**5. Discussion**

#### **5.1. Random irregularity**

The fiber flocks length distribution is always considered of a great importance for textile laps and web processing. Nowadays, it is still a source of statistical interpretations more or less empirical, such as cumulative frequency diagram.

In the following part, the cumulative frequency function *q*(ℓ) of the fiber length ℓ is calculated by taking into account the shape of the most common distributions as isoprobable, equiprob‐ able, and uniform distributions.

Based on textile sciences, these diagrams are usually represented by permuting the coordi‐ nated axes, that is, by plotting the value of the fiber length ℓ in the ordinate and *q*(ℓ) in abscissa, as shown in Table 1.

**Table 1.** Diagrams of distribution

**Table 1.** Diagrams of distribution

The usual empirical criterion for the quality estimation of the fiber web is based on the shape of the diagram. If all fiber flocks have the same length ℓ=ℓ*<sup>m</sup>* <sup>=</sup>ℓ¯ and same *<sup>m</sup>* mass, the diagram of cumulative frequency would obviously be a rectangle (Table 1); ℓ¯ and ℓ*<sup>m</sup>* being, respectively, the mean and the maximum flock fiber length. This type of distribution is designated as "isoprobable". In case of a cumulative frequency of length distribution triangular shape, the distribution is considered an equiprobable distribution. Such a shape shows the presence of a linear unevenness of length. The combination the two previous distributions leads to define a new third distribution called "uniform". In this distribution, the fiber length decreases linearly between two values *b* and *c* as shown in Table 1. Considering the three above-described distributions, the calculated mean lengths are given in Table 1 and are, respectively, mathe‐ matically expressed as follows:

$$
\overline{\ell} = \ell\_{\text{in}}
$$

$$
\overline{\ell} = \frac{\ell\_{\text{in}}}{2} \text{ and }
$$

$$
\overline{\ell} = \frac{c+b}{2}.
$$

Based on the shape of the diagram of cumulative frequency, the function *q*(ℓ) can be defined. On the other hand, the autocorrelation function *ρ*(ℓ=*u*) of the fiber flocks is given by the following equation:

$$\rho\left(u\right) = \frac{1}{\overline{\ell}} \cdot \int\_{\ell - u}^{\ell\_n} g\left(\ell\right) d\ell \tag{19}$$

where ℓ*m* is the length of the longest fiber flocks.

**Table 1.** Diagrams of distribution

66 Non-woven Fabrics

Cumulative frequency diagram of the fiber length

*<sup>m</sup> l L*

*<sup>m</sup> l L*

matically expressed as follows:

**Table 1.** Diagrams of distribution

*<sup>S</sup>*

**Isoprobable distribution Equiprobable distribution Uniform distribution**

**0 0,5 1**

*<sup>m</sup>* <sup>2</sup>

 *q*

**c**

**b**

*bc*

**<sup>0</sup> <sup>1</sup>** *<sup>q</sup>*

*bl <sup>L</sup> <sup>A</sup>* ;

 *u*

*cl <sup>L</sup> <sup>b</sup> <sup>A</sup>* ;

 22 2

<sup>2</sup> *bc uc <sup>u</sup>* 

<sup>1</sup> *u* 1

**<sup>0</sup> <sup>1</sup>**

*u*

 *<sup>m</sup>* 

Mean length of fiber web flock *<sup>m</sup>* <sup>2</sup>

 *q* **(l)**

 *m* 

**Autocorrelation function:**

*<sup>u</sup>* <sup>1</sup> <sup>2</sup>

*<sup>A</sup>* <sup>0</sup> <sup>0</sup> <sup>0</sup>

The usual empirical criterion for the quality estimation of the fiber web is based on the shape of the diagram. If all fiber flocks have the same length ℓ=ℓ*<sup>m</sup>* <sup>=</sup>ℓ¯ and same *<sup>m</sup>* mass, the diagram of cumulative frequency would obviously be a rectangle (Table 1); ℓ¯ and ℓ*<sup>m</sup>* being, respectively, the mean and the maximum flock fiber length. This type of distribution is designated as "isoprobable". In case of a cumulative frequency of length distribution triangular shape, the distribution is considered an equiprobable distribution. Such a shape shows the presence of a linear unevenness of length. The combination the two previous distributions leads to define a new third distribution called "uniform". In this distribution, the fiber length decreases linearly between two values *b* and *c* as shown in Table 1. Considering the three above-described distributions, the calculated mean lengths are given in Table 1 and are, respectively, mathe‐

