A hybrid analytical/Boolean approach to the generation of rack and pinion drives with variable transmission ratio

The present work proposes and details a computerized generation method for variable-ratio rack and pinion drives. Such gears are of special interest to the automotive industry as steering mechanisms. At first, the analytical formulation proposed by the Theory of Gearing is adopted to study the planar generation problem, in particular for the determination of undercut, tooth pointing, and contact ratio in the transverse plane. Afterwards, a novel method of Direct Boolean Generation is proposed to deal with the generation of variable-ratio rack and pinion drives in more involved three-dimensional scenarios. This approach has proved to be more robust and efficient than the analytical method with this type of gears. Two geometric layouts are investigated: (i) parallel-axis rack and pinion provided with arbitrarily shaped tooth curves in the lengthwise (axial) direction, (ii) skew-axis rack and pinion provided with helical teeth. An optimization procedure is also proposed to maximize the variability of the transmission ratio function while avoiding undercut and tooth pointing. A numerical test case and its 3D-printed counterpart are presented to demonstrate the effectiveness of the proposed Boolean generation method and optimization procedure.