With the rapid expansion of the electric vehicle industry, the demand for high-precision internal helical gears has surged. These components are critical in planetary gear reducers due to their compact structure, high reduction ratios, and efficient power transmission. However, traditional machining techniques such as power skiving, honing, and form grinding face significant challenges, including excessive tool wear, inadequate correction of tooth surface errors, and low processing efficiency. These limitations hinder their ability to meet the stringent requirements for short production cycles and high-performance transmissions in electric vehicles. To address these issues, I propose a new generating grinding method for internal helical gears based on a spherical worm grinding wheel. This approach leverages conjugate surface envelopment theory and spatial meshing principles to achieve high-precision machining. In this article, I will systematically outline the mathematical foundations, computational models, and simulation validations for this method, emphasizing the role of helical gears in modern transmission systems.
The core of this method lies in the precise digital characterization of the tooth surfaces involved. I begin by establishing the mapping relationships among the equivalent helical gear tooth surface, the spherical worm grinding wheel tooth surface, and the internal helical gear tooth surface. The spherical worm grinding wheel’s profile is derived through conjugate envelopment theory, where a virtual gear shaper cutter serves as the generating tool. The mathematical formulation involves solving complex equations to define the wheel’s profile and spiral angle, ensuring accurate meshing with the helical gears. The derivation process is rigorous, incorporating coordinate transformations and relative velocity analyses to model the interactions between the components. This foundation enables the efficient grinding of internal helical gears with minimal errors, addressing the bottlenecks of traditional methods.
To compute the spherical worm grinding wheel’s profile, I start with the formation principle based on conjugate surface envelopment. The virtual gear shaper cutter’s end-face profile is represented mathematically, considering its involute geometry. For a point on the involute curve, the position vector in the cutter’s coordinate system can be expressed as follows, where the parameters account for the helical nature of the gears:
$$r_s(\mu, \theta) = \begin{bmatrix} x \\ y \\ z \\ 1 \end{bmatrix} = \begin{bmatrix} r_b [ \cos(\mu + \theta – \delta_0) + \mu \sin(\mu + \theta – \delta_0) ] \\ r_b [ \mp \sin(\mu + \theta – \delta_0) \pm \mu \cos(\mu + \theta – \delta_0) ] \\ p \theta \\ 1 \end{bmatrix}$$
Here, $r_b$ is the base circle radius, $\mu$ is the involute parameter, $\theta$ is the rotation angle, $\delta_0$ is the initial angle, and $p$ is the spiral parameter related to the lead of the helix. The signs $\mp$ and $\pm$ correspond to the left and right flanks of the helical gears, respectively. This equation captures the essence of the gear tooth geometry, which is fundamental for subsequent envelopment calculations.
Next, I define the coordinate systems for the envelopment process between the gear shaper cutter and the spherical worm grinding wheel. The transformation matrix from the cutter’s moving coordinate system to the grinding wheel’s moving coordinate system is derived as:
$$M_{nm} = \begin{bmatrix} -\cos \Phi_m \sin \Phi_d + \sin \lambda_w \sin \Phi_m \cos \Phi_d & -\sin \Phi_m \sin \Phi_d – \sin \lambda_w \sin \Phi_m \cos \Phi_d & \cos \lambda_w \cos \Phi_d & E_{mn} \sin \Phi_d \\ \cos \Phi_m \cos \Phi_d + \sin \lambda_w \sin \Phi_m \sin \Phi_d & \sin \Phi_m \cos \Phi_d – \sin \lambda_w \cos \Phi_m \sin \Phi_d & \cos \lambda_w \sin \Phi_d & -E_{mn} \cos \Phi_d \\ -\cos \lambda_w \sin \Phi_m & \cos \lambda_w \sin \Phi_m & \sin \lambda_w & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}$$
In this matrix, $\Phi_m$ and $\Phi_d$ are the rotation angles of the cutter and grinding wheel, respectively, $E_{mn}$ is the shortest distance between their axes, and $\lambda_w$ is the spiral angle of the grinding wheel. The spherical worm grinding wheel’s tooth surface in its coordinate system is then given by:
$$r_n(\Phi_m, \mu, \theta) = M_{nm}(\Phi_m) r_{ms}(\mu, \theta)$$
with the meshing equation defined as $f_{nm}(\Phi_m, \mu, \theta) = \mathbf{N}_{sm} \cdot \mathbf{v}_{swm} = 0$, where $\mathbf{N}_{sm}$ is the unit normal vector of the cutter tooth surface, and $\mathbf{v}_{swm}$ is the relative velocity between the cutter and grinding wheel. By solving this meshing equation, I eliminate $\theta$ and express the grinding wheel’s surface as a function of $\mu$ and $\Phi_m$:
$$r_n(\mu, \Phi_m) = r_n[\Phi_m, \mu, \theta(\Phi_m, \mu)]$$
The spiral angle $\lambda_w$ of the grinding wheel is critical for proper meshing with the helical gears. It is derived from the relative motion analysis at the meshing point. For a right-handed cutter and left-handed grinding wheel, the angle between their axes is $\gamma_{sw} = 90^\circ – \beta + \lambda_w$, where $\beta$ is the helix angle of the cutter. The meshing condition requires that the normal vector and relative velocity are perpendicular, leading to the following equation for $\lambda_w$:
$$\lambda_w = \arcsin\left( \frac{N_w r_{ps} \cos \beta}{N_s (E_{mn} + r_{ps})} \right)$$
Here, $N_w$ and $N_s$ are the number of threads of the grinding wheel and teeth of the cutter, respectively, and $r_{ps}$ is the cutter’s radius. This calculation ensures that the grinding wheel and cutter maintain continuous contact during the envelopment process, which is essential for accurate generation of helical gears.
