This article mainly introduces the design and manufacturing method of arc cylindrical gears. Based on the study of the basic rack formed by the expansion of the arc cylindrical gear, it details the basic parameters and design calculation of the gear. The article also focuses on the gear’s tooth cutting process, including the tooth cutting methods, the equation of the tooth cutting tool’s tooth surface, and the macro program compilation for the tooth cutting process. Finally, the meshing transmission test results show that the gears meet the expected requirements.
Introduction
Gear transmission has the advantages of large power transmission, high efficiency, and accurate transmission ratio, which is widely used in mechanical transmission. Among them, involute spur gears and helical gears have been standardized in design and manufacturing, and are most widely used in mechanical transmission. However, involute spur gears have large impact and noise and poor transmission stability during meshing, while helical gears have a large axial force during the transmission process. Therefore, the research on the meshing transmission of arc cylindrical gears has attracted extensive attention from scholars at home and abroad. Arc cylindrical gears have the advantages of high meshing 重合度，small axial load, and strong load-carrying capacity. Moreover, the meshing of each pair of teeth is gradual, without the meshing impact of spur gears or the axial force of helical gears, so its transmission performance is better. Additionally, the tooth direction of arc cylindrical gears is an arc tooth line, which allows the gear to be rotated by a small angle along the tooth line during installation without affecting the meshing transmission, and has strong adaptability to the parallelism error of the gear pair installation.
Basic Parameters and Design Calculation of Arc Cylindrical Gear
In general, the study of arc cylindrical gears is based on the basic rack of the arc cylindrical gear. As shown in Figure 1, the basic rack of the arc cylindrical gear, where p and q are the convex and concave surfaces of the gear teeth respectively, which are parts of two cones P and Q. R is the radius of the cone at the pitch circle, t/2 is the distance between the two cone axes, b is the width of the rack, the A-A section is the middle section of the rack, and the B-B section is the indexing plane of the rack. The rack at the A-A section has an involute tooth profile. Studies have found that the tooth profile of the arc cylindrical gear at the middle section of the gear tooth is the same as that of the involute spur gear, so the design calculation of the arc cylindrical gear can refer to that of the involute spur gear. Its basic geometric parameters include module m, number of teeth z, pressure angle α, addendum height coefficient ha, and clearance coefficient c, and the geometric size calculation is the same as that of the involute spur gear. The calculation formulas are shown in Table 1.
The design process of the arc cylindrical gear can refer to that of the involute spur gear, and the design criteria are the same. In the design process of the spur gear, because the main failure forms of the gear are tooth surface fatigue pitting and gear tooth fatigue fracture, according to different gear transmission application scenarios, the design calculation criteria of tooth surface contact fatigue strength or tooth root bending fatigue strength can be selected for design calculation. Therefore, in the design calculation process of the arc cylindrical gear, referring to the design calculation process of the involute spur gear, first select the gear material, heat treatment method, and accuracy grade, and then determine the corresponding design criteria according to the operating conditions, and calculate to determine the number of teeth and module of the pinion. Due to the small number of teeth of the pinion, considering the undercut problem, the number of teeth of the pinion should satisfy zmin ≤ z1 when selected; once the number of teeth of the pinion is determined, the number of teeth of the gear can be determined according to the gear transmission ratio, that is, z2 = i z1; then, calculate the main dimensions of the arc cylindrical gear according to the geometric size calculation formulas in Table 1; finally, check the strength of the arc cylindrical gear.
