Digital generating method for cylindrical helical gear based on indexable disk milling cutter

To solve the problem of low efficiency in digital generating machining for producing small or medium batches of helical gears, a method to improve the machining efficiency with an indexable disk milling cutter is presented in this paper. First, the mathematical model of the tooth profile and the indexable disk milling cutter are established. Second, according to the spatial free-form envelope theory, the overall planning scheme of the tool path is given; the relative position and the relative transformation matrix of the tool and tooth profile during digital generating machining using the indexable disk milling cutter are solved, the simulation cutting and actual cutting experiments are conducted, and the cutting efficiency per unit time and cutting simulation time of the two tools under the same deformation conditions is obtained through a finite element analysis experiment. The results show that the cutting efficiency of the indexable disk milling tool was 2–3 times higher than that of the end mill cutter.


Introduction
Cylindrical gears are widely used for motion and power transmission between parallel shafts and are key basic parts in aviation, automotive, energy, and other fields. Traditional gear machining mainly adopts generating machining methods, such as gear hobbing and gear shaping [1,2]. However, with the gradual development of mechanical products in the direction of small-batch customization, the shortcomings of traditional gear processing methods (requiring special machine tools and special tools, long processing preparation cycles, and insufficient machine tool start-up rates) are increasingly prominent. This has led to the emergence of a digital generating method based on the spatial free-form envelope theory, which is realized by multiple-tool flank milling on a general-purpose multi-axis computer numerical control (CNC) machining center.
Regarding gear processing methods, scholars have conducted extensive research. Zhu and Lu [3] and Bo et al. [4] presented a representation of the envelope surface of the tool sweep body for flank milling envelope processing and found numerous applications in the geometric problems arising from 5-axis flank milling. Harik et al. [5] introduced the principle of 5-axis flank milling in detail and provided some suggestions on its tool path optimization. Lo [6] discovered a method to improve the efficiency of milling surfaces with a flat-end cutter by minimizing the tool path length in a 5-axis machining center. Pechard et al. [7] dealt with tool path optimization in the flank milling of the 5-axis machining center based on the geometric deviation and smoothness of the tool path. Shih and Chen [8] proposed a free-form flank modification based on a 5-axis CNC helical gear profile grinding machine. Gong and Wang [9] illustrated a new tool path generation method for free-form surface flank milling considering the constraints for a ball-end cutter in a 5-axis machining. Guo et al. [10] studied the machining error of a cylindrical gear tooth profile with an end milling cutter and obtained the relationship between the tooth profile accuracy and machining tool type, gear parameters, and feeding strategy of the tool. Chiang and Fong [11] proposed a mathematical method for determining the distribution of the cutter body and inserts, which is used in the rough machining of screws with form milling cutters with multiple insert. Zheng et al. [12] studied the dynamics of spiral bevel gear face-milling with a disk milling cutter. Habibi and Chen [13] derived a semi-analytical formula for the undeformed cutting area and predicted the cutting force in the face-hobbing of bevel gears with a disk milling cutter. Shih et al. [14] proposed a machining method for small or medium batch bevel gears with a disk milling cutter on a 5-axis machine because of the high production cost caused by special machine tools and cutters for face-milling and gear hobbing of bevel gears.
Researchers have extensively worked on digital generating tool path planning and machining accuracy control of gears using end milling cutters, while the research on digital generating machining using disk milling cutters is mostly focused on spiral bevel gear machining; meanwhile, there is a lack of relevant research on digital spreading of cylindrical gears with disk milling cutters. Considering that the diameter of the end milling cutter is limited by the width of the tooth groove when machining small-or medium-sized gears, the strength and material removal rate will be significantly reduced. In this paper, we present a digital generating method for machining cylindrical gears with indexable disk milling cutters and discuss the trajectory planning, tool positioning, and interference avoidance. Thereafter, the accuracy of the tool location is verified by machining simulation and the efficiency of the milling tooth with a disk milling cutter by finite element cutting simulation. Finally, compared to the digital generation of gears with the end milling cutter, this method can improve the material removal rate of the machining gear to a considerable extent while maintaining the quality of tooth surface machining.

