With the rapid growth of the electric vehicle industry, the demand for high-precision internal helical gears has surged. These gears are critical components in planetary gear reducers, which are widely used in wind turbine gearboxes, shield machine main drives, and electric vehicle reduction systems due to their compact structure, high reduction ratios, and efficient power transmission. However, traditional hard gear surface machining techniques for internal helical gears, such as power skiving, honing, and form grinding, face significant challenges, including severe tool wear, inadequate correction of tooth surface errors, and low processing efficiency. These limitations hinder their ability to meet the short production cycles and high-performance requirements of modern electric vehicles. To address these issues, this paper proposes a novel generating grinding method for internal helical gears using a spherical worm grinding wheel. This approach leverages conjugate surface envelopment theory and spatial meshing principles to achieve high-precision machining, offering a potential breakthrough in manufacturing core transmission components.
The internal helical gear is a complex component that requires precise tooth geometry to ensure smooth meshing and high load-bearing capacity. Traditional methods often fall short in terms of efficiency and accuracy. For instance, power skiving involves continuous rotation of the workpiece and tool, enabling high-efficiency cutting but introducing inherent tooth surface errors that compromise topology and precision. Honing can produce unique surface textures that reduce vibration and noise, but it has limited ability to correct tooth profile deviations. Form grinding, while capable of achieving high accuracy (e.g., up to grade 4), suffers from low efficiency due to its single-tooth indexing process, making it unsuitable for mass production. In contrast, the proposed generating grinding method with a spherical worm grinding wheel aims to combine high precision with efficiency by simulating the meshing process between a virtual gear shaper cutter and the internal helical gear. This method facilitates continuous grinding along the tooth slot, reducing processing time while maintaining superior surface quality.
The spherical worm grinding wheel is central to this method, as its profile is derived from the enveloping process of a virtual gear shaper cutter. Based on conjugate surface theory, the worm grinding wheel’s tooth surface is generated by solving the meshing equation between the cutter and the worm. The virtual gear shaper cutter has an end-face profile defined by an involute curve, which can be mathematically represented. For a point on the involute, the position vector in the cutter’s coordinate system is given by the following equation, where parameters describe the geometry and motion. The tooth surface equation of the gear shaper cutter is expressed as:
$$ r_s(\mu, \theta) = \begin{bmatrix} x \\ y \\ z \\ 1 \end{bmatrix} = \begin{bmatrix} r_b \left[ \cos(\mu + \theta – \delta_0) + \mu \sin(\mu + \theta – \delta_0) \right] \\ r_b \left[ \mp \sin(\mu + \theta – \delta_0) \pm \mu \cos(\mu + \theta – \delta_0) \right] \\ p \theta \\ 1 \end{bmatrix} $$
In this equation, \( r_b \) is the base circle radius, \( \mu \) is the involute parameter, \( \theta \) is the rotation angle around the Z-axis, \( \delta_0 \) is the angle between the X-axis and the direction of \( Oe \), and \( p \) is the helix parameter, defined as \( p = p_z / 2\pi \), with \( p_z \) being the lead of the helix. The symbols \( \mp \) and \( \pm \) correspond to the right and left tooth flanks of the cutter, respectively. This equation captures the spatial geometry of the helical gear tooth surface, which is essential for accurate grinding.
