The microscopic shape of an object’s surface is called surface topography. No matter what method is used to process the parts, the profile of the roller and the mold will produce a certain geometry rough shape, which is random. When the roller and the mold are in contact, the contact point is actually on the micro convex body of the micro surface, as shown in Figure 1. There are many parameters used to characterize the surface morphology, such as arithmetic mean deviation of contour, root mean square deviation of contour and so on.

In the study of calculation, The autocorrelation function method can be used to obtain random rough surface:

(1) A two-dimensional random sequence η (x, y) of Gaussian white noise is generated by computer, and its Fourier transform a (ω x, ω y) is calculated;

(2) According to the specified autocorrelation function R (τ x, τ y), the power spectral density g (ω x, ω y) of the output signal of the filter is obtained through Fourier transform, and the power spectral density c of the input sequence is determined at the same time;

(3) The transfer function of the filter is calculated by the formula

(4) The formula is derived to obtain the Fourier transform Z of the output sequence after the input sequence passes through the two-dimensional filter;

(5) The height distribution function Z (x, y) of surface H (W x, W Y) is obtained by inverse Fourier transform of Z (W x, W Y).

The correlation function is as follows

This is a commonly used autocorrelation function in exponential form, and the correlation length is β x = β y = 10. If the root mean square deviation of surface roughness is σ 1 = σ 2 = σ 3 = 25 μ m, the random surface is obtained, as shown in Figure 2.

After considering the surface roughness, the surface profile equation of the roller and the mold can not be expressed by the standard circle equation. Similar to the equation considering the surface shape error, the surface profile equation of the roller and the mold can also be expressed by the formula after considering the surface roughness.