In modern gear design and manufacturing, achieving high precision in tooth surface geometry is critical for ensuring optimal performance, especially in applications such as aerospace, automotive, and marine industries. Straight bevel gears are widely used due to their ability to transmit motion between intersecting shafts, but traditional methods for evaluating their meshing quality, such as rolling tests, often involve significant errors and cumbersome adjustments. To address these challenges, this paper explores the use of Non-Uniform Rational B-Spline (NURBS) surfaces for fitting straight bevel gear tooth surfaces. NURBS offers superior control over surface shape through weight factors and is recognized as the standard mathematical representation for free-form curves and surfaces by the International Organization for Standardization (ISO). By fitting a NURBS surface to the actual tooth surface, we can create a digital twin that facilitates accurate simulation and analysis, thereby reducing reliance on physical prototypes and improving manufacturing efficiency. This approach not only enhances the precision of straight bevel gear design but also paves the way for digital inspection and manufacturing processes.
The foundation of NURBS-based fitting lies in understanding the mathematical representation of curves and surfaces. A cubic NURBS curve is defined as follows:
$$ p(u) = \frac{\sum_{i=0}^{n} w_i N_{i,3}(u) d_i}{\sum_{i=0}^{n} w_i N_{i,3}(u)} $$
where \( w_i \) (for \( 0 \leq i \leq n \)) represents the weight factors, \( d_i \) (for \( 0 \leq i \leq n \)) denotes the control vertices, and \( N_{i,3}(u) \) are the B-spline basis functions determined by the node vector \( U = [u_0, u_1, \ldots, u_{n+4}] \). The basis functions satisfy the recurrence relation:
$$ N_{i,0}(u) = \begin{cases} 1 & \text{if } u_i \leq u \leq u_{i+1} \\ 0 & \text{otherwise} \end{cases} $$
$$ N_{i,3}(u) = \frac{u – u_i}{u_{i+3} – u_i} N_{i,2}(u) + \frac{u_{i+3} – u}{u_{i+3} – u_{i+1}} N_{i+1,2}(u) $$
For the node vector \( U \), the endpoints have a multiplicity of 4, meaning \( u_0 = u_1 = u_2 = u_3 \) and \( u_{n+1} = u_{n+2} = u_{n+3} = u_{n+4} \). The matrix representation of a cubic NURBS curve segment is given by:
$$ p_i(t) = \frac{T^3 N_i H_i}{T^3 N_i W_i}, \quad 0 \leq t \leq 1 $$
where \( t = \frac{u – u_i}{\Delta_i} \), \( \Delta_i = u_{i+1} – u_i \), and \( T = [1, t, t^2, t^3] \). The matrices \( N_i \), \( H_i \), and \( W_i \) are derived from the basis functions and control points.
To interpolate a cubic NURBS curve through given data points, we employ the cumulative chord length parameterization to compute the node vector. For data points \( p_i \) (where \( i = 0, 1, \ldots, n \)), the node vector is defined as:
$$ U = \left[ u_0 = u_1 = u_2 = u_3 = 0, \quad u_{i+3} = \frac{\sum_{j=1}^{i} |p_j – p_{j-1}|}{\sum_{j=1}^{n} |p_j – p_{j-1}|}, \quad u_{n+3} = u_{n+4} = u_{n+5} = u_{n+6} = 1 \right] $$
This method ensures that the parameterization reflects the geometry of the data points. Next, we determine the control vertices \( d_i \) by solving a linear system derived from boundary tangent conditions. Given \( n+1 \) data points and \( n+3 \) weight factors (initially set to 1 for simplicity), the interpolation conditions yield:
$$ a_i d_i + b_i d_{i+1} + c_i d_{i+2} = (a_i + b_i + c_i) p_i \quad \text{for } i = 0, 1, \ldots, n $$
where \( a_i, b_i, c_i \) are coefficients based on the basis functions. The boundary conditions involve the tangent vectors at the endpoints:
$$ p’_0(0) = \frac{3w_1}{w_0} (d_1 – d_0), \quad p’_{n-1}(1) = \frac{3w_{n+1}}{w_{n+2}} (d_{n+2} – d_{n+1}) $$
This leads to a linear system that can be solved for the control vertices. Extending this to surfaces, a bicubic NURBS surface is defined as:
$$ P(u,v) = \frac{\sum_{i=0}^{n} \sum_{j=0}^{m} N_{i,3}(u) N_{j,3}(v) w_{ij} D_{ij}}{\sum_{i=0}^{n} \sum_{j=0}^{m} N_{i,3}(u) N_{j,3}(v) w_{ij}} $$
where \( D_{ij} \) are the control vertices, \( w_{ij} \) are the weight factors, and \( N_{i,3}(u) \) and \( N_{j,3}(v) \) are the basis functions in the u and v directions, respectively. The fitting process involves computing control vertices in one direction (e.g., u-direction) and then using them as data points for the other direction (v-direction).
