# Analysis of a Vibrating Motor Considering Electrical, Magnetic, and Mechanical Coupling Effect
## 2 Analysis Method  
### Electrical and Magnetic Analysis  
During the electrical and magnetic analysis, the vibrating motor was modeled with several different positions of the vibrating part, and current was applied through the coil. The governing equations are [PERSON]'s equations. \[ \begin{array}{l}\
abla\cdot D=\rho\\ \
abla\cdot B=0\\ \
abla\times H=J+\partial D/\partial t\\ \
abla\times E=-\partial B/\partial t\end{array}  \tag{1}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Method** & **Strength** & **Weakness** \\ \hline ECM & Fast & No consideration of nonlinear parameters \\ Commercial software & Easy to use & No consideration of nonlinear parameters \\ Proposed method & Consideration of nonlinear parameters & Requires complex post-processing \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison between the equivalent circuit method (ECM), commercial software, and the proposed method. Figure 1: (**a**) Exploded view of vibrating motor. (**b**) Projection view of vibrating motor. where \(D\) is electric flux density, \(B\) is magnetic flux density, \(J\) is current density, \(H\) is permanent magnetic intensity, and \(\rho\) is the charge density. For each position and current, the flux density was obtained by FEM. After solving [PERSON]'s equations, the flux density located at every node point in the model was acquired. The model used for electrical and magnetic analysis is shown in Figure 2.  
After obtaining the results, the flux density was converted into flux linkage, from which it is easy to calculate the inductance and speedance. The definition of flux linkage is the product of the flux density, coil turns, and the coil's sectional area. The flux density and flux linkage are shown in Figure 3.  
After calculating the flux linkage, inductance and speedance can be calculated using the following method: Inductance is defined as the derivative of flux linkage with respect to the current, and speedance is defined as the derivative of flux linkage with respect to displacement. Additionally, flux linkage, inductance, and speedance are all functions of displacement and current and these parameters can be used to calculate the back EMF. \[ \begin{array}{l}\lambda(y,i)=NS\overline{B}_{x}(y,i)\\ L(y,i)=\frac{\Delta\lambda(y,i)}{\Delta I}\\ K_{emf}(y,i)=\frac{\Delta\lambda(y,i)}{\Delta y}\end{array}  \tag{2}\]  
where \(i\), \(y\), \(N\), \(S\), and \(B_{x}\) are current, displacement, coil turns, coil sectional areas, and average flux density, respectively. \(\lambda(y,i)\), \(L(y,i)\), and \(K_{emf}(y,i)\) are flux linkage, inductance, and speedance, which are described as functions of displacement and current. By using flux density, the force that acts on the vibrating part was calculated using the Maxwell stress tensor method. The Maxwell stress tensor is a second-order tensor used in electromagnetic field analysis to represent the interaction between electromagnetic forces and mechanical momentum,  
\[F_{m}=\frac{1}{\mu_{0}}\iint\limits_{S}\left[(B\cdot n_{e})B-\frac{1}{2}B^{2}n _{e}\right]dS \tag{3}\]  
where \(F_{m}\) is the magnetic force, \(B\) is the flux density, \(n_{e}\) is the unit normal vector, and \(S\) is the closed surface of the selected part. The electromagnetic force (\(F_{m}\)) is the force acting on the magnet of the motor and it is expressed as the total force (\(F_{total}\)). Because of the permanent magnet, magnetic material, and current, the total force (\(F_{total}\)) is composed of the cogging force (\(F_{cogging}\)) and the current force (\(F_{current}\)). Figure 2: Model used for electrical and magnetic simulation. The cogging force is a purely attractive force between the magnetic material and the permanent magnet depending on the position of the vibrating part when there is no current in the coil. The current force is calculated by subtracting the cogging force from the total force on the vibrating part. Additionally, the force factor is calculated as the force divided by the current,  
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