1. Introduction
Gear transmission represents one of the most important forms of mechanical power transmission, and gears serve as core components in numerous mechanical products. The unique combination of compact structure, high transmission efficiency, precise motion transfer, and long service life makes gears indispensable across a wide range of industries. Among the various types of gears, bevel gears hold a particularly important position in automotive, construction machinery, and machine tool manufacturing sectors. Depending on the tooth length characteristics, bevel gears are generally classified into straight bevel gears and curved tooth bevel gears. Although the development of spiral bevel gears has advanced rapidly, straight bevel gears remain widely used in aerospace, military, and defense applications because their design, manufacturing, and installation are relatively straightforward and cost-effective. This is something that spiral bevel gears cannot completely replace.
Compared to the rapid progress in spiral bevel gear machining technology, the processing of straight bevel gears has remained relatively underdeveloped. The commonly used manufacturing methods are still traditional mechanical processes such as gear milling, gear pulling, and gear planing. The machine tools applied for straight bevel gear processing mainly include the Y236 bevel gear planing machine and the Y2726 double-cutter straight bevel gear milling machine. Although these machines have been widely used in the field of straight bevel gear processing, they lag considerably behind the advanced machine tools used for spiral bevel gears. These machines mostly employ mechanical transmission chains with low transmission accuracy, low production efficiency, low machining precision, and difficult adjustment procedures, which greatly restricts the development and application of straight bevel gear machining technology.
Existing straight bevel gear processing equipment cannot meet the demands of high-precision and high-efficiency processing. The aviation and military industries in China urgently require the development of efficient and high-precision straight bevel gear manufacturing technology based on domestic CNC machines. This research focuses on the key technology of machining straight bevel gears using an inner concave cutter. The objective is to establish a cutting calculation model and a tooth surface mathematical model, develop tooth contact analysis software, and verify the entire methodology through actual cutting experiments. By using a single inner concave cutter on a domestic five-axis CNC machine, the straight bevel gear milling process can be realized. Furthermore, with CBN grinding wheels, the grinding process for straight bevel gears can also be achieved, meeting the processing requirements of the aviation and military industries for high-precision straight bevel gears. This research aims to form a complete set of efficient machining technology for straight bevel gears using inner concave cutters and to enhance the manufacturing level of straight bevel gears in China.
1.1 Research Status of Straight Bevel Gear Machining Technology
As early as the 19th century, special machine tools and cutting tools based on the generating principle were introduced, providing more advanced processing means for gears and demonstrating the enormous superiority of involute tooth profiles. Russian scholar Toxmah established the analytical theory of gear meshing and conducted extensive research on meshing theory. Litvin and others combined gear design with mathematics, elaborating on the geometric and practical application theories of gears.
In the early 1980s, China established a number of gear manufacturing enterprises in conjunction with product upgrades and the development of hardened tooth surface gear manufacturing technology, essentially forming a complete gear manufacturing system. The category of straight bevel gear milling methods mainly includes form milling and generating milling. In terms of milling machinery, the main equipment includes the YK2725 double-cutter straight bevel gear milling machine produced by Neijiang Hongchang Machine Tool Co., Ltd., the YK2730 CNC straight bevel gear double-cutter milling machine produced by Tianjin First Machine Tool Plant, and the YE2725 straight bevel gear double-cutter milling machine produced by the Neijiang Machine Tool Plant. These machines are primarily used for cutting straight bevel gears in automotive, aviation, light industrial, and other mechanical manufacturing sectors.
The double-cutter straight bevel gear milling machine employs the generating method to process straight bevel gears. The processing principle assumes an imaginary flat straight bevel gear that meshes with the workpiece gear in a conjugate transmission relationship, meaning that when the flat gear rotates, it drives the workpiece gear to rotate. The left and right milling cutter blades on the two cutter heads form a tooth side that replaces one tooth of the flat straight bevel gear. By maintaining this transmission relationship for generating motion, the milling blades and the gear blank mesh without clearance, thereby cutting out the required tooth space and producing the straight bevel gear. This imaginary flat gear is called the generating gear. During cutting, the cutter head only performs a simple rotary motion, with the working milling blades positioned at one tooth position of the planar generating gear. The workpiece gear rotates about its own axis while simultaneously performing pure rolling on the pitch plane of the planar generating gear, cutting in to full tooth depth and then gradually cutting out. After completing one tooth space, the cutter head withdraws from the cutting plane through a ball screw and nut mechanism. The cradle and workpiece gear quickly return and index to the next tooth space cutting position. After all tooth spaces are completed, the cutter head automatically withdraws and the cradle stops rotating.
For the cutter head used in this cutting process, the main cutting edge constituting the tooth side of the imaginary planar generating gear is not perpendicular to the cutter head rotation axis, but rather forms an inclined angle, meaning the cutting edge lies on an inner concave conical surface. Therefore, for the workpiece gear, the tooth length direction presents an arc with a raised middle, forming a crowned tooth bevel gear. Because the cutter head does not perform infeed motion along the tooth length direction during machining, the tooth bottom is arc-shaped along the tooth length direction. When the cutter head diameter is relatively large, for straight bevel gears whose tooth width is less than one-third of the pitch cone generatrix length, the influence of the tooth bottom arc on transmission quality and accuracy is minimal. Crowned teeth can reduce transmission noise and the sensitivity of the gear pair to installation position errors.
Gleason developed the No.104 milling machine for straight bevel gear double-cutter milling processing, but this machine is a mechanical type with low efficiency. For a long period, straight bevel gear milling technology did not receive corresponding development. It was not until 2007 that Gleason successfully applied the CNC Free-form machine tool to the processing of straight bevel gears, realizing multi-axis linkage CNC machining and greatly improving production efficiency.
