An Approach on V-Shaped Milling for Rotor Balancing of Armatures

Abstract

In order to improve the dynamic balancing accuracy of the micromotor armature, a method of V-shaped milling based on a discrete vector model for unbalance correction is proposed. The discrete vector model is fitted according to the parameters of the milling cutter and rotor, and then all the unit unbalance vectors in the discrete vector model are added to the milling center. The numerical relationship between the milling depth and the removal of the mass unbalance vector is obtained, and the accuracy of the model is verified via comparison with the data of the simulation experiments. The complexity of the integral formula of the numerical milling model makes it difficult to apply in practice. The discrete vector model does not require integration of the numerical formula and only considers the milling area as being composed of countless discrete blocks, which greatly simplifies the process of solving the unbalance vector. In view of the different thicknesses of the tooth surface of the armature, in order to avoid damage to the armature during milling, the unbalanced vector is decomposed at the center of the tooth surface by force decomposition. The experimental results show that this proposed method can effectively improve the dynamic balancing accuracy of the micromotor armature.

1. Introduction

As a crucial component of the mechanical system, rotating machinery plays an important role in many fields. Rotor unbalancing is one of the most common problems in rotating machinery [1] and is the main cause of rotor system faults [2,3,4]. A variety of factors, such as manufacturing, assembly errors, rotor wear, and operating conditions [5,6,7,8], as well as the irresistible centrifugal inertia forces generated by the rotation, will cause variable disturbance forces, which eventually lead to rotor vibration unbalance [9]. This vibration often increases the load of the bearing, aggravates wear, makes noise, and shortens the service life of the equipment. Eventually, fatigue gaps in rotating shafts may occur, resulting in safety accidents [10,11,12,13]. The dynamic balancing of the rotating body to effectively solve the above problems has become one of the essential processes in the manufacturing of motors [14].

According to the state of balance, rotors can be classified into balanced and unbalanced rotors. When the inertia axis of the rotor does not coincide with the rotation axis [15,16,17,18], the rotor is said to be unbalanced. The dynamic balancing technology is to change the rotor from an unbalanced state to a balanced state by adjusting the rotor’s mass distribution [19,20,21]. Ideal rotor balance is difficult to achieve in actual production. Generally, when the rotor operates stably and the amplitude of the rotor is reduced to the allowable range [22]—that is, residual unbalance vector—rotor balance is considered to have been achieved in engineering [23].

Correction is a vital step of rotor dynamic balancing. There are two methods of rotor dynamic balancing: weight removing and weight adding. The weight adding correction is mostly used for manual operation, while the weight-removing correction is mostly for automatic balancing machines, with high weight-removing efficiency. For weight removal correction, problems for the rotor, such as milling position deviations and poor milling accuracy, always exist. Depending on the different milling cutters, the main types of weight removal correction for automatic balancing machines for micromotor armatures are R-shaped milling unbalance correction and V-shaped milling unbalance correction. In addition to accurately measuring the unbalance signal, it is also necessary to obtain the size and position of the rotor unbalance mass through dynamic balancing analysis. Finally, accurate modeling is carried out through the magnitude of the unbalance between the milling material part and the rotor. Liu et al. [24] developed an unbalance weight removal model based on R-shaped milling cutters for the case of the distribution of slots on the circumference of the motor rotor. Liu et.al. [25] conducted integral modeling for V-shaped milling unbalance correction. To apply integral modeling to the programmable logic controller (PLC), they simplified some milling depth formulas using numerical analysis methods. Zeng et al. [26] carried out tooth face modeling for V-shaped milling. To decompose the initial unbalance vector of the rotor based on the maximum weight-removing milling vector, they derived an efficient multi-milling tooth correction milling method applicable to PLC. Fan et al. [27] established a mathematical model of the rotor online active balancing influence coefficient table method for the single-plane rotor online active balancing problem and simplified the influence coefficient table method to improve the online balancing efficiency. For the wear caused by repeated machining operations of milling cutters, Tseng et al. [28] developed a compensation algorithm to estimate the parameter settings required in the milling vector calculation matrix. Zhang [29] et al. proposed an iterative control method with a variable weight-removing rate to improve the unbalance correction accuracy for the structural characteristics of a thin disc component with a high diameter–thickness ratio, leading to a small unbalance mass that can also form a large unbalance vector. The micromotor armature requires extremely high correction accuracy, but the milling depth deviation, angle deviation, and model deviation will have a certain impact on the correction accuracy, so the accuracy of the milling model obtained by simplifying the integration formula cannot easily meet the accuracy requirements. Therefore, this paper proposes a discrete vector model to discretize and integrate the unit unbalance vector. The higher the discretization accuracy, the higher the integration accuracy.