Deduction of tooth profile equation
The tooth surface of a helical gear is a helicoid, as represented by Eq. (1). In the derivation of this equation, it was assumed that the helicoid is generated by the screw motion of an involute curve in the cross profile (or the transverse plane) about the gear axis [15], as shown in Fig. 1. Here, S g = O g , x g , y g , z g is the coordinate system rigidly  The transverse plane δ connected with the gear, in which O g is the center of the base circle of the gear, x g O g y g coincides with the transverse plane of the gear, +z g is the axis of the gear, +x g and +x`g are tangent to involute curves I and II at the base circle in the x g O g y g , respectively. +y g and +y`g are determined according to the right-hand rule. The tooth profile I and I` are respectively generated by the screw motion of involute curves I and I` around the gear axis by an angle . The involute curves I and I` are symmetrical about the +x g , and tooth profile II is obtained by rotating tooth profile I` around +z g by an angle . The parameter lines u and v are generated by the screw motion of the involute in the transverse plane around the z-axis; therefore, the position vector of P and P 0 in the coordinate system S g can be expressed by the parameters and as follows: At point P, takes 0, and at point P 0 , takes the calculated value and u takes −u . Here, r b is the radius of the base circle, the parameter u is measured from the position vector in O g T in the direction shown in Fig. 1, is the angle of rotation around +z g , and is the angle between +x g and +x`g, which can be expressed as follows: where r is the reference circle radius and e is the transverse space width.

Establishment of the mathematical model of indexable disk milling cutter
Indexable disk milling cutters (hereafter referred to as disk milling cutters) mainly comprise a hilt and an insert with two sides (a side edge and the other side edge) that can be machined simultaneously; the two sides are connected by a tip fillet. The insert is fixedly connected to the handle by bolts, which not only avoids the shortcomings of integral cutters that are prone to cracks, but also realizes rapid replacement of blades of different materials according to the material of the blank, making its processing versatile and highly productive. Simultaneously, the large diameter and high strength of the disk milling cutter as well as the small fillet make it suitable for high-efficiency machining of gears with large feeds and the machining of narrow areas of tooth roots. Note that for the cross profile of the disk milling cutter shown in Fig. 2, x t , y t , z t is the coordinate system rigidly connected with the tool, O t is the cutter location of the point, x t O t y t plane coincides with the bottom plane of the disk milling cutter, +z t is the cutter shaft, +y t coincides with the other edge, +x t is determined according to the right-hand rule, r ID is the nominal diameter of the disk milling cutter, O c is the center of the fillet, r is the radius of the fillet, and P 1 , P 2 , and P 3 are arbitrary cutting points on the side edge, the other edge, and the fillet, respectively. Hence, the position vectors of P 1 , P 2 , and P 3 in the coordinate system S t can be expressed as:  The other edge r ID

The bottom plane
Here, t is the insert angle, l is the projection distance from O c to O t at +y t , l 1 is the projection distance from P 1 to O c at the side edge, l 2 is the projection distance from P 2 to O c at +y t , r is the angle between r and +y t , and M(z, ) is the rotation matrix rotated by angle around +z t , which can be expressed as follows:

Cutting of the helical gear in 5-axis machining center
Disk milling cutters are often used for machining tooth grooves by adopting a variable contour milling strategy, as shown in Fig. 3. When machining the tooth groove, the first layer of cutting thickness and cutting tool path are determined by the cutting process and initial tooth surface parameters; after the first layer of machining, the tool feeds along the radial direction of the blank to start the second layer of cutting, and this cycles until the tooth groove processing is completed. In the cutting process of each layer (considering the first layer as an example), a reciprocating tool feeding strategy is adopted, and the tool moves along tool paths I to II. In this process, to ensure that the cutting thickness of the layer is the same, the tool position is changed from position I through position i, gradually transitioned to position II, and in different positions the tool is tangent to the cylindrical surface of the same radius, thereby completing the cutting process of this layer. As indicated above, three steps are required to realize the tooth groove cutting process. First, the first layer of cutting tool paths I and II should be calculated according to the tooth surface grid points and  4 Cutter position calculation for generating helical gear with disk milling cutter cutting conditions. Second, the reciprocating tool path is obtained according to the transformation of tool paths I and II. Finally, the tool radially feeds to repeat the single layer cutting process to realize the complete tooth groove cutting process.

