Mathematical Modeling and Topographic Error Compensation for Plunge-Shaving Cutters Generated by a Grinding Worm

Plunge shaving is a widely used finishing process for high-precision gears due to its high productivity and cost-effectiveness. However, manufacturing the plunge-shaving cutter itself remains challenging, particularly for modified tooth profiles. Because the theoretical cutter flank exhibits a hyperboloid-like geometry in the lead direction, conventional disk-wheel grinding tends to introduce systematic twist-like topographic bias. To overcome this limitation, a comprehensive mathematical framework is developed for the generative grinding of plunge-shaving cutters using an involute-helicoid grinding worm. Based on envelope theory and homogeneous coordinate transformations, the theoretical cutter surface is first derived, followed by the establishment of a complete kinematic grinding model. A linear least-squares optimization algorithm is then formulated to determine the optimal center-distance compensation parameter for minimizing the normal deviation between the generated and theoretical surfaces. Numerical simulations demonstrate that the proposed method significantly suppresses twist-related topographic errors. In a benchmark moderate-helix case, the maximum residual deviation is controlled to approximately 2 µm. For a more demanding large-helix configuration, a two-level optimization strategy—combining machine-setting compensation and grinding-worm helix-angle adjustment—reduces the peak deviation from about 5.5 µm to 4.7 µm, corresponding to an improvement of approximately 15%. This confirms that worm-geometry tuning provides an additional, effective degree of freedom for high-helix cutter applications.