Bogdan Supeł,
Zbigniew Mikołajczyk
Department for Knitting Technology
and Structure of Knitted Fabrics
Technical University of Łódź
ul. Żeromskiego, 90-543 Łódź, Poland
E-mail:katdziew@p.lodz.pl
Model of the Connector for 3D Distance
Knitted Fabric Fastened by Articulated
Joints
Abstract
During the first stage of our considerations, a model of the compression process of a 3D
distance knitted fabric was related to a single connector fastened to it by articulated joints,
and considered as a slender rod with an assumed shape. In our physical and mathematical
models, the compression of a slender, elastic rod was based on assumptions concerning the
knitted fabric’s morphology, as well as the mechanical properties of the threads, particularly monofilaments placed in the internal layer of the fabric. A calculation method was
developed which enables to determine the functional dependencies between the compressing force and bending deflection, as well as a method of determining curves representing
the shape of the compressed rod – connector. A computer simulation of the compressed rod
connecting the two outside layers of the knitted fabric, considering the variable parameters
of the model, was carried out with the use of calculation algorithms elaborated by us, and
MathCad program.
Key words: knitted 3D fabric, distance fabric, compression process, connector, articulated
joint, mathematical model, physical model, simulation.
Introduction
The modelling was carried out with the
aim of finding the optimum procedure
which would facilitate in solving the
problem of 3D distance knitted fabric
compression. The process of compressing such a fabric is accompanied by a
number of physical, particularly mechanical phenomena, such as the bending of
the connectors, which form the knitted
fabric’s structure, and the mutual interaction of the friction between neighbouring connectors, among others. Designing
a distance knitted fabric which would
fulfil the demands of its future user is a
complex procedure. Irrespective of the
shape of such a fabric, described by its
geometry, it should fulfil a number of
usability requirements, including sufficient, mainly high strength, and resistance to the action of external forces. An
analysis of literature carried out by us
concerning the structure and mechanical properties of 3D knitted fabrics and
the modelling of their strength behaviour
led to the conclusion that the problems
of modelling the compression of such
fabrics have hitherto not been described.
Therefore, developing a model description of the fabric’s compression process
will create the possibility of designing
distance fabrics of assumed elasticity
without the need to manufacture a prototype. The designing process concerning
geometric and technological aspects will
be completed by selecting an appropriate material, and next manufacturing the
fabric with the use of a suitable knitting
machine without the need to carry out
initial experimental stages. Such a procedure will allow to significantly shorten
the designing time of 3D distance knitted
fabric.
Assumptions of the model
The object compressed, which is a 3D
distance knitted fabric, is considered as
two rigid planes connected by a network
of rods which form an elastic spatial
trust. The shape of this trust depends on
the knitted structure of the stitches, as
well as on the mechanical properties of
the threads which form the structure [1].
Modelling th compression process of the
knitted fabric was performed during the
first stage’ of the ivestigation, assuming a single connector comprising a rod
joining the outside layers of the knitted
fabric.
a)
The rod compressed we considered as
placed in the xyz co-ordinate system, and
to facilitate the analysis of the phenomenon, we mentally divided the rod – connector into two rods consisting of its
projections on the perpendicular planes
0xz and 0xy, which are the planes of two
cross-sections of the knitted fabric (Figure 1).
The outside planes, p1 and p2, of the knitted fabric are composed of single loops or
loops connected into loop systems. The
loops consist of single systems of the outside planes p1 and p2. We consider them
as basic elements of the planes’ structure
with a dimension of A × B, where B is
the width of the wale, and B is the height
of the course. The segments do not deform, which means that A = constant and
B = constant. The loops of the model
b)
Figure 1. 3D distance knitted fabric; a) design of the fabric, b) photograph of the crosssection of a 3D distance knitted fabric.
Supeł B., Mikołajczyk Z.; Model of the Connector for 3D Distance Knitted Fabric Fastened by Articulated Joints.
FIBRES & TEXTILES in Eastern Europe January / December / A 2008, Vol. 16, No. 5 (70) pp. 77-82.
77
force P acting at the midpoint of the surface of the rigid plane p1 (Figure 3).
The following dependency connects the
unit forces Pi with the summary force P:
P = ∑ · Pi
where: m – is the number of loops closed
by the surface of plane p1.
Figure 2. Physical model of the outside
planes of the 3D knitted fabric.
Figure 3. Equivalent load of the planes of
the 3D distance knitted fabric.
Figure 4. Fastening scheme of the physical
object – a bent elastic rod.
planes are connected by articulated ball
joints (Figure 2).
From a mechanical point of view, we
consider the planes as shells which are
susceptible to deflection and non-susceptible to stretching. For such a model of
the planes p1 and p2 accepted by us, only
small and insignificant bending moments
occur (Mg ≅ 0) at the points of loop joints.
Only the mutual influence of neighbouring elements, e.g. of loops, occurs due to
the reaction forces (F1) in the articulated
joints, which act on the surface of the
shell.
In the case of the compression process of
3D distance knitted fabric, in order to accept a uniformly distributed load caused
by the unit forces Pi acting on the singular loop segments, we assume that this
is equivalent to the impact of a singular
78
The structural model of the plane accepted by us may be related to the process
of compressing the distance knitted fabric with the use of a tensile tester. In this
case, the rigid dished disks connected
to the measuring head of a tensile tester
perform the compression process of the
sample investigated.
The physical model of compressing a
slender, elastic rod in a single plane, 0xz
or 0xy, is based on the following assumptions:
 the planes of the knitted fabric are displaced in the transversal direction,
 the loading by force P acts in a direction perpendicular to the planes of the
knitted fabric,
 the connectors are considered as elastic
rods whose structure has a spatial configuration, as well as elements fastened
by an articulated joint to the outside surfaces of the knitted fabric at both ends,
 the connectors which belong to a single loop are joined at a single point,
the planes p1 and p2 are not deformed
(this is a simplifying assumption of
our analysis, as the connector considered presents a whole set of connectors the fact that the knitted distance
fabric analysed). In the event that
such an assumption would not be accepted, irrespective of all the connector loads acting are of the same value,
the connectors would be characterised
by different deflections, which in turn
would complicate the considerations.
An analysis taking into account loads
causing irregular connector deflections will be the subject of our future
works and publications;
 in the preliminary stage of the compression process, the rods retain the
assumed shape, and next, with an increasing force P1 , a transversal force
S1 of the base reaction appears,
 such factors as temperature and humidity do not influence the proceeding of the deformation process of the
knitted fabric.
In a further analysis of the mathematical
model based on the physical model, the
connector is considered as a slender rod
with an assumed shape, which is affected
by the bending process. The assumed
shape is obtained by accepting the thickness g0 of the knitted fabric at which the
deflection value ∆g takes the zero value,
and the mutual displacement y0 of the
rod’s end in the direction perpendicular
to the deflection, as well as the assumed
length lp of the rod. The rod is considered
in a free state (not loaded) and takes such
a shape that its potential energy will be
the smallest.
Physical model of the
compression process
Taking into consideration all aspects related to the structure of the 3D distance
knitted fabric, as well as the behaviour
of this fabric under the influence of loads
acting on the fabric, and on the basis of
the assumptions mentioned in the previous chapter, below is presented a physical model of a slender, elastic rod of assumed shape. The general assumption is
that the rods are fastened at both ends by
articulated joints.
As far as compressed rods are concerned, the model discussed is related
to the Euler’s theory, the difference being that our assumption, confirmed empirically, considers the rod as a slender
object with assumed preliminary shape;
what connects is existing of additional
forces S1 that occur during the compression process, which are caused by
the reaction of the base i.e., the reaction
of the outside layers of the knotted fabric [2-5]. Figure 4 presents a scheme
of the mechanical model in the form
of a rod mentally liberated from constraints; this presents the outside planes
of the knitted fabric, whose action was
substituted by the forces P1 and S1, and
their motion is limited by both the immovable and slidable articulated joints.
