On the Kinematics of the Closed Epicyclic Differential Gears

The epicyclic differential gear has been known in modern times since 1575 when it appeared as a mechanism in a clock. Artifacts from an ancient shipwreck prove that it was known to the ancient Greeks at least 100 hundred years before Christ. The methods of kinematic analysis by either the relative angular velocity method or the instant center and linear velocity method, as given in the literature, are oriented toward specific solutions rather than general ones; they do not readily allow for parametric trend studies and they require a degree of imagination and intuition which may well be beyond the capabilities of those who are not practitioners of the art. The discussed methodology defines simple and compound epicyclic gears in terms of the overall ratio of a geometrically similar planetary gear. The kinematic analysis is derived in general terms for both the simple and compound epicyclic gear. It is shown that location of the point of zero tangential velocity of the velocity triangle relative to the system datum governs the characteristics of the gearset and whether it will perform as a differential gearset, or as a solar, star, or planetary gear. Simple mathematical relationships are given which define the proportions of the component gears, their speeds (rpm) and directions of rotations, and the resulting power splits. The formulas may be incorporated into simple computer programs oriented toward specific design requirements.