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.

本工作提出并详细介绍了一种用于变比齿条齿轮传动的计算机化生成方法。这种齿轮作为转向机构在汽车工业中具有特殊的意义。首先，采用齿轮理论提出的解析公式来研究平面生成问题，特别是确定横切中的底切，齿尖和接触比。此后，提出了一种新的直接布尔生成方法，以处理涉及更多三维场景的可变比率齿条齿轮传动的生成。与使用这种类型的齿轮的分析方法相比，该方法已被证明更加可靠和有效。研究了两种几何布局：（i）在长度（轴向）方向上具有任意形状的齿曲线的平行轴齿条和小齿轮，（ii）具有螺旋形齿的斜轴齿条和小齿轮。还提出了一种优化程序，以使传动比函数的可变性最大化，同时避免咬边和齿尖。提出了一个数字测试用例及其3D打印副本，以证明所提出的布尔生成方法和优化过程的有效性。