*Article* **Power-Flow and Mechanical Efficiency Computation in Two-Degrees-of-Freedom Planetary Gear Units: New Compact Formulas**

**Essam Lauibi Esmail 1,†, Ettore Pennestrì 2,\*,† and Marco Cirelli <sup>3</sup>**


**Abstract:** The mechanical efficiency is a computed value for comparing the performance of the multi degrees-of-freedom geared transmissions of hybrid vehicles. Most of the current methods for estimating gear trains mechanical efficiency require the decomposition of gear transmissions in basic structural elements or planetary gear units (PGU). These are two degrees-of-freedom components whose mechanical efficiency has a deep influence on the overall device. The authors (E.L.E., E.P.) already evidenced that, under certain kinematic conditions, the classic Radzimovsky's formulas, widely accepted for computing the mechanical efficiency of PGUs, are not adequate. In this paper, more general and reliable formulas for computing the mechanical efficiency are deduced. The proposed formulas herein, exploiting the concept of potential or virtual power, evidence the dependency between kinematics and efficiency. A numerical example compares our results with previous work on the subject.

**Keywords:** planetary gear trains; mechanical efficiency; power-flow; hybrid vehicles
