NIP ROLLER INDUCED CONTACT STRESSES
By
CHUNBAO XU
Bachelor of Science
Beijing University
Beijing, China
1988
Submitted to the Faculty of the
Graduate College of the
Oklahoma State University
in partial fulfillment of
the requirements for
the Degree of
MASTER OF SCIENCE
May, 1995
OKLAHOMA STATE UNrVERSITY
NIP ROLLER INDUCED CONTACT STRESSES
Thesis Approved:
ean of the Graduate College
ii
ACKNOWLEDGMENTS
I wish to express my sincere appreciation to my graduate advisor, Dr. 1. K. Good,
for his tremendous support, guidance and assistance throughout this research project. I
also wish to thank my other committee members, Dr. R. L. Lowery and Dr. C. E. Price,
for sacrificing their time. My thanks also go to Dr. T. Minahen for his counsel at the
beginning of this project.
I extend thanks to the Web Handling Research Center for providing this research
opportunity. And thanks go to graduate students Will, Yunpeng and Nanda for their help
and friendship.
Finally, I give my gratitude and appreciation to my sister Xiao and to my parents,
Fengchuan and Shouren, for their sacrifices and support. I also give sincere gratitude to
my American parents, Ray and Louise, who gave me so much love and supported me all
the way.
iii
TABLE OF CONTENTS
Chapter Page
I. IN'TRODUCTION........................................................................................... 1
II. LITERA. TURE REVIEW ......•.•........•.......•••••.............................•••••......••••...... 3
ill. CONTACT OF ELASTMOMERIC COVERED ROLLER
WITH RIGID ROLLERS..... .......... ... ................ ...................... ...................... 9
Experimental Setup............................................................................. 9
Tekscan Measurement ......................................................................... 11
Her'tz's Theory ..................................................................................... 13
Finite Element Analysis ....................................................................... 16
Comparision of Results ........................................................................ 18
Mooney-Rivlin Materials .................................................................... 23
IV. CONTACT OF A RIGID ROLL WITH A WIN'DIN'G ROLL .......••....••••••.. 27
Experimental Setup ............................................................................. 27
Material Properties of Bond Paper .......................•............................. 28
Pull-Tabs Calibration .......................•.••...............•.•••.......................... 30
Hakiel' s Model .................................................................................... 31
Finite Element Analysis ....................••..............••................................ 33
V. CONCLUSIONS AND FUTURE WORK. ..........................••••••••••...••••.••....... 38
Conclusions .......................................................................................... 38
Future Work ........................................................................................ 39
REFEREN CES .......................................................................................................... 40
iv
LIST OF TABLES
Table Page
I. Comparision of Area Beneath Three Curves .................................................. 23
LIST OF FIGURES
Figures Page
1-1. Pacer or Drive Roll Geometry .................................................................... 3
1-2. A Nip in Contact with a Winding Roll Application ................................... 3
3-1. A Schematic Diagram of the System Experimental.. ................................ l0
3-2. Nip Loads vs. Air Pressure Applied ........................................................... 11
3.3. Contact Rollers ............................................................................................ 13
3-4. Variation of Maximum Pressure with Nip Load(Hard Rubber) .............. 15
3-5. Finite Element Grid .................................................................................... 18
3-6. Variation of Contact Width with Nip Load .............................................. 19
3-7. Variation of Maximum Pressure with Nip Load(Soft Rubber) ................ 20
3-8. Pressure Distribution on Contact Surface -- Nip Load = 6.6Ibs/in ........... 21
3-9. Pressure Distribution on Contact Surface -- Nip Load = 8.8Ibs/in ........... 21
3-10. Pressure Distribution on Contact Surface -- Nip Load = 201bs/in .......... 22
3-11. Correlation with a Straight Line Using the Mooney-Rivlin
Constitutive Equation ............................................................................... 24
3-12. Pressure Distribution on Contact Surface ................................................. 26
3-13. Comparision of Results from FEM and Spengos ...................................... 26
4-1. Layer of Center Winding with Nip Roller on 3M Splicer Winder ........... 28
vi
Figures Page
4-2. Pull Tab Calibration ................................................................................... 31
4-3. Interlayer Pressures Distribution ............................................................... 32
4-4. Pressure Distribution on Contact Surface (Wound Paper Roller) ........... 37
vii
NOMENCLATURE
E modulus of elasticity
EI Young's modulus of nip roller
E2 Young's modulus of rubber
Er radial modulus of a wound roll
Ee tangential modulus of a wound roll
u Poisson's ratio
U 1 Poisson's ratio of nip roller
U2 Poisson's ratio of rubber
R radius of roller
N RPM
ex angle of inner face of the rubber covering
f3 angle of outer face of the rubber covering
P nip load
p(x) pressure distribution at contact area between rollers
a half contact width at contact suface
Rc radius of core
P rn maximum pressure on the contact area between rollers
e tangential direction
viii
p radial pressure
h web thickness
Il coefficient of friction betweensheets of paper
~j material constants of wound roll in Polar coodinates
b·· material constants of wound roll in Cartesian coodinates i)
Gre shear modulus at Polar coordinates
ix
CHAPTER I
INTRODUCTION
1.1 Overview
Flexible, continuous sheet materials, called webs, are often modified in process
machinery to add value to the web. Printing, coating, and laminating are examples of
web processes in which value is added to the web prior to its conversion to a discrete
product, such as a newspaper or possibly a wrapper for a food product. In web
processing machinery, the web will come into contact with rollers called "nip rolls". The
purpose of the nip roll is to provide enough traction to pull the web through the process
machinery. A nip roller typically involves two rollers, one whose surface is
incompressible (i.e., rubber and like materials) and one whose surface is compressible
(i.e., typically a metal material).
There are two common applications of nip rollers in web lines. One application
is that of a pacer or drive roller. The purpose of this type of a roller is to control the
tension and velocity of the web. To prevent slippage, the web passes through two rollers
which are in intimate contact, as shown in Figure 1.1. The second application of a nip
roller is that of a rider or lay-on roller in contact with a winding roll. In this application
the purpose of the nip roller may either be to (1) drive the winding roll or (2) to exclude
air which is attempting to enter the winding roll as shown in Figure 1.2.
