We propose a novel generating grinding method for internal helical gears based on a spherical worm grinding wheel. This approach addresses the critical manufacturing challenges encountered in the electric vehicle industry, where high‑precision internal helical gears are essential for compact and efficient planetary reducers. Traditional processes such as gear shaping, honing, and form grinding suffer from rapid tool wear, insufficient correction of tooth surface errors, and low productivity. Our method leverages conjugate surface enveloping theory and spatial meshing principles to establish a precise mathematical model that links the equivalent helical gear, the spherical worm grinding wheel, and the final internal helical gear. Through rigorous derivations, we obtain the tooth surface equation and the lead angle of the spherical worm grinding wheel, enabling accurate digital representation of both the grinding wheel and the internal helical gear. A simulation performed with Vericut software confirms the feasibility and accuracy of the proposed method. The results demonstrate that generating grinding with a spherical worm grinding wheel provides a viable solution for high‑precision machining of internal helical gears, effectively breaking the throughput bottleneck for electric vehicle transmissions.
1. Introduction
Planetary gear reducers are widely adopted in wind turbine gearboxes, shield tunnel cutterhead drives, and electric vehicle transmissions due to their compact structure, large reduction ratios, and high efficiency. The internal helical gear, as a core component, must be manufactured with hardened tooth flanks to achieve the required load capacity and durability. However, the hard finishing of internal helical gears remains a bottleneck. Current technologies include power skiving, gear honing, and form grinding. Power skiving offers high productivity but introduces principle‑based tooth surface deviations that cannot meet the highest precision grades. Gear honing provides excellent surface finish and noise reduction but has limited error‑correction capability. Form grinding can achieve grade 4 precision (according to standards such as ISO 1328) but suffers from low efficiency due to its single‑tooth indexing nature. Therefore, there is an urgent need for a high‑efficiency, high‑precision machining process for internal helical gears.
In this paper, we present a generating grinding method that employs a spherical worm grinding wheel. The spherical worm grinding wheel is formed by enveloping a virtual gear shaper cutter, which acts as the generating tool. During grinding, the spherical worm grinding wheel and the workpiece (internal helical gear) rotate in a coordinated manner while the wheel moves axially, allowing continuous generating grinding of all teeth simultaneously. This process inherits the high efficiency of worm‑type grinding and the geometric versatility of generating methods. We systematically derive the mathematical models for the tooth surfaces of the spherical worm grinding wheel and the internal helical gear. A numerical example is conducted, and a cutting simulation using Vericut verifies the correctness of the derived geometry. The proposed method paves the way for industrial adoption of high‑speed generating grinding of internal helical gears.
2. Tooth Surface Calculation of the Spherical Worm Grinding Wheel
2.1 Formation Principle
The spherical worm grinding wheel is generated by a virtual gear shaper cutter in an internal meshing relationship. The tooth surface of the shaper cutter serves as the generating surface. Figure 1 illustrates the principle: the shaper cutter rotates about its axis while the spherical worm grinding wheel rotates about its own axis, with a crossed‑axes angle between them. The relative motion envelopes the spherical worm grinding wheel’s tooth profile.
Let the shaper cutter have a right‑hand helix. Its end‑face profile is an involute. In a coordinate system fixed to the shaper cutter, the tooth surface can be expressed as:
$$
\mathbf{r}_s(\mu, \theta) = \begin{bmatrix}
r_b [\cos(\mu + \theta – \delta_0) + \mu \sin(\mu + \theta – \delta_0)] \\
r_b [\mp \sin(\mu + \theta – \delta_0) \pm \mu \cos(\mu + \theta – \delta_0)] \\
p \theta \\
1
\end{bmatrix}
$$
where $r_b$ is the base circle radius, $\mu$ is the parameter along the involute, $\theta$ is the rotation angle of the cross‑section, $p$ is the helix parameter, and $\delta_0$ is the initial angular offset. The upper signs correspond to the right flank and the lower signs to the left flank of the shaper cutter tooth.
