Gear hobbing is a fundamental process in mechanical manufacturing, widely used for producing gears in transmission systems. The iterative methods commonly employed in gear hobbing simulation are complex and challenging, often failing to replicate actual machining errors. Given the irreversible nature of gear hobbing, errors during processing can severely impact workpiece quality and production schedules. In this study, I develop a simulation approach using the AutoLisp platform to model gear hobbing by simulating real-world machining conditions and parameters. By adjusting tool or workpiece parameters during calculations, I can emulate various errors encountered in gear hobbing. Practical machining tests validate the accuracy and reliability of this method, demonstrating its potential to enhance gear machining precision and production efficiency in industrial applications.
The simulation leverages AutoLisp’s integration with AutoCAD for secondary development, enabling precise control over the gear hobbing process. Gear hobbing involves the interaction between a hob tool and a gear blank, where the rotational motion of the hob and the workpiece translates into a generating motion. This process is analogous to the meshing of crossed helical gears, and when the hob’s base circle radius approaches infinity, it simplifies to a rack-and-pinion model. In my simulation, I convert the three-dimensional gear hobbing model into a two-dimensional representation for computational ease, as illustrated in the following equations that govern the relationship between hob rotation and rack movement:
$$ L = n_1 \times L_1 $$
$$ \phi = \frac{n_1 \times K}{Z \times 360} $$
where \( L \) is the rack displacement, \( n_1 \) is the hob rotational speed, \( L_1 \) is the hob pitch, \( \phi \) is the gear rotation angle, \( K \) is the number of hob threads, and \( Z \) is the number of gear teeth. The ratio of rack displacement to gear rotation angle is derived as:
$$ \frac{L}{\phi} = \frac{L_1 \times Z}{360 \times K} $$
This foundational principle allows me to simulate the gear hobbing process by coordinating the linear motion of the rack with the rotational motion of the gear. The secondary development using AutoLisp and DCL (Dialog Control Language) facilitates a user-friendly interface for inputting parameters such as module, number of teeth, pressure angle, and helix angle for the gear blank, as well as pitch, pressure angle, and corner radius for the hob tool. The workflow involves parameter input via DCL, computation in AutoLisp, and graphical output in AutoCAD, as summarized in the design flow below:
| Step | Description |
|---|---|
| 1 | Input gear and hob parameters through DCL dialog |
| 2 | Generate gear blank and hob rack profiles using AutoLisp |
| 3 | Simulate hobbing motion by applying domain subtraction in AutoCAD |
| 4 | Analyze errors by modifying parameters in the simulation |
The gear blank is constructed based on logical conditions defined by equations such as the addendum diameter calculation:
$$ d_a = \frac{M_n \times Z}{\cos \beta} + 2 \times M_n (h_a^* + x_n) $$
where \( d_a \) is the addendum diameter, \( M_n \) is the normal module, \( \beta \) is the helix angle, \( h_a^* \) is the addendum coefficient, and \( x_n \) is the profile shift coefficient. For the hob rack profile, I establish a coordinate system with the gear center as the origin, the hob length direction as the x-axis, and the height direction as the y-axis. Key points on the hob profile are calculated using:
$$ x_1 = \frac{b}{2} + \tan \alpha_1 \times (h – h_1), \quad y_1 = \frac{M_n \times Z}{2 \times \cos \beta} + (h – h_1) $$
$$ x_2 = \frac{b}{2} – \tan \alpha_1 \times (h_1 – h_2), \quad y_2 = \frac{M_n \times Z}{2 \times \cos \beta} – (h_1 – h_2) $$
$$ x_3 = x_2 – \tan \alpha_2 \times (h_2 – h_3), \quad y_3 = \frac{M_n \times Z}{2 \times \cos \beta} – (h_1 – h_3) $$
$$ x_4 = 0, \quad y_4 = \frac{M_n \times Z}{2 \times \cos \beta} – h_3 $$
Here, \( b \) is the hob pitch width, \( \alpha_1 \) is the hob pressure angle, \( \alpha_2 \) is the side edge pressure angle, \( h \) is the total hob tooth height, \( h_1 \) is the hob addendum, \( h_2 \) is the hob protrusion height, and \( h_3 \) is the hob arc height. These equations enable the precise modeling of the hob rack, which is essential for accurate gear hobbing simulation.
