LATERAL DYNAMICS AND CONTROL OF A WEB IN A
HORIZONTAL LOOPER
By
GIM KHUANNG
Bachelor of Science
Oklahoma Slate University
Stillwater, Oklahoma
1999
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, 2003
LATERAL DYNAMICS AND CONTROL OF A WEB IN A
HORIZONTAL LOOPER
Thesis Approved:
Thesis AdvIser
~/9-~~
_~_7~a·~
-----rt;tof the Graduate College
II
ACKNOWLEDGMENTS
I wish to express my sincerest appreciation to my major advisor, Dr. Prabhakar R.
Pagilla for his intelligent supervision. constructive guidance, inspiration, and friendship
throughout the development of this investiga£ion. Wi thout hi s he lp and supervision, I would
not have completed this work.
I would like to extend my warmest thanks to my committee members: Dr. John J.
Shelton and Dr. Eduardo A. Misawa for their support and suggestions in completion of this
research. Their guidance and understanding made the development of this thesis a positive
learning experience.
I would also like (0 thank my colleagues at Oklahoma State University, Ramamunhy
Venkata Dwivedula and Yongliang Zhu. I am grateful for the technical support provided
by Ronald E. Markum.
III
TABLE OF CONTENTS
Chapter
1 INTRODUCTION
1.1 Background.
1.2 Literature Review
1.3 Thesis Contributions
1.4 Thesis Outline. . . .
2 EXPERIMENTAL PLATFORM
2.1 Software.
2.2 Hardware
3 SYSTEM IDENTIFICATION OF FREE SPAN DYNAMICS
3.1 Lateral Dynamics of a Moving Web .
3.2 Identification of the lateral dynamics.
3.2.1 Estimation of impulse responc;;e of G(q)
Page
1
2
3
3
4
4
6
9
9
10
II
3.2.2 Structure of G(q). H(q) and Autoregressive moving average (ARMA)
model fOT {U(l)} ........ 15
3.2.3 Maximum Likelihood algorithm 19
3.2.4 Diagnostic testing 21
3.2.5 Experimental result. 22
3.3 Estimated and theoretical model comparison . 24
4 LATERAL WEB GUIDING WITH VARIABLE ENTERING SPAN LENGTH 29
4.1 Open-loop analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 32
IV
4.2 Catenary Analysis of Web .
4.2.1 Catenary web analysis with respect to horilOntallenglh .
4.2.2 Catenary web analysis with respect (0 the curve .
4.3 Steel Web Process Line with moving carriage .
4.4 Control of variable entering span length guiding system .
5 SUMMARY AND FUTURE RESEARCH
5.1 Summary ...
5.2 Future Research
BIBLIOGRAPHY
A LATERAL DYNAMICS
A.I Assumptions .
A.2 Static Behavior
A.3 Dynamics of a Real Moving Web.
A.3.1 Fixed Roller .
A.3.2 Remotely Pivoted Steering Guide
A.3.3 Displacement Guide .
v
33
3S
38
42
45
S3
53
53
S5
57
57
58
60
61
61
62
Figure
LIST OF FIGURES
Page
2.1 OSU Web Line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 6
2.2 Experiment Setup for identification of the lateral dynamics of a web be-
3.1
tween two rollers. . . . .
A sketch of a web spans.
7
9
3.2 The control to OPG, U(l) and y(t) measured from Sensor I and Sensor 2
respectively ................ 14
3.3 u(t) and y(t) used in identification design 14
3.4 Estimated impulse response for G(q) 15
3.5 Impulse response for W(q) 23
3.6 a(k) and its autocorrelation. 2~
3.7 Prediction error £(r) = y(t) - Wt18) used estimated model and its autocor-relation
.
3.8 v(k): Estimated disturbance and its autocorrelation
3.9 gk: estimated impulse response for G(q) .
3. J0 Experimental input, output and theoretical model output
3.t] Experimental input, output and identified model output .
4.1 A schematic of a continuous metal processing web line is shown
4.2 Response at remotely pivoted steering guide .
4.3 Sketches of a remotely pivoted guide. The remotely pivoted guide consists
of a single roller. .
4.4 Impulse Response of a Steering Guide
VI
25
25
26
27
28
29
30
31
32
4.5 Frequency Response of a Steering Guide.
4.6 Free Body of Catenary Web. . . . . . . .
4.7 Catenary Web with the length, I = a+b, and h - 11 = h
4.8 Catenary web with both rollers at the same level
4.9 Catenary web analysis with respect to the curve
4.10 Catenary web Analysis . . . . . . . . . . . . .
4.11 A sketch of one section of the accumulator in a metal process line
4.12 Swinging supporting rollers.
4.13 Web sag versus span length .
4.14 Wrapped angle versus length
4.15 Response at remotely pivoted steering guide
4.16 Response at fixed roller .
4.17 Simplified diagram of variable entering span.
4.18 Web span length and carriage length . . . . .
4.19 Steering guide response due to step disturbance
4.20 Steering guide response due to sinusoidal disturbance
4.21 Steering guide response due to pulse disturbance
4.22 Steering guide response due to pulse disturbance
Al Normal section of web . . . . . .
A2 Force balance of a section of web.
A.3 Response at fixed roller .
A4 Response at remotely pivoted steering gu ide
A5 Offset Pivoted Guide . . . . . . . . . . ..
VII
33
34
35
37
38
39
42
43
44
44
45
46
47
49
50
51
51
52
58
59
61
62
62
LIST OF TABLES
Table
2.1 NO and D/A channels .
2.2 Experimental conditions.
3.1 Estimated impulse response of G(q), g(k) using Equation (3.11)
3.2 Experimental GPAC for G(q)
3.3 Ex.perimental GPAC for H(q)
3.4 Experimental GPAC for W(q) .
3.5 Experimental parameters
4.] Operating Conditions .
4.2 Simulation parameters
VIII
Page
6
8
13
17
18
18
27
32
43
IX
NOMENCLATURE
A cross-sectional area of web
c width of steering guide
E modulus of elasticity of web
G modulus of elasticity in shear
I moment of inert ia of web (~)
L length of web span
Ll distance of the center of rotation of guide roll
M bending moment in web
n stress-averaging factor
N shear force normal to the web curvature
K a constant for a given web (= JT1£1)
Ki intrgra] gain
KI' proportional gain
s laplace operator
T web tension
web thickness
u, V actuator input
It web velocity
W web width
x, X distance parallel to the original centerl ioe of web
Xl distance from guide to its instant center
y, Y lateral web deflection from original position
Z, Z latera] posit Ion of roller
80 roller angle
e angular position of the motor
't transport lag (Uv)
XI
CHAPTER 1
INTRODUCTION
1.1 Background
A web is any material in continuous flexible strip form which is very long compared
to its width and very wide compared to its thickness. Examples of web include all fOnTIS
of paper, fabric, plastic wrap, adhesive tape, photographic film, and strip metals. Handling
of a web during processing directly affects the quality of the finished web. Lateral control,
also caHed web guiding, involves controlling the web fluctuations in the directions perpendicular
to the travel of the web. Control of web guides to maintain the lateral position of
the web on a roller prior to coating, printing, winding and other processes is essential in the
web processing industry.
Web handling involves unwinding the material, feeding it to a processing plant and
then winding it back onto a roll. Generally, the web must follow a predetermined path
which is in the longitudinal direction with a constant velocity. Deviation from this may
result in inferior product quality. With the need for increased performance and productivity
in the web processing industry, accurate modeling and effective controller design for web
handling systems is essential. Accurate lateral control has always been a key element of
web handling systems. An important objective of the lateral control system is to maintain
web position within the desired limits under a wide range of dynamic conditions such as
speed changes, variations in roll sizes, and web properties. Web latera] position variations
affect printing quality and tend to cause uneven wound rolls.
Lateral control of a web involves the use of web guides at different points in the process
line. The web guides should be located as close as possible immediately preceding the
point in the process where accurate lateral positioning is required. it should be designed
and installed for correction of the lateral deviation. Once the web leaves the web guiding
system, it is free to wander until it reaches the next guiding system. Web properties,
machine and process disturbance may influence web guiding.
In web handling processes. continuous web processing lines are used for maintaining
the rate of prOduction. These web processing lines use an accumulator to store and discharge
web material for unwind and rewind roll changes without disrupting the process
speed. Within a horizonlal accumulator, carriages travel horizontally while accumulating
and discharging web material, resulting in changes of web span length.
As the web span length changes, the distance between the guide roller and the entering
roller is also changing. Whenever this distance becomes large compared to the length of
the contact area between the web and any of the two rollers, web sagging is likely to occur.
In order to reduce the impact caused by web sagging. support rollers are incorporated to
hold the weight of the web. The engagement of these support rollers are triggered when
the web span exceeds a predefined length. However, there is also concern about the usc of
these support rollers. There may be forces thaL cause disturbances to the movi ng web when
the support rollers come into contact with the web.
1.2 Literature Review
This section will first review the statics and the dynamics of a moving web. Modeling of
web lateral dynamics is an important first step in an attempt La improve web lateral control
and end-roll product quality. The model must provide an accurate description of the system
and facilitate the implementation of online parameter and state estimation. Mathematical
modeling of lateral web dynamics was introduced by Campbellll]. Campbell's model was
based on the assumption that the web behaves like a string. This basic assumption led to a
first order model which was not very accurate.
The mOSl signi ficant work on lateral dynamics was done by Dr. J.J. Shelton. In [2], he
2
developed an improved model based on the assumption that the web between two rollers
behaves like a beam. Shelton's model describes accurately the response of the web at a
roller due to an input from a pure displacement guide. A comparison of the first order
model for certain frequ~ncies and operating conditions for lateral dynamics is presented in
[3]. In [4], clear insights into the fundamentals of lateral control of a web is given. In [5J,
state estimation is used to predict the lateral web position on a downstream sensor with the
use of a feed-forward sensor to achieve improved control.
