Hypoid bevel gears are widely used in commercial vehicle drive axles, marine propulsion systems, aerospace transmissions, and precision machine tools because they can transmit motion between intersecting axes with a substantial offset. This offset, known as the hypoid offset, allows the pinion axis to be placed below or above the ring gear axis, which increases the pinion spiral angle, improves the pinion strength and rigidity, and enables a higher reduction ratio. However, the tooth surface of hypoid bevel gears is a complex spatial surface that cannot be represented by simple analytical functions. The machining process involves a cutter head and a virtual generating gear, and the relative motion between the tool and the workpiece determines the final tooth geometry. In actual production, manufacturing errors of the machine tool, cutter errors, thermal deformation, and other factors inevitably cause deviations between the actual tooth surface and the theoretical design surface. These deviations, referred to as tooth surface errors, directly affect the vibration, noise, and fatigue life of hypoid bevel gears. Therefore, correcting tooth surface errors efficiently and flexibly is of great practical importance.
In this work, I focus on the numerical control (NC) machining simulation and tooth surface error correction of hypoid bevel gears. I first establish the mathematical model of the tooth surface for both the formate ring gear and the generated pinion. Then I build a five-axis CNC machining simulation platform based on the kinematic transformation from a traditional mechanical cradle-type gear milling machine. Next, I study the influence of machine tool adjustment parameters on tooth surface errors and propose an improved linear regression analysis method for error correction. Finally, I validate the proposed method through simulation and experiments. The entire study is carried out with the goal of reducing trial cutting cost and improving the efficiency of tooth surface error correction for hypoid bevel gears.
1. Mathematical Model of Hypoid Bevel Gear Tooth Surface
The first step in studying the machining simulation and error correction of hypoid bevel gears is to establish an accurate mathematical model of the tooth surface. The tooth surface of hypoid bevel gears is generated by the enveloping motion between a cutter head and the workpiece. The cutter head rotates about its own axis, and the cradle rotates about the machine center. The workpiece rotates about its own axis according to a predetermined ratio. The relative motion between the cutter and the workpiece creates the tooth flank. In this section, I describe the coordinate systems, the cutter head model, and the tooth surface equations for both the formate method and the generating method.
1.1 Coordinate Systems and Cutter Head Model
To describe the machining process, I define several coordinate systems. The cutter head coordinate system $S_c$ is fixed to the cutter head. The machine fixed coordinate system $S_m$ is fixed to the machine frame. The workpiece coordinate system $S_1$ for the ring gear and $S_2$ for the pinion are fixed to the respective workpieces. For the generating method, additional coordinate systems are introduced to account for the cradle rotation, the eccentric drum, the cutter tilt, and the cutter swivel. The transformation matrices between these coordinate systems are derived using homogeneous coordinate transformations.
The cutter head is a rotating disk with cutting blades. For a face-milling cutter, the cutting edge can be modeled as a conical surface. The position vector of a point on the cutting edge in the cutter head coordinate system is given by:
$$
\mathbf{r}_c(s,\theta) = \begin{bmatrix}
(R \mp s \sin\alpha)\cos\theta \\
(R \mp s \sin\alpha)\sin\theta \\
-s\cos\alpha \\
1
\end{bmatrix}
$$
where $s$ is the cutting depth, $\theta$ is the cutter phase angle, $\alpha$ is the blade pressure angle, and $R$ is the cutter tip radius. The upper sign corresponds to the inner blade, and the lower sign corresponds to the outer blade. The unit normal vector of the cutting edge surface is:
$$
\mathbf{n}_c(\theta) = \begin{bmatrix}
\cos\alpha \cos\theta \\
\cos\alpha \sin\theta \\
\mp \sin\alpha
\end{bmatrix}
$$
The cutter tip radius $R$ is related to the nominal cutter radius $r_0$ and the point width $W$ by:
$$
R = r_0 \mp \frac{W}{2}
$$
For a complete cutter head, the inner and outer blades are alternately mounted. The inner blade generates the convex flank of the gear tooth, and the outer blade generates the concave flank. The geometry of the cutter head is critical because it directly determines the tooth profile and the pressure angle of the hypoid bevel gears.