Tool location of tooth groove cutting
The principle of calculating the tool position point for machining helical gears with disk milling cutters is shown in Fig. 4. The Cartesian coordinate systems S g = O g , x g , y g , z g and x t , y t , z t are rigidly connected to the coordinates of the gear and cutting tool, respectively. According to the principle of generating, the cutting tool surface and tooth surface must meet the condition of overlap and tangency at the cutting point. For tooth profile I, the tangent vector P coincides with the disk milling cutter side edge vector, and the normal vector n P coincides with the reverse at the cutting point P (conjugate point P 1 ) in the gear coordinate system. Profile II satisfies the same conditions at the point P 0 (conjugate point P 2 ). Considering tooth profile I as an example, the equation to satisfy the cutting conditions can be expressed as follows: where r OgOt is the vector from S g to S t , which can be expressed as follows: where P and n P are the tangent and normal vectors of the tooth surface point P , respectively, which can be expressed as: Here, P 1 and n P 1 are the tangent and normal vectors of the cutting point P 1 on the side edge of the disk milling cutter, respectively. When the disk milling tool is machined, the tool rotates around the spindle +z t at high speed, and the vector P1 , normal vector n P1 , and position vector r P 1 at the cutting point P 1 (or P 2 ) should always satisfy the cutting conditions, which can be expressed as where M O g O t is the transfer-matrix from S g to S t , which can be expressed as follows: Here, the column vector a, b, c is the vector expression of +x t , +y t , +z t in the gear coordinate system S g , which can be expressed as follows: The process calculation flow is shown in Fig. 5. Given the initial tooth surface parameters , and the tool cutting edge parameters l 1 , l 2 at the cutting point P , calculate the tangent vector and the normal vector at this point according to Eqs. 10, 11, 12, and 13. Subsequently, solve the system of (12) P 1 = 0 cos t sin t (13)  which is the transformation matrix in the coordinate system S g . Finally, find the vector r O g O t according to Eq. 8, which is the tool location point vector in the coordinate system S g . Similarly, the transformation matrix and the tool location point vector at the cutting point P 0 can be obtained.
As shown in Fig. 4, the depth of cut at cutting points P and P 0 is determined by point P 3 on the fillet. Point P 3 satisfies the coincidence with points A and A 0 and the normal vector reversal. Points A and A 0 are located on a cylindrical surface with a radius of R . The cylindrical surface can be expressed as follows: For posture I, the equation that satisfies the condition can be expressed as follows: In Eq. 17, n P 3 and n A are calculated by Eqs. 6 and 16, respectively, and the radius R of the cylindrical surface is obtained by solving the system of equations. For posture II, to ensure that the layer cutting thickness is the same, points A and A 0 should be located on the cylindrical surface of the same radius; therefore, the tool control point at position II should be moved k times along the vector P 0 . The equation satisfying the condition can be expressed as follows: The tool control point vector after the move is obtained by solving the system of equations and finding the parameter k. The vector r A i of point A i is obtained by interpolating equidistantly between points A and A 0 according to the disk milling cutter cutting width; the transformation matrix M O g O t (i) of posture i is obtained by interpolating transformation matrices I and II. According to Eq. 19, the tool control point position vector r O g O t (i) at position i is obtained by solving the system of equations. Thus, an intermediate reciprocating tool path was obtained.
When the first layer of cutting is completed, the tool feeds along the radial direction, and an equal residual height tool path planning strategy is adopted, as shown in Fig. 6.
When the disk milling cutter is in position I (corresponding to the first layer of tool path I), the cutting point with tooth profile I is B. When the disk milling cutter is in position II (corresponding to the second layer of tool path I), the cutting point with tooth profile I is C. The cutting edge of the disk milling cutter intersects at point E in the two adjacent positions, and a straight line is made through point E perpendicular to tooth profile I, with the vertical foot F. The length of the straight line EF is the residual height of the tooth surface. The calculation method adopted in [6] and [7] can obtain the tooth surface parameters of the adjacent cutting point C according to the tooth surface parameters of the initial cutting point B. According to cutting point C, the second-level cutting tool paths I and II and reciprocating tool paths are obtained, and it is cycled to obtain all the cutting tool paths of the tooth groove.

Interference checking clearance side of the cutter
According to the generation principle of the tooth milling cutter path of the disk milling cutter, the interference in the machining process mainly comes from the interference cutting between the bottom plane of the milling cutter and tooth   Fig. 7.
From the point-to-surface distance theory, it is known that when the normal vector on the tooth surface intersects with the tool bottom, the distance between the two points is the shortest distance from that point to the tooth surface, and the calculation formula can be expressed as follows: In Eq. 20, r D is the position vector of the bottom plane of the disk milling cutter in the gear coordinate system when the tool is in posture i; r d is the radius of the bottom surface with a value range is 0-l; and N is a constant. The point on the tooth surface corresponding to the shortest distance can be determined using Eq. 21, and the shortest distance d from the bottom surface to tooth profile II is determined using Eq. 22. If d is less than 0, interference will occur; otherwise, it does not. When interference occurs, the bottom surface  can be moved away from tooth profile II by reducing the tool angle t .

