Unified mathematical model for external and internal gears with variable hyperbolic circular-arc-tooth-trace

The absence of a comprehensive meshing theory for variable hyperbolic circular-arc-tooth-trace (VH-CATT) cylindrical internal gears has significantly impeded their development and industrial application. This study proposes a novel approach that employs a dedicated generating gear to envelop both external and internal VH-CATT gears, thereby overcoming the limitation of the conventional counterpart rack, which is confined to generating external VH-CATT gears. Based on the dual-blade cutter milling principle for VH-CATT cylindrical gears, the concept of the generating gear is introduced and its mathematical model is established. A unified mathematical model is subsequently developed to generate both external and internal VH-CATT gears through enveloping by the generating gear. A quantitative comparative analysis is conducted to characterize the similarities and differences among the generating gear, external gear, and internal gear in terms of full tooth surface geometric deviations, curvature distribution, and theoretical tooth thickness at the reference circle. Finally, a planetary gear set is manufactured based on the unified model for tooth surface deviation measurement and tooth contact testing. Theoretical analysis reveals that although the generating gear and the external gear exhibit similar geometric features, their tooth thickness variations show an inverse correlation, while the tooth surface of the generating gear and that of the internal gear are complementary in geometry. Experimental results validate the accuracy and feasibility of the proposed mathematical model.