61

論 文

A Mathematical Model for Prediction of

Goniophotometric Curves for

Single-Jersey Knitted Fabrics

Alec Sirikasemlert, Xiaoming Tao, Rup Dhingra

AbstractA mathematical model prediction the distribution curves of the specular reflection light (goniophotometric

curves) of single-jersey knitted fabrics is proposed. The model is based on Nihira et. al.'s approach treating theknitted yarn with a two dimensional feature. Hence the relationship between yarn/fabric structure and theirlight reflection properties can be examined in terms of fiber refractive index, yarn/fiber cross-section , knittingangle and incident light angle. Experimental verification of the model revealed a fair agreement between the

predicted and measured curves. Disagreement has mainly concentrated in the regions of the initial and endreceiving angles.

(Received Aug 23, 1996)(Accepted for Publication May 9, 1997)

1 . Introduction

Techniques for objective measurement of

fabric surface properties related to appearance

have been developed during the last decades.

One of the techniques is goniophotometry,

which involves measuring the light reflectance

from material surfaces by using fixed angle of

incidence and varying angle of detection. Gonio-

photometric curves provide useful information

on fabric surface characteristics such as luster,

gloss, sheen etc. Their application to textiles can

be dated back to 1940s when Quynn1) used the

technique to study gloss of textile fabrics. The

goniophotometer has been adopted as one of

objective tools to characterize luster of fabrics,

others include the multidimensional conversion

luster instrument developed by Yao et. al.8) etc.

In addition, the technique was adopted by Jose

et. al.7) to quantify appearance changes of a

carpet's surface wearing.

Nihira et. al.2-6) developed mathematical models

to predict goniophotometric curves for plain

woven structure, which were based on a theory

of specular reflection on curved surface. A

number of models were developed, gradually im-

proved from simple versions to those with fea-

tures closer to real fabrics. Two separated treat-

ments were used for the yarn which was parallel,

and for the yarn which was perpendicular to the

incident luminous flux imitating weft and warp

yarn respectively. The surface of warp yarn was

simulated by dozen single filaments lying on an

elliptical column, a parallel luminous flux per-

pendicular to the axis of the column was as-

sumed to project onto the surface of the yarn.

The surface of weft yarn consists of several

dozen single filaments curved elliptically, inci-

dent light flux was assumed to be perallel to the

axes of the filaments. Calculation for each group

of yarn was conducted separately and super-

imposed subsequently to obtain the curves for

the whole fabric. The predicted curves were in

good agreement with the experimental results,

Institute of Textiles & Clothing, The Hong Kong Polytechnic University, Hung Horn, Kowloon, Hong Kong

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but their models were limited to plain woven

structure only. There has been little theoretical

work published towards characterizing the re-

flectance behavior of knitted fabrics. Hence the

present paper is concerned with an extension of

the Nihira et. al's approach to analyze light re-

flectance from single-jersey knitted fabrics.

2. Theoretical Development

Symbol Nomenclature

a major axis of the yarn cross-section

b minor axis of the yarn cross-section

ƒÁ rotational angle (•‹) of the yarn

an gradient angle of a normal line at Pn

„qn receiving angle at Pn (•‹)

„U1 angle between the incident light and the

z axis (•‹)

•¢S surface area of the infinitesimally small

unit under consideration

angle between the incident light and the

normal line at Pn (•‹)

•¢w solid angle

I specular reflected luminous intensity

n refractive index of the material

ƒÓ'refractive angle

2.1 Assumptions

In this paper, a mathematical model is to be

developed in order to predict the specular light

reflection from a single-jersey knitted fabric.

The model can be applied to fabrics knitted from

monofilament as well as multi-filament yarns,

with the following assumptions :

1. The yarn shape is elliptical but the con-

stituent filaments in the multi-filament

yarn may have any cross-section shape.

2 . The effect of twist in a multi-filament

yarn may be included by extending the anal-

ysis.