Moving to the internal helical gear tooth surface calculation, I model it as being generated by the virtual gear shaper cutter through a gear shaping process. The coordinate systems for this envelopment are established, and the transformation matrix from the cutter to the gear coordinate system is constructed. The relationship between the rotation angles of the cutter and gear is given by $\Phi_f N_f = \Phi_m N_s$, where $N_f$ is the number of teeth of the internal helical gear. The tooth surface of the gear in its coordinate system is then:
$$r_f(\mu, \theta, \Phi_m) = M_{fm} r_{ms}(\mu, \theta)$$
where $M_{fm}$ is the composite transformation matrix. This equation allows me to derive the numerical mapping between the spherical worm grinding wheel and the internal helical gear, integrating the previous results:
$$r_n(\Phi_m, \mu, \theta) = M_{nm} M_{mf} r_f(\mu, \theta, \Phi_m)$$
with the meshing equation remaining $f_{nm}(\Phi_m, \mu, \theta) = 0$. This comprehensive model links all components, facilitating the simulation and analysis of the grinding process for helical gears.
To illustrate the application of these theoretical developments, I present a numerical example with key parameters for the gear shaper cutter, spherical worm grinding wheel, and internal helical gear. The table below summarizes these parameters, which are typical for automotive applications involving helical gears:
| Parameter | Gear Shaper Cutter | Spherical Worm Grinding Wheel | Internal Helical Gear |
|---|---|---|---|
| Module (mm) | 3 | 3 | 3 |
| Pressure Angle (°) | 25 | 25 | 25 |
| Number of Teeth/Threads | 25 | 1 | 120 |
| Axis Distance (mm) | – | 80 | – |
| Helix Angle (°) | 10 | 10 | 10 |
| Handedness | Right | Left | Right |
| Face Width (mm) | 25 | – | 100 |
| Whole Depth (mm) | 6.75 | – | 6.75 |
Using these parameters, I compute the spherical worm grinding wheel’s tooth surface points and construct its 3D model through numerical simulations. The internal helical gear tooth surface is similarly derived, and the meshing relationship among the cutter, grinding wheel, and gear is visualized. The grinding wheel’s spiral angle is calculated as $\lambda_w = \arcsin\left( \frac{1 \times 3 \times \cos 10^\circ}{25 \times (80 + 3)} \right) \approx 2.29061^\circ$, which ensures proper alignment during machining.
For the grinding process, the installation angle $A$ between the grinding wheel and gear axes is determined by $A = \beta – \lambda_w = 10^\circ – 2.29061^\circ = 7.70939^\circ$. The speed ratio between the gear and grinding wheel is critical for synchronized motion. When the gear rotates one revolution, the grinding wheel’s rotation $n$ is given by:
$$n = \frac{N_f}{N_w} + \frac{S_n}{T} \times \frac{N_f}{N_w}$$
where $S_n = 1$ mm/rev is the axial feed per revolution of the gear, and $T = \pi d_f \cot \beta$ is the lead of the helical gear, with $d_f$ being the root diameter. Substituting the values, $T = \pi \times 360 \times \cot 10^\circ \approx 6513.1062$ mm, and $n = \frac{120}{1} + \frac{1}{6513.1062} \times \frac{120}{1} \approx 120.0185$ revolutions. This ratio ensures that the grinding wheel follows the correct path along the gear teeth, maintaining continuous engagement with the helical gears.
I validate the proposed method through simulation using Vericut software, which models the grinding machine’s kinematics and control system. The machine tool structure includes components such as the workpiece spindle, grinding wheel spindle, and axial feed axes, configured with a Siemens 840D CNC system. The simulation process involves importing the 3D models of the grinding wheel and gear blank, setting up the motion chains, and running the NC program to simulate the grinding process. The results show that the ground gear tooth surface closely matches the theoretical design, with deviations analyzed using the software’s automatic comparison feature.
The simulation results indicate that the tooth flanks of the internal helical gears are accurately generated, with no significant overcut or undercut errors on the working surfaces. However, minor residuals are observed at the tooth root, which is non-functional and does not affect the meshing performance of the helical gears. The table below quantifies the deviation analysis between the simulated and theoretical tooth surfaces, highlighting the precision achievable with this method:
| Tooth Region | Deviation Type | Magnitude (mm) |
|---|---|---|
| Left Flank | Overcut | 0.000 |
| Right Flank | Overcut | 0.000 |
| Tooth Root | Residual | 0.002 |
This analysis confirms that the generating grinding method with a spherical worm grinding wheel can produce high-quality internal helical gears, meeting the demands for accuracy and efficiency in electric vehicle transmissions. The mathematical models and simulation approaches provide a robust framework for further optimization and industrial application.
In conclusion, I have developed a comprehensive generating grinding method for internal helical gears using a spherical worm grinding wheel. The key contributions include the derivation of precise tooth surface equations, the calculation of critical parameters such as the spiral angle, and the validation through advanced simulations. This method addresses the limitations of traditional techniques by offering improved accuracy, efficiency, and adaptability to the needs of modern helical gears in high-performance applications. Future work will focus on enhancing the grinding wheel dressing techniques to overcome challenges related to profile complexity, further advancing the manufacturing capabilities for helical gears in the electric vehicle industry.