A computationally robust solution to the contact problem of two rotating gear surfaces in space

In this study, the realm of three-dimensional geometrical contact problems is delved into, with a specific focus on the scenario involving two C¹ gear surfaces undergoing rotational motion about two fixed axes in the spatial domain. The development of a novel mathematical model for this contact problem is encompassed by our endeavor. The foundation of our model is rooted in a new parameterization technique, which enables the reduction of the conventional system of five generally non-linear equations, each with five unknowns and one independent parameter, into a more manageable system. This streamlined approach results in a system consisting of just two equations, two unknowns, and one independent parameter. Subsequently, a back-substitution methodology is employed to explicitly derive the values for the remaining three unknowns. In a bid to ascertain the efficacy of the newly proposed model, a comparative analysis is conducted with the well-established model formulated by Litvin. Our findings demonstrate that the accuracy and stability of numerical solutions in the context of this intricate geometrical contact problem are not only enhanced by our model but also that it represents a significant advancement in the understanding and computational treatment of such scenarios.