2. Basic Geometric Parameters and Cutting Parameters for Straight Bevel Gears Based on the Inner Concave Cutter
In order to process straight bevel gears on domestic CNC spiral bevel gear milling machines using an inner concave cutter, this section analyzes the basic geometric parameters of the straight bevel gear and the design calculation methods for the machine adjustment parameters and cutter parameters based on the cutting principle of the inner concave cutter.
2.1 Geometric Parameter Design of Straight Bevel Gears
Straight bevel gear pairs are typically used for transmission with a shaft angle of 90°. The transmission principle is that two cones perform relative pure rolling on the pitch cone plane, as shown in the standard bevel gear drive configuration. For a pair of straight bevel gears to mesh correctly, the large-end modulus and pressure angle of both gears must be equal, and the pitch cone generatrices must coincide.
The geometric parameters of straight bevel gears are generally calculated based on the large end as the standard. The main parameters include the large-end modulus m, pressure angle α, tooth width B, clearance c, number of teeth z, shaft angle Σ, large-end tip circle diameter da, full tooth height h, addendum ha, dedendum hf, pitch cone angle δ, face angle δa, root angle δf, dedendum angle θf, outer cone distance R, and crown distance Ak. These parameters have a significant influence on the processing and performance of straight bevel gears.
The straight bevel gear pair can adopt either equal clearance contraction or unequal clearance contraction tooth height forms. The more widely used form is unequal clearance contraction, where the root cone, pitch cone, and face cone vertices coincide, and the clearance of the gear pair gradually increases from the small end to the large end. The manufacturing difficulty of unequal clearance contraction gears is lower than that of equal clearance contraction gears. Table 1 lists the calculation formulas for the basic geometric parameters of straight bevel gears.
| Parameter | Pinion | Gear |
|---|---|---|
| Large-end pitch diameter d (mm) | d₁ = mz₁ | d₂ = mz₂ |
| Pitch cone angle δ (°) | δ₁ = arctan(z₁/z₂) [Σ = 90°] | δ₂ = 90° − δ₁ |
| Outer cone distance R (mm) | R_e = 0.5d₁/sinδ₁ | R_e = 0.5d₂/sinδ₂ |
| Mean cone distance R_m (mm) | R_m = R_e − 0.5B | R_m = R_e − 0.5B |
| Large-end addendum h_a (mm) | h_{a1} = m(h_a^* + x_j) | h_{a2} = m(h_a^* − x_j) |
| Large-end dedendum h_f (mm) | h_{f1} = m(h_a^* + c^* − x_j) | h_{f2} = m(h_a^* + c^* + x_j) |
| Large-end full tooth height h (mm) | h₁ = h_{a1} + h_{f1} | h₂ = h_{a2} + h_{f2} |
| Large-end tip circle diameter d_a (mm) | d_{a1} = d₁ + 2h_{a1}cosδ₁ | d_{a2} = d₂ + 2h_{a2}cosδ₂ |
| Dedendum angle θ_f (°) | θ_{f1} = arctan(h_{f1}/R_e) | θ_{f2} = arctan(h_{f2}/R_e) |
| Root cone angle δ_f (°) | δ_{f1} = δ₁ − θ_{f1} | δ_{f2} = δ₂ − θ_{f2} |
| Face cone angle δ_a (°) | δ_{a1} = δ₁ + θ_{f2} | δ_{a2} = δ₂ + θ_{f1} |
| Large-end circular tooth thickness s (mm) | s₁ = m(π/2 + 2x_j tanα + x_τ) | s₂ = m(π/2 − 2x_j tanα − x_τ) |
| Large-end space width e_d (mm) | e_{d1} = πm − s₁ − 2h_{f1}tanα | e_{d2} = πm − s₂ − 2h_{f2}tanα |
Table 1: Calculation formulas for basic geometric parameters of straight bevel gears
2.2 Cutter Parameters and Machine Adjustment Parameters
The inner concave cutter used for machining straight bevel gears can be divided into two types: one is an integral cutter head, and the other is a cutter head equipped with carbide inserts. This research uses the integral inner concave cutter. The main structural parameters of the cutter head include the concave angle of the cutting edge, the cutter head diameter, and the cutter top width.
The characteristic of the inner concave cutter is that the main cutting edge is concave by a certain angle with respect to the end face of the milling cutter head. When the cutter head rotates, it forms a conical surface with an inner cone angle, causing the tooth side surface of the virtual generating gear to be a concave conical surface. Using such an inner concave cutter for gear milling can produce crowned teeth. The degree of convexity of the tooth line is measured by the distance ΔS from the two endpoints of the tooth line to the vector radius line at the midpoint of the tooth line. The advantage of producing crowned teeth is that it can improve the contact performance of the two tooth surfaces during straight bevel gear transmission, reduce the sensitivity of the gear pair to installation errors, and avoid edge contact.
The calculation formula for the crowning amount is:
$$ \Delta S = \frac{B^2 \cos\alpha_0}{4A_c}\tan\delta \tag{2.2} $$
where B is the tooth width of the workpiece gear, δ is the concave angle of the main cutting edge, α₀ is the cutter pressure angle, and A_c is the cutter head diameter. From the above equation, it can be seen that the crowning amount is proportional to the tooth width B and the concave angle δ, and inversely proportional to the pressure angle α₀ and the cutter head diameter A_c.
Since the cutter head does not move along the tooth width direction during machining, the root line of the workpiece gear is arc-shaped, meaning the tooth space bottom is a concave arc. The cutter head diameter A_c is related to the tooth width B of the workpiece gear, the concave arc depth Δh of the tooth space bottom, and the tooth pressure angle α, and is calculated as follows:
$$ A_c = \frac{B^2 \cos\alpha}{4\Delta h} \tag{2.3} $$
The cutter top width depends on the large-end space width and the small-end space width of the workpiece gear. The cutter top width must simultaneously satisfy the condition of being greater than half of the large-end space width and less than the small-end space width.