> and 2

> > . 2

*c b*

<sup>+</sup> <sup>=</sup>

*m m*

=

ll <sup>l</sup> <sup>l</sup>

l

=

 *dqu <sup>m</sup> <sup>u</sup>* <sup>1</sup> 

> *u*

<sup>2</sup> <sup>1</sup> *u*

It can be highlighted that the autocorrelation function is a double integration of the histogram (frequency distribution function) of a fiber web numerical sample.

The cumulative frequency function *q*(ℓ) and the autocorrelation function exist only if the cut length of a fibrous web *<sup>l</sup>* <sup>=</sup> *<sup>A</sup> <sup>L</sup>* is small, that is, *<sup>l</sup>* ≤ℓ*<sup>m</sup>* as clearly shown in Table 1. Otherwise if *A <sup>L</sup>* <sup>=</sup>*l* is large, *<sup>l</sup>* ≥ℓ*m*, we can demonstrate easily that *ρ*(*u*)=0. Considering these parameters, the between-area-density variance *B*(*A*) can be calculated using the following equation, if *l* ≤ℓ*m*:

$$B\left(A\right) = \frac{2\,\,B\left(0\right)}{l^2} \int\_0^l \left(l-u\right)\cdot\rho\left(u\right)du\tag{20}$$

While, if *<sup>A</sup> <sup>L</sup>* <sup>=</sup>*l* is large, *<sup>l</sup>* ≥ℓ*m*, we have *ρ*(*u*)=0. The function *B*(*A*) takes the following expression:

$$B(A) = B(0)\left[1 - \frac{2}{l^2} \int\_0^{\ell\_m} (l - u) \cdot \left(1 - \rho(u)\right) du - \frac{2}{l^2} \int\_{\ell\_m}^l (l - u) du\right] \tag{21}$$

Then *B*(*A*) and *C B* 2(*A*) can be normalized:

$$\mathcal{B}\left(A\right) = \frac{\mathcal{B}\left(A\right)}{\mathcal{B}\left(0\right)} = \frac{\mathcal{C}\mathcal{B}^2\left(A\right)}{\mathcal{C}\mathcal{B}^2\left(0\right)}\tag{22}$$

with *C B* 2(*A*)= *<sup>B</sup>*(*A*) *E*(*μ*)<sup>2</sup>

For a fiber web having the **isoprobable distribution** of the fiber flocks, the normalized between-area-density variance *β*(*A*) takes the following expressions:

$$\text{if } l \le \ell\_m \quad \beta \left( A \right) = \frac{B \left( A \right)}{B \left( 0 \right)} = \frac{CB^2 \left( A \right)}{CB^2 \left( 0 \right)} = \left[ 1 - \frac{l}{3 \overline{\ell}} \right] \tag{23}$$

$$\text{if } l \ge \ell\_m \quad \beta\left(A\right) = \frac{B\left(A\right)}{B\left(0\right)} = \frac{CB^2\left(A\right)}{CB^2\left(0\right)} = \left[\frac{\overline{\ell}}{l} - \frac{\overline{\ell}^2}{3l^2}\right] \tag{24}$$

The difference between a "short" and a "long" fiber web element is shown by the diagrams of *<sup>β</sup>*(*A*) in Figures 9 and 10. In case of *<sup>l</sup>* ≤ℓ*m*, the functions *B*(*A*) and *C B* 2(*A*) present straight segments in the interval 0, ℓ*m* as mentioned in Figures 9 and 10. The slope *β*(*A*) curve at the origin is <sup>−</sup> <sup>1</sup> <sup>3</sup>ℓ¯ . It can be noticed that the two branches of the *β*(*A*) graph are connected to point B coordinates (ℓ¯, <sup>2</sup> 3 ). **Figure 9.** Short‐term irregularity – between‐area‐density variance vs. fiber web length: Case of length isoprobable distribution

