Modeling and Analysis of Unbalanced Magnetic Pull in Synchronous Motorized Spindle Considering Magneto-thermal Coupling

: The unbalanced magnetic pull (UMP) is one of the main vibration sources of the motorized spindle. A calculation model of UMP in a synchronous motorized spindle considering the magneto-thermal coupling is proposed in this paper. The finite element analysis models of the electromagnetic field and the temperature field of a motorized spindle are first established. Then a two-way coupling analysis method considering the effect of temperature variations on electromagnetic material properties of the motor is proposed. An experiment is conducted to verify the efficiency of this method. The thermal deformations of the spindle are calculated and used to analyze the air-gap variations between rotor and stator of the built-in motor. The 3D finite element electromagnetic model is adopted to calculate the UMP in the motorized spindle. The analysis results show that the eccentricity caused by thermal deformation can generate large UMP in the motorized spindle.

To investigate the effect of UMP on motorized spindle, the UMP must first be calculated accurately. In general, the studies on UMP can be roughly classified into two methods: analytical method and finite element method [11]. The analytical method is relatively simple and can thus be used for study of the influence of the UMP on stability of the rotor motion, but it cannot involve the effect of slot harmonics and saturation effect of a ferromagnetic core of rotor and stator windings [12]. By comparison, the finite element method can consider more parameters and provide more accurate and reliable solutions, but it requires more computational resources and long calculation time [11,12]. Extensive research has been done on UMP due to air-gap eccentricity in recent years, including the static eccentricity [13], dynamic eccentricity [14], inclined eccentricity [15] and mixed eccentricity [16]. However, these studies on UMP are based on the assumption that both the rotor and stator are perfect cylinders. In engineering practice, the shape errors of the rotor are inevitable due to manufacturing deviation. Xu and Han [17] established a general electromagnetic model for the elliptical and corrugated shape deviations in PMSM supported by three-pole active magnetic bearings. In addition, curved dynamic eccentricity occurs to some extent when the shaft is bent under the effect of external forces. Di et al. [18] presented a new method to calculate the UMP in cage induction motors considering the curved dynamic eccentricity.
For motorized spindle, the thermal growth due to temperature raise becomes a critical issue to be considered with the increase of speed and power of motorized spindle [19]. The main heat sources in the motorized spindle are motor and bearings [20]. When the spindle unit is in rotational running state, the power loss of its motor and bearing friction heat cause temperature elevation and lead to thermal deformations of spindle parts [21]. Chen et al. [22] pointed out the air gap length can be altered by radial expansion of the spindle rotor. The experimental research of Chang et al. [23] also shows that the air gap length varies with the thermal deformation of the motorized spindle. Mover, Vaijone et al. [24] found that the variations of air gap length cause the change of stator copper losses and rotor eddy current losses. Li and Wu [25] drew a conclusion that the electromagnetic vibration of the motorized spindle increased with the rise of radial electromagnetic force when the air gap changes. The above researches all indicate that complex coupling relations exist between the air gap length and the thermal deformation of motorized spindle. However, up to now, the effect of magnetic-thermal coupling on UMP has not been well investigated. This paper presents a two-way magneto-thermal coupling model, in which the mutual interaction between the electromagnetic parameters and temperature rise can be considered. The thermal deformations of the spindle are obtained and used to calculate UMP based on a 3D finite element electromagnetic model. Furthermore, the air gap length and UMP of motorized spindle due to thermal deformations are analyzed.