According to the cutting principle of machining straight bevel gears with the inner concave cutter, the relative position relationship between the inner concave cutter and the workpiece gear during machining can be determined. In the relative position vector diagram, O_m is the machine center, O_i is the intersection point of the workpiece axis and the cradle axis. The machine adjustment parameters are: radial cutter setting S, angular cutter setting q, workpiece installation angle δ_M, bed setting X_b, horizontal setting X_p, vertical setting E_m, cutter tilt angle i, cutter swivel angle j, and tooth interval angle λ.
The tooth interval angle λ is the angle between the tool tip motion trajectory and the tooth space bisector passing through the machine center, calculated as:
$$ \lambda = \tan^{-1}\left(\frac{e}{2R_m}\right) \tag{2.4} $$
where e is the mean space width and R_m is the mean cone distance.
The angle between the cutter head axis and the A coordinate axis of the machine coordinate system is called the cutter tilt angle, calculated as:
$$ i = 90^\circ – \alpha_0 – \delta \tag{2.5} $$
where α₀ is the cutter pressure angle and δ is the concave angle.
On the generating gear pitch plane, the distance from the cutter head center to the generating gear center is called the radial cutter setting, calculated as:
$$ S = \sqrt{(R_m \cos\delta_f \cos\lambda)^2 + (R_m \cos\delta_f \sin\lambda + r_i \cos i)^2} \tag{2.6} $$
where r_i is the cutter head radius. On the generating gear pitch plane, the angle between the line connecting the generating gear center and the cutting calculation point and the line connecting the cutter head center and the generating gear center is called the angular cutter setting, calculated as:
$$ q = \arccos\left(\frac{S^2 + R_m^2 – r_i^2}{2 S R_m}\right) \tag{2.7} $$
The angle between the generating gear pitch plane and the axis of the workpiece straight bevel gear is called the workpiece installation angle. Because the characteristic of straight bevel gear tooth form is tapered teeth, the workpiece installation angle is consistent with the root cone angle, that is, δ_M = δ_f.
The distance from the generating gear pitch plane to the cutter tip plane is called the bed setting, calculated as:
$$ X_b = r_i \sin i – \Delta h \tag{2.8} $$
The angle between the projection of the cutter head axis in the X-Y plane and the direction perpendicular to the radial cutter setting is called the cutter swivel angle, calculated as:
$$ j = q – \lambda \tag{2.9} $$
The ratio of the generating gear rotational speed to the workpiece rotational speed is called the roll ratio, calculated as:
$$ i_{12} = \frac{\cos\lambda}{\sin\delta_f} \tag{2.10} $$
3. Mathematical Modeling of the Straight Bevel Gear Tooth Surface Based on the Inner Concave Cutter
Based on the cutting principle of machining straight bevel gears with the inner concave cutter, as well as the relative position and motion relationships among the cutter head, the generating gear, and the workpiece gear during the cutting process, the mathematical model of the straight bevel gear tooth surface is established. Based on the tooth surface mathematical model and the tooth surface grid division principle, the coordinates of discrete points on the tooth surface and the unit normal vectors are calculated, which serve as the standard for tooth profile error detection.
3.1 Virtual Generating Gear Tooth Surface Equation
For the cutting process of straight bevel gears, the concept of the generating gear is usually used to describe the cutting process. The virtual generating gear is formed by the rotation of the cutter head. The coordinate system of the generating gear is established with the apex of the face cone of the generating gear as the origin. Let τ be the half angle of the generating gear tooth thickness, and w and v be the two parameters along the tooth height direction and tooth width direction respectively. In the generating gear coordinate system S_c, the tooth surface position vector r_c and unit normal vector n_c can be expressed as:
$$ \mathbf{r}_c(w,v) = \begin{bmatrix} v \cos\beta \\ \pm (v \sin\beta + w \sin\alpha_0) \\ w \cos\alpha_0 \end{bmatrix} \tag{3.1} $$
$$ \mathbf{n}_c(w,v) = \frac{\partial \mathbf{r}_c/\partial w \times \partial \mathbf{r}_c/\partial v}{\left| \partial \mathbf{r}_c/\partial w \times \partial \mathbf{r}_c/\partial v \right|} \tag{3.2} $$
where α₀ and β are determined from the basic gear parameters:
$$ \beta = \frac{s_e}{2R_e\cos\delta_f} \tag{3.3} $$
$$ \alpha_0 = \arctan\left(\frac{\tan\alpha}{\cos\delta_f}\right) \tag{3.4} $$
where s_e is the large-end tooth thickness, δ_f is the dedendum angle, R_e is the outer cone distance, and α is the pressure angle.
