In modern mechanical transmission systems, the straight bevel gear plays a critical role due to its simplicity in design, manufacturing, and installation, as well as its ability to transmit motion without generating axial forces. These attributes make straight bevel gears widely applicable in agricultural machinery, automotive differentials, and industrial equipment. However, as the demand for higher transmission performance and accuracy in agricultural machinery increases, traditional methods of quality control, such as contact pattern inspection, have become insufficient. Instead, geometric precision of the tooth surface has emerged as a key indicator for assessing the manufacturing quality of straight bevel gears. Gear measuring centers (GMCs) are commonly used for this purpose, as they enable precise measurement of tooth surface errors by comparing theoretical and actual tooth surface data. The accuracy of these measurements heavily relies on the correct acquisition of theoretical tooth surface data, which traditionally derives from mathematical models based on machining principles or approximate involute profiles. Unfortunately, these models often fail to account for tooth surface modifications—such as crowning or profile shifts—applied by manufacturers to enhance meshing performance. Consequently, the measured errors may include both modification deviations and machining inaccuracies, leading to distorted results that do not accurately reflect the actual manufacturing errors.
To address this issue, we propose a novel method for extracting theoretical tooth surface data directly from the solid model of a straight bevel gear. This approach leverages the three-dimensional (3D) CAD data used in the design and manufacturing process, particularly in precision forging, where mold cavity data are derived from the gear’s solid model. By utilizing this existing geometric information, we can obtain theoretical coordinates and normal vectors that exactly match the designed tooth surface, including any modifications. This ensures that the measurement results solely represent machining errors, providing a reliable basis for quality control. Our method involves importing the solid model into a CAD environment, such as UG/NX, and using UG Open programming tools to extract data points and their unit normal vectors based on a predefined measurement grid. This process eliminates the inconsistencies associated with traditional theoretical calculations and offers a flexible solution applicable to various gear types, including straight bevel gears with complex tooth surfaces.
The theoretical foundation of our method begins with the mathematical representation of the straight bevel gear tooth surface. In theory, the tooth profile of a straight bevel gear is defined by a spherical involute. However, due to the complexity of modeling on a spherical surface, an approximate approach using the back-cone involute is commonly adopted in engineering. This simplification facilitates the derivation of the tooth surface equations. Consider the coordinate system $S(X, Y, Z)$ fixed to the axial cross-section of the straight bevel gear, where the origin $O$ is located at the design crossing point, and the $X$-axis coincides with the rotational axis. The back-cone coordinate system $S_1(X_1, Y_1, Z_1)$ is defined such that the $Y_1$-axis lies along the intersection line of the back-cone and the axial cross-section. Another coordinate system $S_2(X_2, Y_2, Z_2)$ is used to describe the generation of the involute profile, where $\theta_k$ represents the involute expansion angle, and $\gamma_0$ is the half-angle of the tooth thickness. The equivalent gear diameters—such as the equivalent tip diameter $D_{av}$, equivalent root diameter $D_{fv}$, equivalent base diameter $D_{bv}$, and equivalent pitch diameter $D_v$—are derived from the gear geometry. Through coordinate transformations and geometric relations, the tooth surface equation $\mathbf{r}_i$ and the unit normal vector $\mathbf{n}_i$ can be expressed as functions of the surface parameters $u_i$ and $\theta_i$:
$$
\mathbf{r}_i = \mathbf{r}_i(\theta_i, u_i), \quad \mathbf{n}_i = \mathbf{n}_i(\theta_i, u_i), \quad \text{for } i = 1, 2
$$
Here, $i=1$ corresponds to the pinion (small gear), and $i=2$ to the gear (large gear) in a gear pair. These equations form the basis for traditional theoretical data calculation. However, as mentioned, they do not incorporate tooth surface modifications, which are common in practice to reduce noise, improve load distribution, or compensate for misalignments. For instance, in agricultural machinery, straight bevel gears often undergo profile modifications to enhance durability under varying loads. Therefore, relying solely on these equations for measurement data can lead to inaccuracies.
