**7. Classical roughness characteristics**

Because the basic output form RCM is set of "slices" (roughness profiles in the cross direction at selected position in machine direction) it is possible to compute all profile roughness characteristics separately for each slice and show the differences between slices. Another possibility is to use the reconstructed surface roughness plane for evaluation of planar roughness.

There are two reasons for measuring surface roughness. First, is to control manufacture and is to help to ensure that the products perform well. In the textile branch the former is the case of special finishing (e.g. pressing or ironing) but the later is connected with comfort, appearance and hand.

From a general point of view, the rough surface display process which have two basic geometrical features:


The random part of roughness can be suppressed by proper smoothing. In this case the only structural part will be evaluated.

From the individual roughness profiles, it is possible to evaluate a lot of roughness parameters. Classical roughness parameters are based on the set of points *R(dj ) j =1.. N*  (SHV) defined in the sample length interval *Ls*. The distances *dj* are obviously selected as equidistant and then *R(dj)* can be replaced by the variable *Rj* . For identification of positions in length scale, it is sufficient to know that sampling distance *ds = dj - dj-1 = Ls/N* for *j>1.*The standard roughness parameters used frequently in practice are (Anonym, 1997):

i. Mean Absolute Deviation *MAD.* This parameter is equal to the mean absolute difference of surface heights from average value *(Ra)*. For a surface profile this is given by,

$$MAD = \frac{1}{N} \sum\_{j} \left| R\_j - \overline{R} \right| \tag{15}$$

This parameter is often useful for quality control and textiles roughness characterization (called SMD (Kawabata, 1980)). However, it does not distinguish between profiles of different shapes. Its properties are known for the case when *Rj's* are independent identically distributed (i. i. d.) random variables. For rough SHV from fig. 4, dependence of SMD on aggregation length L is shown in fig. 19.

ii. Standard Deviation (Root Mean Square) Value *SD*. This characteristics is given by

$$SD = \sqrt{\frac{1}{N} \sum\_{j} (R\_j - \overline{R})^2} \tag{16}$$

Woven Fabrics Surface Quantification 139

than MAD for monitoring certain surfaces having large deviations (corresponding

Variance

am

iii. The Standard Deviation of Profile Curvature *PC*. This quantity called often as waviness

*d Rx PC N dx* 

For rough SHV from fig. 4 dependence of PC on aggregation length L is shown in fig. 22.

Roughness curvature

0 20 40 60 80 100

length L

The influence of teeth profile parameters *am* and *ym* on the PS are shown in the fig. 23.

The curvature is characteristics of a profile shape. The *PS* parameter is useful in tribological applications. The lower the slope, the smaller will be the friction and wear. Also, the

1 ()

<sup>2</sup> <sup>2</sup> 2

*j j*

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.1

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

(17)

distribution has heavy tail).

ym

is defined by the relation

0

0.01

0.02

0.03

mean curvature

0.04

0.05

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Fig. 21. Influence of *am* and *ym* on the variance (square of SD)

reflectance property of a surface increases in the case of small *PC*.

Fig. 22. Dependence of SMD on aggregation length *L* for twill fabric

Fig. 19. Dependence of SMD on the aggregation length *L* for twill fabric

The influence of teeth profile parameters am and ym on MAD is shown in the fig. 20.

Fig. 20. Influence of am and ym on the MAD

Its properties are known for the case when *Rj's* are independent identically distributed (i.i.d.) random variables. One advantage of SD over MAD is that for normally distributed data, it can be simple to derive confidence interval and to realize statistical tests. SD is always higher than MAD and for normal data *SD = 1.25 MAD.* It does not distinguish between profiles of different shapes as well. The parameter *SD* is less suitable than MAD for monitoring certain surfaces having large deviations (corresponding distribution has heavy tail).

The influence of teeth profile parameters *am* and *ym* on the square of SD (i.e. variance) is shown in the fig. 21.

It is visible that the MAD and SD have similar dependence on the tooth parameters *am* and *ym.* SD is always higher than MAD and for normal data SD = 1.25 MAD. It does not distinguish between profiles of different shapes as well. The parameter SD is less suitable

Roughness Kaw SMD

0 20 40 60 80 100

length L

The influence of teeth profile parameters am and ym on MAD is shown in the fig. 20.

Mean absolute error

am

Its properties are known for the case when *Rj's* are independent identically distributed (i.i.d.) random variables. One advantage of SD over MAD is that for normally distributed data, it can be simple to derive confidence interval and to realize statistical tests. SD is always higher than MAD and for normal data *SD = 1.25 MAD.* It does not distinguish between profiles of different shapes as well. The parameter *SD* is less suitable than MAD for monitoring certain surfaces having large deviations (corresponding distribution has heavy

The influence of teeth profile parameters *am* and *ym* on the square of SD (i.e. variance) is

It is visible that the MAD and SD have similar dependence on the tooth parameters *am* and *ym.* SD is always higher than MAD and for normal data SD = 1.25 MAD. It does not distinguish between profiles of different shapes as well. The parameter SD is less suitable

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.1

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

Fig. 19. Dependence of SMD on the aggregation length *L* for twill fabric

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04

ym

tail).

shown in the fig. 21.