3.2 Establishment of the Tooth Surface Equation
Due to the existence of the concave angle δ, the inner concave cutter forms a concave conical surface when rotating around its central axis. Taking the intersection point of the cutter tip plane and the cutter head central axis as the origin of the cutter coordinate system, with the cutter head central axis as the z-axis and the line from the origin to a reference point as the y-axis, a spatial rectangular coordinate system S_t is established by the right-hand rule. In this coordinate system, m is an arbitrary point on the cutting surface of the cutter head, δ is the concave angle, θ is the angle of the cutting edge rotating around the cutter head axis, r is the cutter head radius, and s is the distance from the point along the straight cutting edge to the cutter tip. The cutting surface equation of the cutter head can be written as:
$$ \mathbf{r}_t(s,\theta) = \begin{bmatrix} \sin\theta (r – s\cos\delta) \\ \cos\theta (r – s\cos\delta) \\ s \sin\delta \end{bmatrix} \tag{3.5} $$
Taking the partial derivatives of r_t with respect to s and θ respectively yields:
$$ \mathbf{r}_{ts} = \begin{bmatrix} -\cos\theta \cos\delta \\ -\sin\theta \cos\delta \\ \sin\delta \end{bmatrix} \tag{3.6} $$
$$ \mathbf{r}_{t\theta} = \begin{bmatrix} \cos\theta (r – s\cos\delta) \\ -\sin\theta (r – s\cos\delta) \\ 0 \end{bmatrix} \tag{3.7} $$
Based on the calculation formula for the unit normal vector:
$$ \mathbf{n}_t = \frac{\mathbf{r}_{ts} \times \mathbf{r}_{t\theta}}{\left| \mathbf{r}_{ts} \times \mathbf{r}_{t\theta} \right|} \tag{3.8} $$
the unit normal vector and unit tangent vector of the cutting surface are obtained as:
$$ \mathbf{n}_t = \begin{bmatrix} -\sin\theta \sin\delta \\ -\cos\theta \sin\delta \\ \cos\delta \end{bmatrix} \tag{3.9} $$
$$ \mathbf{t}_t = \begin{bmatrix} -\cos\theta \sin\delta \\ \sin\theta \sin\delta \\ \sin\delta \end{bmatrix} \tag{3.10} $$
When machining the left tooth surface of the workpiece, the gear pitch cone vertex is taken as the origin of the machine coordinate system, and the three coordinate axes are established according to the right-hand rule to form the fixed coordinate system S_m. The generating gear coordinate system S_c, i.e., the cradle coordinate system, has its origin O_c coincident with the machine coordinate system origin O_m, with Z_c coincident with Z_m, and the generating gear rotates around Z_c at angular velocity ω₁. The workpiece coordinate system S_g has its origin coincident with the machine coordinate system origin O_m, and the workpiece rotates around Z_g at angular velocity ω₂. The coordinate systems S_t, S_a, S_b are auxiliary coordinate systems used to determine the relative positions of the cutter head and the machine, as well as the workpiece and the machine.
The coordinate transformation sequence for spatial meshing analysis is from the cutter coordinate system S_t to the auxiliary coordinate system S_a, then to the auxiliary coordinate system S_b, then to the cradle coordinate system S_c, then to the machine coordinate system S_m, then to the auxiliary coordinate system S_d, and finally to the workpiece coordinate system S_g. The corresponding transformation matrices are as follows.
The rotation matrix from S_t to S_a is:
$$ \mathbf{M}_{1t} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & \cos i & \sin i & 0 \\ 0 & -\sin i & \cos i & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \tag{3.11} $$
The rotation matrix from S_a to S_b is:
$$ \mathbf{M}_{21} = \begin{bmatrix} \cos\theta_c & \sin\theta_c & 0 & 0 \\ -\sin\theta_c & \cos\theta_c & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \tag{3.12} $$
The translation matrix from S_b to S_c is:
$$ \mathbf{M}_{c2} = \begin{bmatrix} 1 & 0 & 0 & L \\ 0 & 1 & 0 & A \\ 0 & 0 & 1 & D \\ 0 & 0 & 0 & 1 \end{bmatrix} \tag{3.13} $$
The rotation matrix from S_c to S_m is:
$$ \mathbf{M}_{mc} = \begin{bmatrix} \cos\delta_f & -\sin\delta_f & 0 & 0 \\ \sin\delta_f & \cos\delta_f & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \tag{3.14} $$
The rotation matrix from S_m to S_d is:
$$ \mathbf{M}_{3m} = \begin{bmatrix} \sin\delta_f & 0 & \cos\delta_f & 0 \\ 0 & 1 & 0 & 0 \\ -\cos\delta_f & 0 & \sin\delta_f & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \tag{3.15} $$
The rotation matrix from S_d to S_g is:
$$ \mathbf{M}_{g3} = \begin{bmatrix} \cos\varphi_2 & \sin\varphi_2 & 0 & 0 \\ -\sin\varphi_2 & \cos\varphi_2 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \tag{3.16} $$
In the generating gear tooth surface, a reference point P is given with its position vector expressed by equation (3.1) and the unit normal vector determined by equation (3.2). The position vector and unit normal vector of an arbitrary point m on the cutter head are transformed through the coordinate transformations to coincide with the reference point P on the generating gear. This procedure solves for the five unknowns. The relationship between the generating gear rotation angle and the workpiece rotation angle is:
$$ \varphi_1 = i_{12}\varphi_2 \tag{3.17} $$
Through the transformation formulas derived above, the cutter cutting surface equation is transformed to the workpiece gear coordinate system:
$$ \mathbf{r}_g = \mathbf{M}_{g3}\mathbf{M}_{3m}\mathbf{M}_{mc}\mathbf{M}_{c2}\mathbf{M}_{21}\mathbf{M}_{1t}\mathbf{r}_t \tag{3.18} $$
$$ \mathbf{n}_g = \mathbf{L}_{g3}\mathbf{L}_{3m}\mathbf{L}_{mc}\mathbf{L}_{c2}\mathbf{L}_{21}\mathbf{L}_{1t}\mathbf{n}_t \tag{3.19} $$
where the matrix L is the matrix M with the last row and last column removed.