1.2 Formate Method for the Ring Gear
In the formate method, the ring gear is machined without generating motion. The cutter head is fed into the workpiece to the full tooth depth, and the tooth slot is cut in one plunge. The cradle and the workpiece remain stationary during the cutting process. This method is highly efficient and is commonly used for the ring gear of hypoid bevel gears. The transformation matrix from the cutter head coordinate system $S_c$ to the ring gear coordinate system $S_1$ is:
$$
M_{1c} = M_{1m} M_{mc}
$$
where $M_{mc}$ accounts for the horizontal and vertical cutter positions and the machine center to cutter center distance, and $M_{1m}$ accounts for the machine root angle and the horizontal machine setting. The tooth surface position vector and normal vector are:
$$
\mathbf{r}_1(s_1,\theta_1) = M_{1c} \mathbf{r}_c(s_1,\theta_1)
$$
$$
\mathbf{n}_1(\theta_1) = M_{1c} \mathbf{n}_c(\theta_1)
$$
The formate method produces a tooth surface that is a conjugate of the cutter head surface. The contact pattern of hypoid bevel gears machined by the formate method is controlled by the cutter head geometry and the machine settings.
1.3 Generated Method for the Pinion
The pinion of hypoid bevel gears is usually machined by the generating method, often with cutter tilt. The generating motion is achieved by rolling the cradle and the workpiece according to a predetermined ratio. The cutter tilt mechanism allows the cutter axis to be inclined relative to the cradle axis, which creates a conical generating gear instead of a planar generating gear. This enables the control of the tooth surface curvature and the contact pattern. The transformation matrices for the pinion are:
$$
M_{2c} = M_{2f} M_{fe} M_{ed} M_{dc} M_{cb} M_{ba} M_{ac}
$$
where $M_{ac}$ accounts for the cutter tilt angle $i$, $M_{ba}$ accounts for the cutter swivel angle $j$, $M_{cb}$ accounts for the eccentric drum radius $S_R$, $M_{dc}$ accounts for the cradle angle $q$, $M_{ed}$ accounts for the vertical offset $E_m$ and the sliding base $\Delta B$, $M_{fe}$ accounts for the machine root angle $\gamma_m$ and the horizontal setting $\Delta A$, and $M_{2f}$ accounts for the workpiece rotation angle $\phi$. The tooth surface position vector and normal vector are:
$$
\mathbf{r}_2(s_2,\theta_2,\phi) = M_{2c} \mathbf{r}_c(s_2,\theta_2)
$$
$$
\mathbf{n}_2(\theta_2,\phi) = M_{2c} \mathbf{n}_c(\theta_2)
$$
The meshing equation for the generated pinion is:
$$
\mathbf{n}_2 \cdot \mathbf{v}_{12} = f(s_2,\theta_2,\phi) = 0
$$
where $\mathbf{v}_{12}$ is the relative velocity between the generating gear and the pinion. Solving this equation yields the relationship between $s_2$, $\theta_2$, and $\phi$. Substituting this relationship into the position vector gives the final tooth surface equation:
$$
\mathbf{r}_2(\theta_2,\phi) = \mathbf{h}(\theta_2,\phi)
$$
The mathematical model of hypoid bevel gears is the foundation for all subsequent analyses, including the simulation of CNC machining and the correction of tooth surface errors.
2. Five-Axis CNC Machining Simulation Platform
Traditional mechanical cradle-type gear milling machines are complex and have long kinematic chains, which can introduce errors and reduce machining accuracy. To study the tooth surface error correction of hypoid bevel gears without repeated trial cutting, I build a five-axis CNC machining simulation platform. This platform is based on the kinematic transformation from the traditional mechanical cradle-type machine to a five-axis CNC machining center. The platform is built in VERICUT and CATIA, and it serves as a virtual machining center for subsequent error correction research.
2.1 Kinematic Transformation from Traditional Machine to Five-Axis CNC Center
The traditional mechanical cradle-type gear milling machine has a cradle, an eccentric drum, a cutter tilt body, a cutter swivel body, and a workpiece spindle. The five-axis CNC machining center replaces these mechanical structures with three linear axes (X, Y, Z) and two rotary axes (A, B). The cutter head rotation (C axis) provides the cutting speed and does not participate in the generating motion. The X and Y axes simulate the cradle rotation and the eccentric drum motion. The Z axis controls the feed and retract. The A axis rotates the workpiece to achieve the generating motion and the indexing motion. The B axis tilts the workpiece spindle to simulate the cutter tilt and swivel.
