1 Introduction

Gears are a vital fundamental component in the industry, widely used in aerospace, rail transport, new energy vehicles, and other industries. The processing quality of gears directly determines the service performance of the equipment, so the transmission system of high-end equipment puts forward higher requirements for the precision processing of gears [1]. The method named continuous generation grinding has significant advantages in industrial applications due to its high processing efficiency and machining accuracy [2]. To improve the gear transmission performance (load distribution, noise, vibration, etc.), it is usually necessary to modify the gear [3]. The traditional tooth profile modification is generated by the interpolation movement of the workpiece and the radial direction of the grinding wheel relative to the workpiece gear along the tooth profile modification curve [4]. However, the flank modification of the helical gear will produce a distortion change of the tooth profile angle from the top to the bottom of the gear flank, resulting in deviation of the gear flank. That is the phenomenon of gear flank distortion [5]. When using the generating grinding method to modify the topology of the helical gear flank, it has generally been simplified into tooth lead modification and tooth profile modification in the machining process. In the traditional machining process, when the workpiece gear needs to achieve tooth profile modification, it is necessary to modify the gear flank of the grinding wheel tool to the conjugate gear flank corresponding to the modified gear flank of the gear.

It not only reduces production efficiency but also increases the processing cost. At the same time, it is hard to achieve the unification of design and manufacture. Therefore, how to accurately modify gear tooth flanks of any topology without changing grinding wheels has become a critical issue in gear modification machining technology.

Aiming at the above problems, scholars have carried out relevant research mainly from the computer numerical simulation of tools and workpieces, high-order polynomial modification technology, and gear flank deformation suppression. Among them, in terms of numerical simulation of cutting tools and workpieces, scholars used computer technology to simulate the gear flank of workpiece cutting tools and the forming process of gears. LITVIN et al [6] and LIN et al [7] discussed the necessary and sufficient conditions for the two-parameter envelope of the tool surface and applied the process of machining helical gears with grinding wheels. HE et al [8] proposed a two-parameter point-vector (PV) envelope method, established the point-vector envelope principle and approximation algorithm, and applied the approach to the calculation process of the left and right tooth profiles of grinding wheels. The experimental results show that the method can be used to calculate and process any gear tooth profile with high machining accuracy. CHIANG et al [9] proposed a simplified two-dimensional numerical simulation method that can be used to simulate the grinding process of cylindrical gears and compared it with the three-dimensional numerical simulation method. This method projects the contact trace between the grinding wheel and the workpiece onto the plane. Compared with the 3D numerical simulation method based on CAD software, the calculation efficiency and stability are improved while ensuring the same accuracy. WU et al [10] simulated the grinding process of cylindrical spur gears and helical gears with the radial ray method (RRS). The numerical simulation results show that it still maintains high accuracy in the case of the quadratic envelope, interference, and overcut. SIMON et al [11] proposed a tooth modification calculation method to determine the optimal tooth top trimming and modification parameters for the load and stress distribution problem of modified gears. ZHOU et al [12] proposed a generated method to improve the efficiency by dressing the worm surface with only one path, and a closed-loop manufacturing process is applied to ensure machining accuracy.

