# Regular and Chaotic Vibrations of a Nonlinear
## 2 Model of the System  
In the present paper, a model with six degrees of freedom will be used to describe the vibrations of the system rotor--limiter of motion, as shown in Figure 1. The rotor's motion is described by the coordinates \(x_{1},y_{1}\) of its geometrical centre \(S_{1}\) (centre of exactly balanced mass \(m_{1}\)) and the rotation angles \(\alpha_{1}\) and \(\beta_{1}\), whereas the coordinates \(x_{2}\) and \(y_{2}\) describe the displacements of the limiter of motion. Axial and torsional vibrations of both rotor and limiter are neglected. It has also been assumed that the rotational speed of the rotor \(\Omega\) is constant, and thus, the motor has unlimited power. The gyroscopic effects are accounted for in the model. It has also been assumed that the resilient elements are described by the Voigt-Kelvin model with the damping coefficients (\(\varepsilon_{0}\), \(\varepsilon_{1}\), \(\varepsilon_{2}\)) proportional to the respective stiffnesses (\(c_{0}\), \(c_{11}\), \(c_{12}\), \(c_{2}\)). The location of the limiter of motion is specified by the distance \(d_{0}\) and that of the shaft supports by \(l_{1}\) and \(l_{2}\). For the case \(d_{0}=0\), the limiter of motion is called a stator. There is a gap in the clearance \(\delta\) between the rotor and the limiter of motion. The model takes into account the influence of the inertial forces resulting from the rotor unbalance (through eccentricity \(e_{0}\)) as well as the static forces \(Q_{1}\) and \(Q_{2}\). Using the laws of change of momentum and angular momentum about the rotor centre of mass, the motion of the system is described by the following set of six second-order ordinary differential equations:  
\[m_{1}\ddot{x}_{1}+c_{1}(\varepsilon_{1}\dot{x}_{1}+x_{1})-c_{1}d_{1}( \varepsilon_{1}\dot{a}_{1}+\alpha_{1})+F_{x}=Q_{1}+m_{1}e_{0}\Omega^{2}\cos \Omega t \tag{1}\]  
\[m_{1}\ddot{y}_{1}+c_{1}(\varepsilon_{1}\dot{y}_{1}+y_{1})+c_{1}d_{1}( \varepsilon_{1}\dot{\beta}_{1}+\beta_{1})+F_{y}=m_{1}e_{0}\Omega^{2}\sin\Omega t \tag{2}\]  
\[I_{1}\ddot{a}_{1}-l_{0}\Omega\ \dot{\beta}_{1}+(k_{1}+c_{1}d_{1}^{2})( \varepsilon_{1}\dot{a}_{1}+\alpha_{1})-c_{1}d_{1}(\varepsilon_{1}\dot{x}_{1} +x_{1})+F_{x}d_{0}=0 \tag{3}\]  
\[I_{1}\ddot{\beta}_{1}+l_{0}\Omega\ \dot{\alpha}_{1}+(k_{1}+c_{1}d_{1}^{2})( \varepsilon_{1}\dot{\beta}_{1}+\beta_{1})+c_{1}d_{1}(\varepsilon_{1}\dot{y}_{ 1}+y_{1})-F_{y}d_{0}=0 \tag{4}\]  
\[m_{2}\ddot{x}_{2}+c_{2}(\varepsilon_{2}\dot{x}_{2}+x_{2})-F_{x}=Q_{2} \tag{5}\]  
\[m_{2}\ddot{y}_{2}+c_{2}(\varepsilon_{2}\dot{y}_{2}+y_{2})-F_{y}=0 \tag{6}\]  
Figure 1: Model of the system: 1—rotor, 2—limiter of motion. Here \(m_{1}\) and \(m_{2}\) are, respectively, the masses of the rotor and the limiter of motion, and \(I_{1}\) and \(I_{0}\) are the moments of inertia of the rotor about the \(x\)-axis (or about the \(y\)-axis thanks to axial symmetry) and the \(z\)-axis. The following symbols are used in Equations (1)-(6):  
\[c_{1}=c_{11}+c_{12} \tag{7}\]  
\[d_{1}=(c_{11}l_{1}-c_{12}l_{2})/(c_{11}+c_{12}) \tag{8}\]  
\[m_{1}\ddot{x}_{1}+c_{1}(\varepsilon_{1}\dot{x}_{1}+x_{1})-c_{1}d_{1}( \varepsilon_{1}\dot{a}_{1}+\alpha_{1})+F_{x}=Q_{1}+m_{1}e_{0}\Omega^{2}\cos \Omega t \tag{1}\]0  
The components \(F_{\rm x}\) and \(F_{\rm y}\) of the impact forces take on non-zero values for \(r>\delta\), where the distance \(r=\sqrt{x_{0}^{2}+y_{0}^{2}}\) is calculated using the coordinates \(x_{0}=x_{1}+d_{0}\ \alpha_{1}\ -x_{2}\), \(y_{0}=y_{1}+d_{0}\beta_{1}-y_{2}\) of the vector of the displacement of the rotor axis relative to the limiter of motion (Figure 2). The following relationship is used for the tangential component of the reaction force:  
\[m_{1}\ddot{x}_{1}+c_{1}(\varepsilon_{1}\dot{x}_{1}+x_{1})-c_{1}d_{1}( \varepsilon_{1}\dot{a}_{1}+\alpha_{1})+F_{x}=Q_{1}+m_{1}e_{0}\Omega^{2}\cos \Omega t \tag{1}\]1  
\[f=\left\{ \begin{array}{cc}f_{\rm S}{\rm sgn}V_{\rm S}&if\ |V_{\rm S}|>V_{\rm Smin}\\ f_{\rm S}V_{\rm S}/V_{\rm Smin}&if\ |V_{\rm S}|\leq V_{\rm Smin}\end{array} \right.