## 1 Introduction  
There has been growing interest in new power sources to replace the existing internal combustion engine. Among them, electric motors have been widely used as an alternative power source in industry because of their high-power efficiency and better performance in terms of noise and vibration compared to the internal combustion engine. In industry, electric motors have been mainly used in large plants and robots. Recently, electric motors have also attracted attention in the field of hybrid electric vehicles (HEVs) and electric vehicles (EVs) to overcome the problem of pollutant emission. In particular, electric motors used in the engines of EVs and HEVs operate in a wide range from low to high speeds. During the design of these electric motors, characteristics have to be evaluated not only at the rated speed but also in the entire operating speed range. The vibration of a motor can causes fatigue failure of not only the motor parts but also the system components around the motor. Since the vibration of the system is directly related to the driver's comfort, the analysis of motor vibration characteristics over various operating speed ranges is very important in the design process of electric motors. Electromagnetic force, which causes the rotational motion of the electric motor, is one of the excitation sources of motor vibration. The electromagnetic force, determined by the combination of the slot numbers and the pole numbers of the stator and rotor, causes vibration [1; 2; 3; 4]. In the 1990s, motor vibration reduction design was studied by considering the dynamic characteristics of the stator core only [5; 6; 7; 8; 9]. Then, both the electromagnetic characteristics and the mechanical characteristics of the stator core were considered based on the multiphysics analysis method [10; 11; 12; 13], which analyzes the natural frequency, mode shape, and frequency response with 2-D geometry, instead of the actual 3-D geometry. Since it is difficult to predict or calculate vibration characteristics accurately using 2-D geometry results, vibration analysis has started to use 3-D geometry of the stator core structure with an electromagnetic-structural vibration coupled analysis method [14; 15; 16; 17; 18; 19; 20; 21]. Furthermore, in order to predict the motor vibration characteristics more accurately, the finite element (FE) model includes the housing part attached to the stator and other parts [22; 23; 24; 25; 26], but the rotor is not considered or simplified with a rigid body. This simplified FE model results in a discrepancy in the vibration characteristics, such as the natural frequency and mode shape between the FE analysis results and the test results [27]. In addition, it is very important to investigate the motor vibration characteristics for the entire operating speed range because a motor can be used for many different applications, such as EVs and HEVs with various operating ranges. However, there are insufficient studies to predict the vibration characteristics of motors in a wide range of operating speeds. In order to predict the noise and vibration characteristics of the variable-speed permanent magnet synchronous motor (PMSM), a multiphysics model of stator core and coil was constructed to perform the vibration and noise analysis [28; 29]. Further, to analyze the vibroacoustic characterization of a PMSM motor powertrain system, a 3-D entire model was constructed to perform a multiphysics analysis [30]. However, this considered only the circumference modes of the motor and not the analysis of the influence of other components that made up the 3-D entire model. In the previous study, the influence of the rotor or other end shield components were found by using the comparison of the results between the 3-D entire model and the simplified model [27]. However, there was a limitation as only certain operating speed results could be investigated.