**4. Simulated "teeth" profiles**

The surface of cord fabrics has typical "teeth" in the machine direction. For indication of the influence of geometry of teeth on the values of the roughness characteristics the simulated roughness profile in the cross direction was created. The standard pattern is composed from two parts (see. fig. 12).

The top height of one tooth is selected as 1. Bottom height of one tooth is equal to the value of ym. The tooth size is therefore 1 – ym. The length of distance between teeth is equal to am and tooth thickness is equal 1- am. Total length of standard pattern equal to the 1 and 100 individual values are generated, i.e. the standard pattern is characterized by 100 points with

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

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

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,

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

The surface of cord fabrics has typical "teeth" in the machine direction. For indication of the influence of geometry of teeth on the values of the roughness characteristics the simulated roughness profile in the cross direction was created. The standard pattern is composed from

The top height of one tooth is selected as 1. Bottom height of one tooth is equal to the value of ym. The tooth size is therefore 1 – ym. The length of distance between teeth is equal to am and tooth thickness is equal 1- am. Total length of standard pattern equal to the 1 and 100 individual values are generated, i.e. the standard pattern is characterized by 100 points with

are found to be capable for measuring fabric structural pattern in the whole plane.

for evaluation of results for step thickness meter.

2000; Quinn Hannan, 2001).

complex approach should be used.

**4. Simulated "teeth" profiles** 

two parts (see. fig. 12).

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

Fig. 12. Standard pattern

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

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

$$R\_a = 1 - am(1 - ym) \tag{2}$$

Corresponding standard deviation SD is equal to

$$SD = \sqrt{\frac{1}{100} \left( \left[ 100 \,\mathrm{ym}\,\,\mathrm{am}\right]^2 + \left[ 100(1 - \,\mathrm{ym})\,\mathrm{am}\right]^2 - 100R\_a^{-2}\right)}\tag{3}$$

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 paragraphs.

Woven Fabrics Surface Quantification 131

Before choosing the approach, some preliminary analysis is needed mainly to test the stationarity and linearity. This is important as some kind of stochastic (self affine) processes with power-law shape of their spectrum may erroneously be classified as chaotic processes on the basis of some properties of their non-linear characteristics, e.g., correlation dimension and Kolmogorov entropy. In this sense, the tests for stationarity and linearity may be regarded as a necessary preprocessing in order to choose an appropriate approach for further analysis. Prior to selecting any method for data analysis, some simple tests are useful to apply on the series *R(i)*. The first one may be to observe the *R(i)* distribution e.g. via histogram as simple estimator of probability density function (pdf) or by using kernel density estimator (Meloun Militký, 2011). The histogram of series *R(i)* corresponding to the

histogram

h = 0.024291


In this figure, the solid line corresponds to the Gaussian pdf with parameters: mean = 0.000524 and standard deviation = 0.0358. The dotted line is nonparametric kernel density

In most of the methods for data processing based on stochastic models, normal distribution is assumed. If the distribution is proved to be non-normal (according to some test or

2. the process has linear dynamics, but the observations are as a result of non-linear

It is suitable to construct the histograms for the four quarters of data separately and inspect non-normality or asymmetry of distribution. The statistical characteristics (mean and variances) of these sub series can support wide sense stationarity assumption (when their

estimator wit optimal bandwidth h= 0.0243. The bimodality pattern is clearly visible.

"static" transformation (e.g. square root of the current values)

raw SHV trace of twill weave fabric (shown in the fig. 4) is shown in fig. 14.

0

inspection), there are three possibilities:

1. the process is linear but non-gaussian;

3. the process has non-linear dynamics.

values are statistically indistinguishable).

Fig. 14. Histogram and pdfs of raw SHV for twill fabric

2

4

6

Rel. Freq.

8

10

12

14