**Figure 9.** Short-term irregularity – between-area-density variance vs. fiber web length: Case of length isoprobable dis‐ tribution

Otherwise, for the **equiprobable distribution**, the corresponding diagram has a triangular shape (see Table 1). By a similar calculation to the previous we obtain two expressions for *β*(*A*):

$$\text{if } \begin{array}{c} 0 \le l \le \ell\_m \end{array} \begin{array}{c} \mathcal{B} \begin{pmatrix} A \\ \end{pmatrix} = \frac{\mathcal{B} \begin{pmatrix} A \\ \end{pmatrix}}{B(0)} = \frac{\mathcal{C} \mathcal{B}^2 \begin{pmatrix} A \\ \end{pmatrix}}{\mathcal{C} \mathcal{B}^2 \begin{pmatrix} 0 \\ \end{pmatrix}} = \left[ 1 - \frac{l}{\overline{3} \, \overline{\ell}} + \frac{l^2}{2 \mathbf{4} \, \overline{\ell}^2} \right] \tag{25}$$

$$\text{if } \; l > \ell\_m \; \; \beta \left( A \right) = \frac{B \left( A \right)}{B \left( 0 \right)} = \frac{C B^2 \left( A \right)}{C B^2 \left( 0 \right)} = \left[ \frac{4 \overline{\ell}}{3l} - \frac{2 \overline{\ell}^2}{3l^2} \right] \tag{26}$$

**Figure 10.** Long-term irregularity – between-area-density variance vs. fiber web length: Case of length isoprobable dis‐ tribution

It can be noticed that the graph of *β*(*A*) related to the equiprobable distribution is always above the one corresponding to the associated isoprobable distribution for the same ℓ¯ (Figure. 11).

Finally, the *q*(ℓ) **uniform distribution** that has a trapezoidal shape (Table 1) has the mean and the variation coefficient respectively expressed [22] as follows:

$$
\overline{\ell} = \frac{c+b}{2} \text{ and } \operatorname{CV\%} = \frac{100}{\sqrt{3}} \cdot \left(\frac{c-b}{c+b}\right) \tag{27}
$$

It can be noticed that if *b* =*c* =ℓ¯ the distribution is isoprobable, then if *b* =0 and *c* =2ℓ¯ the distribution is equiprobable.

In this case of uniform distribution, the cumulative frequency function and the autocorrelation function have, respectively, the following expressions:

$$\begin{aligned} \text{if } \; 0 \le \bigwedge\_{L} & l \le b, \; q\_1(\ell) = 1 \; \text{hence:}\\ \rho\_1(u) = \frac{1}{\overline{\ell}} \int\_l^{\epsilon\_u - c} q(u) du &= \frac{1}{\overline{\ell}} \int\_l^b q\_1(u) du + \frac{1}{\overline{\ell}} \int\_b^c q\_2(u) du \end{aligned} \tag{28}$$

After integration *ρ*<sup>1</sup> (*u*) becomes:

( ) ( ) ( )

*l A*

origin is <sup>−</sup> <sup>1</sup>

68 Non-woven Fabrics

B coordinates (ℓ¯, <sup>2</sup>

**0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1**

tribution

*β*(*A*):

 **(A)** 3 ).

of length isoprobable distribution

b

2 2 if 0 0 3 *<sup>m</sup> B A CB A*

( ) ( )

<sup>3</sup>ℓ¯ . It can be noticed that the two branches of the *β*(*A*) graph are connected to point

*B CB l l*

 é ù ³ = = =- ê ú

The difference between a "short" and a "long" fiber web element is shown by the diagrams of *<sup>β</sup>*(*A*) in Figures 9 and 10. In case of *<sup>l</sup>* ≤ℓ*m*, the functions *B*(*A*) and *C B* 2(*A*) present straight segments in the interval 0, ℓ*m* as mentioned in Figures 9 and 10. The slope *β*(*A*) curve at the

