1. Introduction
1.1 Background and Significance
Spiroid face gear transmission, a form of space non – parallel shaft gear transmission, combines a cylindrical gear and a spiroid face gear. Compared with bevel gear transmission, it boasts advantages such as a larger coincidence degree, more sufficient tooth surface contact, smoother operation, and less noise. These characteristics enable its wide application in various fields, including robot joints, vehicle rear – axle reducers, and helicopter reducers.
However, the existing face gear machining methods have limitations. Traditional mechanical generation methods, like hobbing and shaping, are mainly applicable to spur face gears and small – helix – angle face gears. They struggle to process face gear tooth surfaces with large helix angles. For instance, shaping has low processing efficiency due to idle strokes and discontinuous operations, while hobbing can only handle certain types of face gears with significant deviations from the theoretical tooth profile. CNC milling, although avoiding the development of special tools, has issues with low processing accuracy, high cost, and low efficiency when dealing with large – helix – angle face gears.
The skiving method, a combination of hobbing and shaping, has emerged as a potential solution. With the continuous development of CNC machine tools and cutting tools, skiving technology has become more mature. Applying this technology to spiroid face gear machining can significantly enhance processing efficiency and quality, filling the gap in high – efficiency machining of spiroid face gears and thus having crucial engineering application value.
1.2 Research Status at Home and Abroad
1.2.1 Face Gear Machining
- 国外研究现状：国外学者在面齿轮加工研究方面成果颇丰。Litvin 等人提出多种面齿轮传动的设计与分析方法，如偏置非正交蜗杆螺旋齿面齿轮传动的计算机设计与分析方法 ，并开发了 TCA 计算机程序来仿真啮合过程、降低传动误差；还提出圆锥蜗杆和圆柱蜗杆的面齿轮传动机构，建立传动误差模型以降低噪声和振动 。Wenjin Wang 提出基于斜头铣刀的面齿轮几何设计方法，确定加工数学模型并建立仿真模型。Kazumasa Kawasaki 研究了带螺旋小齿轮的面齿轮传动几何设计，分析了螺旋角和偏移距离对齿轮的影响。H.A. Zschippang 提出面齿轮的一般生成方法，对插齿机包络过程进行仿真并提出改进措施。Jonas – Frederick 提出面齿轮主动齿面方程计算的新方法，减少了齿面方程数值后处理。
- 国内研究现状：国内对面齿轮传动的研究起步 later，但也取得了不少成果。朱如鹏团队深入研究面齿轮传动技术，建立根切与齿顶变尖方程，推导齿宽计算公式。李政民卿推导正交对心直齿面齿轮插齿加工齿面方程，通过仿真和试验验证加工方法的可行性。赵宁等人提出在普通滚齿机上改造加工面齿轮的方法，分析误差并进行齿面修形。王延忠等对滚齿刀具方程、压力角影响、磨削加工等方面进行研究。彭先龙提出蝶式砂轮刀具切削面齿轮的方法并进行理论分析。刘玄研究螺旋齿面齿轮剃齿加工，景琦探索直齿圆柱齿轮刀具车齿法加工螺旋齿面齿轮技术，符云龙和张广分别对直齿面齿轮热滚轧和滚轧成形进行建模与仿真分析。
1.2.2 Gear Skiving
- 国外研究历程：Gear skiving, known as a continuous chip – removal method for gear manufacturing, was first described in a patent by Wilhelm von Pittler in 1910 and was industrially applied in 1960. M.Kojima analyzed the geometric relationship between skiving cutters and straight gears and the clearance angle in 1974. Spath and Huhsam developed the first models for simulating the periodic motion of skiving. Volker established a 3D – FEM model to study skiving kinematics and chip – formation mechanisms. Hartmut and Olaf proposed semi – complete cutting methods and devices to improve tool life. Stadtfeld defined the optimal conditions for skiving, and Klocke et al. analyzed the skiving process using technical and simulation methods. Guo et al. studied the cutting mechanism and designed skiving cutters for involute gears. Masatomo Inui conducted geometric simulations and developed visualization software for gear shaping. Komio et al. studied the skiving process of cylindrical internal gears in detail.
