Bevel gears are indispensable components for transmitting motion between intersecting axes. Because straight bevel gears are relatively easier to design and manufacture compared with spiral bevel gears, they are widely used in automotive and machine tool industries. However, the domestic manufacturing of straight bevel gears still relies on traditional mechanical machine tools with intermittent indexing methods, resulting in low production efficiency. The research on new machining technologies for straight bevel gears has been quite limited, which greatly constrains the development of straight bevel gears. In this thesis, based on the principle of cycloid rotational indexing machining, I propose a novel rotational indexing machining method for straight bevel gears that achieves continuous indexing.
The rotational indexing machining technology described in this thesis represents a departure from conventional generating gear concepts. The machining and calculation processes become more intuitive. This technology enables continuous indexing machining of straight bevel gears, significantly improving production efficiency. Meanwhile, this research lays the foundation for designing new types of straight bevel gear machine tools. The implementation and application of this rotational indexing machining technology will improve the domestic manufacturing status of straight bevel gears and promote the advancement of straight bevel gear manufacturing technology.
1. Introduction and Research Background
The research background of this thesis stems from the observation that while spiral bevel gear machining technology has developed rapidly with full CNC capability, the machining of straight bevel gears has remained at a relatively backward level. The commonly used machining methods for straight bevel gears include gear planing, gear milling, and circular broaching, which all rely on intermittent indexing. The corresponding machine tools, such as the Y236 gear planer and Y2726 double-head straight bevel gear milling machine, use mechanical transmission chains with low transmission accuracy and difficult adjustment procedures. These limitations have significantly restricted the application and development of straight bevel gears.
In recent years, some foreign companies have recognized this problem and developed new machining technologies for straight bevel gears. The Gleason Company developed a new machining process that can use Coniflex tools on Phoenix series Free-form machines, achieving a 100% to 120% increase in machining efficiency compared with mechanical machine tools. The Samputensili Company of Italy has also realized continuous machining of straight bevel gears using cutter heads, greatly improving both efficiency and accuracy. However, these monopolistic enterprises never disclose their technical details, and domestic research in this area remains relatively scarce.
Since 2002, our research group has been developing cycloid rotational indexing machining technology with independent intellectual property rights. We have successfully developed rotational indexing chamfering machine tools for machining synchronizer gear hub slider grooves and back tapers. The literature has demonstrated that by using a certain segment of cycloid curve to approximate circular arcs, the machining of synchronizer gear hub slider grooves can be achieved with high efficiency and flexibility. Further research extended the cycloid rotational indexing machining technology to end-face straight groove parts, converting intermittent processing into continuous processing. These previous achievements provide a solid foundation for extending the cycloid rotational indexing principle to the machining of straight bevel gears.
Drawing upon the successful foreign research experience on straight bevel gear machining, learning from mature spiral bevel gear machining technologies, and combining our research group’s expertise in cycloid rotational indexing machining, I propose a novel machining technology for straight bevel gears. This technology adopts a cutter head to realize continuous indexing machining of straight bevel gears, which will greatly improve machining efficiency and quality.
2. Fundamental Principles of Straight Bevel Gear Machining
2.1 Basic Knowledge of Straight Bevel Gears
Straight bevel gears transmit motion between intersecting axes. If cylindrical gear transmission can be considered as pure rolling of a pair of cylinders, then bevel gear transmission can be regarded as pure rolling of a pair of cones. The instantaneous axis of relative motion is the line of intersection of the two pitch cones. The correct meshing condition for a pair of bevel gears requires equal module and pressure angle at the large end, equal cone distances, and coincident cone apices.
The basic geometric parameters of straight bevel gears include: large-end module m, number of teeth z, pressure angle α, pitch diameter d, addendum diameter da, dedendum diameter df, cone distance R, face width B, pitch cone angle δ, face cone angle δa, and root cone angle δf.
The theoretical tooth profile of a straight bevel gear is formed as follows. A circular plane with radius equal to the cone distance of a base cone rolls purely on the base cone. Any radius of this circular plane develops a curved surface called the involute cone surface. The intersection of this involute cone surface with a sphere centered at the cone apex is called the spherical involute. The tooth flank of a straight bevel gear is composed of spherical involutes on spheres of different radii centered at the cone apex.
