**3. Surface roughness**

There exists a vast number of empirical profile or surface roughness characteristics suitable often in very special situations (Quinn Hannan, 2001; Zhang & Gopalakrishnan, 1996). Some of them are closely connected with characteristics computed from fractal models as fractal dimension and topothesy (Davies, 1999). A set of parameters for profile and surface characterization are collected in (Militký Bajzík, 2003, Militký Mazal, 2007). Parameters for profile and surface characterization can be generally divided into the following groups:


General surface topography is usually broken down to the three components according to wavelength (or frequency). The long wavelength (low frequency) range variation is denoted as form. This form component is removed by using of polynomial models or models based on the form shape. The low wavelength (high frequency) range variation is denoted as roughness and medium wavelength range variation separates waviness. The most common way to separate roughness and waviness is spectral analysis. This analysis is based on the Fourier transformation from space domain *d* to the frequency domain 2 / *d* .

Woven Fabrics Surface Quantification 129

constant increment. Teeth profile is then composed from 11 repetitions of standard pattern. The simulated teeth profiles were generated for the set of value 0.1 am 0.9, 0.1 ym 0.9 with increment 0.1. The tooth profile for the case am = ym = 0.1 is shown

a) b)

<sup>1</sup> 2 2 <sup>2</sup> 100 100(1 ) 100 <sup>100</sup>

Some of the other characteristics can be analytically expressed as well but expressions are complicated. Generated teeth profiles were used for computation of profiles characteristics. The dependence of these characteristics on the am and ym are shown in subsequent

*SD <sup>a</sup> ym am ym am R* (3)

*R am <sup>a</sup>* 1 1 *ym* (2)

Fig. 13. Detail of teeth profile for a) am = ym = 0.1 and for b)

Corresponding standard deviation SD is equal to

paragraphs.

It can be easily derived that the mean height Ra of teeth profile is equal to the

in the fig. 13a and for the case of am = ym = 0.9 is shown in the fig. 13b.

Fig. 12. Standard pattern

Data from contact based measurements of roughness often represents height variation on line transects of the surface. Usually, it is possible to obtain structural data for one direction of the fabric, whereas the results on the other direction do not give clear information about the respective structural patterns. Some contacts-less methods based on the image analysis are found to be capable for measuring fabric structural pattern in the whole plane.

Standard methods of surface profile evaluation are based on the relative variability characterized by the variation coefficient - analogy with evaluation of yarns mass unevenness or simply by the standard deviation. This approach is used in Shirley software for evaluation of results for step thickness meter.

Common parameters describing roughness of technical surfaces are given in the ISO 4287 standard (Anonym, 1997). For characterization of roughness of textiles surfaces the mean absolute deviation MAD (SMD as per Kawabata) is usually applied (Meloun Militký, 2011). The descriptive statistical approach based on the assumptions of independence and normality leads to biased estimators, if the SHV has short or long-range correlation. There is therefore necessity to distinguish between standard white Gauss noise and more complex models. For description of short range correlation the models based on the autoregressive moving average are useful (Maisel, 1971). The long-range correlation is characterized by the fractal models (Beran, 1984; Whitehouse, 2001). The deterministic chaos type models are useful for revealing chaotic dynamics in deterministic processes where variation appears to be random but in fact predictable. For the selection among above mentioned models the power spectral density (PSD) curve evaluated from experimental SHV can be applied (Eke, 2000; Quinn Hannan, 2001).

Especially the fractal models (Mandelbrot Van Ness, 1968) are widely used for rough surface description. For these models the dependence of log (PSD) on the log (frequency) should be linear. Slope of this plot is proportional to fractal dimension and intercept to the so-called topothesy. White noise has dependence of log (PSD) on the log (frequency), nearly horizontal plateau for all frequencies (the ordinates of PSD are independent and exponentially distributed with common variance). More complicated rough surfaces can be modeled by the Markov type processes. For these models the dependence of log (PSD) on the log (frequency) has plateau at small frequencies, then bent down and are nearly linear at high frequencies (Sacerdotti et. al, 2000). A lot of recent works is based on the assumption that the stochastic process (Brownian motion) can describe fabrics surface variation ( Sacerdotti et. al, 2000). It is clear that for the deeper analysis of rough surface, the more complex approach should be used.