In the research of higher-order modification technology, scholars realized the modification of gears in the form of higher-order polynomials in the form of additional motions on different motion axes according to the characteristics of various gear processing methods. SHIH et al [13-15] proposed a tooth-face correction method based on hypoid gear machining machine tools, which is divided into two main parts. In the first step: the required modified tooth flanks are obtained based on the preset transmission and bearing ratios; In the second step: the sensitivity matrix of the machine tool’s six-axis kinematic parameters is established, and then the machine tool tuning parameters are determined using a linear regression method. TRAN et al [16, 17] replaced the rotation angle and swing angle of the workpiece gear with the sixth-order McLaughlin formula in the honing process to simulate the gear honing process, and achieved the tooth profile by controlling the rotation angle and swing angle of the workpiece gear reshape. HAN et al [18] proposed a higher-order correction method for standard diamond grinding wheels based on an internal gearing power honing machine, representing the rotary axis of the honing head and the oscillating axis of the honing head holder as fourth-order polynomials concerning the axial motion of the workpiece gears and calculating the sensitive data matrix of the polynomial coefficients based on the deviation of the gear teeth and estimating the polynomial coefficients by the method of least squares, and finally realizing the method of modifying the profile and lead of the gear teeth. TIAN et al [19] proposed a continuous generation grinding tooth topology modification method based on contact traces, expressing the radial feed, tangential feed, and axial feed of the machine tool as a fourth-order polynomial, and the genetic algorithm is used to optimize the polynomial coefficients. Compared with the least square estimation algorithm based on the sensitivity matrix, the genetic algorithm has higher accuracy and stability, and it improves the machining accuracy of the gear flank topology modification.

In the research on gear flank distortion, scholars have conducted research on distortion suppression from the aspects of tool dressing, distortion compensation, and gear design. FONG et al [20] used the diamond wheel to modify the grinding wheel by adjusting the axial feed of the diamond wheel to obtain a variable lead grinding wheel (VLGW) and then used the variable lead grinding wheel to modify the tooth direction of the gear, so as to achieve a reduction of tooth flank distortion. TIAN et al [21] calculated the theoretical gear flank distortion according to the gear parameters and the tooth direction modification amount, took the distortion amount as the gear flank deviation, and used the kinematics inverse solution method to calculate the compensation amount of each feed axis of the machine tool, based on electronic gearbox deviation control for distortion compensation, realizing the reduction of gear flank distortion. YU et al [22] established an approximate mathematical model representing the relationship among gear flank distortion, gear parameters, and modification parameters, and compared the gear flank distortion calculated by the model with the distortion calculated based on the gear meshing principle. The comparison verifies the correctness of the model. On this basis, the influence of gear parameters on the gear flank distortion is studied, which helps to reduce the gear flank distortion in the design process. TANG et al [23] proposed an innovative geometric error compensation method for machining with non-rotary cutters based on two main points. The cutter rotation angle is considered for both the modeling and compensation of the geometric errors.

In summary, there is already deep research on topology modification and tooth profile modification technology. However, the gear flank deviation is mainly controlled by modifying the profile of the grinding wheel. This method requires special modifications to the standard worm wheel grinding wheel based on gear parameters. It reduces the processing efficiency and increases the processing costs, but it is hard to achieve the consistency of design and manufacturing of topological gear surface. At the same time, the research on gear flank modification is only at the stage of computer-aided simulation and calculation. Especially for high-order shape modification technology, how to accurately add the additional motion expressed in the form of high-order polynomials to the corresponding motion axis of the machine tool has not been explored in-depth and specifically.

In order to solve the above challenges, this paper proposes a flexible topological high-order modification technology of gear flank based on an electronic gearbox. Firstly, the grinding model of the double-drum topological gear flank of the gear and the continuous generation grinding of the grinding wheel is established, and the additional motion of each axis of X, Y and Z is expressed in the form of a fifth-order polynomial. During gear grinding, the grinding wheel has the characteristic of point contact with the workpiece gear, and equations are set up based on this characteristic to solve the polynomial coefficients. We use the polynomial interpolation function to establish a gear modification processing model based on the electronic gearbox. And finally, the topology modification of the target gear flank is generated.

2 Calculation of gear flank by continuous generative grinding method

2.1 Building gear flank model

The tooth profile of any axial section of the standard involute helical gear flank is a standard involute, as shown in Figure 1(a). Taking the left gear flank as an example, the curve BC is the standard involute tooth profile of the left gear flank. In order to obtain the standard involute equation, the coordinate system shown in Figure 1(a) is established with the line connecting the center point of the tooth space and the rotation center of the gear as the X-axis. Point M is just any point on the involute line, and the parameters of point M are easy to obtain the equation is as formula (1), where is the radius of the base circle, and is the roll angle of the occurrence line on the base circle, which is equal to the sum of the spread angle at point M and the pressure angle at this point M. is the starting angle of the involute, which is calculated by formula (2), where is the number of gear teeth, and is the pressure angle.