In either application, the web is subjected to normal contact stresses similar to
those first quantitatively described by Hertz[l]. These contact stresses can either
1
improve or degrade web quality. In the paper industry, the paper is forced to transgress
nip rollers called calendar rolls. The paper is intentionally subjected to contact and
shearing stresses which result in a smoother web surface and, therefore, improved
2
quality. In most cases, however, the contact stresses are detrimental to quality. Coating
is a common web process in which value is added to the web. In sections of the web
machinery near the coating operation, a nip roller can spell disaster because the coating
may have higher affinity for the nip rollers than for the web. Relatively little is known
concerning the absolute values of those contact stresses. Some researchers, as discussed
in the literature review, have chosen to model these contact stresses through extensions of
Hertz's contact relationships, which were originally derived for isotropic materials.
1.2 Purpose
The purpose of this research is to remove as many of the simplifying assumptions
as possible, to study how the contact stresses behave in cases in which the roller materials
are (1) incompressible or (2) have complex properties due to being composed of multiple
layers of web. Both analytical and experimental investigations will be performed. The
outcome of this research will aid those engineers who are responsible for transporting
webs, whose quality is affected by contact stresses.
1.3 Organization
A literature survey addresses contact topics in Chapter 2. A plane strain model
for rubber covered rolls in contact with rigid rolls is developed in Chapter 3. A model is
developed in Chapter 4 for the contact of a nip roll with a winding roll. An additional
complexity is involved in this case since the radial modulus of elasticity of the winding
roll is a function of the contact pressure between web layers. In Chapter 5, the
conclusions of this research are presented, followed by possible avenues of future
research.
Fig. l.1
Web
Nip Roller
Rubber-covered
Roller
Pacer or Drive Roll Geometry
Q Nip Roller
~ Winding Roll
Fig. 1.2 A Nip in Contact with a Winding Roll
3
CHAPTER II
LITERATURE REVIEW
Hertz's contact theory of two elastic bodies was published in 1881 [1]. This
problem has received much attention and has been studied by several researchers. Due to
the complexity of the problem and instrumentation difficulties, experimental
investigations have been rather infrequent and most researchers have employed
numerical techniques, typically finite element methods, to solve the problem.
In 1950, Hannah[3] modified Hertz's theory of contact for a roll which consists of
a thin elastic cover on a rigid core. She studied the effects of material, thickness of the
cover, the pressure between rollers and the roller size on the contact area, deformation of
the cover and the contact pressure. She mathematically considered the problem as one of
generalized plane stress in an elastic layer, with given displacement conditions on the
interface between metal and cover, and subjected to pressure by a body of given shape on
its free face.
Hannah found that, next to the roller diameter and elastic modulus, the layer
thickness is the most important factor determining the relationship between loading and
deformation for this type of roller. The pressure distribution over the contact zone is
only slightly affected by layer thickness. However, a change in Poisson's ratio makes an
appreciable difference to the loading necessary for a given contact length but has very
little effect otherwise. Slipping at the inner surface gives results very similar to those
with a fixed surface and a zero value of Poisson's ratio.
4
5
Parish[4], in 1955, conducted an experiment to measure the pressure distribution
between two rollers in contact. He used the method of pressure distribution measurement
which consisted of mounting a pressure transmitting pin in a radial hole in one of the
rollers with its outer end flush with the roller surface and its inner end in contact with
some pressure-sensitive device inside the roller. As the outer end of the pin passes
through the nip, the variations in load are transmitted to the pressure-sensitive device
where they are converted to some form which may be extracted from the roller and
displayed to represent the original distribution of pressure. He compared the load per
linear inch on the nip with that found from the integrated area under the pressure
distribution curve and found a good agreement between them.
In 1958, Parish[5] worked out the theoretical relation between the roller load and
cover tangential surface strain in the cover for Hannah's case, modified to plane strain
conditions. He made the changes by replacing u/(1 +u) by u and E by E/(1-u2) in
Hannah's results. He also made an experimental investigation and compared his results
with that obtained from Hannah's solution after modification for plane strain. A
noticeable discrepancy was observed between his experimental pressure distribution and
the theoretical distribution. Parish suggested that the probable causes of this discrepancy
might have arisen from the following assumptions, none of which were satisfied exactly
by the experiment:
1) the nip width and the cover thickness are small as compared with roller diameter,
2) strains are infinitesimal,
3) the material of the cover possesses linear elastic properties
4) the rollers are stationary.
But, the most likely cause of the discrepancy, as Parish pointed out, was the non-linearity
in the elastic properties of the rubber.
In 1961, Parish[6] supplied an empirical method of finding the indentation under
load based on his own experimental measurements. He extended his research to
6
conditions of uneven load distribution along the length of the rollers. Experiments were
made with both stationary and rotating rollers. He found that the static nip widths are
larger than dynamic ones and the difference generally lies between 5 and 10%.
Foreman[7], in 1964, investigated the speed change of a strip being processed by
pinch rolls and bridles. He showed that the strip velocity through a rubber covered pinch
roll could be calculated using
2 n RNa
Vw p
where alp will have a value of 1.0 to 1.1, depending on the characteristics of the rubber
covering and the roll pressure. This greater velocity is not due to slipping, but to the
indentation of the rubber covering. The covering must speed up if indented. This is
analagous to an incompressible fluid encountering a constriction in a flow channel. In
the vicinity of the constriction, the velocity of the fluid must increase. He tested the
accuracy of his theoretical reasoning by conducting experiments and concluded:
1) increasing the radius of the roll will decrease the value of the factor alP due to a
wider nip and consequently a lower net pressure,
2) unless the covering thickness is made quite thin, there is no benefit from reducing
the thickness,
3) increasing the covering hardness will proportionately decrease the value of alp,and
4) decreasing the roll pressure will decrease the value of alp.
The experimental work of Spengos[8], in 1965, was quite extensive and involved
a wide range of loads, thickness of the rubber layer and speed differences between the
mating rollers. A steel cylinder, driving a rubber-covered cylinder, was instrumented for
a quantitative, as well as qualitative, investigation of the action taking place in the
contact area. Spengos found that the transmission of a tangential force through the
contact alters the radial pressure distribution considerably, particularly when it is
associated with a heavier normal load. This is a consequence of the large tangential
displacement the rubber undergoes in the contact area.