To transform the traditional machine settings to the CNC axes, I derive the position and orientation of the cutter head in the workpiece coordinate system. Let $\mathbf{R}_{p1}$ and $\mathbf{N}_{p1}$ be the position and orientation vectors of the cutter head in the workpiece coordinate system for the traditional machine. Let $\mathbf{R}_{p2}$ and $\mathbf{N}_{p2}$ be the corresponding vectors for the five-axis CNC machine. The transformation conditions are:
$$
\begin{cases}
\mathbf{R}_{p1} = \mathbf{R}_{p2} \\
\mathbf{N}_{p1} = \mathbf{N}_{p2}
\end{cases}
$$
Solving these equations yields the expressions for the CNC axes A and B:
$$
\begin{cases}
A = \phi_c \cdot \text{Ratio} + \arctan\left(\frac{\sin i \sin(q+j)}{\sin i \sin\gamma_m \cos(q+j) – \cos i \cos\gamma_m}\right) \\
B = \arcsin(\cos i \sin\gamma_m + \sin i \cos\gamma_m \cos(q+j))
\end{cases}
$$
where $q = q_0 + \phi_c$, $q_0$ is the initial cradle angle, and $\phi_c$ is the cradle rotation angle. The expressions for the X, Y, and Z axes are obtained by substituting the A and B angles back into the position equation:
$$
\begin{cases}
X = a\cos B – \sin B (b\sin A + c\cos A + a\tan B) \\
Y = b\cos A + (a\tan B – b\sin A – c\cos A – a\tan B \sin^2 B – a\sin B \cos B)\tan A \\
Z = -(b\sin A \cos B + c\cos A \cos B + a\sin B)
\end{cases}
$$
where $a$, $b$, and $c$ are the components of $\mathbf{R}_{p1}$:
$$
\begin{cases}
a = S_R \cos q \cos\gamma_m – \Delta B \sin\gamma_m – \Delta A \\
b = (E_m – S_R \sin q)\cos\phi – (S_R \cos q \sin\gamma_m + \Delta B \cos\gamma_m)\sin\phi \\
c = -(E_m – S_R \sin q)\sin\phi – (S_R \cos q \sin\gamma_m + \Delta B \cos\gamma_m)\cos\phi
\end{cases}
$$
These equations allow the complete transformation from the traditional machine settings to the five-axis CNC axes. The cradle rotation angle $\phi_c$ is discretized into small steps, and each step corresponds to a set of CNC axis positions. The accuracy of the machined tooth surface depends on the step size; a smaller step size yields higher accuracy.
2.2 G-Code Generation and VERICUT Simulation
Based on the derived CNC axis expressions, I generate the G-code program for machining hypoid bevel gears. The program includes the following steps: start the spindle, move the axes to the initial generating position, perform the generating motion for one tooth slot, retract the tool, index the workpiece by $360/z$ degrees (where $z$ is the number of teeth), and repeat until all tooth slots are machined. The G-code program is written with a step size of $1^\circ$ for clarity, but in actual machining, a much smaller step size is used to ensure accuracy.
I build the five-axis CNC machining simulation platform in VERICUT. The machine tool model is created in a 3D CAD software and imported into VERICUT. The control system is selected, and the machine components are assembled according to the kinematic chain. The tool library is created with the inner and outer blades. The workpiece blank and the design model are loaded. The G-code program is imported, and the machining simulation is performed. The simulation results show that the tooth profile is complete without obvious overcut or residual material, which verifies the correctness of the five-axis CNC model and the G-code program.
However, the simulation accuracy in VERICUT is limited by the model import resolution and the discrete approximation of the generating motion. The comparison between the simulated tooth surface and the design model shows deviations of about 0.02 mm, which is not sufficient for studying tooth surface error correction of hypoid bevel gears. The error correction method requires a simulation accuracy better than 0.001 mm to distinguish the artificially introduced errors from the simulation errors. Therefore, I use CATIA for the cutting simulation to achieve higher accuracy.
2.3 CATIA-Based Cutting Simulation
CATIA has powerful 3D modeling capabilities and a macro recording function. I use VBA to control the position and motion of the cutter and the workpiece in CATIA, thereby simulating the cutting process of hypoid bevel gears. The simulation flow is as follows: establish the cutter and workpiece models, set the initial positions according to the machine settings, perform the generating motion by rotating the cutter and the workpiece, and use Boolean operations to remove the material. The process is repeated for all tooth slots.