The following designations are marked
in Figure 4:
g0 – initial thickness of the knitted
fabric,
gi
– thickness of the fabric during
compression,
∆g – deflection of the knitted fabric,
p1, p2 – planes in which the knitted fabric’s loops are placed,
y0 – horizontal displacement of the
rod’s fastening points,
P1 – force compressing the bent rod,
and
S1 – transversal force of the base’s reaction.
FIBRES & TEXTILES in Eastern Europe January / December / A 2008, Vol. 16, No. 5 (70)
ed
fabric,of they0knitted
- horizontal
of the
rod’s fastening
points,
Sinertia
reaction
force ofcross-section.
the transversal, sliding support, where:
1 - is the
eflection
fabric, displacement
respectively;
P1J action,
- is the
modulus
ofDifferentiating
of the
bent
object’s
C41 = 0 ‘x’, we subsequently obtain:
equation
(1)and
twice towards
sec
g
compression,
P
–
force
compressing
the
bent
rod,
and
E
is
the
Young
modulus,
1
– planes in which the knitted fabric’s loops
S1 - isare
theplaced,
reaction force of the transversal, sliding support,
S1 – transversal
forcefastening
ofEDifferentiating
the
reaction.
J - is (1)
the twice
modulus of inertia
of the
bent object’s
cross-section.
ric,
orizontal
displacement
of the rod’s
- is base’s
thepoints,
Young
modulus,
and
equation
‘x’,IVwe
subsequently
obtain:
P1 y′′( x)
P1 y′( x) −towards
S1
′′′
and
where:
ed
fabric’s
loops
are
placed,
y ( x) = y0 ( x) + y1 ( x)
y ( x) =of the bent object’sy cross-section.
( x) =
rce compressing the bent rod, and J - is the modulus of inertia
For the condition which
connects the force
EJ
EJ
Differentiating
equation
(1)
twice
towards
‘x’,
we
subsequently obtain:
the rod’s fastening
points,
ansversal
force of the
base’s reaction.
P1 y′′( x)
P1 y′( x) − S1
IV
,
∆g),
we
substitute t
implicit
function
f(P
1
and y ( x) =
where: y ( x) = y0 ( x) + y1 ( x)
(1.a)
y′′′( x) =
rod, and
Differentiating
equation
(1)finally
twicetakes
towards
‘x’, we subsequently
obtain:
Equation
(1.a)
the
shape:
C
,
C
,
and
C
determined
from
equaEJ
EJ
determined
from
equation
system
(3) into
21
31
41
Mathematical
model
of
the
EquationP(1.a)
the shape:
′′( x) (3) into the condition that
− S1 takes IV
P ysystem
y′( x)finally
’s reaction.
and y ( x)2tion
where:
y ( xdoes
y1 ( x)
) = y0 not
( x) +changing
y′′′IV( x) = 1 2
= 1P
compression process of a
curved
rod,
which
durin
1
EJ
EJ
′( x) −finally
P′′1(yx′′)( x=) 0 where: k1 describes
P1 y(1.a)
S1
the
length
of
the
curved
rod,
=
y
(
x
)
+
k
y
IV
Equation
takes
the
shape:
1
single connector
expression:
and y ( x) =
where: y ( x) =EJ
(1.a)
y0 ( x) + y1 ( x)
y′′′( x) =
which does
not changing during processEJ
EJP1
2
IV
2
The physical model presented above
general
oftakes
equation
(1) may
be we
written
thefollowing
form of:exk1 integral
= finally
y ( xena) + k1 y′′( x) =The
0Equation
where:
ing.
Finally,
obtaininthe
(1.a)
the shape:
n
gi 4.
gi
g
Figure
5. Fastening
scheme of the yphysical
object
–+aCbent
elastic
rod
.cos(
EJ
bles to formulate
parametrical,
functional
pression
Equation
(
x
)
=
C
+
C
x
sin(
k
x
)
+
C
k
x
)
C
(
+
C
(
sin(
k
i
) − sin( k1 (i − 1) i )) + C
P
11 shape:
31
1
1 21
2 41 ∑
IV
2 21
31
1
1
Equation
(1.a)integral
finally takes
the
′′
The
general
of
equation
(1)
may
be
written
in
the
form
of:
k
=
y
(
x
)
+
k
y
(
x
)
=
0
where:
n
n
n
as
well
as
normal
differential
equations
i =1
ure 5. Fastening scheme of the physical object – a bent
elastic integral
rod . 1of equation (1) may 1
The general
EJ
P1 cos(connector
2+
IV(the
2 + C x + Cprocess
of the
fourth order, which
interrelate
the
Mathematical
model
of
compression
of
a
single
y
x
)
=
C
sin(
k
x
)
C
k
x
)
(2)
in the form
of:41
Determining the deflection ∆g consists in
y ( x) + k111 y′′be
( x21written
) = 0 31
where:
1 k1 =
1
The general
integral
of
equationsubstituting
(1) may beeach
written
inofthe
form
load
the rod with its process
deformations.
EJ
value
force
P1, of:
con-consists in s
cal model of
theofcompression
of a The
single connector The integration
Determining
the
deflection
∆g
constants
C
,
C
,
C
,
and
C
we
determine from
11
21
31
41
y (equation
x) formulate
= C11 +(1)
C21constants
xparametrical,
+–Cbe
sin(Ckelastic
)in
Crod
cos(
kaand
xwell
) C41 into
5. Fastening
scheme
ofThe
the
physical
object
a31bent
.31aas,parameter,
rod-connector
isFigure
considered
and
integration
C+sidered
,41as
C
we
determine
from
the b
Theaccepted
general
integral
of
may
written
the
form
1 ,xfunctional
1of:
parameter,
into
equation
(4)
The physical
model
presented
above
enables
to
as
11
21
equation
(4)
and drawin
The
integration
constants
C
,
C
,
C
,
and
C
we
determine
from
the
(2)
conditions:
11
21
31
41
The
integration
constants
C
,
C
,
C
,
and
C
we
determine
fromb
11
21
31
41
by us
as
two
individual
rods
which
are
conditions:
and
drawing
the
curves
of
the
function
n
yequations
as normal
differential
order,
interrelate
the load of the rod
( x) = C11parametrical,
+ofC21the
x +fourth
C
k1 x) +which
C41ascos(
k1 x)
(2)
al model presented
above enables
to formulate
functional
well
gi
gi
31 sin(
conditions:
y
(0)
y
C
C
y
=
−
→
+
=
−
conditions:
0single
0
the
projections
of
the rodmodel
thephysical
0xzthe
and
f ( P1 ,as
gi )and
= ∑ (C21 + C31 ( sin(k1 i ) − sin( k1 (i − 1)
Mathematical
of
process
of11rod
a+determine
5)
y041→we
C
Crod
y 41
=is
−C
=11−connector
with
its
deformations.
The
rod-connector
considered
and
by us
two
Figureequations
5.The
Fastening
scheme
ofon
the
–yy,(0)
a(0)
bent
elastic
integration
constants
C11
, compression
Cobject
and
from (Equation
the
boundary
differential
of the
fourth
order,
which
interrelate
the
the
21, C31
n
n
i =1
y0constants
C41
= y−load
→
+ .C
=C+
−accepted
(0)
C
C,y0410C =, − y0 3
=of
−Cy11
41P
0yz planes.
0 →
11
y
P1 y0
The
integration
,
3
11 1and
21
31 planes.