A discrete vector model is established to solve the problem of position deviation and insufficient milling accuracy, then a set of discrete points according to the edge curve of the milling cutter and rotor to replace the milling space of the milling cutter to the rotor can be generated. Finally, the numerical relationship between milling depth and unbalance vector can be obtained. Combined with the modified influence coefficient method, this model can be applied to the micromotor armature milling correction to effectively improve the rotor correction accuracy.

2. Milling Correction for Rigid Rotor Balancing

A rigid rotor can be regarded as composed of countless particles, each of which generates centrifugal forces. The distance between each mass and the center of the rotor on its surface is different, which leads to different magnitudes. As shown in Figure 1, the principle of dynamic balancing of a rigid rotor is to make the unbalanced forces generated at various places when the rotor rotates at high speed equivalent at one or more positions, where A and B are equivalent surfaces. The equivalent force position is a point with no size, and the milling cutter mills a region around that point.

$$\overrightarrow{F_A}={\displaystyle \sum _{i=1}^n\frac{L_{Bi}}{L_{Ai}+L_{Bi}}\overrightarrow{F_i}}$$

(1)

$$\overrightarrow{F_B}={\displaystyle \sum _{i=1}^n\frac{L_{Ai}}{L_{Ai}+L_{Bi}}\overrightarrow{F_i}}$$

(2)

where $\overrightarrow{F_{\mathrm{i}}}$ is the unbalanced force generated by each face and $\overrightarrow{F_A}$ and $\overrightarrow{F_{B }}$ are the resultant forces distributed to measuring surface A and B, respectively, through $\overrightarrow{F_{\dot{i}}}$

The influence coefficient method is currently the most commonly used method for the dynamic balancing of rigid rotors. This method is suitable for the correction of dynamic unbalance conditions in rigid rotors and is widely used in engineering due to its simplicity. By measuring the linear relationship between the unbalance vector and the amplitude of the selected plane of the rotor, the influence coefficient and the correction mass of the plane are obtained so that the rotor balance can be achieved.

After the unbalance vector of the rotor has been measured and calculated by the influence coefficient method, a V-shaped milling-weight-removing correction is performed. According to the relationship between the size of the rotor unbalance vector and the milling depth of the V-shaped milling cutter, the milling cutter mills the calculated depth at the specified position, thus correcting the unbalance vector of the rotor for the balance. As shown in Figure 2, the milling cutter moves vertically to realize the radial milling of the rotor, thus removing the material at the location of the rotor’s unbalance mass. It is only necessary to know the radius of the milling cutter, the half-edge angle of the milling cutter, the length of the flat edge section of the milling cutter, and the material and radius of the rotor to calculate the depth that the milling cutter should mill for the rotor with a different unbalance vector.

For micromotor armatures, the integral formula for the milling model is too complex, and the simplified model leads to a lack of accuracy. In order to solve the above problems, this paper improves the dynamic balancing accuracy by establishing a discrete vector milling model and correcting the influence coefficients according to the actual milling conditions.

