An Optimization-Based Approach to Twist Control Through Tool Geometry and Feed Coordination in Worm-Type Gear Generation

In precision gear manufacturing, longitudinal crowning on tooth flanks is commonly produced by applying diagonal feed in worm-type generating processes using tools such as variable-tooth-thickness hobs and dressable grinding worms. However, precise twist control remains difficult because the geometric parameters of the generating tool are strongly coupled with the machine feed settings in the underlying generating kinematics. In addition, direct numerical optimization becomes unreliable near the standard tool state, where the sensitivity of the diagonal-feed coefficient degenerates and conventional linearized solvers may lose effectiveness. To address these issues, this study proposes a multi-variable optimization framework for twist-constrained worm-type gear generation. An iterative singular value decomposition (SVD) scheme is developed to construct and update the sensitivity matrix, while a warm-start continuation strategy is introduced to overcome the local singularity and improve numerical robustness. Two closed-form expressions for the diagonal-feed coefficient are also proposed as practically useful initial estimates, corresponding respectively to the minimum SVD topographic residual and the minimum tooth-flank twist. Numerical validation over a 60-case parameter sweep shows maximum relative errors below 1.6% within the tested range. The proposed framework coordinates the tool-geometry design and diagonal-feed selection to generate tooth flanks with prescribed crowning characteristics while satisfying a specified twist requirement and limiting the required diagonal shift. Numerical examples show that the iterative framework reduces the root-mean-square (RMS) topographic error from 1.14 μm to 0.027 μm relative to the analytical setting of Hsu and Fong. These results indicate that the proposed method provides a reliable computational basis for twist control and process-parameter design in advanced CNC gear generation. From a manufacturing standpoint, because the three design criteria are accessed by adjusting only the diagonal-feed ratio on the machine, a single generating-tool design can serve a range of crowning and twist requirements without retooling, reducing setup and tooling efforts in production.