## 2 Equations of Motion
\begin{array}{ll}F_{CY}&=\frac{1}{4}\mu_{0}n^{2}A\cos(\theta)\bigg{[}\frac{8\cos^ {2}(\beta)f_{0}^{2}}{s_{0}^{3}}y-\frac{8\cos(\beta)f_{0}k_{1}}{s_{0}^{2}}y- \frac{4b_{0}k_{1}}{s_{0}^{2}}y+\frac{4l_{0}^{2}}{s_{0}^{3}}y-\frac{4l_{0}k_{2}} {s_{0}^{2}}\dot{y}\\ &-\frac{8\cos(\beta)f_{0}k_{2}}{s_{0}^{2}}\dot{y}+\frac{4l_{1}^{2}}{s_{0}^{3}}y ^{3}+\frac{8l_{0}^{2}}{s_{0}^{3}}y^{3}+\frac{16\cos^{4}(\beta)l_{1}^{2}}{s_{0} ^{3}}y^{3}+\frac{8\cos^{2}(\beta)k_{1}^{2}}{s_{0}^{3}}y^{3}\\ &-\frac{24\cos^{3}(\beta)f_{0}k_{1}}{s_{0}^{2}}y^{3}-\frac{12l_{0}^{2}k_{1}^{ 3}}{s_{0}^{3}}y^{3}+\frac{24\cos^{2}(\beta)k_{1}^{2}}{s_{0}^{3}}y^{2}-\frac{7 \cos^{3}(\beta)f_{0}k_{1}}{s_{0}^{3}}y^{2}\\ &+\frac{48\cos^{2}(\beta)f_{0}^{2}}{s_{0}^{3}}yx^{2}-\frac{24\cos^{3}(\beta) f_{0}k_{2}}{s_{0}^{3}}y^{2}\dot{y}+\frac{16\cos^{2}(\beta)k_{1}k_{2}}{s_{0}^{3}}y^{ 2}\dot{y}-\frac{12l_{0}k_{2}}{s_{0}^{3}}y^{2}\dot{y}\\ &+\frac{8\xi_{1}k_{2}}{s_{0}^{2}}y^{2}\dot{y}+\frac{4k_{2}^{2}}{s_{0}^{3}}y \dot{y}\dot{y}+\frac{8\cos^{2}(\beta)k_{2}^{2}}{s_{0}^{3}}y\dot{y}\dot{y}^{2}+ \frac{32\cos^{2}(\beta)k_{1}k_{2}}{s_{0}^{3}}y\dot{x}\dot{x}-\frac{48\cos^{3} (\beta)f_{0}k_{2}}{s_{0}^{4}}y\dot{x}\\ &+\frac{8\cos^{2}(\beta)k_{2}^{2}}{s_{0}^{3}}y\dot{x}^{2}+\frac{16\cos^{2}( \beta)k_{2}^{2}}{s_{0}^{3}}\dot{y}\dot{x}\dot{x}+\frac{16\cos^{2}(\beta)k_{1} k_{2}}{s_{0}^{3}}\dot{y}x^{2}-\frac{24\cos^{3}(\beta)f_{0}k_{2}}{s_{0}^{3}}y \dot{x}^{2}\end{array}  \tag{17}\]  
Inserting Equations (16) and (17) into Equations (8) and (9), we can obtain the whole dynamical model that governs the non-linear interaction between the rotor and the eight-pole actuator as follows:  