In the machine coordinate system S_m, the position vector of the meshing point can be expressed as:
$$ \mathbf{r}_m = \mathbf{M}_{mc}\mathbf{M}_{c2}\mathbf{M}_{21}\mathbf{M}_{1t}\mathbf{r}_t \tag{3.20} $$
and the unit normal vector of the generating surface can be expressed as:
$$ \mathbf{n}_m = \mathbf{L}_{mc}\mathbf{L}_{c2}\mathbf{L}_{21}\mathbf{L}_{1t}\mathbf{n}_t \tag{3.21} $$
Let the angular velocity of the generating gear during machining be:
$$ \boldsymbol{\omega}_1 = \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} \tag{3.22} $$
Then the angular velocity of the workpiece gear is:
$$ \boldsymbol{\omega}_2 = i_{12} \begin{bmatrix} -\sin\delta_f \\ \cos\delta_f \\ 1 \end{bmatrix} \tag{3.23} $$
In the machine coordinate system, the relative velocity between the generating gear and the workpiece is:
$$ \mathbf{v}_{12} = (\boldsymbol{\omega}_1 – \boldsymbol{\omega}_2) \times \mathbf{r}_m \tag{3.24} $$
According to the gear meshing equation:
$$ \mathbf{v}_{12} \cdot \mathbf{n}_m = 0 \tag{3.25} $$
the parameter s can be solved as a function of θ. Substituting s back into the position vector expression, the contact line on the tooth surface with θ as the parameter can be obtained. By continuously varying θ, different contact lines on the tooth surface can be determined, and the collection of these contact lines forms the complete tooth surface of the gear.
3.3 Tooth Surface Grid Division
Let the distance from a point P on the straight bevel gear tooth surface to the gear axis be R, and the distance from point P along the bevel gear axis to the cone vertex O_m be L. When calculating discrete points on the tooth surface, in the rotary projection plane of the gear tooth, the discrete points on the tooth surface generally take 9 columns in the tooth length direction and 5 rows in the tooth height direction. At both ends of the tooth height direction, each contracts by 5% of the full tooth height, and the contraction amount must not be less than 0.6 mm. At both ends of the tooth width direction, each contracts by 10% of the total tooth width. The relationships for the tooth surface grid division are shown in the following figure.
Based on the geometric relationships of the grid division, the coordinates of the discrete points on the tooth surface can be calculated. The coordinates of the 45 discrete points are expressed as (R_ij, z_ij), where i = 1,…,9 and j = 1,…,5. Combining the tooth surface equation and the grid division rules, a program for calculating the tooth surface discrete point coordinates was written in MATLAB. Using the gear parameters from Table 2 as an example, the coordinates and unit normal vectors of the discrete points on the gear and pinion tooth surfaces were calculated.
| Parameter | Pinion | Gear |
|---|---|---|
| Number of teeth z | 25 | 51 |
| Modulus m (mm) | 2.6737 | 2.6737 |
| Tooth width B (mm) | 4.5 | 4.5 |
| Pitch cone angle δ (°) | 26.11 | 63.88 |
| Outer cone distance R_e (mm) | 75.93 | 75.93 |
| Root cone angle δ_f (°) | 29.24 | 65.60 |
| Cutter radius r (mm) | 57.15 | 57.12 |
| Concave angle δ (°) | 10 | 10 |
Table 2: Gear geometric parameters and cutter parameters
The machine adjustment parameters calculated for this example are listed in Table 3.
| Parameter | Pinion | Gear |
|---|---|---|
| Roll ratio i₁₂ | 2.270896 | 1.112031 |
| Cutter tilt angle i (°) | 60 | 60 |
| Cutter swivel angle j (°) | 20.61 | 19.92 |
| Radial cutter setting S (mm) | 79.24 | 79.16 |
| Angular cutter setting q (°) | 21.60 | 21.45 |
| Vertical setting E_m (mm) | 0 | 0 |
| Horizontal setting X_p (mm) | 0 | 0 |
| Bed setting X_b (mm) | 49.45 | 49.45 |
| Workpiece installation angle δ_M (°) | 24.40 | 60.76 |
Table 3: Machine adjustment parameters
3.4 Three-Dimensional Solid Modeling of the Straight Bevel Gear
Based on the calculated discrete point coordinates of the gear tooth surface, the three-dimensional solid model of the straight bevel gear can be established in SolidWorks software. First, the rotary cross-section sketch of the gear is drawn, including the large-end tip circle diameter, pitch cone angle, root cone angle, face cone angle, cone distance, and crown distance. Then, the calculated discrete points are imported into SolidWorks to generate a series of tooth surface curves. Using the boundary surface command, the curves are connected one by one to form one tooth surface of the gear. The same method is used to create the other tooth surface. After establishing one tooth, the single tooth is arrayed circumferentially according to the required number of teeth, thereby obtaining the complete three-dimensional model of the straight bevel gear.
4. Tooth Contact Analysis of the Straight Bevel Gear Based on the Inner Concave Cutter
In order to judge whether the meshing quality of the straight bevel gear pair meets the requirements, the rolling inspection of the gear pair is generally performed in practical engineering to obtain the contact pattern. With the advancement of technology, gear tooth contact analysis (TCA) is commonly used to simulate the meshing contact process of tooth surfaces using computer technology, obtaining the contact trace, contact area, and transmission error curve of the tooth surface.
During the meshing process of the gear pair, the two cooperating tooth surfaces perform continuous contact motion, and at any instant, the two tooth surfaces have a common normal line, a common point, and a common tangent plane. In the meshing coordinate system, the two tooth surfaces have identical unit normal vectors and equal position vectors at the instant of contact. According to the gear meshing principle, the three-dimensional coordinates and unit normal vectors of the instantaneous contact points can be obtained. By determining all the contact points on the tooth surface, the meshing line of the tooth surface is obtained. The elastic deformation of the tooth surface is set to 0.00635 mm (the thickness of red lead powder), and the contact ellipse at the contact point can be determined. The set of contact ellipses for each contact point constitutes the contact area of the tooth surface.
