Abstract: Based on the study of the basic rack formed by the expansion of the arc cylindrical gear, this paper introduces the basic parameters and design calculations of the arc cylindrical gear. Based on the principle of the rotating cutter head method, a milling processing method for cutting the convex and concave surfaces of the arc gear teeth with a single-edged cutter is proposed. Combined with production practice, a numerical control processing program is compiled to carry out trial cutting of the arc cylindrical gear on a dedicated dual-station gear milling machine. The meshing transmission test of the trial-cut arc cylindrical gear shows that the meshing performance of the gear is good and meets the expected requirements.
1. Basic Parameters and Design Calculation of Arc Cylindrical Gear
Usually, the study of arc cylindrical gears is based on the basic rack of the arc cylindrical gear. Figure 1 shows the basic rack of the arc cylindrical gear, where p and q represent the convex and concave surfaces of the gear teeth, respectively. They are parts of two cones P and Q. R is the radius of the conical body at the pitch circle, t/2 is the distance between the axes of the two cones, b is the width of the rack, the cross-section A-A is the middle section of the rack, and the cross-section B-B is the pitch plane of the rack. The rack has an involute tooth profile at the cross-section A-A. It is found that the tooth profile at the middle section of the arc cylindrical gear is the same as that of the involute straight cylindrical gear, so the design calculation of the arc cylindrical gear can refer to that of the involute straight cylindrical gear. Its basic geometric parameters are module m, number of teeth z, pressure angle α, addendum coefficient h_a*, and dedendum coefficient c*, and the geometric size calculation is the same as that of the involute straight cylindrical gear. The calculation formulas are shown in Table 1.
The design process of the arc cylindrical gear can refer to that of the involute straight cylindrical gear, and the design criteria are the same as those of the straight cylindrical gear. In the design process of the straight cylindrical gear, because the main failure modes of the gear are tooth surface fatigue pitting and tooth fatigue fracture, different design calculation criteria, such as tooth surface contact fatigue strength or tooth root bending fatigue strength, can be selected according to the application scenarios of the gear transmission for design calculation. Therefore, in the design calculation process of the arc cylindrical gear, referring to the design calculation process of the involute straight cylindrical gear, the gear material, heat treatment method, and precision grade are first selected, and then the corresponding design criteria are determined according to the operating conditions. The number of teeth and module of the pinion are determined through calculation. Considering the undercutting problem, the number of teeth of the pinion should satisfy z_min ≤ z1 when selecting. Once the number of teeth of the pinion is determined, the number of teeth of the large gear can be determined according to the gear transmission ratio, that is, z_2 = i * z_1. Then, according to the geometric size calculation formulas in Table 1, the main dimensions of the arc cylindrical gear are calculated. Finally, the strength of the arc cylindrical gear is checked.
Table 1: Basic Parameters and Geometric Size Calculation of Arc Cylindrical Gear
| Name | Formula | Determined According to Strength Calculation |
|---|---|---|
| Basic Parameters | Module (mm) | |
| Number of Teeth: | z is the number of teeth of the gear, z2 is the number of teeth of the large gear, and i is the transmission ratio | |
| Pressure Angle) | The pressure angle at the central section is generally taken as -20° | |
| Addendum Coefficient h_a* | Generally taken as h_a* = 1.0 | |
| Dedendum Coefficient c* | Generally taken as c* = 0.25 | |
| Geometric Size | Pitch Circle Diameter (mm) | d = m * z |
| Addendum Height (mm) | h_a = h_a* * m | |
| Dedendum Height (mm) | h_f = (h_a* + c*) * m | |
| Whole Tooth Height (mm) | h = h_a + h_f = (2 * h_a* + c*) * m | |
| Tip Circle Diameter (mm) | d_a = d + 2 * h_a = (2 * h_a* + z) * m | |
| Root Circle Diameter (mm) | d_f = d – 2 * h_f = (z – 2 * h_a* – 2 * c*) * m |
2.Tooth Cutting Processing of Arc Cylindrical Gear
2.1 Tooth Cutting Method of Arc Cylindrical Gear