0 0yz
individual
rods which
are the projections
of the rod
on
the
0xz
conditions:
eformations. The
rod-connector
is considered
and accepted
by
us
as
two
P
y
P
y
′′′
=y0S1 the
= boundary
→ −C k1 cos(
y ( gi )P
k x)(+PC
,k41gkix1)) ==sin(
l pP1k1 1y0x0 ) =P y (6)
1well
Cenables
( gi determine
) =toS1formulate
= 1from
→−
y41′′′′′′we
Cg031 k133cos(31k1 functional
x) +and
C41 1k1f33sin(
physical
presented
above
as
scheme
of the
physical
object
bent
rod of
. and
1 3
3 x) + C
1−y
athematical
model
of−the
compression
process
aand
single
ydifferential
(0)The
y0 →
C–11
C
yon
=
+the
=elastic
−describes
ods which
are
projections
ofamodel
0yz
) y=′′′S( 1gi=) =g1 Si 10 =→Pparametrical,
sin(
)
=
y (connector
gplanes.
Ci 31→k1−cos(
k
k
k
x
41rod
0 the 0xz
cos(
)
sin(
+
C
k
k
x
C
k1gxi ) = 1 g0i
1
41
1
1
i
Thethe
equation
which
31 1
1
41 100
1
conditions
Equation
3. gwhich interrelate
where:
n
=
is
the number of parts into w
as differential
normal differential
equations
of the fourth
order,
the
load
of
the
rod
where:
n
=
100
is
the
number
g
The
equation
which
describes
deformations
of
the
rod
influenced
by
the
loads
i
y ( gi ) =P10y→
C i + C gi + C31 sin(k1 g i ) + C41 cos(k1 ggi i) = 0ofgi parts
y0 influenced by
deformations of theP1rod
3 the
3)=0→C
0 g 11+ C 21sin(
(
)
cos(
)
0
+
+
=
y
g
C
k
g
C
k
g
into
which
the
interval
the function
is
divied,
which
is
accepted
using intuition
with
its
deformations.
The
rod-connector
is
considered
and
accepted
by
us
as
two
′′′
compression
process
of
a
single
connector
(presented
)the
)=
= following
→
−C31 k1 tocos(
+ C41
yhas
githe
Sfollowing
kthe
k(1g→
xby
31well
i +as
i )+C
i ) = 0 of
form:
e physical
model
enables
formulate
parametrical,
functional
1 = above
1 x )rod
yk(1gi sin(
C11
C21C
ntial
equation
which
describes
deformations
of
influenced
the
loads
231gsin(
loads
has
form:
0 0g+→→
sin(
) cos(
+i CkC
+
=C 0 = 0is acCk131
k1 41gk41i2cos(
Ck1411kisgcos(
k10g→
11
2111gC21,
1 g
i ) =y0
i
i
The
integration
constants
C11,
21C31,
i ) ==
i +k
i )which
(3)
g
′′
(0)
sin(
)
)
−
−
=
y
C
x
C
x
2
2
variability
(2)
divied,
i
i
the
calculation.
The
greater
the
value ‘n’, t
rods
which
are order,
the projections
rod
0xz
and
0yz1kplanes.
The integration
constants
Cwhich
, ofC=the
and
Cload
we
from41k the
boundary
1
1 C
41
11, Cy21
310, →
′′interrelate
(0)
sin(
−the
Con
k41the
x1) −rod
C
normalform:
differentialindividual
equations
of the
fourth
ofk311determine
the
owing
31the
1 2equation
41 k
1 2cos(
1 2xx))==00→
41 ==00
2 x )sysand C41
obtained
from
′′
(0)
0
sin(
cos(
=
→
−
−
→
y
C
k
C
k
C
cepted
using
intuition
by
the
authors
as
′′
(0)
0
sin(
)
cos(
)
0
0
=
→
−
−
=
→
=
y
C
k
k
x
C
k
k
x
C
31
1
1
41
1
1
41
but
significantl
(to
cos(
+′(Cx21) +giS+xC31where
=)0integration
gJy
k1 gyi()x+) C
gThe
dh above
enablesyE
functional
as41ywell
, C121at, Cthe
and Ctime
from
the equa
constants
C1141
1
1
41 this way
311, same
41 obtained
i )formulate
′′=(conditions:
its deformations.
The
rod-connector
issin(
considered
and
by us31 as
= −C
x0) →
P111 yparametrical,
( xintegration
)k+1accepted
yi1)( xequal:
(1)
tem=are
,C
and C
obtained
the equation sys
The
constants
Ctwo
1
0relatively
11, C21giving
31, sufficient
41 accuracy
to from
the
calcula(1)
,
C
,
C
,
and
C
obtained
from
the
equation
sy
The
integration
constants
C
y
(0)
y
C
C
y
=
−
→
+
=
−
computer
program.
The
differential
which
deformations
of
the
rod
influenced
by
the
loads
2 equation
2 describes
ons
of
the
fourth
order,
which
interrelate
the
load
of
the
rod
relatively
equal:
11
2111
3121
41
,
C
,
C
,
and
C
obtained
from
the
equat
The
integration
constants
C
0
11
41
0
31
41
ividual
rods
which
are
the
projections
of
the
rod
on
the
0xz
and
0yz
planes.
The
integration
constants
C
,
C
,
C
,
and
C
we
determine
from
the
boundary
11 krelatively
21
31 C41 = 0 41 (1)
y′′(0) = 0y (→
− P1 y′( x) + S1 x where
x) −=Cy310 (kx1 )sin(
+ y1k(1xx)) − C41 k1 cos(
1 x ) = 0 → equal:
tion. The greater the value ‘n’, the greater
relatively
equal:
the
following
form:
rod-connector ishas
considered
and
accepted
by
us
as
two
(3.a)
relatively
equal:
=
−
C
y
conditions:
P1 y0 C , C 3, C , Cand= −Cy obtained
P y0 equation
11
0
3
where:
the accuracy
of the calculations is, but at
system
are
The
integration
11
31k1 C
0 k sin( k x )from
( gthe
) =−0xz
→
−C
= 1 the
yy′′′on
Syconstants
k21
x1111) += C
− 41
y41
1 =
31
1 cos(
1= − y 1
i=
ofequation
the rod
and
0yz
planes.
The value
of ∆g,
is related to point ‘
C
0 11
(0)
C
C
y
→
+
=
−
eprojections
differentialwhere:
which
describes
deformations
of
the
rod
influenced
by
the
P1as result
0
0
11
41
0
(3) which
the same
time
this way
significantly ing
g(i g loads
y0 (x), and
y1 (x) are the
initial buckling, and the bucklingy occurring
of the
force
relatively
equal:
i
P
EJ
tg
)
y0
(x),
and
y1
(x)
are
the
initial
buckling,
1
i
0
′′
′
Jy
(
x
)
P
y
(
x
)
S
(
x
)
=
y
(
x
)
+
y
(
x
)
=
−
+
E
x
y
we
read
from
the
x-axis
(see
6). Thi
(1)
where
the following form:
y0 EJ
) y PEJ
P1P
1
1
1 ytg
creases the(3.a)
calculation time used by Figure
the
0 ( gi
P1 as
ybuckling
respectively;
1 action,
3
ych
are the
buckling,
the
as++0 yC
result
the
=EJ
+iy()gxiforce
y30 of
tg
)1gi 0 1 )
CP
y0yy(′′′g(ig
0 result
) =of
0and
sin(
cos(
0
→
=
Crod
C
g−
Coccurring
k1kgCloads
k
g
EJ
1 (x)
0 41 C
EJ
tg
(
21
andinitial
the
buckling
occurring
of
11 = −
11 +
21
31
1
i +
ix))=
describes
deformations
the
influenced
by
the
)
cos(
sin(
)
=
=
→
=
S
C
k
C
k
k
0
(5
and
6).