1.3 Tbesis Contributions
The contribution of this thesis can be divided into two parts.
1) The lateral roller dynamics of a web between two rollers is experimentally identified
using the system identification techo iques available in 1Iterature.
2) Investigation was carried out on sagging web behavior between two rollers with variable
span length. Different modeling was considered for a web guide with variable
entering span length. A horizontal looper in a continuous metal processing line is
used in the study. Disturbances due to carriage motion and action of support rollers
were taken into consideration for si mu lation purposes.
1.4 Thesis Outline
The rest of the report is organized as follows. Chapter 2 describes the experimental platform
used to identify the lateral dynamics of a web span. System identification for the web span
dynamics is discussed in chapter 3. In chapter 4, catenary analysis of the web between
two rollers is accomplished and the response of a steering guide with variable entering
span length is investigated. Summary of this report and future work are given in chapter 5.
Dynamic equations of the fixed roller, remotely pivoted guide, and displacement guide are
given in Appendix A.
3
CHAPTER 2
EXPER~ENTALPLATFORM
This chapter gives a detailed description of the experimental platform, on which the identification
data are collected. The experimental platform is divided into two parts: the hardware
and the software. The hardware consists of a closed-loop web line, a signal processor,
drivers for actuators, and a computer for implementing control algorithms in real-time.
The software consists of an open architecture real time program written in C programming
language. In the following sections. details of the hardware and the software of the
experimental platform are presented.
2.1 Software
The computer system consists of a Pentium 1 GHz computer with a digital data acquisition
board. The description of the data acquisition board are as follows:
• DS I l03 PPC controller is a complete real-time system based on Motorola PowerPC
640e processor. It can read from 20 channels through NO (these are numbered
ADCHl - ADCH20 on the board) and can send output to 8 D/A channels (these are
numbered DACH I - DACH8 on the board).
• Base dSPACE software includes:
- Control Desk which features a graphic user interface (GUI) for managing
dSPACE boards. A source code editor is also included. The Microtec PowerPC
C Compiler Version 1.8m is required to run C programs.
- The dSPACE Real-Time Library.
- C Programming Interface.
• C Program File: The C Program file consists of three main modules:
- A module to initialize the dSPACE board.
- Functions for executing various tasks (these are invoked in the interrupt service
routine).
- Interrupt Routine which calls the following functions as needed after each sampling
period.
* NO conversion.
* Digital I/O.
* The Control Algorithm.
* DIA conversion.
• Layout File: This file serves as a GUI to display data on the host computer. This file
uses Virtual Instruments that can be dragged from a panel on the control desk. Thus,
this file serves as a communication device between the dSPACE Board and the host
computer.
• Trace file: This file contains definitions describing the variables used in the C program
so that these can be accessed in the Layout file. Thus, a major part of this file
contains a list of the variables used in the C program file. Thus, this file stands as a
link between the C program file and the Layout File.
Table 2.1 indicates the channel assignment for the experimental platform.
5
AOCHI Kamberoller Sensor
AOCH2 Kamberoller Tachometer
AOCH3 OPG Sensor 2
ADCH4 OPG Sensor 1
ADCH5 Web Velocity
DACHI Kamberoller Motor
DACH3 OPG Motor
I Channels I Description
Table 2.1: AID and 01A channels
2.2 Hardware
Kambemliler GUloe
r~------
Edge Senscr' ' : A):---=:-==-_
,
- -,
t
,•
... _ .. __ 1
Figure 2.1: OSU Web Line.
An open-architecture experimental platform is developed for conducting free span identification
experiments. The platfonn consists of an endless web line shown in Figure 2.1.
The term endless web line refers to a web line without unwind and rewind rolls. This type
of platform mimics most of the features of the process section of a web processing line.
There are four main elements in this closed-loop web line. A remotely pivoted guide sys-
6
tern, an offset pivoled guide, a passive dancer mechanism, and an active dancer mechanism.
This endless web line consists of 21 rollers. A master speed roller. shown as a large
diameter roller, is used to transport the web in the web line. Since the roller width is 8
inches, the maximum web width that can be used in !.he web line is 6 inches. The diameter
of each roller is 5 inches, except for the master speed roller. which has a diameter of 10
inches and the offset pivoted guide rollers, which are 2 inches. A nip roller is used to reduce
slip during start-up. An analog controller for the master speed roller is available to obtain
the desired transport velocity of the web.
The guiding mechanism used for the experiments is an offset pivoted guide. This guide
mechanism consists of two guide rollers on a base which is actuated by a DC motor. An
edge sensor located immediately downstream of the guide roller measures the web lateral
position. A simplified schematic of the experimental platform is shown in Figure 2.2.
OPG
~........~-t--' Sensor 1
Roller A
Figure 2.2: Experiment Setup for identification of the lateral dynamics of a web between
two rollers.
Experiments were conducted on the experimental platform shown in Figure 2.2. Using
the data collected from the ex.perimental platform. the lateral dynamics of the moving web
7
Web velocity 310ft/min
Web tension JOlb
Web material Magnetic disk film
Web thickness 0.00325 in
Web width 5.4 in
Sampling period I milli-seconds
Table 2.2: Experimental conditions.
between Roller A and Roller B was identified. An offset pivoted guide (OPG) is used
to generate the desired lateral displacement where the web enters Roller A. Sensor I was
placed close to the Roller A, and sensor 2 was located close to the Roller B. As both
sensors are optical sensors, their outputs are sensitive to the material properties of the web.
However, the sensor output is proportional to lhe lateral displacement for a particular web.
In this experiment. an opaque magnetic film is used in the experiment. Table 2.2 shows the
experimental conditions used for the lateral identification experiments.
8
CHAPTER 3
SYSTEM IDENTIFICATION OF FREE SPAN DYNAMICS
The aim of this chapter is to identify the lateral dynamics of a moving web through exper-iments
and compare it with the theoretical model from the literature. Theoretical lateral
dynamics of a moving web can be developed by applying beam theory. Assumptions, such
as that the web is massless and no slip occurs between the web and the roller on which
the web travels, and boundary conditions are taken to derive the lateral dynamics. These
assumptions and boundary conditions may vary in real life.
3.1 Lateral Dynamics of a Moving Web
Lateral dynamics of a moving web is derived through beam theory, where boundary con-ditions
are assumed. For example, [4] gave the dynamics of the web between two fixed
rollers shown in Figure 3.1.
I-ly
L
v
~
x
Figure 3.1: A sketch of a web spans
The dynamics given in [4] is
where
(KL)2[cosh(KL) - J1 f(KL)- .
1 - KLsinh(KL) - 2cosh(KL) -+- 2'
f (KL) = KL[KLcosh(KL) - sinh(KL)]
2 KLsinh(KL) - 2cosh(KL) +2'
h(KL) = KL sinh(KL) - KL
KLsinh(KL) - 2cosh(KL) +2
The boundary conditions for Equation (3.1) are
y(O) = Yo, 8(0) = 80, y(L) = YL, 8(L) = eL~
(3. J)
(3.2)
(3.3)
(3.4)
(3.5)
where 80 and 8L are the angle of roller A and roller B, respectively. A mov1ng web creates
its own boundary conditions which may not match to those assumed by the analyst in
deriving the dynamic model.
To develop the lateral dynamics of a moving web, many assumptions are considered.
such as massless web, no slip between web and roller and constant web velocity. These
assumptions may not exist in industrial applications. Hence the mathematical model may
have variations.
3.2 Identification of the lateral dynamics
The lateral dynamics of the web between Roller A and Roller B in Figure 3.1 can be
described in the fonn of the Box~Jenkins model given by
y(t) = G(q)U(I) +H(q)e(t). (3.6)
where y is the measured output, u is the control input and e is the white noise with variance
0 2 .
10
Equation (3.6) can be rewritten as
00
y(t} = L g(k)u(t - k) +vet),
k=O
vet) = H(q)e(t) = L h(j)e(t - j).
j=O
(3.7)
where
B(q)
G(q) = F(q)'
8(q) =bo+blq-l + +bnbq-nb ,
F(q) = 1+ !tq-l + +!njq-nj .
H(q) = C(q)
D(q) ,
C(q) = 1+Clq- t + +cnc q-lI"
D(q) = I +d\q-l + + dndq-TIJ,
q-l denotes the unit time delay operator.
3.2.1 ESlimalion of impulse response of G(q)
Assume g(k) = 0 for k > K, then
yet) =g(O)u(t) +g( I )u(t - 1) +g(2)u(t - 2) +... +g(K)u(t - K) + v(r), (::\.8)
where {g(k)} is the impulse response of C(q). Multiplying both sides of Equarion (3.8) by
u(t - k). simplifying yields
Rlly(k) =g(O)Ru(k) +g( 1)R,,(k - 1) +... +g(K)Ru(k - K) (3.9)
where k = 0, ... IK, Ru)'(k) denotes the cross-correlation between yet) and u(t - k) and
RII(k - j) is the auto-correlation of U(I) and u(t - k) where j = 0, ... ,K.
For example,
Ru(D) RII(l) Ru(K) g(O) Ruy(O)
Ru( I) RII(O) ,,- Ru(K - l) gel) RIIy(I)
- (3.] 0)
Ru(K) Ru(K-I) ... RiO) g(K) Ruy(K)
II
where it is assumed u(t) and e(t) are independent.