**Figure 9.** Short‐term irregularity – between‐area‐density variance vs. fiber web length: Case

**12 cm**

**6 cm**

**4 cm**

**0 5 10 15 20**

**Figure 9.** Short-term irregularity – between-area-density variance vs. fiber web length: Case of length isoprobable dis‐

Otherwise, for the **equiprobable distribution**, the corresponding diagram has a triangular shape (see Table 1). By a similar calculation to the previous we obtain two expressions for

( ) ( ) ( )

b

*l A*

b

l

2 2 if 0 1 (0) 0 3 24 *<sup>m</sup> B A CB A l l l A B CB*

( ) ( ) ( )

4 2 if (0) 0 3 3 *<sup>m</sup> B A CB A*

**1 cm**

**2 cm**

**Web length (cm)**

( )

( )

*B l CB l*

 é ù > = = =- ê ú

 é ù ££ = = = - + ê ú

2 2

2 2 2 2

ë û

ë û l l <sup>l</sup> (26)

<sup>l</sup> <sup>l</sup> (25)

2 2

ë û l l <sup>l</sup> (24)

$$\rho\_1(\mu) = 1 - \frac{l}{\overline{\ell}} \tag{29}$$

$$\begin{aligned} \text{if } & b < \bigwedge\_{L} = l \le c, \, q\_z \left( u \right) = \left( \frac{c - l}{c - b} \right) \text{hence:}\\ \, & \rho\_z \left( u \right) = \frac{1}{\ell} \int\_{l}^{\ell\_w - c} q(u) du = \frac{\left( c - l \right)^2}{c^2 - b^2} \end{aligned} \tag{30}$$

$$\begin{aligned} \text{if} \quad & \bigwedge\_{L} > c, \; q\_{3}(\ell) = 0, \; \text{hence}:\\ \rho\_{3}(\mu) = \frac{1}{\ell} \int\_{a}^{a} q(\mu) d\mu = 0 \end{aligned} \tag{31}$$

According to the values of *<sup>A</sup> <sup>L</sup>* , *β*(*A*) takes the following expressions [28,29]:

$$\text{if } 0 \le \bigwedge^{A} = l \le b \quad \beta\left(A\right) = \left[1 - \frac{l}{\Im \overline{\ell}}\right] \tag{32}$$

$$\text{if } b < \bigwedge\_{L} = l \le c \quad \beta\{A\} = \frac{2}{l^2} \cdot \left[ \frac{lcb}{2\overline{\ell}} - \frac{b^2\left(3c - b\right)}{12\overline{\ell}} + \frac{\left(c - b\right)^3}{24\overline{\ell}} \left( \left(\frac{c - l}{c - b}\right)^4 - 4\left(\frac{c - l}{c - b}\right) + 3\right) \right] \tag{33}$$

$$\text{if } \bigwedge^4 \underline{\mathcal{A}} \bigvee \mathcal{C} \text{ } \mathcal{B}\{A\} = \frac{2}{l^2} \cdot \left[ l \frac{\left(c^2 + b^2 + cb\right)}{\overline{6\ \ell}} - \frac{\left(c + b\right)\left(c^2 + b^2\right)}{24\ \overline{\ell}} \right] \tag{34}$$

As shown in Figure 12, if 0<sup>≤</sup> *<sup>A</sup> <sup>L</sup>* <sup>=</sup>*<sup>l</sup>* <sup>≤</sup>*b*, the functions *B*(*A*) and *C B* 2(*A*) present straight segments in the interval 0, *<sup>b</sup>* . The slope of *β*(*A*) curve at the origin is <sup>−</sup> <sup>1</sup> <sup>3</sup>ℓ¯ . Based on this figure, we can note that if *<sup>b</sup> c* is low while the flocks length variation coefficient is high, the slope of *β*(*A*) decreases more slowly.

