New Equivalent model and Modal Analysis of Stator Core-Winding System of Permanent Magnet Motor With Concentrated Winding

An accurate prediction of modal characteristics of motor stator is essential in order to design a low vibration motor and to operate it quietly. Scholars have done a lot of research on the modal analysis of stator core and winding. However, there are few papers for the 0th-order mode shape and frequency of the stator system. This paper aims to propose a new stator system analysis model, which can effectively analyze the 0th-order mode and improve the model calculation efficiency. Firstly, the modal of stator core and stator core-winding system were analyzed theoretically. Secondly, four stator system finite element analysis models were established. The modal analysis of the stator core and stator core-winding system were carried out using the four models. The validity and computational efficiency of the four models of the stator core and stator core-winding system were compared. A new stator system analysis model was proposed. Thirdly, the modal tests of the stator core and stator core-winding system were carried out by hammering method, and the validity of the finite element analysis model was verified. Compared with the traditional model, the newly proposed model can analyze the 0th-order mode more effectively and improve the computational efficiency.


I. INTRODUCTION
As motor design is key to the development of electric vehicles (EVs) and hybrid EVs (HEVs), it has recently become the subject of considerable interest [1]- [3]. An accurate prediction of modal characteristics of motor stator is essential in order to design a low vibration motor and to operate it quietly [4], [5]. The modal analysis of the stator core and winding is a difficult point in motor modal analysis. Scholars have done a lot of research on the modal analysis of stator cores and windings.
Since the stator core is a typical laminate structure, its stiffness will increase without considering the stacking effect, and then the calculated natural mode frequency deviates greatly from the actual situation. It is not reasonable to use the material properties of silicon steel sheet directly in the modelling The associate editor coordinating the review of this manuscript and approving it for publication was Zhong Wu . process of stator core. In paper [5], a method to identify the physical parameters of laminated core and windings is proposed based on the modal testing of the motor stators with different conditions. Comparing the results of 3-D finite element analysis and experimental measurements, the equivalent Young's modulus of the isotropy laminated stator core and the windings are obtained. The stator is treated as homogeneous orthotropic laminate structure material, the recommended fitting curve is used to obtain material property parameters [6]. In paper [7], the laminated stator core is assumed to be modeled as a continuous solid with composite material, of which the equivalent Young's modulus is calculated by the Voigt-Reuss formula. In addition, since the circumferential modes have a major contribution in vibration generation, the equivalent material properties assigned to the stator core are isotropic. In paper [8], a novel approach of equivalent material identification is developed for multi-layered orthotropic structures. A numerical model of a laminated stack's dynamics applicable to general laminated structures was developed. A simple linear contact model facilitated the computational efficiency and in this way enabled the modeling of the stack's dynamics using a large number of laminas [9]. Paper [10] introduces a simple and nondestructive method for the measurement of Young's modulus; it is then used in a finite-element (FE) program to determine the resonant frequencies of SRM (switched reluctance motor) stator vibration. The effects of mass density and Poisson's ratios are also discussed. The FE results are validated by vibration tests, which show good accuracy. The effects of laminated core on the rotor mode shapes are investigated in paper [11], experimental modal analysis was carried out for shaft and for rotor core mounted on shaft.
Paper [12] examines the effects of the stator windings and end-bells on stator modal vibration frequencies. The effect of the windings is equated to an increase of the pole mass. In paper [13], six kinds of winding equivalent models are established, and then compared with the modal test results. Taking a 48-slot 8-pole permanent magnet synchronous motor as an example, the influence of winding structure, winding immersion paint and winding end on the natural frequency of the stator of the motor is studied in paper [14]. Compared with the stator core structure, the natural frequencies of the low-order radial modes such as the 2nd, 3rd, and 4th steps of the core-and-winding structure are reduced before the varnishing, and the natural frequency is increased after the varnishing. The immersion process can greatly improve the structural rigidity and natural frequency of the motor. Comparing the calculated values with the measured values, the effect of the stator core shape and stiffness of windings in the slots are studied, as a result, in the 2.2kW motor, an equivalent Young's modulus as stiffness of windings in the slots is obtained as being about 1/100th that of copper [15]. In paper [16], the material properties of the orthotropic laminate structure and the windings were determined by finite element analysis and experimental measurements, and compared with the measured results. Studies have shown that for large asynchronous motors, the stator windings have a great influence on the natural frequency, and the elastic modulus is much lower than that of solid copper. The effective part of the winding and the end winding cannot simply calculate the natural frequency in the form of additional mass; the end winding quality and The effect of stiffness on the natural frequency generally cancels each other out, so the effect of the end winding on the natural frequency is relatively small. Paper [17] shows the development of a material model of the complete stator bar. Special attention was paid to the experimental determination of the material characteristics of the orthotropic composite space brackets. The numerical results have been evaluated against measurement.
In summary, scholars have done a lot of research on the modal analysis of stator core and winding. The research focuses on the analysis of the equivalent material properties of the stator core and winding, and the influence of the winding on the stator core mode. However, there are few papers for the 0th-order mode shape and frequency of the stator system. According to the existing literature, when the winding is equivalent to the model of additional mass, the 0th-order mode shape of the stator core shows good, but the contribution of the stiffness of the winding cannot be considered, and the deviation of the calculation result is large; when the winding is equivalent to a continuous entity, a large number of local modalities are generated, which reduces the efficiency of modal analysis.
This paper aims to propose a new stator system analysis model, which can effectively analyze the 0th-order mode, reduce the number of local modes caused by the stator winding equivalent model, and improve the model calculation efficiency. In section 2, the modal of stator core and stator core-winding system are analyzed theoretically and compared with the results of finite element analysis. The results show that the influence of winding on stator core is divided into additional mass and stiffness. The influence of stiffness is not ignore. In section 3, four stator system finite element analysis models are established. It is demonstrated that model 1 and model 3 are effective in paper [4]. In this paper, model 1 and model 3 are improved based on previous research results, model 2 and model 4 are obtained respectively. The modal analysis of the stator core and stator core-winding system is carried out using the four models. The validity and computational efficiency of the four-mode analysis of the stator core and stator core-winding system are compared. The influence of the winding on the stator core mode is analyzed by model 4.
In section 4, the modal tests of the stator core and stator core-winding system are carried out by hammering method, and the validity of the finite element analysis model is verified. In section 5, some conclusions are drawn.