The cutter head model is built according to the nominal diameter, blade pressure angle, point width, and tip radius. The workpiece model is built according to the face cone angle, root cone angle, face width, and other blank parameters. The coordinate origin is set at the design crossing point of the hypoid bevel gears. The motion of the cutter and the workpiece is controlled by transformation matrices that account for the horizontal setting, machine root angle, vertical offset, sliding base, cutter tilt, cutter swivel, eccentric angle, and cradle angle.
The cutting simulation is performed for both the ring gear and the pinion. The simulation accuracy is checked by measuring the distance from the theoretical tooth surface points to the simulated tooth surface. The results show that the maximum error of the simulated tooth surface is less than 0.0011 mm when the step size is $0.644114^\circ$ for the workpiece and $0.1^\circ$ for the cutter. When the step size is reduced to $0.0644114^\circ$ and $0.01^\circ$, the maximum error decreases to 0.0003 mm. This confirms that the CATIA-based cutting simulation is accurate enough for studying tooth surface error correction of hypoid bevel gears.
3. Tooth Surface Error Correction Method
The tooth surface errors of hypoid bevel gears are caused by machine tool manufacturing errors, cutter errors, thermal deformation, and other factors. These errors must be corrected by adjusting the machine tool settings. In this section, I establish the tooth surface mathematical model with machine tool adjustment parameters, study the influence of each parameter on tooth surface errors, and propose an improved linear regression analysis method for error correction.
3.1 Tooth Surface Error Decomposition and Calculation
The tooth surface error describes the deviation between the actual tooth surface and the theoretical tooth surface. It can be decomposed into two parts: the pitch error and the tooth form error. The pitch error is caused by the indexing error of the machine tool and is the same for all tooth slots. It does not affect the tooth surface accuracy and does not need to be corrected. The tooth form error is caused by the combined effect of various errors and describes the shape deviation of the tooth surface. Only the tooth form error needs to be corrected.
To calculate the tooth form error, I first establish the tooth surface mathematical model with machine tool adjustment parameters. Let $k = [i, j, S_R, q_1, E_m, \Delta B, \Delta A, \gamma_m]^T$ be the vector of machine tool adjustment parameters. When these parameters have errors, the actual tooth surface equation becomes:
$$
\mathbf{r}'(s,\theta,\phi) = M’_{2f} M’_{fe} M’_{ed} M’_{dc} M’_{cb} M’_{ba} M’_{ac} \mathbf{r}_c(s,\theta)
$$
The tooth surface error is defined as the distance from the theoretical tooth surface point to the actual tooth surface along the normal direction:
$$
\mathbf{r}’ = \mathbf{r} + E \cdot \mathbf{n}
$$
where $E$ is the tooth surface error. To solve for $E$, I combine this equation with the meshing equation of the actual tooth surface:
$$
\mathbf{n}'(\theta, k’) \cdot \mathbf{v}'(s, \theta, k’) = f(s, \theta, k’) = 0
$$
This system of equations is solved to obtain the tooth surface error at any point. The pitch error is eliminated by rotating the actual tooth surface about the gear axis until the center point coincides with the theoretical center point. The remaining error is the tooth form error.
3.2 Influence of Machine Tool Adjustment Parameters on Tooth Surface Errors
To study the influence of each machine tool adjustment parameter on the tooth surface errors of hypoid bevel gears, I introduce a small error in one parameter while keeping the others unchanged. The tooth surface error distribution is then calculated. The results show that different parameters affect the tooth surface error in different ways. For the ring gear, the cutter horizontal setting, cutter vertical setting, and horizontal machine setting mainly affect the spiral angle error, while the machine root angle mainly affects the pressure angle error. For the pinion, the sliding base, radial cutter position, and cutter tilt angle mainly affect the spiral angle error, while the horizontal setting, vertical offset, cutter swivel angle, machine root angle, and roll ratio mainly affect the diagonal error. The angular cutter position has almost no effect on the tooth surface error.