Considering the three above-described distributions, the random unevenness varies with the variation of the fibrous web length and as shown is Figures 9 and 10, if *l* increases, *β*(*A*) decreases.

This can be explained by a greater possibility of compensating local irregularities in case of longer surfaces of fiber web. Normally, the *β*(*A*) fiber web curve has a decreasing course, tending toward zero for *l* →*∞*.

<sup>70</sup> Non-woven Fabrics **Figure 11.** Comparison between the *A* curves of the isoprobable and equiprobable distributions for the same length Variance Analysis and Autocorrelation Function for 2D Fiber Lap Statistical Analysis http://dx.doi.org/10.5772/61795 71

( )

( )

l

<sup>1</sup> <sup>0</sup> *<sup>a</sup>*

if , 0, hence :

= = ò

3

> =

2

*<sup>L</sup>* , *β*(*A*) takes the following expressions [28,29]:

3

ë û <sup>l</sup> (32)

<sup>3</sup>ℓ¯ . Based on this figure, we can

(30)

(31)

, hence:

è ø -

*c l*

2

æ ö - < =£ = ç ÷

( ) ( ) ( )

*c l b l cq u <sup>A</sup> L c b*

> *<sup>m</sup> c l*

*<sup>A</sup> c q <sup>L</sup>*

l l

1

<sup>=</sup>

3

r

2

<sup>2</sup> if

b

in the interval 0, *<sup>b</sup>* . The slope of *β*(*A*) curve at the origin is <sup>−</sup> <sup>1</sup>

b

2 3

2

if

r

According to the values of *<sup>A</sup>*

As shown in Figure 12, if 0<sup>≤</sup> *<sup>A</sup>*

*c*

tending toward zero for *l* →*∞*.

decreases more slowly.

note that if *<sup>b</sup>*

decreases.

2 2 2

( ) ( )

if 0 ( ) 1

*<sup>l</sup> <sup>A</sup> lb A <sup>L</sup>* bé ù £ =£ = - ê ú

( ) ( ) ( )<sup>3</sup> <sup>2</sup> <sup>4</sup>

é ù - - æ ö æ öæ ö - - < =£ = × - + ê ú ç ÷ ç ÷ç ÷ - + ê ú è øè ø - - ë û è ø ll l (33)

( ) ( ) ( )( ) 2 2 2 2

> =× - ê ú

Considering the three above-described distributions, the random unevenness varies with the variation of the fibrous web length and as shown is Figures 9 and 10, if *l* increases, *β*(*A*)

This can be explained by a greater possibility of compensating local irregularities in case of longer surfaces of fiber web. Normally, the *β*(*A*) fiber web curve has a decreasing course,

6 24

is low while the flocks length variation coefficient is high, the slope of *β*(*A*)

ë û l l (34)

*<sup>L</sup>* <sup>=</sup>*<sup>l</sup>* <sup>≤</sup>*b*, the functions *B*(*A*) and *C B* 2(*A*) present straight segments

é ù ++ + +

if 4 3 2 12 24 *lcb b cb cb cl cl b lc A <sup>A</sup> L l cb cb*

*c b cb c b c b <sup>A</sup> cA l <sup>L</sup> <sup>l</sup>*

l

*u q u du* ¥

*u q u du c b*


**Figure 11.** Comparison between the *β*(*A*) curves of the isoprobable and equiprobable distributions for the same length ℓ¯ **Figure 12.** Short‐term irregularity – between‐area‐density variance vs. fiber web length: Case of length uniform distribution

**Figure 12.** Short-term irregularity – between-area-density variance vs. fiber web length: Case of length uniform distri‐ bution

#### **5.2. Periodic unevenness**

We suppose that the area density of the fibrous web is defined as follows:

$$
\mu \left( t \right) = \mu + a \cos \left( at \right) \tag{35}
$$

where *μ* : average area density, assumed constant *a*: amplitude of the sinusoidal component irregularity *ω* : pulsation of the sinusoidal component irregularity We have seen that *B*(*A*) can be written as follows:

$$B\left(A\right) = E\left[\frac{L}{A}\int\_{x}^{x\cdot s} \mathcal{H}\left(\mu\left(t\right) - E\left(\mu\right)\right)dt\right]^2\tag{36}$$

with *E*(*μ*(*t*))=*μ*.

