DYNAMIC ANALYSIS AND CONTROL OF A
ROLL BENDING PROCESS
by
MICHAEL BRUCE HALE
'B.S., Oklahoma State University
(1983)
SUBMITTED
TO THE DEPARTMENT
OF
MECHANICAL ENGINEERING
IN PARTIAL FULFILLMENT OF THE
REQUIREMENTS OF THE DEGREE OF
MASTER OF SCIENCE IN
MECHANICAL ENGINEERING
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 1985
(E
Massachusetts
Institute
of Technology
1985
The author hereby grants to M.I.T. permission to reproduce and
distribute copies of this thesis document in whole or in part.
Signature of Author
,
,
_-I
%,v
, I
,
v
-
Department of Mechanical Engineering
June 3, 1985
Certified
by
David E. Hardt
Thesis Supervisor
Accepted
by
M~A4A
Ain A. Sonin
TI&partment Graduate Committee
OFTECHNOLOGY
JUL 2 2 1985
LIBRARIES
Archives
DYNAMIC ANALYSIS
AND CONTROL OF A
ROLL BENDING PROCESS
by
MICHAEL BRUCE HALE
Submitted to the Department of Mechanical Engineering
on June 3, 1985 in partial fulfillment of the
requirements for the Degree of Master of Science in
Mechanical Engineering
ABSTRACT
can
Recent research has shown that the roll bending
be automated by the addition of a closed-loop
process
control
system that continuously measures the springback of the metal
of a roll
In the present work the components
workpiece.
bending system, controlled by such a closed-loop scheme, are
analyzed to determine important dynamic characteristics of
Dynamic models are developed for
the roll bending process.
the individual components and for the roll bending process as
a whole. These models are verified using an experimental
A linear control analysis and a
roll bending apparatus.
nonlinear simulation are performed, using the system model,
to determine
The analysis
the limits of the roll bending system response.
shows that very good system response is possible
using a simple proportional controller and proportional-plusThe analysis also indicates that the
feedback.
derivative
derivative feedback, which for roll bending is the rate of
change of unloaded curvature, can be approximated by the rate
of change of the control variable, roll velocity. Experiments were performed which verified the control analysis.
The experiments
also show that workpiece
vibration
is a major
problem as the roll bending system bandwidth is increased
because the control system is unable to distinguish between
workpiece vibration and curvature disturbances. Because the
workpiece dynamics change as the workpiece moves through the
roll bending apparatus, the control system must be designed
for the worst
case.
Workpiece
vibration
is the major
which limits the roll bending system response.
the proposed
the control
control scheme represents a major improvement in
because
of the
system
of the roll bending
increased bandwidth and stability possible.
Thesis
factor
Nevertheless,
Supervisor:
Dr. David E. Hardt
Title: Assistant Professor of Mechanical Engineering
2
ACKNOWLEDGMENTS
I would
guidance
spent
like
and
to
support
in discussion
thank
Professor
throughout
with
David
this
Professor
Hardt
research.
Hardt
for
The
resulted
his
time
in many
valuable insights.
I
would
America for
also
their
like
to
thank
financial
the
Aluminum
support and
Company
of
technical assis-
tance.
Thanks
also
to James
Owen
who
generously
examine the first draft of this document.
and helpful
suggestions
have
made
the
consented
to
His careful review
final
version
a much
more readable and incisive manuscript.
Finally,
a
special
and support and prayers.
thanks
to my family
for
their
love
Thanks especially to Patti whose
companionship remains a comfort and an inspiration.
3
TABLE OF CONTENTS
Page No.
Title
Page
1
Abstract
2
Acknowledgments
3
Table of Contents
4
List of Figures
6
Chapter 1
Chapter 2
Chapter 3
IChapter
4
INTRODUCTION
10
Motivation
10
Previous Research
11
Thesis Overview
15
STATIC ANALYSIS OF BENDING MECHANICS
17
Moment-Curvature Relationship
17
Straightening
23
Unsymmetrical Sections
26
Bidirectional Bending
32
DYNAMIC ANALYSIS AND MODELING
34
Experimental Apparatus
35
Workpiece Model
38
Servo Model
45
Measurement and Filter Model
49
Disturbance Model
49
Controller Model
53
CONTROL
55
Control Objectives
56
Linear Control Analysis
60
4
Chapter
Nonlinear Control Analysis
73
EXPERIMENTATION
79
Experimental Objectives
79
Experiments
83
Step Tests
84
5
Chapter 6
Disturbance Tests
109
CONCLUSIONS
115
Future Research
118
122
Bibliography
Appendix 1
EXPERIMENTAL BENDING APPARATUS
125
Appendix 2
MEASUREMENT ALTERNATIVES
141
Loaded Curvature Measurement
142
Moment Measurement
149
Bending Stiffness Measurement
152
Appendix 3
ERROR ANALYSIS
155
Appendix 4
COMPUTER PROGRAMS
159
Bending Control Program
159
Modeling Program
169
Step and Frequency Response Program
174
5
LIST OF FIGURES
Page No.
Figure No.
1
Pyramid 3-Roll Bender Configuration
12
2
Stress
State in a Loaded Beam
18
3
Moment-Curvature Relationship
4
Momient vs. Workpiece
5
Roll Bender Configuration
20
6
Closed-Loop Control Block Diagram
24
7
Shifted Moment-Curvature Diagram
24
8
Triangular
27
9
K
10
Experimental Roll Bending Apparatus
36
21
Roll Bending System Block Diagram
36
12
Elastic-Perfectly-Plastic Stress-Strain
38
13
1/8" X 1" Aluminum
Workpiece
Model
41
14
1/4" X 1" Aluminum
Workpiece
Model
41
15
Workpiece Hysteresis
43
16
Hysteresis Test
43
17
Velocity Servo Step Response
47
18
Velocity Servo Bode Diagram
47
19
Position
Servo Step Response
48
20
Position Servo Bode Diagram
48
21
Filter Bode Diagram
50
22
Disturbance Model Block Diagram
51
23
Disturbance Detection
51
u
Position
20
Rod
vs. Center Roll Position
6
18
33
24
Discretized Control Output
25
Disturbance
26
Position-Servo
Based Block Diagram
65
27
Velocity-Servo
Based Block Diagram
65
28
Position-Servo
Based Root Locus
66
29
Velocity-Servo
Based Root Locus
66
30
Velocity-Servo
Based Z-Plane Root Locus
69
31
Z-Plane
32
Block Diagram
54
62
Root Locus with a Zero
69
Simulated
System Step Response
77
33
Simulated
System Step Response
78
34
Simulated System Disturbance Response
78
35
Test
#1
Plot
#1
85
36
Test
#1
Plot
#2
85
37
Test
#1
Plot
#3
86
38
Test
#1
Plot
#4
86
39
Test
#2
90
40
Test
#3
92
41
Test
#4
92
42
Test
#5
96
43
Test
#6
96
44
Test
#7
98
45
Test
#8
98
46
Test
#9
Plot
#1
99
47
Test
#9
Plot
#2
99
48
Test #10
100
49
Test #11
103
7
50
Test
#12
Plot
#1
103
51
Test
#12
Plot
#2
104
52
Test
#13
107
53
Test
#14
107
54
Test
#15
Plot
#1
108
55
Test
#15
Plot
#2
108
56
Test
#16
Plot
#1
112
57
Test
#16
Plot
#2
112
58
Test
#17
Plot
#1
113
59
Test
#17
Plot
#2
113
60
Test
#18
Plot
#1
114
61
Test
#18
Plot
#2
114
62
Roll Bender Hardware Relationships
127
63
Outer Roll-Pair Rotation
128
64
Center-Roll
129
65
LVDT Curvature Measurement Device
131
66
LVDT Noise
132
67
Potentiometer
68
Force Transducer
69
F
70
F
x
Housing
Noise
and Potentiometer
132
134
Noise
135
Noise
135
y
71
Roll Bender Geometry
136
72
Curvature Measurement Scheme
137
73
Translated Moment-Curvature Origin
143
74
LVDT Configurations
143
75
Opposing LVDT Curvature Measurement
147
8
76
Measurement
77
Bending Program Flow Chart
Errors
9
157
164
Chapter
1
INTRODUCTION
Motivation
In
an
effort
competitive
to
increase
in the world
market,
productivity
many
companies
ering production process automation.
and
remain
are
consid-
In the metal forming
industry much interest has been generated by the introduction
of
for
automation
brakeforming
automation
position
and
traditionally
roll
for these
bending.
processes
a
operator
to
position
program
The
was
series
of
first
improve
productivity,
generation
of
of a simple
for
On a brakepress,
(servo)
movements
enabled
which
the
could
be
This did not signi-
repeated very quickly and precisely.
ficantly
as
the addition
servomechanism
a
such
operations
and controller.
servomechanism
example,
manual
however,
because
of
the
extensive reworking needed to achieve the necessary shape
accuracy.
controlled
material
Allison
Even
very
property
and
though
the
closely,
final
variations.
Gossard
brakepress
position
part shape
varied
could
be
because
of
More recent work by Hardt
[2], and
Stelson
[3]
has
shown
[1],
that
a
material adaptive control scheme can be developed for the
brakeforming
process
which
will
significantly
improve
productivity by explicitly accounting for material properties
in-process.
A
variation
of
the
brakeforming
10
process,
the
roll
bending process, is used in many metal forming situations
that fit within
bending
the particular
process,
where
hardware
applicable,
restraints.
is much
The roll
faster
than
the
brakeforming process because rather than forming the material
in
discrete
steps,
as
in
brakeforming,
the
workpiece
is
formed continuously as the material is rolled through the
machine.
The advantages of the continuous forming provide
the
motivation
The
roll
cient
bending
the
speed
process
of the
has
presented
great
metal forming
was to develop
roll bending
research
operation
and versatile
research
ent
for
in
potential
process.
an automatic
bender
while
thesis.
as an effi-
The goal of this
control
scheme
that would take advantage
roll
this
for the
of the inher-
incorporating
a closed-
loop control scheme.
Previous Research
The three-roll
a
typical
outer
pyramid
configuration
rolls
roll bender
that
and a movable
center
rolls is driven, and friction
piece
permits
the material
As the workpiece
duce a variable
process
plate
also
is used
moves,
roll.
of
a
One
pair
of
or more
extensively
and
fixed
between the rolls and the work-
to be rolled
the center
to form
1 is
of the
through
the machine.
roll is adjusted
to pro-
bend along the length of the workpiece.
used in boilers
used
consists
shown in Figure
long
flat
reactor
to form
11
workpieces
vessels.
long
thin
This
such as heavy
Roll
bending
workpieces
such
is
as
Final
Shape
Workpiece
Movable
Center
Roller
4
Fixed
Fixed
Outfeed
Roller
Rollnfeed
Roller
Figure 1.
Pyramid 3-Roll Bender Configuration
aluminum extrusions.
is also
of
used for
extrusions.
accuracy
In the aluminum industry roll bending
final
This
straightening
operation
is
and contour
entirely
correction
manual
of the final product shape is dependent
and
the
on the skill
of the operator.
In
recent
automating
the
confronting
roll bending,
years
some
effort
roll
bending process.
researchers has
the metal
has
been
workpiece
been
directed
The
major
toward
obstacle
material springback.
is plastically
the workpiece moves through the machine.
deformed
In
as
As the workpiece
exits the bending apparatus the elastic stresses in the metal
are
relaxed
and
the
metal
"springs
12
back".
Thus
the
metal
does
not
exited
with
obtain
the
its
machine.
the material
[5])
final
used
Some
shape
of
springback
empirical
data
until
the
the earliest
problem
to
various
attempts
(Sachs
analytical
has
to deal
[4] and
characterize
certain materials and workpiece shapes.
([6] to [14]) used
workpiece
Shanley
springback
Other
methods
for
researchers
in an attempt
to predict stress and strain conditions as well as springback
for
various
Later,
materials.
Hansen
and
[15]
Jannerup
developed a more complex model for elastic-plastic bending of
beams
that
could
of
estimate
Trostmann,
be used
to obtain
the workpiece
[16]
'ised the
a
more
Cook,
springback.
beam
model
nearly
in
[15]
accurate
Hansen,
to
and
design
an
open-loop controller for a roll bending machine and demonstrated that good control of the final part shape is possible
if the material properties, including springback, are known
beforehand.
The condition of prior knowledge of material
properties is very restrictive, though, and seriously limits
the
usefulness
machine which
exits
the
of the controller.
measures
bending
the final
Foster
[17]
developed
part shape as the workpiece
apparatus.
This
measurement
can
incorporated into a closed-loop curvature controller.
advantage
of
this
type
a
of control
scheme
is
that
no
be
The
prior
knowledge of material properties is required for controller
implementation.
The curvature measurement, though, is taken
after the workpiece has exited from the bending apparatus.
At this
point
the workpiece
already
13
has
been
formed
to its
final shape
current
so the information
error,
but
final curvature.
[18],
in
research
by Gossard
has
be used to correct
to maintain
a
relatively
the
constant
More recent research by Hardt, Roberts, and
Stelson
forming,
only
cannot
the
area
of
and Stelson
shown
that
it
roll
bending,
[19],
in the area
is
possible
to
and
similar
of brake-
measure
the
important material properties, including springback, during
forming
and
processes.
have
thus
design
Hale
and
extended
the
a closed-loop
Hardt
[20],
approach
straightening process and
show that with
in
controller
and
Lee
[18]
to
and
have conducted
scheme
these
Stelson
include
the
[21]
roll
experiments that
a closed-loop controller, straightening is
nothing more than bending to zero curvature.
control
for
presented
in [18] works
well
Although the
in theory,
the
productivity gains possible with this closed-loop curvature
controller
are
limited
by
feedrate will be very slow.
avoid unwanted
the
feedrates
oscillation
are kept
the
assumption
that
workpiece
This assumption was necessary to
and instability.
below 0.7
in/sec.
In [18] and [20]
In [21] the work-
piece was actually stepped through the bending device and the
forming was performed while the workpiece was stationary.
This closed-loop control method enhances the versatility of
the roll bending process by making one-pass forming of arbitrary
shapes
speed
advantages
the possible
possible,
but it does
of the
productivity
not
roll bending
exploit
process.
gains are lost.
14
the inherent
Thus
much
of
Thesis Overview
The work described in this thesis presents a new control
method
that
response.
roll
greatly
In Chapter
bending
improves
the
2, a complete
process
is
bending cases that merit
roll
bending
system
static
analysis
of the
presented.
In
addition,
three
special attention: straightening,
forming unsymmetrical sections,
and bidirectional bending,
are
determine
examined
approach
cases.
in some
presented
components
are
to
in [18] must be modified
In Chapter
into five primary
detail
3 the roll bending
components.
developed
Models
to
how
using
described
briefly
system
is broken
down
of each
of these
five
determine
their
1 and
[22].
individual
These models are
an experimental bending apparatus
in Chapter
In Chapter
4
3 and in more detail
the
control
to apply to these
influence on system dynamics and response.
verified
the
dynamic
models
that
is
in Appendix
are
used
to
develop a digital controller that satisfies specific control
objectives.
tives
listed
bending,
The relative importance of the control objecin Chapter
so the control
4 varies
analysis
ral terms and then examined
using
for specific types of bending.
according
to the
is presented
first
the experimental
type
of
in geneapparatus
This analysis indicates that
the roll bending system bandwidth can be greatly increased by
a simple modification to the control
[18].
system presented
in
Using a proportional-plus-integral controller with a
15
velocity servo results in a much simplified dynamic system
with good stability and improved bandwidth.
The new control
method
response
presented
maintained
at
productivity
procedures
results
the
much
gains
and
verify
new
allows
the
greater
for
very
results
are
the models
control
method
same
system
feedrates,
little
cost.
presented
developed
proposed
resulting
The
in
in
in
Chapter
be
large
experimental
Chapter
in Chapter
to
5.