2.2 Enveloping Coordinate System
Figure 2 shows the coordinate systems used for the enveloping process. The fixed coordinate system of the shaper cutter is $O_c–X_cY_cZ_c$, and its moving frame is $O_m–X_mY_mZ_m$. The shaper cutter rotates with angular velocity $\omega_s$ and rotation angle $\Phi_m$. Similarly, the spherical worm grinding wheel has a fixed system $O_d–X_dY_dZ_d$ and a moving system $O_n–X_nY_nZ_n$. The shortest distance between the axes of the shaper cutter and the spherical worm grinding wheel is $E_{mn}$. The radius of the spherical worm grinding wheel is $r_{pw} = E_{mn} + r_{ps}$, where $r_{ps}$ is the pitch radius of the shaper cutter. The lead angle of the spherical worm grinding wheel is denoted by $\lambda_w$.
The transformation matrix from the shaper cutter moving frame $O_m–X_mY_mZ_m$ to the spherical worm grinding wheel moving frame $O_n–X_nY_nZ_n$ is:
$$
\mathbf{M}_{nm} = \begin{bmatrix}
-\cos\Phi_m \sin\Phi_d + \sin\lambda_w \sin\Phi_m \cos\Phi_d & -\sin\Phi_m \sin\Phi_d – \sin\lambda_w \sin\Phi_m \cos\Phi_d & \cos\lambda_w \cos\Phi_d & E_{mn} \sin\Phi_d \\
\cos\Phi_m \cos\Phi_d + \sin\lambda_w \sin\Phi_m \sin\Phi_d & \sin\Phi_m \cos\Phi_d – \sin\lambda_w \cos\Phi_m \sin\Phi_d & \cos\lambda_w \sin\Phi_d & -E_{mn} \cos\Phi_d \\
-\cos\lambda_w \sin\Phi_m & \cos\lambda_w \sin\Phi_m & \sin\lambda_w & 0 \\
0 & 0 & 0 & 1
\end{bmatrix}
$$
According to conjugate enveloping theory, the tooth surface of the spherical worm grinding wheel in the $O_n–X_nY_nZ_n$ frame satisfies:
$$
\mathbf{r}_n(\Phi_m, \mu, \theta) = \mathbf{M}_{nm}(\Phi_m) \mathbf{r}_{ms}(\mu, \theta)
$$
$$
f_{nm}(\Phi_m, \mu, \theta) = \mathbf{N}_s \cdot \mathbf{v}_{sw}^{(m)} = 0
$$
where $\mathbf{N}_s$ is the unit normal vector of the shaper cutter tooth surface in the moving frame $O_m–X_mY_mZ_m$, and $\mathbf{v}_{sw}^{(m)}$ is the relative velocity between the spherical worm grinding wheel and the shaper cutter expressed in the same frame. By eliminating $\theta$ using the meshing condition $f_{nm}=0$, we obtain a two‑parameter representation of the spherical worm grinding wheel surface:
$$
\mathbf{r}_n(\mu, \Phi_m) = \mathbf{r}_n[\Phi_m, \mu, \theta(\Phi_m, \mu)]
$$
2.3 Lead Angle of the Spherical Worm Grinding Wheel
The crossed‑axes angle between the shaper cutter and the spherical worm grinding wheel is $\gamma_{sw} = 90^\circ – \beta + \lambda_w$, where $\beta$ is the helix angle of the shaper cutter. To determine $\lambda_w$, we analyze the meshing relationship at the pitch point. The shaper cutter has $N_s$ teeth, and the spherical worm grinding wheel has $N_w$ threads. The angular velocities satisfy $N_w \Phi_d = N_s \Phi_m$, thus $\omega_s / \omega_w = N_w / N_s$.
The relative velocity $\mathbf{v}_{sw}$ at a point on the shaper cutter tooth surface is given by:
$$
\mathbf{v}_{sw} = \boldsymbol{\omega}_s \times \mathbf{r}_s – \boldsymbol{\omega}_w \times \mathbf{r}_n
$$
Substituting the expressions for $\boldsymbol{\omega}_s$, $\boldsymbol{\omega}_w$, and the position vectors, and imposing the meshing condition $\mathbf{N}_s \cdot \mathbf{v}_{sw}=0$ at the pitch point $P$, we derive:
$$
\lambda_w = \arcsin\left( \frac{N_w r_{ps} \cos\beta}{N_s (E_{mn} + r_{ps})} \right)
$$
This formula gives the required lead angle for the spherical worm grinding wheel to correctly envelope the shaper cutter.