To demonstrate the application of NURBS fitting for straight bevel gears, consider a pair of straight bevel gears with the following geometric parameters:
| Parameter | Pinion | Gear |
|---|---|---|
| Number of Teeth | 10 | 16 |
| Module (mm) | 7.65 | 7.65 |
| Pressure Angle (°) | 22.5 | 22.5 |
| Shaft Angle (°) | 90 | 90 |
| Addendum (mm) | 7.84 | 4.4 |
| Dedendum (mm) | 7.84 | 9.28 |
| Face Width (mm) | 20.4 | 20.4 |
| Outer Cone Distance (mm) | 72.17 | 72.17 |
| Face Angle (°) | 39.33 | 62.62 |
| Root Angle (°) | 27.38 | 50.668 |
The selection of data points is crucial for accurate fitting. For the straight bevel gear tooth surface, a grid is defined with 9 points in the tooth length direction (v-direction) and 5 points in the tooth height direction (u-direction). The data points are computed based on the meshing equation of straight bevel gears, ensuring that points are denser in regions of high curvature to capture the surface geometry accurately. The control vertices for the pinion are derived as a 7×11 matrix, and the fitted surface is constructed accordingly. Similarly, the gear surface is fitted using the same methodology. The visualization of the fitted straight bevel gear surface can be represented as follows:
Error analysis is essential to validate the fitted NURBS surface. The accuracy depends on factors such as the density of data points, the node vector, and boundary conditions. Increasing the number of data points generally improves accuracy but may slow down computation. The boundary tangent conditions influence the control points at the edges, and deviations from the actual tangent vectors can cause oscillations in the fitted surface. To quantify the error, we project the NURBS surface \( [x(u,v), y(u,v), z(u,v)] \) and the actual surface points \( [X, Y, Z] \) onto a common rotational plane. The error minimization problem is formulated as:
$$ \min F(1) = |x(u,v) – X| $$
$$ \min F(2) = \left| \sqrt{y^2(u,v) + z^2(u,v)} – \sqrt{Y^2 + Z^2} \right| $$
Using an iterative method, such as the FORTRAN software, we compute parameters \( u_0 \) and \( v_0 \) that minimize \( F(1) \) and \( F(2) \). The overall surface error is then evaluated as:
$$ L = \sqrt{ [x(u,v) – X]^2 + [y(u,v) – Y]^2 + [z(u,v) – Z]^2 } $$
For the straight bevel gears in this study, the maximum fitting error across different sections is calculated. The results show that the error is minimal, with values not exceeding 1 micrometer, indicating that the NURBS surface accurately represents the actual straight bevel gear tooth surface. This low error validates the use of digital surfaces for applications like tooth contact analysis (TCA) and digital rolling tests.
In conclusion, the use of NURBS surfaces for fitting straight bevel gear tooth surfaces offers a robust and precise method for digital design and manufacturing. The mathematical framework of cubic NURBS curves and bicubic NURBS surfaces enables effective control over surface shape, while error analysis confirms the high accuracy of the fitted surfaces. This approach not only facilitates digital simulations and inspections but also supports advancements in CNC machining of straight bevel gears by providing a computable representation of the tooth geometry. Future work could focus on optimizing the weight factors and node vectors for even higher precision and exploring real-time applications in gear quality control.