## 1 Introduction  
Permanent magnet synchronous motors (PMSM) have been used in many fields due to their many advantages, such as their fast response, high efficiency, and so on [1]. With their widespread applications, the vibration and failure of motors has been an important issue, and the failures can be divided into mechanical failure and electrical failure [2; 3]. In general terms, the electromagnetic forces on the rotor surface are balanced when the air-gap distribution between the rotor and the stator is uniform [4]. However, the rotor is often eccentric with respect to the stator due to manufacturing tolerances, bearing wear and incorrect assembly, which means that the air-gap distribution is not uniform and the electromagnetic forces are imbalanced. Consequently, a net radial force, named unbalanced magnetic pull (UMP), is generated, and the UMP pulls the rotor towards to the stator in the direction of the minimum air-gap. The motion of the rotor can cause some vibrations and stability problems for the motor [5; 6], and even cause the motor to fail and affect the normal operation of the entire system. Generally speaking, the eccentricity between the rotor and stator can be classified into static eccentricity, dynamic eccentricity and a combination of the two [7; 8]. With static eccentricity, the axis of the rotor and the stator is misaligned and the rotor rotates about its axis. With dynamic eccentricity, the rotor does not rotate about its own axis. The position of the minimal air-gap is fixed with static eccentricity, while it rotates with the rotor with dynamic eccentricity. The eccentricity is the main cause of UMP, and many researchers have studied UMP for decades. The approaches can be classified into the analytical method and the finite element method. Some researchers use the finite element method to analyze the UMP for its convenience and efficiency, but the analytical method can identify the key factors and reasons in the production of UMP. In the early twentieth century, [PERSON] [9] assumed that the magnetic flux density was in proportion to the air-gap length and obtained the linear expression of UMP. Then, [PERSON] [10] presented a new improved equation of UMP by considering the effects of magnetic saturation. The linear equation of UMP was often used in engineering and it also solved some practical problems, but the accuracy was not reliable if the eccentricity was large. In the 1960s, [PERSON] [11] put forward the idea that the relationship between the UMP and the eccentricity was nonlinear and studied the influence of UMP on a synchronous machine. [PERSON] [12] presented the nonlinear analytical expression of UMP by expanding the air-gap permeance as a Fourier series. In recent years, many researchers have adopted the nonlinear analytical method to study the UMP, and the air-gap permeance approach was a commonly used method for calculating the magnetic flux density to obtain UMP. [PERSON] [13] presented an analytical expression of the UMP with any pole-pairs by expanding the air-gap permeance as a Fourier series and studied the effects of the UMP on the vibration of the rotor system. [PERSON] [14; 15] described a model based on the actual position of the rotor inside the stator and the model was more general than other models. [PERSON] [16] analyzed the dynamic behavior of a brushless direct-current (BLDC) motor when the motor was undergoing a mechanical and electromagnetic interaction due to the air-gap variation between the stator and rotor. [PERSON] [17] studied the circular whirling and stability of a synchronous generator under the UMP and mass eccentric force. [PERSON] [5] presented an analytical model to study the influence of nonlinear magnetic pull on rotor stability for a synchronous hydropower generator. [PERSON] [18] studied the UMP resulting from a non-uniform magnetic field and analyzed the nonlinear phenomenon vibrations for PMSM used in hybrid electric vehicles (HEV).