7
Hahn and Levinson[9], in 1974, gave a two dimensional linear elastic analysis for
the quasistatic, frictionless indentation problem with the exact formulation. They
assumed that the rubber-like layer is made of a Hookean material and its deformation is
within the range of applicability of the linear theory. The problem is solved by using an
Airy stress function and the solution is in terms of double infinite series, one of which
converges slowly.
Batra[10], in 1980, approached the problem of a rubber covered roll, indented by
a rigid cylinder, numerically by using the finite element method(FEM). He assumed that
the material of the rubber-like layer is homogeneous and can be modeled as a MooneyRivlin
material, as discussed in section 3.5. He treated his problem as a plane strain large
deformation situation, and because of this Mooney-Rivlin material like characteristic of
the rubber cover, and based on the experimental finding that the maximum strain
commonly encountered in practice is probably much higher than what is usually thought
to be the range of applicability of the linear theory, he considered his problem to be
nonlinear. To solve his finite element model, he assumed the half width and a form of
pressure profile at the contact area. The results obtained were in good agreement with
the experimental results of Spengos. The only noticeable difference between the two
results was explained as being due to the assumption of plane strain made in his work
was not quite valid for Spengos' experimental set up, wherein a three dimensional state of
strain existed.
In 1993, Diel, Stack and Benson[ll] evaluated the axial variations in nip
parameters such as contact pressure, contact area generated by deformation of the drums
and the elastomeric covering for two general cases; an identical-hollow-drum design and
a classic calendering design. Both cases included the effects of elastomeric coverings.
8
Comparison they made were between modeling the resolution axial variations in nip
parameters by beam effects and shell effects for the identical-hollow-drum design. They
found that the primary cause of axial variations in nip parameters was the axial variation
in the drum's deflection. The shell models predicted much greater axial variation than
the beam models due to localized shell effects. Also, at the end of the drum, the
elastomer was not constrained and tended to behave softer. Therefore, near the free edge
of the nip, edge effects became dominant in the thin elastomeric covering on the drums.
Moreover, for the classic calendering problem, only the free edge effect of the thick
e1astomeric covering caused axial variations in nip parameters. They, also, found that
decreasing the radius of the noncovered rigid roller increases both measures of speed
ratio, i. e., ratio of speed of the undeformed covered roll and that of the deformed covered
roll, and their axial variations.
Thus, to date, no nip roll in which an elastomeric covered roller contacts a rigid
roller has been successfully analyzed without assumptions of contact width pressure
distribution. There is no reference whatsoever regarding the solutions of contact
problems between nip rollers and wound rolls.
CHAPTER III
CONTACT OF ELASTMOMERIC COVERED ROLLERS WITH RIGID ROLLERS
If the indentation of the rubber covered roller in the nip is appreciably greater
than the change in the thickness of the processed material, the presence of this material
will have very little effect on the roller deformation. This allows webs transported
through nip rollers to be treated as two cylindrical rollers in contact. As described in
chapter 2, this type of problem is difficult to solve analytically since the pressure profile
at the contact area and the contact width are unknown. And, due to instrumentation
difficulties, it is not easy to obtain experimental results. Hertz's theory is applied well in
linear elastic, small strain contact problems. Unfortunately, many web handling devices
do not exhibit linear-elastic material behavior and often involve nonlinear deformation.
So, numerical analysis can be considered as an appropriate method to solve this kind of
problem. The finite element method, whose attributes are accuracy, flexibility and
adaptability, is one of the most powerful tools. A FEM model of a rubber-covered roller
has been developed using the finite element code MARC[12]. A new instrumentation,
called Tekscan, is also used in experimental investigation of the contact pressure and
contact area[ 13].
3.1 Experimental Setup
A schematic diagram of the system to be studied is shown in Fig. 3.1.
9
The main components of the experimental setup include a rigid nip roller, a rubber
covered roller and two Bellofram air cylinders, each with a pressure gage.
Air Supply
Pressure Gage t I
,------------{I ~-®_I I
Pressure Gage
Bellofram ReHor",," n
Nip Roller
(.------------1
Rubber-covered Roller
Rubber Steel
Fig.3.1 A Schematic Diagram of the System in Experiment
10
11
A load cell, connected with a digital strain indicator is used to calibrate the
Belloframs. The result is seen in Fig. 3.2. We can see that the nip load varies linearly
with the air pressure supplied to the Belloframs. From the curve fit equations, we can
infer the nip load is based upon the supplied air pressure. The total nip load will include
two Bellofram loads and the nip weight.
120
100
80
:a
:a- 60
ftJ
0
..J
40
20
0
0
Fig.3.2
right curve
y = 3.9757x - 5.8471
R2 = 0.9999
left curve
y = 4.1997x -15.722
R2 = 1
10 20 30
Air pressure(psi)
Nip Loads vs. Air Pressure Applied
3.2 Tekscan Measurement
--.-Ieft cylider
···iIII···· right cylinder
--Linear (left cylider)
--Linear (right cylinder)
The Tekscan sensor is a new kind of Force Sensing Resistor(FSR). It consists of
an array of hundreds of force sensitive resistors and is only 4 thousandths of an inch
thick. The sensor is connected with a computer through interface and software. An FSR
is one of the devices commonly used in measuring pressure. It changes resistance when
the force is applied on its surface. A good description of a FSR is given in
reference[14]. It is very convenient to use but has slow dynamic response, which may
influence the result. The Tekscan array sensor has the same dynamic response, so it is
very important to use this sensor properly. After running several experiments, the
12
following recommendations can be made for using the Tekscan system. Sensor
calibration should be performed prior to each measurement. When this research began,
Tekscan was using a one point calibration routine. On a log-log plot, the resistance of a
FSR is linear with pressure. The early Tekscan calibration software assumed that the
slope of this line was constant and the line was located vertically on the log-log scale by
a one-point calibration (i.e. the sensor was submitted to a known pressure and the data
acquisition system read the change in voltage across the FSR sensors from which the
resistance due to the calibration pressure could be inferred). The slope of the line is
dependent on the quality control of the pressure sensitive inks used in the FSR. The
overall resolution of the Tekscan sensors was 5% of the full scale pressure with one-point
calibration. After pressure from 3M company and OSU, Tekscan begun using a two
point calibration routine which allowed the slope to be determined for each sensor.
