In modern mechanical transmission systems, planetary gear trains comprising sun gears, planet gears, and internal gears are widely utilized due to their compact design and high efficiency. As an internal gear manufacturer, enhancing the manufacturing precision of internal gears is critical for improving the load-bearing capacity and overall performance of these systems. Hard-faced internal gears require precision finishing processes, and grinding stands out as the most reliable method to achieve high accuracy. Form grinding, in particular, offers significant advantages in efficiency, simplified machine structure, and cost-effectiveness, making it a preferred choice for internal gear manufacturers. However, the technical challenges associated with internal gear form grinding, especially in wheel dressing, necessitate advanced algorithmic solutions. This paper addresses these challenges by developing a mathematical model and software for wheel dressing in internal gear form grinding, with a focus on practical applications for internal gears.
The development of internal gear grinding machines has been hindered by high costs and reliance on imported equipment. For instance, grinding machines from countries like Germany and Italy are prohibitively expensive, limiting accessibility for many internal gear manufacturers. Therefore, there is a pressing need to develop cost-effective, locally produced grinding solutions. The key to this development lies in the numerical control (NC) software for dressing form-grinding wheels. This research focuses on the wheel dressing algorithm, which enables precise wheel profiling for internal gears. By establishing a mathematical model and implementing it in software, we aim to provide a viable alternative for internal gear manufacturers.
Mathematical Modeling of Internal Gear Form Grinding
The tooth surface of an internal gear is characterized by an involute helical surface, with the end profile being an involute curve. For internal gears, the base circle radius $r_b$ is defined by the gear parameters. Let us consider the right-side involute of the tooth slot, starting at point $e$ with an initial angle $\delta_0$ relative to the x-axis. $\delta_0$ represents the half-angle of the tooth slot on the base circle. Using the parameter $u = \angle e o a$, which denotes the angle from the start point to any point $M(x_M, y_M)$ on the involute, the coordinates can be derived based on the properties of the involute. The equations are as follows:
$$ x_M = r_b \cos u + r_b u \sin u $$
$$ y_M = r_b \sin u – r_b u \cos u $$
where $r_b = m z \cos \alpha$, $m$ is the normal module, $z$ is the number of teeth, and $\alpha$ is the pressure angle. For internal gears, the coordinate transformation involves rotating the involute curve. The transformed coordinates in the gear coordinate system are given by:
$$ \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} \cos \delta_0 & \sin \delta_0 \\ -\sin \delta_0 & \cos \delta_0 \end{bmatrix} \begin{bmatrix} x_M \\ y_M \end{bmatrix} $$
This results in:
$$ \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} r_b \cos(u – \delta_0) + r_b u \sin(u – \delta_0) \\ r_b \sin(u – \delta_0) – r_b u \cos(u – \delta_0) \end{bmatrix} $$
Here, $\delta_0 = \frac{\pi}{2z} + (\tan \alpha – \alpha) – \frac{2x \tan \alpha}{z} + \frac{\Delta L}{r_b}$, where $x$ is the profile shift coefficient and $\Delta L$ is the tooth thickness reduction. This formulation is essential for internal gear manufacturers to accurately model tooth geometry.
To compute the wheel profile for form grinding, the gear tooth surface, represented as a continuous set of points, is transformed into the wheel coordinate system. The numerical solution involves discretizing the parameter $u$ over the interval $u_{\text{min}} \leq u \leq u_{\text{max}}$, based on equal error principles. The workpiece coordinate system $O-xyz$ and the wheel coordinate system $O_0-x_0y_0z_0$ are related by a transformation matrix. The transformation from the workpiece to the wheel coordinates is given by:
$$ \mathbf{M}_{O_0O} = \begin{bmatrix} 1 & 0 & -a \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} $$
where $a$ is the center distance. The tooth surface equation in the workpiece system is transformed to the wheel system as:
$$ \mathbf{r}_0 = \mathbf{M}_{O_0O} \mathbf{r} $$
Substituting the transformation matrix, the wheel coordinates become:
$$ \begin{bmatrix} x_0 \\ y_0 \\ 1 \end{bmatrix} = \begin{bmatrix} 1 & 0 & -a \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ 1 \end{bmatrix} $$
This yields the wheel profile equation:
$$ \mathbf{r}_0(u) = \begin{bmatrix} r_b \cos(u – \delta_0) + r_b u \sin(u – \delta_0) – a \\ r_b \sin(u – \delta_0) – r_b u \cos(u – \delta_0) \end{bmatrix} $$
This mathematical model provides a foundation for wheel dressing in internal gear form grinding, enabling internal gear manufacturers to achieve precise wheel profiles.
