., \(nT\)-periodic). The number \(n\) is the ratio of the system response period to the excitation period.