- 国内研究进展：国内对车齿加工的研究也在不断深入。金精深入研究车齿工作原理，总结工艺理论、机床、刀具等方面的规律。刘健提出精车加工齿轮的前刀面精加工法，设计圆锥螺旋车齿刀并建立切削数学模型。李旺辉提出车齿成形原理，推导刀具刀刃方程并进行加工试验。陈新春建立无理论刃形误差的车齿刀具模型，优化刀具结构参数。郭二廓提出适用于渐开线齿形的车齿刀刃形计算方法，建立理论数学模型。王长健研究小螺旋角斜齿刀具车齿法加工大螺旋角外斜齿轮技术，分析齿面偏差。刘冰、王凯、赵海洋和徐雷分别利用仿真软件对车齿加工进行仿真分析、研究切屑形成、确定加工参数和开发车齿刀参数设计系统。
1.3 Existing Problems in Spiroid Face Gear Machining
Despite the efforts of scholars at home and abroad, research on spiroid face gear machining remains insufficient. Most studies focus on the design and manufacturing of spur and helical face gears, with few on the mechanical generation manufacturing technology of spiroid face gears. Regarding skiving technology, while significant progress has been made in the machining of cylindrical internal gears, there is a lack of research on high – efficiency skiving technology for spiroid face gears. This research aims to fill this gap by proposing and verifying the skiving method for spiroid face gears, laying a foundation for high – efficiency and high – precision machining.
1.4 Research Content and Technical Route
1.4.1 Research Content
- 齿面设计：Based on the shaping principle of spiroid face gears, establish the tooth surface equation. Build the shaping processing coordinate system, derive the tooth surface equation of involute cylindrical gears, and then obtain the theoretical tooth surface equation of spiroid face gears through the meshing relationship. Use numerical calculation methods to solve the tooth surface equation, perform grid division, and create a 3D digital model. For the skiving method, design the spatial layout of the tool and workpiece, establish the tooth surface equation of the bowl – shaped skiving cutter, and derive the tooth surface equation of the skived spiroid face gear. Establish a tooth surface error analysis model to compare the skived and theoretical tooth surfaces.
- 加工仿真：Analyze the skiving processing motion of spiroid face gears, determine the motion parameters, and establish the processing motion coordinate system. Design the structure of the bowl – shaped skiving cutter, study the modification method of the cutter edge curve, and create 3D models of the cutter and the CNC machine tool. Use VERICUT simulation software to import the models, write CNC programs, conduct skiving processing simulations, and analyze the results to obtain the general rules of CNC programming.
- 加工试验：Select the skiving processing of spur face gears considering processing equipment. Conduct simulation processing and analysis based on the general rules obtained from spiroid face gear skiving simulations. Design a skiving test plan, use a CNC skiving machine tool to process spur face gears, and measure the tooth surfaces to verify the correctness and feasibility of the skiving method for face gears.
1.4.2 Technical Route
The technical route involves establishing the spatial position layout of the skiving cutter and the spiroid face gear, verifying the correctness of the skiving design theory. Then, establish the theoretical tooth surface equation of the spiroid face gear, followed by creating the machining kinematic model and the tooth surface equation of the skiving cutter. Next, use VERICUT software for simulation, conduct error analysis, and finally, perform experimental verification on the CNC machine tool to confirm the correctness of the skiving processing technology .
2. Tooth Surface Design of Spiroid Face Gear Based on Skiving Method
2.1 Design Principle of Spiroid Face Gear Tooth Surface Based on Virtual Shaping
The theoretical tooth surface of a spiroid face gear can be obtained through shaping processing. As shown in Figure 2, during shaping, the shaping tool rotates around its axis with an angular velocity of \(\omega_{1}\), and the spiroid face gear rotates around its axis with an angular velocity of \(\omega_{2}\). The relationship between their angular velocities is \(i_{12}=\frac{\omega_{1}}{\omega_{2}}=\frac{N_{2}}{N_{1}}\), where \(N_{1}\) and \(N_{2}\) are the number of teeth of the shaping tool and the face gear respectively, and \(i_{12}\) is the transmission ratio.
