# Analysis of a Vibrating Motor Considering Electrical, Magnetic, and Mechanical Coupling Effect
## 1 Introduction  
Electronic devices have fascinated people all over the world, with emphasis being placed on improving performance. Especially in the case of vibrating machines, such as vibrating motors, the importance of analyzing the electromagnetic and mechanical systems is becoming increasingly significant. To enable improvement, an accurate method of analyzing a motor is required and there are various suggestions available for motor analysis.  
In previous studies, a novel design of a vibrating machine was analyzed using the finite element method (FEM) for the electromagnetic and mechanical fields [1; 2; 3; 4]. A novel linear impact-resonant actuator was analyzed using a magnetic circuit model, which represented the magnetic circuit using a lumped parameter method [5]. [PERSON] et al. proposed a novel design of a linear actuator with a large magnetic force to reduce the response time, which was analyzed by a transient analysis in the mechanical domain [6]. A resonant piezoelectric vibrator was developed and analyzed by outlining the mechanics of the piezoelectric unimorph [7; 8].  
There are several widely used approaches to conduct a multi-physics analysis, such as the equivalent circuit method (ECM), the lumped parameter method (LPM), and the finite element method. A new linear-electromagnetic actuator used for cellular phones was designed using the voltage equation of the equivalent circuit method that treated electromagnetic parameters such as inductance, speedance, and force factor as constant values [9]. A vibration actuator was developed based on active magnetic springs by taking advantage of polynomial fitting to outline the magnetic force characteristics [10], and [PERSON] developed a horizontal linear vibrating actuator and modeled the mechanical domain by a two-degrees-of-freedom system that considered the electromagnetic and mechanical domains separately [11]. However, in both cases, nonlinear parameters such as inductance, speedance, and force factors were not considered. In other application fields, to make the electromagnetic-mechanical coupling analysis convenient, commercial software (COMSOL Multi-physics) has been developed. This software has been used to analyze a loudspeaker driver [12; 13]. In the software, the mechanical forces are calculated as the product of a force factor and a current. The current is from the voltage equation thatconsiders the back electromotive force (EMF) term and both the current and displacement are calculated by the software.  
In previous research, the characteristics of the electrical, magnetic, and mechanical coupling with respect to the inductance, speedance, and force factor, which are caused by the vibration of the motor, were considered as constant values. In this study, the nonlinear parameters are considered as functions of displacement and current to analyze the performance of the vibrating motor using FEM and a numerical iteration method. The electrical, magnetic, and mechanical coupling analysis considering nonlinear parameters of the vibrating motor is used for securing an accurate motor analysis result. A comparison between the ECM, commercial software, and the proposed method is given in Table 1.  
The vibrating motor used in this research was cylindrical in shape and its outer diameter, height, direct current resistance (DCR), and input power were 10 mm, 4.1 mm, 12.6 \(\Omega\), and 1 W, respectively, as depicted in Figure 1.