Hence by replacing *μ*(*t*) by its value:

$$B\left(A\right) = E\left[\frac{L}{A}\int\_{x}^{x \ast A \parallel L} a \cos\left(\alpha t\right) dt\right]^2\tag{37}$$

and finally after integration:

$$\mathcal{B}\left(A\right) = \frac{a^2}{2} \cdot \frac{\sin^2\left(\frac{a\nu \cdot A/L}{2}\right)}{\left(\frac{a\nu \cdot A/L}{2}\right)^2} \text{ and } \mathcal{B}\left(0\right) = \frac{a^2}{2} \tag{38}$$

$$B(A) = B(0) \cdot \frac{\sin^2\left(\frac{\omega \cdot A / L}{2}\right)}{\left(\frac{\omega \cdot A / L}{2}\right)^2} \text{ and the normalized for } \beta(A) = \frac{B(A)}{B(0)} = \frac{\sin^2\left(\frac{\omega \cdot A / L}{2}\right)}{\left(\frac{\omega \cdot A / L}{2}\right)^2}.$$

Hence we deduce the corresponding variation coefficient:

4 *m*

$$\begin{aligned} \text{CB}\{A\} &= \text{CB}\{0\} \cdot \frac{\sin\left|\frac{\alpha \cdot A/L}{2}\right|}{\left(\frac{\alpha \cdot A/L}{2}\right)} \text{ and} \\ \text{IV}\{A\} &= \frac{\text{CB}\{A\}}{\text{CB}\{0\}} = \frac{\sin\left|\frac{\alpha \cdot A/L}{2}\right|}{\left(\frac{\alpha \cdot A/L}{2}\right)} \text{ (normalized form)} \end{aligned} \tag{39}$$

where *<sup>ω</sup>* <sup>=</sup> <sup>2</sup>*<sup>π</sup> <sup>v</sup> <sup>λ</sup>* is the pulsation of the irregularity sinusoidal component

*v* represents the fiber flow velocity through the sensor

*λ* is the wavelength irregularity

**5.2. Periodic unevenness**

72 Non-woven Fabrics

with *E*(*μ*(*t*))=*μ*.

*B*(*A*)= *B*(0) ⋅

Hence by replacing *μ*(*t*) by its value:

and finally after integration:

sin<sup>2</sup>

( *<sup>ω</sup>* <sup>⋅</sup> *<sup>A</sup>* / *<sup>L</sup>* <sup>2</sup> )

Hence we deduce the corresponding variation coefficient:

( *<sup>ω</sup>* <sup>⋅</sup> *<sup>A</sup>* / *<sup>L</sup>* <sup>2</sup> )

We suppose that the area density of the fibrous web is defined as follows:

mm

where *μ* : average area density, assumed constant

*a*: amplitude of the sinusoidal component irregularity

*ω* : pulsation of the sinusoidal component irregularity

We have seen that *B*(*A*) can be written as follows:

 w

( ) ( () ( )) <sup>2</sup> *<sup>x</sup> <sup>A</sup> <sup>L</sup>*

( ) ( ) <sup>2</sup> cos

é ù <sup>+</sup> = ê ú

2 2 2

*A L*

æ ö <sup>×</sup> ç ÷ è ø <sup>=</sup> × =

<sup>2</sup> and 0 2 2

*x AL x <sup>L</sup> B A E a t dt A*

( ) ( )

w

*a a B A <sup>B</sup> A L*

<sup>2</sup> and the normalized for *β*(*A*)= *<sup>B</sup>*(*A*)

2

æ ö <sup>×</sup> ç ÷ è ø

2

w

sin

m

é ù <sup>+</sup> = - ê ú

 m

w

*x <sup>L</sup> BA E t E dt*

*A*

() ( ) *t at* = + cos (35)

ë û ò (36)

ë û ò (37)

*<sup>B</sup>*(0) <sup>=</sup>

sin<sup>2</sup>

( *<sup>ω</sup>* <sup>⋅</sup> *<sup>A</sup>* / *<sup>L</sup>* <sup>2</sup> )

<sup>2</sup> .