I define the comprehensive influence coefficient of tooth surface error as:
$$
E = \frac{\sum_{j=1}^{45} |\Delta d_i|}{\Delta k_i}
$$
where $\Delta k_i$ is the change in the $i$-th machine tool adjustment parameter, and $\Delta d_i$ is the error at the $i$-th point on the tooth surface. The comprehensive influence coefficients for the pinion are listed in Table 1.
| Parameter | Convex flank | Concave flank |
|---|---|---|
| Horizontal setting $\Delta A$ | 1.651 | 2.247 |
| Sliding base $\Delta B$ | 0.657 | 1.840 |
| Vertical offset $E_m$ | 3.146 | 2.777 |
| Angular cutter position | 0 | 0 |
| Radial cutter position $S_R$ | 5.121 | 5.813 |
| Cutter tilt angle $i$ | 8.015 | 8.136 |
| Cutter swivel angle $j$ | 0.617 | 0.522 |
| Machine root angle $\gamma_m$ | 3.500 | 2.848 |
| Roll ratio | 84.5 | 91.4 |
The roll ratio has the largest influence on the tooth surface error of hypoid bevel gears, followed by the cutter tilt angle, radial cutter position, machine root angle, vertical offset, and horizontal setting. The cutter swivel angle, sliding base, and angular cutter position have relatively small influences.
3.3 Sensitivity Coefficient Matrix and Basic Correction Method
To quantitatively describe the relationship between tooth surface errors and machine tool adjustment parameters, I define the sensitivity coefficient. For a point $M_i$ on the tooth surface, the sensitivity coefficient with respect to the $j$-th parameter $k_j$ is:
$$
s_{ji} = \frac{\Delta m_i}{\Delta k_j}
$$
where $\Delta m_i$ is the error at point $M_i$ caused by the change $\Delta k_j$ in the parameter $k_j$. For all points on the tooth surface, the relationship can be written in matrix form:
$$
\Delta M_{45 \times 1} = S_{45 \times n} \cdot \Delta K_{n \times 1}
$$
where $S$ is the sensitivity coefficient matrix, $\Delta M$ is the tooth surface error vector, and $\Delta K$ is the machine tool adjustment parameter correction vector. Since the number of equations (45) is greater than the number of unknowns ($n$), this is an overdetermined system. The least squares method is used to find the approximate solution:
$$
S^T S \cdot \Delta K = S^T \Delta M
$$
However, the least squares method often produces meaningless solutions when many parameters are corrected, and it lacks selectivity. In actual production, it is desirable to correct the tooth surface errors of hypoid bevel gears with as few parameters as possible.
3.4 Improved Linear Regression Analysis Method
Linear regression analysis is a mathematical tool for studying the relationship between two or more variables. In the context of tooth surface error correction, I use linear regression to select the most relevant machine tool adjustment parameters for correcting the tooth surface errors of hypoid bevel gears. The traditional method works as follows:
- Perform a simple linear regression between each machine tool adjustment parameter and the tooth surface error, and rank the parameters by the coefficient of determination $R^2$.
- Select the parameter with the largest $R^2$ as the first correction parameter. Check if the residual sum of squares meets the requirement.
- If not, add another parameter and perform multiple linear regression. The parameter that gives the largest $R^2$ is selected as the next correction parameter.
- Repeat until the residual sum of squares meets the requirement.
The traditional method has a drawback: when performing multiple linear regression, the parameter with the largest $R^2$ in the simple regression may not be among the best parameters in the multiple regression. This can cause the method to miss the optimal correction parameters and require more parameters than necessary. To address this, I propose an improved method. The improved method keeps the first two steps unchanged, but in the subsequent steps, it directly searches all possible combinations of parameters and selects the combination that gives the largest $R^2$. For example, when selecting two parameters, it evaluates all $C_m^2$ combinations, where $m$ is the total number of parameters. The combination with the largest $R^2$ is chosen as the best correction parameter set. This process continues until the residual sum of squares meets the requirement.
The improved method has two advantages. First, it avoids missing the optimal correction parameters because it does not rely on the ranking from the previous step. Second, it provides more alternative parameter combinations, which is beneficial for practical production. Table 2 compares the traditional and improved methods for a simulation case.
| Method | Correction parameters | Maximum $R^2$ | Maximum error after correction ($\mu$m) | Average error after correction ($\mu$m) |
|---|---|---|---|---|
| Least squares | All parameters | — | 13.0 | — |
| Traditional | $\Delta A$, $\gamma_m$, Ratio | 0.9804 | 6.6 | 2.098 |
| Improved | $E_m$, $\gamma_m$ | 0.9719 | 8.4 | 2.320 |
The simulation results show that the improved method achieves a comparable correction effect with only two parameters, while the traditional method requires three parameters. This demonstrates the flexibility and efficiency of the improved method for correcting the tooth surface errors of hypoid bevel gears.