II. THEORETICAL ANALYSIS
In the conventional method, the stator core-winding system is simplified to a ring having a certain thickness, and the stator teeth and windings are simplified to an additional mass. The natural frequency of the m-order mode in the circumferential direction of the stator system can be expressed as where K m is the concentrated stiffness of the stator system (N/m), M m is the concentrated mass of the stator core-winding system (kg). The concentrated stiffness and concentrated mass of the 0th-order mode (also called the breathing mode) in the circumferential direction can be expressed by equations (2) and (3), respectively.
where, E c is the Young's modulus, h c is the thickness of the core, M c is the mass, D c is the average diameter, ρ c is mass density of stator core, k i is lamination coefficient, k md is additional mass factor. The k md can be expressed by the formula (4).
where, M t is the mass of the stator teeth, M w is the mass of the stator winding, M i is the mass of the insulation. Bringing the formulas (2) and (3) into the equation (1), the frequency of the 0th order mode in the circumferential direction is Equation (5) shows that the natural frequency of the 0th-order mode is inversely proportional to the stator diameter, square root of density, lamination coefficient, and addition mass coefficient. This indicates that motors with smaller diameters, densities, lamination factors, and addition mass coefficient have higher frequencies. The natural frequency of the 0th order mode is proportional to the square root of Young's modulus. This indicates that the motor with higher elastic modulus has a higher 0-order mode natural frequency.
The natural frequency of the mth mode (m ≥ 2) can be expressed as where, In Equation (9), k mrot is the additional mass factor for rotation, s 1 is the number of stator teeth (slots), c t is the tooth width, and h t is the tooth height, I c is the area moment of inertia about the neutral axis parallel to the cylinder axis.
Taking a 2.2kW concentrated winding permanent magnet brushless DC motor for electric vehicle as an example, the modal of the stator core-winding system is calculated according to equation (5) and equation (6). The basic dimensions of the permanent magnet brushless DC motor are shown in Table 1, lamination coefficient k i = 0.96, Young's modulus E c = 185000MPa, mass density of stator core ρ c = 7495kg/m3, mass density of winding ρ cw = 8960kg/m3. The modal calculation results are shown in Table 2. The modal shape (m, n) represents the state of the stator system after deformation, m represents the order of the circumferential deformation, and n represents the order of the axial deformation. According to the analysis results of Table 2, the maximum error between the analytical method and the finite element method is 13.5%. Large error is caused while the stator teeth and windings are simplified to an additional mass. The winding is located in slots which are separated by steel teeth. According to [19], the tooth-slot zone with the winding can be regarded as an additional ring internal to the stator core yoke. Thus, the natural frequency of the stator core with the winding will be In equation (11), the effects of the winding on the equivalent mass and equivalent stiffness of the stator core are simultaneously considered.