( *<sup>ω</sup>* <sup>⋅</sup> *<sup>A</sup>* / *<sup>L</sup>* <sup>2</sup> ) (38)

The *β*(*A*) and *γ*(*A*) curves show a periodic-arches succession with a decreasing amplitude and a *πλ <sup>v</sup>* (Figures 13 and 14). The shape of these curves is totally different from those correspond‐ ing to the random distribution. This result is very significant for the production manager in order to detect the defects origin.

**Figure 13.** Normalized between‐area‐density variance curve – periodic irregularity

**Figure 13.** Normalized between-area-density variance curve – periodic irregularity *λ* =4 *m*

irregularity

4 *m*

**Figure 14.** Normalized between-area-density variance coefficient curve – periodic irregularity *λ* =4 *m*

#### **5.3. Compound unevenness**

In the above-mentioned development, we have intentionally unheeded the random compo‐ nent and assumed the presence of only one type of periodic sinusoidal irregularity. In fact, the random component is always present in the 2D textile fibrous structures and is existing; the additional components are of several sinusoidal types generated by the manufacturing process. That way, the periodic irregularity has to be split according to Fourier series [17] and the composition of the *B*(*A*) variance concerning the various components (random and periodic) of the area density irregularity have to be calculated. The calculation is based on the additivity property of variances of independent variables.

Firstly, only one type of sinusoidal periodic irregularity is taken in account as follows:

$$M(\mathbf{x}) = \mu(\mathbf{x}) + a\cos(\alpha \mathbf{x})\tag{40}$$

where *μ*(*x*)= random component of the area density of the fibrous web

*a*cos(*ω x*)= sinusoidal component of amplitude *a* and pulsation *ω*

Based on the law of additive covariances composition [26], we deduce the expression of the autocorrelation function *ρ*(*u*):

$$\rho\left(u\right) = \left(\frac{\sigma\_{\
u}^{2}}{\sigma\_{\mu}^{2}}\right) \cdot \rho\_{\mu}\left(u\right) + \left(\frac{\sigma\_{\
p}^{2}}{\sigma\_{\mu}^{2}}\right) \cdot \rho\_{\mu}\left(u\right) \tag{41}$$

Let *<sup>η</sup>* =( *<sup>σ</sup><sup>p</sup>* 2 *σμ* <sup>2</sup> ). With *η* being the weighting factor (0≤*η* ≤1).

$$\rho\left(u\right) = \left(1 - \eta\right) \cdot \rho\_{\ast}\left(u\right) + \left(\eta\right) \cdot \rho\_{\rho}\left(u\right) \tag{42}$$

with *σ<sup>μ</sup>* <sup>2</sup> = overall variance of the area density irregularity

*σa* <sup>2</sup> = variance due to the random component

**Figure 14.** Normalized between‐area‐density variance coefficient curve – periodic

**0 5 10 15 20 25 30**

**Web length (m)**

In the above-mentioned development, we have intentionally unheeded the random compo‐ nent and assumed the presence of only one type of periodic sinusoidal irregularity. In fact, the random component is always present in the 2D textile fibrous structures and is existing; the additional components are of several sinusoidal types generated by the manufacturing process. That way, the periodic irregularity has to be split according to Fourier series [17] and the composition of the *B*(*A*) variance concerning the various components (random and periodic) of the area density irregularity have to be calculated. The calculation is based on the

Firstly, only one type of sinusoidal periodic irregularity is taken in account as follows:

*Mx x a x* () () ( ) = + m

Based on the law of additive covariances composition [26], we deduce the expression of the

 m  r

s

 s

( ) ( ) ( ) <sup>2</sup> <sup>2</sup> 2 2 *p a a p uuu*

æ ö æ ö <sup>=</sup> ç ÷×+× ç ÷ ç ÷ ç ÷ è ø è ø

where *μ*(*x*)= random component of the area density of the fibrous web

m

s

s

rr

*a*cos(*ω x*)= sinusoidal component of amplitude *a* and pulsation *ω*

 w

cos (40)