It
should
be
stressed
that
it
is imp
(3.b)
+ 1 y0gi 1
y041
EJ P
1
31 1
1 21
i
computer program. (3)
S1 - Pis action,
the reaction
force
sliding
espectively; the force
gi C+21 =support,
gi 2 of the transversal,
P1 gi 2 g i 1EJ
2C21 =
2+
P
1 y ′′(0) = respectively;
equation
(4)
as
a
great
number
of
solution
0(y→
=2 0 P1EJ (1)
+ -the
y′′( x) = −force
P1 yS′( –xof)E
Sis
x−of
)C=1 31
xsin(
)and
+ ky11 (xx) )− C41 k1 cos(gk1i x) = g0gigi→
where
where:
2 CP
41
1
y Young
EJ tg
g(i modulus,
)yk01(transver1 xtransversal,
the
eaction
sliding
support,
EJ
isthe
y0reaction
1
EJ11the
) =force
0→
sin(
) +and
cos(C
)EJ=g0i occurring
+constants
+ CC
y ( g0and
C
C
ginitial
k1 g, iC
C
k1i gobtained
equation
may
beisfound
graphically. T
value
of
∆g,
which
related only
to point
21
31
41
i
i
i
+
CJ21-=yis
(x),
y
(x)
are
the
buckling,
the
buckling
as The
result
of (3.b)
the
force
EJfrom
,
C
,
and
the
equation
system
are
The
integration
P1
1 of inertia of the bent
11
21
31 cross-section.
41
the modulus
object’s
sal, 0sliding
support,
oung
modulus,
and
g
P
y
EJ
sec(
g
)
P
2
2
2
i
1
0
i
‘A’
(the
point
where
the
function
charts
1
(relatively
xy)′′(0)g=respectively;
ere y ( x) = y0 ( x) +P1y1action,
(1)
y
EJ
sec(
g
)
P
→ −C
01 x ) = 0 →
i C411 = 0 P1EJ
i 0 equal:
ere: of inertia
E – isofthe
Young
31 k1 sin( k1 x ) − C41 k1 cos( ky
odulus
the
bentmodulus,
object’s
cross-section.
−y gEJ
EJ
sec(
EJ )gi
EJ and
)
31 =
0C
cross), we read from the x-axis (Figure 5,
0 i sec(
C31 =‘x’,
− sliding
EJof the
Sthe
-CThe
is
the
reaction
force
of
the
transversal,
support,
P1 EJ
(3.a)
1 modulus
=
−
y
J –are
is the
of
inertia
of
the
bent
Differentiating
equation
(1)
twice
towards
we
subsequently
obtain:
(x), and y1 (x)
initial
buckling,
and
the
buckling
occurring
as
result
force
,
C
,
C
,
and
C
obtained
from
the
equation
system
integration
constants
C
− C31 = − 41
11
0
(3.c)
11
21C31 =31
P
g
see
page 80).
This isare
a graphical solu1
i
P1 modulus, and
gi
P
E
is
the
Young
EJ
object’s
cross-section.
P
1
y
EJ
sec(
g
)
action,
respectively;
1
equal:
0 towards
i ‘x’,
ing equation (1) twicerelatively
we
subsequently
obtain:
g
P
EJ
tion
of
the
equation
system
(5 and 6). It
g
i
1
(3.c)
EJ
i
) of the
the
object’sCcross-section.
i as result
C31force
=J −- is
ial
and
the
buckling
of1 ybent
the
EJ
P1impossible1 to
′′( x)force
−y0 S1 tgof( ginertia
(0yx0transversal,
) occurring
P
P1=the
yy−′modulus
where:
- isbuckling,
the reaction
of
sliding
support,
= 0 EJ
(3.a)
(3.b)
IV
EJ
C
should
be
stressed
that
it
is
41
11
P
1
)=
gi
andtwice
(1.a)P1sec(
y ( x) = y0 ( x) +where:
ywhere:
+P1
y′′′( x) C=21 =gequation
C41 where:
=0
y ( xto)=
1 ( x)
Differentiating
(1)
= directly
g
P1EJ 1 from equa1
P1
1) )
where:
P1 P
obtain asec(
point
− S1Young modulus,
(isx)the
P1and
y′′(igxEJ
EJ C41 = 0 C41 = 0 where:
2
i )EJ
sec( gi i (gEJ
=
IV
i, P1)g
P1cos( gi
)
sec(
)
=
g
1
and
where:
(1.a)
i
y
x
y
x
y
x
(
)
=
(
)
+
(
)
wards
‘x’,
we
subsequently
obtain:
=
y
x
(
)
i tg ( g
Differentiating
equation
(1)
twice
towards
‘x’,
we
subsequently
obtain:
he
transversal,
sliding
support,
EJ
y
EJ
)
0
1
g
cos(
)
P
EJ of
i
0
isEJthe modulus of inertia EJ
of ythe
bent
object’s
cross-section.
tion (4) as a great number
of1EJ
P
i solutions
1
EJ
(3.b)
g
cos(
)
EJ
EJ
0
g
cos(
)
P
1
i
(3.d)
where:
C41 = 0 C21 = +
1
d
EJ deformation
(3.d) connects
this equation
The
solution
ofi this
sec( gi For
) = the condition which
EJ
the exists.
force P
the
∆g in
Equation (1.a)
1 with
gi finally2 takes
PP11 ) the shape: For the
EJ
condition
which
connects
the
force
P
with
the
deformation
∆g in the for
P
1
EJ
sec(
g
′
′′
−
(
)
P
y
x
P
y
x
S
(
)
1
g
fferentiating
the bent object’s
cross-section.
0
i
equation
may
be
found
only
graphically.
IV
1
1i
1
For
the
condition
which
connects
the
force
P
with
the
deformation
∆g
in
the
fo
g
cos(
)
equation
(1)
twice
towards
‘x’,
we
subsequently
obtain:
1
P
(3.c)
For
the
condition
which
connects
the
force
P
with
the
deformation
∆g
in
,
∆g),
we
substitute
the
integration
constants
C
,C
implicit
function
f(P
i
EJ
′′′(C
and
where:
(1.a) 1
y ( x)f(P
y ( x) + y11( x) substitute the integration
= EJ
.a) finally takes the
EJ y (2x) = 1 implicit
constants
C11, C21, C1131,
function
k1 =
y IVy shape:
( xx))31+==k−12 y′′EJ
( x) = 0 where:
1,0 ∆g), we
The
reasonthe
forintegration
this the
is also
that a symEJ
,
∆g),
we
substitute
constants
C
,
C
,
C
implicit
function
f(P
P
determined
from
equation
system
(3)
into
condition
that
describes
1
11
21
31t2,
substitute
the integration
constants
C11
,C
implicit
function
f(P , ∆g), we
P1
1
EJforce
P1
2 condition
2
gwhich
determined
from
equation
(3)form
into
theancondition
that and
describes
lengt
For kwe
the
connects
theFor
Pcondition
deformation
∆g
in the
of
the
which
connects 1system
the
metrical
deflection
line exists
that the
i ′′ obtain:
1 with the
twice
‘x’,
subsequently
=
y′′( x)Ptowards
=
0
where:
y
EJ
sec(
g
)
′
−
(
)
P
y
x
y
x
S
(
)
determined
from
equation
system
(3)
into
the
condition
that
describes
the
leng
0
i
1
EJ
IV
curved
rod,
which
does not
changing
during
processing.
Finally,
we obta
exists
‘different’
buckling
shapes
occur,
which
will(3.c)
be
1 The general
1
from
equation
system
(3)
into
the
condition
that
describes
th
of equation
(1)
may
bexwith
in
the
form
of:
EJ and
anddeflection
where:
(1.a)
y ( xforce
ydetermined
)so-called
= ycurved
)written
+integration
( x) deformation
(1.a)that
( x) = 1
= line
EJ( x)integral
rod,
which
does∆g
not
processing.
Finally,
in
the
‘different’
buckling
shapeswe
oc-obtain the fo
0P(1the
1the
we
substitute
constants
Cchanging
, C21so-called
, Cduring
, and
C
implicit
Cyfunction
1, ∆g),
11
31
41
Equation
finally
takes
the
shape:
31 = −(1.a)f(P
curved
rod,
which
does 1not
changing
during
processing.