| Name | Calculation Formula | |
|---|---|---|
| Basic Parameters | Module (mm) | Determine according to strength calculation |
| Number of Teeth | zmm ≤ 21, 22 – i z1 (z1 is the number of teeth of the pinion, z2 is the number of teeth of the gear, and i is the transmission ratio) | |
| Pressure Angle (°) | Generally take a = 20° for the pressure angle at the central section | |
| Addendum Height Coefficient h*a | Generally take h*a = 1.0 | |
| Clearance Coefficient c* | Generally take c* = 0.25 | |
| Geometric Sizes | Pitch Circle Diameter d (mm) | d = m z |
| Addendum Height (mm) | h = h*a m | |
| dedendum Height (mm) | hy = (ha + c) m | |
| Whole Tooth Height h (mm) | h = h + hy = (2 ha + c) m | |
| Tip Circle Diameter d0 (mm) | d0 = d + 2 h = (2 h*a + 1) m | |
| Root Circle Diameter df (mm) | df = d – 2 hy = (1 – 2 ha – 2 c) m |
Tooth Cutting Process of Arc Cylindrical Gear
2.1 Tooth Cutting Methods of Arc Cylindrical Gear
The tooth cutting process of the arc cylindrical gear is different from that of the spur gear. Currently, the more typical tooth cutting methods mainly include the rotating cutter head method and the parallel connecting rod method. Due to the high efficiency and easy realization of the rotating cutter head method in the tooth cutting process, this method is widely used in the tooth cutting process of the arc cylindrical gear. The principle is shown in Figure 2. When using the rotating cutter head method to cut the arc cylindrical gear, there is an accurate generating motion relationship between the cutting cutter head and the gear blank, that is, the pitch circle linear velocity of the gear blank to be processed is equal to the linear velocity of the pitch line of the cutting cutter head. In the process of cutting the gear teeth, in addition to the main rotational motion of the cutting cutter head, there is also a feed motion in the radial direction of the gear blank. When using the generating method for processing, because only one tooth slot can be cut each time, after each tooth slot of the gear tooth is cut, the gear blank must be precisely indexed, and then the second tooth slot is cut. After all the tooth slots are cut in sequence, the processing of the arc gear can be completed. In theory, the tooth slot of the arc cylindrical gear can use a double-edge cutter to cut a complete tooth slot at once to form the convex and concave surfaces of the gear teeth. However, because the double-edge cutter is full-edge cutting in the tooth cutting process, the cutting amount when cutting the tooth slot is large, the cutter wears quickly, and the accuracy of the cut tooth surface is not high. It is difficult to cut the arc gear that meets the requirements in actual tooth cutting. Therefore, based on the tooth cutting practice of the double-edge cutter, a tooth cutting processing method using a single-edge cutter is proposed, and the tooth cutting tool for the arc cylindrical gear is shown in Figure 3. In Figure 3, R and Ri are the cutter head radii of the inner and outer single-edge cutters respectively, and α1 is the cutter pressure angle. Combined with the practical experience of the double-edge cutter tooth cutting, in the process of cutting the tooth slot of the arc gear, the tooth slot can be roughly cut out using a double-edge cutter first, and then the convex and concave surfaces of the arc cylindrical gear can be precisely cut using single-edge cutters respectively to ensure the tooth cutting accuracy.
The advantage of using single-edge cutters to precisely cut the convex and concave surfaces of the arc cylindrical gear is that the inner and outer single-edge cutters can be designed as cutters with equal radii, and the cut convex and concave surfaces of the gear teeth have higher accuracy. However, because the convex and concave surfaces need to be cut separately, the problems of gear blank installation and cutter replacement during the processing must be considered. Once the cutter needs to be replaced or the gear blank needs to be disassembled and assembled during the processing, it will inevitably affect the processing accuracy of the gear teeth. Therefore, a single-sided double-milling processing method for the arc cylindrical gear is proposed, and its processing principle is shown in Figure 4. In the process of cutting the gear teeth of the arc cylindrical gear, two stations are designed on a special milling machine to cut the convex and concave surfaces of the gear teeth respectively. In Figure 4, 1 and 4 represent the convex and concave single-edge milling cutters located at the left and right stations respectively, 2 and 5 represent the gear blanks to be processed when milling the convex and concave tooth surfaces respectively, and 3 and 6 represent the convex and concave single-edge milling cutters respectively. It can be seen that the tooth cutting process of the arc cylindrical gear requires a dedicated CNC milling machine to complete.
2.2 Tooth Surface Equation of the Tooth Cutting Tool for Arc Cylindrical Gear
In the process of milling the arc cylindrical gear, a special milling cutter is used. To design and manufacture a special milling cutter, the tooth surface equation of the tooth cutting tool can be studied and established based on the principle of the rotating cutter head method for tooth cutting. Figure 5 shows the coordinate system of the tool and the gear blank to be processed. First, establish the coordinate system [O1, x1, y1, z1] fixed on the gear blank to be processed, and the coordinate system [Od, xd, yd, zd] is fixed with the milling cutter head, which rotates with the gear blank and the milling cutter head respectively; then establish the fixed coordinate system [O, x, y, z] and [O1, x10, y10, z10] as the auxiliary coordinate system. If the pitch circle radius of the gear blank is R1 and the average radius of the milling cutter head is r0, then the inner edge radius of the milling cutter is r0 – πm/4, and the outer edge radius is r0 + πm/4. In addition, in Figure 5, ω and ω1 are the angular velocities of the tool and the gear blank respectively, and φ1 is the rotation angle of the gear blank. It can be known that in the coordinate system [Od, xd, yd, zd] fixed with the tool, the tool surface parameter equation can be established as follows:
where u is the distance from a point along the generatrix of the cone to the reference point, and θ is the rotation angle of the tool holder from the middle section of the gear blank to the end face.