3. Analytical Model of V-Shaped Milling Weight Removal

Through a discrete vector modeling approach, all centrifugal forces generated in each unit area are concentrated and equivalent to the center line of the weight removal model. Since there exists a linear relationship between the centrifugal force generated in the milling area and the unbalance vector, by combining the equivalent mass approach with the results of the influence coefficient method, the milling position and depth are obtained to correct the balance of the rotor. The equivalent mass approach is that the milling part mass produces the same balancing effect at a point as the equivalent mass at this point.

As established in the V-shaped weight-removing vector model, Figure 3 shows the schematic diagram of the rotor and the V-shaped milling cutter during milling. The rotor is tangential to the milling cutter in the initial position, and the depth of mill is t = 0; with the increase of the depth of mill, the intersection area in Figure 3 gradually increases, where Figure 3a is the cross-sectional view of the milling cutter milling the rotor on the X–Z plane and Figure 3b is the sectional view of the milling cutter milling the rotor on the X–Y plane.

When z = 0, Figure 3b is a cross-section on the X–Y plane of the milling center, in which the left envelope in the shaded area shows the inner edge curve of the milling cutter after milling the rotor:

$$x(y)=\left\{\begin{array}{cc}R-t& 0\le \left|y\right|\le d/2\\ R-t+(\left|y\right|-d/2)cot\alpha & d/2<\left|y\right|<D/2\end{array}\right.$$

(3)

The right envelope of the shaded area is the edge curve of the outermost milling area of the rotor:

$$x(y)=\sqrt{R^2-y^2}$$

(4)

where R is the rotor radius; r is the radius of the milling cutter; α is the half-edge angle of the milling cutter; d is the length of the flat edge section of the milling cutter; t is the depth of mill; and D is the width of the milling cutter.

y₁ is the maximum value of the shaded area along the Y direction when z = 0 shown in Figure 3b. The expression for y₁ concerning the depth of mill t is obtained by combining Equation (2) with Equation (3) as:

$$y_1(t)=\left\{\begin{array}{c}\sqrt{R^2-{(R-t)}^2}0\le t\le R-\sqrt{R^2-d^2/4}\\ \begin{array}{l}\frac{-cot\alpha (R-t-cot\alpha \times d/2)+\sqrt{R^2\left(co{t}^2\alpha +1\right)-{(R-t-cot\alpha \times d/2)}^2}}{co{t}^2\alpha +1}\\ R-\sqrt{R^2-d^2/4}\le t\le R-\sqrt{R^2-D^2/4}+\left(D-d\right)\cot \alpha /2\end{array}\end{array}\right.$$

(5)

The shaded area of the cross-section on the X–Z plane in Figure 3a is related to the value of the Y-axis coordinate point. The left envelope of the shaded area is the inner edge curve of the milling cutter milling area:

$$x_1\left(z\right)=\left\{\begin{array}{cc}R+r-t-\sqrt{r^2-z^2}& 0|\le \left|y\right|\le d/2\\ R+r-t-\sqrt{\left[r-(\left|y\right|-d/2)cot\alpha {)}^2\right]-z^2}& d/2\le \left|y\right|\le D/2\end{array}\right.$$

(6)

The left envelope of the shaded area is the peripheral curve of the rotor:

$$x_2(z)=\sqrt{R^2-y^2}$$

(7)

where z₁ is the maximum value of the shaded area along the Z direction. The expression for z₁ concerning y is obtained by combining Equation (5) with Equation (6):

$$z_1(y)=\left\{\begin{array}{cc}\sqrt{r^2-{\left(R+r-t-\sqrt{R^2-y^2}\right)}^2}\hfill & \hfill 0\le \left|y\right|\le d/2\\ \sqrt{{\left(r-cot\alpha \left(\left|y\right|-d/2\right)\right)}^2-{\left(R+r-t-\sqrt{R^2-y^2}\right)}^2}\hfill & \hfill \left|y\right|\ge d/2\end{array}\right.$$

(8)

