In the field of gear design and manufacturing, achieving high precision in tooth surfaces is critical for optimal performance. Traditional methods, such as tooth contact analysis (TCA) simulations and rolling tests, have been widely used for evaluating straight bevel gears. However, these approaches often suffer from machining errors and practical limitations, leading to inconsistencies between design and inspection benchmarks. To address this, researchers have explored fitting actual tooth surfaces using mathematical models. Previous studies have employed techniques like error surface superposition and B-spline interpolation, but these methods often result in significant fitting errors and poor control over surface shape. In this paper, I propose a novel approach using Non-Uniform Rational B-Spline (NURBS) surfaces to fit straight bevel gear tooth surfaces. NURBS offers superior shape control through weight factors and high fitting accuracy, making it ideal for applications in aerospace, automotive, and other industries. By deriving linear equations for control points and analyzing factors like data points and boundary conditions, I demonstrate that NURBS fitting can effectively replace physical tooth surfaces, facilitating digital design and manufacturing.
The foundation of this work lies in the mathematical representation of NURBS 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 $U = [u_0, u_1, \dots, u_{n+4}]$. 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 $U = [u_0, u_1, \dots, u_{n+6}]$, the end nodes have a multiplicity of 4, meaning $u_0 = u_1 = u_2 = u_3$ and $u_{n+1} = u_{n+2} = u_{n+3} = u_{n+4}$. To express the curve in matrix form, I introduce notation such as $\Delta_i = u_{i+1} – u_i$ and $t = (u – u_i) / \Delta_i$. The matrix representation for the $i$-th segment of the curve is:
$$p_i(t) = \frac{T^3 N_i H_i}{T^3 N_i W_i}, \quad 0 \leq t \leq 1$$
where $T = [1, t, t^2, t^3]$, $N_i$ is the basis matrix, and $H_i$ and $W_i$ are related to control vertices and weights.
For interpolating a cubic NURBS curve, I use cumulative chord length parameterization to determine the node vector. Given data points $p_i$ (for $i = 0, 1, \dots, n$), the node vector is computed as:
$$u_0 = u_1 = u_2 = u_3 = 0$$
$$u_{i+3} = \frac{\sum_{k=1}^{i} |p_k p_{k+1}|}{S}, \quad \text{for } i = 1, 2, \dots, n-1$$
$$u_{n+3} = u_{n+4} = u_{n+5} = u_{n+6} = 1$$
where $S = \sum_{k=1}^{n-1} |p_k p_{k+1}|$ is the total chord length. To solve for the control vertices $d_i$ (for $i = 0, 1, \dots, n+2$), I apply boundary tangent conditions. Assuming initial weight factors $w_i = 1$ for simplicity, the interpolation equations lead to a linear system. For instance, the boundary conditions include $p_0 = d_0$, $p_n = d_{n+2}$, and tangent vectors at the endpoints:
$$p’_0(0) = \frac{3w_1}{w_0} (d_1 – d_0)$$
$$p’_{n-1}(1) = \frac{3w_{n+1}}{w_{n+2}} (d_{n+2} – d_{n+1})$$
This results in a system of equations that can be represented in matrix form:
$$\begin{bmatrix}
a_0 & b_0 & & & \\
a_1 & b_1 & c_1 & & \\
& a_2 & b_2 & c_2 & \\
& & \ddots & \ddots & \ddots \\
& & & a_{n+1} & b_{n+1} & c_{n+1} \\
& & & & b_{n+2} & c_{n+2}
\end{bmatrix}
\begin{bmatrix}
d_0 \\
d_1 \\
d_2 \\
\vdots \\
d_{n+1} \\
d_{n+2}
\end{bmatrix}
=
\begin{bmatrix}
e_0 \\
e_1 \\
e_2 \\
\vdots \\
e_{n+1} \\
e_{n+2}
\end{bmatrix}$$
where coefficients like $a_i$, $b_i$, and $c_i$ are derived from the basis functions and weights. Solving this system yields the control vertices for the curve.
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 control vertices, $w_{ij}$ are weight factors, and $N_{i,3}(u)$ and $N_{j,3}(v)$ are basis functions in the $u$ and $v$ directions, respectively. To fit a surface, I first compute control vertices for curves in one direction (e.g., $u$-direction) and then use these as data points for the perpendicular direction (e.g., $v$-direction).
As an example, I apply this method to a pair of straight bevel gears with the following 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 |
| Tip Angle (°) | 39.33 | 62.62 |
| Root Angle (°) | 27.38 | 50.668 |
For the pinion, I select data points based on surface curvature to capture the tooth shape accurately. A grid of 9 points in the tooth length direction (v-direction) and 5 points in the tooth height direction (u-direction) is used. The coordinates of these points are derived from the meshing equation of the straight bevel gear. The control vertices are computed as a 7×11 matrix, ensuring a precise fit. A similar process is applied to the gear.
Error analysis is crucial for validating the fitted surfaces. I examine the impact of data point density, node vector selection, and boundary conditions on fitting accuracy. For instance, denser data points in high-curvature regions reduce errors but increase computational cost. The boundary tangent conditions also play a significant role; if the tangent magnitudes do not match the actual curve, it can lead to oscillations and larger errors. To quantify errors, I define a NURBS surface as $[x(u,v), y(u,v), z(u,v)]$ and compare it to actual points $[X, Y, Z]$. By projecting both onto a common rotational plane, I minimize the functions:
$$\min F(1) = |x(u,v) – X|$$
$$\min F(2) = \left| \sqrt{y^2(u,v) + z^2(u,v)} – \sqrt{Y^2 + Z^2} \right|$$
Using an iterative method, I find parameters $u_0$ and $v_0$ that minimize these functions. The overall surface error is computed as:
$$L = \sqrt{[x(u,v) – X]^2 + [y(u,v) – Y]^2 + [z(u,v) – Z]^2}$$
For the straight bevel gear example, I calculate the maximum error per column for both the pinion and gear. The results show that errors are very small, on the order of micrometers, indicating that the NURBS-fitted surfaces can effectively replace the actual tooth surfaces. This approach not only enhances accuracy but also supports digital simulations like TCA, providing a foundation for advanced manufacturing and inspection techniques.
In conclusion, I have developed a method for fitting straight bevel gear tooth surfaces using bicubic NURBS, addressing key factors such as data point selection and boundary conditions. The mathematical derivations and error analyses confirm the reliability of this approach. By enabling digital representation of gear surfaces, this work paves the way for improved design and manufacturing processes in the industry.