# Analysis of a Vibrating Motor Considering Electrical, Magnetic, and Mechanical Coupling Effect
## 2 Analysis Method
### Electrical, Magnetic, and Mechanical Coupling Analysis  
To perform the coupling analysis, two differential equations, which are the forced vibration equation and the voltage equation, must be solved simultaneously. However, it is difficult to solve these correlated equations by hand. Therefore, numerical analysis is needed to solve these two equations. The numerical fixed-point iteration method was chosen to solve the electrical, magnetic, and mechanical coupling analysis. Figure 7 shows a flow chart that describes this method. The fixed-point method was chosen because it is easy to program in an application and the flow of the method can be seen and understood intuitively. Through this procedure, the forced vibration equation and the voltage equation can be solved simultaneously and the simulation displacement and current can be obtained. There are many other numerical methods that can be used to find the solution, such as the Newton-Raphson method or the modified secant method. These may provide the solution, but these methods require a more advanced level of programing skill. In the electrical and magnetic analysis, the voltage equation is expressed with nonlinear parameters such as inductance, speedance, and force factor as Equation (5). Input voltage and current are waveforms and can be written using [PERSON]'s formula. By dividing the input voltage into the input current, the impedance can be obtained as:  
\[\frac{V}{I}=Z_{e}=R+j\omega L(y,i)+j\omega K_{emf}(y,i)\frac{y}{i} \tag{8}\]  
where \(y\) and \(i\) are the initial displacement and current and \(Z_{e}\) is the impedance. Because the transfer function from Equation (7) is maintained, Equation (9) can be obtained:  
\[y=f(y,i)h. \tag{9}\]  
Figure 6: Frequency-based transfer function. The actual force depending on current and displacement can be expressed as follows:  
\[f(y,i)=K_{f}(y,i)i. \tag{10}\]  
By substituting Equations (9) and (10), Equation (11) can be obtained:  
\[Z_{e}=R+j\omega L(y,i)+j\omega K_{emf}(y,i)K_{f}(y,i)h. \tag{11}\]  
Here, the electrical, magnetic, and mechanical coupling effects are derived and, finally, the calculated impedance can be obtained using nonlinear parameters from the electrical and magnetic analysis and the transfer function from the mechanical analysis. By using this calculated impedance, the simulated current is obtained as follows:  
\[i_{simulated}=\frac{V}{Z_{e}}. \tag{12}\]  
The simulated displacement and impedance can also be obtained:  
\[y_{simulated}=K_{f}(y,i)hi_{simulated}, \tag{13}\]  
\[Z_{e_{e},~{}simulated}=\frac{V}{i_{simulated}}. \tag{14}\]  
Figure 7: Flowchart of analysis procedures. When using the numerical iteration method, the simulation was run until the initial values and simulated values converge. Before solving the voltage equation, the initial values of current and displacement, which are determined from the mechanical analysis, were set. For solving the voltage equation, the simulation procedure was as follows. In the beginning, frequency was given as an initial value, and the corresponding voltage and mechanical force were given. The initial values of current and displacement were fed into the simulation. By using the initial values, the initial impedance, which includes nonlinear parameters, was obtained. The value of the calculated current was acquired by dividing the voltage by the initial impedance. The calculated and initial current values were then checked to see whether they fall within the margin of error. The simulation was run until the current converged. After the value of the calculated current was obtained, the displacement iteration, which includes current information, was started using the calculated current. Because of the electrical, magnetic, and mechanical coupling effects, both the current and displacement iterations should be considered at the same time. When the iteration was complete, the simulated impedance, current, and displacement were attained.