4. Experimental Verification
To further verify the feasibility and correctness of the proposed tooth surface error correction method for hypoid bevel gears, I design a measurement and correction experiment. The experiment is carried out on a commercial vehicle drive axle pinion. The tooth surface errors are measured using a M&M 3525 measurement center. The measurement principle is based on comparing the actual tooth surface coordinates with the theoretical tooth surface coordinates. The tooth surface is discretized into a grid of 5 points in the tooth height direction and 9 points in the tooth length direction, resulting in 45 measurement points. The boundary points are moved inward to avoid collision with the probe.
The measurement scheme uses the center point of the tooth surface as the reference point, and the probe moves in an “S” shape from the center to both ends. This scheme has high measurement accuracy because the rotary table rotates in one direction. The gear is mounted on the measurement center with the design crossing point as the origin. The measurement results before correction are shown in Table 3.
| Flank | Maximum absolute error ($\mu$m) | Average absolute error ($\mu$m) |
|---|---|---|
| Convex | 106.9 | 30.76 |
| Concave | 110.6 | 31.60 |
The measured errors show significant diagonal errors on both flanks. Using the improved linear regression analysis method, I calculate the correction values for the machine tool adjustment parameters. The correction parameters for the convex flank are the vertical offset, cutter swivel angle, and machine root angle. The correction parameters for the concave flank are the radial cutter position, cutter swivel angle, and roll ratio. The correction values are listed in Table 4.
| Flank | Parameter | Before correction | After correction | Correction value |
|---|---|---|---|---|
| Convex | Vertical offset (mm) | 34.6100 | 34.4188 | -0.1912 |
| Cutter swivel angle ($^\circ$) | 281.4170 | 283.1732 | +1.7562 | |
| Machine root angle ($^\circ$) | -4.0000 | -3.6655 | +0.3345 | |
| Concave | Radial cutter position (mm) | 127.39791 | 127.23881 | -0.1591 |
| Cutter swivel angle ($^\circ$) | 261.2330 | 258.9299 | -2.3031 | |
| Roll ratio | 5.82073 | 5.83093 | +0.0102 |
After inputting the corrected machine tool adjustment parameters into the gear milling machine, a new gear is machined and measured. The tooth surface errors after correction are shown in Table 5.
| Flank | Maximum absolute error ($\mu$m) | Average absolute error ($\mu$m) |
|---|---|---|
| Convex | 8.2 | 3.376 |
| Concave | 7.6 | 3.620 |
The experimental results show that the maximum absolute errors of the tooth surface measurement points are reduced to below 8.2 $\mu$m, and the average absolute errors are reduced to about 3.5 $\mu$m. The theoretical correction residuals are in good agreement with the measured results after correction. This confirms the feasibility and correctness of the improved linear regression analysis method for correcting the tooth surface errors of hypoid bevel gears.
5. Conclusion
In this work, I have studied the CNC machining simulation and tooth surface error correction of hypoid bevel gears. The main contributions and findings are summarized as follows:
- I established the mathematical models of the tooth surface for the formate ring gear and the generated pinion of hypoid bevel gears. The models are based on the cutter head geometry, the meshing principle, and the coordinate transformation. The tooth surface discrete points are solved, and the 3D models of the hypoid bevel gears are built.
- I built a five-axis CNC machining simulation platform based on the kinematic transformation from the traditional mechanical cradle-type gear milling machine. The platform is implemented in VERICUT and CATIA. The simulation accuracy in CATIA is better than 0.0011 mm, which is sufficient for studying tooth surface error correction.
- I proposed an improved linear regression analysis method for correcting the tooth surface errors of hypoid bevel gears. The method selects the optimal correction parameters by searching all possible combinations, avoiding the drawback of the traditional method that may miss the optimal parameters. Simulation results show that the improved method achieves the same correction effect with fewer parameters.
- I designed a tooth surface error measurement and correction experiment. The experimental results show that the maximum absolute error is reduced to below 8.2 $\mu$m, and the average absolute error is reduced to about 3.5 $\mu$m. This validates the feasibility and correctness of the proposed method.
The proposed method provides a flexible and efficient way to correct the tooth surface errors of hypoid bevel gears. It reduces the number of trial cuts and the production cost, and it is suitable for practical production. Future work will focus on considering the coupling effects between machine tool adjustment parameters and extending the method to direct CNC axis correction for hypoid bevel gears.