When the depth of mill t is determined, the X–Z cross-sectional area is divided along the Y-axis to obtain the expression for the volume of the milling area V as follows:

$$V={\displaystyle {\int }_{-y_1}^{y_1}dz}{\displaystyle {\int }_{-z_1}^{z_1}dy{\displaystyle {\int }_{x_1}^{x_2}dx}}$$

(9)

The rotor unbalance is generally expressed as U, the value of which is equal to the product of the mass of a point on the correction plane of the rotor and the distance from that point to the rotor axis, corresponding to the expression for the unbalance of the milling area U as follows:

$$U={\displaystyle {\int }_{-y_1}^{y_1}dz}{\displaystyle {\int }_{-z_1}^{z_1}dy{\displaystyle {\int }_{x1}^{x_2}x\rho dx}}$$

(10)

where ρ is the density of the rotor milling area.

4. Modification of V-Shaped Milling Correction Theory

4.1. Discrete Model of V-Shaped Milling Weight Removal

Due to the above Formulae (8) and (9), it can be seen that the integral formula is very complex, and it is difficult to obtain an accurate integral analytical formula. Therefore, a set of discrete points is generated according to the edge curve of the milling cutter and rotor to replace the milling space of the milling cutter on the rotor, on which the coordinates of each discrete point are (x_ijk, y_i, z_ij). The density of the rotor milling area is ρ, the discrete spacing is K, a single point occupies the space as K³, and the resulting unbalance effect is:

$$\overrightarrow{e}=\rho K^3\overrightarrow{r}$$

(11)

As shown in Figure 4, the milling area is centrosymmetric, and the two red squares in Figure 4a are all at the same value from the X-axis and Y-axis. In different X–Y sections, the centrifugal forces in the perpendicular direction to the axis of rotation counterbalance each other, leaving only the centrifugal force in the X-axis direction. The forces of the two discrete blocks are combined in the X-axis direction, and the magnitude is:

$$e_1+e_2=\rho K^{3}x+\rho K^{3}x=2\rho K^{3}x$$

(12)

The two discrete blocks in Figure 4b are symmetrical with the X-axis and Z-axis. The centrifugal force of the discrete blocks in Figure 4a is synthesized in the X-axis direction of the circular section in the above manner, and then the centrifugal force on the opposite symmetrical Z-axis is synthesized on the X-axis. At this time, the synthetic axis is the center line of the milling area, and the magnitude of the synthetic force is:

$$e_1+e_2+e_3+e_4=2\rho K^{3}x+2\rho K^{3}x=4\rho K^{3}x$$

(13)

Similarly, superimposition of all discrete spaces is performed, with the resulting unbalance effect being obtaining the volume V of this milling area with the unbalance vector U.

$$V={\displaystyle \sum _{i=1}^a{\displaystyle \sum _{j=1}^b{\displaystyle \sum _{k=1}^c{K}^3}}}$$

(14)

$$U={\displaystyle \sum _{i=1}^a{\displaystyle \sum _{j=1}^b{\displaystyle \sum _{k=1}^c\rho K^3{x}_{ijk}}}}$$

(15)

where a, b, and c are the number of discrete nodes corresponding to the Y-axis direction, Z-axis direction, and X-axis direction, respectively:

$$\begin{array}{ll}a=INT\left(\frac{2y_1\left(t\right)}{K}\right)\hfill & \\ b=INT\left(\frac{2z_1\left(y_i\right)}{K}\right)\hfill & i=1,2,\dots ,a\\ c=INT\left(\frac{x_2\left(z_{ij}\right)-x_1\left(z_{ij}\right)}{K}\right)\hfill & j=1,2,\dots ,b\end{array}$$