Since spherical curves cannot be developed into plane curves, traditional manufacturing often replaces spherical involutes with projections on the back cone. However, this inevitably introduces manufacturing errors. To maximize machining accuracy, I directly use the theoretical tooth profile of the straight bevel gear as the evaluation standard in this thesis.
2.2 Tooth Flank Equation of Straight Bevel Gear
To obtain the involute cone surface equation, I establish a fixed coordinate system O-xyz with the cone apex O as the origin, the z-axis coinciding with the base cone axis, and a moving coordinate system O-x₁y₁z₁ with the z₁-axis along the instantaneous rotation axis ON. Through coordinate transformation, the involute cone surface equation of a straight bevel gear can be expressed as:
$$ \begin{cases} x = r(\cos\psi \sin\gamma \cos\phi + \sin\psi \sin\phi) \\ y = r(\cos\psi \sin\gamma \sin\phi – \sin\psi \cos\phi) \\ z = r\cos\psi \cos\gamma \end{cases} $$
where r is the parametric variable, γ is the base cone angle satisfying sinγ = cosα·sinδ, ψ and φ are angular parameters. When r = R, the spherical involute at the large end is obtained.
2.3 Concept of Rotational Indexing Machining
Rotational indexing machining, abbreviated as rotational indexing, refers to a machining method in which the workpiece rotates while the tool rotates synchronously, realizing cutting and indexing simultaneously during rotation. By changing the rotational speed ratio between the workpiece and the tool, different workpieces can be machined. When the cutting point trajectory is made to follow a cycloid trajectory, the method becomes cycloid rotational indexing machining.
2.4 Principle of Epicycloid Bevel Gear Machining
The machining process of epicycloid bevel gears can be viewed as a meshing process between an imaginary plane generating gear with a 90° cone angle and the workpiece. The rotation of the cutter head around its own axis combined with the revolution of the cutter head center around the generating gear axis realizes the motion of the generating gear. In actual machining, the imaginary generating gear does not exist. The entire process is completed by the composite motion of three movements: cutter head rotation, cutter head axis revolution around the generating gear axis, and workpiece rotation.
2.5 Principle of Rotational Indexing Machining for Straight Bevel Gears
The formation of cycloid trajectories essentially involves a generating circle of radius r rolling purely on a base circle of radius R. A point fixed to the generating circle at distance e from its center traces a cycloid trajectory. By adjusting the parameters R, r, and e, the cycloid can approximate a straight line to a certain degree. If the axis of the generating circle is kept at an angle to the axis of the base circle while maintaining the same motion relationship, the relative trajectory becomes a spatial curve. A certain segment of this spatial curve can be found near the cone generatrix of the workpiece, and by further adjusting parameters, these curve segments can approximate straight lines.
The key to machining straight bevel gears with this rotational indexing method lies in how to make the spatial curve segments approximate straight line segments. Utilizing the rotational indexing principle and the epicycloid bevel gear machining principle, I first set the cutter head plane tangent to the cone surface, with the workpiece axis forming an angle θ with the cutter head plane, defined as the workpiece swing angle. A point M on the cutting edge starts cutting from the small end of the conical workpiece blank. Since the surface being machined is conical rather than planar, as the cutter head rotates, point M will inevitably move farther away from the cone surface. This contradiction can be resolved by giving the cutter head an inclination angle β for position compensation.
The main motions required for completing the rotational indexing machining of straight bevel gears include: rotation of the cutter head around its own axis at angular velocity ω₁; rotation of the workpiece around its own axis at angular velocity ω₂, maintaining a fixed speed ratio relationship with the cutter head, η = ω₁/ω₂; and reciprocating oscillation of the cutter head around the reference axis l, causing the inclination angle β to vary linearly. A cutter head equipped with multiple groups of cutter teeth is adopted, where each group consists of three blades for rough machining and machining of both tooth flank surfaces. The workpiece and cutter head rotate according to the gear ratio to achieve continuous indexing.
3. Mathematical Model of Rotational Indexing Machining
3.1 Cutting Point Trajectory Equation
To establish the mathematical model of rotational indexing machining for straight bevel gears, I define the coordinate systems as follows. S₁ = {O₁; X₁, Y₁, Z₁} is the initial coordinate system of the cutter head. S₂ = {O₂; X₂, Y₂, Z₂} is the workpiece coordinate system. S₃ = {O₃; X₃, Y₃, Z₃} is the coordinate system after the cutter head rotates about the reference axis l by the inclination angle β.