Figure 1全屏查看

(a) Involute gear profile; (b) Involute gear flank and topological gear flank

(1) (2)

The above-mentioned involute tooth profile is spirally moved along the gear axis direction to obtain the standard involute gear flank. The transformation matrix M is multiplied by the formula (1) to the left to obtain the standard involute flank which is shown in formula (3); the matrix M is shown in the formula (4), where is the rotation angle of the involute tooth profile along the gear axis and , is the helix angle of the base circle.

The topological gear flank modification can be simplified as the superposition of the axial modification and the profile modification. Taking crowning as an example, the axial modification equation is taken as formula (5), where is the maximum modification amount; and is the tooth width. The tooth profile modification equation is shown in formula (6), where is the maximum modification amount in the direction of the tooth profile, is the pressure angle of the addendum circle, is the pressure angle of the root circle, and is the pressure angle at any point M on the involute line. Considering the tooth profile modification equation and the axial modification equation comprehensively, the topological tooth flank equation is established as shown in formula (7), where , and the tooth flank normal vector can be obtained according to formula (8). The schematic diagram of topological tooth surface is shown in Figure 1(b).

(3) (4) (5) (6) (7) (8)

In order to describe the macroscopic morphology of the tooth flank, it is necessary to manually set the mesh surface. This paper uses the traditional mesh division method as shown in Figure 2(a). In order to simplify the representation of the modified gear flank compared to the standard gear flank variation, the involute helicoid of a standard helical gear is usually represented as a mesh plane. As shown in Figure 2(b), the grid numbers A to E correspond to the equally divided points in the tooth profile direction, and the grid point numbers 1 to 9 correspond to the equally divided points in the tooth axial direction. Based on the standard helical gear involute helicoid, when the grid point deviation (i=0, 1, …, 45) is negative, it is measured toward the tooth surface, indicating that the modified gear surface is cut more than the standard gear surface; when the grid point deviation is positive, it faces the outside of the gear surface, which means that the modified gear surface is cut less than the standard gear surface. (i=0, 1, …, 45) is calculated by formula (9), where (i=0, 1, …, 45) represents the coordinates of the actual gear surface grid points; represents the theoretical gear surface grid point coordinates; and represents the normal vector of the theoretical gear surface grid points.

(9)

Figure 2全屏查看

Mesh definition method of gear flank

2.2 Mathematical model of continuous generating gear grinding

As shown in Figure 3(a), there are 6 axes that mainly participate in the gear grinding process, which are the A1 axis, B1 axis, C1 axis, X1 axis, Y1 axis, and Z1 axis. The A1 axis is the swing axis of the grinding wheel, which is determined by the helix angle of the grinding wheel and the gear, and remains unchanged during the processing. The X1 axis is the radial feed axis of the grinding wheel, and the Y1 axis is the tangential feed axis of the grinding wheel which is installed on the B1 axis. The workpiece gear is installed on the C1 axis, and the grinding wheel grinds the entire tooth flank through the movement on the Z1 axis. The spatial relationship between the grinding wheel and the workpiece gear is shown in Figure 3(b), and the movable coordinate systems and are respectively fixed on the grinding wheel and the workpiece gear. and are fixed coordinate systems. During the gear grinding process of the generating method, the movement of the B1 axis, C1 axis, Y1 axis, and Z1 axis has a strict linkage relationship, which is controlled by the electronic gearbox [24]. The specific linkage relationship is shown in formula (10):

