Fractal Fract. 2025 , 9 , 123
6of 14
as an ISO standard method to determine the friction of tissue products according to ISO 12625-18:2022 [27]. 2.3. Fractal Geometry Dimension Analysis of Surface Roughness Profiles The FD values of the surface roughness profiles were calculated using the periodogram method, which analyzes the relationship between the PSD ( J ( ω )) and angular frequency ( ω ) in the frequency domain. Surface roughness profiles, represented by one-dimensional height variations h ( x ) measured across the surface at evenly spaced intervals, were used as input data for the analysis. To compute the FD, the roughness profile h ( x ) (Figure 3) was first transformed into the frequency domain using the Fourier transform as follows: H ( ω )= ∞ − ∞ h ( x ) e − i ω t dx , (8) where H ( ω ) represents the Fourier-transformed amplitude at each angular frequency ω . ThePSD( J ( ω ) ) was then calculated as the squared magnitude of the Fourier coefficients as follows: J ( ω )= | H ( ω ) | 2 . (9) The periodogram method assumes a power–law relationship between the PSD and angular frequency: J ( ω ) ∝ ω − β . (10) Logarithmic transformation was applied to both J ( ω ) and ω , producing a linear relationship in the log–log plot. log ( J ( ω ))= − β log ( ω )+ C , (11) where C is a constant. Linear regression was performed on the log–log plot to estimate the slope ( − β ), which is related to the FD: FD = β + 1 2 (12) Figure 6 shows the spectral density of P&W1 on a log–log scale, with linear regression applied. This is based on autocorrelation from surface roughness profiles, such as those shown in Figure 3.
Figure6. Spectral density of P&W1 on log–log scale.
Made with FlippingBook flipbook maker