The precise modeling of the real tooth surface of hyperboloidal gears is a critical foundation for predicting their dynamic performance, transmission error, and service life. These gears are pivotal components in demanding applications within the aerospace and automotive industries. During actual operation, factors such as machining inaccuracies, assembly errors, and suboptimal lubrication inevitably lead to wear and modification of the active tooth flanks. Traditional digital modeling approaches, often designed for pristine, ground surfaces, struggle to accurately represent these worn regions. They either sacrifice accuracy for smoothness or fail to efficiently handle the irregular data associated with wear zones. This paper presents a robust and efficient methodology for constructing high-fidelity digital models of real tooth surfaces for hyperboloidal gears, explicitly accounting for the presence of wear, by integrating advanced data segmentation and refined interpolation algorithms.
The core challenge lies in processing the discrete point cloud data obtained from Coordinate Measuring Machines (CMM). A scan-line measurement strategy is typically employed, resulting in a set of points that represent the as-measured, potentially worn, gear flank. The objective is to reconstruct a continuous digital surface from this sparse and possibly non-uniform data, faithfully capturing wear features without introducing undue computational complexity or smoothing out critical real-world deviations.
1. Mathematical Foundation: Theoretical vs. Real Tooth Surface Models
1.1 Theoretical Tooth Surface Model
To establish a reference frame, we first define the theoretical (nominal) tooth surface of the hyperboloidal gear. This surface is generated as the envelope of the cutter blade surface during the gear manufacturing process. In the gear coordinate system \( S_g \), the theoretical surface \( \Sigma_g \) is represented by its position vector \( \mathbf{r}_g \) and unit normal vector \( \mathbf{n}_g \), which are functions of the cutter surface parameters \( u \) and \( \theta \):
$$ \mathbf{r}_g = \mathbf{r}_g(u, \theta), \quad \mathbf{n}_g = \mathbf{n}_g(u, \theta) $$
The surface is derived via the coordinate transformation from the cutter system \( S_t \) to the gear system \( S_g \), governed by the meshing equation:
$$ \mathbf{r}_g(u, \theta) = \mathbf{M}_{gt}(\phi) \cdot \mathbf{r}_t(u_t, \theta_t), \quad f(u_t, \theta_t, \phi)=0 $$
$$ \mathbf{n}_g(u, \theta) = \mathbf{L}_{gt}(\phi) \cdot \mathbf{n}_t(u_t, \theta_t), \quad f(u_t, \theta_t, \phi)=0 $$
Here, \( \mathbf{M}_{gt} \) and \( \mathbf{L}_{gt} \) are the coordinate transformation matrices for position and orientation, respectively. \( \mathbf{r}_t \) and \( \mathbf{n}_t \) are the position and normal of the cutter surface, and \( \phi \) is the gear rotation angle. The meshing equation \( f=0 \), expressed as \( \mathbf{N}_t \cdot \mathbf{V}_{tg} = 0 \), ensures tangency between the cutter and the generated gear surface, where \( \mathbf{N}_t \) is the common normal and \( \mathbf{V}_{tg} \) is the relative velocity.
For measurement, the gear coordinate system \( S_g \) is related to the CMM machine coordinate system \( S_m \) by a fixed displacement \( \lambda \) and an installation angle \( \delta \). The theoretical model in the measurement frame is:
$$ \mathbf{r}_m(u, \theta) = \mathbf{M}_{mg} \cdot \mathbf{r}_g(u, \theta) $$
where \( \mathbf{M}_{mg} \) is the transformation matrix from \( S_g \) to \( S_m \).