3. Parallel light flux is projected on the

fabric.

4 . The axis of incident light flux as well as of

photo detector lie in the same plane, which is

perpendicular to the running direction of

fabric.

5. The reflected light considered for calcula-

tion includes only the specular reflection

from the surface interaction and excludes

the diffused reflection from within the mate-

rial.

6. The knitted loop is simplified first by a

V-shaped unit, which is formed by rotating

two vertical legs in the fabric plane.

7 . The V-shaped unit is further simplified as

series of cylinders with infinitesimally small

thickness. Thus the reflected light off the

incident light/detector plane is ignored.

2.2 Description of the knitted yarn cross-

-section

The single-jersey knitted loop on the fabric

surface can be first simplified as a V-shaped

structure as shown in Figure 1. The V-shaped

unit, consisting of a left and a right leg, can be

represented by the rotation of the two yarns

(parallel to the fabric running direction) by y and

-ƒÁ- degrees respectively around the axis of sym-

metry as shown in Figure 2.

Considering two coordinate systems, XYZO

(warp yarn) and X'Y'Z'O (knitted yarn), as shown

in Figure 3, the system X'Y'Z'O is obtained by

rotating XY axes around Z axis in the XY plane.

The rotational angle ƒÁ is directional.

2.2.1 Monofilament yarn

Following the above-mentioned assumptions,

the cross-section being observed is perpendicu-

Fig. 1 Knitted loop as V-shaped units

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lar to the running direction of the fabric. An

arbitrary point on the yarn surface can be descr-

ibed in X'Y'Z'O system as

(1)

The two systems can be related by the following

equations :

(2.1)

(2.2)

(2.3)

As the observing system XYZO system, Equa-

tion 1 becomes : -

(3)

For the other leg of the knitted loop, the rotatio-

nal angle is negative, -ƒÁ ; thus

(4)

x is treated as a constant C hence the yarn

cross-section considered is in the same plane

with the light projector as well as the detector.

2.2.2 Multi filament yarn

The transformation can be applied for knitted

loops using multi-filament yarns. The calcula-

tion takes into account a fiber cross-sectional

shape (arising from yarn twist). The cross-sec-

tion of a surface fiber is described as

(5)

2.3 Mathematical model for specular re-

flection of knitted fabric

2.3.1 Monofilament yarn

This model is based on Nihira et. al. approach5)

by treating the knitted yarn as two-dimensional

object, a series of cylinders with infinitesimally

small thickness as shown in Figure 4. The series

of cylinders have a cross-section of an elliptical

ring. Thus the model is an approximation of the

real knitted yarn, as it calculates the reflected

light as parallel light in the detector plane, hence

the reflected light off the plane between the

incident light and detector is ignored.

As shown in Figure 5, at each point of yarn

cross section, reflected angle is calculated by

applying the theory of light interaction on the

curved surface5). Relative intensity of reflected

light is then calculated in terms of the following

variables : Fresnel's coefficient, cosine of inci-

dent angle at the point, and area of interaction.

The gradient angle a. of a normal line on an

Fig. 2 Trasformation of knitted loop

Fig. 3 Trasformation of knitted yarn

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(6)

The receiving angle is

(7)

Assuming the length (1) of the knitted loop is

unity, the surface area (•¢S) at the point pn is

(8)

The incident light angle at point pn is

(9)

The spread angle converts to the solid angle is

(10)

If the incident parallel luminous flux per unit

area is standardized as unity the specular refle-

cted luminous intensity of the infinitesimally

small AS at point pn is

(11)

where f (ƒÓn, n) is Fresnel's coefficient, which is

determined by following function :

(12)

the refraction angle ƒÓ' is given by

where n is refractive of the material and assum-

ing the refractive index of air is unity.