2.2 Tooth Surface Design of Spiroid Face Gear in Virtual Shaping Processing
2.2.1 Cylindrical Gear Tooth Surface Equation
An involute cylindrical gear is formed by a straight – tooth involute profile and a helix on the base cylinder. The position vector of a moving point N on the tooth side Ⅰ of the involute helical cylindrical gear can be expressed as \(\vec{O_{S}N}=\vec{O_{S}K}+\vec{K M}+\vec{M N}\). Through geometric relationships, the coordinates of the moving point N can be obtained as \(\left\{\begin{array}{l}x_{s}=r_{b s}\cos\theta_{s}+\mu_{s}\cos\lambda_{b s}\sin\theta_{s}\\y_{s}=r_{b s}\sin\theta_{s}-\mu_{s}\cos\lambda_{b s}\cos\theta_{s}\\z_{s}=-\mu_{s}\sin\lambda_{b s}+p_{s}\theta_{s}\end{array}\right.\), where \(r_{b s}\) is the base – circle radius, \(\theta_{s}\) and \(\mu_{s}\) are tooth – surface parameters, \(\lambda_{b s}\) is the lead angle of the helix, and \(p_{s}=r_{b s}\tan\lambda_{b s}\) is the spiral parameter. The unit normal vector of the tooth surface Ⅰ is also calculated. Similarly, the equations for the tooth side Ⅱ can be derived.
In the end – section of the involute cylindrical gear (\(Z_{s} = 0\)), the tooth profile equations and unit normal vectors of tooth profiles Ⅰ and Ⅱ can be obtained by substituting relevant parameters. The relationship between the tooth – profile angle \(\alpha_{t}\), the lead angle \(\lambda_{p s}\) on the pitch circle, and the lead angle \(\lambda_{b s}\) on the base circle, as well as the relationship between the spiral parameter \(p_{s}\) and the pitch – circle radius \(r_{p s}\) and helix angle \(\beta\), are also determined.
2.2.2 Meshing Equation
According to the principles of differential geometry and gear meshing, for two meshing gears in space, the relative velocity \(v_{n}\) at the contact point must be perpendicular to the common normal vector n to ensure continuous contact. For an involute cylindrical gear and a spiroid face gear, \(n\cdot v_{n}=0\). The angular velocity vectors of the involute cylindrical gear and the spiroid face gear in their respective coordinate systems are determined. By substituting relevant vectors into the equation, the meshing equation \(f\left(\mu_{n k},\theta_{n k},\phi_{n k}\right)=n_{n k}\cdot v_{n k}^{(12)} = 0\) is obtained, where the signs in the equation correspond to different tooth sides of the involute cylindrical gear.
2.2.3 Face Gear Tooth Surface Equation
The tooth surface of a face gear consists of a working tooth surface and a transition tooth surface at the root. The working tooth surface is formed by the swept surface of the involute cylindrical gear tooth surface through coordinate transformation. By combining the tooth surface equation of the cylindrical gear and the meshing equation, the working – tooth – surface equation of the spiroid face gear is derived as \(r_{2}\left(\theta_{1},\phi_{1}\right)=\left\{\begin{array}{l}r_{2}\left(\mu_{1},\theta_{1},\phi_{1}\right)=M_{21}\left(\phi_{1}\right)\cdot r_{1}\left(\mu_{1},\theta_{1}\right)\\f_{21}\left(\mu_{1},\theta_{1},\phi_{1}\right)=n_{1}\cdot v_{1}^{(12)} = 0\end{array}\right.\).
The transition tooth surface is formed by the top of the shaping tool. By determining the parameter \(\theta_{1}^{*}\) at the top of the tool, the transition – tooth – surface equation \(r_{2}\left(\mu_{1},\theta_{1}^{*},\phi_{1}\right)=M_{21}\left(\phi_{1}\right)\cdot r_{1}\left(\mu_{1},\theta_{1}^{*}\right)\) is obtained.
