## 1 Introduction  
In recent years, the demand for electric vehicles has increased rapidly as opposed to internal combustion engine vehicles, not only with increased interest in the performance of electric vehicles but also with increasing new problems. Electric vehicles are driven by a drive motor instead of an internal combustion engine, which generates less noise than an engine does. However, in electric vehicles, a new problem of hearing subtle noises that are not audible owing to engine noise is emerging. For this reason, the problem of noise, vibration, and harshness (NVH) of a vehicle is emphasized and is an important factor in the design of vehicles. The DC motors used in vehicles are divided into brushed DC motors and brushless DC (BLDC) motors. Brushed DC motors have the advantage of being easy to control in terms of speed and direction and are relatively inexpensive; however, there is a brush in contact with the commutator, meaning that high-speed rotation is difficult and the lifecycle of the engine is short. BLDC motors eliminate the brush and compensate for the shortcomings of DC motors, resulting in longer lifecycles and higher efficiencies. Owing to these advantages of BLDC motors, the utilization of BLDC motors is increasing, with several studies on the noise and vibration of BLDC motors currently being conducted [1; 2; 3; 4; 5]. BLDC motors, however, require additional controllers, and because of their high price compared to DC motors, traditional brushed DC motors are still widely used in many vehicles. Therefore, various studies have been conducted on the causes of noise and vibration in brushed DC motors [6; 7; 8; 9; 10]. In particular, there are research results that indicate that the electromagnetic force (EMF) generated by electromechanical machines causes noise and vibration; however, these studies did not identify the cause using modeling [11; 12; 13]. Several studies have been conducted to solve the noise and vibration problems in motors. For example, [PERSON] et al. [14] and [PERSON] et al. [15] conducted studies to solve the motor vibration problem analytically; [PERSON] et al. [16] and [PERSON] et al. [17] solved the noise and vibration problem by changing the housing outside the motor, and [PERSON] et al. [18] studied the reduction of noise and vibration by changing the internal structure of the motor. In addition, [PERSON] et al. [19] suggested a vibration reduction method for brushed DC motors using the sawtooth edge pole (SEP) method, and [PERSON] et al. [20] reduced motor vibration by attaching a copper ring to the air gap of a permanent magnet. However, all of these studies dealt with methods to reduce rather than identify noise vibration components. To develop a model for predicting the vibration component of a motor, a control model should be considered, and various studies on effective control methods have been conducted. [PERSON] [21] modeled a DC motor using MATLAB Simulink for the effective control of wheeled mobile robots. [PERSON] et al. [22] demonstrated various types of motor modeling using state space and transfer functions. [PERSON] et al. [23] modeled a DC motor to control the motion of a robot arm, and [PERSON] [24] combined the functions of Powersys and Simulink to model the motor. Recently, research has been conducted using methods such as linear quadratic regulator (LQR) control [25], Fuzzy Logic (FL), and artificial neural networks (ANNs) [26]; however, these studies did not consider the noise and vibration of the DC motor. On the other hand, [PERSON] et al. [27] revealed that the window motor is an important part of the noise and vibration of the vehicle as it is located closest to the vehicle user and studied the method of reducing noise by analyzing the vehicle door with ANSYS structural analysis. Instead, the noise generated by the plate was reduced rather than directly reducing the noise of the motor. This study proposes a new 1D model that can analyze noise and vibration by adding a vibration component to a brushed DC motor model.