2.3.2 Multi-filament yarn

The gradient angle a- of a normal line on an

arbitrary point of intersection pnro is equal to the

derivative of the fiber cross-section with respect

to y, hence

where

(13)

From equation (13), a similar mathematical

treatment to that for monofilament yarn is ap-

plied for calculation of the reflected angle and

relative intensity of light.

3. Computer simulation

A computer program has been written in Mat-

hematica software to compile calculation from

each point of the cross-section and then to

derive the goniophotometric curve of the whole

surface based on the model. The computer simu-

lation program takes into account the effects of

such factors as the degree of flatness of the yarn

cross section, knitted leg angles, incident light

angles and refractive index of the textile fibers.

The program makes it possible to investigate the

relationship of yarn/fabric surface geometry

and light reflection from the fabric surface. Two

example of computer simulated curves for mon-

ofilament yarn are illustrated as follows.

3.1 Effect of knitted yarn angle

Figure 6 shows the simulated curves with con-

stant values of incident light at 60 degrees, re-

Fig. 4 Knitted yarn treated as a series of cylinders

with infinitesimally small thickness and el-

liptical cross-section

Fig. 5 Specular reflection on elliptical cylinder

arbitrary point of intersection pr, is equal to de-

rivative of the knitted yarn cross-section with

respect to y, hence

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fractive index at 1. 5, and the degree of flatness

of yarn cross-section as 0.5. The degree of flat-

ness is the ratio of the major to the minor axis of

the elliptical cross section. The knitted yarn

angles (rotated angle from the warp yarn posi-

tion) vary from 15° to 60°.

It is evident that by increasing the knitted

yarn angle, the curves change from flat curve

with the peak near to the high end of the receiv-

ing angles, to the curve with a high peak at the

receiving angles equal or close to the specular

reflection angle. As luster can be defined as the

ratio of light intensity in specular reflection to

the light intensity in diffused reflection ,thus it

can be interpreted that the yarn which is knitted

with a wider angle will result in a fabric that has

higher luster than the fabric knitted with a nar-

rower knitted yarn angle.

The predicted curves shown in Figure 6 were

calculated for a right leg of the knitted loop,

having the rotational angle (r) as positive value.

Calculation of the goniophotometric curves for a

left leg of the knitted loop having the rotational

angle as negative value (-r) has resulted in the

identical goniophotometric curves. The symmet-

rical knitted legs arisiting from same rotational

angle (r) in opposite directions give identical

goniophotometric curves.

3.2 Effect of yarn flatness

Figure 7 shows simulated curves with con-

stant values of incident light at 60 degrees, re-

fractive index at 1.5, the knitted yarn angle 15

degrees, with varied flatness of yarn cross-sec-

tion. The degree of flatness of the yarn cross-sec-

tion ranges from 0.07 to 0.5. As the degree of

flatness of yarn cross-section decreases, the

curves with a high peak at the receiving angles

equal or close to the specular reflection angle,

which can be interpreted as a flatter yarn will

knit into a fabric that has a higher level of luster

than a round yarn.

4. Experimental verification

4.1 Sample preparation and apparatus

Two experiments have been conducted in

order to verify the mathematical model for mon-

ofilament yarn. In the first experiment, a black

wire which followed our assumptions, that is a

constant elliptical cross-section along its length

and black color absorbs the light pass through

the surface. The wire had a degree of flatness of

0.722. In the second experiment, a nylon mono-

filament yarn (80 dtex) was knitted into a singl-

e-jersey knitted fabric using circular knitting

machine. It was then dyed black (acid dyes) in

fabric form, and heat set into three fabrics with

different knitted yarn angles, by dry air at 180°C.