(16)

y₁ and -y₁ are the two edge points of the rotor milling area in the Y-axis direction, and y_i is the discrete point generated by y₁ and -y₁ along the inner side of the milling area in the negative direction of the Y-axis:

$$y_i=y_1\left(t\right)-(i-1)\times Ki=1,2\dots ,a$$

(17)

z₁ and -z₁ are the two edge points of the rotor milling area in the Z-axis direction, and z_ij is the discrete point generated by z₁ and -z₁ along the inner side of the milling area in the negative Z-axis direction:

$$z_{ij}=z_1\left(y_i\right)-(j-1)\times Kj=1,2\dots ,b$$

(18)

x₁ and x₂ are the two edge points of the rotor milling area in the X-axis direction, and x_ijk is the discrete point generated by x₁ and x₂ along the inner side of the milling area in the positive X-axis direction:

$$x_{ijk}=x_2\left(z_{ij}\right)-x_1\left(z_{ij}\right)-\left(k-1\right)\times Kk=1,2\dots ,c$$

(19)

To verify the accuracy of the discrete model, we use the 3D software-generated milling volume as a standard, which is compared to the volume generated by the simplified integration and the volume of the discrete model. The parameters of the micromotor armature and milling cutter used for the simulation experiments are shown in

Table 1. Micro-motor armature and milling cutter parameters.

	Description	Value
R	Rotor radius	11.5 mm
r	Milling cutter radius	13 mm
α	Milling cutter half-edge angles	60°
d	The length of the flat section of the milling cutter	0.4 mm
D	The width of the milling cutter	5 mm

. As shown in

Table 2. Comparison of milling volume.

Depth of Mill (mm)	3D Software Milling Volume (mm3)	Discrete Model Calculation Volume (mm3)	E1 (%)	Simplified Integral Model Volume (mm3)	E2 (%)
0.2	0.399	0.399	0	0.411	2.886
0.4	1.568	1.568	0	1.635	4.233
0.6	3.661	3.662	0.027	3.869	5.680
0.8	6.780	6.799	0.279	7.285	7.126
1.0	11.080	11.079	0.009	12.030	8.588

, E1 is the error between the volume generated by the 3D software and the volume calculated with the discrete model, and E2 is the error between the volume generated by the 3D software and the volume calculated with the simplified integration. From the numerical point of view, the error between the discrete model and the 3D model is extremely small, and the error between the integral model and the 3D model increases as the depth of mill increases. The main reason for this is that the simplified integral model simplifies the outermost arc as a straight-line tangent to it, which then forms a triangular area with the two diagonal lines of the milling model. As the depth of mill becomes greater, the gap to be filled becomes greater. Moreover, since the unbalance force induced by the unbalance mass is proportional to the distance from the center point of the mass to the axis of rotation. In the integral model, the position to be filled is furthest from the center of the rotor, which has a greater error in unbalance vector than its volume error.

4.2. Weighting and Milling Plane Conversion

By measuring the linear relationship between the unbalance vector and amplitude of the selected plane of the rotor, the influence coefficient and correction mass of the plane are obtained. Thus, the method of achieving the rotor dynamic balancing is called the influence coefficient method. As shown in Figure 5, during the trial weighting of the rotor, the weighting center does not coincide with the center of the milling area, and the trial weighting is a necessary step to obtain the rotor balancing mass using the influence coefficient method. The balancing amount and the trial weighting center are on the same plane.

The balancing mass obtained by the influence coefficient is the added weight, while the milling cutter milling is the removed weight, with a difference of 180° between the two. The dynamic balancing principle of the rotor is used to transform the unbalance vector in the trial weight center of the rotor towards the plane where the center of the milling area is located to obtain the new unbalance vector.

