This article focuses on the design and manufacturing of arc cylindrical gears. It begins with an introduction to the significance and advantages of arc cylindrical gears in mechanical transmission. Then, it elaborates on the basic parameters and design calculation methods, compares different cutting methods, and details the manufacturing process and programming. Through practical examples and analysis, it demonstrates the feasibility and application potential of arc cylindrical gears. The aim is to provide a comprehensive understanding of arc cylindrical gears for researchers and practitioners in the field of mechanical engineering.
1. Introduction
Gear transmission is a crucial component in mechanical systems, playing a vital role in transferring power and motion. Arc cylindrical gears, with their unique tooth profile and meshing characteristics, have attracted increasing attention in recent years. Unlike traditional involute gears, arc cylindrical gears offer several advantages, such as higher contact ratio, smaller axial load, and stronger load-carrying capacity. These features make them suitable for various applications where smooth and efficient power transmission is required.
2. Basic Parameters and Design Calculation of Arc Cylindrical Gears
2.1 The Basis of Design
The design of arc cylindrical gears is typically based on the basic rack. As shown in Figure 1, the basic rack consists of convex and concave surfaces (p and q), which are parts of two cones (P and Q). The radius at the pitch circle is R, and the distance between the axes of the two cones is t/2. The width of the rack is b, and the A – A section is the middle section of the rack, while the B – B section is the pitch plane. The tooth profile at the A – A section has an involute shape.
| Name | Calculation Formula |
|---|---|
| Module (mm) | Determined by strength calculation |
| Number of teeth | , (where is the number of teeth of the pinion, is the number of teeth of the gear, and is the transmission ratio) |
| Pressure angle () | The pressure angle at the central section is usually |
| Addendum coefficient | Generally |
| Clearance coefficient | Generally |
| Pitch circle diameter (mm) | |
| Addendum (mm) | |
| Dedendum (mm) | |
| Whole depth (mm) | |
| Tip circle diameter (mm) | |
| Root circle diameter (mm) |
2.2 Design Process
The design process of arc cylindrical gears is similar to that of involute spur gears. First, appropriate gear materials, heat treatment methods, and accuracy grades need to be selected according to the application requirements. Then, based on the working conditions, the design criteria, either the tooth surface contact fatigue strength or the tooth root bending fatigue strength, are determined. After that, the number of teeth and module of the pinion are calculated, taking into account the minimum number of teeth to avoid undercutting. Once the number of teeth of the pinion is determined, the number of teeth of the gear can be calculated using the transmission ratio. Finally, the main dimensions of the arc cylindrical gears are calculated, and the gear strength is checked to ensure that the designed gears can meet the performance requirements.
3. Cutting Methods of Arc Cylindrical Gears
3.1 Rotating Cutter Head Method
The rotating cutter head method is one of the commonly used cutting methods for arc cylindrical gears. In this method, there is a precise generating motion relationship between the cutting cutter head and the gear blank. The pitch line velocity of the gear blank is equal to the pitch line velocity of the cutting cutter head. During the cutting process, the cutter head rotates and has a radial feed motion along the gear blank. Although theoretically, a double-edge cutter can be used to cut a complete tooth space at once, forming the convex and concave surfaces of the tooth, in practice, due to the large cutting amount and rapid tool wear, it is difficult to achieve the required gear accuracy.
3.2 Single-Edge Cutter Cutting Method
To improve the cutting accuracy, a single-edge cutter cutting method has been proposed. In this method, a double-edge cutter can be used for rough cutting of the tooth space first, and then single-edge cutters are used to finish cutting the convex and concave surfaces of the arc cylindrical gears separately. By designing the inner and outer single-edge cutters with the same radius, higher tooth surface accuracy can be obtained. However, this method requires careful consideration of the installation of the gear blank and the replacement of cutters to avoid affecting the machining accuracy.
