## 3 Derivation of Theoretical Force Model
### Theoretical Force Analysis of Motor  
The side view used for theoretical analysis is showed in Figure 7. An idealized model was assumed in which the end effects were neglected. The idealized model is thus similar to an idealized rotating machine. A distribution equation of the electromagnetic system can be expressed using [PERSON]'s equation and [PERSON]'s law. The magnetic vector potential A is defined as B =\(\
abla\times\text{A}\). Their solutions have previously been studied in related research areas [20; 21; 22; 23]. Therefore, we briefly describe the theoretical analysis of the flux density of the air gap. To establish analytical solutions as a function of the air gap, variations in the r-direction are ignored, and \(\
abla\cdot\text{A}=0\) can also be assumed, and six terms of g, h, t, p, \(\mu_{0}\) and \(\mu_{r}\) are air gap, the pole width, thickness of the stator, pole pitch, the permeability of air gap and the permeability of the material, respectively. The governing equation of the electromagnetic phenomena in the wobble motor can be calculated as  
\[\
abla^{2}\text{A}=\mu\sigma\bigg{\{}\frac{\partial\text{A}}{\partial t}- \textbf{v}\times(\
abla\times\text{A})\bigg{\}} \tag{5}\]  
Figure 7: Geometric model for theoretical analysis. where \(\mu_{0}=4\pi\times 10^{-7}N/A^{2}\) is the permeability of the free space, v is the speed of the moving rotor (m/s), and \(\sigma\) is electrical conductivity (\((\Omega\cdot m)^{-1}\)). \(B\) is expressed from the Fourier series as  
\[B=\sum_{n=1,3,5\cdots}^{\infty}B_{m}e^{i(\omega t-\beta x)}\widehat{j} \tag{6}\]  
where \(B_{m}=4B/n\pi\sin(\beta d/2)\) and \(\beta=(n\pi/p)\). \(\beta\) and \(\omega\) are the wave number and angular velocity, respectively. The y-direction flux density in the air gap can be calculated using the following equation:  
\[ \begin{array}{l}\text{B}=\
abla\times\text{A}=\left(\frac{\partial A_{y}}{ \partial y}\right)\widehat{i}-\left(\frac{\partial A_{y}}{\partial x}\right) \widehat{j}=B_{x}\widehat{i}+B_{j}\widehat{j}\\ B_{x,II}=\frac{\partial A_{II}}{\partial y}=\sum_{n=1,3,5,\cdots}^{\infty} \left(\lambda_{1}C_{II}e^{\lambda_{1}y}+\lambda_{2}D_{II}e^{\lambda_{2}y} \right)e^{i(\omega t-\beta x)}\\ B_{y,II}=-\frac{\partial A_{II}}{\partial y}=\sum_{n=1,3,5,\cdots}^{\infty} j\beta\left(\lambda_{1}C_{II}e^{\lambda_{1}y}+\lambda_{2}D_{II}e^{\lambda_{2}y} \right)e^{i(\omega t-\beta x)}\end{array}  \tag{7}\]  
where \(\widehat{i}\) and \(\widehat{j}\) indicate x-and y-axis unit vectors, respectively, and \(II\) represents the layer of the air gap. where \(\lambda_{1,2}=\left(\gamma\pm\sqrt{\gamma^{2}+4\alpha^{2}}\right)/2\), \(\alpha^{2}=\beta^{2}+j\mu_{0}\sigma v_{s}\beta\), \(\gamma=\mu_{0}\sigma v_{x}\), and \(s\) and \(v_{s}\) indicate slip and a synchronous speed, respectively. The boundary conditions are as follows:  
\[ \begin{array}{l}B_{II}\big{|}_{y=0}\,,\quad B_{II,y}=B_{m}e^{i(\omega t- \beta x)}\,;\quad B_{II}\big{|}_{y=g}\,,\,B_{II,g}=B_{III,g}\,\text{and}\,\mu B_ {II,x}=\mu_{r}B_{III,x};\quad B_{III}\big{|}_{y=\infty}\,,\,A_{III}=0\end{array}  \tag{8}\]  
The coefficients in Equation (8) form boundary conditions that can be solved, and the coefficients is as follows:  
\[ \begin{array}{l}C_{II}=\frac{B_{m}(B_{0}+\lambda_{2}y)e^{\lambda_{2}y}}{j \beta(-e^{\lambda_{1}\delta x_{1}}+e^{\lambda_{2}\delta x_{2}}-\Delta)}\\ D_{II}=\frac{B_{m}(B_{0}+\lambda_{1}y)e^{\lambda_{1}\delta x}}{j\beta\left(e^{ \lambda_{1}\delta x_{1}}-e^{\lambda_{2}\delta x_{2}}+\Delta\right)}\end{array}  \tag{9}\]  
where \(\Delta=(e^{\lambda_{1}\delta}-e^{\lambda_{2}\delta})\beta\mu_{0}/\mu\). Finally, in order to calculate the Equation (7) on the basis of the change in the air gap [24], the air gap of the wobble motor can be simplified by representing it as a rectangular coordinate system as shown in Figure 8.  
The air gap, g, between the rotor and stator as a function of \(\theta\) is expressed and simplified. \(R_{s}\) and \(R_{r}\) are the radius of stator and rotor, respe  
Figure 8: Simplified geometry.