The line contact problem of elastohydrodynamic lubrication - I. Asymptotic structure for low speeds
The line contact problem of elastohydrodynamic lubrication - I. Asymptotic structure for low speeds
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弹流润滑的线接触问题——I.低速渐进结构
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发表时间：
1989
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通讯作者：
E. J. Bissett
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作者：
E. J. Bissett
本文报告了弹性流体动力润滑（EHL）线接触问题的第一个正式渐近解，这是一个描述润滑滚动元件（例如滚子轴承、齿轮齿和其他类似几何形状的接触）的弹性变形的基本问题。所考虑的渐近方案是小 λ 的渐近方案，这是一个与轧制速度、粘度和弹性模量成比例的无量纲参数。该解决方案具有四个区域：润滑膜既薄又缓慢变窄的区域，与没有润滑剂时发生的接触区域密切相关，低压的上游入口区域，以及接触区域两侧的两个薄层。前两个区域的解由简单的解析表达式给出。两个薄层中的解是由 Bissett & Spence (Proc. R. Soc. Lond. A 424, 409 (1989)) 获得的两个通用函数获得的。尽管这两个与局部膜厚度相关的函数是由 Bissett & Spence 通过数值技术获得的，但应该强调的是，所考虑的渐近方案中的所有情况都在此得到明确解决，而无需进一步计算。尽管其他解决方案已经提出了该结构的一些特征，但一般来说，这些都是数值或临时近似值。请参阅 Johnson（Contact Mechanics，第 328 页（1985））和 Dowson & Higginson（弹性流体动力润滑（1977））的文本，这项工作为理解 EHL 的大多数主要特征提供了正式的数学基础。该解决方案提供了最小油膜厚度的简单公式，并显示了出口附近薄层中润滑膜的急剧变窄。在此提供的基本渐近解中，无量纲压力粘度系数 α 假定为 O(1)，并且在此参数方案中，不会出现压力尖峰。通过与 Hooke 的工作进行比较 (J. mech. Engng Sci. 19(4), 149 (1977))，我们可以表明当 α 变得与 O(λ-1/5)一样大时，会出现初始压力尖峰。然而，后一个参数方案中的渐近解需要针对每种感兴趣的情况提供新的数值解，这里不进行讨论。
This paper reports the first formal asymptotic solution to the line contact problem of elastohydrodynamic lubrication (EHL), a fundamental problem describing the elastic deformation of lubricated rolling elements such as roller bearings, gear teeth and other contacts of similar geometry. The asymptotic régime considered is that of small λ, a dimensionless parameter proportional to rolling speed, viscosity and the elastic modulus. The solution is shown to possess four regions: a zone where the lubricating film is both thin and slowly narrowing and which is closely related to the contact area that occurs in the absence of lubricant, an upstream inlet zone of low pressure, and two thin layers on either side of the contact zone. The solutions in the first two just-mentioned zones are given by simple analytical expressions. The solutions in the two thin layers are obtained from two universal functions obtained by Bissett & Spence (Proc. R. Soc. Lond. A 424, 409 (1989)). Although these two functions, related to the local film thickness, are obtained by numerical techniques by Bissett & Spence, it should be emphasized that all cases in the asymptotic régime considered are hereby solved definitively without recourse to further computation. Although some features of this structure have been suggested by other solution approaches, generally, these are numerical or ad hoc approximations. See the texts by Johnson (Contact Mechanics, pp. 328 (1985)) and Dowson & Higginson (Elasto-hydrodynamic lubrication (1977)), this work provides a formal mathematical basis for understanding most of the principal features of EHL. The solution provides a simple formula for minimum film thickness and displays the sharp narrowing of the lubricating film in the thin layer near the exit. In the basic asymptotic solution provided here, the dimensionless pressure-viscosity coefficient, α, is assumed to be O(1), and in this parameter régime, no pressure spike will occur. By comparing with the work of Hooke (J. mech. Engng Sci. 19(4), 149 (1977)), we can show that an incipient pressure spike occurs when α becomes as large as O(λ-1/5). However, asymptotic solutions in this latter parameter régime require new numerical solutions for each case of interest and are not pursued here.