Tooth Cutting Process and Macro Program Compilation of Arc Cylindrical Gear
Based on the analysis of the previous tooth cutting process, the tooth cutting process of the arc cylindrical gear can be divided into three steps: The first step is to lower the cutter in the middle of the tooth slot of the arc cylindrical gear and use a three-edge tooth slot milling cutter to mill out the tooth slot, removing most of the gear blank material; the second and third steps are to use single-edge cutters to process the concave and convex tooth surfaces of the arc cylindrical gear respectively. When processing on a special milling machine, the indexing in the processing process is programmed using two methods: absolute indexing and relative indexing. For example, if the indexing angle of the gear according to the number of teeth is δ, the starting angle of the processing program starts from δ/2 in the first step, that is, to roughly process the tooth slot from the middle of the gear tooth slot; start from δ = 0 in the second step, and start from δ in the third step, and use two single-edge cutters respectively for the corresponding concave and convex tooth surface processing; after each cutter completes the entire tooth slot and tooth profile processing of the gear for one week, the processing of the entire arc cylindrical gear can be completed. Since the cutter cuts in according to the involute, the cutter’s cutting trajectory is also in accordance with the involute feed. For example, the basic parameters for processing the arc cylindrical gear are as follows: module m = 3 mm, number of teeth z = 25, addendum height coefficient ha = 1.0, and clearance coefficient c = 0.25. A macro program for a tooth slot cutting cycle is compiled as follows:
N10 R1 = 3 / Module m
N20 R2 = 25 / Number of teeth z
N30 R3 = R1 * R2 / 2 / Pitch circle diameter d
N40 R4 = 1 / Addendum height coefficient ha
N50 R41 = 0.25 / Clearance coefficient c
N60 R42 = 0 / Modification coefficient
N70 R5 = 1 / Dedendum height coefficient
N80 R6 = R3 / 2 + R4 * R1 / Tip circle radius
N90 R7 = R3 / 2 – R4 * R1 / Root circle radius
N100 R8 = 1 / Initial value of involute opening degree 0 to 1 decrement
N110 R9 = 0 / Increment of t
N120 R10 = 60 * R8 / α value
N130 R11 = 3600 / R2 / Indexing angle per tooth 3600 / R2 – atan((pi * R2 * R10 / 180) / R2)
N140 R12 = 0 / R12 = A Gear rotation angle A = R10
N150 R13 = R3 * cos(R10) + pi * R3 * R10 / 180 * sin(R10) / Assignment of R13 = X
N160 R14 = R6 + 5 / Starting position of the cutter in the X feed direction to move 5 mm away from the tooth tip
N170 G90 G54 G0 A0 X = R14 / The workpiece rotates quickly to A0, and the cutter moves quickly
AA:
N180 G01 X = R13 A0 F300 / Move to the starting point
N190 R8 = R8 – 0.002 / Assignment of R8
N200 R10 = 60 * R8 / α angle value
N210 G01 X = R3 * cos(R10) + pi * R3 * R10 / 180 * sin(R10) A = R10 – atan((pi * R2 * R10 / 180) / R2) F300 / Movement point X, A
N220 IF R8 > 0 GOTO AA / Loop
N222 BB: / BB loop
N230 R9 = R9 – 0.002
N240 R101 = 60 * R9 / R101 reverse feed motion
N250 G01 X = R3 * cos(R101) + pi * R3 * R101 / 180 * sin(R101) A = R101 – atan((pi * R2 * R101 / 180) / R2) F600 / Reverse rotation return
N260 IF R9 < 1 GOTO BB
The milling program of the arc cylindrical gear is fed in the direction of the involute, and the tool withdrawal is also withdrawn in the direction of the involute. When the tool moves along the involute direction, attention should be paid to avoid overcutting the unprocessed tooth surface, so as to ensure that there is sufficient machining allowance on the surface to be processed to ensure the processing quality of the arc tooth surface. Figure 6 shows the arc cylindrical gear obtained by trial cutting, and through the meshing transmission test, it is proved that the cut arc cylindrical gear meets the expected requirements.
Conclusion
The meshing transmission performance of the arc cylindrical gear is good, but due to the need for dedicated processing equipment and milling cutters for tooth cutting, it leads to the fact that the arc cylindrical gear cannot be widely promoted and applied in mechanical transmission. Based on the study of the design calculation of the arc cylindrical gear, this article focuses on the in-depth study of the tooth cutting process of the arc cylindrical gear, proposes a single-sided double-milling processing method to cut the convex and concave surfaces of the arc gear teeth, and conducts tooth cutting processing based on a dedicated milling equipment. In the process of milling the convex and concave surfaces of the gear teeth of the arc cylindrical gear, the tool cuts in and exits along the involute trajectory. After adjusting the machine tool to complete the processing of the convex or concave surface of the first gear tooth, the subsequent milling processing of the gear teeth only requires precise indexing of the gear blank, and then repeat the previous cutting feed. By compiling the NC machining macro program and trial cutting the arc gear on a dedicated milling machine, the accuracy of the cut arc cylindrical gear is basically at grade 6 to 7, and the meshing transmission test proves that the cut arc gear meets the expected requirements. In addition, if it is necessary to further improve the accuracy of the arc gear, a gear grinder can be used to further grind the tooth surface to improve the gear accuracy.