III. MODELING AND MODAL ANALYSIS A. EQUIVALENT model OF STATOR CORE
Accurately establishing the equivalent model of the stator core and defining the equivalent material properties of the stator core are the key to accurately analyzing the stator core mode. The stator core of the permanent magnet brushless DC motor is laminated by silicon steel sheets. In paper [4], the stator core is equivalent to the continuous entity composed of orthotropic materials, and the modal analysis is carried out by JMAG-Designer. The results are verified by experiment. In this paper, two kinds of model of the stator core are established.
(a) model 1: Stator core is equivalent to the continuous entity, the material type is set to orthotropy; (a) model 2: Stator core is equivalent to the continuous entity, the material type is set to anisotropy.
The equivalent model of stator core is shown in Fig. 2. The equivalent material property settings for model 1 have been elucidated in [4]. In this paper, the equivalent material property settings for model 2 are explained. Each components for the matrix displayed in the following formula should be specified when anisotropy is set according to paper [17].  The calculation results of model 1 and model 2 are shown in table 4. The calculation error between model 1 and model 2 is within 1%, and the calculation accuracy is similar. model 1 cannot calculate the frequency of the (0,0)th order mode, and model 2 can calculate the frequency of the (0,0)th order mode.

B. EQUIVALENT model OF WINDING
As shown in Fig. 3(a), in the paper [4] the coil of stator winding is equivalent to a conductor that has the same shape as the stator slot and is in good contact with the stator slot. The analytical accuracy of this equivalent model is high. However, the model has many local modes of the winding, which greatly reduces the calculation efficiency of modal analysis. At the same time, it is difficult to analyze the mode shape of the 0th-order mode because of the interference of a large number of local modes. To solve these problems, a new winding equivalent model is proposed. As shown in Fig. 3(b), the coil of stator winding is equivalent to a thinwalled tubular body that is in good contact with the stator slots. The thickness of the thin wall tube should be less than the length of the grid, in this paper, the thickness of the thin wall tube is 1.0 mm. As described in equation (11), the effect of this equivalent winding on the stator core mode has two aspects: (1) increase the equivalent mass of the stator core; (2) increase the equivalent stiffness of the stator core.

C. EQUIVALENT model OF STATOR CORE-WINDING SYSTEM
As shown in Fig. 4, based on the previously proposed models of the stator core and the winding, two models of the stator core-winding system are established.
(a) model 3: The stator core is equivalent to a continuous entity, and the material property is set to an anisotropic material; The windings are equivalent to mutually independent columnar bodies, the interface shape of the columnar body is consistent with the shape of the slot as shown in Fig.3(a), the material property is set to an orthotropic material. (b) model 4: The stator core is equivalent to a continuous entity, and the material property is set to an anisotropic material; The windings are equivalent to thin-walled tubular bodies that is independent of each other, and the tubular body is closely connected to the stator slots, the material property is set to an orthotropic material. The material parameters of the stator core are shown in table 3, and the equivalent material parameters of the winding are shown in table 5. The calculation results of model 3 and model 4 are shown in table 6. The analysis results show that the modal analysis results of the second to fifth orders are similar, and the error is controlled within 5%. model 3 cannot calculate the frequency of the (0,0)th order mode, and model 4 can calculate the frequency of the (0,0)th order mode.