Finally,
we
obtain
the f
EJ
EJ
expression:
discussed
in
the
subsequent
parts
of
this
article.
curved
rod,
which
does
not
changing
during
processing.
Finally,
we
obtain
P
(3.d)
where:
P
C
=
0
1
y
(
x
)
=
C
+
C
x
+
C
sin(
k
x
)
+
C
cos(
k
x
)
(2)
l integral
(1)41 11may
be
the
form41
of:P
form
of1 an
implicit
cur,
whichof
will
expression:
21 written
311 in system
1
sec( gi function
)that
= f(Pdescribes
1, ∆g), we the
determined
from
(3)
into
the
condition
length
thebe discussed in the subseP1 y′′(of
x) equation
gequation
2
IV y ( x ) = y
2 ( x )i + y ( x )
1 expression:
(1.a)theexpression:
EJ (2)
PC
EJ
′′
integration
constants
quent
parts
of
this article.
=
y
(
x
)
+
k
(
x
)
=
0yx
10 where: k1 substitute
1 11,
+( xC) 21=x +EJ
C31 sin(where:
k
x
)
+
C
cos(
k
)
1
curved
rod,
does not changing during
processing.
Finally,
the following
cos( gi we obtain
)
1
41
1
n
uation (1.a) finally
takes
thewhich
shape:
EJ
g41
g i boundary
gi
g 2
EJ g i
The
integration
constants
C
,
C
C
we
determine
from
the
n 21, C31, and P
i
1
11
where:
C
=
0
1
g
g
g
g
gik1 (i2 − 1)gi i )))
+ C31 (i sin(k1 i ) − sin(ik1 (i − 1) )) + C(3.d)
) − cos(
expression:
41
n
i g (C21 ) =
41 (icos( k1 i
sec(
n∑
i in
g
g
g
g
g
gni) 2 2 =
C
(
+
C
(
sin(
k
i
)
−
sin(
k
(
i
−
1)
))
+
C
(
cos(
k
i
)
−
cos(
k
(
i
−
1)
)))
+
(
TheFor
general
integral
of
equation
(1)
may
be
written
the
form
of:
P
2
n
n
n
n
g
g
g
g
g
∑
i
i
i
i
i
2
V
2
21
31
1
1
41
1
1
conditions:
the
which connects the
force
of
EJ
i the
(C21∑ni =1P+
( sin(
i(deformation
) k−1sin(
−∆g
1)kn1 in
+1)C41iform
())cos(
i cos(
− 1)kn1 (i)))
k1 condition
= 1
) + k1 y′′( x) = 0 where:
nsin(
nan) k−1cos(
(1CCwith
+ Ck31
iP1 )ik)1 −(isin(
(i))−the
+ Ck41
i i k)1−(icos(
− 1)+ (ni ))))2 +=
1cos(
1(
i∑
=1
he( xshape:
2131
g
i
n integration
nconstants
n
y ( ximplicit
)y=(0)
C11= +−function
C
x +CC
x)y+0we
C41 substitute
cos(
i =1k1 x ) i =1the
n
n
n
nn
y021EJ
C , k=
→
+sin(
31
1−
∆g),
Cg11,nC21, Cn31, (2)
and Cn41
n
gi
g11i f(P141
gi
gi
gEJ
P1
2
i
i 2
2
(4)
+the
C31 ((1)
sin(may
k1 iP ybe
) −equation
sin( k1 (i −in
1)
+ the
C(3)
cos(
k1the
iP with
)condition
− cos(
k1the
(P
i −that
1)
))) + ( ∆g
) the
= lthe
e kgeneral
integral
of(Cequation
written
the))form
deflection
∆g
consists
inofsubstituting
each value of forc
∑
21
41 (of:
determined
from
system
intoDetermining
describes
length
For
condition
force
the
deformation
form
ofthe
an
1 =
1nthe
3 n
1n 0 which connects
1 y0 n
n in
∆g
consists
in psubstituting
each value of force P1, con
i =1
cos(kDetermining
sin(
=CS1 cos(
=
→
−Cnot
= Finally,
ynk′′′(xg)i )+rod,
x) +
CDetermining
k13aprocessing.
k1deflection
x) the
the
deflection
∆g
consists
in
substituting
each
value
31 k1changing
1Determining
41as
parameter,
into
equation
(4)
and
drawing
the
curves
of of
theforce
function:
1, co
x) = C11EJ
+ C21 x + C31 sin(
k
x
)
(2)
curved
which
does
during
we
obtain
the
following
deflection
∆g
consists
in
substituting
each
value
of Pforce
1
41
1 f(P1, ∆g), we substitute
the integration
C11drawing
, C21, C31
, and
C41 of the function:
implicit
function
(3)
gconstants
as a parameter,
inton equation
(4) and
the
curves
i
(3) the
ion (1) may be writtenexpression:
in the form of:gi
as
a
parameter,
into
equation
(4)
and
drawing
the
curves
of
the
function:
g
g
g
g
as
a
parameter,
into
equation
(4)
and
drawing
the
curves
of
function:
i
i
i
determined
from
system
(3) C
intof n(the
that
the
length
of the
g31i( sin(Pk11 ,i considered
g k (i − 1)
gi k1 (2i − 1)g
gcondition
+ force
Cdescribes
) − sin(
)) + C41g(i cos( k1 i i ) − cos(
deflection
∆g consists
in substituting
of
∑
i value
i )g=g
nP1 ,k
0(31C(21sin(
=0→
y ( gi ) the
C11 + Cequation
x) + C41 cos(k1 xDetermining
)
21 g i + C31 sin( k1 gf(2)
1 21
i n)gi ==
(i )P1+
, gi )41= cos(
(each
C
C
− sin(gk1 n(i − 1) gi i))1 + C41g( cos(
n k1 i gi)we
n k1 i gi) − cos(gki1n(i − 1) g)))
2+ ( gg
∑
gsin(
i+
i
1
i
i
i
curved
rod,
which
does
not
changing
during
processing.
Finally,
obtain
the
following
f
(
P
,
g
)
=
(
C
+
C
(
k
i
)
−
sin(
k
(
i
−
1)
))
+
C
(
cos(
k
i
)
−
cos(
k
(
i
−
1)
n the
n
n
n
∑
(
(
C
+
C
(
sin(
k
i
)
−
sin(
k
(
i
−
1)
))
+
C
(
cos(
k
i
)
−
cos(
kn1 ()))
i − 1)+ (ni
i f curves
1 the
31
1
1
41
1
1
iP
=11 , g i ) =21
as a parameter,
into
equation
(4)
and
drawing
of
function:
2
2
∑
21
31
1
1
41
1
n ′′
n
n
n
n
n
Figure
6.
Determination
of
one
of
the
points
(P
,
g
).