Hence, we can get g'(k), the estimate of g(k), from Equation (3.10), i.e.,
-1
g(O) Ru(O) Ru( I) Ru(K) Ruy(O)
g( 1) Ru( I) Ru(O) Ru(K - 1) Ruy ( I)
(3.11 )
g(K) Ru(K) Ru(K -1) Ru(O) Ruy(K)
In this experiment, K is selected to be 50. From the identification point of view, the control
sequence {u(t)} in the system described by Equation (3.6) is set to be white noise so that
all modes are excited. In practice, it is difficult to satisfy this requirement. As shown in
Figure 3.2, the input sequence {u(t)} generated by the OPO cannot be white noise even
through a white noise voltage U(t) is applied to the OPG. Furthermore, u(t) does not vary
around the zero value, that is, it is not zero mean. This is because the OPG consists of a
DC motor which has a low bandwidth (about 8 Hz) and the associated guide mechanism
may have nonlinear behavior, such as non-symmetric friction and backlash. It can also be
observed that the signal u(t) has two spikes at time instances near 3000 and 9000. These
spikes are caused by the adhesive tape which is used for joining both ends of the web in
order to create a continuous web loop in the endless web platfonn. The sensor output near
the region of spikes does not show the true lateral displacement because the thickness of
the web near this region is different from the actual thickness of the web. Hence, only data
within the sample time interval [4000,8500] are used in the identification process.
Figure 3.3 shows u(t) and y(t) with initial values set to zero. Figure 3.4 shows the estimated
impulse response for G(q) by applying Equation (3.11). The results of estimated g(k) are
also shown in the following table.
12
k 0 1 2 3 4 5
g(k) 0.0221 -0'()()23 -0.0345 0.0181 -0.0158 -0.0238
k 6 7 8 9 10 I I
g(k) 0.0146 -0.0246 -0.0012 -0.0160 0.0400 -0.0239
k 12 13 14 15 16 17
g(k) -0.0170 -0.0437 -0.0045 -O.OJ 82 0.0420 -0.0081
k 18 19 20 21 22 23
g(k) 0.0240 -0.0202 0.0183 -0.0119 0.0073 0.0048
k 24 25 26 27 28 29
g(k) -0.0084 -0.0090 -0.0097 0.0094 -0.0531 -0.0157
k 30 31 32 33 34 35
8(k) -0.0123 0.0211 0.0005 -0.0133 0.0546 0.0074
.. - . - -. ..... - -- . -~ - --- -
k 36 37 38 39 40 41
-_._----- L...-._...
g(k) 0.0091 0.0008 -0.0115 0.0044 -0.0163 -0.0123
k 42 43 44 45 46 47
g(k) -0.0012 -0.0025 0.005] 0.0076 -0.0020 0.0219
Table 3.1: Estimated impulse response of G(q), g(k) using Equation (3.11)
]3
Control input 10 the OPG
10
2
7
S-O
ec
~ 0u
-5
-10
0 50 100 150 200 250 300
Samples k
Input and outpul sequenoe of the system
~ o~
~-2t
-4r----~.....-----.........--....".......,..,---------
_6'-----'---------'-----'---------'----"---------'-----L-----' o 2000 4000 6000 8000 1 סס oo 12000 14000 16000
Samples k
Figure 3.2: The control to OPG, u(t) and y(t) measured from Sensor I and Sensor 2 re-spectively
lnput sequence for identification
-0.5 L-_----'-__---'--__--'---__L-_----'-__--'---__--'---__L-_--l.__--'
o 500 1000 '500 2000 2500 3000 3500 4000 4500 6000
Samples k
Output sequenoe lor IdenlillcaUon
0.6,------,------,---,-----,------,------,---,-----,------,-----;
0.4
-0.4
1000 '500 2000 2500 3000 3500 4000 4500 SOOO
Samplas k
500
-0.6
-0.8 L-_----'-__---'--__--'---_---.JL-_----'-__--'---__--'---__L-_--l.__--'
o
Figure 3.3: u(t) and y(t) used in identification design
14
Eslima!8d impuI&e response lor G(q)
0.04
0.02
~ a
-0.02
-0.04
-0.06 '-------'--_----'---_---'-----_--'-----_'-------'--_-----'---_-L-_---L.------l
o 10 15 20 25 30 35 4{) 45 50
k
Figure 3.4: Estimated impulse response for G(q)
3.2.2 Structure of G(q), H(q) and Autoregressive moving average (ARMA) model
for {u(t)}
In Equation (3.11), the impulse response of G(q) is approximated in polynomial of q-1
with finite number of estimated gck). Although this approximation is quite course, g(k)
can be used to estimate the order of the numerator and the denominator of G(q) and H(q).
From Equation (3.8) and Equation (3.11), the estimated output yu(t) due to the input u(t)
can be obtained by
K
Yu (t) = L g(k) u(t - k).
k=O
From Equation (3.7), v(t) is estimated by
V(t) = y(t) - yu(t),
where V(t) is the estimate ofv(t). The ARMA model for the noise v(t) is given by
v(t) = H(q)e(t) = ~~~~ e(t).
15
(3.12)
(3.13)
(3.14)
The orders of C(q) and D(q) can be estimated by the Generalized partial autocorrelation
function (GPAC) in [6J for H(q). The GPAC for H(q) is computed by using VCt) with the
U,k)-th element given by
Rr;(j - I)
Rr;(j)
RvU - 2)
R>;U - 1)
Rr;(j)
Rv-(j + 1)
RvU+k-2) Rv(j +k - 3) ... R;;(j +k - 1)
djk = (3.15)
Rf.(j - 1) Ry(j - 2) Rr:(j - k)
Rr;(j) Rv(j - 1) ... Rv(j - k+ I)
RoU+k-2) Ro(j+k-3)... R~(j-I)
where Ry(t) is the autocorrelation of V, i.e., Rv(-r) = E[V(t)V(t +-r)].
The orders of the numerator and the denominator of G(q) can also be estimated by the
GPAC for G(q). The (j,k)-lh element of the GPAC matrix for G(q) is given by
g(j - I)
g(j)
KU - 2)
gU -- I)
RU)
g(j + I)
g(j +k - 2) g(j +k - 3) ... g(j +k - I)
djk = (3.16)
g(j - I) g{j-2) g(j - k)
g(j) g{j - I) g(j - k+ I)
g(j +k - 2) g(j +k - 3) g{j - 1)
When (he system described by Equation (3.6) is causal. g(k) is chosen to be zero for k < O.
The ARMA model for {U(l)} is given by
u(t) = W(q)rt(t) = ~~~~ aCt)
16
(3.17)
-0.11 -1.6 0.49 -3 0.57 -5.6 0.38 -9.9 -1.5
15 -1.6 -9.2 -2.9 -29 -5.6 -150 -10 99
-0.52 0.18 0.89 -0.12 0.027 -0.31 0.59 -1.6 1.3
-0.87 3.1 0.92 0.067 -1.4 -0.26 -0.27 -l.2 0.036
1.5 -1.2 0.93 29 -2.1 -0.046 1.5 -1.2 210
-0.61 0.47 0.65 1.2 -1.7 -55 1.3 -1.6 0.11
-1.7 1.7 0.4 2.1 -7.8 -8.3 -11 -1.7 24
0.049 0.63 -3 1 0.059 0.31 1.4 0.94 0.59
13 0.78 -1.2 1.\ -5.3 0.028 1.2 -0,42 -0.65
Table 3.2: Ex.perimental GPAC for G(q)
where aCt) is the white noise signal, Seq) = SO+Slq-l +... + sn,q-n" and R(q) =ro +
(Iq-l + ... +(n,q-nr • The orders n\ and nr of Seq) and R(q) are estimated by the GPAC
for u(t). The (j,k)-th element of this GPAC has the same form as in Equation (3.15). The
only difference is that Rv(m) is replaced by Ru(m). Ru(m) is the autocorrelation ofu(t), i.e.,
Ru(m) = E[U(t)U(l +m)]. The experimental GPAC for G(q), H(q) and W(q) are shown in
Tables 3.2, 3.3, and 3.4, respectively.
From the GPAC for H(q) and W(q), it can be observed that nc = I and nd = I. nJ· = 1
and nr = I is one option. GPAC for G(q) does not show a clear pattern for determining nc
and nd. n<; and nd are both chosen to be two based on the knowledge of the theoretical analysis
of the lateral dynamics. It is not certain if the orders of B(q), C(q), D(q), F(q), R(q).
Seq) are correct or not after the first trial. The parameter estimation algorithm (described in
Section 3.2.3) estimates the unknown parameters and the CHI-·Square statistics (described
in Section 3.2.4) to check jf the estimated result is reasonable. If it is not, the orders will
be changed and the parameter estimation algorithm and CHI-Square statistics are applied
agalO.
Other options of model structure are also reasonable from GPAC, such as nc == 2 and
17
0.44 0.27 0.22 0.17 0.14 0.15 0.12 0.078 0.12
0.94 -0.056 0.019 -0.0036 -0.04 0.046 0.016 -0.093 0.086
0.32 0.0089 -0.22 -0.046 0.056 0.26 -0.062 0.04
0.99 0.14 4.1 -0.11 -0.01 0.013 0.048 -0'()()61 -0.022
0.99 -17 2.5 -0.31 -0,12 0.041 0.05 -0.17 0.011
1.2 1.7 1.6 -0.26 0.046 -0.25 -1.9
0.98 -0.33 1.3 -5.2 4.8 1.1 -0.065 -0.078 0.079
0.96 4.7 1.1 0.089 0.47 2.5 -3. I -0.12 -0.013
1.1 lA 0.99 -5.7 0.035 1.7 1.6 -0.32 0.68
Table 3.3: Experimental GPAC for H(q)
0.97 0.44 0.27 0.2 0.2 0.16 0.15 0.15 0.1
I -0.038 -0.03 -0.057 0.039 -0.018 -0.0023 0.041 0.03
-0.89 0.05 -0.084 0.018 -0.023 -0.32 0.042 0.041
1 -2.2 -3.9 -0.089 -0.037 0.057 0.1 0.033 0.058
0.41 -0.065 -0.19 -0.45 0.29 0.078 -0.008 0.085
0.6 -1.3 -0.37 0.9 -0.2 0.085 0.092
-0.35 2 2.5 0.65 0.44 -0.13 0.031 0.061
8.5 3.1 -1.5 0.19 l.J 0.14 0.28 0.071
-0.67 0.69 -0.9 6.3 1.\ -4.3 0.56 -0.064
Table 3.4: Experimental GPAC for W(q)
18
nd = 1, or ns =2 and nr = 1. By observation, the pattern of the GPAC. nb = 2, nf = 2,
Me = ], nd = 1. ns = 5 and n, = 3 is selected in the following sections.