The
3 as well
4.
Chapter
as
6
contains the conclusions and also some suggestions for future
research.
The
major
conclusion
is
that
there
is
a
rather
severe limitation imposed on maximum system bandwidth because
of workpiece vibration.
More research is needed to charac-
terize completely the effect of the workpiece vibration on
the
final
curvature
detailed
study
insight
into
of the
workpiece.
of the mechanics
the
problems
and
In addition,
of bending
could
possibilities
a more
yield
of
two-dimensional and three-dimensional bending.
some
one-pass
The
appen-
dices contain more detailed information on the experimental
hardware
and software
as well as an error
analysis
ware concerns and measurement alternatives.
16
and hard-
Chapter
2
STATIC ANALYSIS OF BENDING MECHANICS
The
closed-loop
curvature
roll bending process
control
scheme
and the brakeforming
for
both
the
process is based on
real-time measurement of the material springback.
The devel-
opment of this control scheme is repeated below.
Moment-Curvature Relationship
A
work?iece
modeled
Figure
as
2.
stressed
stress
when
a
a
beam
beam
regain
If the
is always
is unloaded,
its
original
bending
three-point
is loaded,
elastically.
in the
three-roll
under
As the beam
the beam
and
in
machine
loading
the material
beam
is
loaded
below
the
yield
the
material
shape.
as
If,
will
however,
can
be
shown
in
is initially
so
that
stress,
"spring
the
the
then
back"
beam
is
loaded so that some of the fibers are loaded past the elastic
limit,
then
the
beam
is
permanently
deformed
stress
in a beam
The
state
final workpiece
initial
stress
when
unloaded.
that is loaded
shape
state,
plastically
and
depends
deformed
Figure
the
curvature
which
can
stress
state
2
on the initial
the stress
will
shows
past the yield
distribution,
function of position along the workpiece.
between
and
of a workpiece
be
the
point.
shape,
the
all as a
The relationship
and
the
resulting
can
be seen
in the moment-curvature
relationship,
be
derived
from
relationship.
the
17
stress-strain
_
·
_
zone
Stress State in a Loaded Beam
Figure 2.
C
M
c
E
B
0
dM
dK
D
A
KL
Figure 3.
Curvature
Moment-Curvature Relationship
18
Figure
3 is a general
tially
flat
workpiece.
This
along
the length
of the
point
is needed
for
completely
every
the
moment-curvature
point
workpiece
greatly simplified
diagram
diagram
is drawn
workpiece.
curvature.
for
A similar
on the workpiece
by considering
for
The
an inia single
diagram
to characterize
situation
the geometry
can
be
of the three-
roll bending process.
For a workpiece
that is loaded in a roll bending appara-
tus, the exact moment distribution along the sheet is complex
but can be approximated
assuming
that no moment
and the workpiece
as
a linear
function
is generated
(Figure
4).
between
of arc length
the drive
As indicated
for
rolls
three-point
bending, the bending moment applied to the workpiece increases from zero
at the
roll
point
roll.
contact
This
input
and then
loading
curvature diagram
roll
roll the workpiece
flat
workpiece,
curvature
until the yield
can
(Figure 3),
at the
to zero at
be traced
on
diagram
curvature
and
the
the
moment-
diagram
(Figure 5). At the input
(Point
workpiece
point is reached
center-
the output
the moment-position
has zero moment and, assuming
zero
increase
decreases
sequence
(Figure 4), and on a machine
to a maximum
A).
The
deforms
(Point B).
an initially
moment
and
elastically
The slope of the
elastic loading line in Figure 3 is the effective bending
stiffness.
at Point
As the moment
C
the
sheet
increases
deforms
from Point
plastically.
B to a maximum
The
moment
and
curvature decrease linearly as the sheet moves from maximum
19
Mmax
C
B
,,4-
c
E
o
d2
Position
Figure 4.
Moment vs. Workpiece Position
Workpiece
Movemenf
Figure 5.
inch
Roller
Roll Bender Configuration
20
loading
(Point
moment
to
the
output
zero
but
the
C)
is again
(Point
curvature
plastically
is not
because
the
slope
of
the
unloading
line in Figure 3 is the same as the elastic loading
springback
The
the
has
The workpiece
deformed.
where
D)
workpiece
line.
been
roll
as a function
of distance,
s,
along the workpiece is the difference between the maximum
loaded
curvature
and
the
unloaded
final
curvature
and
is
found from Figure 3 to be:
= KL(s) -
AK(s)
K
(s)
(1)
U
where
KL
unloaded
is
the
maximum
curvature.
The springback
terms of the moment.
As shown
function
is a linear
loaded
L
curvature
K
is
~~~~~~~~u
can also be expressed
in Figure
of the moment
and
3 the unloading
and so the springback
the
in
path
can
be expressed:
AK(s)
where
M(s)
dM/dK
is
is
the
the
distribution
moment
slope
of
(2)
= M(s)/(dM/dK)
the
elastic
along
loading
equation
points out one of the major advantages
bending
process
has
over
the
brakeforming
the
sheet
line.
and
This
that the roll
process.
For
brakeforming, where the workpiece is formed at discrete loca-
21
tions,
it is necessary
distribution
along
the
predict
accurately
forming,
however,
controlling
the
workpiece
in the
the
distributed
every
to the maximum moment
by
to obtain
full
moment-curvature
forming
region
springback.
point along
For
the workpiece
moment,
each
roll
is loaded
as it passes under the center
the maximum
to
roll, and
point
along
the
and
the
workpiece can be formed individually.
As
shown
springback
curvature
in
are
[22],
known
if
for
the
a given
is found by rearranging
K
This can be combined
u
loaded
= KL
L
curvature
point,
then
Equation
the
unloaded
1:
- AK
(3)
O
with Equation
2 to yield:
(4)
Ku = KL - Mmax/(dM/dK)
where
point
it is understood
along
possible
piece
to
while
the
that these two equations
workpiece.
calculate
the
the workpiece
Equation
unloaded
4
shows
curvature
is in the loaded
apply to each
that
of
it
the
condition
is
workif the
moment and curvature at the contact point under the center
roller are known together with the bending stiffness of the
workpiece.
In
other
words
it
22
is
possible
to
measure
the
material
springback
in real time.
Because the measurement
made at the center roll where the forming occurs,
measurement
presented
lag as there is with post-forming
in [17].
using the unloaded
A closed-loop
curvature
there is no
measurements
controller
as the loop feedback.
scheme is presented
6.
the
springback
shapes
based
on
has many advantages
measurement
primary
system
or springback
advantage
is
is possible
over
real-time
systems
prediction,
that
shown
based
forming
because the controller
curvature
in Figure
servo.
yield point, which
even
and
along
the
curvature
cally.
Roberts,
6 can be thought
Changes
in
of
material
workpiece,
measurements
This was demonstrated
and Stelson
are
and
on delayed
properties
arbitrary
Thus the
such
to workpiece
compensated
static
in [18] and Roberts
The
of as a closed-loop
reflected
for the
of
can use the actual
can differ from workpiece
same
in Figure
measurement
system output, unloaded curvature, for feedback.
system
A general
such as in [16].
one-pass
as
can be designed
block diagram of such a control
A control
is
in
as
and
the moment
for
automati-
case
by Hardt,
in [22].
Straightening
Another significant advantage of real-time measurement
of
unloaded
curvature
application.
The
effect
diagram
is
evident
in
the
straightening
Consider a workpiece that is initially curved.
of the
is shown
initial
in Figure
curvature
7, which
23
on the moment-curvature
is drawn
for
a single
I
t
4F
Controller
iS
Workpiece
Rolls
Roll
Error
Desired
ervo
Work piece
II
I
I
I
- .
Ku Measured
I
KL Measured
±&
,&
K
I
I
Moment
Measured
d M/d K
Figure 6.
Closed-Loop Control Block Diagram
..-
E
0
Yield Point
I
Initial
I
Curvature
I
I
I
I
r
i
Figure 7.
U.
y
U
Curvuture
Shifted Moment-Curvature Diagram
24
-
point on the workpiece
tive.
The initial
curve.
where
the
curvature
initial
simply
curvature
is nega-
shifts the origin
of the
In the unloaded state (zero applied bending moment)
the loaded
curvature
to reflect
this initial
is no longer
zero and the curve
condition.
The
actual
shifts
shape
of the
nonlinear portion of the curve depends on the initial stress
state in the workpiece.
ing lines
does
not
The slope of the loading and unload-
change
though,
unaffected by initial stresses.
ler, initial
bance.
curvature
A model
in Chapter
3.
can
of this
so the
scheme
is
In the closed-loop control-
be described
disturbance
The closed-loop
control
as a system
is developed
control
scheme
disturbance and react to eliminate it.
distur-
more
will
fully
sense
the
Because the distur-
bance is measured under the center roll, the control system
can
react
the
closed-loop controller, straightening
than
while
bending
the
workpiece
to zero curvature
is still
loaded.
Thus,
is no
with
different
and can be accomplished
in a
one-pass operation without prior knowledge of the disturbance.
This is clearly not possible using predictive methods
of calculating the springback unless the disturbances are
fully known beforehand.
Prior knowledge of the disturbances
requires a separate operation and additional instrumentation,
and even then, multiple
controller based on
passes may
be required
because a
predicted springback is open-loop and
cannot compensate for material property variations.
25
Unsymmetrical Sections
For
a
symmetric
constant
workpiece
with
a
cross-sectional
about a given axis and material
along
pure moment
the workpiece
acting
in the
of
about
symmetry
deflection only, in the plane of symmetry,
are
very
number
restrictive
though,
of industrially
three-dimensional
and
relevant
bending
is
apply
cases.
much
that
properties
and symmetric
plane
area
that are
the axis, a
will
produce
a
These conditions
only
The
more
is
to
a
limited
general
case of
complex.
For
an
arbitrary cross-section with a bending moment applied in an
arbitrary
direction,
it
is
possible
for
the
workpiece
to
twist about the longitudinal axis and bend about two orthogonal
transverse
bending
and
axes.
the
In other
twisting
are
words,
all
coupled
the
two-dimensional
and
may
all occur
from a one-dimensional loading.
Consider
about
the
a
triangular
the two orthogonal
xy plane
is
related
rod
(Figure
transverse
to
the
axes.
loading
8)
that
The
is
loaded
deflection
by Equation
in
5 (see
[23] and [24]):
a MzI +M T YZ
as
where
is the
distance
along
local
E(I
beam
I
yy zz
angle
-
I
yz
2
in the
the beam, E is the modulus
26
(5)
xy plane,
s is the
of elasticity,
MM
Figure 8.
and
M
are
the
moments
I
and I
Rod
Triangular
applied
about
the
z
and
y
axes
y
respectively,
zz
yy
are the area moments
of inertia
is the area product
the z and y axes respectively,
and I
inertia for the y and z axes.
The deflection
yz
for
of
in the xz plane
is:
a
As
where
is the local
to see that a moment
- M I
-MzIY
E(I
I
I
yy zz
beam angle
applied
-
I
YY
yz
(6)
2)
in the xz plane.
It is easy
only in the z (or y) direction
27
will
produce
coupling
in two
from the product
dimensional
process
bending
because
controlled
general
a deflection
in
coupling
of inertia
term, Iyz .
of the
This multi-
yz
undesirable
for
the
roll
three
dimensions
must
be
measured
to
control
Therefore
effect
because
is
order
case.
directions
to
it
the
is
determine
final
important
how
it
shape
to
can
be
bending
and
for
the
analyze
the
prevented
or
reduced.
From Equations
uncoupled
when
principal
axes,
I
5 and 6 it is obvious
= 0.0.
yz
such
as
This
the
that the bending
is the case
axes
through
if y and z are
the
centroid
parallel to the edges of a rectangular section.
about
a principal
axis
will
tion.
For many sections
cannot
be met.
however,
Even if coupling
can be minimized.
involve
result
plastic
is
and
Thus bending
in an uncoupled
deforma-
this restrictive
is unavoidable,
condition
the effects
Because permanent changes in curvature
deformation,
if
a
moment
in
one
direction
produces plastic deformation in one direction and elastic
deformation
in
all
other
tively uncoupled.
direction,
curvature
the
bending
is effec-
Although bending occurs in more than one
plastic
changes,
directions,
deformation,
will
occur
and
in
one
therefore
direction
permanent
only.
The
following analysis demonstrates the effect of coupling for a
two-dimensional case.
Suppose
tially
that
flat and
the
workpiece
is loaded
shown
in a single
in Figure
direction
8
(M
Y
28
is
ini-
= 0.0).
Equation
Rewriting
6 gives:
-M
I
- I 2)(7
as E(II
Rewriting Equation 5 yields:
MzIzz
Da
Substituting
Equation
E(I
this
governed
equation
it
yy
Izz
zz
-
IZ
yz
8 into Equation
as
From
2
"
=
as
(8)
)
7 yields:
(as)(acz)'1(9)
as
\
to see
is easy
that
the
coupling
is
by the ratio of the product of inertia to the moment
of inertia.
the coupling
A similar
development
cn
be used to show
that
= 0.0 is given by:
that occurs when M
z
as
Now let K
xz
be the maximum
I
asIz
Iyy
elastic
29
loaded curvature
(10)
in the xz
plane and K
xy
xy
be the maximum
the
Then
plane.
elastic
loaded curvature
that
conditions
will
ensure
in the
uncoupled
bending are given by:
QCz)()as
Kxz(l
(11)
z/
in the xy plane and
for bending
(I)( as)
2xy
YY
for
bending
triangular
b =
in
the
cross section
1.25 in,
E =
Consider,
plane.
xz
10x106
and
psi
proportional
the
limit,
the proportional
fibers are
then:
zz
I
.4
=
0.186
in
=
0.095
in
= -0.066
in
.4
Yy
I
K
K
yz
xz
xy
the
is
If the maximum elastic loaded
outermost
I
limit
by the loaded curvature
can be approximated
workpiece when
example,
shown in Figure 8 where h = 1.75 in,
35x103 psi (Aluminum 6061-T6).
curvature
for
.4
= 0.004
in
= U.u03
in
30
stressed
of the
to
the
Thus
the
largest
that
curvatures
S --
straightening
keep
the
-
minimized
above.
curvature
is sure
coupling
by
By
(0.004) = 0.011
each
in
(13)
below
are
to
beam,
bending
the
very
occur.
careful
0.0043 in 1
(0.003)
--
the triangular
magnitudes
curvature
in
\ yz
as
to
formed
are:
as
For
be
the unloaded curvature in the
direction without affecting
other direction
can
the
application
first
For
in
because
bending,
though,
the
effect
coupling
of
the
direction
be possible
values
indicated
small.
But
it might
(14)
can
be
analysis
shown
which
least
is
affected by bending in the opposite direction, the required
number
method
of passes
used.
is
calculate,
using
underbend
needed
defeats
the
will
Better
yet,
Equations
to
purpose
if an iterative
be minimized
9
it
might
and
10,
compensate
of
the
for
control
be
possible
to
the
overbend
or
the
coupling.
control
closed-loop
This
scheme
however, because the coupling is estimated and not measured
explicitly.
If
the
closed-loop
curvature
could be applied to two or three dimensions
effects would
be measured
and compensated
31
control
scheme
then the coupling
for automatically,
much
like
initial
straightening.
curvature
disturbances
The work presented
one-dimensional
uncoupled
are
handled
in
here will be restricted
to
bending to simplify the
eperi-
mental apparatus.