(41)

**Figure 14.** Normalized between-area-density variance coefficient curve – periodic irregularity *λ* =4 *m*

additivity property of variances of independent variables.

irregularity

74 Non-woven Fabrics

**0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1**

**5.3. Compound unevenness**

autocorrelation function *ρ*(*u*):

 **(A)** 4 *m*


The figure [28] shows that the standardized form of the variance composition can be written as follows:

$$
\beta\left(A\right) = \eta \cdot \beta\_p\left(A\right) + \left(1 - \eta\right) \cdot \beta\_s\left(A\right) \tag{43}
$$

The extension of the above results to the more general case of the superposition of n sinusoidal type irregularities can be written as follows:

$$M\left(\mathbf{x}\right) = \mu\_a\left(\mathbf{x}\right) + \sum\_{1}^{n} a\_i \cos\left(\alpha\_i \mathbf{x} + \phi\_i\right) \tag{44}$$

where *μa* (*x*) is the random component and the following term (∑ 1 *n ai* cos(*ωix* + *ϕ<sup>i</sup>* ) is the element due to Fourier series developments of the periodic components. Hence

$$\mathcal{A}\left(A\right) = \sum\_{1}^{n} \left[\eta\_{i} \cdot \mathcal{J}\_{p\_{i}}\left(A\right)\right] + \left(1 - \sum\_{1}^{n} \eta\_{i}\right) \cdot \mathcal{J}\_{a}\left(A\right) \tag{45}$$

*ηi* being the weighting factor for the periodic component of rank i.

Figures 15 and 16 show some simulation examples of compositions of one or two sinusoidal irregularities with a random irregularity (isoprobable distribution). The interpretation of the curves *β*(*A*) respectively *γ*(*A*) becomes much more difficult. Of course the presence of "arches" or "sinuosities" indicates the presence of periodic irregularity.

**Figure 15.** Normalized between‐area‐density variance curve – compound irregularity

**Figure 15.** Normalized between-area-density variance curve – compound irregularity

**Figure 16.** Normalized between-area-density variance coefficient curve – compound irregularity.

#### **6. Conclusion**

Nonwoven industries are fiber-based industries. Based on their physical and mechanical properties, the nonwoven applications are increasing inducing a strong growth of the non‐

**Figure 16.** Normalized between‐area‐density variance coefficient curve – compound irregularity.

woven production. The physical and mechanical properties of these products strongly depend on the evenness of laps or webs. Thus, the nonwoven manufacturers absolutely need tools to evaluate these properties. Hence, an accurate knowledge of the web and lap formation as well as the associated theoretical approach is absolutely necessary. In this paper we propose a theoretical approach simulating the real faults of the fiber-web forming-step during the industrial process. Thanks to this theory, we were able to calculate from measurements both the random and periodic components of the real defects for all textile types as fibrous webs and nonwovens. The periodic component depends on the production machine and the random component depends mainly on the characteristics of the fibers. In this study, we are not interested in uniformity of the visual appearance (2-D uniformity) of the fiber web but we assess the 3-D uniformity from the area density. This uniformity is determined from small surface-elements and not from image analysis [16,24]. We did not analyze the correlation between the irregularity of the yarns and the woven fabric [20] but measured directly surface and mass irregularities thereof. This involves the use of specific sensors (such as capacitive, ultrasonic, and radioactive sensors). To highlight these irregularities (random or periodic), the fiber web is divided into several elements that have the same surface. Then the between-areadensity variance function is defined with the help of the autocorrelation function. The common distribution functions of the fiber flocks are used to carry out the random component of the equation. Finally, according to the law of additive covariances composition, periodic defects have been added to the random component in order to determine the theoretical equation. This information is fundamental to the production manager in order to detect the earliest possible manufacturing breakdown and to optimize the machine settings and textile produc‐ tion quality.