Comparative analysis of cutting efficiency
To verify that the disk milling cutter has a higher machining efficiency, the end milling cutter is used as the experimental comparison object. In the process of tooth milling, tool deformation is an important factor that affects the quality of tooth surface machining. Therefore, to ensure the quality of tooth surface machining, the strength and cutting efficiency per unit time of the disk milling cutter and end milling cutter were compared under the same deformation and linear cutting speed conditions. According to the tool material (both disk milling cutter insert and end milling cutter are carbide steel), the cutting line speed is 94 m/min. According to the basic gear parameters, the maximum diameter that can be used for rough milling gears with an end milling cutter is 6 mm and the number of cutting blades is 2. Using a finite element analysis software, the cutting force load is added to the cutting edge of the tool model, and the deformation results of the load applied to the disk milling cutter and end milling cutter were obtained through analytical calculations, as shown in Fig. 8. The deformation data were extracted and processed to obtain the relationship between the load and deformation of the two tools, as shown in Fig. 9. Figure 9 shows that under the same deformation conditions, the disk milling cutter bears a load of 1508 N and the end milling cutter has a bearing capacity of 77.8 N. The former is approximately 20 times greater than the latter, indicating that the strength of the disk milling cutter is much greater than that of the end milling cutter, and the disk cutter can bear a larger cutting amount.
When performing cutting simulations, cutting simulation experiments are carried out on only one insert cutting process of the disk milling cutter in order to reduce the cost of analysis time, which is the usual method used in finite element analysis. The simplified cutting model was simulated using the special metal cutting simulation software Advant-Edge. Figure 10a, b, and c represent the cutting simulation experiment of the disk milling cutter, and Fig. 10d, e, and f represent the cutting simulation experiment of the end milling cutter. Figure 10a and d show the relative motion relationship between the tool and workpiece in the simplified cutting model. Figure 10b and e show the simulation results obtained using AdvantEdge. Figure 10c and f show the changes in cutting forces Fx, Fy, and Fz in the process of cutting simulation.
By synthesizing the cutting force, the cutting force F under different cutting parameters is obtained, and the results of the finishing experiment are listed in Table 1.
The deformation of the tool in Table 1 is obtained by interpolating the load-deformation diagram. The volume of material removed per unit time in Table 1 is obtained by multiplying the removal volume per tooth of the tool by the rotational speed per second. The removal volume per tooth was obtained by Boolean operation using the tool model and the blank model, as shown in Fig. 11. After the Boolean difference operation between the (i-1)th blade and the workpiece is completed, the blade moves f z along the feed direction to the ith blade position and performs the same Boolean operation. The cutting volume per tooth can be obtained by comparing the volume of the workpiece obtained twice. Table 1 indicates that the deformation of the disk and end milling cutters increases with the increase in feed per tooth and cutting volume under the condition of constant back feed ( a P ).
As shown in Fig. 12, through the interpolation calculation, we find that in the deformation amount U = 0.05 mm, Fig. 11 Removal volume of each tooth obtained by Boolean operation. a Relative position between (i-1)th blade and workpiece. b Blade moves f z along the feed direction. c Relative position between ith blade and workpiece Fig. 12 Relationship between tool removal volume per unit time and tool deformation the removal volume per unit time is 57.17 mm 3 for the disk milling cutter and 25.59 mm 3 for the end milling cutter, and the former is 2.2 times of the latter. With an increase in the deformation amount, the ratio of removal volume per unit time of both tends to increase. However, to ensure the quality of tooth surface machining, the deformation amount of the tool should be within a reasonable range. According to Table 1 and the removal volume per unit time, the approximate cutting parameters a p and f z are determined. The tool path planning strategy described in Sect. 3 is adopted to plan the tool path of the disk milling cutter and end milling cutter, and the tool path is imported into 3D modeling software, as shown in Fig. 13. The left and right figures show the motion path of the control point of the end milling cutter and the disk milling cutter in Fig. 13, respectively. Through the given feed rate f z and post-processing, the simulation processing time of the disk milling cutter and the end milling cutter in 3D modeling software can be obtained, as shown in Fig. 13. The results of the simulation processing time demonstrate that, compared with the end milling cutter, the time of machining a tooth groove with the disk milling cutter is reduced by 2.91 min, and the efficiency is increased by 54%.