The cutting point trajectory refers to the path traced by point M on the cutting edge in the workpiece coordinate system. According to the coordinate transformation principle, the parametric equation of point M’s trajectory is:
$$ \begin{bmatrix} x_2 \\ y_2 \\ z_2 \\ 1 \end{bmatrix} = \mathbf{M}_{21} \cdot \mathbf{M}_{13} \cdot \begin{bmatrix} x_c \\ y_c \\ z_c \\ 1 \end{bmatrix} $$
where (xc, yc, zc) are coordinates of point M in the cutter head moving coordinate system, with xc = e·sinφ, yc = e·cosφ, zc = 0. The transformation matrices are:
$$ \mathbf{M}_{21} = \begin{bmatrix} \sin\theta & 0 & \cos\theta & -H\sin\theta \\ 0 & 1 & 0 & -e\cos\varphi \\ \cos\theta & 0 & -\sin\theta & -H\cos\theta + R_1 \\ 0 & 0 & 0 & 1 \end{bmatrix} $$
$$ \mathbf{M}_{13} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & \cos\beta & -\sin\beta & e\cos\varphi + L \\ 0 & \sin\beta & \cos\beta & -S \\ 0 & 0 & 0 & 1 \end{bmatrix} $$
After simplification, the cutting point trajectory equation is obtained. The trajectory curve is a spatial curve, and the segment AB is used to approximate the cone generatrix.
3.2 Parameter Analysis
From the cutting point trajectory model, the parameters required for completing the rotational indexing machining include: cutter head rotation angle α₁, workpiece rotation angle α₂, cutter head radius e, cutter position angle φ, cutter head inclination angle β, workpiece swing angle θ, workpiece cone distance R, and workpiece face width B.
The parametric relationships are as follows. The workpiece rotation angle relates to the cutter head rotation angle by α₂ = η·α₁. The cutter head inclination angle varies linearly: β = βmax·α₁/αmax. The workpiece swing angle θ ranges from the root cone angle to the face cone angle: δf ≤ θ ≤ δa. When approximating the pitch cone generatrix, θ = δ.
The rotational speed ratio η is determined by the number of workpiece teeth z, the number of cutter head blade groups z₁, and the number of teeth crossed per index kz:
$$ \eta = \frac{\omega_2}{\omega_1} = \frac{z_1}{z \cdot k_z} $$
The cutter head radius e must satisfy: π·z₁·B/2 ≤ e ≤ 4B. The maximum cutter head rotation angle is given by αmax = kα·B/e, where kα is the maximum rotation angle coefficient, with 1 ≤ kα ≤ (π/2 – φ)·e/B.
Through geometric analysis, the cutter position angle φ can be expressed as:
$$ \varphi = \arccos\left(\frac{(2e\sin\frac{\alpha_{max}}{2})^2 + B^2 – (2R\eta\sin\alpha_{max}\sin\theta)^2}{4e\sin\frac{\alpha_{max}}{2} \cdot B}\right) – \frac{\alpha_{max}}{2} $$
Similarly, the maximum cutter head inclination angle is:
$$ \beta_{max} = \arcsin\left(\frac{R\sin\theta\cos\theta[1-\cos(\eta\alpha_{max})]}{2e\sin\frac{\alpha_{max}}{2}\sin(\frac{\alpha_{max}}{2}+\varphi)}\right) $$
The horizontal relative position between the cutter head center and the workpiece cone apex is O₂A = e·cosφ, and the vertical relative position is O₂C = R – B – e·sinφ.
3.3 Straightness Error Evaluation
I select the straightness error to evaluate how well the cutting point trajectory curve segment approximates the cone generatrix. Taking the cone generatrix as the axis, a cylinder with radius μ that exactly encloses the trajectory curve segment gives the straightness error. The smaller the μ value, the better the approximation.
