Abstract Non-conforming contact pairs, such as gears, roller bearings and rail-wheel systems, have free-end surfaces, and the classical Hertz theory generally is based on the assumption of half space, which neglects the influences of free-end surfaces. In this paper, the equivalent load method is presented to investigate the effect of free-end surfaces of finite-length elastic space on internal stresses and von-Mises stress. This method can effectively solve the free-edge problems and multi-direction stresses problem. The results obtained by the presented method are evaluated and validated by the results from quarter space model and finite element method. The distinctions among the half space model, the quarter space model and the finite-length space model are discussed. It can be found that when the edge distance is larger than 2a, the half space model can be approximately applied. And when the distances between two free end surfaces is less than 4a, the effect of two free end surfaces should be considered simultaneously.

中文翻译：

---

考虑两个自由端面的等效载荷法在有限长弹性空间中的应用

摘要 齿轮、滚子轴承、轨道轮系等非协调接触副具有自由端面，经典赫兹理论一般基于半空间假设，忽略了自由端面的影响。本文采用等效载荷法研究有限长度弹性空间的自由端面对内应力和von-Mises应力的影响。该方法可以有效解决自由边问题和多向应力问题。通过四分之一空间模型和有限元方法的结果对本文方法获得的结果进行了评估和验证。讨论了半空间模型、四分之一空间模型和有限长度空间模型之间的区别。可以发现，当边缘距离大于2a时，可以近似应用半空间模型。当两个自由端面之间的距离小于4a时，应同时考虑两个自由端面的影响。