As shown in Figure 6a, the specific removed unbalance vectors $\overrightarrow{U_1}$ and $\overrightarrow{U_2}$ are obtained using the modified influence coefficient method The unbalanced forces generated by the removed weight part of the rotor during its high-speed rotation are $\overrightarrow{F_1}$ and $\overrightarrow{F_2}$, which are located on the same cross-sectional circle as the center of the rotor trial weight. F_a and F_b are the combined forces generated during the rotation of the rotor milling area, located on the same cross-sectional circle as the center of the rotor milling area, and they are not on the same cross-sectional circle. As shown in Figure 6b, $\overrightarrow{F_1}$ and $\overrightarrow{F_2}$ are transformed to the center face of the milling area, and finally, the resultant unbalance vector of the milling area is obtained:

$$\begin{array}{l}\overrightarrow{U_a}=\frac{a+x}{x}\overrightarrow{U_1}-\frac{b}{x}\overrightarrow{U_2}\\ \overrightarrow{U_b}=-\frac{a}{x}\overrightarrow{U_1}+\frac{b+x}{x}\overrightarrow{U_2}\end{array}$$

(20)

When the unbalance vector is large, the required milling depth is larger. If the center of the mill is close to the shaft end, this results in a half-length z₁ of the milling area exceeding the distance between the center of the mill and the shaft end. To avoid empty milling, the position of the milling center must be readjusted.

4.3. Milling Times of Micromotor Armatures

4.3.1. Small Unbalance

For the five-slot micromotor armature, as shown in Figure 7, the milling depth is inconsistent in different areas, with the maximum milling depth being when the center of milling is at the center of the tooth face. $\overrightarrow{F_a}$ is decomposed at the center of the nearest two adjacent tooth faces.

$$\overrightarrow{F_a}=\overrightarrow{F_{t1}}+\overrightarrow{F_{t2}}$$

(21)

There is a positive proportional relationship between the force and the balancing vector, and the decomposition of the force can be directly replaced by the decomposition of the balancing vector, that is:

$$\overrightarrow{U_a}=\overrightarrow{U_{t1}}+\overrightarrow{U_{t2}}$$

(22)

For the five-slot micromotor armature, the angle between every two slots is 2π/5. Assume that the angle between $\overrightarrow{U_a}$ and $\overrightarrow{U_{t1}}$ is θ and that the angle between $\overrightarrow{U_a}$ and $\overrightarrow{U_{t2}}$ is 2π/5-θ. The magnitudes of $\overrightarrow{U_{t1}}$ and $\overrightarrow{U_{t2}}$ are:

$$\begin{array}{l}\left|\overrightarrow{U_{t1}}\right|=\frac{\sin \left(\frac{2\pi }{5}-\theta \right)}{\sin \frac{3\pi }{5}}\left|\overrightarrow{U_a}\right|\\ \left|\overrightarrow{U_{t2}}\right|=\frac{\sin \theta }{\sin \frac{3\pi }{5}}\left|\overrightarrow{U_a}\right|\end{array}$$

(23)

After obtaining the magnitudes of $\overrightarrow{U_{t1}}$ and $\overrightarrow{U_{t2}}$, the actual milling is performed according to the corresponding relationship between the milling depth t and the unbalance vector $\overrightarrow{U}$.

4.3.2. Unbalance Exceeds a Certain Value

When the unbalance exceeds a certain value, if the milling cutter only mills at the position of the maximum tooth face thickness of the rotor because the two tooth face centers furthest away from the milling area are more than 90° to the direction of the milling force, only three tooth face centers can be weight removal balanced. The tooth face center closest to the milling force is milled with the maximum milling depth t_m, and the remaining residual force is decomposed to the other two tooth face centers.

For rotors with fewer slots, there are limitations in decomposing the milling forces only to the center of the tooth face. When the milling center is rightly the center of the tooth face, the milling depth range becomes larger. However, due to the low number of slots, the angle difference between the centers of the tooth surface is larger, resulting in the large milling force decomposed at the center of the tooth face. When the unbalance vector exceeds a certain value, the decomposition of two and three sides can no longer meet the balance requirements, so it is necessary to consider milling at the non-surface center.