For a straight bevel gear, this straightness error corresponds to the tooth trace error. The tooth trace tolerance values for bevel gears with different precision grades are shown in Table 1.
| Face width B/mm | Grade 5 | Grade 6 | Grade 7 | Grade 8 | Grade 9 | Grade 10 |
|---|---|---|---|---|---|---|
| <5 | 4 | 5 | 6 | 8 | 10 | 12 |
| 5~10 | 6 | 8 | 10 | 12 | 16 | 20 |
| >10 | 10 | 12 | 16 | 20 | 25 | 32 |
For calculating the straightness error, I use the distance from points on the trajectory curve segment to the cone generatrix. The cone generatrix equation in the workpiece coordinate system is:
$$ \begin{cases} x – z\tan\theta = 0 \\ (R-B)\cos\theta \leq x \leq R\cos\theta \\ y = 0 \end{cases} $$
The distance from any point A(x, y, z) to the cone generatrix is:
$$ d = \frac{1}{\sqrt{1+k^2}} \sqrt{(x-kz)^2 + y^2} $$
where k = tanθ. The maximum value of d over the trajectory curve segment is the straightness error μ.
3.4 Parameter Optimization
From the parameter analysis, the distance d can ultimately be expressed as a function of three unknown parameters: the rotational speed ratio η, the cutter head radius e, and the maximum rotation angle coefficient kα. The optimization objective is:
$$ \min \max_{\alpha_1} d(\alpha_1, \eta, k_\alpha, e) $$
subject to the constraints:
$$ \begin{cases} 0 \leq \alpha_1 \leq \alpha_{max} \\ \eta = z_1/(z \cdot k_z), \quad k_z = 1, 2, 3, 4 \\ \pi z_1 B/2 \leq e \leq 4B \\ 1 \leq k_\alpha \leq (\pi/2 – \varphi)e/B \end{cases} $$
This is a minimax optimization problem. I use the fminimax function in MATLAB to perform the optimization calculation.
Before performing the optimization, I analyzed the influence of each parameter on the straightness error μ. The results show that:
(1) As the rotational speed ratio η increases, the straightness error μ first decreases and then increases, with an optimal value existing.
(2) As the cutter head radius e increases, the straightness error μ first decreases rapidly and then becomes relatively stable. When e reaches a certain value, μ can essentially satisfy the accuracy requirement.
(3) In the initial range, as kα increases, the straightness error μ changes little. If kα continues to increase, μ increases rapidly. Therefore, kα should be limited to approximately 1 to 1.15.
Comparing the influences of η and e on μ, when e varies from 40 to 70 (a range ratio of 0.27), the minimum μ value changes by a ratio of 10.87. When η varies from 0.56 to 0.72 (a range ratio of 0.22), the minimum μ value changes by a ratio of 32.09. This indicates that η has a greater influence on μ than e. The parameter kα has the least influence.
4. Tooth Flank Formation and Cutting Edge Modification
4.1 Tooth Flank Formation Process
The surface formation methods include trajectory method, forming method, tangent method, and generating method. The simplest is the forming method, where the cutting edge shape is a cutting line that perfectly matches the generatrix shape. During machining, the tool does not need generating motion; it only needs to follow the directrix trajectory.
In the rotational indexing machining of straight bevel gears, I design the cutting edge profile as the spherical involute at the small end of the theoretical tooth flank. Using this spherical involute as the generatrix and the cutting point trajectory as the directrix, the tooth flank formation is realized. The workpiece is installed and positioned according to the root cone angle (θ = δf), and the trajectory swept by the spherical involute cutting edge forms the machined tooth flank.
4.2 Spherical Involute Cutting Edge
From the tooth flank equation, when r = R – B, the spherical involute at the small end of the straight bevel gear is obtained. For the tooth slot left flank, the equation is:
$$ \begin{cases} x = (R-B)(\cos\psi \sin\gamma \cos\phi + \sin\psi \sin\phi) \\ y = -(R-B)(\cos\psi \sin\gamma \sin\phi – \sin\psi \cos\phi) \\ z = (R-B)\cos\psi \cos\gamma \end{cases} $$
This equation is in the workpiece coordinate system and needs to be transformed to the cutter head coordinate system. The coordinate transformation yields the spherical involute cutting edge equation in the cutter head coordinate system.
4.3 Tooth Flank Error Evaluation
The tooth flank error refers to the maximum distance between the machined tooth flank and the theoretical tooth flank. I calculate the distance from points on the machined tooth flank to the theoretical tooth flank, and the maximum value is the tooth flank error Δ.
