Kinematic synthesis and design of non-circular gears through a symbolic-numeric modelling approach

The paper presents a computational approach to the kinematic synthesis of non-circular gears. The two gears are assumed to be external, with parallel axes and with an average transmission ratio of -1. The symbolic expression adopted for the definition of the variable ratio input-output relation is first introduced. Then, by using as a computing environment a computer algebra software, the parametric expressions of the two pitch curves and their internal and external offset curves are directly generated. The tooth conjugate profiles are obtained through an extension of the cycloidal tooth gearings to the case of non circular pitch curves. The implementation of the proposed method requires the solution of several mathematical problems. For example, the generation of the tooth profile equation requires the explicit parametric representation of the arc length of the pitch curves; the definition of each tooth height is determined by solving a system of two nonlinear equations obtained by intersecting the tooth profile curve with the curve offset from the pitch curve. Case by case, the paper discusses in detail the mathematical formulation and the adopted solution procedure: in general they take advantage of the chosen computing environment that allows a mixture of symbolic-numeric algorithms. Criteria for the choice of design parameters (number of teeth, tooth height, radius of the rolling circle) and their influence on the resulting gears properties (contact ratio, pressure angle) are then discussed. Finally, some numerical examples are reported and manufacturing aspects discussed.