To derive the spherical worm grinding wheel’s tooth surface, the coordinate systems for the enveloping process must be established. The transformation between the gear shaper cutter and the worm grinding wheel involves multiple coordinate frames, including static and moving systems for both components. The homogeneous transformation matrix from the cutter’s moving coordinate system \( O_m – X_m Y_m Z_m \) to the worm grinding wheel’s moving coordinate system \( O_n – X_n Y_n Z_n \) is given by:
$$ M_{nm} = \begin{bmatrix} -\cos \Phi_m \sin \Phi_d + \sin \lambda_w \sin \Phi_m \cos \Phi_d & -\sin \Phi_m \sin \Phi_d – \sin \lambda_w \sin \Phi_m \cos \Phi_d & \cos \lambda_w \cos \Phi_d & E_{mn} \sin \Phi_d \\ \cos \Phi_m \cos \Phi_d + \sin \lambda_w \sin \Phi_m \sin \Phi_d & \sin \Phi_m \cos \Phi_d – \sin \lambda_w \cos \Phi_m \sin \Phi_d & \cos \lambda_w \sin \Phi_d & -E_{mn} \cos \Phi_d \\ -\cos \lambda_w \sin \Phi_m & \cos \lambda_w \sin \Phi_m & \sin \lambda_w & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} $$
Here, \( \Phi_m \) and \( \Phi_d \) are the rotation angles of the cutter and worm grinding wheel, respectively, \( E_{mn} \) is the shortest distance between their axes, and \( \lambda_w \) is the lead angle of the worm grinding wheel. The worm grinding wheel’s tooth surface in coordinate system \( O_n – X_n Y_n Z_n \) is then described by the vector equation \( r_n(\Phi_m, \mu, \theta) = M_{nm}(\Phi_m) r_{ms}(\mu, \theta) \), subject to the meshing equation \( f_{nm}(\Phi_m, \mu, \theta) = N_{sm} \cdot v_{swm} = 0 \), where \( N_{sm} \) is the unit normal vector of the cutter tooth surface in the moving system, and \( v_{swm} \) is the relative velocity between the worm and cutter. By solving the meshing equation, the parameter \( \theta \) can be eliminated, resulting in a two-parameter expression for the worm grinding wheel’s tooth surface: \( r_n(\mu, \Phi_m) = r_n[\Phi_m, \mu, \theta(\Phi_m, \mu)] \).
The spiral lead angle of the spherical worm grinding wheel is crucial for proper meshing with the virtual gear shaper cutter. For a right-handed cutter and a left-handed worm grinding wheel, the angle between their axes is given by \( \gamma_{sw} = 90^\circ – \beta + \lambda_w \), where \( \beta \) is the helix angle of the cutter. The relative motion at the meshing point must satisfy the condition \( N_s \cdot v_{sw} = 0 \), with \( v_{sw} = v_s – v_w \), where \( v_s \) and \( v_w \) are the velocities of the meshing point on the cutter and worm, respectively. By analyzing the kinematics, the lead angle \( \lambda_w \) can be derived as:
$$ \lambda_w = \arcsin \left( \frac{N_w r_{ps} \cos \beta}{N_s (E_{mn} + r_{ps})} \right) $$
In this formula, \( N_w \) is the number of starts of the worm grinding wheel, \( N_s \) is the number of teeth of the gear shaper cutter, \( r_{ps} \) is the radius of the cutter, and \( E_{mn} \) is the center distance. This equation ensures that the worm grinding wheel and cutter maintain correct contact during the grinding process, which is vital for achieving high-quality tooth surfaces on the internal helical gear.
The tooth surface of the internal helical gear is generated through the enveloping process of the gear shaper cutter. The coordinate systems for this process include static and moving frames for both the cutter and the gear. The transformation matrix from the cutter’s moving system \( O_m – X_m Y_m Z_m \) to the gear’s moving system \( O_f – X_f Y_f Z_f \) is constructed as \( M_{fm} = M_{fe} M_{ea} M_{am} \), where \( M_{am} \), \( M_{ea} \), and \( M_{fe} \) are transformation matrices accounting for rotations and translations. 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 internal helical gear in coordinate system \( O_f – X_f Y_f Z_f \) is then expressed as \( r_f(\mu, \theta, \Phi_m) = M_{fm} r_{ms}(\mu, \theta) \). This equation maps the cutter’s tooth surface to the gear’s tooth surface, enabling the numerical simulation of the grinding process.