Make sure that the two points in pressure used in the two point calibration are within the
load range of the experiment. While doing calibrations, leave the dead weight for a short
certain time, like 20 seconds, on the surface of the sensor. Then, record the data after the
same certain time while taking measurements.
Another important point is surrounding the sensor with the same material as that
used in the experiment during calibration. In this experiment, the sensor will be inserted
into the contact area between the nip and rubber-covered rollers. The rubber is soft
material and experiences large deformations. If the sensor is calibrated on a flat and hard
surface, it will have an error of one-third percent of integrated load in measurement.
This becomes another contact problem. In these experiments, the Tekscan pressure
sensors were calibrated by inserting them into a flat stack of material, which was of the
same thickness and material properties as those materials to be tested, and subjected to
known pressures.
13
3.3 Hertz's Theory
The first analysis of contact stress of two elastic bodies was made by Hertz[l].
His linear theory is satisfactory. He made the following assumptions:
1) the surfaces are continuous and non-conforming,
2) the strain is small. The dimensions of the contact area are small compared with
the dimensions of the bodies and with the relative radius of curvature,
3) the contact bodies are considered homogenous and isotropic bodies,
4) the surfaces of contact bodies are frictionlesss,
5) the problem is a plane strain problem.
When two cylindrical rollers, (radius R 1 and R2), with their axes both lying
parallel, one pressed in contact by a force P per unit length, they make contact over a
long strip of width 2a lying parallel to the cylinders' axes (see Fig. 3.3). Then, based on
the above assumptions, through mathematical derivation, we have the following forms.
p
x
Fig. 3.3 Contact Rollers
14
Hertz assumed that the pressure distribution along x direction is:
p(x ) -_ -2 -P a(2 -x 2 )112 -_ 2- P -(1 -x- 2 )1/2
1ta 2 1ta a 2
(3.1)
where x is the present distance measured from the center line of the nip(-asxsa) and
p(x) is the pressure at point x.
The half of the contact width is :
a ( 4 PR ) 1 /2
1t E •
1 1 1
where -=-+-
R 1) R2
and
I-VI 2 1- v 2 2 -1
E* = ( + )
El E2
In our case, the nip is a comparatively rigid body, which has a much higher
modulus of elasticity (30* 106 psi) than the rubbers (700 psi). SO,we can assume
I-v 2
_----"1'-.- ~ 0, when E 1 ~oo.
El
That is:
E*= E2
I-v 2
2
where E2 is the rubber's Young's modulus and V2 is the rubber's Poisson's ratio
Therefore,
a = 4 PR (1 - vi)
1tE2
and the maximum pressure Pm is obtained at x=O, by Hertz was:
Pm
2 P
1t a
(3.2)
(3.3)
(3.4)
In this experiment, a hard rubber covered roller with a durometer of 80 is used to verify
the Hertz theory. The radius of the nip is 2 inches, and the rubber roller has the
following properties:
Young's modulus: E2=707.64 pSI
Poisson's ratio: V2=0.45
Radius: R2=2.72 inches
The comparison of the results from Hertz and Tekscan is quite close and is shown in
Fig.3.4. The maximum error of 14.4% was computed at 4.8 load level.
- 70
'iii
-Q. 60 aE. 50 e ~Hertz
:::J 40 C/) __ exp1
C/) ae. 30 -*-~2
E 20 :::J
E 'x 10 cu
~ 0
0 5 10 15 20
Nip Load p(lbslin)
Fig. 3 .4 Variation of Maximum Pressure with Nip Load as Predicted
by Hertz and via Experiments
15
Because the contact area between the two rollers is small compared with the FSR density
upon the Tekscan sensor, tests in which the contact width was studied as a function of nip
load could not be performed with this material.
16
3.4 Finite Element Analysis
A 2-D plane strain FEM model was composed by using MARC, a powerful finite
element package. The MARC system contains a series of integrated programs that
facilitate analysis of engineering problems in the fields of structural mechanics, heat
transfer, and electromagnetics. The computer code includes algorithms for large strain,
linear or nonlinear material laws, and general contact between flexible and rigid bodies.
In MARC, the contact problem is considered to be a nonlinear problem. There
are three sources of nonlinearity: material, geometric and nonlinear boundary conditions.
Contact leads to nonlinear boundary conditions. A problem is nonlinear if the forcedisplacement
relationship depends on the current state (i.e., current displacement, force,
and stress-strain relations).
P=K(P,u)u
where P is a generalized force vector, K is the stiffness matrix and u is a generalized
displacement vector. Nonlinear analysis is usually more complex and expensive than
linear analysis. Also, a nonlinear problem can never be formulated as a set of linear
equations. In general, the solutions of nonlinear problems always require incremental
solution schemes and sometimes require iteration ( or recycles) within each load/time
increment to ensure that equilibrium is satisfied at the end of each step.
[K] [du] = [dp]
All contact problems require a time step and the number of time steps must be enough to
insure a good result. The procedure is often very slow and tedious.
The Herrmann incompressible plane strain element is used here since Poisson's
ratio of rubber used in the experiment is nearly equal to one-half. The Herrmann
formulation allows the correct treatment of incompressible behavior without the
17
numerical difficulties associated with conventional displacement formulation elements.
In particular, conventional elements should not be used in plane strain, axisymmetric or
3-D continuum analysis if Poisson's ration is close to 0.5. MARC element 80 is used for
this cylinder model. The element is a 5-node isoparametric, quadrilateral plane strain
element. The filth node is the extra pressure node, meaning it only has a pressure degree
of freedom which is not shared with other elements. This extra node doesn't require the
usual spatial coordinates to be defined.
If the pressure distribution is found, then the load pressing the rolls together can
be calculated as :
P = 2f; p(x)dx (3.5)
In practice, the load P is specified. Contact width 2a and pressure p(x) at the contact
surface are unknown and are to be determined as part of the solution. One of the
methods to solve the problem is to assume the half nip width and the pressure
distribution p(x), and find the load pressing the two rollers. One must iterate on the
pressure distribution to arrive at the actual distribution. Another approach is to prescribe
the indentation between the two rollers and to compute the necessary load. In the MARC
program, all contact problems are solved by assuming an indentation.