In the simulation program, I utilize AutoLisp to define curves for the gear and hob, create parameter input interfaces, and establish communication between AutoLisp and AutoCAD. By invoking commands in AutoCAD, the program automates the generation of gear blanks and hob racks, followed by the simulation of the hobbing process through iterative domain subtraction. For each rotation of the gear by an angle \( \phi \), the rack moves a distance \( L \), gradually forming the gear tooth profile. It is important to note that the domain subtraction command in AutoCAD converts entities into regions; therefore, for post-simulation analysis, these regions must be exploded to access individual geometric elements.
Error analysis in gear hobbing is critical due to the cumulative effects of inaccuracies in the hob, workpiece, gear hobbing machine, and fixtures. My approach involves classifying errors and simulating them within the AutoLisp environment. One common error is hob mounting eccentricity, which affects the initial tool engagement point. This eccentricity can be modeled by varying the distance between the rack and the gear center during simulation, following a sinusoidal pattern:
$$ x = \delta \times \cos \phi, \quad y = \delta \times \sin \phi $$
where \( \delta \) is the eccentricity distance and \( \phi \) is the hob rotation angle. This simulation reveals that the tooth profile deviates in a sinusoidal manner, with peaks above the theoretical profile and troughs coinciding or below, depending on the engagement point. For instance, if the tool starts at the highest eccentricity point, the generated teeth exhibit elevated sections, potentially reducing machining allowances and accuracy.
Another significant error is hob tooth profile angle deviation, which may arise from manufacturing or regrinding processes. I simulate this by altering the side edge pressure angle \( \alpha_2 \) in the hob profile equations. When \( \alpha_2 \) exceeds the design value, the simulated tooth profile shifts above the theoretical curve; conversely, a smaller \( \alpha_2 \) results in a profile below the ideal. This method provides a straightforward way to visualize and quantify the impact of hob geometry errors on the final gear quality.
To validate the simulation model, I conducted practical gear hobbing tests using a Liebherr LC1200 gear hobbing machine. The gear parameters are listed in the table below:
| Parameter | Value |
|---|---|
| Module | 9 mm |
| Number of Teeth | 23 |
| Pressure Angle | 20° |
| Profile Shift Coefficient | 0.32 |
| Addendum Coefficient | 1 |
| Dedendum Coefficient | 0.4 |
The hob tool, a double-cut hob from LMT with AA grade accuracy, had the following parameters:
| Parameter | Value |
|---|---|
| Hob Pressure Angle | 20° |
| Side Edge Pressure Angle | 10° |
| Total Tooth Height | 23.7 mm |
| Addendum | 11.1 mm |
| Protrusion Height | 6.9 mm |
| Arc Height | 2.6 mm |
| Pitch Width | 12.2 mm |
I introduced an eccentricity of 0.1 mm in the hob arbor and set the gear hobbing machine to operate at a hob speed of 70 rpm and a feed rate of 6.2 mm/min. The workpiece was aligned within 0.005 mm. After machining, I measured the tooth profile using a Klingelnberg P350 gear inspection machine. The simulation results, processed in AutoCAD, showed that the eccentricity-induced profile deviations followed a sinusoidal curve with amplitudes between +0.06 mm and -0.06 mm over a range of 233 mm to 266 mm on the x-axis. The actual inspection data correlated closely with these predictions, confirming the simulation’s accuracy in replicating real-world gear hobbing errors.
In conclusion, the AutoLisp-based simulation platform offers a robust method for analyzing gear hobbing processes and associated errors. By modeling complex interactions in a simplified 2D environment, I can efficiently predict outcomes and identify potential issues before physical machining. This approach not only reduces the risk of workpiece scrapping but also optimizes production efficiency in gear manufacturing. The flexibility of the simulation allows for extensive error analysis, including those related to the gear hobbing machine dynamics, hob wear, and alignment inaccuracies. Future work could expand this methodology to other gear machining processes, such as shaping or grinding, further enhancing its applicability in industrial settings.