3.3 Single Double Milling Method
The single double milling method is designed to address the issues of cutter replacement and blank installation in the single-edge cutter cutting method. In this method, two workstations are set up on a special milling machine to cut the convex and concave surfaces of the gear teeth separately. As shown in Figure 4, the left and right workstations are equipped with single-edge milling cutters for the convex and concave surfaces, respectively, and the gear blanks are processed in corresponding positions. This method can improve the machining efficiency and accuracy while ensuring the smooth progress of the cutting process.
4. Gear Cutting Tools and Their Tooth Surface Equations
4.1 Special Gear Cutting Tools
Arc cylindrical gear milling requires special cutting tools. These tools are designed based on the principle of the rotating cutter head method. To accurately design and manufacture these tools, it is necessary to establish the tooth surface equation of the cutting tools.
4.2 Tooth Surface Equation of Cutting Tools
By establishing a coordinate system fixed on the gear blank and a coordinate system fixed on the milling cutter head , as well as auxiliary coordinate systems and , the tooth surface parameter equation of the milling cutter can be derived as follows:
where is the distance of a point along the generatrix of the cone from the reference point, and is the rotation angle of the tool holder from the central section of the gear blank to the end face.
5. Gear Cutting Process and Macro Program Compilation
5.1 Cutting Process Steps
The cutting process of arc cylindrical gears can be divided into three steps. First, a three-edge slot milling cutter is used to mill the tooth space in the middle of the arc cylindrical gear tooth slot, removing most of the gear blank material. Then, single-edge cutters are used to process the concave and convex tooth surfaces of the arc cylindrical gears respectively.
5.2 Programming Method
When programming on a special milling machine, two indexing methods, absolute indexing and relative indexing, are used. For example, if the gear is indexed by the number of teeth with an indexing angle of , the program starts from for the rough machining of the tooth slot in the first step. In the second step, it starts from , and in the third step, it starts from , using two single-edge cutters to process the corresponding concave and convex tooth surfaces. The following is an example of a macro program for cutting a tooth slot of an arc cylindrical gear with parameters such as module , number of teeth , addendum coefficient , and clearance coefficient :
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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 /addendum coefficient
N60 R42 = 0 /shift coefficient
N70 R5 = 1 /dedendum 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 0 - 1 decrement
N110 R9 = 0 /t increment
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)
R13 = X assignment
N160 R14 = R6 + 5 /starting position of the tool in the X feed direction
moving 5mm away from the tip
N170 G90 G54 G0 A0 X = R14 / the workpiece rotates quickly to A0, and the tool moves quickly
AA:
N180 G01 X = R13 A0 F300 /move to the starting point
N190 R8 = R8 - 0.002 /R8 assignment
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 return movement
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 cutter of the arc cylindrical gear feeds and retracts along the involute direction. During the movement of the cutter along the involute direction, attention should be paid to avoiding overcutting of the unprocessed tooth surface to ensure sufficient machining allowance for the surface to be processed and guarantee the machining quality of the arc tooth surface.
6. Application and Prospect of Arc Cylindrical Gears
Although arc cylindrical gears have excellent meshing performance, their application in mechanical transmission has not been widely promoted due to the need for special processing equipment and milling cutters. However, with the continuous development of manufacturing technology and the increasing demand for high-performance gears, the research and application of arc cylindrical gears are expected to receive more attention. In some high-precision and heavy-duty transmission systems, arc cylindrical gears can play an important role in improving transmission efficiency and reducing noise. Future research can focus on further improving the manufacturing accuracy and efficiency of arc cylindrical gears, exploring new materials and heat treatment methods to enhance the performance of gears, and promoting the wide application of arc cylindrical gears in various fields.
In conclusion, the design and manufacturing of arc cylindrical gears involve multiple aspects such as basic parameter calculation, cutting method selection, tool design, and programming. Through continuous research and innovation, arc cylindrical gears are expected to have a broader application prospect in the field of mechanical engineering.