3.2.3 Maximum Likelihood algorithm
To estimate the unknown parameters, the Maximum LikeHhood (M-L) algorithm is applied
here. Considering the general case. the unknown model parameter vector in Equation (3.6)
IS
(3.18)
where 6i. i = ], ... , n is the unknown parameter in Equation (3.6); n is the number of the
unknown model parameters. Choose the likelihood function as
N
1(8) = F(8) = ~e2(t,e) = eT(8)£(8),
/=1
£(8) = [£(1,8), ... ,c(N,8)f,
where N is the number of the measurement data; c(t, 8) is the predictlon error with
E(t , 8) = y(I) - Y( t 18) ,
where nfl'S) is the estimate of yet) with
where H(q) and G(q) are the estimates of H(q) and G(q). respectively.
(3.19)
(3.20)
(3.21 )
(3.22)
Using Levenberg-Marquardt algorithm in [7), the increment of the estimated parameter
vector is found to be
(3.23)
where k is the k-th step used for estimated parameter vector, 8k is the estimate of eat k-th
step, Jik denotes a parameter to used in the algorithm at k-th step and I denotes a identity
19
matrix of n x n, and
... , (3.24)
where 8ki is the i'h estimated parameter of 8 at the k-th step, and a~t 18k
) is a column vector
8ki
with N elements and is computed numerically by
Y(tlekl, .. · ,8ki+Ski, ... ,8kn) -Y(t I8k)
Ski
(3.25)
where 8ki is small real number.
The procedure for implementing the Levenberg-Marquardt algorithm is given below.
1. Input 8(0), minimal gradient Gradmin, f-lO, and v;
2. For k = 0, I, ... , K repeat the following
7. TestifF(8k+d ~F(8k).lfitis,setf-lk=fJk*vandgot05;
8. Setf-lk+l =f-lk;
9. Test if II2JT (8k+l)c:(8k+dll < Gradmin or f-lk > f-lmax (f-lmax is maximal allowed f-l).
Otherwise go to step 2;
10. Compute covariance matrix for the estimated e, i.e., cr;[JT (8k+1 )J(8k+ 1)] -1.
20
(3.26)
3.2.4 Diagnostic testing
Consider the following error signal
V(r) = y(t) - G(q)u(t)
= (G(q) - G(q))u(t) +H(q)e(r).
Multiplying botb sides of the above equation by ~i:j and simplifying yields
f' = ~~:~V(t) = (G(q) - G(q))et(t) +~~;j H(q)e(t). (3.27)
It can be seen from Equation (3.27) that, if G(q) = G(q) then E'(t) and a(r) are uncor-related
because a(r) and e(t) are both white noise signal and are thus uncorrelated. This
observation can be used to test if G(q) is an accurate estimate of G(q). Also, from
€(t) = y(t) - Y(t 18)
= y(t) - (H(q)-IG(q)u(t) +(1 -ii-I(q))y(t)]
= jj-\(q)(G(q) - G(q))u(t) +B-l(q)H(q)e(l),
(3.28)
it can be seen that, if G(q) = G(q) and H(q) = H(q) then £(t) = e(t) should be white noise.
The above two observations give some clues to adjust the order of G(q) and H(q). This is
useful because the GPAC for G(q) and H(q) may exhibit several patterns for deciding the
structure of G(q) and H(q). If €'(t) and a(t) are correlated, the structure of G(q) should
be changed; if g'(t) and a(t) are uncorrelated and €(t) is not a white noise signal, and the
structure of H(q) should be changed.
The X 2 statistics can be used to determine the confidence level of two sets uf data are
uncorrelated or one set of data is white noise. If £'(1) and a(t) are uncorrelated and £(t) is
white noise, the following statistics are introduced:
~ (R((e) (k)) 2 2
Q= N L.J R (0) f'.J XK-n£-nd'
k=! e(El)
S = N.f. ( R()£f(8) ) 2 "-' X 2
£.J K-nb-n/,
k=O a~'(e) ()~
21
(3.29)
(3.30)
where 0-;/(8) and ~ are the variance of £'(8) and a(t), respectively; RE{s)(k) is the autocorrelation
of £(8); Ra.£l(e) is the cross correlation between o.(t) and £'(8). As the degrees
of freedom (DOF) of the Chi-square variables in Equation (3.29) and Equation (3.30) are
known (K - nc - nd and K - nb - nl. respectively). given a confidence 1- E, we can look
up the CHI-Square distribution table to find a threshold of the alann level~. If Q < j3
and S < ~ then we can say that G(q) and H(q) are accurate estimates of G(q) and H(q).
respectively, with a confidence level of 1 - £.
3.2.5 Experimental result
For the system under consideration in this thesis, the parameter vector 8 is
(3.31 )
The parameter vector for ARMA model of {U(l)} IS
(3.32)
The following values are chosen in the MATLAB program, p = 0.01, ,uma.r=10 1O , v = 10,
Gradmin = 10- 15, bk = 0.001, and £ = 0.05.
The ARMA model for {u(t)} is
_ I - 2.3348q-l + 2.1981q-2 - 0.7566q-3 - O.0866q-4 +O.0062q-5 (r)
u (t) - ) _ 2.4584q-1 +2.4589q-2 _ l.OOO8q-3 a ,
S(q) = I - 2.3348q-1 +2.1981q-2 - O.7566q-3 - 0.0866q-4 +O.OO62q-5,
R(q) = 1 - 2.4584q-1 +2.4589q-·2 - 1.0008q-3.
(3.33)
(3.34 )
(3.35)
Qo = NIf", (~:~~D' = 21.1. The threshold level ~o for 12 (K - n, - n.,· = 20 - 3 - 5 =
12) OOF Chi-square distribution with 95% confidence is 21. As Qa ~ ~Ct.. the ARMA
model for {U(l)} is validated, i.e., a(k) is white signal. Figure 3.5 shows [he impulse
response for W(q) = S(q)jR(q). Figure 3.6 shows a(k) sequence and its autocorrelation.
22
Impulse response lor W(q)
09
0.8
0.7
0.6
i o.5
0.4
0.3
0.2
0.1
aa 2 6 8 10
k
12 14 18 18 20
Figure 3.5: Impulse response for W(q)
While dJalumance a(kl
-0.1
-0.15
-0.2
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000
Sample. k
X10~ AUIOCOmllBlion 01 a(l)
a
4
a: 2 ~~
0
-2a 20 40 60 60 100 120
Figure 3.6: a(k) and its autocorrelation
The estimated model is
( )
= 0.0490 +O.0246q-l - 0.0346q-2 u( )
yz I-O.1517q-I_0.3679q-2 Z
1- 0.98l4q-1
+ I - O.9998q-1 e(z).
23
(3.36)
The thresholds of alann level for 16 (=K - Tlb - n/) DOF and 18 (=K - nc - nd) DOF
Chi-square distribution with 95% confidence are St = 29 and ~, = 26.3. respectively.
Q, ~ NIfd (~:i~D2 =39.1. As Q, > ll.. =29, the prediction error is white with less
confidence « 95%). One reason why QE. is large is that the output y(t) measured by sen-sor
2 has some external signals. which points to a large noisy peak as shown in Figure
3.3. Normally, the sensor output signal with noise having derivation around 0.04 volls.
However, at exception points, the derivation reaches 0.5 volts. The samples with large
noise level contributes to large prediction error as shown in Figure 3.7. which plots the
prediction error sequence and its autocorrelation. It can be observed from Figure 3.7 that
the prediction error Elk] distributes around zero. Further computation yields that the expectation
of Elk] is E(£[k]) = 2.9746 x JO-4. and variance is var(£[k]) = 1.4 x JO-'3; the
autocorrelation of E[k] at '[ =0 is almost 20 times larger than at other instants. From this
point of view one can assume elk] to be white noise although Chi-square error testing fails
here. SCJ£.1 = N rf=o (./0:/(8) 2) 2 = 8.1462, As SCJLI < ~Ct£" the input a(k) and the es-crr'(
8)lJll
timated disturbance E' are uncorrelated with quite high confidence (> 95%). Figure 3.8
shows the sequence and amplitude of the frequency response of the estimated disturbance
qr) (= y(t) - G(q)u(t)) introduced into the system. The disturbance includes measuremenl
noise and the ex.ternal disturbance. Figure 3.8 shows that v(k) is almost white noise except
having a 70 Hz disturbance. This 70 Hz signal is produced either by the imperfect sensor 2
or the mechanical resonance. The impulse response of the estimated web lateral dynamics
C(q) is shown in Figure 3.9. Figure 3.9 shows that C(q) is a stable system.
3.3 Estimated and theoretical model comparison
Ex.perimental parameters are obtained from practical experimental steps shown below•
• Web thickness and width are measured.
• The web is stretched 50 feet and laid on a flat surface with one end taped and weights
24
4500 5000
Autocorrelation of proolCUon a"or uSIng estimated model
0.6 ,-----,--,---r---,-----,--,----.----,-~-_r_-___,
0.4
-0.6
-0.8 L-_-"--__-'----_---'-__..L-_----'__--'---__L-_-'-__--'---__
o 500 1000 1500 2000 2500 3000 3500 4()()()
Samples k
x 10"'"
15,------,-----.-----,--,-----,---,---.----··,----------,
10
~e 5
a:
0 - ) l - ~ ~
-5
0 10 15 20 25 30 35 40 45
Figure 3.7: Prediction error E(t) = y(t) - y(tI8) used estimated model and its autocorrela-tion
Estimated combined dislurbance vlk)
0.6,-----,--,--------.----.-----,----,---,----.,------,-----,
0.4
-0.4
-0.6
-08
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000
Samples k
20 ,--r--t
:=0: 15 g
~
~ 10
~
Ci
E «
50 100 150 200 250 300 350 400 450 500
Frequency (Hz)
Figure 3.8: v(k): Estimated disturbance and its autocorrelation
are placed to prevent it from moving.