The tooth cutting processing of arc cylindrical gears is different from that of straight cylindrical gears. At present, the more typical tooth cutting processing methods mainly include the rotating cutter head method and the parallel connecting rod method. Since the rotating cutter head method is efficient and easy to implement in the tooth cutting process, this method is widely used in the tooth cutting processing of arc cylindrical gears, and its principle is shown in Figure 2. When using the rotating cutter head method to cut arc cylindrical gears, there is an exact generating motion relationship between the cutting cutter head and the gear blank, that is, the pitch circle linear velocity of the processed gear blank is equal to the linear velocity of the pitch line of the cutting cutter head. During the tooth cutting process of the arc gear, in addition to the main rotation motion, the cutting cutter head also has a radial feeding motion along the gear blank. In the processing using the generating method, because only one tooth groove can be cut at a time, after cutting each tooth groove, the gear blank must be accurately indexed, and then the second tooth groove is cut. After all the tooth grooves are cut, the processing of the arc gear can be completed. In theory, the tooth groove of the arc cylindrical gear can be cut out with a double-edged cutter at one time to form the convex and concave surfaces of the gear teeth. However, since the double-edged cutter is full-edge cutting during the tooth cutting process, the cutting amount is large, the tool 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 practice of tooth cutting with double-edged cutters, a single-edged cutter tooth cutting processing method is proposed, and the tooth cutting tool for arc cylindrical gears is shown in Figure 3. In Figure 3, R_e and R_i are the radii of the inner and outer single-edged cutter heads, respectively, and a1 is the pressure angle of the tool. Combining the practical experience of tooth cutting with double-edged cutters, during the tooth groove cutting process of the arc gear, the tooth groove can be roughly cut with a double-edged cutter first, and then the convex and concave surfaces of the arc cylindrical gear can be precisely cut with single-edged cutters to ensure the cutting accuracy.
The advantage of using single-edged cutters to precisely cut the convex and concave surfaces of the arc cylindrical gear is that the inner and outer single-edged cutters can be designed as cutters with equal radii, and the accuracy of the cut convex and concave surfaces of the gear teeth is higher. However, since the convex and concave surfaces need to be cut separately, the installation of the tooth blank and the replacement of the tool during the processing must be considered. Once the tool needs to be replaced or the tooth blank needs to be disassembled and assembled during the processing, the processing accuracy of the gear teeth will inevitably be affected. To this end, a single-sided double-milling processing method for arc cylindrical gears is proposed, and its processing principle is shown in Figure 4. During the tooth cutting process of the arc cylindrical gear, two stations are designed on a dedicated milling machine to cut the convex and concave surfaces of the gear teeth respectively. In Figure 4, 1 and 4 represent the single-edged milling cutters for the convex and concave surfaces located at the left and right stations, respectively, 2 and 5 represent the gear blanks of the arc cylindrical gear being processed when milling the convex and concave tooth surfaces, and 3 and 6 represent the single-edged milling cutters for the convex and concave surfaces, respectively. It can be seen that the tooth cutting processing of the arc cylindrical gear requires a dedicated numerical control milling equipment.
2.2 Tooth Surface Equation of the Tooth Cutting Tool for Arc Cylindrical Gear
During the milling process of the arc cylindrical gear, a dedicated tool is used. To design and manufacture the dedicated milling tool, the tooth surface equation of the tooth cutting tool can be established based on the principle of the rotating cutter head method for tooth cutting processing. Figure 5 shows the coordinate system of the tool and the processed gear. Firstly, the coordinate system [O1 x1, y1, z1] is fixed on the processed gear blank, and the coordinate system [Od, xd, yd, zd] is fixed with the milling cutter head, which rotate with the processed gear blank and the milling cutter head, respectively. Then, the fixed coordinate system [O, x, y, z] and [O1 x10, y10, z10] are auxiliary coordinate systems. 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 blade radius of the milling tool is r0 – πm/4, and the outer blade 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.
Thus, in the coordinate system [Od, xd, yd, zd] fixed with the tool, the parametric equation of the tool surface can be established as follows:
{x_d = (±u * sin α1 + r0 ± π/4 * m) * cos θ
y_d = (±u * sin α1 + r0 ± π/4 * m) * sin θ
z_d = u * cos α1}
where u is the distance from the point on the tool surface along the generatrix of the cone to the reference point, and θ is the rotation angle of the tool holder from the central section of the gear blank to the end face.