2.3 Digital Modeling of Spiroid Face Gear
2.3.1 Tooth Surface Mesh Generation of Spiroid Face Gear
The tooth surface of a spiroid face gear is complex. To obtain a high – precision 3D model, it is necessary to solve the tooth – surface equation and divide the tooth surface into discrete grid points. The 3D coordinate points on the tooth surface are projected onto a 2D plane. The coordinate transformation relationship between the 3D and 2D coordinate systems is \(\left\{\begin{array}{l}R_{h}=\sqrt{x_{2}^{2}+y_{2}^{2}}\\Z_{h}=z_{2}\end{array}\right.\).
The tooth surface is meshed with an appropriate grid density. According to the gear measurement standard, in this study, the tooth surface of the spiroid face gear is divided into 15 rows and 23 columns, with a total of 345 nodes. The non – linear equations of the tooth – surface parameters are solved using the Newton – Raphson method and the fsolve function in MATLAB. The calculation process involves defining the initial value, iteratively solving the equations, and saving the discrete – point coordinates.
2.3.2 3D Modeling of Spiroid Face Gear
Pro / E software is used for 3D modeling. First, the discrete – point coordinates of the tooth surface are loaded, and then surface modeling is performed to generate a single tooth. Finally, solid array operations are carried out to obtain the entire gear model. The specific steps include creating a new part, inserting independent geometry, selecting data files, setting curve and point tolerances, using the 造型工具 to connect curves, performing boundary blending, merging tooth surfaces, solidifying the single – tooth shape, and finally creating the entire gear by arraying the single tooth.
2.4 Tooth Surface Design of Spiroid Face Gear by Skiving Method
2.4.1 Spatial Layout Design of Skiving Method for Machining Face Gears
Skiving is based on the principle of spatial crossed – axis gear meshing. In skiving, the axes of the tool and the workpiece are inclined at an angle \(\sum\), which is related to the helix angles of the tool and the workpiece (\(\sum=\left|\beta_{1}\pm\beta_{2}\right|\)). The sign in the formula depends on the meshing type (external or internal gear meshing) and the helix – direction relationship.
To achieve skiving of spiroid face gears, multiple motions are required, including the rotation of the skiving cutter and the workpiece, and the radial feed motion of the cutter. When considering the radial feed, an additional rotation angle is added to either the tool or the workpiece to ensure correct tooth – shape processing. In this study, an additional rotation angle is added to the workpiece, and the speed – ratio relationship \(\omega_{2}=i_{21}\omega_{1}+\frac{2v\sin\beta_{2}}{m_{n}\cdot N_{2}}\) is obtained.
2.4.2 Establishment of Skiving Tool Tooth Surface Equation
The skiving cutter can be approximated as a helical cylindrical gear. By adding the rake angle and clearance angle to the helical cylindrical gear, the skiving – cutter structure is obtained. The 端面齿廓 of the skiving cutter is a standard involute. The coordinates of points on different sections of the tooth profile are calculated. When the point \(p_{a}\) is on the \(S_{1}S_{2}\) section, its coordinates in the \(S_{1}\) coordinate system are \(r_{1}^{i}(\varphi)=\left[\begin{array}{c}r_{b}(\cos\varphi+\varphi\sin\varphi)\\K_{a}r_{b}(\sin\varphi – \varphi\cos\varphi)\\0\\1\end{array}\right]\), and when on the \(S_{2}S_{3}\) section, the coordinates.
2.4.3 Skiving Motion Meshing Equation
Based on the skiving method for machining spiroid face gears, a processing coordinate system is established, as shown in Figure 3. This coordinate system includes the fixed coordinate system of the processing tool \(S_{p}(O_{p}, x_{p}, y_{p}, z_{p})\), the rotating coordinate system \(S_{1}(O_{1}, x_{1}, y_{1}, z_{1})\) fixed to the tool, the fixed coordinate system \(S_{g}(O_{g}, x_{g}, y_{g}, z_{g})\) of the helical face gear, the rotating coordinate system \(S_{2}(O_{2}, x_{2}, y_{2}, z_{2})\) fixed to the spiroid face gear, and the auxiliary coordinate systems \(S_{m}(O_{m}, x_{m}, y_{m}, z_{m})\) and \(S_{p}(O_{p}, x_{p}, y_{p}, z_{p})\) connected to the frame. The angles \(\varphi_{1}\) and \(\varphi_{2}\) represent the rotation angles of the processing tool and the helical face gear respectively, and L is the distance between the coordinate origins of the tool and the gear coordinate systems.