Figure 3全屏查看

(a) Machine tool structure; (b) Coordinate system for the worm wheel and standard cylindrical helical gear

(10) (11) (12)

where

,,. (13) (14) (15) (16)

In formula (10): and are the angular velocity of C1 axis and B1 axis, respectively, and the unit is ; are the Y1 axis and Z1 axis speed, respectively, and the unit is mm/s; z_w and z_g are the number of the grinding wheel heads and gear teeth, respectively; and are the helix angle of the gear and the helix angle of the grinding wheel, respectively; and are the pitch circle radius of the gear and the pitch circle radius of the grinding wheel, respectively; is the lead angle of the grinding wheel, and is the modulus of gear; , values are +1 or -1, which are determined by the direction of rotation of the grinding wheel and the gear, respectively. When the grinding wheel is left-handed or right-handed, is respectively +1 or -1. When the gear is left-handed or right-handed, is respectively -1 or +1.

The grinding model proposed in this paper is based on the principle of gear meshing, and the grinding point of the workpiece gear flank is solved by the helical surface of the grinding wheel. First, the tooth surface of the grinding wheel is transformed from the coordinate system to the coordinate system through coordinate transformation. The expressions are shown in formulas (11-13). Among them: and are the two-parameter expressions of the tooth flank coordinates and normal vector of the grinding wheel in the coordinate system ; and are the two-parameter expressions of the tooth surface coordinates and the normal vector of the grinding wheel in the coordinate system , respectively; is the coordinate transformation matrix from coordinate system to coordinate system ; is the upper triangular matrix of .

Then, according to the principle of gear meshing, the meshing equations of the grinding wheel and the gear are listed as formulas (14-16).

Simultaneous formulas (14-16) can solve the parameters and , and bring the double parameters into formulas (11-13) to obtain the coordinates and normal vectors of the gear grinding point.

3 High-order flexible topology modification technology

3.1 Solution of motion of each axis of machine tool based on the kinematics model

According to the content described in Sections 2.1 and 2.2, the parameter matrix of the topological gear flank information and the parameter matrix of the flank of the grinding wheel are established. As shown in formulas (17-18), where and are the coordinate and normal vector of the gear flank in the coordinate system respectively; and are the coordinate and normal vector of the grinding wheel tooth surface in the coordinate system, respectively. According to the principle of generative grinding, during the machining process, the contact form of the flank of the grinding wheel meshes with the gear flank is point contact. During the grinding process, the pose of the grinding wheel and the gear is shown in Figure 4. The grinding points of the flank of the grinding wheel correspond to the grinding points of the gear surface, so the and matrices satisfy formula (19). A1 is the installation angle of the grinding wheel, which is determined by the helix angle of the grinding wheel and the gear. The calculation formula is shown in formula (20), where is the intersection angle of the gear and the axis of the grinding wheel. During the gear processing process, the size of A1 remains unchanged. In order to obtain the relationship between the topological gear flank and the motion of each axis, and are respectively expressed in the form of fifth-order polynomials with respect to time , such as formulas (21-23). When the grinding wheel parameter matrix and the target gear flank parameter matrix are known, the exact value of (i=1, 2, …, 18) can be obtained by solving formulas (17-23).

Figure 4全屏查看

Schematic diagram of grinding wheel and gear pose

(17) (18) (19) (20) (21) (22) (23)

In order to obtain the additional feed of each axis of the machine tool when the gear is modified, the feed of each axis of the machine tool corresponding to the standard involute gear surface (, , ) should subtract from the feed of each axis of the machine tool corresponding to the topological gear surface (), as shown in formula (24).

