Power losses in two-degrees-of-freedom planetary gear trains: A critical analysis of Radzimovsky's formulas

Two-degrees-of-freedom gearing are key elements of any planetary gear train. Thus, the full understanding of their basic mechanics and mechanical efficiency evaluation is not only of significant theoretical interest, but important in many industrial applications. Some previous investigations on the estimation of mechanical efficiency in two-degrees-of-freedom differential gearing rely upon the relationships deduced by Radzimovsky. However, to the best of authors' knowledge, a critical analysis on the validity conditions of these relationships is not available yet in technical literature. To fill the apparent gap, in this paper a broad physical interpretation of mechanical efficiency analysis of two-dof planetary gear trains is offered. The discussion allowed to outline the theoretical limits of the Radzimovsky formulas. The current approach provides three different results for the power loss of a two-input gear pair entity with the planet carrier as the output link. This leads to the conclusion that even for the same PGT and for the same input and output links the power loss has, for each valid sequence of angular velocities, a peculiar mathematical expression.