An indicator to quantify the complexity of signals and surfaces based on scaling behaviors transcending fractal

Abstract

In the well-established fractal scheme, the fractal dimension (D) is a central indicator of the complexity of fractal features. The D values of non-fractal signals and surfaces are 1 and 2, respectively, while there can be varieties in their complexities. In this study, the scaling characteristics of root-mean-squared roughness could exhibit a continuous variation transcending the boundary between fractal and non-fractal by using the roughness scaling extraction (RSE) method proposed in previous study, and an universal indicator (HRSE, Hurst exponent calculated by RSE method) to quantify the complexity of both fractal and non-fractal features is demonstrated. The actual signals (milling vibration) and surfaces (silver thin films) together with the artificial ones generated through Weierstrass–Mandelbrot (W–M) function were analyzed. Within the fractal scope, the calculated results with RSE method could be close to the ideal D values of W–M function with an accuracy higher than those of the traditional fractal methods (including Box-Counting, Higuchi, Katz, power spectral density, structure function, and autocorrelation function methods). For the non-fractal features, the complexity could also be quantified effectively by HRSE. Chatter could be recognize by HRSE of milling vibration signals, because it was larger than 1, from 0.5 to 1, and less than 0.5 in idling, stable milling and chatter milling states, respectively; For thin film surfaces, HRSE increased monotonically from 0.79 to 1.32 along with Sq increasing, indicating a strong positive correlation. The findings indicated that the scaling analysis could be utilized for both fractal and non-fractal features, which would be beneficial for various engineering applications.