The theoretical tooth flank equation is given by the involute cone surface equation. Given a point P(x₀, y₀, z₀) on the machined tooth flank, I need to find the minimum distance to the theoretical tooth flank. The minimum distance conditions lead to a system of equations that can be solved for the parameters r and φ. Substituting the solution back gives the distance from the point to the theoretical tooth flank.
4.4 Cutting Edge Modification Using Backtracking Method
From the tooth flank error evaluation, I found that the machined tooth flank with the spherical involute cutting edge cannot achieve the required accuracy. The error regions with large deviations are formed by the trajectories of corresponding cutting edge points. Therefore, I modify these cutting edge points to reduce the tooth flank error.
The modification procedure is as follows. First, I discretize the spherical involute cutting edge into many cutting edge points. I observe the trajectory curve of each point and calculate its error to the theoretical tooth flank. Points with errors within the allowable range are designated as retained points, while the remaining points are designated as points to be modified. The retained points remain unchanged. For each point to be modified, I find the point B on its trajectory curve with the maximum error to the theoretical tooth flank. Let D be the foot of the perpendicular from B to the theoretical flank surface. I take the midpoint B₁ of BD and back-calculate a new cutting edge point A₁ to replace the original point A. This process is repeated iteratively until the error of the new cutting edge point trajectory reaches the allowable range.
4.5 Fitting of the New Cutting Edge
The new cutting edge points obtained from the modification procedure are discrete points that cannot be directly used in machining. I use the least squares method to fit these points into a continuous cutting edge curve. For three-dimensional data points (xc, yc, zc), I use zc as the intermediate variable and separately fit xc and yc as polynomial functions of zc:
$$ \begin{cases} x_c = a_0 + a_1 z_c + a_2 z_c^2 + \cdots + a_n z_c^n \\ y_c = b_0 + b_1 z_c + b_2 z_c^2 + \cdots + b_n z_c^n \end{cases} $$
The coefficients are determined by solving the normal equations of the least squares problem.
4.6 Machining Parameters and Machine Tool Adjustment Parameters
From the rotational indexing machining principle and mathematical model, the parameters required for completing the machining include machine tool adjustment parameters and machining parameters.
The machine tool adjustment parameters include: workpiece swing angle θ, cutter position angle φ, and the horizontal position O₂A and vertical position O₂C of the cutter head center relative to the workpiece cone apex.
The machining parameters include: rotational speed ratio η, cutter head radius e, maximum cutter head inclination angle βmax, number of cutter head blade groups z₁, and number of teeth crossed per index kz.
Table 2 Summary of machining parameters for straight bevel gear rotational indexing
| Parameter Category | Parameter Name | Symbol | Determination Method |
|---|---|---|---|
| Machining | Rotational speed ratio | η | Optimization with pitch cone approximation |
| Cutter head radius | e | Optimization with pitch cone approximation | |
| Adjustment | Workpiece swing angle | θ | θ = δf, given by part drawing |
| Cutter position angle | φ | Computed from geometric relationships | |
| Max inclination angle | βmax | Computed from geometric relationships | |
| Horizontal/vertical positions | O₂A, O₂C | O₂A = e·cosφ, O₂C = R – B – e·sinφ |
4.7 Implementation on Free-form Type Machine Tools
Theoretically, any motion required for tooth flank machining can be realized on Free-form type spiral bevel gear machine tools. The Free-form type six-axis CNC machine tool includes three linear motions (X, Y, Z) and three rotational motions (A, B, C). The tooth flank is formed by the coordinated motion of the six axes.
The rotational indexing machining method proposed in this thesis requires three rotational motions: workpiece rotation, cutter head rotation, and reciprocating oscillation of the cutter head around the reference axis l. Since existing CNC machine tools do not have cutter head oscillation, I designed a motion transformation. Since the oscillation axis l is parallel to the x-axis, this reciprocating oscillation can be transformed into the combined motion of linear motions along the y-axis and z-axis.
