A novel method for calculating the fractal dimension of three-dimensional surface topography on machined surfaces

In this paper, a new fractal dimensioning method (3D-Sa) for three-dimensional surface morphology is proposed. First, the fractal models of isotropic and anisotropic surfaces are established based on Weierstrass-Mandelbrot (W-M) functions. By calculating the surface parameters of the fractal surfaces, the mapping relationship between the fractal dimension D and the regular surface parameters was obtained. Combining the surface arithmetic mean height equation and the definition of fractal dimension, the fractal dimension equation is derived with the surface arithmetic mean height Sa as the main body. Then, the equation is optimized using simulation fitting to achieve the correction of fitting parameters. The proposed 3D-Sa method for calculating the fractal dimension of the simulated surface is compared with the difference box dimension (DBC) method and the structure function method, respectively. Finally, the surface morphology of the turned, milled and ground surfaces of the end face was collected using a white light interferometer. The feasibility of the proposed 3D-Sa method in characterizing the topographic parameters of real machined surfaces is demonstrated by calculating the fractal dimension of the collected surface data. It provides a new approach to accurately calculate the fractal dimension for the theoretical and applied studies of fractalization of machined surfaces.