Machining experiment
To verify the solution equation for the location of the tool position using the disk cutter and the high cutting efficiency of the disk milling tool compared to the end milling tool, a cylindrical gear milling process experiment (gear parameters, disk milling cutter, and end milling cutter parameters are shown in Table 2) was conducted. The experiment included a cutting simulation experiment and an actual machining experiment.
A 5-axis simultaneous gear milling simulation platform (AC linkage) was established based on the machine structure of the KMC600SU machining center, as shown in Fig. 14a. The tool location of tooth groove cutting is calculated as described in Sect. 3, and then rewrites it to NC programs. Subsequently, the NC program was imported into the milling simulation platform to obtain the cutting results. The process of cutting simulation is shown in Fig. 14b and c, and the movement of each axis of the machine tool in the process of cutting simulation is shown in Fig. 15. The error between the gear model obtained from the machining simulation and the theoretical model was obtained, and the results are shown in Fig. 14d. Figure 14d shows that there is no error point on the tooth surface, implying that the simulated cutting surface and the theoretical tooth surface completely coincide, and all  the blue error points are distributed at the root, which may be because of the straight-line transition in the simulation machining of the root, while the theoretical root transition is fillet. In the actual cutting experiment, the KMC600SU 5-axis machining center was used for tooth cutting and the actual machining parameters and cutting times of individual tooth groove for the disk milling cutter and the end milling cutter, respectively, as shown in Fig. 16. The machine motion and machining results during the machining process are the same as the machine motion and results during the simulation, as shown in Fig. 16. Due to the limitations of the actual machine construction, the long overhang of the workpiece and tool and the weak rigidity of the installation, the feed rate was reduced in equal proportions in the actual experiment to ensure safety. Figure 16a and b shows that the cutting time for a single tooth groove is 13.6 min for the disk milling cutter and 27.8 min for the end milling cutter. The actual cutting time is 5 times more than the cutting simulation time. The main reason is that the time consumed by the machine tool's non-cutting motion is not considered during the simulation cutting. The results of the cutting processing time demonstrate that, compared with the end milling cutter, the time of machining a tooth groove with the disk milling cutter is reduced by 13.8 min, and the efficiency is increased by approximately 51%. Therefore, the actual machining efficiency comparison experiment and the cutting efficiency simulation experiment are in good agreement. In addition, during the machining of the gear with the end milling cutter, the tool was worn more severely and it took four replacement end milling cutters to complete the rough machining of the entire gear, while the disk cutter did not replace the insert in completing the rough machining of the entire gear.
To illustrate the acceptable accuracy of roughing gears with disk milling cutters, while increasing efficiency, the gear profile deviation and helix deviation are measured by a gear measuring center, as shown in Fig. 17. The maximum value of profile form deviation is 8.3 μm , and the accuracy grade is ISO-0006; the maximum value of profile slope deviation is − 14.4 μm , and the accuracy grade is 8, as shown in Fig. 18. The maximum value of helix form deviation is 9.5 μm and the accuracy grade is 7; the maximum value of helix slope deviation is − 10.3 μm , and the accuracy grade is 7, as shown in Fig. 19. Considering there is a finishing process after this process, this machined gear meets the rough machining accuracy requirements. The deviation comes from the following aspects: (1) the overhang of the fixture is too long due to the structural limitation of the machine tool; (2) during the roughing process, the larger cutting parameters a p and f z are adopted; (3) there is an eccentric deviation between the workpiece axis and the machine C-axis axis. Furthermore, both profile slope and helix slope deviations are higher than profile form and helix form deviations, indicating that the mounting deviation had a greater influence on the machining. If the mounting conditions are improved and the machining process parameters are changed, the deviations in rough machining of gears can be effectively reduced, and the machining method can be applied to the gear finishing process.

Conclusions
Aiming at the development trend of gear processing to small-scale customization, a gear processing method using large-diameter indexable disk milling cutters instead of end milling cutters for digital generating milling of tooth is proposed in this study to solve the problem of low efficiency of end milling cutters for processing small-and medium-sized gears. By establishing a unified mathematical model of the helical gear and disk milling cutter, the relative position of the tool and tooth face and the relative transformation matrix relationship during the gear milling process are derived, and a general scheme of tool path planning is provided. Thereafter, the tool position location solution equation is also described, and the efficient digital generating machining of helical gears with disk milling cutters is realized. The results obtained in this study were as follows: (1) The results of the finite element analysis and cutting experiments show that the milling efficiency of disk milling cutters is higher than that of end milling cutters under the same cutting line speed and tool deformation conditions. The material removal per unit time of the disk milling cutter is 2-3 times that of the end milling cutter, and the cutting time of the former is reduced by 54% compared with the latter. (2) The high strength of the disk milling cutter facilitates faster removal of the workpiece material in rough machining and increases the machining efficiency. The digital generating method of the disk milling cutter helps to reduce the number of tool movements and improves the machining efficiency, which is ideal for single-piece and small-batch gear machining.