Analyzing the motion transformation, the y-axis and z-axis motions during machining are:
$$ \begin{cases} y = e[\cos\varphi – \cos(\varphi + \omega_1 t)] \cdot [1 – \cos(\beta_\omega t)] \\ z = e[\cos\varphi – \cos(\varphi + \omega_1 t)] \cdot \sin(\beta_\omega t) \end{cases} $$
where t is the machining time. During the return stroke, both y and z axes move in the negative direction to return to their initial positions. Through this motion transformation, the rotational indexing machining method changes from three rotating axis linkages to two rotating axis and two linear axis linkages, enabling the use of Free-form type machine tools for machining straight bevel gears.
5. Calculation Example
5.1 Workpiece Parameters
To verify the correctness and feasibility of the proposed rotational indexing machining method for straight bevel gears, I present a specific calculation example. The workpiece is a straight bevel gear with the following geometric parameters determined from the drawing:
Table 3 Geometric parameters of the straight bevel gear workpiece
| Item | Parameter | Symbol | Unit | Value |
|---|---|---|---|---|
| 1 | Number of teeth | z | – | 25 |
| 2 | Large-end module | m | mm | 3 |
| 3 | Pressure angle | α | ° | 20 |
| 4 | Pitch cone angle | δ | ° | 45 |
| 5 | Face width | B | mm | 18 |
| 6 | Addendum coefficient | ha* | – | 1 |
| 7 | Clearance coefficient | c* | – | 0.2 |
| 8 | Pitch diameter | d | mm | 75 |
| 9 | Addendum | ha | mm | 3 |
| 10 | Dedendum | hf | mm | 3.6 |
| 11 | Cone distance | R | mm | 53 |
| 12 | Root cone angle | δf | ° | 40°25′ |
| 13 | Face cone angle | δa | ° | 48°15′ |
5.2 Cutting Point Trajectory and Parameter Optimization
Taking the pitch cone generatrix as the approximation target with θ = δ = 45°, I performed the parameter optimization using MATLAB. The optimal machining parameters and resulting straightness error are:
Table 4 Optimal parameters for pitch cone generatrix approximation
| kα | η | e / mm | φ / ° | βmax / ° | μ / mm |
|---|---|---|---|---|---|
| 1.1053 | 0.72 | 54 | 19.39 | 5.39 | 0.0068 |
From Table 1, the tooth trace tolerance for this straight bevel gear is δxB = 20 μm. The calculated straightness error μ = 6.8 μm is smaller than the tooth trace tolerance, indicating that the straightness error at the pitch cone generatrix can satisfy the accuracy requirement.
5.3 Tooth Flank Formation and Cutting Edge Modification
According to the rotational indexing machining principle, the workpiece is positioned at the root cone angle. Using the spherical involute as the initial cutting edge profile for machining, the cutting edge sweeps along the cutting point trajectory to form the machined tooth flank. The error trend of the machined tooth flank is: from the root to the tip, the error first decreases and then increases. The error at the root region deviates inside the theoretical tooth flank, while the error at the tip region deviates outside the theoretical tooth flank.
The tooth flank error obtained with the spherical involute cutting edge is 0.0564 mm, which is larger than the typical requirement of 0.02 mm. Therefore, cutting edge modification is necessary. I discretized the involute cutting edge into 13 points, numbered from root to tip as points 1 to 13. Calculating the trajectory curve error for each point, only the 4 points near the root (points 1 through 4) have errors exceeding 0.02 mm and require modification.
For fitting accuracy, I further refined the segment from point 1 to point 4, obtaining 8 points to be modified. Using the backtracking method to modify these 8 points, the errors before and after modification are:
Table 5 Errors of the points to be modified before and after modification
| Point | Error before / mm | Error after / mm |
|---|---|---|
| 1 | 0.0514 | 0.0182 |
| 2 | 0.0476 | 0.0164 |
| 3 | 0.0423 | 0.0155 |
| 4 | 0.0367 | 0.0132 |
| 5 | 0.0304 | 0.0115 |
| 6 | 0.0265 | 0.0097 |
| 7 | 0.0248 | 0.0086 |
| 8 | 0.0226 | 0.0080 |
All modified points satisfy the error requirement. I obtained a total of 17 cutting edge points (8 new points plus 9 retained points). Using the least squares method with fourth-degree polynomial fitting, the new cutting edge curve equation is:
$$ \begin{cases} x_c = 16.5158 – 0.0532z_c – 0.0246z_c^2 – 0.0006z_c^3 – 4.4 \times 10^{-5}z_c^4 \\ y_c = 51.4338 + 0.15z_c – 0.0642z_c^2 – 0.0028z_c^3 – 1.38 \times 10^{-4}z_c^4 \\ z_c \in [-7.8513, 0.0013] \end{cases} $$
Using this modified cutting edge, the new machined tooth flank has a tooth flank error of 0.0188 mm, which is smaller than 0.02 mm and satisfies the accuracy requirement as shown in Figure 4.