When $\overrightarrow{U_{t1}}$ is larger than the maximum malleable unbalance vector $\overrightarrow{U_m}$, according to the angle occupied by the maximum malleable volume as β. As shown in Figure 8 and Figure 9, when milling multiple positions on the same tooth face, the angle between the center line of the tooth face and the center line of the position is γ. The magnitude of this unbalance vector $\overrightarrow{U_t}$ is:

$$\left|\overrightarrow{U_t}\right|=\frac{\left|\overrightarrow{U_{t1}}\right|-\left|\overrightarrow{U_{\max }}\right|}{2\cos \gamma }$$

(24)

For rotors with a large number of slots, the angle between the centers of the tooth faces is small. The periphery of the micromotor armature is composed of tooth surfaces and slots. The area of unbalance vector that can be milled by the milling cutter is the part where the slot is removed. If the milling center is not set at the thickest position of the tooth face, the milling area may be partially at the slot, resulting in empty milling, which will affect the actual dynamic balance accuracy of the rotor. The principle of milling unbalance vector distribution for multi-slot rotors is the same as for five-slot micromotor armatures.

5. Experiment Verification

The above analysis has completed the conversion between the discrete modeling of the milling area of the V-shaped milling cutter and the influence coefficient method of balancing vector and splitting times. The balanced accuracy of the method is verified by the weight removal pass rate of specific experiments. As shown in Figure 10, the experimental machines are dynamic balancing machine for measuring unbalance vector and milling unbalance mass machine. The rotor dynamic balancing machine uses the Keen KV-8000 PLC as the controller for the actual milling study. The selected milling cutter parameters are milling cutter radius r = 13 mm, milling cutter half edge angle α = 50°, and milling cutter flat edge section length d = 0.4 mm. The selected rotors are five-slot micromotor armatures, as shown in Figure 11. The rotor parameters are rotor radius R = 11.5 mm, rotor density ρ = 7.85 g/cm³, and rotor length L = 22.5 mm. Rotor milling is carried out according to the correspondence between the milling depth of the milling cutter and the magnitude of the unbalance vector mill. The ratio between the difference of the rotor unbalance before and after milling and the initial unbalance before milling is used as the basis for judging the balancing effect, i.e., the residual unbalance vector rate. The residual unbalance vector rate is one of the most important indexes for the comprehensive performance evaluation of a dynamic balancing machine and it is customary to convert this ratio into a percentage.

$$p_{ij}=\frac{\left|\left|A_{ij}\right|-\left|B_{ij}\right|\right|}{A_{ij}}\times 100\%$$

(25)

where A_ij is the initial unbalanced mass of the rotor, B_ij is the unbalanced mass of the rotor after milling, and P_ij is the residual unbalance vector rate of the rotor.

As shown in Figure 12, the above method and the rotor are applied to the dynamic balancing machine for testing. The micromotor armature balancing accuracy adopts G2.5 class, the armature speed is 2100 r/min, and the power rating of the electric machine is 288 W when the unbalance masses of both the left and right side of the rotor are below 20 mg, and all rotors qualified.

The result of the influence coefficient method is a mass that can be compared and balanced in two ways: one is by finding the corresponding unbalance vector by multiplying the mass found by the influence coefficient by the rotor radius, as shown in Figure 13a, to obtain the milling depth; the other is by obtaining the equivalent mass by dividing the unbalance vector by the rotor radius, as shown in Figure 13b, where the blue curve is the mass of the milling area and the red curve is the equivalent mass of the milling area.

As the milling machine has an accuracy of 0.01 mm, it can be seen in Figure 13b that as the milling depth becomes greater, the difference in mass between each 0.01 mm becomes greater. The larger the rotor unbalance vector, the larger the resulting error. There is also a slight difference in rotor material density from one manufacturer to another, which also affects the dynamic balancing accuracy.

