Parametric Investigation of Gielis’ Superformula-Based Non-Circular Gears: Geometric Features and Transmission Ratio Functions

Current design approaches for non-circular gears are generally restricted to predefined pitch curve geometries, lacking a unified methodology for systematically exploring wider families of admissible curves. To overcome this limitation, the paper proposes a generalized parametric design methodology capable of generating and evaluating an entire family of feasible pitch curves from a single analytical formulation—the Gielis’ superformula. Numerical analysis and CAD-based implementation are employed to identify a sub-family of generalized ellipses within the infinite family of Gielis curves that meet the geometric requirements for non-circular gear centrodes, namely closed and periodic profiles, smooth curvature, the absence of angular discontinuities and undercutting during tooth generation. The proposed methodology integrates the Gielis curves into gear design by (i) introducing gear-specific parameters, (ii) generating the conjugate pitch curve through the rolling without-slip equations, (iii) determining the center distance, (iv) determining the tooth number relation associated with the revolution ratio, (v) considering the curvature-based undercutting criterion and (vi) validating the resulting gears through CAD-based tooth generation simulations. This approach extends the geometric design space while providing a unified framework for the non-circular gears synthesis. Appropriate combinations of the Gielis parameters allow the amplitude, frequency and waveform of the transmission ratio function to be tailored to the kinematic requirements of variable-speed applications. Rather than identifying universal optimal values, the proposed methodology provides a framework for systematically evaluating Gielis parameter combinations against the geometric and gear-design requirements.