\[\begin{array}{ll}m\ddot{x}+c\dot{x}+k_{l}x+k_{n}(x^{3}+xy^{2})&=me\omega^{2} \cos(\omega\tau)+\frac{1}{4}\mu_{0}n^{2}A\cos(\theta)\bigg{[}\frac{8\cos^{2}( \beta)f_{0}^{2}}{s_{0}^{3}}x\\ &-\frac{8\cos(\beta)f_{0}k_{1}}{s_{0}^{2}}x-\frac{4l_{0}k_{1}}{s_{0}^{2}}x+ \frac{4l_{0}^{2}}{s_{0}^{3}}x-\frac{4l_{0}k_{2}}{s_{0}^{3}}\dot{x}-\frac{8\cos (\beta)f_{0}k_{2}}{s_{0}^{2}}\dot{x}\\ &+\frac{4k_{1}^{2}}{s_{0}^{3}}x^{3}+\frac{8l_{0}^{2}}{s_{0}^{3}}x^{3}+\frac{ 16\cos^{4}(\beta)f_{0}^{2}}{s_{0}^{3}}x^{3}+\frac{8\cos^{2}(\beta)k_{1}^{2}}{ s_{0}^{3}}x^{3}\\ &-\frac{24\cos^{3}(\beta)f_{0}k_{1}}{s_{0}^{4}}x^{3}-\frac{12l_{0}k_{1}}{s_{0}^{3 }}x^{3}+\frac{24\cos^{2}(\beta)k_{1}^{2}}{s_{0}^{3}}xy^{2}\\ &-\frac{72\cos^{3}(\beta)f_{0}k_{1}}{s_{0}^{4}}xy^{2}+\frac{48\cos^{4}(\beta) f_{0}^{2}}{s_{0}^{3}}xy^{2}-\frac{24\cos^{3}(\beta)f_{0}k_{2}}{s_{0}^{3}}x^{2} \dot{x}\\ &+\frac{16\cos^{2}(\beta)k_{1}k_{2}}{s_{0}^{3}}x^{2}\dot{x}-\frac{12l_{0}k_{1} }{s_{0}^{4}}x^{2}\dot{x}+\frac{8k_{1}k_{2}}{s_{0}^{3}}x^{2}\dot{x}+\frac{4k_{2 }^{2}}{s_{0}^{3}}\dot{x}\dot{x}^{2}\\ &+\frac{8\cos^{2}(\beta)k_{1}^{2}}{s_{0}^{3}}x\dot{x}^{2}+\frac{32\cos^{2}( \beta)k_{1}k_{2}}{s_{0}^{3}}x\dot{y}\dot{y}-\frac{48\cos^{3}(\beta)f_{0}k_{2}}{ s_{0}^{3}}x\dot{y}\dot{y}\dot{y}\dot{y}\dot{y}\dot{y}\dot{y}\dot{y}\dot{y\[ \begin{split}\ddot{u}+\mu\dot{u}+u+\lambda u^{3}+\lambda v^{2}u& =E\Omega^{2}\cos(\Omega t)+\beta_{1}u+\beta_{2}\dot{u}+\beta_{3}u^{3}+\beta_{4 }uv^{2}+\beta_{5}u^{2}\dot{u}+\beta_{6}u\dot{u}^{2}\\ &+\beta_{7}uv\dot{v}+\beta_{8}u\dot{v}^{2}+\beta_{9}u\dot{v}\dot{ v}+\beta_{10}\dot{u}v^{2}\end{split}  \tag{20}\]  
\[ \begin{split}\ddot{v}+\mu\dot{v}+v+\lambda v^{3}+\lambda u^{2}v& =E\Omega^{2}\sin(\Omega t)+\beta_{1}v+\beta_{2}\dot{v}+\beta_{3}v^ {3}+\beta_{4}vu^{2}+\beta_{5}v^{2}\dot{v}+\beta_{6}v\dot{v}^{2}\\ &+\beta_{7}v\dot{u}+\beta_{8}v\dot{u}^{2}+\beta_{9}\dot{v}u\dot{ u}+\beta_{10}\dot{v}u^{2}\end{split}  \tag{21}\]  
Equations (20) and (21) are the generalized dimensionless equations of motion of the considered control system, where the coefficients \(\beta_{1},\ \beta_{2},\dots,\ \beta_{10}\) are given in Appendix A.