(24)

3.2 Polynomial interpolation function

Electronic gearbox is widely used to replace the internal transmission chain of machine tools, maintain multi-axis transmission, and realize accurate linkage relationships between multiple axes. In order to implement the flexible topology modification technique of gear flank proposed in this paper, we study the polynomial interpolation function in the Siemens 840Dsl system. Taking the tangential feed axis X1 as an example, the polynomial interpolation function is defined in the NC code of the machine tool as shown in formula (25). The polynomial form in its polynomial interpolation is as formula (26). indicates the end position; an (n=0, 1, …, 5) indicates the coefficient of the polynomial defined by the program, where is automatically generated by the system; PL indicates the parameter interval defining the polynomial coefficient; is a parameter variable between 0 and . In order to use the polynomial interpolation function in the Siemens 840Dsl system to realize the movement of the additional motion amounts ΔX, ΔY and ΔZ of the X1 axis, Y1 axis, and Z1 axis in formula (24), we convert the polynomial formulas (21-23) about time to a polynomial function of the parameter in the 840Dsl system. The conversion process is shown in Figure 5.

(25) (26)

Figure 5全屏查看

Polynomial interpolation function coefficient calculation flow

3.3 High-order flexible topological modification of gear flank based on electronic gearbox

The Y1, Z1, B and C axes strictly follow the linkage relationship in the process of gear grinding by generating method, so the modification compensation axis must be added to the standard electronic gearbox model to realize the modification function. The S axis is not equipped with a private motor, but it can accept position and speed instructions, and the following axis of the drive shaft is also the C axis. The C axis is rotated by the modified compensation axis which is named the S axis, which solves the problem that the following axis can not directly increase the compensation amount in the standard electronic gearbox. We call the modified electronic gearbox model for topological modification of the tooth surface as a modified electronic gearbox (MEGB). The working principal diagram of the modified electronic gearbox is shown in Figure 6. ΔX, ΔY, ΔZ refer to the additional movement of the X1 axis, Y1 axis and Z1 axis, respectively, when the gear needs to be modified. The linkage relationship of the modified electronic gearbox is shown in formula (27).

Figure 6全屏查看

Structural diagram of modified electronic gearbox

In order to realize the above function, it is necessary to convert the compensation amount solved in Section 3.1 into ΔY, ΔZ to the S axis of the modification compensation axis. Since the linkage coefficient of the S axis of the modification compensation axis and the C axis of the following axis is 1, formula (28) is directly used to convert the compensation amount.

Finally, the above modified electronic gearbox model is brought into the grinding mathematical model described in Section 2.2 to realize the flexible topology modification method based on the modified electronic gearbox.

(27) (28)

4 Results and discussion

In this section, an experiment example is used to verify the effectiveness of the flexible topology modification method proposed in this paper. The experiment example is based on the YW7232 universal CNC gear grinding machine (Chongqing Machine Tool). The workpiece gear parameters, worm wheel parameters, and main machine tool setup parameters are listed in Table 1. The gear profile deviation and lead deviation of the gear are measured by the measurement center (JINGDA JE32).

According to the method described in Section 2.1, this section uses the normal deviation between the tooth surface of the actual modified gear and the gear surface of the standard unmodified gear to express the effect of the method proposed in this paper. The formula for calculating the gear surface deviation is shown in formula (7). In order to better describe the modification ability of the method proposed in this paper, the rate of gear surface error is defined as shown in formula (29), where represents the maximum normal deviation between the actual modified gear surface and the target gear surface.

(29)

The conventional (variable radial feed) method and the flexible high-order topology modification method based on the modified electronic gearbox proposed in this paper are respectively adapted for grinding.

When the traditional method is used to grind the target gear flank, the radial feed is in the form of a parabola, and the machine tool generates 106 interpolation points. The linear interpolation command is used for grinding during the grinding process. There are twice grinding in the process.

When using the high-order flexible topology modification method proposed in this paper to process the gear, the radial feed is still in the form of a parabola. However, it is different from the linear interpolation command used in the traditional method. We use the contents of Section 3.1 to calculate the polynomial interpolation coefficient, and then write polynomial interpolation instructions according to the contents of Sections 3.2 and 3.3, and perform modified grinding on the machine tool. For the CNC instructions shown in formula (25), the specific values of x_e, , , , are 0.0006, -0.7192, 0.0023, 0.0027, -0.0027, respectively.