Theoretically, for a symmetrical plane strain model, 180 degrees in the hoop
direction should be modeled. However, for the contact solutions presented here, the
deformation is localized near the contact area. And, in order to decrease the
computational costs and computer memory space limit, 90 degrees is modeled. The
mesh is finer in the contact area. This provides a more accurate measure of nip width
and other nip parameters. The mesh plot is shown in Fig 3.5.
Rc
x
Fig 3.5 Finite Element Grid
3.5 Comparision of Results
A soft rubber covered roller used in the FEM model and the experiment has the
following properties:
Radius: R2=2.053 inches
Young's modules: E2=240 psi
Poisson's ratio: u2=0.45
18
19
The radius of the steel core (Rc) is 1.47 inches and the radius of the nip roller is 2 inches.
In the FEM analysis, the following assumptions are made for the ease of
computation:
1) the rubber covered material is assumed to be linear and isotropic,
2) the rollers are stationary and the problem is static,
3) the contact surface between the rollers is frictionless.
Fig. 3.6 shows the variation of nip load with contact width. Results from the
finite element solution are compared with those from Hertz's theory and experimental
data from the Tekscan sensor.
0.4
0.35
:2 u 0.3
s:::::
(ij" 0.25
-..s::::: "'C 0.2
~
- 0.15 u .a s::::: 0.1
0 u 0.05
0
0 10 20 30
Nip Load P(lbs/inch)
-+- Hertz theory
... -8 ..... exp 1
--.-exp2
.... ;:{ ..... exp3
~exp4
""'"*- FEM
Fig 3.6 Variation of Contact Width with Nip Load
20
The FEM result is close to the Hertz theory, which is calculated by Eq.3.3.
Measurements from the Tekscan sensor are higher than the Hertzian result. The reason is
that this rubber is soft and has more deformation than the hard one.
The relation between the nip load and the maximum pressure is shown in
Fig. 3.7. The Hertzian calculation is made using Eq.3.4. The three results agree well.
- 50
til
-C. E 40
D-
~
::::J
30 -+- Hertz
til ............... exp
til
~ 20 D- ----.- FEM
E 10 ::::J
E
';( 0 n:I
:i!: 0 10 20 30
Nip Load P(lbs/inch)
Fig. 3.7 Variation of Maximum Pressure with Nip Load
30
25
- 20 VI c..
! 15
::::I
VI
VI 10
~
Q.
5
0
0 0.05 0.1
x (in)
0.15 0.2
-.-FEM
.........a- Tekscan
---.- Hertz
Fig 3.8 Pressure Distribution on Contact Surface -- Nip Load = 6.6 lbs/in
35
30
- 25
VI c.. 20 'E
::::I 15 VI
VI E 10
Il..
5
0
0
'''-'''''-'''''''-'---l
0.05 0.1 0.15 0.2 0.25
x (in)
-+-Tekscan
............... Hertz
----.- FETIII
Fig. 3.9 Pressure Distribution on Contact Surface -- Nip Load = 8.8 lbs/in
21
50 .................................................................................... .
-'Cij
c.. 30
l
; 20
en
~
Il.. 10
O+----+----+----lllHk------4
o 0.1 0.2 0.3 0.4
x (in)
Fig. 3.10 Pressure Distribution on Contact Surface -- Nip Load = 20 lbs/in
Figs. 3.8, 3.9 and 3.10 show the pressure distribution along the contact area. The FEM
and Hertz produce similar results. All methods agree well from x=O.2 to 0.25 in
Fig. 3.10, and their curve shapes are similar in the area close to the end of the contact
22
width in Figs. 3.8 and 3.9. However, in the maximum pressure area, measurement from
Tekscan is different from the two analytical methods. From Eq. 3.5 we can see that in
the plot of pressure profile vs. half contact width, the area beneath each curve should be
about half of the total nip load. Spreadsheet software can be used to do approximate
calculations of area by using the data in Figs 3.8 , 3.9 and 3.10. The areas beneath the
three curves are very close to half nip load, except the value from Tekscan is larger in
these figures (see Table 1).
Half nip load Area beneath Area beneath Area beneath
(lbs/inch) Hertz cmve FEMcmve Tekscan cmve
Fig. 3.8 3.3 3.27 3.36 3.59
Fig. 3.9 4.4 4.38 4.55 4.78
Fig. 3.10 10.0 9.96 10.89 9.96
Table 1 Comparision of Area Beneath Three Curves
3.6 Mooney-Rivlin Materials
As described in Chapter 2, Batra analyzed the indentation of an elastomer layer
covered roller by a rigid body. He assumed the rubberlike material to be homegenous
Mooney-Rivlin material, a kind of elastromeric and incompressible material.
23
Elastomeric materials are elastic in the classical sense. Upon unloading, the stress-strain
curve is retraced and there is no permament deformation. Elastomeric materials are
initially isotropic. An elastomer is a polymer which shows nonlinear elastic stress-strain
behavior. One of the two elastomer modes used in MARC has the following strain
energy function:
where
W is the strain energy function,
ClO, COb Cll , C20, C30 are material constants obtained from experimental data,
and Ib 12 are the first and second invariants of the elastic strain.
Mooney-Rivlin materials have a reduced form of the above strain energy function
For Mooney-Rivlin materials, ClO and COl are the only two material constants. These
values can only be obtained from experimentation. The force and deformation for a
uniaxial test speciman may be related as
where
P is the force of the specimen,
Ao is the original area of the specimen, and
Al is the uniaxial stretch ratio.
