# A Multi-Physics Modeling-Based Vibration Prediction Method for Switched Reluctance Motors
## 3 The Proposed Multi-Physics Modelling-Based Method
### The Electromagnetic Module  
There are three main numerical methods available for the electromagnetic field simulation, including the finite element methods, boundary element methods, and finite difference methods. In this work, we adopt the most widely used finite element method based on the following assumptions [28]:  
1. Neglect both magnetic flux leakages from the excitation pole to the stator yoke through the stator pole space and from the excitation pole to the adjacent stator teeth. 2. The eddy current effect is negligible. 3. Neglect the magnetic field and motor end effect outside the boundary formed between the outer diameters of stator and rotor. 4. The equation \(B_{s}=\
abla\times\mathbf{U}\) is satisfied when the vector magnetic potential \(\mathbf{U}\) is introduced in the solution domain. Considering the above assumptions, the electromagnetic field problem in the plane \(\Omega\) can be expressed as the following boundary value problem [29]:  
\[\left\{ \begin{array}{l}\Omega:\frac{\partial}{\partial x}\big{(}\gamma\frac{ \partial U_{z}}{\partial x}\big{)}+\frac{\partial}{\partial x}\Big{(}\gamma \frac{\partial U_{z}}{\partial y}\Big{)}=-J_{z}\\ \Gamma_{1}:U_{z}=U_{0}\\ \Gamma_{2}:\gamma\frac{\partial U_{z}}{\partial n}=-H_{t}\end{array} \right., \tag{3}\]  
where \(\gamma\) is the magnetoresistance of the stator and rotor cores, \(J_{z}\) is the current density, \(U_{z}\) is the vector magnetic potential, \(H_{t}\) is the tangential component of the magnetic field, and \(\mathbf{\Gamma}_{1}\) and \(\mathbf{\Gamma}_{2}\) are the first and second boundary conditions, respectively. Equation (3) can be equivalent to a problem of the conditional variation as follows:  
\[\left\{ \begin{array}{l}W(U_{z})=\int_{\Omega}\!\left(\!\!\!\!\int_{0}^{B_{s} }\gamma B_{s}dB_{s}-J_{z}U_{z}\right)\!dxdy-\int_{\Gamma_{2}}\!\left(-H_{t} \right)U_{z}\!dl=\min\\ B_{s}=\sqrt{B_{s}{}_{x}^{2}+B_{s}{}_{y}^{2}}=\sqrt{\left(\frac{\partial U_{z} }{\partial x}\right)^{2}+\left(\frac{\partial U_{z}}{\partial y}\right)^{2}} \\ \Gamma_{1}:U_{z}=U_{0}\end{array} \right. \tag{4}\]  
Figure 3: Controller of SRM. After a finite element division of the entire solution domain of the electromagnetic field, the difference function for each small element can be given by:  
\[U=N_{i}U_{i}+N_{j}U_{j}+N_{m}U_{m}, \tag{5}\]  
where \(\mathbf{U}\) is the magnetic vector potential at any point in the element, \(N_{i}\), \(N_{j}\), and \(N_{m}\) are the shape functions that are related to the element node coordinates; \(U_{i}\), \(U_{j}\), and, \(U_{m}\) are the magnetic vector potentials on the element nodes. The algebraic equations of the nodal function can be obtained through converting the variational problem to another problem of finding the extremum of the energy function W. We can get the nodal magnetic potential by solving the algebraic equations using the Newton -Raphson method. Then, the field quantity can be obtained by a finite element post-processing. According to [30], the flux linkage \(\psi\) and inductance \(L\) through the phase winding can be expressed as  
\[\left\{ \begin{array}{l}\psi=l_{st}\int_{N_{r}}^{N_{t}}B_{sy}dx=l_{st}(U_{r}- U_{l})\\ L=\frac{\psi}{l}=\frac{l_{st}(U_{r}-U_{l})}{i}\end{array} \right. \tag{6}\]  
where \(l_{st}\) is the axial length of the motor stator. \(U_{r}\) and \(U_{l}\) represent the magnetic vector potentials on the left and right sides of the coil, respectively. The radial force \(F_{r}\) and tangential force \(F_{t}\) are calculated by Maxwell tensor method as follows:  
\[\left\{ \begin{array}{l}F_{r}=\frac{1}{2l_{0}}\oint_{S}\left(B_{sr}^{2}-B_{st }^{2}\right)dS=\frac{1}{2l_{0}}\frac{D}{2}l_{st}\sum_{i=1}^{n}\left[(B_{sri}^{ 2}-B_{sti}^{2})(\theta_{i}-\theta_{i-1})\right]\\ F_{t}=\frac{1}{\mu_{0}}\oint_{S}\left(B_{sr}B_{st}\right)dS=\frac{1}{\mu_{0}} \frac{D}{2}l_{st}\sum_{i=1}^{n}\left[(B_{sri}B_{sti})(\theta_{i}-\theta_{i-1}) \right]\end{array}