Fibres & Textiles in Eastern Europe 2000, 8, 1(28), 57-58
MODELLING OF DYNAMICAL PROPERTIES OF TEXTILES UNDER COMPRESSION
Jerzy Zajaczkowski
Department of Mechanics of Textile Machines
Technical University of Lodz
Abstract
The paper is concerned with the problem of determination of constants characterising dynamical
properties of textiles under compression. Proposed model of a textile specimen is composed of a serially connected
three elements: a third degree nonlinear spring, a third degree nonlinear spring connected in a parallel manner to a
linear damper and p-th degree nonlinear spring connected in a parallel manner to a linear damper. The model is
characterised by the following properties. For small contraction the elastic force is so small that mainly the damper
resists the motion. When the contraction becomes large enough a large force is required to overcome the nonlinear
elasticity and the spring becomes almost rigid. In this case the motion is maintained mainly due to another serially
connected elements. Differential equations describing behaviour of a textile specimen placed at a test set-up are
derived and solved. The results are presented graphically. It is found that the electromagnet driving force depends
on tested specimen properties and it cannot be assumed a priori.
Keywords: geotextiles, mathematical modelling of textiles under compression.
Introduction
Equations of motion
The studying of dynamical properties of a textile specimen requires a suitable
environment revealing those properties. It is usually a test set-up. In case of the mathematical
modelling, as opposed to experimental study, we don’t have to make a set-up. It is sufficient to
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write the equations describing its motion. A schematic diagram of a test set-up for textiles is
shown in Figure 1. The loading mass m4 is attached to the presser board of mass m3 with a flat
spring of stiffness a43. The tested specimen, represented by set of springs and dashpots, is located
between the presser board and the motionless press table. The presser board is driven by the
electromagnet whose inductance L is a function of its position. The coil circuit has resistance R.
It is supplied with voltage u=u(t) being a function of time. The gravity forces are denoted as
F3=m3.g and F4=m4g
Figure 1. A schematic diagram of a set-up for testing textiles under compression.
The relationships between the compressive forces and the specimen contraction are assumed as
F1  a1sx1  b1 ( sx1 ) 3 ,


3
dx 
 dx
F21  a21s x2  x1   b21 s x2  x1   A21  2  1  ,
 dt
dt 
2


F32  e32 s x3  x2 
p
 dx3 dx2 
 A32 

.
 dt
dt 
(1)
Here, F denotes the force, x -the displacement, a, b, e denote the stiffness constants and A is the
coefficient of viscus damping. In the test set-up only the compressive forces can occur.
Consequently for x1<0 the force F1=0. In order to clarify the meaning of stiffness constants the
multiplier s is introduced. Taking s=1000 for compression x1=0.001m one finds that the stiffness
constants are equal to the forces: a1sx1=a1, b1(sx1)3=b1 .
Assume that in the expression defining the force F32 the exponent p is odd and greater
then three. Note, that raising a fraction to a power greater then one gives a fraction of lower
value. On the other hand, when a number raised to that power is greater then one then the number
of higher value is obtained. By reason of the p-th power, for small relative displacement s(x3x2)<1, the force associated with the stiffness e32 may be negligible in comparison with the force
associated with the damping constant A32. In this case the behaviour of the system relies on the
damping force. On the other hand, if the relative displacement is such that s(x3-x2)>1 then the
force associated with the stiffness e32 may become large. Consequently, the corresponding
element (e32, A32 in Figure 1) becomes almost rigid and therefore it has little influence on the
motion of the system.
The restoring forces F1=F21=F31 are equal. The equations of the motion of the system may
be written in the following form
dx 
p
 dx
e32 s p  x3  x2   A32  3  2   a1sx1  b1s3 x13  0,
 dt
dt 
3
dx 
3
 dx
A21  2  1   a 21 s x2  x1   b21 s 3  x2  x1   a1 sx1  b1 s 3 x13  0,
 dt
dt 
d 2 x3
p
 dx dx 
m3 2  e32 s p  x3  x2   A32  3  2   a43s x4  x3   FE  F3  0,
 dt dt 
dt
d 2 x4
m4
 a 43 s x4  x3   FE  F4  0.
dt 2
(2)
The driving force FE of the electromagnet is given by the expressions (3). Here, the intensity
of the current i can be found from the voltage equation (4)
1 dL
FE   i 2 ,
2 dx
x  x4  x3 ,
L
D
,
xd
dL
D

,
dx  x  d 2
(3)
d
 L x i   Ri  u.
dt
(4)
D, d are the electromagnet constants, R -the circuit resistance, u -the feed voltage, L -the
electromagnet inductance. It is convenient to are rewrite equations (2,4) as a set of first order
equations

dx1
1
p

e32 s p  x3  x2   A32 x5  a1sx1  b1s3 x13
dt
A32




1
3
a21s x2  x1   b21s3  x2  x1   a1sx1  b1s3 x13 ,
A21
4


dx2
1
p

e32 s p  x3  x2   A32 x5  a1sx1  b1s3 x13 ,
dt
A32
dx3
 x5 ,
dt
dx4
 x6 ,
dt


dx5
1
0.5Di 2
p
p

e s  x3  x2   a43s x4  x3  
dt
m3 32
 x4  x3  d 2 m3

 