Bidirectional Bending
To form arbitrary
workpiece
shapes
in a single
pass the
bending apparatus must be capable of forming both positive
and
negative
(bidirectional) curvatures.
more complex hardware,
(see
Appendix
1).
such as opposing
The
closed-loop
This
means
roll pairs,
control
is needed
scheme
applied to bidirectional bending with no changes.
that
can
be
The major
difference between unidirectional and bidirectional bending
is due
to the effect
of the elastic
deadband
3 show that there
tion 4 together with Figure
region.
Equa-
is no change in
the unloaded curvature while the workpiece is loaded elastically.
system
Thus for a certain
output
does
not
range
change.
called the deadband region.
of center-roll
This
range
movement
of
movement
the
is
Figure 9 shows the relationship
between center-roll displacement and unloaded curvature for a
workpiece that is stationary in the bending apparatus.
The
points noted in Figure 9 correspond to the loading conditions
shown
piece.
the
in Figure
Notice
complete
3 for the special
that a large
unloading
deadband region.
path
case of a stationary
portion
shown
of the loading
in Figure
3 are
work-
path
and
in
the
For unidirectional bending with no over-
32
shoot, the system passes through the deadband region only
once
at the
response
start
is
of
very
bending
small.
and
For
the
effect
on the
bidirectional
system
bending,
and
especially for straightening where curvature levels are very
small,
most
of the center-roll
band region.
movement
Thus it is possible
dominate the system response.
for static
may
be in the dead-
for the deadband
region
to
This is of little importance
bending but becomes
very important
for the dynamic
case where the workpiece moves through the roll bending apparatus.
The
stability
large.
dynamics
deadband
and
The
region
error,
effect
will
be
has
a
especially
of
the
large
as
deadband
explored
in
effect
the
feedrate
region
more
on
detail
on
in
system
becomes
the
the
chapter.
Ku
/
Deadband
T
_-mm~
I
D
/
-o
C~~~~
B
Center Roll
-. NPosition
=
'Deadband
/F
/~~~~~~~~~~~~
Figure 9.
P
K
U
vs. Center-Roll
33
Position
system
next
Chapter
3
DYNAMIC ANALYSIS AND MODELING
The
research
presented
in
[18],
[20],
[21],
and
[22]
demonstrated that the closed-loop control scheme described in
Chapter
2 works
the workpiece
The
is rolled
assumption
ignore
any
workpiece.
very well
of very
for bending
through
low
associated
In
chapter
a
straightening
apparatus
feedrate
dynamics
this
the
and
allowed
with
the
model
of
very
slowly.
the
authors
to
servo
system
or
the
roll
bending
system that includes dynamic effects will be developed.
model will
be used to analyze
roll
bending
Then
the
operation
dynamic
algorithm.
The
hardware
model
predict
will
be
system dynamics
tus will depend
and
and
the dynamic
used
performance
to develop
This
of the
limits.
a
control
for any roll bending
appara-
on the particular
under
performance
the
if
roll bending
consideration.
A
general
configuration
roll
bending
system must contain several key components to implement the
closed-loop controller.
nents
are presented
These
models
are
configuration
Chapter
to
5.
examine
The
the
in general
used
that
The dynamic models of these compoterms
to analyze
was
used
experimental
validity
to
perform
the
bending
dynamic
roll
bending
experiments
apparatus
models
develop and evaluate different controllers.
34
in this chapter.
a particular
roll
of the
later
and
in
is used
also
to
A short descrip-
tion
of
the
experimental
detailed
information
presented
in Appendix
apparatus
about
the
is
given
below.
experimental
More
hardware
is
1 and [22].
Experimental Apparatus
The experimental bending apparatus (Figure 10) consists
of two outer
Bridgeport
roll-pairs
milling
and a center
machine.
fixed
roll and an adjustable
ratus
is capable
All
of
opposing
of bidirectional
roll-pair
the
mounted
roll-pairs
roll.
bending
Thus
of
on a
have
a
the appa-
various
sized
workpieces.
The outer roll-pairs are fixed to the milling
machine
bed
which
screw.
The
movement
is
driven
by DC
of the outer
motors
rolls
through
with respect
center roll provides the forming action needed.
roll
is mounted
workpiece
opposing
in and
is clamped
center
driven
between
roll.
by the milling
the
Friction
driven
between
ballto the
The center
spindle.
center
the
a
roll and
rolls
and
The
the
the
workpiece causes the workpiece to move through the apparatus.
Feedrate
can be varied by adjusting
As shown in Equation
the spindle
4, the loaded curvature
speed.
and the maximum
moment must be measured to implement the closed-loop control
scheme.
Loaded curvature is measured on the experimental
apparatus by two linear variable differential transformers
(LVDT's)
which
distance
moment
are
between
is
mounted
the
measured
next
LVDT's
with
a
is
to
the
adjustable.
strain-gage
35
center
force
roll.
The
The
maximum
transducer.
Figure 10.
Experimental Roll Bending Apparatus
Ku
K1
+
Figure 11.
Roll Bending System Block Diagram
36
arm
the
to calculate
used
be
must
rotation
changes
of the roll housings
rotation
The
piece.
moment
maximum
Rotation
measured.
of
the moment
therefore
and
the
center-roll
the
The
potentiometer.
housing is measured with a
the work-
the rolls and
between
is generated
that no moment
are free to rotate so
housings
that all the roll-pair
Notice
outer-roll
rotations are negligible for the bending experiments reported
complete
are
5 and
in Chapter
not
analysis
error
to a computer
where
then uses this information
is sent to a General
controller
they
are
The
digitized.
an error
to generate
servo-controller.
Electric
is connected
before
and filtered
are amplified
signals
Analog
to the moment and curvature measurements.
sent
of neglecting
the effect
which shows
a
contains
Appendix 2 presents some alternatives
various measurements.
from all transducers
3
Appendix
measured.
computer
signal
The
to the DC motor that adjusts
being
which
servo-
the roll
position and the loop is completed.
Analysis
process
of
reveals
the
five
description
machine
components
system
and
that
contribute significantly to the system dynamics.
components
and
are the workpiece,
filters,
disturbances,
the servo
and
the
system,
system
the
bending
appear
to
These five
measurements
controller.
A
block diagram of the roll bending system which includes all
these
components
is shown in Figure
11.
A model
for each of
these components is developed below and verified using the
experimental apparatus described.
37
Workpiece Model
As
shown
anything
loaded
necessary
to measure
to know
each point along
path.
how
Chapter
2
it
is
not
about the plastic deformation
is being
only
in
the workpiece
purposes
deforms,
curvature.
moment
and
know
as it
It
curvature
is
at
and the slope of the unloading
however,
both
of the center-roll
model we will assume
to
of the workpiece
unloaded
maximum
the workpiece
For modeling
as a function
the
the
necessary
it is necessary
elastically
position.
that the workpiece
and
to know
plastically
For the workpiece
material
is elastic-
perfectly-plastic, which means that the stress-strain relationship
is as shown in Figure
12.
In the elastic
region the
0
I.
-
C')
Strain
Figure 12.
Elastic-Perfectly-Plastic Stress-Strain
38
moment and
curvature are
linearly related by
stiffness.
If we
assume
further
that
the
the bending
workpiece
has
a
rectangular cross section that is constant along the length,
then
the
relationship
between
moment
entire loading path can be described
M
=
and
curvature
for
the
as shown in [23] by:
K(dM/dK)
K
< K
Y
(15)
M
=
1.5M
(1
-
(K
y
where M
K
y
at yield.
Y
The curvature
a
K
are the moment and curvature
and K
Y
/K)2/3)
y
of the workpiece
linear function
of
roll
can also be expressed
position
using
as
the deflection
equation of a beam under three-point loading,
K = 3z/(L
where
z
is center-roll
on linear
tion well
equation
beyond
is
not
bending stiffness
beam
as shown
dependent
and
roll.
but
theory,
yield
(16)
displacement
between the center and outer
based
)
on
L
is
This
the
relationship is
is a very good
in Figures
material
distance
approxima-
13 and 14.
properties
or yield point and is therefore
This
such
as
very useful
for a general model.
Substituting
Equations
15
39
and
16
into
4
yields
the
following relationship between unloaded curvature and centerroll displacement (assuming an initially flat workpiece):
K
0.0
=
z
<
z
y
u =°°
(17)
K
u
where
zy is the
L2
workpiece
(Point
C
a
L2z2
y
position
center-roll
(unloaded curvature).
the
3
2
the input
relates
9zz
Y +
-
at the
yield
point
of
Equation 17 is the desired workpiece model
the workpiece.
that
3z
-
(center-roll
to the output
position)
This equation applies to the point on
that
is
in Figure
5).
in
contact
As
shown
with
the
center
roller
2, this
in Chapter
point
corresponds to the maximum moment and curvature which means
that
the
point.
final
Applying
curvature
Equation
as it is rolled through
workpiece
as a function
Figures
of
the
workpiece
17 to each
the apparatus
point
is
at
this
on the workpiece
provides
of the distance
set
a model
of the
along the workpiece.
13 and 14 are plots of measured
moment
scaled by
the bending stiffness, loaded curvature, and unloaded curvature versus
roll position
for a 1/8" X 1" 2024 aluminum
and a 1/4" X 1" 2024 aluminum
15, 16, and
and yield
17 are also
Equations
strip, respectively.
plotted
using
the
bending
point obtained by experimentation.
40
strip
stiffness
These
plots
E-02
I0
14
12
:r
:z
U
10
to
N
8
w
cr,
M
6
z
I-
n3
U
4
2
0
-2
0
2
4
6
8
10
12
14
16
20
1
EE-01
CENTER ROLL POITION
Figure
13.
INDO
1/8" X 1" Aluminum
Workpiece
Model
E-03
_
__
__
____
____
o"n
Ou
70
60
x
U
50
z
-
40
w
30
20
10
to
0
I
_inlIU
0
1
2
3
5
4
6
7
8
CENTER ROLL POSITION (INCH)
Figure
14.
1/4" X 1" Aluminum
41
Workpiece
9
10
E-o01
Model
show that the workpiece
model developed
above
is a reasonable
approximation of the actual workpiece.
There
workpiece
are
three
model.
workpiece
is
deadband
region
ing
stiffness
the
elastic
very small.
region.
important
First,
features
there is a deadband
elastically
depends
loaded.
and
Flexible
The
note
For
therefore
workpieces
from
significance
very
the
the
region while the
on the type of bending
of the workpiece.
region,
to
the
and the bend-
stiff
deadband
have a much
of
workpieces
region,
is
larger deadband
Forming circular shapes or bending workpieces with
curvatures in only one direction involves passing through the
deadband
region only once.
For bidirectional
bending
or for
straightening, the deadband region may dominate the operating
region for a particular
workpiece.
The second important feature of the workpiece response
is that
the
response
has
two
different
forms
depending
on
whether the workpiece is stationary or moving through the
rolls.
For a stationary
is hysteresis
or very slow moving workpiece,
in the model
as shown
in Figure
15.
there
This
because the unloaded curvature does not
change while
workpiece
or during
ing.
of
is in the elastic
If the workpiece
the
workpiece
is
loading
state
is stationary,
formed
during
a
then the
forming
is
the
unload-
same portion
cycle.
For
a
moving workpiece, each point along the workpiece is formed
only
once
since
new
bending apparatus.
material
is
continually
fed
into
the
Figure 15 shows that a moving workpiece
42
Stationary
Moving
Workpiece
Workpiece
Ku
Position
Figure 15.
Workpiece Hysteresis
E-02
acl
7
8
5
x
-
4
3
.J
1y
m
xy
2
1
0
-1
-2
0
1
2
3
TIME
Figure 16.
(SEC)
Hysteresis Test
43
4
5
E*00
does
not
have
hysteresis
but
still
has
a
deadband
region.
Figure 16 shows the dramatic effects of the workpiece hysteresis.
and
This figure shows the loaded and unloaded curvatures
the
scaled
workpiece
to
a
for
such as the
step
region
moment
ne described
curvature
is
apparent
a stationary
command
in
the
of
1/4"
X
in Figure
0.01
first
in
0.25
-1
1" aluminum
14 in response
.
The
sec,
deadband
but
then
the
unloaded curvature responds quickly to move to the commanded
value.
The
position
system
has
decreases
a very
in an attempt
small
overshoot
to eliminate
and
the
roll
the error.
But
even though the moment and loaded curvature decrease, the
workpiece is unloading elastically as indicated in Figure 3
and
the
unloaded
workpiece
unloaded
is
loaded
curvature
eliminated.
This
curvature
This
response
hysteresis
be compared
will
occurs
is
shown
past
the
not
and
at
3.75
in Figure
results
about
the
15.
decrease.
When
the
elastic
limit
the
negative
decrease
exactly
to the
does
the
sec
response
in
will
Figure
indicated
The results
of Test
error
by
n Figure
10 shown
be
16.
the
16 can
in Chapter
5,
which is the same test with a moving workpiece.
The third
model
is
important
that
time
unloaded curvature
only.
This
means
nonlinear gain.
the workpiece
is
feature
not
a
to note about
variable
is a function of
that the workpiece
in
the
the workpiece
model.
center-roll
can
The
position
be thought
of as a
This is, of course, an approximation because
does actually
have mass,
44
compliance,
and damp-
ing,
will
which
all
affect
the
dynamics.
system
In
the
workpiece model these effects are assumed to be negligible.
assumption
This
to be a critical
out
turns
conditions
which is valid only under certain
model,
in Chapter
of the
feature
as shown
5.
Servo Model
The
roll
bending
center-roll position
works
apparatus
change
to
the
adjusting
by
loading in
the
the
work-
Closed-loop control of the roll bender requires auto-
piece.
matic control of the center-roll position by a servo system.
several different
There are
roll
positioning alternatives
Actuation can
available on commercial roll bending machines.
be achieved
using
a hydraulic
be modeled
servo.
The
position
as a standard
for
equation
or a DC motor and
At low frequencies the servo systems
leadscrew, for example.
can all
servo system
a
servo
servo,
position
or velocity
expressed
in
Laplace transform notation, is:
z
w
=
z
C
where
z
frequency,
is
and
the
s
position
+ 2w
n
-
45
(18)
2
+ w
n
command,
is the damping
velocity servo is:
2
n
ratio.
wn
is
The
the
equation
natural
for a
z
K
T
zc
C
where
z
is
command,
the
(19)
Ts + 1
center-roll
K is the
servo
velocity,
gain constant,
zc
is
the
velocity
and
T is the
system
time constant.
The GE servo-control
apparatus
can be configured
position
servo.
velocity
in response
Figure
system
18
response
is
Figure
a
servo
as either
17
is
a
Bode
diagram
the
electronics
are
frequency
is listed in Appendix
4.
servo
center-roll
the
servo.
frequency
function
response
or a
used to
was
gener-
This was done to ensure that
included
The program used to determine
the
shows
The forcing
ated from the control computer.
all
of
bending
to the velocity
which
servo.
and determine
a velocity
plot
to a step input
of the velocity
drivre the
on the experimental
in
the
dynamic
response.
the step and frequency
Figures
response
17 and 18 indicate that the
velocity servo is a first-order system with a time constant
of 0.042
very
The
Thus the servo model given by Equation
good approximation
servo
control
stant
are
sec.
gain
constant,
system.
is assumed
represented
For
to the
actual
system
K, is adjustable
modeling
purposes
to be 1.0 so that all
by the
controller
46
gain.
19 is a
if T = 0.042.
on the GE servo-
the
servo
gain
of the system
Figures
congains
19 and
20
E-O1
o10
9.5
9
a
7
C
E
E
6
00
U
oa >
3.
U
5
0
t0
4
3
2
I
0
0
5
10
12.6
15
20
25
TIME (SEC)
Figure 17.
Velocity Servo Step Response
a
0
-i
.
.
Frequency
Figure 18.
(Hz)
Velocity Servo Bode Diagram
47
30
E-02
E-01
'U
9
8
7
C
E
E
6
.oo
5
.
03
0
a.