2.4.4 Face Gear Tooth Surface EquationThe tooth surface of the tool expressed in the (S_{1}) coordinate system is transformed through the coordinate transformation matrix (M_{21}) to obtain a set of surfaces with motion parameter (\varphi_{1}), which is expressed as (r_{2}(\theta_{s},\gamma_{s},\varphi_{1})=M_{21}(\varphi_{1})\cdot r_{s}(\theta_{s},\gamma_{s})). The points on this set of surfaces that are also on the spiroid face gear tooth surface must satisfy the meshing equation. By combining these two equations, the working – tooth – surface equation of the spiroid face gear is obtained as (\left{\begin{array}{l}r_{2k}(\theta_{s},\gamma_{s},\varphi_{1})=M_{21}(\varphi_{1})\cdot r_{s}(\theta_{s},\gamma_{s})\f(\theta_{s},\varphi_{1},\gamma_{s})=n_{s}\cdot v^{(1,2)} = 0\end{array}\right.).The root transition surface of the spiroid face gear is formed by the part of the surface swept by the tooth – tip line of the involute skiving tool in the fixed coordinate system of the spiroid face gear. By replacing the angle parameter (\theta_{s}) in the tool – tooth – surface equation with the parameter (\theta_{sa}) on the tooth – tip circle ((\theta_{sa}=\sqrt{\frac{r_{as}^{2}}{r_{bs}^{2}} – 1}), where (r_{as}) is the tooth – tip – circle radius of the involute tool and (r_{bs}) is the base – circle radius of the tool), the equation of the tooth – tip line is obtained. Then, in the (S_{2}) coordinate system of the spiroid face gear, the tooth – surface equation of the transition surface is (r_{2}(\gamma_{s},\varphi_{1})=M_{21}(\varphi_{1})\cdot r_{s}(\theta_{sa},\gamma_{s})).2.5 Comparison and Analysis of Tooth Surfaces Generated by Shaping and Skiving Methods for Face GearsTo verify the correctness of the tooth surface generated by the skiving method for spiroid face gears, the tooth surface obtained by the shaping method is regarded as the theoretical tooth surface. The tooth – surface error is defined as the normal distance between the two tooth surfaces. The grid of the two tooth surfaces is divided into the same number of rows and columns to facilitate error comparison. The error (\delta) is calculated as (\delta=n_{sk}\cdot(r_{2k}-r_{1k})), where (n_{sk}) is the normal vector, and (r_{2k}) and (r_{1k}) are the vectors of the corresponding points on the skived and theoretical tooth surfaces respectively.Taking a large – helix – angle spiroid face gear as an example, with the parameters shown in Table 1, the tooth – surface equations of the theoretical and skived tooth surfaces are solved to obtain a series of discrete – point coordinates. Then, the tooth – surface error is compared according to the established error – analysis model.NameParameter SpecificationValueSkiving ToolHelix Angle (\beta_{1})(15^{\circ})Hand of HelixLeft – handNumber of Teeth (Z_{1})28Normal Module (m_{n})(3.25mm)Normal Pressure Angle (\alpha_{n})(20^{\circ})Normal Addendum Coefficient (h_{an})1Normal Clearance Coefficient (c_{n})0.25Involute Cylindrical GearHelix Angle (\beta_{g})(25^{\circ})Hand of HelixLeft – handNumber of Teeth (Z_{g})23Normal Module (m_{n})(3.25mm)Normal Pressure Angle (\alpha_{n})(20^{\circ})Normal Addendum Coefficient (h_{an})1Normal Clearance Coefficient (c_{n})0.25Spiroid Face GearNumber of Teeth (Z_{2})84Inner Radius (r_{1})144mmOuter Radius (r_{2})171mmShaft Angle(90^{\circ})Table 1. Parameters of Skiving Tool, Cylindrical Gear, and Spiroid Face GearThe error comparison results are shown in Figure 4. On the concave surface, most of the discrete points of the working tooth surface fluctuate around 0, and there is obvious root – cutting in the transition – tooth – surface part, resulting in a relatively large error. On the convex surface, the tooth – surface error range is (0.001 – 0.010mm). Overall, the main error exists on the concave surface. To reduce the error, the cutter needs to be modified to avoid root – cutting. Through the error analysis and comparison, the correctness of the skiving method for machining spiroid face gears is preliminarily verified.