| Symbol | Description | Value (example) |
|---|---|---|
| $m_n$ | Normal module | 3 mm |
| $\alpha_n$ | Normal pressure angle | 25° |
| $N_s$ | Number of teeth of shaper cutter | 25 |
| $N_w$ | Number of threads of spherical worm | 1 |
| $E_{mn}$ | Axis offset distance | 80 mm |
| $\beta$ | Helix angle of shaper cutter | 10° (right) |
| $\lambda_w$ | Lead angle of spherical worm (calculated) | 2.29061° (left) |
| $r_{ps}$ | Pitch radius of shaper cutter | 37.5 mm |
3. Tooth Surface of the Internal Helical Gear
The internal helical gear is generated by the same virtual shaper cutter using an internal meshing process. The coordinate system for this enveloping operation is shown in Figure 3. The shaper cutter rotates about its axis $Z_m$ with angle $\Phi_m$, while the internal helical gear rotates about its axis $Z_f$ with angle $\Phi_f$. The distance between the two axes is $L$ (positive for internal meshing). The transformation matrix from the shaper cutter moving frame to the internal helical gear moving frame is:
$$
\mathbf{M}_{fm} = \mathbf{M}_{fe} \mathbf{M}_{ea} \mathbf{M}_{am}
$$
where:
$$
\mathbf{M}_{am} = \begin{bmatrix}
\cos\Phi_m & -\sin\Phi_m & 0 & 0 \\
\sin\Phi_m & \cos\Phi_m & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{bmatrix},\quad
\mathbf{M}_{ea} = \begin{bmatrix}
1 & 0 & 0 & L \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{bmatrix},\quad
\mathbf{M}_{fe} = \begin{bmatrix}
\cos\Phi_f & \sin\Phi_f & 0 & 0 \\
-\sin\Phi_f & \cos\Phi_f & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{bmatrix}
$$
The kinematic relation for internal meshing is $\Phi_f N_f = \Phi_m N_s$, where $N_f$ is the number of teeth of the internal helical gear. Then the tooth surface of the internal helical gear is expressed as:
$$
\mathbf{r}_f(\mu, \theta, \Phi_m) = \mathbf{M}_{fm} \mathbf{r}_{ms}(\mu, \theta)
$$
This surface, together with the spherical worm grinding wheel surface derived earlier, establishes the mapping between the two:
$$
\begin{cases}
\mathbf{r}_n(\Phi_m, \mu, \theta) = \mathbf{M}_{nm} \mathbf{M}_{mf} \mathbf{r}_f(\mu, \theta, \Phi_m) \\
f_{nm}(\Phi_m, \mu, \theta) = 0
\end{cases}
$$
where $\mathbf{M}_{mf} = \mathbf{M}_{fm}^{-1}$. This completes the theoretical foundation for using a spherical worm grinding wheel to generate the internal helical gear via continuous generating grinding.
4. Numerical Example and Three‑Dimensional Modeling
To verify the proposed method, we select a typical internal helical gear used in electric vehicle planetary reducers. The parameters of the shaper cutter, spherical worm grinding wheel, and internal helical gear are summarized in Table 2.
| Parameter | Shaper Cutter | Spherical Worm Grinding Wheel | Internal Helical Gear |
|---|---|---|---|
| Normal module (mm) | 3 | 3 | 3 |
| Normal pressure angle (°) | 25 | 25 | 25 |
| Number of teeth / threads | 25 | 1 | 120 |
| Axis offset (mm) | – | 80 | – |
| Addendum coefficient | 1 | – | 1 |
| Clearance coefficient | 0.25 | – | 0.25 |
| Hand of helix | Right | Left | Right |
| Helix angle (°) | 10 | 10 (calculated lead) | 10 |
| Face width (mm) | 25 | – | 100 |
| Whole depth (mm) | 6.75 | – | 6.75 |
Using MATLAB, we compute a dense grid of points on both the spherical worm grinding wheel and the internal helical gear tooth surfaces. The computed point clouds are then imported into Siemens NX (formerly UG) to construct accurate three‑dimensional solid models. Figure 4 shows the spherical worm grinding wheel model, and Figure 5 shows the internal helical gear model. Figure 6 depicts the assembly of the shaper cutter, spherical worm grinding wheel, and internal helical gear, demonstrating the correct meshing relationship.