This equation and the associated test provide a simple way to determine the MooneyRivlin
constants. The Mooney-Rivlin constitutive equation is applicable if the plot of
1
P / 2 A 0 (1 - -3-)
Al
versus the stretch ratio results in an approximately straight line. See the Fig. 3.11.
p
2Ao(1-1/A~)
Fig. 3.11 Correlation With A Straight Line Using the Mooney-Rivlin
Constitutive Equation
24
If only the Young's modulus E is supplied, and full uniaxial data are not available, then
COl == 0.25 ClO
is a reasonable assumption. The constants then follow from the relation:
6 (ClO+ COl) == E
25
Mooney-Rivlin materials are defmed in the MARC material library. Here a
material model is made to compare the result with Batra's work. The rollers used in this
model correspond to run number 30 of Spengos. That is, R1 =7.62 cm, R2=6.07 cm and
Rc=4.72 cm. The nip load is 92Ibs/inch, and the rubber modulus is 250 psi. The results
are shown in Fig. 3.12. We can see the maximum pressure and contact width from the
FEM analysis is close to Spengos' work. The pressure distribution in the contact area
has light difference with Spengos' measurement since the maximum pressures are
different. Batra's work, in reference [10], has good agreement with Spengos. However,
he only showed the relation between pressure/peak value and x/a, which couldn't show
how the pressure is distributed. This relation is shown in Fig. 3.13. The result from
MARC is quite close to Spengos' and Batra's works. The maximum pressure and
contact width calculated by Hertz's formula are 86 psi and 0.68 inch, which vary from
the values measured by Spengos. Therefore, Hertz's theory does not provide accurate
results for this kind of material. Although the material Spengos used has almost the same
modulus of elasticity as that used in section 3.5, the materials behave differently because
of the different loads. The maximum load used in section 3.5 is only 25 lb/in, but
Spengos used 92 lb/in. So, when the load is small, the soft rubber might still behave
linearly. But, when the load is large, the soft rubber behaves nonlinearly.
140
120
- 100
'iii c.. Q...) 80 :::J 60 VI
VI
.G.J. 40
Il..
20
0
0 0.2
x (in)
0.4 0.6
-+--FEM
............... Spengos
Figure 3.12 Pressure Distribution on Contact Surface
~-~-~~~~~~~
I
~
0.8
I
:::J I VI
VI I 1::'- ~ 0.6 Il.
~ I
IU
CI) 0.4 I-l. ~
:::J 0.2 VI
VI
~
Il. 0
0 0.5 1.5
x/a
Fig. 3.13 Comparision of Result from FEM, Spengos and Batra
26
CHAPTER IV
CONTACT OF A RIGID ROLL WITH A WINDING ROLL
The prediction of stresses in the wound rolls has received much attention during
the past decade. There are a number of wound roll models that predict stress distribution
for center winding, the most rigorous of which are made by Hakiel[ 15], Pfeiffer[ 16], and
Willett & Poesch[ 17]. One feature of these three models is that they allow the radial
modulus, ED of a wound roll to be a function of radial stress. Laboratory experiments
have demonstrated that the radial modulus is indeed a function of pressure. As described
in Chapter 1, the nip roller affects the Wound-On-Tension, and further, affects the
residual stresses in the wound roll. Since the paper roll is considered as orthotropic,
nonlinear material, the classical linear theory and general nonlinear solution are not
suitable for the contact problem of a nip roll with a winding roll. The material properties
have to be obtained from experiments. Hakiel's model is the one most often used in web
handling and is the one used in this study. A 2-D nonlinear FEM model, with Er as a
function of pressure, has been configured in this chapter. An experimental investigation
is made using the Tekscan sensor.
4.1 Experimental Setup
The experiments were performed using the 3M splicer winder. Figure 4.1 shows
the layout of the center winding with the nip roller on this machine. Center winding is
27
28
implemented by driving the roll core using a motor. The nip load is is applied by
attaching dead weights to a pully system which is attached to the pivot bar. The tension
sensor was calibrated by hanging a dead weight at the beginning of each test.
Dead
Weight
Idler Roller
Nip Roller
Tension
Sensor
Idler Roller
Unwind Roll
Fig. 4.1 Layout of Center Winding with Nip Roller on 3M Splicer Winder
4.2 Material Properties of Bond Paper
Bond paper was used in this study. The properties of bond paper, which include
web thickness, Kinetic coefficient of friction between sheets of paper, radial modulus Er
and tangential modulus Ea, etc., were determined.
There are a couple of ways to measure the web thickness. One is to use the
Schaevitz L VDT noncontact caliper measurement gage. As a piece of bond paper moves
from point to point under the air sensor, the meter will show the thickness. The average
value of the thickness at 10 to 20 randomly selected points is used as the bond paper
29
thickness. Another way to determinate web thickness is to use a caliper to apply between
7 and 9 psi on a stack of paper which contains about twenty sheets [ 18]. The thickness of
the stack divided by 20 is the bond paper thickness.
The friction coefficient was measured by a device[19] which consists of a motor,
strain gage, gage indicator, computer data acquisition system, carriage and plate. Web
sheets were stuck to the carriage and plate by tape. A dead weight of 2 lbs was put on
top of the carriage. As the carriage moved, the data acquisition system processed the
data and gave the friction coefficient.
The test in which Er was measured was conducted using the Instron 8502 servo
controlled load frame. In the experiment, a 111 thick stack of 6"x6" paper was used. The
ramp speed was set very slow so that rate effects due to the expulsion of entrained air
were minimized. The data acquisition was done using a GPIB interface and the load and
stack displacement were stored on a data file which can be processed by spreadsheet
software. The load and displacement data were used in a spreadsheet in calculations of
stress and strain. The slope of the stress/strain data was then evaluated at various stresses
which produced a table of Er values at various stresses or equivalently, pressures. Then
the mathematical representation of the radial modulus,Er, was determined using a 3rd
order polynomial curve fit in pressure.
The measurement of Ee was performed using an Instron 4202 loading frame and
a 1 inch wide by 10 inches long strip of bond paper cut along the tangential direction of
the roll. The ends of paper strip were gripped by two clamps and then one end was
pulled. The clamps should be chosen properly and the ends of the strip should be
clamped correctly to ensure the strip is subjected only to uniform tensile stress.
Otherwise, the measured tangential modulus would be less than the real value. The data
acquisition for Et was done using the Labtech notebook, a commercial data acquisition
software package, which was used to collect the raw data for the Er calculation as well.
From the tests described above, the parameters of bond paper are:
Thickness: h = 2.6 mil
Kinetic coefficient of friction between sheets of paper: j...L= 0.255
Radial modulus: Er=50.38*p-0.321 *p2+0.001 *p3 pSI
Tangential modulus: Eo=51 0,910 pSI
4.3 Pull Tab Calibration
30
A pull tab, which consists of a brass shim stock and stainless steel feeler gauge, is
an effective tool to measure the wound roll radial stress. Calibration of the pull tabs is
very important to ensure accurate results. The test used a stack of bond paper 6I x6",
with the same thickness as the paper to be wound. The core IS radius is 1.7 inches and
the outside radius of the paper roll is 5.1 inches. Therefore, the thickness of the bond
paper stack is 3.4 inches. The stack, with a pull tab inserted, is put on an Instron 8502
and is compressed at various loads to produce various calibration pressures.