The mathematical models for the spherical worm grinding wheel and internal helical gear tooth surfaces are interconnected through coordinate transformations and meshing equations. Specifically, the relationship is given by \( r_n(\Phi_m, \mu, \theta) = M_{nm} M_{mf} r_f(\mu, \theta, \Phi_m) \) and \( f_{nm}(\Phi_m, \mu, \theta) = N_{sm} \cdot v_{swm} = 0 \), where \( M_{mf} = M_{fm}^{-1} \). This mapping allows for the precise calculation of discrete points on the tooth surfaces, facilitating digital modeling and simulation. To validate the method, a numerical example is provided with specific parameters for the gear shaper cutter, spherical worm grinding wheel, and internal helical gear, as summarized in the table below.
| Parameter | Gear Shaper Cutter | Spherical Worm Grinding Wheel | Internal Helical Gear |
|---|---|---|---|
| Module (mm) | 3 | 3 | 3 |
| Pressure Angle (°) | 25 | 25 | 25 |
| Number of Teeth/Starts | 25 | 1 | 120 |
| Center Distance (mm) | – | 80 | – |
| Addendum Coefficient | 1 | – | 1 |
| Dedendum Coefficient | 0.25 | – | 0.25 |
| Handedness | Right | Left | Right |
| Helix Angle (°) | 10 | 10 | 10 |
| Face Width (mm) | 25 | – | 100 |
| Whole Depth (mm) | 6.75 | – | 6.75 |
Using these parameters, the tooth surface of the spherical worm grinding wheel is computed and modeled. The discrete points obtained from the equations are processed in software like MATLAB to generate a 3D model, which is then imported into CAD tools for visualization. Similarly, the tooth surface of the internal helical gear is derived and modeled, ensuring that the meshing relationship between the virtual cutter, worm grinding wheel, and gear is accurately represented. The installation angle and speed ratio for the generating grinding process are critical for synchronization. The installation angle \( A \) is calculated as \( A = \beta – \lambda_w \), which for the given parameters is approximately 7.70939°. The speed ratio defines the rotation of the worm grinding wheel per revolution of the gear blank. Specifically, for one revolution of the gear blank, the worm grinding wheel rotates \( n \) times, given by:
$$ n = \frac{N_f}{N_w} + \frac{S_n}{T} \cdot \frac{N_f}{N_w} = 1.00015353 \, \text{r} $$
where \( S_n \) is the axial feed per revolution of the gear blank (e.g., 1 mm/r), and \( T \) is the lead of the internal helical gear, calculated as \( T = \pi d_f \cot \beta \), with \( d_f \) being the root diameter of the gear. This ensures that the grinding motion follows the required helical path.
To verify the proposed method, a simulation is conducted using Vericut software, which models the grinding process on a virtual gear grinding machine. The machine components are assembled in a kinematic chain, and a Siemens 840D CNC system is configured to control the axes. The simulation involves setting up the tool path based on the derived mathematical models, and running the program to grind the internal helical gear from a blank. The results show that the simulated gear tooth surface closely matches the theoretical design, with minimal deviations. An automatic comparison function in Vericut is used to evaluate overcuts and undercuts, revealing that the tooth flanks have no significant errors, while the tooth root area may have some residual material, which is acceptable as it does not affect meshing performance. This confirms the accuracy and feasibility of the spherical worm grinding wheel method for internal helical gear manufacturing.
In conclusion, the generating grinding method using a spherical worm grinding wheel offers a promising solution for high-precision machining of internal helical gears. By establishing precise mathematical models for the tooth surfaces of the virtual gear shaper cutter, worm grinding wheel, and internal helical gear, this method enables efficient and accurate grinding. The simulation results demonstrate that the tooth surfaces are generated with high fidelity, addressing the limitations of traditional techniques. Future work will focus on improving the dressing technology for the worm grinding wheel to enhance precision further and facilitate experimental validation. This approach has the potential to revolutionize the production of internal helical gears for electric vehicles and other high-performance applications, meeting the demands for shorter cycles and superior quality.
The internal helical gear is a key component in many mechanical systems, and its manufacturing precision directly impacts overall performance. The proposed method not only improves accuracy but also increases efficiency by enabling continuous grinding along the tooth slot. As the industry moves towards more advanced transmission systems, such as those in electric vehicles, the ability to produce high-quality helical gears quickly and reliably becomes increasingly important. The mathematical derivations and simulations presented here provide a solid foundation for practical implementation, and ongoing research will refine the process for broader adoption. The spherical worm grinding wheel technique represents a significant step forward in gear manufacturing technology, particularly for complex internal helical gears.