4
3
2
I
0
0
10
5
15
20
30
25
E-02
TIME (SEC)
Figure 19.
Position Servo Step Response
___
0.n
.v
I
Of-
__
o
C
41
J
_,
-0.5
I
.
d
9/dc
I
de
-I0
0
\
0
-90
I
0
-180
a
I
0. II
I
0.2
-
I
I
05
I
10
I
2.0
I
I
37 5.0
I
I
10
20
---
Frequency (Hz)
Figure
20.
Position Servo Bode Diagram
48
50
are the step response
servo.
cally
These
damped
plots
and frequency
show that the position
second-order
a very good approximation
The
is a criti-
system with a bandwidth
of 3.7 Hz.
if
servo given
by Equation
= 1.0 and w
18 is
= 23.2 rad/sec.
and 'Filter Model
transducers needed to measure
loaded curvature
will have a very large
the servo system,
ignored.
of the position
servo
Thus the model for the position
Measurement
response
so the dynamics
maximum moment and
bandwidth
compared
of the measurements
to
can be
The signals must be filtered to eliminate high-
frequency noise from the transducer signals and reduce signal
aliasing.
frequency
of
the
bending,
can
The amount of filtering necessary and the break
of the filters
transducers,
the
and the sampling
be modeled
differential
will
vary
required
The
on the quality
accuracy,
frequency
as a combination
equations.
depending
the
type
of
In any case the filters
of first-
filters
and
used
second-order
on
all
of
the
experimental transducers are second-order Butterworth filters
with a break
frequency
of 130 Hz.
quency response of the filters.
order and can be modeled
w
= 820
Figure
The
using Equation
21 shows the fre-
response is second12 with
C
= 0.707
and
rad/sec.
Disturbance Model
Disturbances in the system such as initial curvatures in
the
workpiece
can
be
modeled
49
as
a
shift
of
the
moment-
z
0
.C
O
a
0
-j0
I.
0
C
4
U1
Ca
Frequency (Hz)
Filter Bode Diagram
Figure 21.
as
in Figure
shown
in the
change
the
disturbance
tion
in Figure
mined
unloaded
piece
The
22.
model.
This
which
is shown
16.
28
is
the
workpiece and curvature disturbance.
in the simulation
a
plot
of
scaled
program
loaded
50
point
the input
equal
a
to
addi-
is deter-
as an addito the work-
model
combined
for
the
This is the method used
listed in Appendix
moment,
causes
as a direct
This is modeled
22 which affects
Equation
is exactly
shift in the yield
Equation
in Figure
the yield point
curvature
disturbance
curvature
curvature.
by applying
tive term
The
22.
shift is incor-
This
model by changing
into the workpiece
porated
7.
curve, as shown in Figure
curvature
4.
curvature,
Figure 23 is
and
unloaded
Disturbance
I Kd
Zy
z
+
+
Figure
Ku
Disturbance Model Block Diagram
22.
E-03
60
45
z
30
-i
15
0
-15
0
1
2
3
4
5
8
7
8
9
10
E-01
TIME (SEC)
Figure 23.
Disturbance Detection
51
curvature
as
workpiece
a
straightened by the
shows
figure
detected
how
which
an
bend
initial
the
initial
indicates
that
disturbance
curvature
control
the
is
This
experimental bending apparatus.
by the closed-loop
disturbance
has
Detection
scheme.
disturbance
is
is
of the
within
the
control loop, which means that the disturbance can possibly
be eliminated with an appropriate controller.
there
is an apparent
in Figure
curvature
such a shift
23.
which
change
means
is
relationship
that
the
any change
a constant
origin
stationary.
the
of
If
the
seem
that
in the moment
in the curvature.
has
if the workpiece
and the loaded
3 it would
Figure
because
is impossible
is true only
the moment
between
From
a corresponding
requires
ture,
shift
Notice that
But
initial
this
curva-
moment-curvature
initial
curvature
changes, then the moment-curvature curve shifts as in Figure
7.
This
(from
moment
shift
zero
to
can
some
(unloaded
be
described
initial
in both
as
a
curvature)
cases),
thus
change
in
without
a
the
curvature
change
apparent
in
shift.
Figure 23 indicates a change in moment without a change in
curvature.
This
is
because
bending apparatus which
of
the
geometry
of
the
roll
causes the apparatus to sense the
disturbance in the moment measurement rather than the curvature measurement.
As stated earlier,
the curvature
strong function of center-roll position.
is a very
In fact the loaded
curvature measurement in Figure 23 is actually proportional
to center roll position
as given in Equation
52
16.
When a flat
workpiece which contains a kink is rolled through the bending
apparatus,
the
following
is observed.
the input roll and begins
tends
to
straighten
stationary,
and
remain constant.
the
to enter
because
curvature
of
rolls
the
response
reaches
the workpiece
are
initially
workpiece
tends
to
7 and
increases
This situation corresponds to
to the
more
first
0.1
than the
shown
in
23 up
Figure
sec
in Figure
This corresponds to
to 0.6
system has nearly reached steady state.
23.
the system
curvature,
responds to eliminate the disturbance.
the
kink
The moment however, changes to reflect the
A in Figure
As the moment
the
the system,
the
occurrence of the disturbance.
Point
As
sec,
where
the
This example demon-
strates that the closed-loop roll bending system will always
detect curvature disturbances with the moment measurement.
Controller Model
A specific controller model will
next chapter,
here.
taken
One
but the general form of the controller
important
into account
a digital
signal.
be developed in the
computer,
feature
of the controller
if the controller
is the
discrete
is given
that must
be
is to be implemented
by
nature
of
the
control
Even though the mechanical system is a continuous
system, the control signal from the computer will change only
at discrete intervals.
This discretization can be modeled by
a zero-order hold equivalence.
This model assumes that the
control signal is held constant over
53
each
interval until
updated
at
the
next
sampling
time
as
shown
in Figure
24.
Using Laplace transform notation the zero-order hold (ZOH)
can be described
by:
e
-TS
=
ZOH
(20)
5
where
T
control
is
time
the
signals
seconds
on
the
the
variable
defined
T varied
amount
of
from 0.009
computation
required
controller.
-
O
00.
-
c
40-
C
00sO
O
screte
T
between
interval
in the
For the computer and controllers used with
the experimental apparatus,
depending
of
s is a complex
and
Laplace operator.
in
2
3
5
4
6
7
8
9t
Time
Figure 24. Discretized Control Output
54
to 0.012
by
the
Chapter
4
CONTROL
The models
developed
in Chapter
3 indicate
that the roll
bending system is very much like a standard servo system.
The
servo
is the major
component
most of the significant
dynamics.
Equation
17
is
a
perfect
between servo movement
of the system
and
If the non-zero
description
and unloaded
of
the
curvature,
contains
portion of
relationship
and if all the
parameters are known exactly, then the roll bending control
design nearly reduces to a standard servo controller.
the
details
of
the
servo/workpiece
interaction
But
introduce
unique dynamic effects which require more detailed analysis
than the standard
servo
control
problem.
In particular,
the
deadband region represented by the zero portion of Equation
17 and
the
presence
of disturbances
in the
form of initial
curvature are unique properties of the roll bending system.
Although the workpiece model is nonlinear, there is some
insight
to be gained
from a linear
control
workpiece
can be described
as a variable
analysis
should
the
response
indicate
to a specific
general
controller.
analysis.
If the
gain, then a linear
form
of
the
For this reason
system
the roll
bending system will first be analyzed using linear control
techniques.
without
These techniques are well known and will be used
derivation
control schemes
(see
suggested
for
instance
[25]
by this analysis
55
or
[26]).
The
will then be eval-
uated
by
using
a
computer
simulation
of
the
roll
bending
system which includes the nonlinear workpiece model and the
discrete controller.
This nonlinear control analysis should
produce a more realistic
simulation of
the actual system
response.
Control Objecti'es
The
bending
feasibility
operation
control
closed-loop
in
this
control
of
has been demonstrated using
in [18], [20], and
analysis
Chapter
of
chapter
[21].
and
5 is to determine,
The purpose
the
roll
rudimentary
of the control
experiments
if possible,
the
described
the ultimate
in
limits
of the roll bending system response and the practical factors
unique
to the
roll bending
process
that limit
the response.
The control objective is to determine what control scheme or
schemes
can
be used
to attain
this ultimate
response.
The
problem is defining "ultimate response", because on a practical level the required
the
type of bending,
ance.
The control
system
response
part shape,
objective
will vary depending
feedrate,
can be stated
on
and error
toler-
in general
terms,
however, with individual objectives taking on more or less
importance
for a specific
application.
To compare
the vari-
ous control schemes the following control criteria will be
used to evaluate system response.
1) Stable system
Obviously
any
controller
should be designed
56
so that the
roll
bending system is always
areas of particular concern.
stability problems.
stable.
There
are
several
First, the nonlinearities cause
Because the workpiece is a variable gain
dependent on bending stiffness and servo position, a control
scheme
that is stable
for a particular
workpiece
and
bending
condition might not be stable for different workpieces and
conditions.
Second, unmodeled dynamics can cause instability
if the system bandwidth is large enough.
dynamics
in
all
components,
system and workpiece.
but
There are unmodeled
particularly
in
the
servo
The servo system has unmodeled dynam-
ics associated with friction, backlash, and compliance in the
coupling
between
the DC motor
piece
is modeled
mass,
compliance, and
and
as a variable
the ballscrew.
gain,
damping.
but
These
The
actually
unmodeled
work-
contains
workpiece
dynamics are especially troublesome as shown in Chapter 5.
2) Zero steady-state
This
objective
applications
degree
error
of
or
for
accuracy
is
very
other
is
important
types
of
required.
for
bending
straightening
where
Steady-state
generally associated with a particular input.
a
high
error
is
The steady-
state error properties of the control schemes developed in
this
chapter
Linear
will
control
open-loop
guarantee
be evaluated
theory
shows
that
transfer function
zero steady-state
in response
a
(Type I
free
to a step
integrator
in
system) is enough
error to a step input.
57
input.
the
to
Note that
a free integrator does not necessarily guarantee zero steadystate error
where
the
in response
disturbances
integrator.
operation,
This
since
to system
disturbances,
enter
system
becomes
initial
the
important
curvature
depending
in relation
on
to the
for the straightening
is a
disturbance
to the
closed-loop roll bending system.
3) High bandwidth
The high bandwidth
criterion
is really a measure
well the system can follow a command.
of the system
is determined
of
command
the
input
bending system
the
bances
have a constant
quency
is variable
the
of the distance
roll
are in
along the
the input commands and distur-
spatial
depending
For
content
and the disturbances
as a function
In other words
bandwidth
by the frequency
disturbances.
the input command
the form of curvature
workpiece.
and
largely
The required
of how
frequency,
on the
but
feedrate.
the
For
time
fre-
example,
consider a workpiece that contains an initial curvature that
is a sine wave
of unit amplitude
frequency along the workpiece.
and
1.0 cycle/ft
of length
If the workpiece is rolled
through the bending apparatus using a feedrate of 1.0 ft/sec,
then
the
controller
frequency
through
the
will
measure
of 1.0 cycle/sec.
the
apparatus
disturbance
2.0 cycle/sec.
using
a disturbance
which
If the same workpiece
a
feedrate
seen by the controller
of 2.0
has
a
is rolled
ft/sec,
is at a frequency
then
of
This shows that the feedrate actually deter-
58
mines
the temporal
bance.
purely
frequency
The frequency
time based
feedrate.
The
content
response
which
system
means
will
of the roll
that
respond
manner regardless of the feedrate.
of
the
spatial/time
bending
system
to a command
is
of the
in the
same
Consider the implications
Any
increase
in
system
a proportional increase in feedrate.
Reverse the example above.
is capable
or distur-
it is independent
relationship.
bandwidth will allow
bandwidth
of the input
A system with a 1.0 cycle/sec
of forming a unit magnitude,
1.0 cycle/
ft frequency curvature with maximum feedrate of 1.0 ft/sec.
If
the
feedrate
is
increased
for
this
system,
curvature will have less than unity magnitude.
bandwidth
is increased
to 2.0 cycle/sec,
the
output
If the system
then the system
can
form a unit magnitude, 1.0 cycle/ft frequency curvature using
a feedrate
of up to 2.0 ft/sec.
Conversely,
a high
spatial
frequency curvature can be formed by a small bandwidth system
if the feedrate
oped
in this
is small
chapter
enough.
will
The
be analyzed
control
on
the
systems
basis
develof time
frequency only, with the intention of increasing productivity
of the roll bending system by increasing feedrate.
4) Disturbance rejection
Initial curvature,
friction, and external
forces all
enter the roll bending system as disturbances which cause the
unloaded curvature to deviate from the commanded curvature.
The closed-loop control scheme must be able to reject these
59
disturbances.
This objective is most important in straight-
ening applications where the only requirement of the controller
is to eliminate
tions
such
as
disturbances.
bending
an
But
initially
some
curved
other
applica-
workpiece,
also
require good disturbance rejection.
Linear Control Analysis
The
transducer
of
filters
the
have
filters
been
designed
is very
large
break
frequency
break
frequency of the position or
so
that
the
compared
with
velocity servo.
This
means that the filter dynamics should be negligible compared
with
the
ignored
servo
in
control
the
analysis
dynamics.
control
The
filter
analysis.
and design
All
is done
negligible filter dynamics.
poles
under
of
can
the
then
be
following
the assumption
of
The validity of this assumption
will be examined in the next chapter.
The only nonlinearity
Chapter
are
3
is in the workpiece
well
analysis
in the component
represented
by
it is necessary
model.
linear
All
models derived in
other
models.
to linearize
the
For
components
the
linear
workpiece
model.
Because the nonlinearity is going to be included explicitly
in the computer
rough
simulation,
approximation
of
it is sufficient
to obtain
the nonlinear model.
only a
Because
the
unloaded curvature is a nonlinear function of roll position
only,
the
operating
simplest
linearized
on the roll position.
60
model
is
a
constant
gain
Consider the difference between a roll bending system
based
on position
If the center
servo
and
one
roll is controlled
with the workpiece,
the deadband region.
workpiece
gain,
by a velocity
then a step change
change
does
in curvature
This
occur,
but
correlates
well
of unloaded
controller
with
the
the center roll
and the workpiece
in servo velocity
the
then a
if the system is
If, however,
servo
in the rate of change
servo,
servo.
result, through interaction
model of a pure gain.
is controlled
this
in a change
on a velocity
by a position
change in servo position will
past
based
should
is a pure
result in a
curvature.
is still
In fact
measuring
un-
loaded curvature and not rate of change of unloaded curvature.
This means that an integration has occurred somewhere
in the velocity servo interaction with
the workpiece.
In
other words, because the unloaded curvature is a function of
roll
position
operation
on
integration.
to include
only
the
and
servo
not
roll
velocity
velocity,
the
is actually
a
workpiece
gain
and
an
Therefore the workpiece model must be modified
a free
integrator
as well
as a
bending system based on a velocity servo.
gain
for
a roll
The location of
the integrator is important because the location has implications
bance.
for
the
steady-state
Consider
Figure 25.
error
in response
to
a
distur-
the general second-order system shown in
The disturbance D 1 occurs before the integrator.
This disturbance might correspond to a torque disturbance on
the velocity servo of the experimental roll bending apparatus
61
Disturbance Block Diagram
Figure 25.
caused by friction
workpiece.
transfer
or to the load applied
D2
Disturbances
initial curvature
The
might
D3
and
y the
to the servo
to
correspond
an
disturbance in the roll bending system.
of the disturbances
from each
functions
output are given
D3()s)
D2(s)
D(s)
below in Equations
Dl(s)
Ts
Y(s)
D2(s)
2
21 to 23.
