渐开线齿轮的接触分析

Tooth contact analysis of involute gear

以Hertz应力表达式为基础,将单齿啮合的渐开线齿轮等效为相互挤压的两个圆柱体,推导了齿轮接触应力的理论表达式.建立一套单齿对啮合的有限元分析方法,包括引入渐开线和齿根过渡曲线方程及对应的自变量区间,建立参数化齿廓;模型轮缘厚度取3倍的模数,周向宽度取3倍的齿厚;在接触面上进行网格细化处理;在对应的主、从动齿轮内缘分别施加均匀切向力和固定约束等.计算结果显示,有限元解和理论解吻合较好,最大偏差不超过5%,该方法适用于不同的啮合轮齿参数.由有限元方法得到的渐开线轮齿的接触应力符合Hertz理论中的半椭圆分布规律,Mises应力和剪应力分布也符合接触力学理论,但应力分布的对称中心存在偏离或者偏斜现象,分析认为,是由于轮齿弯曲变形造成的.

Based on Hertz stress expression, one pair of meshing teeth was equivalent to two contacting cylinders and then the theoretic expression of gear tooth contact expression was derived. A finite element method （FEM） for one pair of meshing teeth was established. This method established parameterized tooth profile according to involute and root transitional curve equations with defined intervals. In this method, the rim thickness was 3 times of module and the circumferential width was 3 times of tooth thickness, too. Grid refining process was adopted on contacting edge and equalized tangential forces and fixed constraints were applied respectively on the inner rims of driving and driven gears. The results from FEM and theoretical formula were in excellent agreement with less than 5% errors, and FEM was proved to be suitable for gear pairs with different tooth parameters. From FEM, the contact stress obeys the semi-ellipse distribution and the Mises stress and shear stress are also consistent with contact mechanics theory, hut there exists offset or deflection in the symmetric lines of stress distribution, which originates from tooth bending deformation.