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There are several commercial off-the-shelf software available to study transmission and driveline dynamics. Many of these software require a faithful representation of the transmission topology in order to carry out the analyses. These modeling techniques utilize several redundant degrees of freedom which may not be necessary for studying low frequency (< ~30 Hz) dynamics and may be computationally inefficient. The dual-clutch model has been proposed as a generic 2-DOF model that overcomes some of these drawbacks. In this paper, the dual-clutch model is initially derived from first principles, starting with the equations of motion for a planetary automatic transmission. The model coefficients - the inertia matrix and the matrix of clutch coefficients - are then derived using a more intuitive approach based on energy considerations.

Understanding planetary gear efficiency is more involved than understanding efficiency of external gears because of the recirculating power that is inherent in planetary gear operation. There have been several publications going back several decades on this topic. However, many of these publications are mathematical in their approach and tend to be overlooked by practicing engineers. This paper brings a new, more visual and more intuitive approach to the problem. It uses lever diagrams, which have been a standard tool in the transmission engineer’s arsenal for almost four decades, to visualize the power flow and develop analytical expressions for the efficiency of simple and compound planetary gears. It then extends the approach to more complex gear trains.