2.6 3D Modeling of Spiroid Face Gear
The discrete – point coordinate file of the spiroid face gear obtained by the skiving method is imported into Pro / E for 3D modeling. The operation steps are the same as those for generating a spiroid face gear by shaping, and the 3D model of the spiroid face gear processed by skiving is obtained, as shown in Figure 5.
2.7 Chapter Summary
In this chapter, taking the meshing of an involute cylindrical gear and a spiroid face gear as an example, the tooth – surface equation of the cylindrical gear is established based on the shaping principle of the spiroid face gear. Then, the theoretical tooth – surface equation of the spiroid face gear is derived according to the spatial gear meshing principle, providing a basis for subsequent research. Through numerical calculation methods, the discrete – point coordinates of the tooth surface are obtained, and 3D digital modeling is carried out using 3D software. The modification method of the involute skiving – cutter edge curve is studied, and the tooth – surface equation of the cutter is established. The spatial position relationship between the skiving cutter and the workpiece is designed, and the tooth – surface equation of the face gear processed by skiving is derived. A tooth – surface error – analysis model is established to compare and analyze the skived and theoretical tooth surfaces, preliminarily verifying the correctness of the skiving – processing method for spiroid face gears. Finally, 3D modeling of the tooth surface generated by the skiving method is performed to obtain the 3D digital model of the spiroid face gear processed by skiving.
3. Skiving Machining Simulation of Spiroid Face Gear
3.1 Skiving Machining Motion Analysis of Spiroid Face Gear
3.1.1 Establishment of Machining Motion Coordinate System
Based on the spatial position relationship and skiving – machining principle of the spiroid face gear, the motion parameters during skiving machining are determined. The machined face gear rotates around its own axis, the skiving cutter rotates at an angle \(\sum\) in the coordinate plane of the spiroid face gear and around its own axis, and also has a feed motion along the radius direction of the face gear, as shown in Figure 6.
3.2 Design of Skiving Tool
3.2.1 Structural Parameter Design of Bowl – Shaped Rake – Face Skiving Cutter
According to the different rake – face grinding methods of skiving cutters, they can be roughly divided into petal – shaped and bowl – shaped skiving cutters. The petal – shaped skiving cutter forms the rake face by tilting the rake angle in a plane perpendicular to the helix, while the bowl – shaped skiving cutter uses a conical surface as the rake face. The petal – shaped skiving cutter has balanced wear on the two cutting edges but has low grinding efficiency and high post – use costs. The bowl – shaped skiving cutter is more convenient for grinding and has relatively low post – use costs.
The structure of the bowl – shaped skiving cutter is designed, which is similar to a helical cylindrical gear. Each rake face of the bowl – shaped cutter is a conical surface, and all the cutting edges of the teeth are located on this conical surface. The rake angle of the cutter is \(\gamma_{0}\), and the clearance angle (cone angle) is \(\alpha_{0}\). The basic parameters of the selected cutter are shown in Table 2.
| Name | Parameter Specification | Value |
|---|---|---|
| Skiving Tool | Helix Angle \(\beta_{1}\) | \(15^{\circ}\) |
| Hand of Helix | Left – hand | |
| Number of Teeth \(Z_{1}\) | 28 | |
| Normal Module \(m_{n}\) | \(3.25mm\) | |
| Normal Pressure Angle \(\alpha_{n}\) | \(20^{\circ}\) | |
| Normal Addendum Coefficient \(h_{an}\) | 1 | |
| Normal Clearance Coefficient \(c_{n}\) | 0.25 | |
| Rake Angle \(\gamma_{0}\) | \(0^{\circ}\) | |
| Clearance Angle \(\alpha_{0}\) | \(20^{\circ}\) | |
| Table 2. Basic Parameters of Skiving Tool |