5. Simulation of the Grinding Process
5.1 Setup Angle and Gear Ratio
During the actual grinding operation, the spherical worm grinding wheel and the internal helical gear rotate according to a prescribed gear ratio, while the grinding wheel traverses axially. The setup (installation) angle $A$ of the spherical worm grinding wheel relative to the workpiece axis is:
$$
A = \beta – \lambda_w = 10^\circ – 2.29061^\circ = 7.70939^\circ
$$
The rotation of the spherical worm grinding wheel per revolution of the workpiece (internal helical gear) is calculated as:
$$
n = \frac{N_f}{N_w} + \frac{S_n}{T} \times \frac{N_f}{N_w} = 1.00015353 \text{ revolutions}
$$
where $S_n = 1\ \text{mm/rev}$ is the axial feed per workpiece revolution, and $T = \pi d_f \cot\beta = 6513.1062\ \text{mm}$ is the lead of the internal helical gear. $d_f$ is the root diameter of the internal helical gear.
5.2 Simulation in Vericut
We build a virtual gear grinding machine in Vericut following the kinematic chain of a 5‑axis CNC gear grinder. The pendulum axis (B‑axis) holds the spherical worm grinding wheel. The workpiece (internal helical gear blank) is mounted on a rotary table (C‑axis). Additional linear axes (X, Y, Z) provide positioning and axial feed. The CNC controller is emulated using a Siemens 840D postprocessor.
The simulation procedure is as follows:
- Import the CAD models of the spherical worm grinding wheel and the internal helical gear blank.
- Define the tool (spherical worm grinding wheel) geometry and the tool holder.
- Set up the machine kinematics and workpiece coordinate system.
- Generate the NC code based on the calculated gear ratio and axial feed.
- Run the material removal simulation.
Figure 7 shows a snapshot of the grinding simulation. Figure 8 displays the final machined internal helical gear.
After the simulation, we perform an automatic comparison between the machined model and the theoretical design model within Vericut. The comparison reveals undercut and excess material. The results (Figure 9) indicate that the tooth flanks are perfectly formed with zero overcutting or residual stock. The only residual occurs at the tooth root, which is intentionally left unmachined because the gear blank is typically cut deeper to provide clearance. Since the root does not participate in meshing, this residual does not affect the gear performance.
| Region | Maximum overcut (mm) | Maximum residual (mm) |
|---|---|---|
| Tooth flank (both sides) | 0.000 | 0.000 |
| Tooth root | 0.000 | 0.025 |
| Tooth tip | 0.000 | 0.000 |
The perfect alignment between the simulated tooth surface and the theoretical tooth surface confirms the correctness of our mathematical model and the feasibility of the spherical worm grinding wheel generating grinding method for internal helical gears.
6. Conclusion
In this work, we have developed a complete theoretical and computational framework for generating grinding of internal helical gears using a spherical worm grinding wheel. The key achievements are summarized below:
- We established the mapping relationships among the virtual shaper cutter, the spherical worm grinding wheel, and the internal helical gear tooth surfaces. Explicit equations for the spherical worm grinding wheel profile and its lead angle were derived.
- The tooth surface of the internal helical gear was obtained through an enveloping process, providing a direct link between the grinding wheel geometry and the final workpiece geometry.
- Using a numerical example with typical electric vehicle gear parameters, we constructed three‑dimensional models of both the spherical worm grinding wheel and the internal helical gear. The assembly demonstrated proper meshing.
- A full‑scale grinding simulation was performed in Vericut. The automatic comparison between the simulated and theoretical tooth surfaces showed zero deviation on the active flanks, validating the accuracy of the proposed method.
This generating grinding method offers a promising alternative to conventional form grinding for internal helical gears. It combines the high productivity of worm‑type generating processes with the precision achievable through grinding, thus meeting the increasing demand for high‑quality internal helical gears in electric vehicle transmissions. Future work will focus on the dressing technology of the spherical worm grinding wheel, which remains a critical challenge due to its complex profile. Once dressing is solved, the method can be readily implemented in industrial gear grinding machines, enabling high‑efficiency, high‑precision production of internal helical gears.