1
Y(s)
to the
-2~~ ~~(21)
+ s + K
Ts + 1
Ts
62
-2~~~
+ s + K
~(22)
2+
Ts
Y(s)
D 3 (s)
(23)
Ts 2 + s + K
From these equations the steady-state (s = O) response of the
system
to each of the disturbances
can be determined:
Y(s)
1
Dl(S)ss
K
(24)
1
(25)
K
D2 (s)s
= 0.0
(26)
D3()ss
As
indicated above, if the disturbance enters before
integrator, the
which enter
completely
output has
a finite
the system after
and
do not
cause
the
any
the
error.
Disturbances
integrator
are rejected
steady-state
error.
It is
not obvious where an initial curvature disturbance enters the
velocity-based roll bending system in relation to the workpiece integrator.
The disturbance model in Figure 22 shows
that the curvature disturbance operates on center-roll position indicating that an integration must occur before the
63
disturbance.
This assumption will be explored in more detail
in the experiments
Figures
system
5.
26 and 27 are block diagrams
based
on
respectively.
to emphasize
servo
of Chapter
a
position
servo
a
velocity
the fact that the servo integrator
integrator,
for zero steady-state error.
26 is operating
free
and
servo,
The position servo block in Figure 26 is drawn
loop and is not a free
integrator.
of the roll bending
directly
integrator
which
model
means
which
the
is necessary
The workpiece model in Figure
on position
The workpiece
is within
and
cannot
in Figure
that
the
contain
an
27 does contain
system
based
a
on
a
velocity servo has the required zero steady-state error.
A
free
integrator
could
position-servo
be included
based
system,
in the
but
controller
this
degrades
of
the
system
response and decreases the relative stability of the system.
This can be seen more easily on the root-locus diagrams shown
in Figures
28 and
the same scale
29.
The
root-locus
in the S-plane.
plots
The numbers
are
drawn
with
used are from the
experimental apparatus models in Chapter 3.
The controller
is actually discrete, as discussed earlier, but these continuous
root-locus
diagrams
are
sufficient
to
point
out
the
differences between the position- and velocity-servo based
systems.
Figure
28 is the root
locus
of
the
based system with an integral controller.
root
locus
proportional
of the
velocity-servo
controller.
Notice
64
based
that
Figure 29 is the
system
the
position-servo
with
addition
only
of
a
the
Position Servo
.
Ku de
Figure 26.
Position-Servo Based Block Diagram
Workpiece
Model
Ku di
Figure 27.
Velocity-Servo Based Block Diagram
65
m
20
15
10
5
-25
Figure 28.
-20
-15
-10
Re
-5
Position-Servo Based Root Locus
Im
A
- 20
- 15
I
-10
-5
I
-
-
!I ^
-25
Figure
29.
I _
-
I
-20
Ve
I
,
·
V
I!
I
-15
-10
-15
-I0
I
-
a
_
-5
_
---
0
I
!
I
_- -
*0
-
- -J
locity-Servo Based Root Locus
66
w
Re
.
integrator
in Figure
28 increases
the order
of
the
system.
Notice also that the higher-order system becomes unstable as
the
gain
second
is
increased.
order
stability
and will
of both
linear analysis.
on a velocity
bending
in
Figure
29
is
Remember
is overstated
in
this
only
the
that
simplified
Nevertheless, the roll bending system based
appears
based
system
this
to have many advantages
system.
state error and stability
roll
system
go unstable.
never
systems
servo
position-servo
The
It
properties.
based
point
has
on
discarded
at
and
continued
for the velocity-servo
a
the
good
over the
inherent
steady-
For these reasons,
position
control
the
servo
will
be
analysis
will
be
based system only.
As indicated above the continuous controller assumption
overstates the actual system stability.
using
discrete-time
control
This can be shown by
analysis.
For
the
discrete
analysis the continuous physical system must be converted to
an equivalent discrete system and described using Z transforms.
There
are
many
different
methods
used
continuous systems to discrete form (see [25]).
method
is a mapping
whereby
technique
the poles
to
convert
The easiest
and zeros
in
the continuous S-plane are mapped to the discrete Z-plane
according
to the rule z = e
of the pole or zero
and T is the cycle
is also a scale
final values
sT
where
in the Z-plane
z and s are the location
and S-plane
time for the discrete
factor
applied
of the discrete
to the
controller.
Z transform
and continuous
67
respectively
models,
There
to match
but
that
will
be
ignored
continuous
sample
system
here.
Figure
shown
in Figure
time of T = 0.01 sec.
30,
in the
the
Z-plane
system
instability
only
is different.
is the unit circle.
becomes
The control
updated
the
at
map
29 in the Z-plane
unstable
at
is caused by the discrete
and control.
are
shows
of
the
using
The shape of the root locus
the same but the interpretation
limit
30
signal
discrete
a
is
The stability
As seen in Figure
higher
nature
gains.
This
of the sampling
and the transducer
intervals.
As
readings
the
system
bandwidth approaches the sampling frequency, the controller
receives
similar
and
sends
to a phase
outdated
information.
lag in continuous
systems
The
effect
and
the result
is
is instability.
If the velocity servo bandwidth is high enough, it might
be possible
to achieve an acceptable
roll
bending system
bandwidth using a simple proportional controller at very low
gains.
zero
Better
to the
system response
system
and
could
drawing
the
be attained
poles
by adding
to a position
a
of
higher bandwidth and greater stability as shown in Figure
31.
A zero can be added
in the controller
output
ever,
or-by
to the system either
feeding
as well as the output.
there
can never
back
the
rate
In a discrete
be more zeros
by including
of change
controller,
than poles
because
it
of
howthis
would require information from future time steps that the
computer
does
not
including
a very fast pole,
have.
This
problem
can
but this increases
68
be
solved
the order
by
of
la
Figure 30.
Velocity-Servo Based Z-Plane Root Locus
1.0
0.75
0.5
T= 0.01
0.25
Re
-1.0
-0.75
-0.5
Figure 31.
-025
0.5
O.
Z-Plane Root Locus with a Zero
69
the system and introduces greater complication.
since
a zero implies
that the controller
vative of the input signal, any noise
In addition,
is taking the deri-
in the system can cause
large fluctuations in the controller output.
Measuring
attractive
is
less
the
method
affected
response
has
rate
of change
of the
output
of adding
a zero to the system
by
and
less
noise
overshoot
also
for
because
similar
is a more
because
the
it
transient
systems.
For
the
roll bending system the output is unloaded curvature, which
is measured
as detailed
in Appendix
1. The rate of change
unloaded curvature is much more difficult to measure.
of
It is
possible, however, to obtain a reasonably good approximation
of the rate of change
methods.
One method
of unloaded
is to divide
curvature
by either of two
the difference
between
two
subsequent unloaded curvature measurements by the time interval between the measurements
as shown in Equation
27:
0
K (nT) = [Ku(nT) - K ((n-1)T)]/T
where
K (nT) is the unloaded
is the rate of change
This
curvature
U
is actually
of the unloaded
not a measurement
output but a backward difference.
a zero
in
problems.
the
controller
In addition
and
at time nT and K (nT)
U
curvature
at time nT.
of the rate of change
of
Therefore it is more like
is subject
it is always
70
(27)
to the
delayed
same
noise
by half a time
step because
current
it uses unloaded
and the previous
curvature
time steps.
information
More
from the
elaborate
differ-
entiation schemes are available, but they require considerably more computation.
roll velocity
unloaded
A better measurement
as an approximate
curvature
similar
measure
given
roll
by
included,
position,
Equation
17.
the equation
is to use
of rate-of-change
of
to the way that roll position
be used to estimate unloaded curvature.
between
method
z,
and
unloaded
If
the
effect
The
relationship
curvature,
of
can
Ku,
disturbances
is
is
becomes:
K
=
Kd
z
<
(d
+ z )
(28)
K
3
-(z
u =
(z
9z
-
z ) -- 2Y +
- zd)d
2L
3z
2L 2 (z
3
+K
Zd)
z
(z + z )
d
y
d~~~~~~~~~~~
where
Kd
is the curvature
disturbance
zd = KL
Taking
the derivative
Taking
the derivative
and zd is given by:
2/3
of Equationd 28 gives:
of Equation
71
28 gives:
(29)
u
=d
<
Z
+
(Zd
y)
(30)
·
3
K
u
The
;-
L2
U
L
rate
of
variables:
3z
d
Zd)
-
+K
L 2 Z _ Zd)3
~L2(z
change
the
3.
(z
of
roll
z (z +z)
d
d
unloaded
curvature
position,
z;
y
depends
roll
on
velocity,
four
z;
the
curvature disturbance, Kd; and rate-of-change of curvature
disturbance,
Kd .
terms are cubed
Notice
that in the second
in the
denominator
which
term the z and Zd
indicates
that the
relative magnitude of the second term will quickly become
much
smaller
curvature
is
distribution
bending
Therefore
estimate
linear
than
the
determined
is
Kd
likely
rate of change
term.
by
of curvature
applications
it
first
the
rate
feedrate
the workpiece.
will
be
that
roll
of unloaded
small
of
as
along
change
well
But
compared
velocity
can
curvature
be
using
as
of
the
for most
with
z.
used
to
only the
terms.
.
·
Ku = 3/(L
Notice
The
that a controller
using
2
)
Equation
(31)
31 as an estimate
of
Ku should provide slightly more damping and greater stability
in the deadband
region
than if the true K
U
72
were
used.
This
is because although
K
is zero), Equation
curvature
If
31 indicates
proportional
Equation
is zero in the deadband
31
region
a rate of change
(if Kd
in unloaded
to the roll velocity.
is
an
accurate
model
of
K
and
u
if
the
workpiece does actually perform an integrating function on
the center-roll
velocity,
then a control
scheme
which
uses a
proportional controller and KU plus KU feedback should result
in a roll bending
Figure 31.
be
system
which
has a root locus
as shown
in
Figure 31 indicates that the system bandwidth can
increased
to
the
Nyquist
frequency,
but
practical
considerations such as transducer noise, servo saturation and
unmodeled
dynamics
will
limit
the
maximum
bandwidth
considerably less than the maximum theoretical limit.
to
Never-
theless the control scheme as described is quite attractive
because, with proper placement of the zero, the system will
have very good stability robustness, zero steady-state error,
high bandwidth, and good disturbance rejection.
Also, a wide
range of second-order system responses is possible by manipulation of the gain and the zero location.
analysis
good
indicates
system
that
response.
this
The
control
next
step
The linear control
scheme
is
will
give
very
a
non-
to perform
linear analysis to determine the effect of the nonlinearities
on system response.
Nonlinear Control Analysis
A computer
program
has
been
73
developed
to
simulate
the
roll bending system and to include the nonlinear workpiece
and the discrete
model
4,
Appendix
a
uses
to model
technique
Runge-Kutta
fourth-order
continuous
the
detailed
The program,
controller.
integration
The
system.
in
discrete
controller is modeled using difference equations exactly as
they
appear
would
signal
in
only
is updated
actual
the
is given
by Equation
models
the loading
This
hysteresis.
28.
deadband
that this
Notice
cor-
equation
but does not include
region,
that
means
steps
used in the simulation
The nonlinear workpiece model
program
time
control
of the controller.
interval
to the sampling
responding
Runge-Kutta
at the
The
controller.
program can
the
any
simulate a
moving workpiece assuming that each point on the workpiece is
formed
only
once,
but
cannot
used to model stationary
bending
by
or
slow
the objective
the productivity
the
increasing
moving-workpiece model
but because
workpieces
is to increase
process
stationary
A more complex workpiece model could be
moving workpiece.
of this research
a
model
maximum
feedrate,
Equation
is adequate.
of the roll
28 must
the
be
calibrated for a specific workpiece since the bending stiffness and the yield point
als
and
workpiece
the
workpiece
are different
shapes.
modeled
For
for different
materi-
the
simulations
shown
X
1" aluminum
strip.
is a 1/4"
below
The
calibration factors used are taken from the experiment shown
in
Figure
14.
The
form
of
the
discrete
controller
which
would implement the control scheme described above is given
74
by:
desired - K
U = G[K
u
step
time
controller
discrete
The
31.
by Equation
given
gain,
of the zero, and K
the location
G2 is used to determine
(32)
is the controller
G
output,
U is the controller
where
- (G2)(K)]
UUU
is
is
assumed to be T = 0.01 sec in the simulation.
Figure
32
G
on
figure.
two
the
between
initial
is zero
response
the
the
response
shown in Figure
The response
overdamped.
The
so
poles
of 0.01 in
controller
simulation
this
For
bending
roll
input curvature
and G2 in the simulated
factors
the
of
simulation
to a step command
response
the
is a
because
deadband
commanded
predicted.
region,
verge
of a
32 is as expected.
moves
Once past
to
quickly
the
is zero,
at steady-state
error
as the
of the deadband
as
can limit cycle or even become
unstable
in Figure
32 in on
The fastest
limit
result
in a limit
large
increase
unstable.
system
be
always
should
The speed of response increases with increasing
at very high gains.
the
The
curvature.
but the system
gain,
the
The
located
is
workpiece moves through the linear elastic region.
the
-1
1.
as listed
are
zero
system
of
Figure
A
cycle.
cycle
around
G
will
33
shows
response
small
the commanded
cause
even
75
increase
the
of G
will
curvature.
system
faster system
to
A
become
response,
although
zero
the
deadband
location
region
has
not
for this simulation
been
eliminated.
The
is as shown in Figure
31.
This system shows a faster response because the zero does not
trap
the
tion.
slow
There
location
is a
and
overshoot
pole,
in
draws
both
possibility
fact
of about
the plot.
but
the
of
poles
overshoot
response
1% even though
to a
for
faster
with
G
=
loca-
this
350
zero
shows
it is difficult
an
to see on
The system remains stable for larger gains with
the zero location
as shown in Figure 31.
Again,
higher gains
result in faster response and also in smaller steady-state
errors to some disturbances.
provisions
for
responses
control
modeling
are possible
power
needed
The computer simulation has no
saturation
effects,
so
very
if very large gains are used.
to achieve
this faster
response
fast
But the
is very
large which means that the velocity servo system would have
very large
power
theoretical
scheme
if
system
has
requirements.
limits
the
In other
words
there
to system response using
models
infinite
used
are
power.
accurate
Figure
34
and
this
if
shows
are no
control
the
the
servo
system
response to an initial curvature step disturbance of -0.005
in
-1
the
.
The curvature
same
as
for
command
Figure
is zero.
32.
The
The
system
zero
again
location
shows
is
good
response and zero steady-state error.
The
nonlinear
control
analysis
does
not
reveal
any
stability problems with the control scheme developed in the
linear control section.
The analysis also does not predict
76
any
limits
analysis
to system
assumes
response.
perfect
But
as mentioned
conditions.
The
above
roll
the
bending
experiments detailed in the next chapter provide more insight
into
the
practical
limits
of
10
15
the
roll
bending
system
response.
E-03
Iz
10
a
2
8
4
2
0
0
5
20
25
TIME (SEC)
Figure 32.
Simulated System Step Response
77
30
E-01
E-03
IZ
10
8
2
a
N.-
4
2
0
0
1
2
3
4
5
8
7
8
9
I Mt (SeL)
Figure
33.
Simulated System Step Response
2
3
10
E-01
E-03
I
0
-1
-2
-3
-4
-5
-a
w
0
1
4
5
TIME (SEC)
Figure 34.
