Modeling Roll Contact Curves of a Squeezing Machine
Kuvondar Bektoshev
University of Tashkent for Applied Sciences, Gavhar Str. 1, Tashkent 100149, Uzbekistan
https://doi.org/10.5281/zenodo.10471311
Keywords:
Roller squeezing, Moisture filtration, Hydraulic pressure, Squeezing area.
Abstract:
Mathematical models of roll contact curves of leather squeezing machines were developed. The
phenomenon of contact interaction of leather with working rolls is considered taking into
account the water filtration during the squeezing process and the strain properties of leather and
cloth. Analysis of the obtained models showed that the contact curves of the rolls depend on the
thickness, humidity and deformation properties of the leather, the radii of the rolls and the
deformation properties of the cloth, as well as on the forces acting on their support.
1 INTRODUCTION
Roll mechanisms are an integral part of many
machines. They are especially widespread in the
leather industry. They operate in squeezing,
spreading, rolling, and other machines for the
mechanical processing of semi-finished leather
products.
In leather production, product quality is associated
with environmental safety, since the squeezing
process is associated with the problem of wastewater
disposal.
In the theory of leather roller pressing, a coupled
solution to two types of problems is necessary -
contact interaction of leather with working rolls
(contact problem) and water filtration (hydraulic
problem).
References [1-5] are devoted to solving contact and
hydraulic problems of roller pressing of wet
materials.
During the squeezing process, the cloth (an elastic
covering of rolls) and the leather are deformed under
the influence of pressure, transmitted to the skin along
the roll contact curves.
In [6,7], the shapes of roll contact curves, which do
not take into account the phenomenon of water
filtration during the squeezing process, are
analytically described.
The article is devoted to modeling the contact curves
of the rollers of leather squeezing machines. This
takes into account changes in moisture during the spin
process and the deformation properties of leather and
cloth.
2 Materials and Methods
Figure 1 shows a diagram of a roller squeezing
mechanism, in which rolls with radii
R
have
covering of cloth with a thickness of
H
, and the
leather has an initial of
1
. This scheme is most
typical for leather squeezing machines and has
advantages over other options in terms of size, metal
consumption, simpleness, and quality of processing
[7].
The skin deformation zone is divided relative to
the line of centers into zones of compression I and
deformation recovery II. Here, zone I is not equal to
zone II, since the skin deformation does not obey
Hooke’s law [8,9]; therefore, the symmetry of the r
oll
contact curves relative to the line of centers is not
observed.
According to [7], in the roll contact curves, there
are zones of lag, sticking, and advance. The sticking
zone is divided into two parts. The first refers to the
compression zone and the second - to the deformation
recovery zone. In this regard, the contact curves of
each roll have sections 1, 2, 3, and 4, corresponding
to segments
,
2
1
A
A
,
3
2
A
A
4
3
A
A
, and
5
4
A
A
(Fig. 1).
Let the equation of the contact curve of each roll be
given in polar coordinates
)
(
i
i
i
r
r
=
,
4
,
1
=
i
,
−
i
is
the index standing for the number of the section.
According to Fig. 1:
,
3
1
1
−
−
,
0
2
3
−
4
3
0
,
2
4
4
,
where
−
2
1
,
are the contact angles,
,
3
−
4
angles separating the sliding and sticking zones.
According to [6], the dependence between contact
voltages is written like this:
,
)
sin(
)
cos(
)
cos(
)
sin(
i
i
i
i
i
i
i
i
i
i
n
r
r
r
r
t
+
+
+
+
−
+
=
(1)
where
;
Q
F
arctg
=
−
F
Q
,
are the pressure force
of the clamping devices and the horizontal response
of the roll supports,
−
i
i
n
t
,
are the shear and
normal stresses distributed over the roll sections.
From equality (1), it follows:
.
)
sin(
)
cos(
)
cos(
)
sin(
i
i
i
i
i
i
i
i
i
i
t
n
t
n
r
r
+
+
+
+
−
+
=
(2)
By research conducted in [6,7], it was established
that in the lag sliding zone, the Amonton law is
observed
,
1
1
1
1
tg
n
f
t
=
=
where
−
1
1
,
f
are the
coefficient and angle of friction of the skin on the
surface of the cloth under compression.
Then from equality (2), for the lag sliding zone, that
is, for the 1st section of the roll, we have:
or
).
(
1
1
1
1
−
+
=
tg
r
r
Having integrated the latter and conducting some
transformations taking into account the initial
condition
R
r
=
1
for
1
1
−
=
, we obtain the
equations for the contact curve of the 1
st
section of
the roll:
,
)
cos(
)
cos(
1
1
1
1
1
R
r
−
+
+
−
=
.
3
1
1
−
−
(3)
From Fig. 1, it follows that the relative deformation
of skin in the compression zone has the following
form:
(
)
.