• A tension meter is pulled from the other end.
25
O.
0.04
003
0,02
''"5
0.01 ~. a
-0.01
-002
0 5 10 15 20 2S )() 35 40 45 sa
I<
Figure 3.9; gk: estimated impulse response for G(q)
• A series of prescribed tensions are applied and the stretch in web length correspond-ing
to each tension is recorded.
From the experiment, the following data is processed in spreadsheet shown in Table 3.5The
theoretical model and the estimated model are shown below.
• Theoretical model for lateral dynamics of a fixed roller
-O.004153q-l +0.004179q-2
y(z) = 1 _ 1.9915q-l -O.9915q-2 u(z)
• Experimentally identified lateral dynamics
( ) = 0.0490 +O.0246q-l - O.0346q-2 u( )
yz 1-O.1517q-J-O.3679q-2 Z
(3.37)
(3.38)
Figures 3.lO and 3.1 J show the comparison between experimental output, estimated
output and theoretical output.
Those comparison plots show the estimated output and the theoretical output are obtai
ned from a same set of experi mental input which is an offset pivoted guide. Figure 3.10
26
Tensi.on (Ib) DisplacemeOl (in) Strain Stress (psi) E (psi)
5.0 0.272 0.0005 285 628000
10.0 0.505 0,0008 570 677000
15.1 0.697 0.0012 860 741000
20.2 0.881 0.0015 1151 784000
25.2 1.1l3 0.0019 1436 774000
30.2 1.318 0.0022 1721 783000
Average 731000
Table 3.5: Experimental parameters
Compo""", ot EAperwnenlll aI'd 'I'heo<etcal """"'I WIll 0011 ...... 1
'4
..
H)
- -3[ ~. ~I""'l- ]
r~: i4'~' JIb
o 2 4 6 8 10 12 "
j~":':~~-~-~~~_-I.~-~---
o 2 • 6 8 '0 12 04 .-J 12
time -->
Figure 3.10: Experimental input, output and theoretical model output
and 3.11 shown the theoretical model output and the estimated model output seem to fol-low
the same pattern of the input. The two spikes in each individual plots are caused by the
joint of the endless web line. The model structure of the fixed roller dynamics is identified
by the experiments conducted. The accuracy of the experimental resul t may be affected
by several physical issues. such as the offset pivoted guide chat is being used has a non
27
'I S -02 i-= ~ -0.3
-0.35o
Comparison 01 Experirnenlal and ldenttfied Mod91 with Data se"l
i~~J
1.4o 2 4 6 8 10 12 14
..;;;:;;; :. 1
2 10 12 14
time -->
Figure 3.11: Experimental input, output and identified model output
symmetrical pivoting angle and sensors being used are optical sensors which may be noise
sensitive and have a small range.
28
CHAPTER 4
LATERAL WEB GUIDING WITH VARIABLE ENTERING SPAN LENGTH
[n this chapter. steering guide behavior with variable entering span length is analyzed. For
a steering-type guide, the guide roller moves laterally and angularly to accomplish lateral
displacement correction of the web. The displacement correction action steers the web
laterally in the entering span to maintain the desired position in the exiting span.
Figure 4.1: A schematic of a continuous metal processing web line is shown
The web in the process section in Figure 4. I continues at a constant velocity when an
unwind or rewind core change takes place. This is made possible by the the accumulation
of web in horizontal loopers via two moving caniag~s. one downstream of the unwind and
the other immediately upstream of the rewind. In some continuous processing lines. such
as aluminum strip processing lines, vertical loopers are used; the carriage moves in the
vertical direction.
Figure 4.1 shows a continuous processing web line with carriages that travd horizon-tally
to accumulate or discharge web, which makes the web span length change when the
carriage is moving. The distance between the two rollers are large in comparison to the
length of the contact area between the web and the entering roller, which causes the web to
sag between two rollers. Support rollers that are automatically enabled when the length of
the span exceeds a prescribed length are used to support the web and to minimize web sag.
29
A steering guide must be installed after a long, free entering span. The length of the
span is determined by the required maximum correction to be made and the web tension, as
well as the characteristics and dimensions of the material to be guided. such as modulus of
elasticity, web width, and web thickness, It is desirable that the web wrap around the fixed
entering idler roller be large enough so that the web will not slip laterally as the steering
guide makes corrections.
L
Instant Center
v
z
Original
Center
Figure 4.2: Response at remotely pivoted steering guide
Figure 4.2 and 4.3 show a schematic of a steering guide which is commonly used in
the web handling industries. The steering guide is mainly composed of a roller, an actuator
and a sensor. The sensor measures the position of the web, and the actuator provides the
required correction. The motion of the guide roller can be considered as a rotation about
an "instantaneous center", Usually, the center of rotation of the steering guide is located
between 2/3 to 3/4 of the entering span length from the guide roller as shown in Figure
4.3.
For a remotely pivoted guide shown in Figure 4.3, the transfer function from the guide
correction, Z(s), to the web displacement, Yr(s) , is
(4.1 )
30
_..... .-- ~._._~.~_.
Figure 4.3: Sketches of a remotely pivoted guide. The remotely pivoted guide consists of a
single roller.
where
(KL)2coshKL- I
II (KL) = KLsinhKL- 2(cosh KL _ I)'
KL [ KLcoshKL-sinhKL ]
h(KL) = KLsinhKL - 2(cosh KL - 1) .
and
K = constant for a given web (j"l;),
L = length of the entering span,
L
L = transpol1 lag (-).
v
v = web velocity,
Xl = distance from guide to its instant center.
31
Web velocity: 600 FPM Web tension: 7500 lb
Web material; AISI 304 Steel Web thickness: 0.25 in
Web width: 60 in Span length: Varies from 20 ft to 100 fl
Young's Modulus: 2.4ge7 pSI Roller width: 72 in
Roller Diameter: 30 In lnst. center: SO f[
Table 4.1: Operating Conditions
4.1 Open.loop analysis
In this section. an open loop steering guide response of the web lateral dynamics is investigated.
A Fife Kamberoller® guide is used for this investigatIon. Conditions of the steering
guide used are given in Table 4.1.
The impulse response of Equation (4.1) is simulated with the above conditions and
shown in FIgure 4.4.
I
I
J
-3 / ~ l
~V -l-~' j
-5~~---------'---1"~
o 0.5 1 5 ~ 2 ~
Figure 4.4: Impulse Response of a Steering Guide
32
Bode Diagram
_ l=l00f1
iDOg~~~~~~
~ '~ ]-5F------------/-~__.
too '-- ,.""
_t5'-cc-~~~~ ..........._~~~~-L.. ~~__'___~ ___J
10" to" 100 10' 10'
F'Q(luem:y(redisec)
-10
-20'----~~---'-!__~__....I!_~~_~~L_~~_ _______...J
10" to" to· to" 10'
F'Q(lU6ncy(rlldl~ec)
Figure 4.5: Frequency Response of a Steering Guide
Figure 4.5 shows the bode plot of Yds) I Z(s). When the entering span length of the
guide roller is equal to 100ft, L = 100ft. there is a phase lag in the frequency range of I to
10 fad/sec. These plots show the effect of entering span length of the guide roller, L. as the
length varies from 20 ft co 100ft by the increment of 10ft.
Sagging of the web can affect the lateral guiding performance. In such a case, catenary
analysis of the web is impoJ1ant for a better understanding of the sagging behavior of the
web, as done in the next section.
4.2 Catenary Analysis of Web
Sagging of a web span can be analyzed like a flexible suspension cable as described in [8].
As an initial study, sagging web spans can be analyzed in the static ca5e as a flexible cable
suspended between two SuppOJ1S at their ends and subjected to the action of a vertical load
continuously distributed along the length.
33
y
A
A' ,--------i-,....,.--n---r-,-----'t---,B'
b
Q
(a)
y
I'-'-----x
Q
(b) (c)
Figure 4.6: Free Body of Catenary Web
Consider a web ACDB simply supported at its ends A and B and a vertical load distributed
along the horizontal as represented by the load diagram AIahB' in Figure 4.6 (a).
Let the lowest point C of the curve be the origin of the coordinates axis x and y. D can be
any point on the curve with coordinates x and y.
A free body diagram of a portion of [he web CD is shown in Figure 4.6 (b). This web
segment is in equiIibri urn under the action of three forces; a venica1 force Q representing
the distributed load on the curve CD and two tensile forces Hand S representing the reactions
by the unconsidered portions of the web on either side. The forces H and S are
tangential to the curve at point C and point D. The three forces H, Sand Q form a closed
34
triangle as shown in Figure 4.6 (c). Relationship among these forces ean be written as
Ssin(B) = Q,
Seos(S) = H,
S= JH2+Q2.
(4.2)
(4.3)
(4.4)
Also
lan(S) = ~, (4.5)
and since lanCe) = dy/dx, we obtain
d.Y' Q
dx H
(4.6)
Equation (4.6) is the di fferential equation of the curve assumed by the web under the act! on
of the load that it carries. Equation (4.4) defines the tension at any poi nt on the curve.
4.2.1 Catenary web analysis with respect to horizontal length
y
1
f,
-.l....------;-------:==---:oc:+---=---------i---<----x
a b
Figure 4.7: Catenary Web with the length, I = a +h, and 12 - II = h
Figure 4.7 shows a web span supported by rollers at points A and B. The weight of the
web can be considered as a veJ1ical load of intensity of which is unifonnly distributed with
respect to the horizontal length of the span. The slope Equation (4.6) can be written as
dy qx
dx H
(4.7)
35
where q/H is a constant.