8
7
8
9
10
E-01
Simulated System Disturbance Response
78
Chapter
5
EXPERIMENTATION
Experimental Objectives
There
are
several
key
assumptions
in
the
control
analysis developed in Chapter 4 which need to be examined
experimentally. , First,
performs
an integrating
steady-state.
function
The position
loop is also important
to
disturbances.
assumption
position-servo
the
for choosing
system.
then the control
If
analysis
the workpiece
for zero error at
of the integrator
Furthermore,
based
that
is crucial
for the analysis
is the basis
incorrect
the assumption
in the control
of steady-state
integrating
a velocity-
the
workpiece
rather
workpiece
is invalid.
error
than
model
is
Second,
the
control scheme developed uses a measurement of rate-of-change
of unloaded curvature in the feedback signal.
earlier,
K
cannot be measured
u
directly,
proportional to center-roll velocity.
As mentioned
but is assumed
Since the K
to be
feedback
u
is used
to increase
system
damping
and stability
as well as
increase the system bandwidth, if the assumption is incorrect
there
might
be
experiments
assumptions
If
stability
detailed
problems.
below
is
One
to
objective
determine
of
if
the
these
are valid and if so, under what conditions.
the
assumptions
listed
above
are
valid,
then
the
simplified control analysis in Chapter 4 implies that the
upper
limits
on
system
performance
79
will
be
determined
by
A second experimental objective will be to
model accuracy.
the
determine
practical
There
system response.
limit
system
upper
limits
are many
performance.
For
of
roll
the
bending
which
practical factors
roll
the
bending
system,
possible limitations are imposed by transducer bandwidth and
noise,
and power,
servo bandwidth
and controller
In
design.
many real systems these factors define the limits of system
performance, because the cost of high quality transducers and
servos
often
not
does
because the objective
the
roll
bending
justify
is to explore
process,
are
limits of
imposed
by
the
The results provide some insight into the process
maximum
limitations
to
designed
hardware.
the
possible,
the performance
experiments
But
gain.
as
or
as
the
performance
eliminate,
limitations
far
the
system
performance
reveal
nearly
optimal,
the robustness
dynamics.
then
the
can
If the conditions
attained given optimal conditions.
indeed
that
experiments
scheme
of the control
should
be
are
also
to the unmodeled
The implementation details given below show some
of the compromises made to reduce hardware limitations.
The
Chapter
velocity
3 and
workpieces
servo
Appendix
formed
in
actuation
system,
1, is oversized
the
experiments.
described
in
to
the
that
the
in
relation
This
means
problem of servo power saturation is greatly reduced and will
not
be
a
limiting
factor
in
the
experiments.
The
major
hardware limitations are the computer speed and transducer
noise.
The computer
speed is a problem
80
because
the discrete
controller update frequency and the sampling frequency should
be at least
the
roll
twice,
bending
and
preferably
system
several
bandwidth.
times as large as
Any
reduction
in
the
computation needed will increase the controller frequency,
but simplifying the computations reduces system accuracy (see
Appendix
2).
For
major concern.
these
experiments
the
accuracy
is
not
a
The objective is to determine the limits of
the control scheme assuming optimal conditions, so for the
purpose of these experiments the simplified calculations can
be assumed valid.
lations
should
The results using the simplified calcu-
provide
some
information
simplifying assumptions are acceptable.
problem is more difficult
about
when
the
The transducer noise
to deal with
because the noise
contains significant energy across a large frequency range
(see Appendix
1).
The transducers
can be filtered,
filtering places limits on system bandwidth.
possibilities the measurement method with
will be preferable
even if some accuracy
but heavy
Of all
the
the least noise
is lost.
There are several ways to measure loaded curvature, each
with specific advantages and disadvantages (see Appendix 2).
In some of the experiments
described
below,
loaded curvature
is measured using the center-roll displacement according to
Equation
16.
The remaining
experiments
are based on an LVDT
loaded curvature measurement calculated as shown in Equation
39.
The
center-roll
displacement
encoder mounted on the DC motor
81
is measured
shaft.
by
a
rotary
Loaded curvature
determined
other
by this measurement
methods,
but
measurement.
has
is not as precise
no
noise
since
it
as by some
is
a
digital
In addition, the loaded curvature measurement
based on center-roll position requires much less computation
in
the
discrete
measurement
curvature
controller
method.
But
does
Therefore
measurement
experiments.
than
is
the
the
this
ideal
for
experiments
LVDT
method
the
show
curvature
for
purposes
that
loaded
of
in some
these
cases,
the inaccuracy of the center roll position loaded curvature
measurement has very large effects on system response.
these
cases
the LVDT
curvature
measurement
results
In
in more
predictable system response even though this measurement is
noisy.
The
moment
stiffness
shown
measurement
in order
in Equation
nearly constant
increases
reason,
possible
to
4.
is
scaled
determine
the
Since
force
the
for all workpieces,
dramatically
the bending
with
the
for
experimental
unloaded
the
bending
curvature
transducer
the signal
stiffer
experiments
by
noise
to noise
workpieces.
use the
stiffest
hardware.
The
as
is
ratio
For
this
workpieces
force
trans-
ducer signals are also filtered to eliminate high frequency
noise.
130 Hz.
The low pass filters
This frequency
ing according
transducers
aliasing
have a break frequency
is much too high to eliminate
to the sampling
is
errors
limited
will
of about
to
theorem,
a
be small.
82
much
but the input
lower
Again,
the
frequency
objective
aliasto the
so
the
is to
achieve
maximum
Placing
system
bandwidth,
not
to compute
the maximum
moment
The sine and cosine terms are
in the moment
stiff
ams
workpieces
negligible.
speed
accuracy.
the filters at such a high frequency will ensure that
they do not affect system dynamics.
used
ultimate
due
to low
The
retains
curvatures
included
to reflect
of the
roll
curvatures,
34.
the change
pairs.
these
For
terms
are
Ignoring these terms increases the computation
considerably.
experiments
is given by Equation
to rotation
formed
The complete equation
moment equation employed
the sine term for D
are relatively
large.
in
the
because
the command
In the computer
program the
sine term is not calculated using the trigonometric function,
but
a
first-order,
instead.
small-angle
approximation
is
used
This compromise works well because the inclusion of
the sine approximation
slow the computation
increases
the precision,
but does not
down noticeably.
Experiments
The
experimental
procedure
straightening workpieces
Appendix
A
short
below.
4
1 and
the control
description
of
bending
and
described
in
in Appendix
4.
the apparatus
program
the
of
described
experimental
procedure
is given
The procedure is presented in more detail in Appendix
along
logic.
using
consisted
with
The
workpieces
complete discussion
tests
with
were
performed
a rectangular
of
the
using
cross-section
83
control
2024-T6
program
aluminum
1.0 in wide
and
0.25 in thick.
The workpiece
length
varied
from 3 to 6 ft.
The experimental procedure for forming a workpiece is:
- Load the workpiece
in the bending apparatus
and start
the computer program.
- Enter the control gains and other necessary input from
the computer keyboard.
- Measure the bending stiffness of the workpiece (see
Appendix '2).
- Turn on the center-roll
drive motor to feed the
workpiece through the bending apparatus.
- Begin real-time control of the unloaded curvature
using the control algorithm developed in Chapter 4 and
shown in the program listing in Appendix
4.
- At the end of the workpiece, turn off the drive motor
and return the apparatus to the zero position.
- Store the experimental data for future evaluation.
Step Tests
Figures
test where
0.01
in
.
35 to 38 show the system
the
input
The
workpiece
the loaded curvature
tion according
The
command
is a step
is initially
is measured
to Equation
16.
using
The
results of all the experiments
this test are
flat.
change
In this
the center
of
test
roll posi-
is 3.3 in/sec.
shown in this chapter
are
The controller gains
shown
on Figure
35 and
gains for the simulation
in Figure
32.
zero is located between
for the first
curvature
feedrate
plotted using every other data point.
for
response
correspond
With these
to the
gains the
the open loop poles at z = 0.84.
The
simulated response from Figure 32 is plotted in Figures 35,
84
E-03
&IC
to
8
10
2
a
N
4
::
2
0
-2
0
5
10
15
20
25
30
35
TINE (SEC)
Figure 35.
Test
#1
Plot
40
E-01
#1
E-03
ou
50
40
1%
z
30
N
20
x
t10
'0
-10
0
5
10
15
20
25
3o
TIME (SEC)
Figure
36.
Test #1
85
35
40
E-01
Plot #2
E-09
~dh
au
50
40
s0
20
t0
I0
-10
- -
0
5
10
s15
20
T IE
Figure 37.
25
so
35
40
F 01
(SEC)
Test #1
Plot #3
E-O1
an~
au
8
U
a
I..
4
2
0
-2
0
5
t10
15
20
25
30
Test #1
86
40
E-01
TIME (SEC)
Figure 38.
35
Plot #4
37 and
38 for comparison.
has essentially
are
three
the same
areas
significantly.
is not
The
zero
first
while
simulated
form as the actual
where
the
predicted.
factor
the
the
two
response
response,
responses
there
differ
First the measured initial unloaded curvature
as
transducers.
Although
is
For
sheet
This
the
is caused
initializing
by two
procedure
this test the transducers
was
stationary.
When
factors.
for
the
were initialized
the
drive
motor
is
started, a small moment is developed between the center drive
roll
and
the
workpiece.
This
small
moment
results
in
an
increase in measured moment from the "zero" position which
results in a small negative unloaded curvature measurement.
This error was eliminated
in subsequent
tests by initializing
the transducers after the drive motor was started.
A second factor which results in errors in the deadband
region
is the
error
due
to
simplifying
assumptions
moment and loaded curvature measurements.
the loaded
this test.
and unloaded
According
and the scaled moment
elastic
region.
and scaled moment
sec.
and the
the
Figure 36 shows
scaled
moment
for
to bending theory,
the loaded curvature
should
the same through
Figure
be exactly
36 shows
are indeed
that
nearly
the
loaded
the
curvature
the same until
about 0.6
But notice that where there are differences between the
two measurements
ture reflects
small
curvature
in
error
in the
the error.
between
the
elastic
And
region,
because
of the steep
two measurements
87
the unloaded
results
curva-
slope,
a
in a large
error
in
the
unloaded
unloaded
curvature
curvature.
error
will
The
magnitude
increase
as
of
the
speed
the
of
response increases because the slope of the loaded curvature
and
scaled
moment
will
increase.
This
error
is due
to the
simplifying approximations made for the measurement calculations
of
and can oly
the
be eliminated
measurements.
The
by increasing
effect
of
the
the precision
error
on
system
response is small, although the damping is reduced somewhat
because
the
negative
while
these
initial
change of
the command
experiments,
the
the
unloaded
is positive.
error
in the
For
curvature
is
the purposes
of
deadband
region
is not
significant and will be acknowledged and ignored.
A second area where the actual response does not conform
exactly
The
to the simulated
actual
response
0.0005 in
1
entering
the
integrator
or 5%.
roll
as shown
response
shows
a
is the steady-state
steady-state
error
error.
of
about
This could be caused by a disturbance
bending
system
in Chapter
before
4 or possibly
of a workpiece integrator is inaccurate.
the
workpiece
the assumption
The velocity servo
model in Chapter 3 was developed under the assumption that
the
internal
workpiece
servo.
caused
small
grator
are
friction
very
and
small
compared
This is essentially
by
the
friction
steady-state
error
in the open-loop
external
and
true,
though
transfer
88
to
the
applied
by
the
inertia
of
the
but the small
external
even
load
load
there
function.
will
disturbances
result
in a
is a free
inte-
If this
is the
case, then there will be a finite steady-state
command to the
servo which results in a torque which exactly offsets the
torque
applied
by
the
disturbances.
Figure
there is indeed a finite steady-state
servo,
This
but
means
system.
as seen
in Figure
command
35 the
The
steady-state
velocity
do
torque
command
shows
that
to the velocity
system
that there is a disturbance
38
not
acting
move.
on the
in Figure
38 is
about 0.075 in/sec.
Experiments with the velocity servo show
that in the unloaded
state,
in/sec
is required
to overcome
system is measuring
the
low
gain,
a command
the
an error
command
velocity
friction.
Thus in Test 1 the
at steady-state,
input
is
of about 0.12
not
but
large
because of
enough
to
overcome friction and the system remains stationary.
There are several steps which can be taken to reduce the
error
due
eliminate
appears
to
the
disturbances
although
the
only
the error is to move the free integrator
before
the
disturbances
in
the
system
way
to
so that it
as
shown
in
Chapter 4. Including an integrator in the controller would
eliminate
bances,
the
but
steady-state
this
would
error
create
due
two
to
free
the
torque
distur-
integrators
in
the
velocity-servo based roll bending system which would degrade
system
response
velocity-servo
system.
shows
Other
and
nullify
of the
advantages
based system
over
the position-servo
options are
more
attractive.
that the error
loop gain.
many
Increasing
is inversely
proportional
the controller
89
gain will
of
the
based
Equation
24
to the opendecrease
the
error.
steady-state
which
was
conducted
that
except
335.
Also
the
the
Figure
39 shows
the results
using
the
same
conditions
gain
was
controller
curvature
loaded
increased
is measured
state
error
noise
in
the
is
later.
much
if
loaded
curvature
not
Test
from
100
the
2
1
to
LVDT
The reasons for
show that the steady-
The results
reduced,
as
using
measurement method according to Equation 39.
this are explained
of Test
zero.
There
so
measurement
the
is
some
unloaded
curvature appears to oscillate, but the oscillation is around
the commanded
gain
does
bances.
might
value and it is easy to see that increasing
reduce
the
steady-state
But the increased
be undesirable,
error
due
gain does have side
to the
distur-
effects
such as the large overshoot
which
observed
E-03
12
10
a
X
I'l
6
v
4
2
0
-3
0
5
10
15
20
25
TIME (SEC)
Figure
39.
90
Test #2
30
35
the
40
E-01
in
Figure
39.
Velocity
feedback
could
also
be
used
to
reduce
steady-state error by closing the loop around the velocity
servo.
This
systems,
is
a
standard
but is not
item
included
on
most
commercial
on the velocity
servo
these experiments because of noise problems.
instead of using
compensation
for
accomplished
by
adding
an
effects
amount
used
an open-loop
in software.
which
for
It is possible,
velocity feedback, to make
the friction
servo
This
represents
is
the
frictional effects to the velocity command.
Figure 40 shows
the
using
results
of
Test
3
which
was
conducted
the
same
conditions as Test 1 except that loaded curvature is measured
using the LVDT method and an open-loop software compensation
was used to eliminate
state
error.
reduced
there
The
even though
the effects
results
show
of friction
that
the controller
is some oscillation
on the steady-
steady-state
gain
due to noise
error
is the same.
in the
loaded
is
Again
curva-
ture measurement but the oscillation is around the commanded
value.
due
This open-loop compensation does not affect the error
to the disturbance
workpiece.
Figure
from the external
41 shows the effect
load applied
by the
of using a higher gain
and the open-loop compensation for frictional effects.
controller
gain
is the same as for Test
meters are unchanged.
state error
and
2: all other
load
para-
Figure 41 shows nearly zero steady-
to a step input even in the presence
external
The
disturbances.
91
The
of friction
open-loop
friction
E-03
12
10
a
x
z
a
N
4
2
0
-2
0
5
tO
15
20
25
30
35
TIME (SEC)
Figure 40.
Test
40
E-01
#3
E-03
Id
10
B
I
U
z
B
N
ZA1
fx
4
2
0
-2
0
5
10
15
20
25
92
Test
35
40
E-01
TINE (SEC)
Figure 41.
30
#4
compensation
Tests
is
2, 3, and
integrating
suggested
used
4
show
all
subsequent
the workpiece
predicted and
4 will
if the effect
ciently reduced.
the
that
function as
in Chapter
requirement
in
satisfy
of
does
the
the
experiments.
perform
an
control
scheme
steady-state
error
the disturbances
can
be suffi-
There is still some question whether the
initial curvature disturbances will enter the system before
or after
the workpiece
integrator.
This question
is explored
in more detail later.