Parameters of motorized spindle
The schematic of the motorized spindle investigated in this paper is illustrated in Fig.1. The shaft of the motorized is supported by two pairs of angular contact ball bearings. A built-in permanent magnet synchronous motor is adopted to drive the shaft to rotate. A water circuit cooling jacket is placed around the stator of the built-in motor. Oil-air lubrication is used for the bearings.
The parameters of the motor and bearings are listed in Tab. 1 where ns is the rotating speed of the spindle, M is the total frictional torque of the bearing.
The total frictional torque M is the sum of the load torque Ml and viscous friction torque Mv lv MMM = + The load torque Ml can be approximated by the following equation where fl is a factor related to the bearing type and load, pl depends on the magnitude and direction of the bearing load, and Dm is the mean diameter of the bearing.
The viscous friction torque Mv can be given by the following equation where fv is a factor related to bearing type and lubrication method, 0 v is the kinematic viscosity of the lubricant.
The main power losses occurring in the permanent magnet synchronous motor are stator winding losses, stator and rotor core losses [28]. Although the rotor eddy current loss is relatively small compared with the stator copper and iron losses, it may cause significant heating of the permanent magnets and cannot be ignored [29].
The stator winding losses PCu can be expressed by where s is the number of phases, I is the stator current, R is stator winding resistance and Ks is the skin effect coefficient for stator conductors.
The losses of the stator and rotor core losses PFe can be calculated by [28] = where ka is the factor taking the increase in losses due to metallurgical and manufacturing processes into account, B is the magnetic flux density, mc is the mass of the core and △p1/50 is the specific core loss at B = 1T and f = 50Hz.
The eddy current loss density distribution We can be obtained from [29] where γ is the electrical conductivity of the permanent magnet, hm is the thickness of the NdFeB material, an and bn are the harmonic amplitude of the magnetic flux density due to the effect of the armature reaction field, and n is the harmonic order.

Convection boundary condition
The coefficient of convection heat transfer c h can be computed by the following equation where Nu is the Nusselt number, λ is the thermal conductivity of the fluid, and Dh is the hydraulic diameter. A free convection can be assumed for ambient air around a stationary surface like the spindle housing and λ can be obtained as 9.7W/(m 2 K).
The Nu is determined based on different heat convection conditions [30]. For calculating Nu, the following equations are used [31,32] Where ufluid is the velocity, vfluid is the kinematic viscosity, cfluid is the specific heat capacitance and kfluid is the dynamic viscosity of the air. Under normal temperature, vfluid equals to16mm 2 /s, Pr equals to 0.701.

Electromagnetic field model of the motorized spindle
With the increase of temperature in the built-in motor, the resistance of the winding and the permanent magnet will change correspondingly, and the winding loss and the eddy current loss of the permanent magnet will change. In this paper, the influence of temperature rise on the material property of the motor was considered. The electromagnetic field model of the motor is established based on FEM in Ansoft Maxwell software, as shown in Fig.3. The model here is mainly to solve the electromagnetic loss of the built-in motor, and the accuracy requirement is not too high, therefore the adaptive mesh was adopted to save computation time. The detailed parameters of the motor are listed in Tab.2.

Magneto-thermal coupling analysis
A two way magneto-thermal coupling analysis was conducted based on the above models. In the coupling analysis, the electromagnetic losses obtained from the electromagnetic field model are imported into the temperature field model to calculate the temperature distribution, and the electromagnetic material property parameters are changed according to the temperature rise and the electromagnetic losses are then updated accordingly. The coupling iteration is repeated until the temperature value of two consecutive iteration steps show minor deviation, which is set as 0.1°C in this paper. In this way, the data of electromagnetic field and temperature field can be two-way transmitted. The scheme of the two-way coupling process is shown in Fig.4

Fig.4 Scheme of two way magneto-thermal coupling analysis
When the rotating speed of the spindle is 2,000 rpm and the ambient temperature is 18℃， the temperature field distributions of the motor are shown in Fig.5. Experiments were conducted on a motorized spindle test rig to verify the simulation method, as is depicted in Fig.6. The experimental motorized spindle is the corresponding physical entity of the model in Fig.1. To measure the temperature, a PT100 sensor was embedded in the motor. Since it is difficult to install sensors on the rotor, only the temperatures of the end winding of the motor were measured and used for comparison. As can be seen in Fig. 7, the simulated temperature values at a given time of 2400s are in good agreement with the experimental results. shaft to the other side as showed in Fig. 8(a). The figure also illustrates the deformations of the permanent magnet that occur due to the temperature rise of the motor. In Fig. 8(b), 30 uniformly distributed nodes of each layer are selected, and their corresponding deformations are obtained from the thermal-structure coupling model. The multilayer 3D model is shown in Fig. 8(c) and Fig. 8(d).    deformations, and the air gap length of the motorized spindle are obtained. It is found that the spindle generates thermal deformations that occur in three-dimensional space, which will cause uneven air gaps between the rotor and stator. The analysis results also reveal that complex coupling relations exist between the air gap length and the thermal deformation of the motorized spindle. Therefore, the thermal deformations must be considered in the calculation of UMP. The proposed method in this paper will be helpful for the further investigation of vibration in motorized spindle with curved dynamic eccentricity.

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