To overcome this limitation, our method bypasses the need for explicit tooth surface equations and instead uses the solid model directly. The solid model, typically exported in formats like STL, IGS, or STP, contains accurate geometric information of the gear, including any modifications applied during design. This model serves as the reference for extracting theoretical data. The process starts with the planning of measurement points on the tooth surface. A common approach is to define a grid of points in the axial cross-section, which are then projected onto the 3D tooth surface using a rotational projection model. For a straight bevel gear, the grid consists of points distributed along the tooth height and tooth length directions. Typically, the tooth height is divided into 5 lines with constant pressure angles, and the tooth length into 9 lines with varying intercepts, resulting in a 5 × 9 grid of 45 points. The coordinates of these grid points in the axial cross-section are calculated based on the gear geometry parameters, such as module, number of teeth, pressure angle, and cone distance.
Let $M(x, y)$ be a point in the axial cross-section coordinate system $S(X, Y, Z)$, with $R_x$ as the intercept and $\alpha_x$ as the pressure angle. The corresponding point on the 3D tooth surface $M^*(x^*, y^*, z^*)$ must satisfy the rotational projection relation:
$$
x = x^*, \quad y = \sqrt{(y^*)^2 + (z^*)^2}
$$
This equation represents a nonlinear system that can be solved iteratively using optimization techniques to find the parameters $(\theta_i, u_i)$ for each grid point. Substituting these into the tooth surface equations yields the theoretical coordinates and unit normal vectors. However, in our method, we avoid this computational complexity by directly interacting with the solid model.
The extraction process begins by importing the solid model of the straight bevel gear into a CAD software environment, such as UG/NX. The coordinate system must be carefully defined to ensure consistency with the gear’s design specifications. The installation distance—defined as the distance from the crossing point of the axes to the axial positioning face of the gear—is used to determine the origin of the absolute coordinate system. This is crucial because the same point may have different coordinates in relative and absolute systems. By setting the origin based on the installation distance, we establish a unified reference frame for all subsequent operations. Additional geometric parameters, such as the pitch cone angle and cone distance, are used to define the axial cross-section position. The predefined grid points in the axial cross-section are then imported, and for each point, a rotational projection line is constructed. This line is a circle that passes through the grid point and is perpendicular to the axial cross-section. In UG/NX, this can be achieved using the UG Open API function UF_CURVE_create_arc_3point(), which creates an arc through three points—the grid point $M$ and two auxiliary points $M_1$ and $M_2$ derived from the geometric relations.
The intersection of this rotational projection line with the tooth surface uniquely defines the corresponding point $M^*$ on the 3D surface. The UG Open API function UF_CURVE_intersect() is employed to compute this intersection. Once the point is obtained, its coordinates and the unit normal vector are extracted. The unit normal vector is derived based on differential geometry principles, using the face properties of the solid model. The function int UF_MODL_ask_face_props() retrieves the UV parameters of the point on the surface, which are then used to calculate the normal vector. This process is repeated for all 45 grid points on each tooth flank (left and right). To facilitate user interaction, a custom interface is developed using UG Open programming, allowing users to import files, select tooth flanks, and export the extracted data.
To validate our method, we conducted a case study using a straight bevel gear pair with a pinion-to-gear ratio of 10:18. The geometric parameters of the pinion are summarized in Table 1. The solid model of the pinion was provided by a manufacturer, and we extracted the theoretical data for the left tooth flank. For comparison, we also calculated the theoretical data using the traditional tooth surface equations. The results are presented in Tables 2, 3, and 4. Table 2 shows a subset of the axial cross-section grid points, Table 3 lists the coordinates and normal vectors obtained from the traditional equations, and Table 4 displays the data extracted from the solid model. The comparison reveals that the differences between the two sets of data are within the tolerance limits of gear measuring centers, confirming the accuracy of our method. For example, at grid point 41, the X-coordinate from the traditional method is 45.853874, while the extracted value is 45.8539, resulting in an error of 0.000026, which is negligible for practical purposes.