A force gage is used to pull the tab and measure the values corresponding to
different pressures. Figure 4.2 shows the calibration curve of one of the 10 pull tabs used
in these experiments.
-+- Stack Test
-Curve-fit
Fig. 4.2 Pull Tab Calibration
The calibration was done according to Cai's [19] recommendation. He suggests
that only one pull tab be inserted in a stack in each calibration test. The tab's radial
position in the paper stack will the same as the position of the pull tab in the winding
roll. The tab should be placed at the bottom position in the web stack, when the tab is
going to measure the wound roll stress close to the core.
4.4 Hakiel's Model
In 1987, Hakiel[15] developed a model to predict the wound roll stress based
upon the theory of elasticity. In his model, he made several assumptions:
1). the winding roll is treated as a geometrically perfect cylinder,
2). the roll is an orthotropic cylinder which has a linear elastic modulus in the
circumferential direction and a nonlinear elastic modulus in the radial direction,
31
32
3). the stresses in the wound roll are assumed to be functions of the radius, but
not of the axial or circumferential position,
4). the plane stress condition is assumed.
By solving a set of algebraic equations, which resulted from the recursive
application of a second order ordinary differential equation with non constant
coefficients and two boundary conditions, the distribution of radial pressure and
circumferential stress can be obtained. In order to solve a set of algebraic equations, the
finite difference method and the Gaussian elimination method are applied in this model.
Using the finite difference method, the derivative terms in the ordinary differential
equation are simplified by central difference approximations. The program, called
Winder, which was developed at the Web Handling Research Center (WHRC) and
contains Hakiel's model, is used to compare the pressure from Hakiel's model with the
results from the pull tab experiment. The comparison is shown in Figure 4.3. The
results match well. This stress status is used as the initial status of the FEM model.
70
60
50
·iii CI. 40
f
::::I 1/1 30
1/1
I!!
a... 20
10
0
1.5 2
R1Rc
2.5 3
-+- Haikel's
•••• !SIl .••• Pull tabs
Fig. 4.3 Interlayer Pressures at Center Winding with 500 psi Tension and 4pli Nip Load
-Or = cosS, ax
-Or = s.I nS, ay
as = -sinS as = cosS, ax ' ay
oz' -=0 ox '
oz' -ay= 0,
or = °
OZ '
oS = °
OZ '
OZ'
-=1.
OZ
where r is the radial direction and S is the tangential direction.
Therefore, the transformation matrix is:
I cosS
l-s~ns
°°11
cosS
sinS
°
(3) Subroutine Anelas allows the user to define the anisotropic elastic law. In
plane stress, the linear orthotropic constitutive equations in local coordinates are:
-Uer / Ee
1/ Ee
°
The inverse form is the elastic law used in MARC, which is:
where the aij, commonly termed the reduced stiffness coefficients, are given by:
a E = S
22 1- utBU Sr
34
where the shear modulus can be obtained from reference [21],
From strain energy constraints:
In Hakiel's model, Poisson's ratio is defined as U = uer.
The material properties ES, Er have already been shown in Section 4.2 and u=O.Ol.
As mentioned above, MARC only uses global constants, which have the form:
35
where the bij are related to the aij by the following equations[22] if c denotes cosS and s
denotes sinS:
bll c4 2c2s2 s4 4c2s2
bl2 c2s2 c4 +s4 c2s2 _4c2s2 all
b22 s4 2c2s2 c4 4c2s2 = * al2 b16 c3s -cs(c2 - s2) -cs3 -2cs(c2 - s2) a22
b26 cs3 cs(c2 - s2) -c3s 2cs(c2 - s2) a66
b66 c2s2 -2c2s2 c2s2 (c2 _ s2)2
Since the subroutine Orient has defined the transformation matrix between two
coordinates, MARC will do the material properties transformation automatically.
(4) The value of cosS and sinS could be obtained either from the mesh
dimensions or from subroutine Intcrd, which outputs coordinates of every element at
each increment. Then, we have:
x
co~ = ----;====
~x2+1
sinS= I Y
vx2+1
36
In order to see how the nip roller influences the outer regions of the paper roller,
the elements of outerlayers of the roller surface in this model have the dimension as
small as the web thickness which is 0.0026 inch. However, this value is so small that it
can cause numerical difficulties for the program. Also, it will generate a mesh plot with
a larger number of elements than is allowed by the computer memory. In order to
overcome these disadvantages, a 45 degree model using only the outer 0.5 inch thickness
of the paper roller was considered in this study, since the radial modulus of elasticity on
the outside of this layer is 40 times less than that on the inside of the layer.
The results from the FEM analysis is compared with the Tekscan measurements
and is shown in Fig. 4.4. Because the nip roller used in the experiment is misaligned, the
pressures on one side of the center line of nip length is higher than the pressures on the
other side, during the pressure measurement using Tekscan sensors. The pressures on
both sides, which have the same distance to the center line of nip length, were compared
in this figure. We can see that the result from the finite element solution is close to the
average value of experimental data from the Tekscan sensors.
50
45
40
- 35
fI'I C- 30 a..r. 25 ;:,
fI'I 20 fI'I
QI ... 15
D..
10
5
0
Fig. 4.4
-r--~""-~-"""""-""""""~"""""'-
0 0,02 0.04
X(in)
0.06 0.08
-+-FEM
---.- Tekscan(left)
~ Tekscan(right)
-.-Avg.
Pressure Profile Along Half Contact Width with 4 pli Nip Load
37
CHAPTER V
CONCLUSIONS AND FUTURE WORK
5. 1 Conclusions
A nonlinear contact problem of two cylinders is solved by using the finite
element package MARC and the Tekscan sensor in this study. The analyses were done in
two contact cases: a rubber-covered roller with a rigid roller and a winding paper roller
with a rigid roller. From this study we can conclude:
1. Since hard rubber has a small deformation and satisfies Hertz's assumptions,
his theory is applied well to the analysis of the contact of a hard rubber roller with a rigid
roller.