1.2 Real Tooth Surface Model from CMM Measurement
The real, worn tooth surface \( \Sigma_r \) deviates from the theoretical one. When measuring with a spherical probe of radius \( \rho \), the center of the probe contacting the theoretical surface is at:
$$ \mathbf{R}_m(u, \theta) = \mathbf{r}_m(u, \theta) + \rho \cdot \mathbf{n}_m(u, \theta) $$
However, when the probe contacts the real surface, the center position is offset by a distance \( \lambda(u, \theta) \) along the normal direction:
$$ \mathbf{R}^*_m(u, \theta) = \mathbf{r}_m(u, \theta) + \lambda(u, \theta) \cdot \mathbf{n}_m(u, \theta) $$
This deviation \( \lambda(u, \theta) \), which encapsulates both form errors and wear, is what we aim to capture from the CMM point cloud and embody in our digital reconstruction. The discrete measurement points correspond to \( \mathbf{R}^*_m \) at various (u, θ) parameters.
2. Core Reconstruction Technology Based on Non-Geometric-Feature Segmentation
To accurately model worn hyperboloidal gears, we propose a pipeline that first intelligently segments the measured point cloud to identify worn regions and then applies a tailored local interpolation scheme. This approach prioritizes accuracy in critical zones while maintaining global surface fairness.
2.1 NURBS Surface as the Modeling Framework
The Non-Uniform Rational B-Spline (NURBS) surface is adopted as the final digital representation due to its standardization and flexibility. A bi-cubic NURBS surface is defined as:
$$ \mathbf{P}(u, v) = \frac{\sum_{i=1}^{m} \sum_{j=1}^{n} B_{i,3}(u) B_{j,3}(v) \omega_{ij} \mathbf{V}_{ij}}{\sum_{i=1}^{m} \sum_{j=1}^{n} B_{i,3}(u) B_{j,3}(v) \omega_{ij}}, \quad u,v \in [0,1] $$
where \( \mathbf{V}_{ij} \) are control points, \( \omega_{ij} \) are their associated weights, and \( B_{i,3}(u) \), \( B_{j,3}(v) \) are the cubic B-spline basis functions defined recursively on knot vectors.
2.2 Delaunay Triangulation of Spatial Discrete Data
The raw scan-line data lacks explicit topological connectivity. We perform a direct 3D Delaunay triangulation to create a triangular mesh approximating the real surface. This is superior to projecting points onto a plane, as it preserves spatial angles. The process connects corresponding points between adjacent scan lines \( \lambda_k \) and \( \lambda_{k+1} \), creating a series of spatial polygons which are subsequently subdivided into triangles using the max-min angle criterion to ensure mesh quality. This mesh serves as the initial discrete model of the real tooth surface of the hyperboloidal gear.
2.3 Non-Geometric-Feature Based Segmentation
Traditional segmentation relies on predefined geometric features (edges, holes), which are absent in free-form worn surfaces. Our novel segmentation algorithm identifies regions with abrupt normal vector changes—characteristic of wear scars, pits, or severe deviations—without any prior geometric assumption.
For each triangle (CenTri) in the mesh with normal \( \mathbf{n} \), we examine its three adjacent triangles (NeiTri_i) with normals \( \mathbf{n}_i \). The angles \( \delta_i \) between \( \mathbf{n} \) and each \( \mathbf{n}_i \) are computed. A smooth region exhibits small variations between these angles. We calculate a metric \( T \) for each central triangle:
$$ T = \frac{\max(|\delta_1-\delta_2|, |\delta_2-\delta_3|, |\delta_1-\delta_3|)}{\min(|\delta_1-\delta_2|, |\delta_2-\delta_3|, |\delta_1-\delta_3|)} $$
A triangle with a low \( T \) value lies in a smooth region. The algorithm starts from the triangle with the minimal \( T \) value (StaTri) and iteratively grows a region by adding adjacent triangles that maintain smooth connectivity (i.e., satisfy a low \( T \) threshold). Triangles that cannot be added to any smooth region are identified as belonging to “non-smooth” or feature zones—these are the potential wear regions on the hyperboloidal gear tooth flank. This step effectively partitions the point cloud into “normal” and “wear” zones for targeted processing.
2.4 Revised Local Interpolation Algorithm
Instead of applying a global interpolation which smooths out wear features, we focus computational effort on the segmented wear zones. A local interpolation algorithm is used to densely estimate points within these regions based on the surrounding measured data, often employing area-weighted schemes. The key innovation is integrating this with a compensation step in the NURBS construction.