2
,
1
,
cos
cos
2
1
1
=
−
=
i
R
r
i
i
i
(4)
The leather is pressed after chrome tanning. It has
been established [8,9] that the nature of skin
deformation is described by empirical dependencies
of the form:
,
1
1
1
*
s
i
in
m
i
i
W
W
A
=
(5)
where
−
*
i
is the compression stress,
−
1
1
1
,
,
s
m
A
are the coefficients characterizing the properties of
leather under compression,
−
i
W
is the skin moisture
content,
−
in
W
is the initial skin moisture content,
that is, skin moisture before squeezing.
Substituting the expression for relative deformation
into equation (5), for the 2
nd
section
from equation (4), we obtain:
.
cos
cos
(
2
1
1
2
1
2
2
1
1
*
2
s
in
m
W
W
R
r
A
−
=
(6)
In the 2
nd
section, we select an element with length
2
dl
directed along the normal of the curved contact
(Fig. 1). The selected element of the skin is acted
upon by forces
2
dN
and
2
dT
and the reaction of the
cut off parts of the skin.
The components of force
2
dN
and
2
dT
are
balanced by force
2
*
2
dx
(Fig. 1):
1
1
1
1
1
1
1
)
sin(
)
cos(
)
cos(
)
sin(
tg
tg
r
r
i
+
+
+
+
−
+
=
Fig. 1. Scheme of the roller
squeezing mechanism
0
0
sin
0
cos
0
2
2
2
*
2
=
−
−
dT
dN
dx
o
,
or
),
cos(
2
2
*
2
*
2
−
=
n
(7)
where
−
2
is the angle between
2
dN
and
.
2
r
Taking into account expression (6), from (7), we
obtain:
1
*
2
1
2
2
1
1
2
( cos
cos )
m
n
A
r
R
=
−
1
2
2
2
cos(
).
s
in
W
W
−
(8)
An analysis of literature sources [6,8] showed that
the strain properties of cloth are also described by a
power law. Using the power law for cloth, we
obtain:
,
cos
)
(
1
2
2
1
2
1
u
R
r
H
B
n
−
=
(9)
where
−
1
1
,
u
B
are the coefficients characterizing the
properties of cloth under ompression.
At the points of the curved contact the condition is
satisfied
2
*
2
n
n
=
(Newton's law).
Then, according to expressions (8) and (9), we
obtain:
1
1
1
2
2
1
1
2
( cos
cos )
s
m
in
i
W
A
r
R
W
−
1
2
2
1
2
2
1
cos(
)
(
cos
,
u
B
R r
H
−
=
−
or assuming that
0
sin
2
2
tg
1
1
2
2
1
1
1
2( cos
cos )
1
m
S
in
r
R
A
W
−
−
−
1
1
2
2
1
2
cos
1
u
s
H
r
R
B
W
H
+ −
=
−
(10)
Expanding the binomials in brackets into power
series, and limiting ourselves to the first terms of the
series since
1
cos
2
cos
2
1
2
2
1
1
−
+
r
R
and
1
2
−
+
H
R
r
H
, after transformations we obtain the
equations for the contact curve of the 2
nd
section of
the roll:
1
1 1
1
2
1
(
(
))
s
B
H
u R H W
M
+
−
=
,
1
1
1
1
2
1
(
)
cos
s
in
A H m
W
N
−
=
,
1
1
2
1 1 1
2
1
1
2
1
2
cos
s
s
in
B u W
A HmW
K
+
=
,
1
1
2
1
M
N
r
K
+
=
,
.
0
2
3
−
(11)
Similarly to (11) and (3), we find the equations for the
contact curves of the 2
nd
and 4
th
sections of the roll:
2
2 2
2
3
2
(
(
))
s
B
H
u R H W
M
+
−
=
,
2
2
2
2
3
2
(
)
cos
s
r
A H m
W
N
−
=
,
1
1
2
2 2 2
3
2
2
3
2
2
cos
s
s
r
B
u W
A Hm W
K
+
=
,
2
2
3
2
M
N
r
K
+
=
,
4
3
0
, (12)
,
)
cos(
)
cos(
1
4
2
2
4
R
r
+
−
−
+
=
2
4
4
, (13)
where
−
2
2
2
,
,
s
m
A
are
the
coefficients
characterizing the properties of leather during
deformation recovery,
−
2
2
,
u
B
are the coefficients
characterizing the properties of cloth during
deformation recovery,
−
r
W
is the residual moisture
of the skin, that is, skin moisture after squeezing,
−
2
is the final thickness of the leather layer,
−
2
is the angle of friction of the skin on the surface of the
cloth during deformation recovery.
RESULTS
Mathematical models of the roll contact curves of
leather squeezing machines were developed. At that,
the phenomenon of contact interaction of leather with
working rolls was considered taking into account the
phenomenon of water filtration during the squeezing
process and the strain properties of leather and cloth.
The study of the squeezing process based on these
models made it possible to find with sufficient
accuracy the appropriate parameters for roller
squeezing of leather, necessary for the rational design
and operation of squeezing machines.
CONCLUSIONS
Analysis of the obtained models showed that the
contact curves of the rolls depend on the thickness,
humidity and deformation properties of the leather,
the radii of the rolls, the thickness and deformation
properties of the cloth, as well as on the forces acting
on their supports.
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