4.1 Tooth Contact Analysis Method for Straight Bevel Gears
Based on the tooth surface mathematical model of the pinion and gear established in the previous chapter, which are respectively established in the gear coordinate systems S₁ and S₂, the tooth contact analysis requires the tooth surface equations and unit normal vectors of both the pinion and the gear to be represented in the same coordinate system. A meshing coordinate system S_f is established, where S_f coincides with the pinion coordinate system S₁, and only the gear tooth surface equation and unit normal vector in the gear coordinate system S₂ need to be transformed into the meshing coordinate system S_f.
In the meshing coordinate system, the position vector r_{fP} of a point on the pinion tooth surface and r_{fG} on the gear tooth surface can be expressed as:
$$ \mathbf{r}_{fP} = \mathbf{M}_{f1}\mathbf{r}_1 \tag{4.1} $$
$$ \mathbf{r}_{fG} = \mathbf{M}_z\mathbf{M}_{f2}\mathbf{r}_2 \tag{4.2} $$
where M_{f1} and M_{f2} are the rotation matrices of the pinion and gear from their respective initial positions, and M_z is the transformation matrix related to the shaft angle. The unit normal vectors are correspondingly transformed as:
$$ \mathbf{n}_{fP} = \mathbf{L}_{f1}\mathbf{n}_1 \tag{4.3} $$
$$ \mathbf{n}_{fG} = \mathbf{L}_z\mathbf{L}_{f2}\mathbf{n}_2 \tag{4.4} $$
When the large and small gears mesh, the contact points on the tooth surfaces must satisfy:
$$ \mathbf{r}_{fP} = \mathbf{r}_{fG} = \mathbf{r}_f \tag{4.5} $$
$$ \mathbf{n}_{fP} = \mathbf{n}_{fG} = \mathbf{n}_f \tag{4.6} $$
The above equations contain six scalar equations. Since n_{fP} and n_{fG} are unit normal vectors, only five independent equations exist. The unknowns to be solved are the surface parameters of the pinion (u₁, v₁), the surface parameters of the gear (u₂, v₂), and the rotation angles (φ₁, φ₂) of the pinion and gear around their axes during meshing, totaling six unknowns. With only five equations, the system cannot be solved directly. Therefore, φ₁ is treated as the known quantity to solve for the remaining five unknowns, thereby obtaining one meshing contact point on the tooth surfaces of the pinion and gear. By changing the value of φ₁ with a certain step size, new tooth surface contact points can be obtained. The iteration process continues until all contact points are determined, with each point being checked to determine whether it lies on the tooth surface.
Before solving the contact points, two preliminary steps must be completed. First, the point on the pinion tooth surface that meshes with the gear tooth surface calculation point at the theoretical transmission ratio must be determined. Second, the gear tooth surface equations transformed into the meshing coordinate system are rotated around their respective axes by certain angles so that the two gears mesh correctly at the calculation point.
In the first step, a point on the gear tooth surface is generally selected as the calculation reference point. Three initial calculation points are typically designated on the gear tooth surface at the large end, middle, and small end, located at distances of 0.25B, 0.5B, and 0.75B from the large end respectively, positioned at the middle of the tooth surface at that location. The contact trace, contact area, and transmission error curve at these three positions are then calculated.
In the second step, after the positions of points G and P on the gear and pinion tooth surfaces are determined, the tooth surface equations are rotated around their respective axes by certain angles so that they mesh correctly in the meshing coordinate system. The installation adjustments of the gear pair in the V/H direction can then be determined.
In the meshing coordinate system, the unit vectors e₁ and e₂ along the tooth height direction of the pinion and gear tooth surfaces are both located in the common tangent plane at the contact point, but they are not necessarily coincident. The relative direction coefficient between the two surfaces is obtained by calculating the principal curvatures and torsions.
The principal curvature A, the secondary curvature B, and the geodesic torsion C of the gear tooth surface at the contact point can be calculated as:
$$ A_2 = \frac{\sin\delta}{r – s\cos\delta} \tag{4.7} $$
$$ B_2 = 0 \tag{4.8} $$
$$ C_2 = 0 \tag{4.9} $$
where δ is the concave angle. The corresponding curvature components for the pinion are computed in a similar manner.
The angle A between the tooth height direction vectors e₁ and e₂ on the pinion and gear tooth surfaces is:
$$ \Lambda = \arccos(\mathbf{e}_1 \cdot \mathbf{e}_2) \tag{4.10} $$
Using Euler’s formula and the Bertrand formula, the normal curvatures and geodesic torsion of the pinion tooth surface along the gear tooth width and tooth height directions can be calculated. The induced normal curvatures along the tooth width and tooth height directions, and the induced geodesic torsion are:
$$ \Delta A = A_2 – A_1′ \tag{4.11} $$
$$ \Delta B = B_2 – B_1′ \tag{4.12} $$
$$ \Delta C = C_2 – C_1′ \tag{4.13} $$
Let e_n be an arbitrary tangent direction in the common tangent plane with an angle ψ to the gear tooth width direction. The induced normal curvature ΔK along the e_n direction can be expressed by Euler’s formula as:
$$ \Delta K = \frac{\Delta A + \Delta B}{2} + \frac{\Delta A – \Delta B}{2}\cos 2\psi + \Delta C \sin 2\psi \tag{4.14} $$
The maximum and minimum induced normal curvatures of the gear tooth surface are:
$$ \Delta K_{\max} = \frac{\Delta A + \Delta B}{2} + \frac{1}{2}\sqrt{(\Delta A – \Delta B)^2 + 4\Delta C^2} \tag{4.15} $$
$$ \Delta K_{\min} = \frac{\Delta A + \Delta B}{2} – \frac{1}{2}\sqrt{(\Delta A – \Delta B)^2 + 4\Delta C^2} \tag{4.16} $$
The length of the major semi-axis and minor semi-axis of the contact ellipse can be calculated by:
$$ l_a = \sqrt{\frac{\rho}{\Delta K_{\min}}} \tag{4.17} $$
$$ l_b = \sqrt{\frac{\rho}{\Delta K_{\max}}} \tag{4.18} $$
where ρ is the elastic deformation of the tooth surface, taken as 0.00635 mm. The contact ellipses at all contact points can be obtained and constructed on the gear tooth surface to show the complete contact pattern.