5.4 Machining Parameters and Machine Tool Adjustment Parameters
Based on the optimization and geometric analysis, the final machining parameters and machine tool adjustment parameters for the example workpiece are:
Table 6 Final machining parameters and machine tool adjustment parameters
| Parameter | Symbol | Value | Unit |
|---|---|---|---|
| Number of blade groups | z₁ | 9 | – |
| Teeth crossed per index | kz | 2 | – |
| Rotational speed ratio | η | 0.72 | – |
| Cutter head radius | e | 54 | mm |
| Workpiece swing angle | θ | 40°25′ | ° |
| Cutter position angle | φ | 17.48 | ° |
| Max inclination angle | βmax | 5.68 | ° |
| Horizontal position | O₂A | 51.5 | mm |
| Vertical position | O₂C | 18.78 | mm |
The calculation results show that the straightness error and tooth flank error after optimization and modification can both satisfy the gear accuracy requirements. The numerical simulation results in MATLAB demonstrate the correctness and feasibility of the proposed rotational indexing machining method for straight bevel gears.
6. Conclusions and Outlook
In this thesis, I addressed the problems of intermittent indexing and low production efficiency in traditional straight bevel gear machining methods. Based on the cycloid rotational indexing machining principle, I proposed a novel rotational indexing machining method for straight bevel gears that achieves continuous indexing.
In this work, I used the mathematical method of spatial curve approximating straight line to explain the machining principle of straight bevel gear rotational indexing. On this basis, I established the mathematical model of rotational indexing machining, obtaining the cutting point trajectory curve. I optimized and adjusted the parameters according to the degree of straightness approximation of the curve segment to the cone generatrix. Following the forming method, I obtained the tooth flank of the straight bevel gear and reduced the tooth flank error through cutting edge profile modification. The research demonstrates that the spatial trajectory curve can approximate a straight line to a certain degree with acceptable accuracy.
The main conclusions of this thesis are:
(1) The proposed spatial curve approximating straight line method extends the plane cycloid rotational indexing principle to three-dimensional space and can be applied to straight bevel gear machining.
(2) The cutter head inclination angle β provides position compensation, keeping the cutting edge close to the workpiece during the relative rotation of the cutter head and workpiece.
(3) The backtracking method for cutting edge point modification effectively reduces the tooth flank error, achieving the required machining accuracy.
(4) The rotational indexing machining method for straight bevel gears achieves continuous indexing, significantly improving production efficiency compared with traditional intermittent indexing methods.
The novel contributions of this thesis include: the spatial curve approximating straight line approach for extending the cycloid rotational indexing principle to space; the cutter head inclination angle compensation mechanism; and the backtracking method for cutting edge modification.
This research represents a preliminary investigation into the rotational indexing machining technology for straight bevel gears. There are still some deficiencies that need further investigation. The parameter optimization was performed with the rotational speed ratio as the optimization variable, whereas in actual machining practice, the number of teeth crossed per index and the number of cutter blades should be used, which may cause some deviation in the optimization results. The research mainly completed the calculation for one side of the tooth slot, and the machining calculation for the other flank requires further study. Additionally, the gear pair contact and transmission conditions were not considered in the evaluation of tooth flank accuracy, which could potentially result in mismatched gear pairs. To truly realize and apply this technology, more systematic and deeper research is needed, including gear pair contact analysis, tooth flank modification, and detailed cutter design.
Nevertheless, the proposed rotational indexing machining technology for straight bevel gears breaks away from the conventional generating gear concept, making the machining and calculation processes more intuitive. The technology realizes continuous indexing machining of straight bevel gears, which significantly improves production efficiency. This research lays the foundation for the design of new straight bevel gear machine tools and will contribute to improving the domestic manufacturing status of straight bevel gears.