= g0 → −C31 k1 sin(
y (0)
x)i ==1 0f→
n
n
g k x) − C k1 cos(
g k and
i =C
1 g = 0
=n l pk (i −11)g gi i )))n2 +g( gi ) 2 = l n(4)
expression:
n
(C21 i g+i C31 ( sin(
k1 i gi )i 1− sin( k1 (41i −and
1) gii ))f +(1C
( cos(
cos(
=(k=1P
P
li1lpg,41i g)i )−i )−cos(
2
1
i
i 2
(5) p
141,(,g
i ))k
f ( P1 , gi )∑
=
(C21
k1 i n ) − C
sin(
k
(
i
−
1)
))
+
C
cos(
i
k
(
i
−
1)
)))
+
(
)
(
f
P
g
n + C31 ( sin(
n
n
n
n
and
1
41
1
1
(
,
)
=
f
P
g
l
i =1 ∑ integration
,
C
,
C
,
and
C
obtained
from
the
equation
system
are
The
constants
1
i 41n1 =np100
11
21 n 31 and
i
where:
isp the number
ofnparts into which the interval of the functio
n
n
n
i =1
where:
ni, P
=1100
the
number of Pparts
into
which
the interval of the function variab
)100
of is
the
dependency
Irrespective
Inrelatively
nthis way we obtain one of the points
1 g= f(∆g).
gi equal:
gi
g i (gwhere:
g
gwhich
2 parts
where:
n
=
is
the
number
of
parts
theby
interval
of (4)
theasof
function
varia
i
i
i into
is
divied,
which
is
accepted
using
the
giving
suffic
n k=is1 i100
iscos(
thekusing
number
ofinto
theauthors
interval
the function
(4)
( sin(dependency
k1 i ) − sin( kP
−=1)divied,
)) + C
( cos(
)−
i − 1) intuition
)))
+ ( intuition
) 2(6)
= l which
, g(Ci )21 = l+p C31the
Pdetermining
and fof(∑
p authors
1 (i is
41is
1 (draw
which
accepted
by
the
giving
sufficient
acc
1
f(∆g),
it
also
possible
to
deflection
lines
for as
1
(3.a)
Determining
the deflection
,
considered
∆g consists
substituting
each
value
of
force
P
Ci =111 = − y0 n
n
n inisthe
n
n
n
1
is divied,
which
is
accepted
using
intuition
by
the
authors
as
giving
sufficient
calculation.
greaterusing
the value
‘n’, by
thethe
greater
theasaccuracy
of acc
the
divied,
which
isThe
accepted
intuition
authors
giving
suffici
the
calculation.
The
greater
the
value
‘n’,
the
greater
the
accuracy
of
the
calcula
bent
rods
by
substituting
the
points
(g
,
P
)
obtained
into
the
rod
deflection
equation
where:the
n
=
100
is
the
number
of
parts
into
which
the
interval
of
the
function
variability
(2)
i
1the curves
as a parameter, into equation
(4) and
of
the
function:
thedrawing
calculation.
Thesame
greater
the
value
the‘n’,
greater
the accuracy
of the of
calcul
butcalculation.
at the
time
this
way‘n’,
significantly
increases
the
calculation
ti
P1
the
The
greater
the
value
the
greater
the
accuracy
the
but
atthe
theauthors
same time
thisg way
significantly
calculation time used
yn0 way
EJ
tg (we
ggi obtain
)
(2).
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a∆g
rod
deflection
related
to the
point
(gi, Pincreases
)i 2in(3.b)
the the
is divied,
which
accepted
using
intuition
as
giving
sufficient
accuracy
1increases
gi consists
gequation
gsignificantly
gto
butby
at
the
time
this
way
significantly
the calculation
time
use
i
i same
i value
i P21, considered
Determining
the
deflection
in
substituting
each
of
force
computer
program.
(5)
EJ
but
at
the
same
time
this
way
increases
the
calculation
tim
0) =
f
(
P
,
g
(
C
+
C
(
sin(
k
i
)
−
sin(
k
(
i
−
1)
))
+
C
(
cos(
k
i
)
−
cos(
k
(
i
−
1)
)))
+
(
)
(5)
C 1= i +The
∑ 21 n the
31
1
1 greater
1
computer
program.
following
form:
the calculation.
value
thedrawing
the41curves
accuracy
of function:
the1 calculations
is,
n‘n’,
nprogram.
n
n
computer
gi i =1 greater
as21a parameter,
and
the
ofnthe
P1 equation (4)
computer
program.
2 into
g
but at and
the same
way significantly
increases
the calculation time used
by the
n
f ( P ,time
g )i =this
lEJ
g
g
gi value of ∆g, gwhich
g to2 point
gi 2 ‘A’
(6)
The
is (related
point where the functi
i
(5)(the
P1 , gi ) =1 ∑ i (C21 pi + C31 ( sin(k1 iP i ) −The
sin( kvalue
+ CP41which
( cos( k1 iis related
) − cos( k1to
i −point
1) i )))
) point
‘A’+ ((the
where the function chart
computerf (program.
1 (i − 1) of))∆g,
1n
1 which
The
value
of
∆g,
is
related
to
point
‘A’
(the
point
where
thesolution
function
n
n
n
n
n
we
read
from
the
x-axis
(see
Figure
6).
This
is
a
graphical
ofchar
the
P
i =1
The
value
of
∆g,
which
is
related
to
point
‘A’
(the
point
where
theequation
functio
)interval
y0 EJ
tg ( gi of )parts
yread
gi the
1 number
where: yn0 =
into
which
the
the function
variability
(2) solution
0 EJ sec(
from
(see
6). This
is a graphical
of the
EJ100
sec(
)
y0 gisi the
P1 ofFigure
EJ ) we
EJx-axis
(7)
we
read
from
the
x-axis
(see
Figure
6).
This
is
a
graphical
solution
of
the
equatio
(3.c)
(5
and
6).
It
should
be
stressed
that
it
is
impossible
to
obtain
a
point
(g
(7)
EJ
(
)
(
sin(
)
=
−
+
+
−
y
x
y
x
x
we
read
thestressed
x-axis
(see
Figure
6).
This is
aobtain
graphical
solution
thei, eP
( P01which
, g i ) =isisl paccepted
isand
using
intuition
thefrom
authors
as
giving
accuracy
to
C31divied,
= −f∆g,
and(the
6).by
It
should
be
thatsufficient
it is impossible
to(6)
a point
(gi, P1of
) direc
The value
of
which
point(5
‘A’
point
where
the
function
charts
cross),
gi P related
EJ
Pto
P
2
1
1
(5
and
6).
It
should
be
stressed
that
it
is
impossible
to
obtain
a
point
(g
,
P
)
(4)should
as aaccuracy
great
number
of
of this
exists.
The
i
1 (gdirec
(5equation
andas6).the
be
stressed
that
itsolutions
is
to equation
obtain
point
1
gnumber
i, P1
calculation.
The
theofvalue
‘n’,
the
the
ofsolutions
the
calculations
is,
i
i(4) greater
g100
equation
a Itsolution
great
number
ofimpossible
this equation
exists.aThe
solution
where:
n =x-axis
is
thegreater
parts
into
which
interval
ofthe
theof
function
variability
(2)
i
we readthe
from
the
(see
Figure
This
is
a ggraphical
of
equation
system
EJ 6).
EJ
equation
(4)
as
a
great
number
of
solutions
of
this
equation
exists.
The
solutio
EJ
equation
may
be
foundnumber
only time
graphically.
The
reason
for this
is also
tha
equation
(4)
as
a
great
of
solutions
of
this
equation
exists.
The
but
at
the
same
time
this
way
significantly
increases
the
calculation
used
by
the
equation
be authors
found
only
reason to
for this is also that a syms
is divied,
which
is accepted
intuitionmay
by
as
giving
accuracy
(5 and 6).
It should
be stressed
that using
it is impossible
to the
obtain
a point
(ggraphically.
, P1)sufficient
directlyThe
from
igraphically.
equation
may
be
found
only
The
reason
for
this
is
also
that
a
sym
P
1
equation
may
be
found
only
graphically.
The
reason
for
this
is
also
that
(3.d)
where:
C
=
0
1
computer
Equations
3,
4,
and
7.
41 5calculation.
the
greater
the value of
‘n’,this
thegequation
accuracy
of
theconsists
calculations
sec(
) = the
equation
(4)of
aspoints
aprogram.
great
number
of solutions
exists.
The solution
of this of theis,
i greater
way
The row
(PThe
1, ∆g = g0 – gi) determined in the
EJ above-described
P1
but
at
the
same
time
this
way
significantly
increases
the
calculation
time
used by the
)
equation may characteristic
be found onlyP graphically.
Thecompressed
reason for thiscos(
isgialso that
a symmetrical
frelated
(∆g)
(Figure
7).charts cross),
FIBRES &mechanical
TEXTILES
in Eastern
Europe
Vol. 16, of
No.to
5the
(70)
EJ the
79
The
value
ofJanuary
∆g,/ December
which1/ A=is2008,
point ‘A’ (the monofilament
point where
function
computer
program.
