Duality between the kinematics of gear trains and the statics of beam systems

This paper provides geometric insight into the duality between the first-order kinematics of gear trains and the statics of beam systems. The two devices have inherent geometrical relationships that will allow the angular velocities of the gears in a gear train to be investigated from a knowledge of the forces acting on the beams of the dual beam system, and vice versa. The primary contribution of the paper is the application of this duality to obtain the dual beam system for a given compound planetary gear train, and vice versa. The paper develops a systematic procedure to transform between the first-order kinematics of a gear train and the statics of the dual beam system. This procedure provides a simple and intuitive approach to study the speed ratios of a planetary gear train and the force ratios of the dual beam system. The speed ratios are expressed in terms of kinematic coefficients, which are a function of the position of the input gear and provide insight into the geometry of the gear train. Two numerical examples of simple and compound planetary gear trains are presented to demonstrate the simplicity of the proposed approach. A third example is included to illustrate the systematic procedure to transform from a given beam system to a planetary gear train. The examples take advantage of the principle that the speed ratios for gearing are dual to the force ratios for beam systems, and vice versa.