Let the initial NURBS curve/surface built from estimated and measured points have an error \( \Delta E_i \) at the \( i \)-th data point. To force the final NURBS model to exactly interpolate these critical points (compensating for the construction error), we solve for adjustments to the control points \( \Delta V_i \) or their weights \( \Delta \omega_i \).
For control point adjustment (\( \Delta \omega_i = 0 \)), the system for a cubic curve is derived as:
$$
\begin{bmatrix}
b_{1,2} & b_{1,3} & 0 & \cdots & 0 \\
b_{2,2} & b_{2,3} & b_{2,4} & \ddots & \vdots \\
0 & \ddots & \ddots & \ddots & 0 \\
\vdots & \ddots & b_{n, n-1} & b_{n, n} & b_{n, n+1} \\
0 & \cdots & 0 & b_{n+1, n} & b_{n+1, n+1}
\end{bmatrix}
\begin{bmatrix}
\Delta V_2 \\
\Delta V_3 \\
\vdots \\
\Delta V_{n}
\end{bmatrix}
=
\begin{bmatrix}
a_1 \Delta E_1 \\
a_2 \Delta E_2 \\
\vdots \\
a_n \Delta E_n
\end{bmatrix}
$$
where \( a_j = \sum_i B_{i,3}(u_j) \omega_i \) and \( b_{j,i} = \sum_i B_{i,3}(u_j) \omega_i \). A similar linear system can be formulated for weight adjustments (\( \Delta V_i = 0 \)). This localized compensation ensures the digital model passes through the estimated wear zone data, preserving its geometric fidelity.
2.5 Fairing Algorithm with Point Retention
Traditional fairing removes “noise points” based on curvature, which could erroneously delete valid data points from a wear scar. Our revised algorithm identifies potential noise points but instead of deleting them, it relocates them to a position that improves local smoothness while retaining their presence in the dataset.
A point is flagged if its curvature sign differs from both its neighbors:
$$ \Delta \kappa_{i-1} \cdot \Delta \kappa_i < 0 \quad \text{and} \quad \Delta \kappa_i \cdot \Delta \kappa_{i+1} < 0, \quad \text{where} \quad \Delta \kappa_i = \kappa_i – \kappa_{i-1} $$
For a flagged control point \( V_j \) corresponding to parameter \( t_j \), a new, faired position \( V’_j \) is calculated using its neighboring control points and knot vector parameters to minimize local curvature variation:
$$ V’_j = \frac{(t_{j+2} – t_j) l_j + (t_j – t_{j-2}) h_j}{t_{j+2} – t_{j-2}} $$
$$ l_j = \frac{(t_{j-1} – t_{j-3})V_{j-1} + (t_{j+1} – t_{j-1})V_j}{t_{j+1} – t_{j-3}}, \quad h_j = \frac{(t_{j+3} – t_{j+1})V_{j+1} + (t_{j+1} – t_{j-1})V_j}{t_{j+3} – t_{j-1}} $$
This “point moving” strategy enhances the overall surface fairness of the hyperboloidal gear model without discarding the measured evidence of wear.
3. Case Study and Comparative Analysis
The methodology was validated using a pair of hyperboloidal gears. Key geometric and machine setting parameters are summarized below:
| Parameter | Pinion | Gear |
|---|---|---|
| Number of Teeth | 6 | 37 |
| Hand of Spiral | Left | Right |
| Shaft Angle | 90° | |
| Offset | 31.75 mm | |
The gears were run on a test rig under load (160 N·m, 4000 rpm) for extended periods (2000 km and 5000 km) with abrasive particles introduced into the lubricant to accelerate wear. Subsequent measurement was performed using a specialized gear measuring instrument.
The digital model was constructed using two methods for comparison: 1) The conventional area-weighting interpolation method (Method A), and 2) The proposed Non-geometric-feature Segmentation and Revised Interpolation method (Method B). The wear zone on the gear tooth flank after 5000 km of testing was selected as the critical validation region.
