章动式双圆弧螺旋锥齿轮齿面曲率与运动特性研究

Tooth surface curvature and motion characteristics of nutation drive with double circular-arc helical bevel gears

为了准确评估章动式双圆弧锥齿轮副两处啮合区域之间的啮合特点和差异,对表征齿轮传动质量好坏的曲率和相对运动参数进行了理论计算研究。首先,基于齿轮空间啮合原理,推导了双圆弧内、外锥齿轮的齿面参数方程;接着,在此基础上建立了章动传动啮合坐标系,采用齿面接触分析(TCA)方法,得到了无装配误差情况下接触点的运动轨迹;然后,根据微分几何理论推导出了接触点处各类曲率和速度的数学表达式,并根据赫兹(Hertz)弹性接触理论确定了齿面接触区形态;最后,采用算例分析了接触点的主曲率、诱导法曲率、卷吸速度、相对滑动速度和滑滚比等变化趋势及其反映的特性。研究结果表明:椭圆长轴方向诱导法曲率接近0,短轴方向诱导法曲率绝对值较小,由齿面公法矢量方向可知,两啮合区域在传动过程中不发生曲率干涉现象,且具有良好的接触强度;相对滑动速度远小于卷吸速度,这有利于减小磨损;两区域的滑滚比均趋近0,表明传动接近纯滚动;瞬时接触点和齿面接触区均位于啮合齿面中部并沿整个齿宽方向分布,有限元结果与理论计算结果一致。该研究结果可为章动式双圆弧锥齿轮的弹流润滑和磨损分析提供理论基础。

In order to accurately evaluate the meshing characteristics and differences between the two meshing regions of nutated double-arc bevel gear pairs,the curvature and relative motion parameters that characterized the quality of gear transmission were theoretically calculated and studied.Firstly,the tooth surface mathematical equations of double circular-arc internal and external bevel gears were derived based on gear spatial meshing theory of gear meshing.Secondly,a meshing coordinate system of nutation drive was established and the meshing contact points under ideal installation conditions were obtained by using tooth contact analysis(TCA).Then,the mathematical expressions of the curvature and velocity at the contact point were derived based on differential geometry,and the contact region was determined with Hertz elastic contact theory.Finally,the principal curvature,induced normal curvature,entrainment velocity,relative sliding velocity and slide-roll ratio of the meshing gear pair were analyzed by an example.The results indicate that the induced normal curvature of the elliptic long-axis direction is close to 0,and the absolute value of the induced normal curvature of the short-axis direction is small,combining with the public law vector direction of the tooth flank,it is known that the curvature interference phenomenon doesn t occur in the meshing process and the gear pair has good contact strength.Meanwhile,the relative sliding velocity is much smaller in value than the entrainment velocity,which is beneficial to reduce wear of tooth.The slide-roll ratio in both meshing areas approaches 0,indicating that the transmission is approximately pure rolling.The contact points and regions are located in the middle of the tooth surface and distributed along the direction of the tooth width;the finite element results are consistent with the theoretical calculations.The results of this study provide a theoretical basis for the elastohydrodynamic lubrication and wear analysis of nutated double-arc bevel gears.