Integrating once, we obtain
q~
y=-+c.
2H
(4.8)
For the coordinate axis chosen as in the Figure 4.7, W~ have y = 0 when x =O. The
integration constant C is equal to zero. Hence
qXL
Y= - .
2H'
(4.9)
Equation (4.9) is the equation of the curve of equilibrium assumed by the web. For such a
loading, the web shape is a portion of a parabola.
Since Q= qx, Equation (4.4) can rewritten as
(4.10)
The tension in the web is minimum at the lowest point C, where it is equal to H. It increases
toward the ends of the web and being a maximum at the highest support.
The forces at the ends of point A and point B of the web can be expressed as
(4.1 J)
(4.] 2)
(4.] 3)
The distance a and b of locating point C with the reference to the supports, point A and
point B, can be determined by using Equation (4.9). Figure 4.7 shows that when x = -a.
y = fl and when x = b, Y = h, which yields
qa2
/1 = 2H'
qb2
12= 2H'
Let the height difference between the two supports be denoted by h = h - fl. By subtract-ing
Equation (4. ]2) from (4.13) we obtain
qb2 qa2
---=h
2H 2H '
2hH = q(b2 _ Q2).
36
(4.14)
Since the lOtallength is I = a+b, combining it with Equation (4.14) yields
I hH
a=---
2 ql'
1 hH
b=-+-.
2 ql
(4.15)
(4. I 6)
When the supports at point A and point B are on the same level, we have a =b = 1/ 2.
Substituting Equation (4. I 5) into Equation (4. 12) and (4.13), H is obtained as follows
(
I hH) 2 ( I hH) 2 h "2-qi =fl "2+qi
2 2ql2 ( h) l/4
H -- 12-- H+-=O h2 2 4h2 I
from which
(4.17)
The minus sign in Equation (4.17) should be used for all cases where the vertex of the
parabola corresponding to the configuration of equilibrium of the web 1ies between the
supports, which is the curve ACB shown in Figure 4.7. The plus sign should be used for all
cases where the vertex of this parabola lies to the same side of both supports as shown hy
the dotted curve AB in Figure 4.7. Figure 4.8 shows a web span supported by two rollers
fr
-'------7----------==-----;,:i-----'""""""=--------r----'-- x
~--____=_-----l....-------,
8 b
Figure 4.8: Catenary web with both rollers at the same level
at the same height. In the case where the two supports are on the same kvd, II = 12 = f.
a = b = //2, Equation (4.12) or (4.13) yields
ql2
H=-
8f
37
(4.18)
For practical cases, the span length I, the height of the supports II and 12, and the
intensity q of the uniformly distributed load will be known. Therefore, the vertex of the
curve, which is the lowest point C, can be calculated.
4.2.2 Catenary web analysis with respect to the curve
y
a b
f,
_.L..---T--_"::::::::"_~_-=O_------+----L_X
Figure 4.9: Catenary web analysis with respect to the curve
Assuming that the catenary web shown in Figure 4.9 hangs freely in the gravity field
and is subjected only to its own weight uniformly distributed along the curve, the slope
Equation (4.6) can be rewritten as
dy qs
dx H
(4. J9)
where q is the weight per unit length of the web and s is the length of the arc CD. To
integrate Equation (4.19), the length of CD needs to be expressed as a function of the
coordinates x and y.
Figure 4.6 shows the sum of the forces in the x and y direction. The equilibrium equa-tion
can be written as
Ssin(8) = qs, (4.20)
Scos(8) = H. (4.21 )
Division of Equation (4.20) by Equation (4.2]) yields the following Equation
dy qs
tan(8) = - = -.
dx H
(4.22)
38
y
H-----=...,...,"""1==++-I---+.-...f.....,..------X
qs
Figure 4.1 0: Catenary web Analysis
Differentiating Equation (4.22) with respect to x results in
d(dY) q (dS)
dx dx = H dx
From Figure 4.10. the differential length of the web can be described as
ds= 11+(dy )2
dx V dx
Using Equation (4.23), we obtain
_ ~(qS)2
ds - Vt + ~Hr·
Integration of Equation (4.24) gives
: sinh- I (~) = x +c) ,
where Cl is a constant. Since s =0 when x =0, C, = O. This yields
H. (qX) s= qsmh H .
39
(4.23)
(4.24)
(4.25)
Substituting Equation (4.25) into (4.19), we obtain
dy = sinh (;;) dx,
which when integrated, results in
H (qX) Y = -qcosh H +C2·
The origin of the coordinate system is placed at the lowest point C of the web where the
slope is zero at this point. Substituting x =0 and y =0 gives C2 = -H / q, which yields
Y = qH [cosh( qHx) - J ] .
Recall that Q= qs. Therefore Equation (4.25) can be written as
Q=Hsinh (;;) I
and when substituted into Equation (4.4) gives
Substituting Equation (4.27) into Equation (4.26) gives
s= H+qy
(4.26)
(4.27)
(4.28)
From Equation (4.28), note that the tension S is a minimum at the low point C where it is
equal to H and it increases toward the ends of the web. being a maximum at the highest
suppon. Forces at the ends of point A and point B can be obtained as follow
Equation (4.26) represents the curve of the equilibrium when the web hangs freely under
its own weight and shows a catenary with vertical axis. From Equation (4.26), the value of
H can be obtained as follows. The expressions for 11 and 12 are
11 = Hq [cosh( qHa ) - 1] ,
12 = qH [cosh( qHb ) - 1] .
Equation 4.29 can be written as
For a + b = l, we obtain,
Differentiating Equation (4.26) with respect to x and evaluating at x = 1/2 gives
(4.29)
(4.30)
(4.31 )
(4.32) dyl . ql
dx x=~ = smh(2H)'
The sag angle can be obtained by solving 8. To determine the relationship between T and
S, combining Equation (4.32) with (4.5) gives,
tan(8) = Sinh(;~),
lan'(8) = ['inh(;~)r
l-cos2(8) [. (ql)]2 -----=----'- = smh - ,
cos2 (8) 2H
1- co"(8) = co,'(8) ['inh( ;~)r
2 1
cos (8) = 2'
1+ [sinh( {# )]
1
cos(8) = 2'
1+ [sinh({k)]
41
(4.33)
H=S
Combining Equation (4.3) and (4.33), we obtain
2 \
1+ [Sinh( tk)]
S = H COSh(2q~).
(4.34)
(4.35)
The catenary analysis is useful in analyzing a heavy metal web supported by two rollers.
With the web entering span length 1, web tension S. and roller height 11 and h. we can
obtain the sag and sag-angle from the analysis.
4.3 Steel Web Process Line with moving carriage
Figure 4.11 shows a steel web process line with a moving carriage. The carriage moves
on rails and the support rollers swing for every 100 feet separation. "KR" in the figure
refers to a Fife "Kamberoller Guide", which is a remotely pivoted steering guide. When
the carriage is at its left most position. the steering guide has an incoming span length of
20 fr and when the carnage is at its right most position, the steering guide has a variable
incoming span length of 200 ft when the carriage moves.
TO P'OOO"
-1 ~.~.~
~IOOh----
SUOl)O,,;ng ,oll.",
/ ~
Figure 4.11: A sketch of one section of the accumulator in a metal process line
For initial analysis, the location of the first supporti ng roller is assu med to be at I()() ft
apart from the guide roller. As the carriage travels from 20 ft to 100ft, the web forms a
sag due to its own weight. The support rollers are swinging in and out when the carriage
moves to a particu lar distance. All the su pport rollers are tilted as shown in Figure 4.12.
42
Tomw"d 'Oll (''It!~It
", .. I .•.. I
,.- ;'" I
.......... ... ~ . _••••• __ • I ..
Figure 4.12: Swinging supporting rollers
Web material AISI 304 steel
Web density 8030 kg/mJ
Web thickness 0.25 in
Web width 60in
Web tension 7500 Ib
Table 4.2: Simulation parameters
Table 4.2 shown the numerical values that are used in obtaining the pIal shown in Figure
4. 13 and Figure 4.14.
The sag shown in Figure 4.7 is computed for varying the distance betwt".en A and B.
The sag is plotted as a function of [he separation distance between the guide roller and
the upstream roller as shown in Figure 4.13. The sag increase when the carriage travels
from 20 ft to JOO ft. The solid line in Figure 4.13 represents the sag of catenary when the
web is supponed by two rollers. A and B, at the same level. The dotted line shows the sag
of catenary web supported by two rollers with a height difference, h. The sag is seen to
increase for both case when the carriage travels from 20 ft to JOO ft.
Figure 4.14 shows the wrap angle on the steering guide roller with variable span length
from 20 ft to 100ft. The wrap angle on the steering guide roller that is used for lateral
correction is decreasing with increasing span length. This means the guide may not be
10,..------,----.,.-----.----.,------------,------.------.----------,
8
6
I
I
3
2
o'--------'--=---==--=-....:.L-----'-------'-------'--------'-------'--~
20 :lO ~ 50 60 70 60 00 tOO
lenglI'o I~)
Figure 4. 13: Web sag versus span length
able to provide the desired lateral correction when guiding heavy metal web with variable
entering span length. The solid line shown in Figure 4.14 represents the sag of catenary
Wfi\Ilped anglo
681----,--~----,-----.--r=======:=======;
- 8i>tll I<>le'" 01 ume 1.......1
i-Roller1l willi 2 " ...~ ICJ'l Io6pat:ll~Ol\
86
74
n
70
oe'----_--'-__----'--.__---'-__.L..-_----'__-----'-__-L-_----'
20 30 40 50 60 70 80 90 100
l$I1glh (hi
Figure 4.14: Wrapped angle versus length
when the web is supported by two rollers at the same level. The dotted line represents the
44
sag angle of catenary SUppOI1ed by two rollers with a height difference, h. The wrap angle
on the guide roller that is used for lateral correction is decreasing with increasing span
length.