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Fig. 20. Influence of am and ym on the MAD

Kaw SMD

than MAD for monitoring certain surfaces having large deviations (corresponding distribution has heavy tail).

Fig. 21. Influence of *am* and *ym* on the variance (square of SD)

iii. The Standard Deviation of Profile Curvature *PC*. This quantity called often as waviness is defined by the relation

$$PC = \sqrt{\frac{1}{N} \sum\_{j} \left( \frac{d^2 R(\mathbf{x})}{d\mathbf{x}^2} \right)\_j^2} \tag{17}$$

The curvature is characteristics of a profile shape. The *PS* parameter is useful in tribological applications. The lower the slope, the smaller will be the friction and wear. Also, the reflectance property of a surface increases in the case of small *PC*.

For rough SHV from fig. 4 dependence of PC on aggregation length L is shown in fig. 22.

Fig. 22. Dependence of SMD on aggregation length *L* for twill fabric

The influence of teeth profile parameters *am* and *ym* on the PS are shown in the fig. 23.

Woven Fabrics Surface Quantification 141

The primary tool for evaluation of periodicities is expressing of signal *R(d)* by the Fourier series of sine and cosine wave. It is known that periodic function given by equally spaced values *Ri, i = 0, ..., N – 1* can be generally expressed in the form of Fourier series at Fourier frequencies *fj = j/N, 1 ≤ j ≤ [N/2]*. If *N* is odd with *N = 2m + 1*, the Fourier series has form

cos( ) sin( ) 0,.. - 1

(18)

1,..

 

> 

(21)

(22)

(20)

(19)

 

*f k Nk m k = 1, ..., m* are angular frequencies. The eqn. (17) is

<sup>0</sup>

*Ra a ib i i N* 

for known frequencies harmonic linear regression model with *2m + 1* parameters (intercept and *2m* sinusoids amplitudes at the *m* Fourier frequencies). The sinusoid with the *j-th* Fourier frequency completes exactly *j* cycles in the span of the data. Due to selection of Fourier frequencies all regressors (*sin(.)* and *cos(.)* terms) are mutually orthogonal, so that

*i k kk k*

1 1

*N N*

*i i*

2 cos( ) 2 sin( )

*i k i k*

Basic statistical characteristic in the frequency domain is power spectral density PSD defined

The simple estimator of power spectral density is called periodogram. The periodogram of

0 0 1 1 ( ) cos( ) sin( ) *N N*

> 2 2 ( ) ( ) 1,.. <sup>4</sup> *k kk <sup>N</sup> I ab k m*

The periodogram ordinates correspond to analysis of variance decomposition into *m* 

*k i*

 *R R* 

1 0 ( ) 0.5 *m N*

*k i*

*I* 

*i i I R i Ri N N*

*i i*

2 2 1 1

 <sup>1</sup> <sup>2</sup>

*R i Ri a b k m N N*

0 0

1

standard least-squares method leads to estimates 0*a R* and

*k k*

an equally spaced series *Ri, i = 0, ..., N – 1* is defined by equation

For rough SHV from fig. 4 periodogram is shown in fig. 25.

orthogonal terms with *2* degrees of freedom each because,

*k*

*m*

**8. Spectral analysis** 

(Quinn Hannan, 2001)

where 2 2 / 1,.. *k k*

 

as Fourier transform of covariance function.

and can be expressed in the alternate form

The normalized periodogram with ordinates

Fig. 23. Influence of *am* and *ym* on the PC

It is visible that the growing of *am* (i.e. decreasing of tooth thickness) leads to the increase of PC. Lowest values of PC are around *ym* equal to the 0.5. This behavior is "inverse" to the behavior of MAD and SD. The PC parameter is useful in tribological applications. The lower the slope the smaller will be the friction and wear. Also, the reflectance property of a surface increases in the case of small PC.

The *MAD* and *PC* characteristics for all slices for cord fabric (see fig. 11) are shown in the fig. 24.

Fig. 24. The a) MAD and b) PC values for all slices of cord fabric (fig. 11)

In the case of MAD, a systematic trend is visible. The variation of PC is nearly random.

For the characterization of hand, it will be probably the best to use waviness *PC*. The characteristics of slope and curvature can be computed for the case of fractal surfaces from power spectral density, autocorrelation function or variogram.
