Metrology and numerical characterization of random rough surfaces—Data reduction via an effective filtering solution

Abstract

Random rough surfaces appear in measurements as noisy signals varying spatially. Mathematically, there is no theoretical difference between such and time-varying signals. Hence, the extensive array of methods and analysis tools that have been developed for signal processing are available also for rough surfaces characterization. In both, the objective is to reduce the vast amount of data to just a few meaningful parameters that allow the application of other physical concepts. Particularly in contact mechanics, it is well known that the Greenwood–Williamson model requires three parameters for the calculation of the elastic deformation of rough surface asperities. The parameters are the roughness standard deviation, the equivalent asperity radius, and the asperity density. These parameters are byproducts of the spectral moments. The spectral moments have been employed for decades in many fields of engineering and science. For rough surfaces, for example, the work by McCool outlines a mathematical blueprint procedure on how to straightforwardly reduce the entire roughness data into the said three spectral moments. It is commonly claimed, however, that the said procedure inherently suffers from resolution problems, that is, a given surface shall have much different spectral moments depending on the sampling rate (or spacing). To study these issues, synthetic surfaces are generated herein using a harmonic waveform precisely as McCool had done. However, here the signals are contaminated by a white noise process with various magnitudes. A signal-to-noise ratio is defined and used to assess the quality of the signal, and the spectral moments are evaluated for various magnitudes of the noise. Since closed-from solutions are available for the spectral moments of the uncontaminated signal, the contaminated signals are evaluated vis-à-vis the exact anticipated values, and the errors are calculated. It is shown that using the common techniques (such as those outlined by McCool) can lead to enormous and unacceptable errors. Resolution is studied as well; it is shown to have an effect only in the presence of noise, but by itself it has no independent influence on the spectral moments. The venerable Savitzky–Golay smoothing filter is used on the noisy signals, showing some improvements, but the resulting spectral moments predicted still contain objectionable errors. A generalized exponential smoothing filter, G-EXP, is constructed, and it is shown to markedly moderate the errors and reduce them to acceptable levels, while effectively restoring the underlying surface physical characteristics. Moreover, the filtered signals do not suffer from resolution problems, where results, in fact, improve with higher (i.e., finer) resolutions. Fractal-generated signals are likewise discussed.