The rotor has a maximum milling depth of 1 mm, an angle of 17.11°, and a milling mass of 86.42 mg. The milling center is set at a distance of 5 mm from the outermost circle of the rotor so that when the rotor has a milling depth of 1 mm, the half-length of the milling area in the direction parallel to the rotor axis is 5 mm and no empty milling occurs. The angle occupied by each tooth of the five-slot rotor is π/3. The farther away from the center of the tooth, the more fragile the part is, so the milling center of the splitting part must be as close to the center of the tooth as possible while also avoiding touching the tool. The unbalanced mass on the left side of rotor 2 is 93.75 mg at 252°. The unbalanced mass is greater than 83.83 mg, and the remaining 9.92 mg must be distributed at 234° and 270°. The rest of the rotor does not need to be split. As can be seen from the experimental results in

Table 3. Residual unbalance vector rate of the rotor under the discrete model.

Rotor Number	Initial Unbalance		Milling Unbalance	Milling Depth	Unbalance after Milling	Residual Unbalance Vector Rate
1	Left	60 mg 68°	43.79 mg 36°	0.74 mm	7 mg 91°	88.3%
			42.79 mg 108°	0.73 mm
	Right	29 mg 36°	19.98 mg 36°	0.51 mm	4 mg 65°	86.2%
			9.36 mg 324°	0.35 mm
2	Left	95 mg 263°	5.22 mg 234° 883.83 mg 252° 5.22 mg 270°	0.26 mm 1 mm 0.26 mm	12 mg 283°	87.4%
			16.06 mg 324°	0.46 mm
	Right	79 mg 273°	58.17 mg 252°	0.84 mm	2 mg 337°	97.5%
			32.77 mg 324°	0.65 mm
3	Left	111 mg 145°	74.46 mg 108°	0.95 mm	11 mg 146°	90.1%
			72.42 mg 180°	0.93 mm
	Right	84 mg 153°	32.58 mg 108°	0.64 mm	7 mg 11°	91.7%
			60.27 mg 180°	0.86 mm
4	Left	61 mg 135°	45.77 mg 108°	0.76 mm	10 mg 130°	83.6%
			29.79 mg 180°	0.62 mm
	Right	58 mg 134°	43.45 mg 108°	0.74 mm	10 mg 105°	82.8%
			26.07 mg 180°	0.58 mm
5	Left	90 mg 8°	78.09 mg 36°	0.97 mm	12 mg 343°	86.7%
			50.81 mg 324°	0.79 mm
	Right	35 mg 0°	9.28 mg 36°	0.35 mm	7 mg 323°	80.0%
			15.25 mg 324°	0.45 mm

, the residual unbalance vector of the rotor after milling is 87.4% with the use of the correction model. The residual unbalance mass on both the left and right sides of the rotor is below 20 mg, and all rotors qualified. This solution can achieve high accuracy weight-removing correction.

6. Conclusions

Focusing on the V-shaped milling weight-removing modeling, this paper analyzed the impact of the rotor milling area center and the trial weight center not being in the same plane on the rotor milling mass and milling position and improved the one-time weight-removing rate and weight-removing qualification rate of the rotor unbalance. With the objective of rotor dynamic balancing accuracy, the residual unbalance mass on both the left and right sides of the rotor is below 20 mg, and all rotors qualified.

The simplified integral model generates errors in cases when the depth of mill is high or the radius of the milling cutter is significantly larger than the radius of the rotor. In comparison with the simplified integral model, the discrete model is effectively able to avoid errors arising from filling the gap between the triangle and the arc. For small rotors, the discrete accuracy can reach 0.01 mm. For large rotors, the discrete accuracy should be selected so that the relative accuracy is the same as for small rotors, depending on the size of the rotor. If the discrete precision chosen is too small, the program will run too slowly or even fail to operate properly.

However, in actual motor operation, the milling of the eccentric effect can also cause changes in motor performance due to changes in the core state. In order to improve rotor performance, we will consider more rotor application scenarios in our future research to improve the eccentricity effect while avoiding negative effects on the application scenarios due to changes in stress, temperature, and magnetic flux.