Kinematic analysis of epicyclic bevel gear trains with matroid method

The paper presents a novel technique for the kinematic analysis of bevel gear trains using the incidence matrices of an edge-oriented graph of the mechanism. The kinematic equations are then obtained in matrix form using a cycle basis from a cycle matroid. These equations can be systematically generated, and allow for an efficient computation of the angular velocities of the gears and planet carriers of the mechanism without employing time derivative operations. As illustrated in the paper, the method is applicable to bevel gear trains of any number of gears or degrees of freedom

A GENERAL METHOD OF KINEMATIC ANALYSIS OF PARALLEL AXES EPICYCLIC GEAR TRAINS BASED ON GRAPH-CYCLE MATROID THEORY

A novel method for kinematic analysis of parallel-axes epicyclic gear trains (EGT) is presented, called the incidence and transfer method (IT), which uses the incidence matrices associated to the edge-oriented graph attached to the mechanism and the transfer joints (teeth contact joints). Relative to such joints, a set of independent equations can be generated for calculating the angular positions, velocities, and accelerations. Complete kinematic equations are obtained in matrix form using a base of circuits from a cycle matroid. The analysis uses the relationships between the number of mobile links, number of joints, and number of circuits in the base of circuits, together with the Latin matrix (whose entries are function of the absolute values of the partial gear ratios of the transmission). Singularities, like groups of gears that rotate as a whole, can be identified by calculating the rank of the Latin matrix. Relationships between the output and input angular velocities and accelerations are then determined in a matrix-based approach, without using any derivative operations. The proposed method has general applicability and can be employed for systems with any number of gears and degrees of freedom, as illustrated by the numerical examples presented