A goniophotometer (VGS-1D) was used to

measure light flux as a function angles of illumi-

nation or observation at a series of receiving

angles (0-85•‹). Following the recommended pro-

cedure, the calibration was carried out by, firstly

Fig. 6 Effect of knitted yarn angle

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selecting the light receiving angle to be as same

as the incident light angle, putting cap 0-ADJ on

the sample bed, adjusting the 0-ADJ knob on the

front panel to obtain 0.0 on the digital indicator,

placing the stand plate on the sample bed, ad-

justing STD-ADJ knobs on the front panel of themeasuring unit so that the digital indicator read

the same value as labeled on the standard plate,

i. e. 89.0 at 60 degrees. A correction coefficient

was used at 85° receiving angle, as only 85% of

the detector area is available to receive the light.

4.2 Comparison between the theoretical

and experimental results

4.2.1 The wire experiment

The wire was measured for their goniophot-

ometric curves at rotational angles (A) of 15°,

30°, 45° and 60°. The predicted curves were

calculated by computer with corresponding

yarn and fabric parameters. Hence the degree of

flatness of yarn cross-section was as 0.722, re-

fractive index was 1.5, and incident light angle

60 degree. The theoretical and experimental res-

ults are shown in Figures 8 and 9, respectively.

The Predicted and the measured curves of the

wires have a number of similarities. Both pre-

dicted and measured curves show that by in-

creasing the knitted yarn angle, to the curve

with a high peak at the receiving angles equal or

close to the specular reflection angle. At both

ends of the receiving angle (i. e. 0-30•‹ and 70-

85•‹), the predicted curves with higher knitted

yarn angles show higher valuer of relative inten-

sity of reflected light than the lower knitted

Fig. 7 Effect of flatness of yarn cross-section

Fig. 8 Predicted curves for the wires

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yarn angle. However, the measured curves have

much lower values in those two regions. This

phenomenon may be explained by that the refle-

cted light at the initial and high ends deviate

more from the incident/detector plane, and the

higher knitted yarn angle leads to higher refle-

cted light coming off the incident/detector

plane.

Figure 10 shows the measured curves ob-

tained by using the same knitted yarn angles

with rotation in clockwise direction (-r). The

curves have shown remarkably similar as the

curves obtained with rotation in anti-clockwise

direction (r). Both theoretical and experimental

Fig. 9 Measured curves for the wires

Fig. 10 Measured curves for the wires (opposite

rotational direction)

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results confirmed that the symmetrical effect of

the knitted legs have resulted in identical gonio-

photometric curves.

4.2.2 The fabric experiment

Comparison was conducted between the pre-

dicted curves and measured curves obtained

from the experiment of black single-jersey kni-

tted fabrics made of monofilament yarn. Figures

11 and 12 illustrate the predicted and measured

curves respectively. The knitted yarn angles

were 15°, 20° and 25°. The curves from predic-

tion and experiment have a reasonable agree-

ment. At lower and higher receiving angles, the

discrepancies became more obvious, which may

again be attributed to the reflected light off the

incident/detector plane.

5. Conclusion

A mathematical model has been developed for

prediction of goniophotometric curves of singl-

e-jersey knitted fabric. The model is based on

Nihira et. al.'s approach treating the knitted yarn

Fig. 11 Predicted curves for the single-jersey kni-

tted fabrics

Fig. 12 Measured curves for the single-jersey kni-tted fabrics

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with a two dimensional feature. Hence the rela-

tionship between yarn/fabric structure and their

light reflection properties can be examined in

terms of fiber refractive index, yarn/fiber cross--section , knitting angle and incident light angle.

Experimental verification of the model revealed

a fair agreement between the predicted and mea-

sured curves. Disagreement has mainly concen-

trated in the regions of the initial and end receiv-

ing angles. Further work on the mathematical

model considering three-dimensional feature of

knitted yarn., which includes calculation of light

reflection off the incident/detector plane, is

being carried out to overcome the limitation of

the present model and results will be reported

separately.

Acknowledgment

One of the authors (A. Sirikasemlert) wishes to

acknowledge a scholarship received from the

Hong Kong Polytechnic University Postgradu

ate Block Grant for the work reported here.

References

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