D. COMPARATIVE ANALYSIS OF 0TH ORDER MODE OF FINITE ELEMENT MODELS
The 0th-order mode has an important influence on the vibration noise of the permanent magnet motor for electric vehicle driving according to paper [21], [22]. In this paper, the 0th-order mode of the stator core is analyzed based on model 1 and model 2. The analysis results are shown in Figure 5 and Figure 6.  It can be seen from the figure 5 that the 0th-order mode analyzed by model 1 is close to the (0, 2) order mode, and the mode shape of the (0, 0) order mode of the stator core cannot be well displayed. Based on the model 2, the (0, 0) order mode is well displayed as shown in the figure 6. In the commercial analysis software JMAG, the stator core equivalent material model has three types: isotropic material, orthotropic material and anisotropic material. When using an orthotropic material model, you need to input the equivalent Young's modulus in the three directions x, y, and z. If the equivalent Young's modulus in the z direction is too small, the 0-order modal shape is (0, 2) displayed as Figure5; when the anisotropic material model is used, the effect is improved, and the modal shape is (0, 0), as shown in Figure 6.
The analysis results of the 0th-order mode of the stator core-winding system analyzed based on the model 3 are shown in Fig. 7. It can be seen from the figure that the 0thorder mode of the stator system analyzed by the model 3 is approximated as the (0, 1) order mode shown as figure 7(a), and the stator winding has a very obvious local mode shown as figure7(b), which reduces the calculation efficiency. That because the winding uses orthotropic material model, the axial equivalent Young's modulus of the winding is too small. The analysis results of the 0th-order mode of the stator core-winding system based on the model 4 are shown in Fig. 8. As can be seen from the figure, the modal shape of the 0th-order mode analyzed by the model 4 is very ideal, and the winding has no local mode, That because the winding model is changed to a thin walled tube model, the axial equivalent Young's modulus increases, and the effect is improved, the modal shape is (0,0) as shown in figure 8(a). Which is beneficial to improve the calculation efficiency.

E. COMPARATIVE ANALYSIS OF CALCULATION EFFICIENCY OF FINITE ELEMENT MODELS
In order to verify the computational efficiency of Model 1, model 2, model 3 and Model 4, the modal of the stator core-winding system is analyzed using the same computer. The CPU of the computer is Inter(R) Core(TM) i7-4810MQ, the memory capacity is 16GB, and the hard disk model is faspeed K5-256G SCSI.
When using model 1 for analysis, the analysis frequency range is 200∼8184Hz, which takes 46 seconds. When using model 2 for analysis, the analysis frequency range is 200∼8236Hz, and the consumption time is 39 seconds. The calculation speed of model 2 is higher than that of model 1 for using material type anisotropy instead of orthotropy.
When using model 3 for analysis, the analysis frequency range is 200∼8530Hz, which takes 29 minutes and 49 seconds. When using model 4 for analysis, the analysis frequency range is 200∼9039Hz, and the consumption time is 12 minutes and 28 seconds. The calculation speed of model 4 is significantly higher than that of model 3 because the winding model in model 3 produces a large number of local modalities.

F. INFLUENCE OF WINDING ON MODAL OF STATOR CORE
The comparison results between the stator core and the stator core-winding system modality are shown in Fig. 9. It can be seen from the figure that the modal frequency of the stator core is reduced after considering the winding. The frequency of (0,0) mode is reduced from 8236Hz to 6934Hz, which is reduced by 1302Hz. Therefore, when the motor power is constant, it is important to improve the stiffness of the winding and reduce the winding quality to increase the frequency of the 0th-order mode of the motor.