The accuracy was evaluated by comparing the position vector magnitude \( r = \sqrt{x^2+y^2+z^2} \) of points on the constructed digital surface with the corresponding actual CMM measurement points \( r^* \) in the wear zone. The absolute error is \( \Delta e = |r – r^*| \). A subset of data points along a critical scan line within the wear zone is presented.
| Point | Real Surface \( (x’, y’, z’) \) /mm | Method A \( (x, y, z) \) /mm | Absolute Error \( \Delta e \) /μm |
|---|---|---|---|
| 1 | 63.3807, -26.2622, 171.2455 | 63.3807, -26.2622, 171.2455 | 0.245 |
| 2 | 61.4428, -21.3597, 166.5882 | 61.4428, -21.3597, 166.5882 | 4.051 |
| 3 | 59.4737, -16.6866, 161.7131 | 59.4737, -16.6866, 161.7130 | 15.553 |
| 4 | 57.4743, -12.2529, 156.6297 | 57.4743, -12.2529, 156.6297 | 21.798 |
| 5 | 55.4455, -8.0690, 151.3496 | 55.4455, -8.0690, 151.3496 | 17.128 |
| 6 | 53.3881, -4.1443, 145.8840 | 53.3881, -4.1443, 145.8840 | 11.975 |
| Point | Real Surface \( (x’, y’, z’) \) /mm | Method B \( (x, y, z) \) /mm | Absolute Error \( \Delta e \) /μm |
|---|---|---|---|
| 1 | 63.3807, -26.2622, 171.2455 | 63.3807, -26.2622, 171.2455 | 0.224 |
| 2 | 61.4428, -21.3597, 166.5882 | 61.4428, -21.3597, 166.5882 | 0.539 |
| 3 | 59.4737, -16.6866, 161.7131 | 59.4737, -16.6866, 161.7131 | 3.146 |
| 4 | 57.4743, -12.2529, 156.6297 | 57.4743, -12.2529, 156.6297 | 3.265 |
| 5 | 55.4455, -8.0690, 151.3496 | 55.4455, -8.0690, 151.3496 | 3.176 |
| 6 | 53.3881, -4.1443, 145.8840 | 53.3881, -4.1443, 145.8840 | 2.550 |
The comparative results are striking. In the worn region, the maximum error for the conventional Method A exceeds 20 μm. In contrast, the proposed Method B maintains sub-4 μm accuracy across all validated points within the wear zone. This demonstrates a significant improvement in modeling fidelity specifically where it matters most—the areas of the hyperboloidal gear tooth surface that have undergone wear-related modification.
4. Conclusion
This paper has detailed a comprehensive and effective methodology for constructing high-fidelity digital models of real tooth surfaces for hyperboloidal gears, with explicit capability to handle wear and localized surface damage. The integration of several key innovations provides a robust solution:
1. Non-Geometric-Feature Segmentation: This technique automatically identifies wear zones on the tooth flank based on surface normal discontinuity, eliminating the need for manual region selection and enabling targeted data processing. This step drastically reduces unnecessary computation on well-behaved surface areas.
2. Revised Local Interpolation with Error Compensation: By focusing interpolation and a subsequent control point/weight adjustment step on the segmented wear zones, the method ensures the final NURBS model accurately passes through the estimated data representing the worn topography, preserving critical geometric information often lost in global fitting schemes.
3. Point-Retentive Fairing Algorithm: The modified fairing strategy improves the overall smoothness of the digital model without discarding the measured data points that characterize wear. This balances the often-conflicting demands of accuracy (retaining wear data) and smoothness (necessary for analysis).
The case study confirms that this methodology not only enhances computational efficiency by reducing the scope of intensive interpolation but, more importantly, yields a digital twin of the hyperboloidal gear with superior accuracy in critical wear regions. This accurate digital representation of the as-worn state is an essential prerequisite for conducting meaningful simulations of dynamic performance, contact stress under load, and predicting the remaining useful life of these critical power transmission components. The framework establishes a reliable foundation for the digital twin paradigm in gear system maintenance and performance optimization.