The measurement results of the gears processed by the two methods are shown in Figure 7. It is not difficult to see from the measurement data that the total deviation of the helix has decreased by an average of 1.4 μm. The total deviation of the gear profile has been reduced by an average of 2.87 μm. The accuracy of the gear flank has also been improved by nearly one level.

Figure 7全屏查看

Tooth surface measurement report: (a) Spiral line (traditional method); (b) Spiral line (high-order flexible modification method); (c) Tooth profile (traditional method); (d) Tooth profile (high-order flexible modification method) fHβ is the helix slope deviation; Fβ is the total helix deviation; ffβ is the helix form deviation; fHα is the profile slope deviation; Fα is the total profile deviation; ffα is the profile form deviation. All of the above are the grade indicators of gear accuracy. The number ±13.0, 18.0 and 13.0 are the tolerance value for accuracy grade 7 taking Figure 7(a) as an example. cβ is the amount of flank line crowning.

When the gear was processed, the X axis motion data of the machine tool were collected and analyzed. The sampling frequency is 5 Hz. As shown in Figures 8-10, the comparisons of acceleration and velocity between the two methods during the first grinding and second grinding are very obvious. In Figure 8, the standard deviations of the X axis acceleration of the first and second grinding of the traditional method are 0.0014×10^-4 and 5.5955×10^-4, and the mean values are 0.0028×10^-4 and 7.0890×10^-4, respectively. The standard deviations of the X axis acceleration of the first grinding and the second grinding are 1.5477×10^-4 and 1.8031×10^-4, and the mean values are 0.0025 and 6.0662×10^-4, respectively. In addition, traditional methods have sudden acceleration changes during the grinding process, which may be due to the fact that the linear interpolation command does not perform additional acceleration optimization processing at the interpolation point. As shown in Figures 9 and 10, it is a comparison of the X axis speeds during two machining processes. It can also be clearly seen that the high-order flexible topology modification method has a more stable speed during the gear flank grinding process.

Figure 8全屏查看

(a) The acceleration in the direction of X-axis (first set of data); (b) The acceleration in the direction of X-axis (second set of data)

Figure 9全屏查看

The speed in the direction of X-axis (first set of data): (a) Traditional method; (b) High-order flexible modification method

Figure 10全屏查看

The speed in the direction of X-axis (second set of data): (a) Traditional method; (b) High-order flexible modification method

5 Conclusions

In this paper, we establish a continuous generation grinding model of grinding wheels based on the principle of gear meshing theory and propose a flexible topological modification method of gear flank based on the modified electronic gearbox. Combined with the polynomial interpolation function of Siemens 840Dsl, the X, Y and Z axes are attached to the electronic gearbox to realize the topological modification of the gear flank. According to the experimental example, we can draw the following conclusions:

1) The method proposed in this paper is suitable for gear profile modification and topological modification. It does not need to continue the special dressing of the grinding wheel, which improves the machining accuracy of the modified gear and reduces the distortion of the gear surface.

2) Through the kinematic model of gear grinding, this paper explores the mapping relationship between the modified gear surface and the feed rate of each axis, and it also solves the problem of polynomial coefficient calculation in high-order modification technology.

全屏查看

(i=0, 1, …, 45)	Normal vector of theoretical grid points
	Lead angle of the grinding wheel
	Modulus of gear
	Normal vector of the grinding wheel
(i=0, 1, …, 45)	Actual tooth surface deviation
(n=0, 1, …, 5)	Coefficient of the polynomial

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3) This paper combines the polynomial interpolation function of Siemens 840Dsl with high-order flexible modification technology to provide a new method for the application of high-order flexible modification technology on machine tools. In the experimental example, the feasibility of this method is verified.