A. MODAL TEST AND ANALYSIS OF STATOR CORE
In order to verify the validity of the finite element analysis, the modal tests of the stator core were carried out by the moving force hammer method. The test arrangement is shown in Figure 10. The test uses a 48-channel LMS-SCL220 data acquisition system, a PCB-356A33 acceleration sensor (frequency bandwidth 5120Hz), and 60 measurement points (5 rows in the circumferential direction and 12 columns in the axial direction) evenly arranged on the outer surface of the stator core. The test results are shown in Figure 11.
The peak of the frequency response curve in Fig. 11 is the natural frequency of each mode of the stator core. In the experimental test results, the modal order of the stator core includes (2, 0), (2, 1), (3,0), (3,1), (4, 0), (4,1) and (5, 0). The same order can be obtained by model analysis. Due to the accuracy of the sensor, the order (5, 1), (6, 0), (0, 0) obtained  in the model cannot be obtained through experiments. However, we usually think that the obtained order comparison can verify the correctness of the model. In order to verify   The comparison results are shown in Figure 12. It can be seen from Fig. 12(a) that the absolute error of the finite element analysis results of the model 1, the model 2 and the modal test analysis results are small. It can be seen from Fig. 12(b) that the relative errors of Model 1, model 2 and the experimental analysis results are all controlled within 1.5%, the calculation accuracy is high. model 2 has higher accuracy than model 1.

B. MODAL TEST AND ANALYSIS OF STATOR CORE-WINDING SYSTEM
In order to verify the validity of the model 3 and model 4, the modal test of the stator core-winding system is carried out. The test method is the mobile hammer method, and the test sensor arrangement is shown in Figure 13. The test uses a 48-channel LMS-SCL220 data acquisition system, a PCB-356A33 acceleration sensor (frequency bandwidth 5120Hz), and 60 measurement points (5 rows in the circumferential direction and 12 columns in the axial direction) evenly arranged on the outer surface of the stator core.
The test results of stator core with winding are shown in Figure 14. The peak of the frequency response curve in Fig. 13 is the natural frequency of each mode. In the experimental test results, the modal order of the stator core includes (2, 0), (2, 1), (3, 0), (3, 1), (4, 0), (4,1) and (5, 0). The same order can be obtained by model analysis. Due to the accuracy of the sensor, the order (5, 1), (6, 0), (0, 0) obtained in the model cannot be obtained through experiments. However, we usually think that the obtained order comparison can verify the correctness of the model.A comparison of the test results with the finite element results is shown in Fig. 15. It can be seen from Fig. 15(a) that the absolute error between the analysis results of model 3 and model 4 and the experimental results is small. From Figure 15(b), it can be seen that the relative errors are all controlled within 4%. The maximum absolute error value of model 3 is 3.26%, and the maximum absolute error value of model 4 is 3.24%. model 4 not only improves the calculation efficiency, but also improves the calculation accuracy.
The comparison of analytical analysis and experiment is shown in table 7. It can be seen from the table that the maximum value of the absolute value of the analytical analysis is 13%, and the analysis accuracy is poor. The maximum value of the absolute value of the finite element model error is 2.8%, and the accuracy is high. The analytical analysis and calculation speed is fast, suitable for rapid calculation in the early design period, but accurate analysis is suitable for the final check and optimization of the model.

V. CONCLUSION
This paper aims to propose a new stator system analysis model. Firstly, the modal of stator core and stator core-winding system were analyzed theoretically. Secondly, four stator system finite element analysis models were established. The modal analysis of the stator core and stator core-winding system were carried out using the four models. The validity and computational efficiency of the four-mode analysis of the stator core and stator core-winding system were compared. Thirdly, the modal tests of the stator core and stator core-winding system were carried out by hammering method, and the validity of the finite element analysis model was verified. Some conclusions are drawn: (1) The influences of winding on stator core are divided into additional mass and stiffness. When ignoring the equivalent stiffness of the winding, the modal analysis of the stator system produces a large error; (2) A new stator system analysis model is proposed. As described in model 4, the stator core is equivalent to a continuous entity, and the material property is set to an anisotropic material; The winding is equivalent to thin-walled tubular bodies that are independent of each other, and the tubular body is closely connected to the stator slots; (3) Compared with the traditional model, the newly proposed model can analyze the 0th-order mode more effectively; (4) The newly proposed model can effectively reduce the local modal interference of winding and improve the computational efficiency.