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TECHNICAL BRIEFS

Determination of the Fractal Scaling Parameter From Simulated Fractal-Regular Surface Profiles Based on the Weierstrass-Mandelbrot Function

[+] Author and Article Information
Shao Wang

School of Mechanical and Aerospace Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798, Singaporemswang@ntu.edu.sg

Ji Shen, Wai Kin Chan

School of Mechanical and Aerospace Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798, Singapore

J. Tribol 129(4), 952-956 (Apr 28, 2007) (5 pages) doi:10.1115/1.2768617 History: Received March 15, 2006; Revised April 28, 2007

A fractal dimension and a fractal roughness parameter are usually used to characterize a fractal surface. For a fractal-regular surface, a fractal domain length is also included. Such a formulation is based on an approximation using a constant value of the fractal scaling parameter that represents the ratio of the spatial frequencies of adjacent harmonic components in the Weierstrass–Mandelbrot (W-M) function. Although there were some reasons for assuming a constant value of 1.5 for the fractal scaling parameter, it is still left more or less arbitrary to adopt this assumption in fractal modeling of solid contact. In the present study, the fractal scaling parameter was treated as a variable rather than a constant by using a form of the W-M function with randomized phases based on a random walk formulation. A simple numerical scheme with clear graphical interpretation was developed to determine the value of the fractal scaling parameter. The fractal dimension, fractal roughness parameter, and fractal scaling parameter were all recovered with reasonable accuracy from numerically generated surface profiles.

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Copyright © 2007 by American Society of Mechanical Engineers
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Figures

Grahic Jump Location
Figure 1

Simulated fractal-regular surface profiles for γ=2 based on (a) a deterministic W-M function and (b) a random W-M function

Grahic Jump Location
Figure 2

Power spectral density data for a simulated fractal-regular surface (D=1.5, G=1×10−12m, γ=1.5) with the peak line and the reference Q-line for determining the fractal scaling parameter

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