4.2 Transmission Error Curve
When the gear and pinion mesh at the calculation reference points G and P, the actual transmission ratio is equal to the theoretical transmission ratio. However, at other points, the transmission ratio may not equal the theoretical value. The transmission error curve is introduced as an indicator to describe the difference between the actual transmission ratio and the theoretical transmission ratio. Taking points G and P as the initial position, the initial rotation angles of the pinion and gear are φ₁₀ and φ₂₀ respectively. After the gear pair rotates around their respective axes by a certain angle, the rotation angles of the pinion and gear are φ₁ and φ₂. If the transmission ratio is constant, then:
$$ \varphi_2 – \varphi_{20} = \frac{z_1}{z_2}(\varphi_1 – \varphi_{10}) \tag{4.19} $$
When the transmission ratio is not constant, the transmission error function is defined as:
$$ \Delta\varphi_2 = (\varphi_2 – \varphi_{20}) – \frac{z_1}{z_2}(\varphi_1 – \varphi_{10}) \tag{4.20} $$
where z₁ and z₂ are the numbers of teeth of the pinion and gear respectively. The transmission error curve is plotted with φ₁ as the abscissa and Δφ₂ as the ordinate.
4.3 TCA Analysis Example
Based on the MATLAB software platform and the tooth surface equations of the gear and pinion established in the previous chapter, a tooth contact analysis program for the straight bevel gear pair was developed according to the principles and methods described above. Using the gear parameters, cutter parameters, and machine adjustment parameters listed in Table 2 and Table 3, the contact areas and transmission error curves at the large end, middle, and small end of the gear pair were obtained.
The analysis results show that when the straight bevel gear pair contacts at different installation positions, the contact area size along the tooth width direction accounts for about 1/2 of the tooth surface width, consistent with the preset contact area length. The transmission error curve is shaped like an opening-downward parabola. Except for the initial calculation point, the transmission error values corresponding to all other contact points are less than zero, indicating that at other meshing positions, the actual rotation angle of the gear is smaller than the theoretical value. This result is due to the tooth length curvature modification (crowned tooth) produced by machining the straight bevel gear with the inner concave cutter.
5. Cutting Experiments of the Straight Bevel Gear Based on the Inner Concave Cutter
In order to verify the correctness of the geometric parameters, cutter parameters, machine adjustment parameters, tooth surface discrete point coordinate calculations, and tooth contact analysis method for machining straight bevel gears with an inner concave cutter, milling experiments were conducted on the H350C spiral bevel gear milling machine developed by Changsha Jinyi Kaishuai Machinery Co., Ltd. The tooth profile error and tooth pitch error of the experimental workpiece were measured, and the contact area was inspected.
5.1 Main Parameters of the Experimental Workpiece and Cutter
Using the software developed for calculating the basic geometric parameters, cutter parameters, and machine adjustment parameters of straight bevel gears machined with the inner concave cutter, the experimental gear parameters were calculated as listed in Table 4 and Table 5.
| Parameter | Pinion | Gear |
|---|---|---|
| Number of teeth z | 35 | 35 |
| Modulus m (mm) | 2.5 | 2.5 |
| Tooth width B (mm) | 15 | 15 |
| Pressure angle α (°) | 20 | 20 |
| Pitch cone angle δ (°) | 45 | 45 |
| Root cone angle δ_f (°) | 42.22 | 42.22 |
| Face cone angle δ_a (°) | 47.78 | 47.78 |
| Addendum h_a (mm) | 2.5 | 2.5 |
| Dedendum h_f (mm) | 3.0 | 3.0 |
| Full tooth height h (mm) | 5.5 | 5.5 |
| Cutter radius r (mm) | 72.78 | 72.78 |
| Concave angle δ (°) | 2 | 2 |
Table 4: Basic parameters and cutter parameters of the experiment
| Parameter | Pinion | Gear |
|---|---|---|
| Roll ratio i₁₂ | 1.412554 | 1.412554 |
| Workpiece installation angle δ_M (°) | 42.22 | 42.22 |
| Radial cutter setting S (mm) | 61.12 | 61.12 |
| Angular cutter setting q (°) | 27.18 | 27.18 |
| Cutter tilt angle i (°) | 68 | 68 |
| Cutter swivel angle j (°) | 26.37 | 26.37 |
| Vertical setting E_m (mm) | 0 | 0 |
| Horizontal setting X_p (mm) | 0 | 0 |
| Bed setting X_b (mm) | 67.12 | 67.12 |
Table 5: Machine adjustment parameters of the experiment
The discrete point coordinates of the left and right tooth surfaces of the straight bevel gear were calculated using the tooth surface discrete point coordinate calculation program. The theoretical contact area and transmission error curve were obtained by substituting the basic geometric parameters, cutter parameters, and machine adjustment parameters into the tooth contact analysis software.
5.2 Milling Experiment of the Straight Bevel Gear
The H350C milling machine is a six-axis five-linkage CNC spiral bevel gear processing machine tool. The spindle and workpiece spindle adopt direct drive technology, and the machine tool is controlled by the SIEMENS 840DSL CNC system. This machine can process Gleason tooth system spiral bevel gears by face milling, and also Oerlikon tooth system cycloidal bevel gears by face hobbing. It can use new structure cutter heads for dry cutting, and can also use traditional cutter heads for wet cutting with the cooling oil supply device. In addition, the machine can also process straight bevel gears using an inner concave cutter.