| Parameter | Value |
|---|---|
| Number of Teeth (Pinion) | 10 |
| Module | 5.4000 |
| Face Width (mm) | 17.0000 |
| Pressure Angle (°) | 22.50000 |
| Installation Distance (mm) | 57.0000 |
| Node Sequence | X (mm) | Y (mm) |
|---|---|---|
| 1 | 31.8329 | 22.21358 |
| 2 | 32.54619 | 21.01083 |
| 3 | 33.26744 | 19.81236 |
| 4 | 33.98513 | 18.61179 |
| 5 | 34.68293 | 17.40026 |
| … | … | … |
| 41 | 45.85368 | 24.10629 |
| 42 | 46.88134 | 30.26516 |
| 43 | 47.92027 | 28.53882 |
| 44 | 48.95407 | 26.80946 |
| 45 | 49.95922 | 25.0643 |
| Sequence | X (mm) | Y (mm) | Z (mm) | X_n | Y_n | Z_n |
|---|---|---|---|---|---|---|
| 1 | 31.832898 | -1.29758 | -22.175648 | -0.2842225 | -0.810849 | -0.511605 |
| 2 | 32.546192 | -2.234023 | -20.891719 | -0.235741 | -0.874281 | -0.424333 |
| 3 | 33.267439 | -2.936866 | -19.593475 | -0.176659 | -0.931491 | -0.317987 |
| 4 | 33.985131 | -3.38366 | -18.301629 | -0.098791 | -0.979091 | -0.177824 |
| 5 | 34.682931 | -3.495263 | -17.045588 | 0.085685 | -0.984312 | 0.154234 |
| … | … | … | … | … | … | … |
| 41 | 45.853874 | -1.869106 | -31.943035 | -0.284225 | -0.810849 | -0.511605 |
| 42 | 46.881342 | -3.21801 | -30.093592 | -0.235741 | -0.874281 | -0.424333 |
| 43 | 47.920266 | -4.230424 | -28.223529 | -0.176659 | -0.931491 | -0.317887 |
| 44 | 48.95407 | -4.874012 | -26.362682 | -0.098791 | -0.979091 | -0.177824 |
| 45 | 49.95922 | -5.034771 | -24.553412 | 0.085685 | -0.984312 | 0.154234 |
| Sequence | X (mm) | Y (mm) | Z (mm) | X_n | Y_n | Z_n |
|---|---|---|---|---|---|---|
| 1 | 31.8329 | -1.2976 | -22.1756 | -0.2842 | -0.8108 | -0.5116 |
| 2 | 32.5462 | -2.2340 | -20.8917 | -0.2357 | -0.8743 | -0.4243 |
| 3 | 33.2674 | -2.9369 | -19.5935 | -0.1767 | -0.9315 | -0.3180 |
| 4 | 33.9851 | -3.3837 | -18.3016 | -0.0988 | -0.9791 | -0.1778 |
| 5 | 34.6829 | -3.4953 | -17.0456 | 0.0857 | -0.9843 | 0.1542 |
| … | … | … | … | … | … | … |
| 41 | 45.8539 | -1.8691 | -31.9430 | -0.2842 | -0.8108 | -0.5116 |
| 42 | 46.8813 | -3.2180 | -30.0936 | -0.2357 | -0.8743 | -0.4243 |
| 43 | 47.9203 | -4.2304 | -28.2235 | -0.1767 | -0.9315 | -0.3180 |
| 44 | 48.9541 | -4.8740 | -26.3627 | -0.0988 | -0.9791 | -0.1778 |
| 45 | 49.9592 | -5.0348 | -24.5534 | 0.0857 | -0.9843 | 0.1542 |
The advantages of this method are manifold. First, it ensures that the theoretical data used in gear measurement centers exactly match the designed tooth surface, including any modifications. This is particularly important for straight bevel gears in agricultural applications, where custom modifications are common to handle dynamic loads and environmental conditions. Second, the method reduces computational complexity by avoiding the need for iterative solutions to nonlinear equations. Instead, it relies on robust CAD operations, which are inherently accurate and efficient. Third, the approach is flexible and can be extended to other types of gears, such as spiral bevel gears or helical gears, where tooth surface modifications are even more complex. The use of UG Open programming allows for automation, making the process repeatable and user-friendly.
In conclusion, the extraction of theoretical tooth surface data for straight bevel gear measurement based on solid modeling offers a reliable and practical alternative to traditional methods. By leveraging the existing 3D CAD data, we can obtain accurate coordinates and normal vectors that reflect the true design intent, including modifications. This enables precise measurement of machining errors on gear measuring centers, leading to improved quality control in the manufacturing of straight bevel gears. Future work could focus on extending this method to real-time measurement systems or integrating it with cloud-based CAD platforms for broader accessibility. Additionally, the principles discussed here can be applied to other complex gear types, further enhancing the accuracy and efficiency of gear metrology in various industrial sectors.