The third significant variation of the actual response
from the
predicted
response
shown
in Figure
actual response has some overshoot.
shoot
is small,
it is significant
35
is that
the
Even though the over-
because
the gain and
zero
location were picked specifically so that the system would be
overdamped.
As
discussed
in
Chapter
4,
if
the
zero
is
located between the poles then the system should always be
overdamped and the response should never overshoot.
are several factors which could
One
is
that
predicted
the
actual
although
gain
this
be causing the overshoot.
and
is
There
zero
location
are
not
as
unlikely
because
the
actual
response so closely resembles the simulated response and the
parameters
for the test are taken from the simulation.
also possible
band region
earlier
the
region causes
that the unloaded
is contributing
negative
curvature
error
to the overshoot.
unloaded
the control command
93
curvature
in
to be larger
It is
in the deadAs mentioned
the
deadband
than it would
be
Another
otherwise.
that
center-roll
possibility
velocity
can
be
is
that
used
to
the
assumption
estimate
is used
to increase
but if the Ku measurement
is wrong
then the damping
affected.
37 suggest
The
invalid.
36 and
another
the
fast
enough
so that
each
shown
will be
for
that if the workpiece
moment
and
in Figure
independently.
loaded
3 and
But
these
Figure
is initially
curvature
two
are
curvature do change independently.
that
along
the
the
the
Essentially
flat,
always
measurements
36 indicates
through
point
workpiece is unaffected by any previous forming.
scaled
damping,
cause
is moving
is that the workpiece
apparatus
this means
is
The underlying assumption used for the simulated
response
bending
feedback
u
Figures
overshoot.
system
K
K
then the
related
cannot
as
change
the moment
and
The system responds as
predicted until the scaled moment and loaded curvature level
off at about
1.0 sec.
point in the simulation
At this
in Figure 37 the moment and curvature
system reaches steady-state.
36 shows
that
decreases
after
the
does not change.
scaled
reaching
remain constant
shown
and the
The actual response in Figure
moment,
rather
than
leveling
off,
a peak at 1.0 sec but the curvature
The decrease in the scaled moment without a
corresponding decrease in the loaded curvature results in an
increase in the unloaded curvature which accounts for the
observed
overshoot.
The
problem
is
determining why
the
moment and curvature move independently.
The most likely reason for the independent movement is
94
that the moment
do
not
Figure
using
and
curvature
reflect
the
true
42 shows
the
results
exactly
the
same
state
of
are in error
the
of a test
parameters
workpiece is stationary.
predicted in Figure
measurements
as
loaded
and
workpiece.
which
was
Test
1
conducted
except
the
The results are nearly exactly as
37.
Obviously
the
measurement inac-
curacy noted in Figure 36 which causes the overshoot only
occurs when the workpiece
is moving.
Thus it is likely
that
the moving workpiece assumption made for the system simulation is not valid for these forming conditions.
Another possibility is that the step command actually
causes a kink to be formed in the workpiece.
This kink could
affect the forming of the following workpiece until the kink
exits
from
in/sec
and a roll spacing
would
the
bending
take nearly
outer roll.
apparatus.
outer
The curious
necessary
roll.
If
a
feedrate
a
behavior
3.3
roll to the
of the moment and curvature
1.0 and 2.5 sec which is the same
to move
kink
a
of 6 in, a point on the workpiece
2 sec to move from the center
in Figure 35 occurs between
time frame
With
is
a point from
being
the center
formed,
this
to the
should
be
reflected in both the moment and loaded curvature measurements.
In
proportional
a kink
Test
the
to center
to exist
loaded
curvature
roll position,
in the forming
region
is
assumed
to
but it is possible
which
would
be
for
be unde-
tected by the center roll loaded curvature measurement.
The
LVDT loaded curvature measurement should be able to detect
95
E-03
7U
60
50
40
i
N
En
N
30
..i
20
x
10
0
-to
5
0
10
15
20
25
30
35
TIME (SEC)
Figure
42.
40
E- 1
E-1
Test #5
E-03
.
An
50
40
U
z
'
30
U)
-J
10
20
w -110
-to
0
5
10
15
20
25
TIME (SEC)
Figure
43.
96
Test #6
30
35
40
E-01
the kink since the LVDT's measure the curvature closer to the
point
of interest,
which
is the contact
point
on the center
roll.
Test 6 was conducted
except
that
the LVDT
The key result
using
curvature
the same parameters
as Test 1
measurement
method
was
43.
Again,
of Test 6 is shown
in Figure
used.
the
scaled moment decreases after reaching a maximum at 1.0 sec,
but the loaded curvature also decreases.
Thus the moment and
curvature do not actually move independently as suggested in
Test
1, but
loaded
only
appear
curvature
position.
to do
measuremeit
Notice
so
method
that the system
not have any overshoot.
because
Notice
based
response
also
of error
that
on
in the
center-roll
in Figure
the two
43 does
curvature
measurement methods converge once the kink has passed through
the
system.
reduce
This
the effects
so that
suggests
that
of the kink
it
might
be
if the feedrate
the kink is past the outer
roll before
possible
to
is increased
the unloaded
curvature reaches the commanded value.
For
increased
7 which
Test
7 and
all
to 13 in/sec.
used
the
same
subsequent
Figure
tests
44 shows
parameters
the
feedrate
the results
as Test
1 except
faster feedrate and the LVDT curvature measurement.
45 shows
the results for a similar
gain is increased
to 335.
is
of Test
for
the
Figure
test except the controller
Both tests show that the effect of
the kink is nearly eliminated with the higher feedrate and
neither test shows any overshoot.
97
In other words the actual
E-03
50
50
40
Z
w
N
30
U)
-I
20
1y
:
10
-10
-in Q0
5
15
10
20
30
25
35
40
E-01
TIME (SEC)
Figure 44.
Test
E-03
5
50
40
X
v
30
U)
N
1 20
-1N0
-inf
0
2
4
6
6
10
12
45.
98
Test
16
18
E-01
TIME (SEC)
Figure
14
#8
E-03
OU
50
40
=
C..
30
UI
z
i
20
Jxy
:S
10
-in
-.
0
2
4
6
9
10
12
14
16
TIME (SEC)
Figure 46.
Test #9
18
E-01
Plot #1
E-03
1U
8
=
U
z.-r
LJ
Z
4
w,
20
-2
0
2
4
6
8
10
12
TIME (SEC)
Figure 47.
Test #9
99
Plot #2
14
16
1
E-03
Ow
50
40
N
-
30
NV
I
-
z
20
-J
X
10
0
-in
0
2
4
8
8
t10
12
14
1t
18
E-01
TI ME (SEC)
Figure
system
response
predicts
is
when
parameters
of
very
the
48.
Test #10
nearly
the
simulation
Tests
7
and
same
as
assumptions
8
are
used
the
are
in
simulation
valid.
Tests
9
The
and
10
respectively except that the center roll position curvature
measurement
Test
is used in the later
9 shown
the simulated
47.
The
in Figures
response
response
due to the
sec
pass
to
unloaded
to
settle
shown in Figure
kink
through
curvature
which
46 and 47 correspond
is well
effects
two tests.
since
the
takes
means
damped
results
of
very well with
37 and also in Figure
and
there
the kink
system.
The
Figure
takes
46
are
no
less
shows
visible
than 0.5
that
the
about 0.8 sec (from 0.6 to 1.4 sec)
that
the
100
kink
is out
of
the
system
before
10
the system
the
speed
smaller
settles.
of
As the gain is increased
response
than the
increases.
time it takes
If
the
in Test
rise
time
for the kink to move
through
the system, then the effect of the kink will be visible.
response
in
Figure
48
is
fast
effects of the kink are visible
moment
very
even with
much
the
reduced
very
from
enough
feedrate.
of Test
which used the same parameters
that
in the decrease
fast
that
so
2
The
some
small
of the scaled
The
shown
is
overshoot
in
Figure
is
39
as Test 10 except for a slower
feedrate.
The
conclusion
from
these
experiments
is
that
the
overshoot observed in Test 1 is caused by measurement errors
in the loaded
indicate
exactly
curvature
that the K
approximation
u
as
measurement.
predicted
in
the
Tests
5, 7, 8, and 9
is not at fault, but works
simulations.
The
center
roll
approximation contains significant error during the transient
response
when
time found
the rise
by dividing
shown in the tests
command
causes
presence
region
a kink
ously formed
the
points
roll
spacing
is the worst
of the kink
between
time of the system
by feedrate.
case
to be formed
causes
point
some
being
because
in
the
interaction
formed
than
The
error
the large
workpiece.
in the
the
step
The
forming
currently
and
previ-
region.
The
analy-
still in the forming
sis cf the kink effects
is less
is quite complex,
but it is of little
practical interest since the step command is a rather extreme
command
and
the
kink
affects
101
the
transient
response
only.
The following
tests were all conducted
using
the center
roll
position curvature measurement and the fast feedrate.
curvature
noise
measurement
allows
faster
is
a better
feedrate
more
reading
reduces
desirable
because
of steady-state
the overshoot
the
error
This
lower
and
the
due to the approxima-
tion error.
Although Test 10 shows excellent system response, Figure
33
suggests
using
that
a zero
this zero
even
better
location
system
as shown
response
in Figure
is
31.
possible
In addition
location allows larger controller gains
without
stability problems which will decrease the steady-state error
due
to
disturbances.
Figure
response
using
controller
the
location
zero
rise
There
time
have
the
same
is moved
both
49
to
decreased
is some overshoot,
shows
gain
the
as in Test
z = 0.77.
in
this
system
The
test
but this is expected
Increasing
speed
the
essentially zero
controller
of response
Test 12 in Figure
even
50.
more
to 670
as shown
The results
very close to the predicted
settle.
gain
predicted.
The system
steady-state error.
should
increase
the
in the simulation
for
do show a faster
response,
and
with the center
roll position curvature measurement and ignored.
quickly settles with
10, but
deadband
as
step
but the system
response
does not
After the transient response the unloaded curvature
oscillates
around
oscillation
appears
the
commanded
to be at
two
curvature.
different
Moreover
frequencies.
the
A
higher frequency oscillation occurs between 0.6 and 1.0 sec
102
E-03
12
a
xA
8
Lb.
Zv-
4
M
2
-2
0
2
4
8
8
10
12
14
18
TIME (SEC)
Figure 49.
18
E-01
Test #11
E-03
* A
ie
8
z
U
z
8
:3
O
4
2
-2
0
2
4
a
9
10
12
14
50.
Test
103
#12
19
E-01
TIME (SEC)
Figure
16
Plot
#1
E-03
/U
60
50
z
U
z
40
x
-
x
20
10
0
-1o
0
2
4
6
10
8
12
14
16
18
E-01
TIME (SEC)
Figure
and
a
lower
sec.
This
51.
frequency
Test #12
oscillation
oscillation
is
false
readings
through
the
on the
moment
unloaded curvature.
the
indicated
indicated
apparent
loaded
from
1.2
unmodeled
to
1.8
dynamics,
The vibration of the workpiece
force transducer
measurement
as
that
apparatus
are
causes
interpreted
fluctuations
in
the
The actual workpiece shows no sign of
fluctuation
in the measurements,
curvature
by
side of the bending
curvature
disturbances.
occurs
caused
vibration of the workpiece.
that is on the outfeed
Plot #2
The
but,
the controller
fluctuation
measurements
is
in
obvious
because
it
responds
to the
is
the
moment
and
in
Figure
51.
Remember that the curvature measurement is proportional to
104
center-roll
visible
position
in this
so the
response
measurement.
Notice
of the
that
system
the
system
width is fast enough so that the system responds
the
lower
frequency
oscillation
"disturbance"
passes
experiment
shows
considered
in
through
that
the
the
system
model,
can
be used
the
system
the workpiece
system bandwidth is increased.
workpiece
but
band-
to eliminate
high
frequency
unchanged.
dynamics,
become
is also
This
which
are
not
significant
as
the
A simple dynamic model of the
to explain
the high and low
frequency
vibration shown in Test 12.
The
outfeed
simply
dynamics
side
as
workpiece
number
of
the
of
the
mode and natural
modes
frequency
experimental
results
frequency
the
of
the
apparatus
beam,
and
there
workpiece
can
of a cantilever
continuous
of vibration
portion
bending
vibration
is a
of
the
be modeled
beam.
will
frequencies.
The
in
fundamental
Figure
mode
for
50.
a
the
very
Because
be an
will be sufficient
shown
on
the
infinite
fundamental
to explain
The
cantilever
the
natural
beam,
using the Euler beam model and assuming a uniform rectangular
beam, is given by:
{EI
w=
where
w
ness,
p is the material
is the natural
1\
3.52
(33)
frequency,
density,
105
EI is the bending
stiff-
and L is the length
of the
beam.
The
workpiece
lightly damped
tion 33.
beam
that the natural
This
means
section of the workpiece
changes
as the
explains
through
the
be modeled
why
the
is fed
oscillation
overhanging
frequency.
The
of
given by Equa-
frequency
of
of the
side of the apparatus
through
the
frequency
machine.
changes
At the beginning
workpiece
pair
is a function
natural
on the outfeed
workpiece
by a
frequency
frequency
that the
the experiment.
natural
can
poles with a natural
Notice
length.
dynamics
of the
This
partway
experiment
is very short and has a very high
dynamics
of
the workpiece
are at a
frequency much higher than the system bandwidth and therefore
the
oscillation
of
the
moment
workpiece
vibration
is
vibration
has
little
very
measurement
sufficiently
effect
caused
attenuated
on
the
by
the
so that
the
system.
As
the
workpiece moves through the apparatus, the natural frequency
of
the overhanging
portion
decreases
and
at some
point
the
lower frequency vibration becomes significant and interferes
with the dynamics
of the bending apparatus
The zero location
z = 0.67.
Test
From
that moving
response
Figure
match
the results
that
gain in Test
shown
In Test
in Test
by the
results
an
gain
increase
is shown in Figure 53 but vibration
106
52 it is obvious
in faster
simulation,
14 the controller
12. Again
13 is the same as in
in Figure
the zero in this manner
just as predicted
52.
response
in Tests 13, 14, and 15 is shifted to
The controller
11.
and controller.
also
system
shown in
is increased
in
the
speed
to
of
is not nearly as
E-03
14
12
10
8
U
z
"I
8
4
2
0
-2
0
2
4
8
8
10
12
14
1
TIME (SEC)
Figure 52
is
E-01
Test #13
E-03
14
12
10
8
U
z
N1
:r
4
2
0
0
2
4
8
8
10
12
14
'2~~~
TIME (SEC)
Figure 53.
107
Test #14
18
~
18
E-01
E-03
14
12
10
8
X
a
w
"I:
I-
4
2
0
-2
0
2
4
8
8
10
12
14
18
TIME (SEC)
Figure
54.
Test #15
Plot
18
E-01
#1
E-03
'U
s0
50
xU
z
40
N
z
Jx
20
10
0
-to
--
0
2
4
8
8
10
12
14
55.
Test #15
108
18
E-01
TIME (SEC)
Figure
16
Plot #2
-
large as in Test
total
the
moment
of
vibration
the
of
effects
the
measurement,
percentage
be a small
may
the vibration
to
54 and 55 show
Figures
increases.
1000 in Test 15, vibration
that even though
gain is increased
As the controller
12.
can easily dominate the unloaded curvature measurement.
11
Tests
15
through
all
to
respond
in
changes
the
discrete control algorithm just as predicted by the control
is
model
system
bending
This means that the roll
in Chapter 4.
theory and simulation
contains
and
accurate
all
significant dynamics except for the workpiece dynamics.
the workpiece
above,
shown
bandwidth
system
bending
increases
workpiece
of the overhanging
length
important
become
dynamics
increases.
The
the
As
as the
and as the roll
unanswered
one
question is how the closed-loop control system will respond
to initial curvature disturbances.
