## 2 Twelve-Pole System Nonlinear Model  
Figure 2 depicts a schematic diagram for the twelve-pole rotor system. The rotor is investigated as a two-degree-of-freedom system that oscillates in \(X\) and \(Y\) directions. The twelve electromagnetic poles are responsible for controlling the system vibrations via generating controllable attractive forces \(R_{X}\) and \(R_{Y}\) in both the horizontal and vertical directions, respectively. Therefore, the differential equations that govern the system motion can be expressed as follows [38; 39]:  
\[m\ddot{x}(t)=me\psi^{2}\cos(\psi t)+R_{X} \tag{1}\]  
\[m\ddot{y}(t)=me\psi^{2}sin(\psi t)+R_{Y} \tag{2}\]  
where \(m\) is the rotor mass, \(\frac{d^{2}x}{dt^{2}}=\ddot{x}(t)\) and \(\frac{d^{2}y}{dt^{2}}=\ddot{y}(t)\) are the horizontal and vertical accelerations of the rotor system, \(e\) is the eccentricity of the rotating disc, \(\psi\) is the disc angular speed, and \(R_{X}\) and \(R_{Y}\) are the net attractive forces of the twelve poles in \(X\)- and  
Figure 1: Closed-loop control system of active magnetic bearings system. \(Y\)-direction. By considering the symmetric design of the twelve-pole system, the attractive force of each opposite pair of the poles (\(f_{j},\,j=1,2,\ldots,6\)) can be expressed as follows [39]:  
\[f_{j}=\frac{\mu_{0}N^{2}A\cos(\varphi)}{4}\left[\frac{\left(I_{0}-I_{j}\right)^ {2}}{\left(c_{0}-\delta_{j}\right)^{2}}-\frac{\left(I_{0}+I_{j}\right)^{2}}{ \left(c_{0}+\delta_{j}\right)^{2}}\right]\,,j=1\cdots 6 \tag{3}\]  
where \(\mu_{0}\) is the free space permeability, \(N\) the number of turns of each coil for the twelve poles, \(A\) is the cross-sectional area of each pole, \(I_{0}\) is the permanentized electrical current, \(I_{j}\) is the control current in the \(j\)th pole, \(c_{0}\) is the nominal air gap size as shown in Figure 2b, (\(c_{0}-\delta_{j}\)) is the instantaneous air gap of the \(j\)th pole. Accordingly, for the small displacements \(x\) and \(y\) of the rotor system, \(\delta_{j}\) can be expressed as follows:  
\[\left. \begin{array}{l}\delta_{1}=x\cos(\alpha)-y\sin(\alpha),\quad\delta_{2 }=x,\quad\delta_{3}=x\cos(\alpha)+y\sin(\alpha)\\ \delta_{4}=x\sin(\alpha)+y\cos(\alpha),\quad\delta_{5}=y,\quad\delta_{6}=-x \sin(\alpha)+y\cos(\alpha)\end{array} \right\} \tag{4}\]  
where, \(\alpha=30^{\circ}\) is the angle between every successive two poles (See Figure 2b). The control currents \(I_{j}\) are designed such that the forces \(f_{1},\,f_{2},\) and \(f_{3}\) are proportional to the displacement \(x(t)\) and velocity \(\dot{x}(t)\), while \(f_{4},\,f_{5},\) and \(f_{6}\) are proportional to the displacement \(y(t)\) and velocity \(\dot{y}(t)\). Therefore, the control currents are designed as follows:  
\[I_{1}=I_{2}=I_{3}=k_{1}x+k_{2}\dot{x},\quad\,I_{4}=I_{5}=I_{6}=k_{1}y+k_{2} \dot{y} \tag{5}\]  
where \(k_{1}\) is known as proportional gain, while \(k_{2}\) represents the derivative gain. Now, by substituting Equations (4) and (5) into Equation (3), and then expanding the resulting equations up to the third order using Maclaurin series, we can obtain the forces \(f_{i},\,(i=1,2,\ldots,8)\) as given in Appendix A. Accordingly, the net attractive forces \(R_{X}\) and \(R_{Y}\) of the twelve-pole system in the horizontal and vertical directions can be expressed as follows:  
\[R_{X}=f_{2}+(f_{1}+f_{3})\cos(\alpha)+(f_{4}-f_{6})\sin(\alpha) \tag{6}\]  
\[R_{Y}=f_{5}+(f_{4}+f_{6})\cos(\alpha)+(f_{3}-f_{1})\sin(\alpha) \tag{7}\]  
Substituting Equations (6) and (7) into Equations (1) and (2), and then replacing the original variables and parameters with the following dimensionless variables and parameters: \(x^{*}=\frac{x}{c_{0}},\,y^{*}=\frac{y}{c_{0}},\,t^{*}=\xi t,\,\dot{x}^{*}= \frac{\dot{x}}{\xi c_{0}},\,\dot{y}^{*}=\frac{\dot{y}}{\xi c_{0}},\,\dot{x}^{*}= \frac{\ddot{x}}{\xi^{2}c_{0}},\,\dot{y}^{*}=\frac{\ddot{y}}{\xi^{2}c_{0}},\,p=\]  
Figure 2: Twelve-pole rotor system: (**a**) three-dimensional 12-pole system, and (**b**) the 12-pole system schematic diagram.