二维分形轮廓微凸体面积分布函数的研究

doi:10.3901/JME.2025.21.329

摘要/Abstract

摘要： 微凸体面积分布函数是对Korcak公式微分获得，其表达形式直接决定分形接触模型的准确性。根据Korcak公式建立了微凸体截面直径与数量的关系式N(Λ>l)=λ(l_max/l)^B，采用数值方法分别对Weierstrass-Mandelbrot函数、傅里叶变换和机械加工生成的二维分形轮廓进行分析计算，拟合出参数λ、B和l_max与分形参数、截断高度的数学表达式，并推导出修正的微凸体面积分布函数。结果表明：参数B仅与轮廓分形维数D和截断高度有关，在任意分形轮廓平均高度附近，B的值近似等于D?1；参数λ和l_max取决于基准取样长度、取样长度、轮廓参数等因素，不同的分形轮廓对应的λ和l_max表达式互不相同；对比Weierstrass-Mandelbrot函数、机械加工生成的二维分形轮廓的实测截断面积，修正的微凸体面积分布函数能较为精确获得不同类型分形轮廓的真实截断面积，而经典微凸体面积分布函数适用于取样长度为γ^-n_min的WM函数模拟的分形轮廓真实截断面积的计算。

关键词: 分形轮廓, 微凸体, 微凸体面积分布函数, 取样长度, 截断高度

Abstract: The accuracy of fractal model of contact between rough surfaces depends on the truncated asperity size distribution function which is obtained by derivative of the Korcak formula. The form of relationship between number of truncated asperities and diameter of section of truncated asperities is developed based on the Korcak formula, namely N(Λ>l)=λ(l_max/l)^B. The fractal profiles generated by Weierstrass-Mandelbrot function, Fourier transform and machining treatment, respectively are used to validate the developed form. The expressions for the parameters λ、B and l_max covering whole height of fractal profile are obtained. Then a revised truncated asperity size distribution function is given. The results show that the parameter B is related to the fractal dimension D and truncated interference. As the truncated interference is equal to the mean height of fractal profile, the value of B is D?1 approximatively. The parameters λ and l_max depend on the characteristics of profile, such as the reference sample length, the sample length, the parameters of profile etc. The expressions of λ and l_max are different for different profiles. The revised truncated asperity size distribution function and classic truncated asperity size distribution function are applied respectively to calculate the real truncated area of fractal profiles generated by Weierstrass-Mandelbrot function and machining treatment, and the values of real truncated area obtained by the revised truncated asperity size distribution function are really close to measured values of truncated area. As the sample length of fractal profile simulated by Weierstrass-Mandelbrot function equals γ^-n_min, the results of real truncated area obtained by the classic truncated asperity size distribution function are valid.

Key words: fractal profile, asperity, the truncated asperity size distribution function, sample length, truncation height

中图分类号:

TH117

刘帅, 原园, 刘晨茜, 乔路群. 二维分形轮廓微凸体面积分布函数的研究[J]. 机械工程学报, 2025, 61(21): 329-344.

LIU Shuai, YUAN Yuan, LIU Chenxi, QIAO Luqun. Research on the Truncated Asperity Size Distribution Function for the Two-dimensional Fractal Profiles[J]. Journal of Mechanical Engineering, 2025, 61(21): 329-344.

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