2. For the soft rubber roller, when the load is small and the material behavior is
linear, Hertz's theory still can be used. The only difference is that the real contact area is
a little larger than Hertz assumed since soft rubber has more deformation. The maximum
pressure produced by the nip loads in the experiments agrees with those based on Hertz's
theory.
The results from finite element analysis, which considers the material as linear
and isotropic, have good agreement with Hertz's theory, although the problem is only
solved as a nonlinear boundary condition problem in the MARC. The pressure
distributions away from center line are similar in the experiment, FEM analysis and
Hertz's theory, but not in the area close to the center line.
38
39
3. If the rubber behaves nonlinearly under high load, namely, has a large strain
like Mooney-Rivlin materials, FEM is adequate to analyze the problem. Hertz's theory
can't be used in this case since his assumption that the material behaves linearly and has a
small strain doesn't exist.
4. The pressure distribution at the contact area between a wound paper roller and
a rigid roller, as determinded by the FEM analysis, is close to the average value of the
experimental investigation. This gives a good and effective way to analyze the contact
problem with radial modulus as a function of pressure.
5.2 Future Work
Areas of future study should include:
(1) The deformation of the core and the indenting roller need to be considered for
very thin rubber covers. The effect of the rubber layer thickness on the roller is not
involved in this research. It may influence the relation between loading and deformation,
as described in Chapter 2.
(2) Since axial variations in contact pressure and deformation can cause web
wrinkling, and uneven material transfer, axial variations with radial modulus as a
function of pressure need to be studied.
(3)The effect of friction at the contact surface between the roller, roller rotation
and its dynamic effects should be studied.
REFERENCES
1. Hertz, H. "Uber die Beruhrung fester elastischer Korper," Journal fur die reine und
AngewandateMathematik, Vol. 92, 1881,pp.156-171.
2. Good, J. K. and Wu, Z. "The Internal Stresses in Wound Rolls with the Presence of a
Nip Roller," Tappi Journal, 1992
3. Hannah, Margaret, "Contact Stress and Deformation in a Thin Elastic Layer,
Quart. Journal of Mechanics ad Applied Math, Vol.4,Pt.l, 1951,pp.94-105.
4. Parish, G. J., "Measurement of the Pressure Distribution Between Roller in Contact,
British Journal of Applied Physics, Vo1.6, July 1955,pp.256-261
5. Parish, G. J., "Measurement of the Pressure Distribution Between Metal and Rubber
Cover Roller," British Journal of Applied Physics,Vol.6,July 1955,pp.256-261
6. Parish, G. J., "calculation of the Behavior of Rubber-Covered pressure Rollers,"
British Journal of Applied Physics,Vol.I2,July 1961,pp.333-336
7. Foreman, A. R, "Application of Rubber Covered Rolls to Pinch Rolls and Bridles,"
Iron and Steel Engineer Year Book, 1964,pp.646-656.
8. Spengos, A. c., "Experimental Investigation of Rolling Contactll , Journal of Applied
Mechanics,Voi. 32,December 1965,pp.859-865.
9. Hahn, H. T. and Levinson, M., "Indentation of An Elastic Layer(s) Bonded to a Rigid
Cylinder- I. Quasistatic Case Without Friction", International Journal of Mechanical
Science,Voi. 16,pp489-502, 1974.
10. Batra, R c., "Rubber Covered Rolls-The nonlinear Elastic Problem",
Journal of Applied Mechanics,Vol. 47,December 1965,pp.82-86.
11. Diehl T., Stack K.D.,and Benson RC., IIA Study of 3-Dimensional on-Linear Nip
Mechanics", Second International Conference on Web Handling,June,1993
12. MARC Manul ,Version 6.1, MARC Analysis Research Corporation, 1994.
40
13. Tekscan Operating Manul, Version 1.3, Tekscan,Inc.,1991.
14. Good, 1. K. and Fikes M.W.R., "Predicting Internal Stresses in Centerwound Rolls
with an Undriven Nip Roller," Tappi Journal,Vol,74,no.6,p101, 1991.
15. Hakiel, Z., "Nonlinear Model for Wound Roll Stress," Tappi Journal, Vol.70,no.5,
pp113-117,May 1987.
16. Pfeiffer, J. D., "An Update of Pfeiffer's Roll-Winding Model," Tappi Journal, Vol.
70, no.l0, 1987.
41
17. Willett, M.S. and Poesch, W.L., "Determine the Stress Distributions in Wound Rolls
of Magnetic Tape using a Nolinear Finite Difference Approach," Journal of Applied
Mechanics, Vo1.55, pp. 365-371,1988.
18. Ireland, G.H., "Paperboard on the Multi-vat Cylinder Machine," Chemical
Publishing Company,lnc., 1968.
19. Ducotey, K.S., "Traction Between Webs and Rollers in Web Handling
Applications," PhD. thesis, Oklahoma State University, May, 1993.
20. Cai Ning, "The effect of Nip Roll Compliancy Upon Center and Surface Winding",
MS. Thesis, Oklahoma State University, December, 1992.
21. Baum G.A., Brennan D.C. and Habeger C.C., "Orthotropic Elastic Constants of
Paper", Tappi, August, Vol. 64, No.8, 1981.
22. Lekhnitskii, S.G., Anisotropic Plates, New York, McGraw-Hill,1968.
VITA
Chunbao Xu
Candidate for the Degree of
Master of Science
Thesis: NIP ROLLER INDUCED CONTACT STRESSES
Major Field: Mechanical Engineering
Biographical:
Personal Data: Born in Fujian, P.R.China, April 17, 1968, the son of Shouren Xu
and Fengchuan Zhu.
Education: Graduated from NO.3 Fuzhou Middle School, Fuzhou, Fijian,
P.R.China, in July 1984; received Bachelor of Science Degree from
Beijing University, Beijing, P.R.China, in July, 1988. Completed the
requirements for the Master of Science degree with a major in Mechanical
Engineering at Oklahoma State University in May, 1995.
Experience: Assistance administration in Natural Science Division of Beijing
University; research assistant, Web Handling Research Center, School of
Mechanical and Aerospace Engineering, Oklahoma State University,
June, 1993, to December, 1994.