3. Tooth Cutting Process of Arc Cylindrical Gear and Macro Program Compilation
Combined with the analysis of the aforementioned tooth cutting process, the tooth cutting process of the arc cylindrical gear can be divided into three steps: The first step is to cut in the middle of the tooth groove of the arc cylindrical gear and use a three-edged tooth groove milling cutter to mill out the tooth groove and remove most of the tooth blank material. The second and third steps are to use single-edged cutters to process the concave and convex tooth surfaces of the arc cylindrical gear, respectively. When processing on a dedicated milling machine, the indexing of the processing process is programmed using both 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 for the first step, that is, to cut in the middle of the gear tooth groove for rough processing of the tooth groove. The second step starts from δ = 0, and the third step starts from δ to use two single-edged cutters for the corresponding concave and convex tooth surface processing. When each cutter completes the entire tooth groove and tooth profile processing for one week of the gear, the processing of the entire arc cylindrical gear can be completed. Since the tool cuts in along the involute, the cutting trajectory of the tool is along the involute. For example, the basic parameters for processing the arc cylindrical gear are as follows: module m = 3 mm, number of teeth z = 25, addendum coefficient h_a* = 1.0, dedendum coefficient c* = 0.25, and a macro program for one tooth groove cutting cycle is compiled as follows:
N10 R1 = 3 / m Module
N20 R2 = 25 / z Number of Teeth
N30 R3 = R1 * R2 / 2 / d Pitch Circle Diameter
N40 R4 = 1 / ha Addendum Coefficient
N50 R41 = 0.25 / Dedendum Coefficient
N60 R42 = 0 / Modification Coefficient
N70 R5 = 1 / Tooth Root Height Coefficient
N80 R6 = R3 / 2 + R4 * R1 / Tip Circle Radius
N90 R7 = R3 / 2 – R4 * R1 / Root Circle Radius
N100 R8 = 1 / t Initial Value of Involute Opening Degree 0 – 1 Decrement
N110 R9 = 0 / t Increment
N120 R10 = 60 * R8 / α Value Taking
N130 R11 = 3600 / R2 / Indexing Angle per Tooth 3600 / R2
N140 R12 = 0 / R12 = A Gear Rotation Angle A = R10 – atan((π * R2 * R10 / 180) / R2)
N150 R13 = R3 * cos(R10) + π * R3 * R10 / 180 * sin(R10) R13 = X Assignment
N160 R14 = R6 + 5 / Starting Position of the Tool in the X Feeding Direction
N170 G90 G54 G0 A0 X = R14 / The workpiece quickly rotates to A0, and the tool quickly moves to 5 mm away from the tooth tip
The milling program for the arc cylindrical gear is fed in the direction of the involute, and the tool also retracts along the involute. When the tool moves along the involute, attention should be paid to avoiding overcutting the unprocessed tooth surface to ensure sufficient machining allowance for the arc tooth surface and the machining quality of the arc tooth surface. Figure 6 shows the arc cylindrical gear obtained by trial cutting. Through the meshing transmission test, it is proved that the cut arc cylindrical gear meets the expected requirements.
4. Conclusion
The meshing transmission performance of the arc cylindrical gear is good, but due to the need for dedicated processing equipment and milling tools for tooth cutting, 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 paper focuses on the in-depth study of the tooth cutting processing of the arc cylindrical gear, proposes to use the single-sided double-milling processing method to cut the convex and concave surfaces of the arc gear teeth, and carries out the milling processing based on the dedicated milling equipment. During the milling process of the convex and concave surfaces of the arc cylindrical gear teeth, 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 tooth, for the subsequent tooth milling, only the precise indexing of the tooth blank is required, and then the previous cutting feed is repeated. By compiling the numerical control processing macro program and trial cutting the arc gear on the dedicated milling equipment, the accuracy of the cut arc cylindrical gear is basically between grade 6 and grade 7. The meshing transmission test proves that the cut arc gear meets the expected requirements. In addition, if the accuracy of the arc gear needs to be further improved, a gear grinding machine can be used to further grind the tooth surface to improve the gear accuracy.