According to the basic geometric parameters and cutter parameters listed in Table 4, an inner concave cutter was custom-made and the gear blanks were manufactured. The cutter and gear blank were installed on the machine tool respectively, and the CNC machining program was imported into the machine for cutting. Figure 1 shows the machining process on the H350C machine.
After the cutting process was completed, a pair of straight bevel gears machined with the inner concave cutter was obtained. The tooth surfaces of the machined straight bevel gears showed good surface quality without obvious chatter marks or burns.
5.3 Tooth Profile Error Measurement
In order to verify the accuracy of the machining principle, tooth surface mathematical model, and tooth surface discrete point coordinate calculation method for machining straight bevel gears with the inner concave cutter, tooth profile error detection was performed on the machined straight bevel gears. The measurement was carried out on a gear measuring center. The left and right tooth surface discrete point coordinate files were imported into the gear measuring center, and the tooth profile error was measured according to the discrete point coordinates.
The tooth profile error chart of the measured straight bevel gear and the pitch error chart are shown in the measurement results. The maximum error between the actually machined straight bevel gear tooth surface and the theoretical tooth surface was 11.9 μm, which meets the actual engineering requirements and the error value is within the allowable range. The machined straight bevel gear reached the national standard grade 6 accuracy. The main causes of tooth profile errors include the positioning error of the CNC axes, fixture error, cutter error, and gear blank error. The experimental results verify the correctness of the tooth surface equation and tooth surface discrete point coordinate calculation software for machining straight bevel gears based on the inner concave cutter.
5.4 Tooth Surface Contact Area Inspection
Inspecting the tooth surface contact area is aimed at comparing the actual contact area with the theoretical contact area, so as to judge whether the position, shape, and size of the actual contact area match the theoretically analyzed contact area. The straight bevel gear pair was installed on a rolling inspection machine. In order to ensure correct gear meshing, the assembly relationship was adjusted according to the calculated V/H values. Red lead powder was evenly applied to the gear teeth for subsequent observation.
After rolling inspection, the contact areas at the large end, middle, and small end of the gear pair were obtained and compared with the theoretical contact areas calculated by the TCA program. The actual contact areas at the large end, middle, and small end positions were basically consistent with the theoretical contact areas, which verifies the correctness of the gear basic parameters, cutter parameters, and machine adjustment parameters. The experimental results confirm the feasibility and correctness of the basic geometric parameter design, the inner concave cutter processing principle, the cutter parameter calculation, the tooth surface mathematical model, the tooth surface discrete point coordinate calculation, and the tooth contact analysis method for machining straight bevel gears with the inner concave cutter.
6. Conclusions and Outlook
The study of straight bevel gear machining technology based on the inner concave cutter is of great significance for improving the manufacturing level of straight bevel gears in China. The following conclusions can be drawn from this research.
First, the geometric parameter calculation methods for the straight bevel gear and the inner concave cutter were studied, and the design calculation methods for the straight bevel gear and inner concave cutter parameters were established. The calculation formulas for the straight bevel gear geometric parameters were provided to ensure that the machined straight bevel gear parameters meet the design requirements.
Second, based on the cutting principle of the inner concave cutter, the machining method for straight bevel gears using an inner concave cutter was studied. By changing the concave angle, the curvature of the straight bevel gear in the tooth length direction was modified to produce crowned teeth.
Third, according to the relative position relationship among the inner concave cutter, the generating gear, and the gear blank in the machine tool during the cutting process, the tooth surface equation of the straight bevel gear machined with the inner concave cutter was established. Through the discretization of the tooth surface and the calculation of the discrete point coordinates, a theoretical three-dimensional model of the straight bevel gear was established in SolidWorks software.
Fourth, based on the straight bevel gear tooth surface equation and the tooth contact analysis principle, the meshing coordinate system of the straight bevel gear pair was established. The tooth surface contact areas at the large end, middle, and small end and the transmission error curves were obtained.
Fifth, based on the MATLAB software development platform, a complete set of software was developed, including the calculation software for straight bevel gear geometric parameters, the calculation software for inner concave cutter parameters, the calculation software for machine adjustment parameters of the gear cutting process, the calculation software for tooth surface discrete point coordinates and unit normal vectors, and the tooth surface contact analysis software.
Sixth, based on the above research results, actual cutting experiments were carried out on the domestic H350C CNC spiral bevel gear milling machine. The tooth profile error detection of the machined straight bevel gear pair was performed on the gear measuring center, and the gear pair was installed on the rolling inspection machine for contact area inspection. The results of the tooth profile error detection met the actual engineering requirements, and the actual contact areas at the large end, middle, and small end were consistent with the theoretically analyzed contact areas from the TCA analysis. This validates the correctness and feasibility of the straight bevel gear geometric parameter design, the inner concave cutter parameter design, the machine adjustment parameters of the gear cutting process, the tooth surface equation, the tooth surface discrete point coordinates, and the tooth contact analysis method.
Due to time limitations and the lack of relevant knowledge of the author, subsequent work has not been completed. In order to further study the method of machining straight bevel gears with the inner concave cutter, the following aspects still need to be investigated. First, this paper studied only the milling process of the inner concave cutter. In the future, the grinding process using CBN grinding wheels can be studied. Second, during the cutting process of the straight bevel gear, there exist cutter errors, machine errors, gear blank fixture errors, etc., and further tooth surface error correction work is required. Third, the calculation and simulation of the tooth surface contact strength, tooth root bending strength, and load-carrying capacity of the straight bevel gear pair machined with the inner concave cutter can be performed.
In summary, the research on the design and machining technology of straight bevel gears based on the inner concave cutter provides a solid foundation for achieving high-precision and high-efficiency manufacturing of straight bevel gears on domestic CNC machine tools in China.