Sagging of a web span may deteriorate lateral guiding performance in a continuous web
processing line. Sagging of a web may result in web slip on the guide roller and a<; a result
the guide may not accurately position the web.
4.4 Control of variable entering span length guiding system
For a continuous web processing line, the carriage moves whenever there is an unwind!
rewind core change. The steering guide shown in Figure 4. J5 transfer function can be
expressed as
Inslant Center
l
v
2
QriQlnal
Center
Figure 4. J5: Response at remotely pivoted steering guide
(
h(~L) ) ( _fJ(KL) + /dKL) )
YL(S) = 2 h(KL~ fl(KL) 80(s) + 2 ~ ~~K.L) YO(s)
s +--S+ --;;r- S + S 1--- T ~ t t L
(
s2 + h(KLl s + Lh~KL))
~ , Xl Z( )
+ 2 + h(KL) + fl(KLJ S s -,-5 ~
(4.36)
(4.37)
where
fL(sJ : response at the Steering guide;
Yo (s) : position disturbance from upstream roller;
80(s) : angular disturbance at upstream roller;
Z(s) : steering guide input
The transfer function at a fixed roller shown in Figure 4.16 can be written as
~o.L ._._.__ .....---I---+------T--------, ._. _, __ . ._... y
~-,
Figure 4.16: Response at fixed roller
(
bJ§:l ) ( _fJ(KL)S+ M~l )
YL(S) = j (KL~' J (KL) 80(s) -+ -_.... ~ ~ Yo(s)
S2 + _2__, s+ _1_ ::;2 + - s+ 1
t ~r-- T T
where 1: = L / v is the transport lag.
(4.38)
As the carriage moves beyond the first pre-defined distance, the support rollers swing
in to support the web. The lateral guiding dynamics are changed due to the introduction of
the second span by the support rollers. To improve the performance of the lateral guiding
system, different modeling is considered.
Figure 4.17 shows a simplified diagram of a web span with a support roller I. between
the guide roller g and the carriage roller O. The first span consists of a guide roller, g and
a support roller, 1, and the second span is between a support roller and a carriage roller, O.
To rewind roll
So{s)
8, (s) 'to(S)
1 - - - - - •• - - - -. - - - - - - - - - -. - - - - •• - -- - - --B
~
From unwind roll
Figure 4.17: Simplified diagram of variable entering span
As the carriage travels away from the steering guide roller which is from 20 ft to 100ft,
there is no support roller in between. In this case, the single span steering guide dynamics
can be wrirten as
(4.39)
The span length between Gde and Gdy is varying due to the carriage motion. From Equation
(4.39), the transfer function of the steering guide can be expressed as
f2(KL) h{KL) /1 (KL) h(KL) h(KL) .
where al = , a.2 = 2,01 = 2,02 - 2 • ~3 - respectively.
l' 't 't l' l'
In time-domain, Equation (4.40) is
(4.41)
where Yg, z. 80, and Yo are the time functions corresponding to the Lapl ace transforms Y,t: (s).
Z(s), 80(5), and Yo(s), respectively.
Define U(s) = Z(s). From Equation (4.40) are obtains
a2 - 01 03 -025+ ~l
Y,ii (s) = U + s2 +ats+ 0 V + 2 0 80 (s) + 2 j) Yo (s) . 1 S +CXjS+ , s +a,s+ J
Denote y = y~ - u. In time-domain, Equation (4.41) IS
Regarding (0 the system given by Equation (4.42), simulations are conducted using a PI
controller, that is,
(4.43)
where e = y, - Yg and y, is the reference.
When the carriage moves beyond 100ft, a set of support rollers swing in to prevent the
heavy web from sagging. By assuming that L'1e support rollers behave like a set of fixed
rollers, the propagation of disturbances generated in the span between the support rollers
and the carriage roller into the entering span of the guide is investigated in the following.
The lateral guiding dynamics now becomes
(4.44)
where f means that the span length is fixed and the displacement YI (s), which is a distur-banee
to the entering span is given by
(4.45)
Combining Equation (4.44) and (4.45), the guide dynamics with support rollers can be
written as
Yg(S) = c{(s)Z(s) +G~e(s)eJ (s) + G~/s)Gde(S)eO(s) + G~yGdY(S}YO(s)
, " vY
d
(4.46)
From Equation (4.46), the guide dynamics with support rollers can be expressed as follows:
(4.47)
In time-domain. Equation (4.47) can be written as
where Yd is the output of the combined disturbances.
There are two different kinds of span length variation in the system. First span length
consists of a variable emering span length between the guide roller and the carriage roller.
Second span length consists of the distance of the carriage which it travels from 20 ft to
200 ft when the web material is feeding into the accumulator and travels back to its initial
position when the web material is leaving the accumulator. Figure 4.18 shows two different v_~"erl"ll ep"n Ie1IQI/l
100
80~
€
t :r
20a \00 200 300 400 soc eoo
c."1ioge leI\glh :l 1 --
~
"- €~,oo~ ~
~ I
SOt
a ----'
__ .---.L-__
0 100 200 300 ~oo ~ 000
T..... (aeo\
Figure 4.18: Web span length and carriage length
variable span lengths in the system. The top plot shows that the entering span length at the
guide ToHer is varying from 20 ft to 100ft. The span length is constant for a period of time
due to the support rollers were in position when the carriage travels beyond 100ft. The
lower plot shows the carriage distance from the guide roller which is the carriage travels
from 20 ft to 200 ft and travels back to its initial position.
The steering guide response was simulated by using three different kinds of disturbances.
A step and a sinusoidal disturbance indicates that the web is disturbed by the
carriage motion. A pulse disturbance indicates the web is disturbed by the support rollers
at the transition at 150 seconds and 450 seconds.
D~
~~~ , ~
-~~'---,__--', -'-c-__---:-'-:. .'..... ,'-_~
o ,00 200 300 400 500 600
Sle.1ng gu;de o"'PU'
~~}=-------'-:-~, -,~-'----~-,-3 o 100 200 :JOO 400
lime
FIgure 4.19: Steering guide response due to step disturbance
Figure 4. J9 shows the steeri ng guide response due to a step disturb anee. The d j stu rbance
is assuming that the displacement of the web on the carriage roller and the support
rollers. The first plot shows the disturbances at the carriage roller and t.he support rollers.
The second plot shows the steering guide response to the disturbances and the third plOl
shows the steering guide output. Notice that the disturbance dynamics relies on the web
span length. As the span length changes, the disturbance also changes.
Figure 4.20 shows the steering guide response due [0 a sinusoidal disturbance. The firsl
plot shows the disturbances at the support rollers and the carriage roller. The second plot
shows the steering guide response to the disturbances. The third plot shows the sleering
guide output for disturbance attenuation.
Figure 4.21 shows the response of the steering guide to the disturbances that cause by
the initial contact of the support rollers at J50 seconds and 450 seconds. Figure 4.22 gives
a closer view tho the response. Those disturbances are assuming that the support rollers
swing in to support the heavy metal web as the carrIage moves. The action of (he support
rollers may create a force against lhe web moving direction and affect the steering guide
50
Disturbance :,-::E- : :-==:1 o 100 200 300 400 500 600 :·::E, :.~~-- : :]
o 100 200 300 400 500 600
Sleering guide output
~-::E : : : 3 o 100 200 300 400 500 BOO
time
Figure 4.20: Steering guide response due to sinusoidal disturbance
Disturbance
:,-::l : n: : :1,: J
o '00 200 300 400 500 600
Steering guide reeponse
:.J ,,': : :: :
o 100 200 300 400 500 600
Steering guide oulpul
~~:llll: : : : :
o 100 200 300 400 500 600
lime
Figure 4.21: Steering guide response due to pulse disturbance
perfonnance.
In this chapter, the catenary analysis of a web and the steering guide response on the
variable entering span length are investigated. First, a study on catenary of web is carried
5]
"I
O,S~
:o_::r : • v: : ~ ~_~~_~ "__j 140 142 144 14(\ \46 160 i62 ISo! 1(>6 lsa 100
St~ng guklo ""'_
: :': : : 'V: : I 1~ \44 I. I. 1~ 162 lW \~ 1~ 1M
<:l • :~ "-""~~-":~ I
1~ lQ 1M I. 146 l~ 162 lW I~ lY 100
lime
Figure 4.22: Steering guide response due to pulse di~turbance
out to show that when the carriage moves and the web span length starts increasing. without
a proper tension adjustment will result in sag in the web spans. It is also desirable that the
web wrap around the pre-entering roller and the guide roller be enough so that the web will
not slip laterally as the steering guide makes corrections. When the sag occurs, the result is
that the wrap angle at the guide roller and the fixed entering roller will reduce. 111e web may
slip laterally at the steering guide roller when making corrections. TIle web must not slip on
the guide roller if the guide is to positi on the web. There must be sufficient frict ion between
the guide rollers and the web such that when the guide moves, the web is transported with
the rollers. Guiding accuracy will be influenced if the tension in the web is not sufficient
to prevent slippage of the web on the guide rollers. Second, sjmulations were carried out
to investigate the steering guide response with variable entering span length. The moving
carriage and support rollers' action are taken into consideration in the simulations.
52
CHAPTERS
SUMMARY AND FUTURE RESEARCH
5.1 Summary
The identification results in chapter 3 shows that the lateral dynamics of a moving web
between two fixed rollers is a second-order system. The estimated model obtained from
the experiments and the theoretical model seem to follow the same pattern of the input
signal generated by the offset pivoted guide.
Guiding with variable span length requires additional attention due to sag in the web
spans and lateral slip at the guide roller. Carriage motion and support rollers' action will
vary the guide dynamics and introduce disturbances into [he lateral web guiding system.