Wheel Dressing Algorithm Based on Equal-Distance Curves
The dressing of the grinding wheel using a diamond roller requires calculating the dressing trajectory as an equal-distance curve to the wheel profile. For internal gears, the dressing trajectory must account for the wheel’s profile and the diamond roller’s geometry. The equal-distance curve vector is expressed as:
$$ \mathbf{r}_R = \mathbf{r}_0 – R \boldsymbol{\beta} $$
where $R$ is the corner radius of the diamond roller, and $\boldsymbol{\beta}$ is the unit normal vector. The unit normal vector is derived from the unit tangent vector $\mathbf{T}$ and the z-axis unit vector $\mathbf{K}$:
$$ \boldsymbol{\beta} = \mathbf{T} \times \mathbf{K} $$
The unit tangent vector $\mathbf{T}$ is given by:
$$ \mathbf{T} = \frac{ [dx, dy, 0] }{ \sqrt{dx^2 + dy^2} } $$
and $\mathbf{K} = [0, 0, 1]$. Substituting these into the cross product:
$$ \boldsymbol{\beta} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \frac{dx}{\sqrt{dx^2 + dy^2}} & \frac{dy}{\sqrt{dx^2 + dy^2}} & 0 \\ 0 & 0 & 1 \end{vmatrix} $$
This results in the components of $\boldsymbol{\beta}$, which are used to compute the equal-distance curve coordinates. The planar coordinates of the equal-distance curve are then obtained, ensuring accurate dressing paths for internal gears. This algorithm is crucial for internal gear manufacturers to maintain wheel accuracy during dressing.
Software Development for Internal Gear Form Grinding
Based on the mathematical model and dressing algorithm, we developed a software tool using VC++ 6.0 to facilitate wheel dressing for internal gear form grinding. The software workflow, as illustrated in the flowchart, involves inputting gear parameters, computing key dimensions, and generating dressing trajectories. The process begins with the input of basic gear parameters, such as tooth number, module, pressure angle, addendum coefficient, profile shift coefficient, and base tangent length reduction. The software then calculates the base circle radius, root circle radius, addendum circle radius, initial involute angle, final involute angle, and tooth slot half-angle.
Using the involute angle, the tooth surface points are computed and transformed into the wheel coordinate system to determine the dressing trajectory points. The software saves these points and iteratively calculates subsequent points based on the normal allowable error until the involute angle reaches the addendum circle value. The output includes a text file with coordinate data, gear parameter storage, and an animation simulating the dressing process. This software enables internal gear manufacturers to generate NC programs for wheel dressing efficiently.
The table below summarizes the input parameters and computed values for a typical internal gear:
| Parameter | Symbol | Value |
|---|---|---|
| Number of Teeth | $z$ | 90 |
| Normal Module | $m$ | 4 mm |
| Pressure Angle | $\alpha$ | 20° |
| Addendum Coefficient | $h_a^*$ | 1.0 |
| Profile Shift Coefficient | $x$ | 0.2 |
| Base Circle Radius | $r_b$ | 169.65 mm |
| Tooth Slot Half-Angle | $\delta_0$ | 0.0349 rad |
The software interface allows users to input parameters, compute dressing trajectories, and visualize the process. For internal gear manufacturers, this tool simplifies the generation of NC code, reducing reliance on expensive imported systems.
Experimental Validation and Results
To validate the theoretical model and software, we conducted a form grinding experiment on an internal gear with module 4, 90 teeth, and a pressure angle of 20°. The software generated the wheel dressing and grinding NC programs, which were executed on a grinding machine. The experimental setup involved aligning the wheel and workpiece centers to ensure accuracy. The grinding process was monitored, and the resulting gear was inspected for deviations.
The inspection report indicated significant total profile deviation, primarily due to misalignment between the wheel and workpiece centers, which affected the pressure angle consistency on both tooth flanks. However, the profile form deviation was minimal, demonstrating the accuracy of the wheel dressing algorithm. The table below summarizes the inspection results:
| Deviation Type | Value (μm) |
|---|---|
| Total Profile Deviation | 25 |
| Profile Form Deviation | 5 |
| Profile Slope Deviation | 20 |
These results highlight that while the wheel dressing software produces precise profiles, operational factors like alignment can impact overall accuracy. For internal gear manufacturers, this underscores the importance of integrating accurate dressing algorithms with robust machine setup.
Conclusion
In this study, we developed a comprehensive mathematical model and software solution for wheel dressing in internal gear form grinding. The model derives the wheel profile and dressing trajectory based on involute geometry and coordinate transformations. The software, implemented in VC++ 6.0, generates NC programs for dressing and simulates the process, providing a practical tool for internal gear manufacturers. Experimental validation confirmed the algorithm’s correctness and software feasibility, though alignment issues highlighted areas for improvement. Future work will focus on optimizing alignment techniques and extending the software to handle varied internal gear types. This research contributes to the advancement of internal gear manufacturing by offering a cost-effective alternative to imported grinding systems.
The integration of advanced algorithms into grinding processes is essential for internal gear manufacturers to achieve high precision and efficiency. By leveraging mathematical modeling and software development, we can overcome the technical barriers associated with internal gear form grinding. The continued refinement of these tools will empower internal gear manufacturers to produce high-quality gears competitively in the global market.