Forread
the from
condition
which(see
connects
the deformation
the form
of an
we
the x-axis
Figurethe
6).force
This P
is1 awith
graphical
solution of∆g
theinequation
system
,
∆g),
we
substitute
the
integration
constants
C
,
C
,
C
,
and
C41
implicit
function
f(P
(5
and
6).
It
should
be
stressed
that
it
is
impossible
to
obtain
a
point
(g
,
P
)
directly
from
1
11
21
31
i
1 charts cross),
The value of ∆g, which is related to point ‘A’ (the point where the function
Simulations of the compression
process on the basis
of the mathematical model
Figure 5. Determination of one of the
points (P1, gi).
Figure 6.
Mechanical
characteristic
P1 = f (∆g) of the compressed monofilament.
Figure 7. Dependencies P1 = f (∆g) for various
values of the rod’s length: 1-for lp = 12 mm,
2-for lp = 15 mm, 3-for lp = 20 mm, 4-for
lp = 30 mm, 5-for lp = 50 mm.
Figure 8. Dependencies P1 = f (∆g) for various
values of rod rigidity: 1-for EJ = 20,0 cN mm2,
2-for EJ = 8,8 cN mm2, 3-for EJ = 4,4 cN mm2,
4-for EJ = 2,2 cN mm2, 5-for EJ = 1,1 cN mm2.
In this way we obtain one of the points
(gi, P1) of the dependency P1 = f(∆g). Irrespective of determining the dependency
P1 = f(∆g), it is also possible to draw deflection lines for the bent rods by substituting the points (gi, P1) obtained into the
rod deflection equation (2). In this way
we obtain a rod deflection equation related to the point (gi, P1) in the following
form (Equation 7).
accepted, we can obtain a general characteristic of the whole spatial knitted
fabric. The force compressing the whole
knitted fabric should only be divided by
the number of existing connectors, and
for the force thus obtained, the mechanical characteristic P1 = f (∆g) would be
designed. Data from comparable results
attained by the mathematical simulation
and by experimental tests obtained with
the use of a measuring stand specially
designed by us are not included in this
article, as the stand is actually used later
in the investigation; however they will
be the subject of the next publication.
The row of points (P1, ∆g = g0 – gi) determined in the above-described way
consists of the mechanical characteristic
P1 = f (∆g) of the compressed monofilament (Figure 6).
The mechanical characteristic was calculated and drawn for the following parameter values: knitted fabric thickness
g = 10 mm, rod rigidity EJ = 4.4 cNmm2,
rod length lp = 15 mm, mutual displacement of the monofilament fastenings in
the planes of the knitted fabric y0 = 2
mm. The curve has a strongly non-linear character with a large increase in
the bending force P1 at the greatest rod
deflections. The mechanical characteristic P1 = f (∆g) has a similar character
to the one obtained empirically [10].
Notwithstanding that the mathematical analysis carried out by us concerns
a single monofilament, and thanks to
the assumptions for the physical model
80
For our simulation we accepted that the
thickness of the compressed knitted fabric would be g0 = 10 mm, and the rodconnector would consist of a polyamide
monofilament (JPA) of d = 0.14 mm. The
tensile elasticity modulus was assessed
empirically with the use of tensile tester
from Hounsfield. The value of the Young
modulus characterises the material of
the rod tested . An increase in the Young
modulus makes material more rigid,
which means that it is not so susceptible
to deformation, whereas a decrease in
rod (polyamide monofilament) diameter
results in an increase in the modulus of
inertia J of the cross-section of the bent
object, and consequently an increase in
rod rigidity.
Aiming at the simulation of the compression process of a monofilament placed
in the internal layer of a spatial knitted
fabric, two calculation algorithms were
elaborated, one for calculating the compression characteristic P1 = f (∆g), and
the second for determining the curves
representing the shape of the compressed
rod. The calculations were carried out
with the use of the MathCad computer
program. As a result of the computer
simulation, the deflection dependencies
were determined as a function of the rod
length lp and rod rigidity EJ, illustrating
the influence of these parameters on the
values of forces and deflections (Figures
7 and 8). Figure 9 presents the curves
P1 = f (∆g) for different values of the rod
length. Only small differences are visible
in the shape of the particular dependencies. This results from the fact that the
functions representing the deflection lines
of the rod in the model described have a
similar shape, notwithstanding relatively
great differences in the lengths (significantly, the curvatures of the deflections
do not mutually differ).
The examples shown in Figure 7 indicate that for various lengths lp and for
the same value of the force P1 different
deflections ∆g occur, or the opposite,
formulated for the same deflection ∆g
different forces P1 ,is also indicated. For
example, if the deflection of a rod of
length lp = 12.5 mm equals ∆g = 7.5 mm,
then the bending force is P1 ≅ 7.72 cN,
whereas for a rod of length lp = 50 mm
the force is P1 ≅ 6,71 cN. Thus we can
conclude that the mutual relations between the force and deflection depend on
the length of the rod.
On the other hand, Figure 8 presents the
dependencies P1 = f (∆g) for various rigidity values EJ. In this figure it is clearly
visible that for various rigidities EJ of
the rod and the same force P1, different
deflections ∆g occur, or in the opposite
case, for the same deflection ∆g, different
forces P1 are indicated. For example, if for
a rod with a rigidity of EJ = 1.1 cNmm2
the deflection equals ∆g = 7mm, then the
bending force is P1 ≅ 1.2 cN, whereas for
a rod with a rigidity of EJ = 20 cNmm2,
the bending force will be P1 ≅ 1.2 cN.
Thus means that the rigidity of the rod
increases nearly 18 times, causing the
resistance to mechanical loading or load
capacity of the rod to increase 30 times .
FIBRES & TEXTILES in Eastern Europe January / December / A 2008, Vol. 16, No. 5 (70)
Thanks to the analysis carried out, it is
visible that changes in the rigidity of the
rod-connector have an significant influence on the character of the dependencies P1 = f (∆g), as they cause a low
number of changes in the shape of the
dependencies; a significant displacement to the left of the chart results in the
same forces P1 causing a much smaller
deflection ∆g.
Deflection x, mm
8
6
4
2
10
8
Buckling y,mm
6
4
2
0
Figure 9. Simulation of the rod’s deflection
lines under the impact of force P1, obtained
using the mathematical program.
Figure 10. Families of curves representing
the shape of monofilaments with the same
deflections ∆g, and differentiated by
parameter y0.
Figure 11. Different shapes of rod deflection
caused by the action of force P1.
A simulation of the shape of monofilaments consisting of 3D knitted fabric,
which are under the influence of the compression force P1, was also carried out.
The first of the simulations performed
illustrates the shape of a monofilament
under the influence of force P1.
From observation of the connector’s
geometry, as well as the mathematical
analysis performed, it can be concluded
that the rod representing the monofilament takes the shape of a fragment of a
sinusoid. The shape of this sine curve depends on the value of the force P1. With
an increase in the force value, an increase
in the amplitude of the curve also occurs,
but the it tends to be narrower. The cause
of P1 increasing is, from a physical point
of view, a decrease in the curvature radius
of the sinusoid’s apex. Figure 9 presents
a broad spectrum of deflections caused
by the action of force P1.
The next simulation was carried out with
the aim of finding the influence of the
distance knitted fabric’s geometry on the
value of the compression force P1. It is
important here to determine the influence
of this force on the mutual displacement
of the connector’s (monofilament’s) ends
in the direction of the y-axis (parameter
y0 is designated in Figure 4). Parameter
y0 has three different values:
 for the symmetrical system y0 ≈ 0 mm,
and
 for
the
asymmetrical
system
y0 = ± 2 mm.