Sagging of a web may cause the wrap angle at the guide roller to become small and result
in lateral slip as the guide roller makes a correction. Without sufficient t~nsion in the weh
process line, the web span will sag. The web guiding response with variable c:::ntering web
span length is simulated with the PI controller.
5.2 Future Research
Future research should focus on ways to incorporate the sag angle into the guide dynamics.
Further study on different roller setup such as additional pre-entering roller or different
kinds of guide needs to be undertaken. Detailed study and modelling on carriage motion
and support rollers' action will also be a challenging task for the future. Future research
should also focus on experimental implementation on web guiding system with varying
entering span length. Current experimental setup is not amenable to implement varying
53
entering span length. Different guides like offset pivoted guide need to be addressed.
There is a need to conduct experiments on a processing line with a well designed ex.perimental
procedure. This can better validate the concepts developed and provide direction
to iterate on these concepts to better model and control the dynamic behavior. Further investigation
of the effect of different kinds of controller on the lateral control of web with
variable span length needs to be undertaken in the future. An active dancer can be taken
into consideration to compensate the sagging web spans or a different kind of setup configuration
will resolve the situation.
54
BIBLIOGRAPHY
[J) D. P. Campbell, Dynamic Behavior of the Production Process, Process Dynamic:.~.
New York: John Wiley and Sons, Inc., led., 1958.
[2] J. 1. Shelton, wteral Dynamics ofa Moving Web. PhD thesis, Oklahoma State University,
1968.
[3] J.1. Shelton and K. N. Reid. "uteral dynamics of a real movi ng web," ASME Journal
of Dynamics Systems, Measurement, and Control, vol. 93, no. 3. pp. 18~ 192, 1971.
[4] G. E. Young and K. N. Reid, "Lateral and longitudinal dynamic behavior and control
of moving webs," ASME Journal of Dynamic Systems. Measurement, and Control.
vol. J 15, pp. 309-317, June 1993.
[5] G. E. Young. J. J. Shelton, and C. Kardamilas, "Model ing and control of multiple web
spans usi ng state estimation," ASME Journal ofDynamic Systems, Measurement, and
Control. vol. Jll, pp. 505-510, September 1989.
[6] D. Abdul-AI-Nadi. "ARMA order determination," Master's thesis. Ok lahoma State
University. July 1991.
[7] M. Hagan. H. Demuth. and M. Beale, Neural Network Desi!?n. Boston, MA: PWS
Publishing. 1996.
[8] S. Timoshenko and D. Young, Engineering Mechanics. New York: Mcgraw-Hill.
3 ed., 1951.
[9] L. Ljung, S)stem Identification: Theory for the User. New Jersey: Prentice Hall.
1990.
55
[ 10) P. R. Pagi Ha. R. V. Dwivedula, Y. Zhu, and L. P. Perera, "Periodic tension disturbance
attenuation in web process lines uSing active dancers," ASME Journal of Dynamic
Systems, Measurement, and Control, to appear.
56
APPENDIX A
LATERAL DYNAMICS
A.I Assumptions
Lateral control of a web considers maintaining lateral position of the web before entering
the web processing line. Mathematical modelling of lateral web dynamics was introduced
in [1 l, which is derived under the assumption that the web behaves Iike a string. Then the
first study of the lateral dynamics of a moving web was done in [2], which modelled lateral
dynamics of web with different types of guides. The following are assumed while deriving
the lateral dynamics of a web.
• Standard beam theory assumptions:
- The web is initially straight and uniform,
- All deAections are small with the maximum correction angle is 50 for displace-ment
guide and 2° for steering guide.
• Area of contact between the web and the roller is small compared to the length of the
web span.
• Friction between the web and the roller is sufficient to prevent slippage and moment
transfer, so that a moment in one span is isolated from the upstream span.
• Stress distribution in the nonnal section of web is linear.
57
T MU
Figure A.J: Normal section of web
A.2 Static Behavior
An accurate understanding of the lateral static behavior of web is essential due to the eri tical
conditions under which web lateral deflection occurs. To obtain the static behavior, which
is given by the curvature equation of a real moving web, at least three factors should be
considered. They are bending effects, shear effects, and velocity effects, The hasic idea
towards understanding this problem is:
• selection of a section of moving web and setting up the force balance equations,
• building up the relationship between deflection and force based on Timoshenko bt:a.JTl
theory.
Figure A.2 shows the force balance of a section of a web. The curvature equation of a
moving web can be derived from the force balance of a section of web given by
[
dY] v2pWt dy
NL - N + T 8L - dx - 386 dx = 0, (A.I)
where NL is the shearing force exerted on web. Based on beam theory, the shearing force,
N, is given by
58
(A.2)
~ INERTIAL LOADING
T
Figure A.2: Force balance of a section of web
Differentiating both sides of Equation A.I and substituting Equation A.2. the curvature
equation of a moving web can be expressed as
where
The general solution 10 Equation A.3 can be obtained as
(A.)
(AA)
where the conSlant coefficients CI ,C2,C3, and C4 are obtained using the boundary condi-tions.
Considering the mOSl general case. which combines the effect of translation and
rotation of web, the boundary conditions are given as follows:
ylx=o = YO,
dyl I - =Yo
dx x=o '
ylx=L = YL,
dvl I .....::... =vL'
dx x=L -
59
The coefficients C" e2, C3, and C4 under the above conditions are
CI = 8L(coshKL - I) + 8o(KLsinhKL - coshKL+ J) - (YL - Yo)KsinhKL
K[KLsinhKL- 2(coshKL- 1)1
C _ 8dKL - sinhKL) +8o(sinhKL - coshKL) + (YL - yo)K(coshKL - 1)
2- K[KLsinhKL-2(coshKL-l)] ,
C3 = eo-clK,
A.3 Dynamics of a Real Moving Web
Considering the deflection of a real moving web, a second-order transfer function of the
web,
(A.5)
where d2z/dt 2 is the lateral acceleration of the web guide, which is zero for a fixed roller
and end-pivoted guide. v is the transport speed of the web, and
d
2
dxy I 2 x=L = K2
[CI sinh(KL) +C2 cosh (KL)]
I 1 1
= L2/1 (KL)(yo - yL,) + "i,h(KL)8L+ L!1(KL)80 , (A.6)
where
(KLf(cosh(KL) - 1)
II (KL) = [KLsinh(KL) _ 2(cosh(KL) _ I))'
KL(KLcosh(KL) - sinh(KL)
h(KL) = [KLsinh(KL) - 2(cosh(KL) - I)]'
f
KL(sinhKL- KL)
3(KL) = .,....-----------[
KLsinh(KL) - 2(cosh(KL) - 1)]
Applying Laplace transfonns on both sides of Equation A.6, a second-order transfer fune-tlon
of a real moving web under general boundary conditions can be obtained. Lateral
dynamics for different types of web guiding are given in the following sections.
60
A.3.1 Fixed Roller
Figure A.3 shown a schematic of a set of fixed roller in a web process line. For a fixed
x
~'_1 ------,r----I- ,---..- _
l
!!.o.t _
--f-------l-------~--------~------
--._--
--~---:=.-:=:=:~._.
--~-----------I-:T~~~=~--'0---- ---
Y,
Figure A.3: Response at fixed roller
roller shown in Figure A.3, the transfer function is
(
!J(~L) ) ( _J3(KL)s+ fI(KL) )
YL(s) = ') ~ ~ 80(s) + ~ ~ YcJ(s)
s~ + 2 S+ I s2 + 2 S+ I
!!O !.
(A.7)
where 't = L/v is the transport lag.
A.3.2 Remotely Pivoted Steering Guide
For a displacement guide shown Figure A.4, the transfer function can be expressed as,
(
~ ) ( _h(KL)s+~ )
fL(s) = 2 h(KL~ ~ 80(s) + 2 ~ ~(KL) Yo(s)
s + -.-s+ • s + • s+ ~
(
S2 + h(KL) S+ LhJKL ))
+ s2+~s+~ 2(s)
(A.8)
YL(s) = G] (s)2(s) +G2(s)80(s) +G3(S)YO(S) (A.9)
61
Instant Center
L
v
z
Original
Center
Figure A.4: Response at remotely pivoted steering guide
A.3.3 Displacement Guide
rw
/TAxesot
Rotation
i i Guide Roller
~~~~
r----~---~~-._._._. . ._.L.t
~ ~ i-I
Pivot
I
D, = Entering Span
Pivot Carrier
----------1-- ,¥ Sensor
Fixed
Exiting
Roller
D
2
= Exiting Span
Figure A.S: Offset Pivoted Guide
62
For a displacement guide shown in Figure A.5, the transfer function can be expressed
by
(A.IO)
where, t3 = L3/u, t = Llu
Lateral web guiding system play an important role in any web handling process system.
63
VITA
GIMKHUAN NG
Candidate for the Degree of
Master of Science
Thesis: LATERAL DYNAMICS AND CONTROL OF A WEB IN A HORIZONTAL
LOOPER
Major Field: Mechanical Engineering
Biographical:
Personal Dat.a: Born in Penang, Malaysia. on November 15, 1976, the son of Chin
Huat Ng and Kim Eng Teh.
Education: Transferred from Rima College. Penang. Malaysia; received the B.S.
degree from Oklahoma State University I Stillwater. Oklahoma. in December
1999. in Mechanical Engineeri ng. Completed the requi rements ror the Master
of Science degree with a major in Mechanical Engineering at Oklahoma State
University in May. 2003.
Experience: Research Assistant at Oklahoma Slate University from 2000 to present;
Technician at Web Handling Research Center, Stillwater, Oklahoma, from 20002002;
Laboratory Technician at Food and Agricultural Product Research and
Technology Center, Stillwater, Oklahoma, from 2002 to present.
Professional Memberships: American Society of Mechanical Engineers.