A32
p
x5  e32 s p  x3  x2   A32 x5  a1sx1  b1s3 x13  g ,
m3
dx6
1

dt
m4


Di 2
 a s x  x   1

2   g,
3
 43 4
2 x  x  d  

4
3
 x  x d
di 
D
3
 4
 u
.
2  x6  x5 i  Ri 

dt 
D
 x4  x3  d 

(5)
Results and discussion
Calculations were carried out for the masses m3=3.3kg, m4= 26.1kg, the stiffness
coefficients a1=a21=200N/m, b1=b21=1000N/m3, multiplier s=1000, the damping oefficients A21=
5
A32=8132.4Ns/m, a43=825.75N/m, the electrical constants D=0.0015Hm, d=0.00141m, R
the feed voltage u=Umsint, Um=220V, =100
p=5, 7, 9 and the zero initial conditions.
Elastic Fe32 =e32(s(x3-x2))p and viscous FA32 =A32d(x3-x2) /dt components of the internal force F
versus total specimen contraction x3 and displacement x32=x3-x2 versus time t for (e32=0.1, p=5)
are shown in Figure 2.
From Figure 2 one may see that at the initial period of time, when specimen contraction
is small, the viscous part of the force FA32 =A32d(x3-x2) /dt is dominant while elastic part Fe32
=e32(s(x3-x2))p is negligible. When the specimen contraction x3 is high enough then the elastic
force becomes significant. The system after the initial period approaches the attractor. The
electromagnetic force FE during initial period depends on specimen properties and therefore it
cannot be assumed a priori.
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Figure 2. Elastic Fe32 =e32(s(x3-x2))5 [N] and viscous FA32 =A32d(x3-x2) /dt [N] components of the
internal force F=F32 versus total specimen contraction x3 [m] and displacement x32[m] versus
time t[s] for e32=0.1N/m5, p=5.
Acknowledgement
This work was sponsored by KBN under Grant No. 3T09B12913. The subject of this study
originated from the research work carried at COBR CENARO, where the phenomenon of fast
settling of vibration isolators under dynamic loading was observed.
References
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1. M. Matsudaira, Q. Hong: Features and Characteristic Values of Fabric Compressional
Curves, International Journal of Clothing Science and Technology. Vol. 6, no. 2/3, 1994, pp.
37-43.
2. P. Potluri, I. Porat, J. Atkinson: Low-stress fabric testing for process control in garment
assembly, Application of robotics. International Journal of Clothing Science and Technology.
Vol. 8, no. , 1996, pp. 12-23.
3. W. Więźlak, K. Gniotek: Indices characterising rheologic properties of geo-nonwovens under
compression.Fibres and Textiles in Eastern Europe. Vol. 5, no. 3(26) 1999, pp. 40-44.
4. B. Bolanowski, W. Tarczyński: Test stand for testing cyclically loaded geo-nonwovens.
Fibres and Textiles in Eastern Europe. Vol. 8, No 2(29) 2000.
Streszczenie
Praca dotyczy zagadnienia określania stałych charakteryzujących właściwości
dynamiczne tekstyliów poddanych ściskaniu. Zaproponowany model składa się z trzech
szeregowo połączonych elementów: sprężyny z nieliniowością trzeciego stopnia, sprężyny z
nieliniowością trzeciego stopnia połączonej równolegle z tłumikiem oraz sprężyny z
nieliniowością p-tego stopnia połączonej równolegle z drugim tłumikiem. Model charakteryzuje
się następującymi właściwościami. Dla małych odkształceń siła jest tak mała, że głównie
równolegle dołczyny tłumik przeciwstawia się ruchowi. Gdy skrócenie staje się wystarczająco
duże wymagana jest duża siła dla pokonania nieliniowej sprężystości. W konsekwencji element
może być traktowany jako sztywny. W tym przypadku ruch podtrzymywany jest głównie dzięki
szerogowo dołączonemym innym elementom. Omówione tu właściwości elementu nieliniowego
osiągnięto przez wykorzystanie faktu, że liczba ułamkowa podniesiona do potęgi daje mniejszą
vartść, natomiast liczba większa od jedności podniesiona do potęgi daje większą wartość.
Wyprowadzono równania różniczkowe ruchu próbki włókniny umieszczonej na stanowisku
badawczym. Wyniki obliczeń przedstawiono na wykresach. Stwierdzono, że przebieg czasowy siły
elektromagnesu zależy od właściwości badanej próbki i nie może być przyjęty a priori.
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