Figure 10 presents 4 families of curves
representing the shape of the monofilament with the same deflections ∆g but
differentiated by the parameter y0.
 lines 1 – force P1 = 0.5 cN for yo = 2 mm,
P1 = 0.4933 cN for y0 = 0 mm
and P1 = 0.4862 cN for y0 = -2 mm,
 lines 2 – force P1 = 1.2 cN for y = 2 mm,
P1 = 1.193 cN for y0 = 0 mm
and P1 =1.043 cN for y0 = -2 mm,
 lines 3 – force P1 = 6 cN for y0 = 2 mm,
P1 =5.327 cN for y0 = 0 mm
and P1 = 4.74 cN for y0 = -2 mm,
 lines 4 – force P1 =100 cN for y0 = 2 mm,
P1 =85.96 cN for y0 = 0 mm
and P1 = 74.41 cN for y0 = -2 mm,
The differences in the bending forces,
which cause the same deflections, result
from the various geometries of the rods
(parameter y0). The rods are differentiated by shape (different functions describe
the shape of the curves), and with them
are connected different curvature radii
FIBRES & TEXTILES in Eastern Europe January / December / A 2008, Vol. 16, No. 5 (70)
of these lines, which finally leads to the
situation that different forces cause the
same deflection. The differences in the
force P1 values change with an increase
in these values:beginning with small
differences of P1 ,causing deflections
of ∆g = 0.615 mm, to relatively high
differences in P1, which bring about deflections of e.g. ∆g = 9.287 mm. This
directly results from the dependency
P1 = f (∆g); as for greater deflections the
curve is steeper. The analysis also allows to state that the greatest value of
P1 related to the same deflection falls
when the value of parameter y0 = 2 mm.
From the point of view of geometry, this
can be explained by the curvature radius
of the sinusoid apex related to the shape
of the connector, which reaches its
smallest value at this parameter value.
The smaller the curvature radius of the
bent rod fragment, the higher the bending tensions are in this fragment. On
the other hand, the force values, which
cause this deflection, depend only on the
value of this spectacular (smallest) curvature radius.
The third simulation concerns the influence of the monofilament’s shape on the
force compressing it. This is a problem
of the so-called buckling form that takes
place. This phenomenon consist in them
that at equal deflections ∆g of the same
rod with the same geometry of end fastenings some different values of force P1
may be created.
Figure 11 presents the following different forms of rod deflection created by the
impact of force P1:
 1-3 lines of the rod deflection under the
impact of bending force P1 with values
of – 1.315 cN, 5.25 cN, amd 11.79 cN,
causing deflections ∆g of 4 mm;
 2-3 lines of the rod deflection under
the impact of bending force P1 with
values of – 3.065 cN, 12.25 cN, amd
27.6 cN, causing deflections ∆g of
6 mm;
 3-3 lines of the rod deflection under
impact of bending force P1 with values
of –12.7 cN, 50,07 cN, amd 116 cN,
causing deflections ∆g of 8 mm.
From observation of the connector’s geometry, as well as from the mathematical analysis, it can be concluded the
force causing a deflection in the case of
the first buckling form is more than four
times smaller than that in the case of the
second buckling form, and eight times
smaller than for the third case.
81
The practical application importance of
the above-described simulation results
from the fact that the rod-connector, being an element of the knitted fabric, is
not isolated. The neighbouring connectors interact by contact, and the greater
their number in the given volume of the
knitted fabric, the greater the probability
of such interaction is. Also the increase
in deflection ∆g causes a decrease in
the volume of compressed rods. This,
in turn, led to the mentioned increase in
the probability of interaction between
neighbouring monofilaments. Contact
between neighbouring monofilaments
causes the occurrence of higher forms of
buckling, and this, in turn, increases the
value of the compression force P1, causing the same deflection ∆g. From the
curves in Figure 11, it is visible that the
trajectories related to equal deflections
∆g are mutually tangential , whereas the
beginnings origins of the trajectories are
tangential only for odd or even numbers.
Summary
This article presents a physical and
mathematical model of the process of
compressing an elastic, slender rod of
assumed shape fastened both sides by
articulated joints. The model presented
is related to the process of compressing
a 3D distance knitted fabric, which we
consider as a spatial trust.
1. The physical (mathematical) model is
based on the following assumptions:
the knitted fabric planes p1 and p2 are
displaced in a perpendicular direction
without deformation, the connectors
are considered as elastic rods with a
structure of spatial configuration , and
there are elements fastening both of
the sides to the outside planes of the
knitted fabric by articulated joints.
On the basis of the physical model, a
mathematical model was developed
which is a differential normal equation of the 4th order. The solution of
this equation at fulfilled boundary
conditions and the length condition
of the rod (lp = constant) is the deflection line of the compressed rod.
The boundary conditions (integration
coefficients C12, C21, C31, and C41) determine the physical (mathematical)
model accepted. As a result of the
mathematical analysis carried out, we
obtained the following two equations:
 an equation determining the set of
points comprising the mechanical
82
characteristic of the compressed rod,
and
 an equation determining the shape of
the compressed rod.
2. A computer simulation was carried
out of the compression process of a
model connector of the knitted fabric
which included:
 elaboration of the mechanical characteristic of the compressed monofilament P1 = f (∆g),
 simulation of the influence of rod
length lp, and such structural parameters as rigidity EJ. the system’s characteristic,
 the influence of the fastening geometry of a singular monofilament on the
value of force P1, causing the monofilament’s deformation,
 Influence of different forms of buckling of the compressed monofilament
which may occur on the value of deflection ∆g at the impact of compression force P1.
The simulation carried out demonstrated
that:
 The mechanical characteristic obtained is a strong non-linear curve; at
this stage of the investigation we did
not find a mathematical equation of
this curve which would connect the
parameters of the system together.
 The change in parameter lp causes
small changes in the mechanical characteristic P1 = f (∆g); for example, for
the compression force P1 = 5 cN, the
deflection difference ∆g between the
lengths lp = 50 mm and lp = 12.5 mm
amounts to 0.2 mm.
 A change in the parameter EJ causes
great changes in the mechanical characteristic P1 = f (∆g); for example,
for the compression force P1 = 5 cN
between rigidities EJ = 20 cN and
EJ – 1.1 cN, the deflection changes
from ∆g = 5.7 mm to ∆g = 1.6 mm,
respectively.
 A change in the parameter y0 (mutual
displacement of the connector’s fastening) influences the value of force
P1, e.g. from 3% of the P1 value for a
deflection ∆g = 0.6 mm to 34% of P1
for ∆g = 9.3 mm.
 For the third form of the buckling of
the bent rod, the force P1 is more than
200% greater in comparison with its
value for the second buckling form,
and more than 800% greater than for
the first form for the same deflection
∆g (these values differ slightly for different deflection values).
Conclusions
The investigations carried out should
be treated as an introduction to further
research aimed at determining an optimum calculation model for 3D distance
knitted fabrics. This involves determining the mechanical properties of such
knitted fabrics, but firstly their susceptibility or resistance to compressing
must be evaluated. In further parts of
our research, we will design and build
an empirical testing model with the aim
of verifying the correctness of the mathematical models already elaborated, as
well as future, new models. The next
stage will be the adaptation of the calculation mechanism developed to solve
real problems occurring while designing
3D distance knitted fabrics., e.g. a case
of spherical loading will also be considered.
This investigation carried out by us and
presented herein of an element of 3D
knitted fabric in the form of a single connector, which means the elaboration of
the mechanical characteristic of the compression process for this connector and
the determination of its shape will be the
basis for further research – more general
and dedicated to designing 3D distance
knitted fabrics
References
1. K. Kopias, „Technologia dzianin kolumienkowych”, ed. WNT, Warszawa 1986.
2. W. Żurek, K. Kopias, „Struktura płaskich
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Received 20.08.2007
Reviewed 30.10.2007
FIBRES & TEXTILES in Eastern Europe January / December / A 2008, Vol. 16, No. 5 (70)