Disturbance Tests
Tests
roll
the
disturbance
16,
17, and
bending
18 show
system.
The
11, 12 and 14 respectively.
tests
the workpieces
in
tests,
ing apparatus
of the
curvature
109
these
that were formed in
curvature.
for each
were
and the disturbance
for
of
Thus for these disturbance
corresponding
the workpieces
response
used
workpieces
have an initial
of the initial
the results
forming
disturbance
tests are the same workpieces
Tests
bution
the
step
The distri-
workpiece
tests.
reinserted
experiments
is shown
After
the
into the bendwere
run using
a zero command
meters
for unloaded
curvature.
The controller
used were the same as in the corresponding
Figures
56 and
57 show the results
that the scaled moment
dently
as predicted
Figure
57 shows
by the disturbance
that the disturbance
tests.
of Test
and loaded curvature
16.
do move
model
para-
Notice
indepen-
in Chapter
increases
initially
3.
even
though the system is moving to reject the disturbance because
the system
workpiece
system
that
still must move through
reaches
the
yield
the deadband
point
at about
settles to zero steady-state
this
shown
response
in Test
smoother
11.
than
appears
This
much
the
in Test
04
The
sec and the
in 1.0 sec.
cleaner
is because
the step command
that the system should
error
region.
than
Notice
the
response
disturbance
is much
11. It is reasonable
respond much better to the disturbance
since the disturbance is really the step command "filtered"
by the system dynamics.
Figures
Test 17 shows faster response in
58 and 59 because of the increased
gain.
Although
it
might appear that the oscillation in Test 17 occurs because
the
system
workpiece
is
in
oscillation
rejecting
Test
shown
12,
disturbances
this
in Test
is
12
not
formed
actually
is the
result
into
true.
of
the
The
measurement
error and does not reflect any physical curvature changes.
The
oscillation
error
in
in
Test
due to vibration
Test
results
18
in
is
shifted
Figures
60
17
is
also
caused
of the workpiece.
to
and
match
61
110
that
show
by
measurement
The zero
in
that
Test
the
14
location
and
vibration
the
is
reduced
just
as
in
Test
14.
The
disturbance
response
is
again smoother than the corresponding step response because
the input has been filtered
by the system dynamics.
The disturbance tests demonstrate excellent disturbance
response
zero
steady-state
actual
response
caused
characteristics with
system
error.
response
good
These
is
much
transient response and
tests
closer
if the system is not hampered
by the step command.
the experiments
The
are summarized
111
also
major
to
show
that
the
predicted
the
by the "kink effects"
conclusions
in the next chapter.
from
all
E-03
4
2
0
x
-2
z
(3
Z
-4
-6
-8
-10t
-12
0
2
4
8
8
10
12
14
16
TIME (SEC)
Figure 56.
Test #16
18
E-01
Plot
#1
E-03
J'bd'
45
x
U
3:
z
NJ
Z
30
v
U()
z"I
J
15
0
-15
0
2
4
6
8
10
12
14
57.
Test #16
112
1i
E-01
TIME (SEC)
Figure
16
Plot
#2
E-03
4
2
0
I
U
z
m
-2
-4
-8
-e
-10
0
2
4
6
8
10
12
14
o
TIME (SEC)
Figure 58.
Test #17
18
E-01
Plot #1
E-03
DU
40
U
30
z
I
20
2
tO
0
-10
0
2
4
a
8
10
12
14
TIME (SEC)
Figure
59.
Test #17
113
Plot #2
18
18
E-01
-
-
~~
E-03
4
2
I
-2
"'-
z
N
V.
Z3
wd
-4
-8
-18
-10
0
2
4
8
8
10
12
14
16
TIME (SEC)
Figure 60.
Test #18
18
E-01
Plot
#1
E-03
60
50
40
xU
z
30
20
-i
10
0
--tit
-.-
0
2
4
6
8
10
12
14
1
19
E-01
TIME (SEC)
Figure
61.
Test #18
114
Plot
#2
~
~
~
~
_
Chapter
6
CONCLUSIONS
The
experiments
in Chapter
5 demonstrate
that the
roll
bending system models developed in Chapter 3 are an accurate
representation
of the low order dynamics
process.
overall experimental
sponds
are
The
very well with the simulated
small
because
differences.
This
simulation
is based
the
properties
while
the
actual
is
of the roll bending
system response
response,
to
be
corre-
although
there
expected
however,
on fixed workpiece
material
workpiece
properties
vary.
The
material property variance is, in fact, a primary motivation
for
this
research,
because
if
the
material
properties
are
constant and well defined, an open-loop control system would
work just
as well and
be much simpler
to implement
than
closed-loop controller presented in this research.
fore,
the
control
analysis
in
Chapter
4
and
the
the
Therecomputer
simulation in Appendix 4 are valuable for determining trends
and comparing
controllers,
but
not
for
final
tuning
of the
controller to a particular bending process or workpiece.
Nevertheless, the control scheme developed in Chapter 4
worked
essentially
as
predicted
in
the
simulations.
The
primary assumptions used to develop this control scheme are
that the workpiece performs an integrating function when the
roll bending system is based on a velocity
center-roll
velocity
servo and that the
is a good approximation
115
of the rate
of
The experiments show that the
change of unloaded curvature.
to a step input
goes
which
a
in response
error
steady-state
if the
disturbances
are
small
grator
in the
oop
transfer
open
integrator
free
gain.
The
always
will
inte-
verifies
cause
finite
a
the
show
experiments
that
the
at
error
the open-
is inversely proportional to
steady-state which
loop
and
function
free
Disturbances which enter the system before
workpiece model.
the
indicates
to zero
error
to
due
disturbances is insignificant.
S
The use of center-roll
was
feedback
experiments.
very
also
feedback
velocity
as
successful
the same manner
feedback,
increasing
by
demonstrated
as would
u
the
the
bandwidth
and
is somewhat
less
successful
is pushed
performance
at
to the
with
stability
higher
limit
true K
with
be expected
judicious choice of controller parameters.
system
K
The system responded to the velocity feedback
in exactly
tion
to simulate
u
a
This approximagains
or as the
because
the error
inherent in the approximation becomes more significant, as do
all errors at higher
step
tests
feedback
is
due
although
gains.
in part
the
The
small overshoot
to the
majority
of
errors
the
from
seen in the
the
overshoot
velocity
is
due
to
errors in the measurements.
Another result of the experiments which has practical
applications
is
the
of
effect
loaded curvature measurements.
ment
based on center-roll
the
in
the
The loaded curvature measure-
position
116
approximations
works
very well
for most
applications but this measurement method contains significant
and
but is much noisier
fast transients
during
accurate
command
The LVDT curvature measurement method is
changes rapidly.
more
the input
during the transient response if
error
more difficult to implement.
show
also
experiments
The
unmodeled
the
that
higher-
order dynamics, such as workpiece vibration, are significant
except
at
vibration
The
gains.
low
relatively
problem
appears to be a rather severe process-related limitation on
the
many
by
hardware.
the
the
of more
use
earlier,
can
limitations which
hardware-related
overcome
noted
As
response.
system
maximum
(and
powerful
there
are
generally
more
be
expensive)
Higher-order dynamics are physical limitations of
particular
In most
process.
system designs,
control
these higher-order effects can safely be ignored because they
occur above a certain high frequency.
The controller can be
designed so that the system bandwidth is always safely below
this
higher-order
As
frequency.
critical
explained
effects due to workpiece
in
Chapter
vibration
5
the
do not occur
at a constant frequency, because the natural frequencies of
are a function
the workpiece
the
dynamic behavior of
bending operation.
the
system
of beam length.
the workpiece
This means that
changes during
To safely ignore the vibration effects,
bandwidth
must
be
kept
well
below
natural frequency of the overhanging workpiece.
the
controller
must
the
be
designed
117
for
the
worst
the
lowest
Therefore
case.
This
imposes severe restrictions on
which
cannot
Discovery
be
of
response
removed
this
by
the
use
of
of
the
roll
limitation
accomplishes
maximum system
the major
performance
better
hardware.
bending
objective
of
the
system
control
analysis, which was to determine the upper limits of system
response.
The
particular
control
scheme
used
was able to meet the four specific
in Chapter
4. The
linear
control
in
the
experiments
control
objectives
listed
analysis
suggested
that
a
simple proportional controller in a velocity-servo based roll
bending system would give the required performance if centerroll
K
u
velocity
feedback
feedback.
velocity
The
is an accurate
nonlinear
is indeed
a valid
approximation
simulation
approximation
showed
to true
that
to K . The
the
simula-
u
tions
also
this
showed
control
model.
The
that
scheme,
experiments
the
system
has
even
with
the
proved
that
produce the expected response.
in very
good
system
good
stability
nonlinear
the control
with
workpiece
scheme
does
This control scheme results
response,
with
the roll
bending
system
bandwidth limited ultimately only by the workpiece dynamics.
Future Research
The
designed
ses
system
response
in Chapter
reported
in
possible
using
4 is a vast improvement
[18],
[20],
and
[21].
the
controller
over the responThe
maximum
roll
bending system response seems to be limited ultimately by
118
vibration
scheme
that
of the workpiece,
used
system
natural
at least with
the simple
in the experiments.
As noted
bandwidth
be greater
frequency
can
of
never
the
overhanging
above,
this means
than
beam.
control
the
It
lowest
might
be
possible, however, to improve system response somewhat
using
a
model .reference adaptive
self-tuning regulator.
to be changed
controller
(MRAC)
by
or a
The MRAC allows the controller gains
depending
on changes
in the
physical
system.
Rather than being designed for the worst case, the controller
can be designed
to give
the whole bending
piece
that
model
the
are a function
gains
some
process.
in Chapter
system
gains,
controller
system
3 and
and
adaptive
for
the
the
different
in Chapter
controller
4
gains,
Thus the controller
workpieces.
might
workpieces
throughout
from the work-
simulations
therefore
controller
to different
response
It is also obvious
of the bending stiffness.
must be changed
other
the best
be able
so that
An MRAC
to
the
response can be attained for each workpiece.
adapt
best
or
the
system
An adaptive
controller might also give better response for bidirectional
bending
could
if
the
increase
deadband
the gains
region
is
large.
in the deadband
The
region
controller
to move
the
system quickly through the region and then reduce the gains
outside
the region.
use of scheduled
A possible
gains, which
variation
can
of the MRAC
be used to detune
is the
the con-
troller as the overhanging workpiece natural frequency lowers
and becomes
significant.
This would
119
have the same effect
as
to implement.
the MRAC but might be simpler
Another control technique which could be explored in
The experiments show
future research is disturbance preview.
that
posed
is
controller
characteristics
of the
response
disturbance
the
But
good.
system
using
the
response
disturbance
the
for the straightening
are most important
pro-
pro-
cess, which also generally requires the greatest precision of
all the
Therefore,
processes.
roll bending
to increase
the
system response for straightening and still retain the precision,
to have
it is necessary
possible.
the best
disturbance
response
A control system that includes disturbance preview
holds promise for improving disturbance response for those
applications which
best disturbance rejection
the
require
possible.
In addition to control system improvements, there are
some
other
areas
As
response.
which
can
explored
be
one
earlier,
explained
to
system
improve
objective
of
the
experiments was to determine limitations on maximum system
response.
To
do
that
tradeoffs
certain
hardware and measurement methods.
for
both
to be a linear
elastic
approximation
result
and
function
plastic
in errors
made
in
the
One of the major tradeoffs
is the measurement of loaded curvature.
is assumed
were
The loaded curvature
of center-roll
bending.
of the
Errors
same magnitude
position
in
this
in the
unloaded curvature, so a better approximation or a better
measurement of loaded curvature will result in a more precise
120
final
product.
Thus there
is a need for more detailed
study
of the bending process from an engineering mechanics perspective.
In particular,
a simple
or computationally
efficient
relationship between loaded curvature and center-roll displacement
would
for
require
bending
tion,
the
a
in both
a more
more
the
a complete
include
full
loading
detailed
plastic
study
detailed
of
and
the
look
range
would
study
of
elastic
mechanics
in order
to expand
the
research
ideal.
the
of
This
mechanics
regions.
at the effects
the bending of unsymmetrical workpieces.
step
be
of
In addi-
bending
should
of coupling
in
This is a necessary
presented
to
include
control of two- and three-dimensional bending, which is a
natural continuation of this research.
Closed-loop control
of three-dimensional roll bending would make one-pass forming
of arbitrary workpiece shapes possible.
increase
the
productivity
of
the
This would not only
current
roll
bending
process, but would also greatly increase the versatility of
the process.
121
BIBLIOGRAPHY
1,
Hardt, D.E., "Shape Control in Metal Bending Processes:
The Model Measurement Tradeoff", editor, D.E. Hardt,
Information Control Problems in Manufacturing
Technology 1982, Fourth IFAC/IFIP Symposium, pp. 35-40.
2.
Allison,
B.T., and Gossard,
D.C., "Adaptive
Brakeforming"
Proc. 8th North American Manufacturing Research
Conference, May 1980, pp. 252-256.
3.
Stelson,
K.A.,
"Real Time Identification
of Workpiece
Material Characteristics From Measurements During
Brakeforming",Trans. of ASME L Journal of Engineering
for Industry,
Vol. 105, No. 1, February
1983, pp. 45-53.
4.
Sachs, G., Principles and Methods of Sheet Metal
Fabricating, 2ed., Reinhold, New York, 1966, p. 103.
5.
Shanley, F.R., "Elastic Theory in Sheet Metal Forming
Problems", Journal of the Aeronautical
No. 9, July 1942, pp. 313-333.
6.
Sciences,
Vol. 9,
Lubahn, J.D., and Sachs, G., "Bending of an Ideal Plastic
Metal", Trans. of ASME, Vol. 72, No. 1, January 1950, pp.
201-208.
7.
Gardiner, B.F.J., "The Springback of Metals", Trans.
of ASME, Vol. 79, 1957, pp. 1-9.
8.
House, R.N., Jr., and Shaffer, B.W., "Displacements
Wide Curved Bar Subjected to Pure Elastic-Plastic
in a
Bending", Trans of ASMEL Journal of Applied Mechanics,
Vol. 24, No. 1, March
9.
Marshall,
1957, pp. 447-452.
J., and Woo, D.M., "Spring-Back
Forming of Sheet Metal"
and Stretch-
The Engineer, August 28, 1959,
pp. 135-136.
10.
Shaffer,
B.W., and Ungar, E.E., "Mechanics
of the Sheet
Bending Process", Trans. of ASME, Journal of Applied
Mechanics,
11.
Bassett,
Vol. 27, No. 1, March 1960, pp. 34-40.
M.B.,
and Johnson,
W., "The Bending of Plate
using a Three-Roll Pyramid Type Plate Bending Machine",
Journal
of Strain Analysis,
Vol. 1, No. 5,
966,
pp.398-414.
.12.
De Angelis,
R.J., and Queener,
C.A., "Elastic
Springback
and Residual Stresses in Sheet Metal Formed by Bending",
122
Trans. of
757-768.
13.
15.
1968, pp.
Palazotto, A.N., and Seccombe, DA.,
Jr.,
"Springback of
Journal
Vol. 95, No. 3,
Wire Products Considering Natural Strain", Trans. of ASME,
of En gineering
August
14.
ASME., Vol. 61, No. 4, December
1973,
for IndustrE,
pp. 809--814.
Kobayashi,
S.,
Mechanical
Sciences,
and Oh, S.I.,
"Finite
Element Analysis
Plane-Strain Sheet Bending", International
Vol. 22, No. 9, 1980,
Journal of
pp., 583--594.
Hansen, N.E. 'and Jannerup, O.E., "Modelling of
Elastic-Plastic
Bending of Beams Using a Roller
Machine",
Trans.
of ASM.
of
Bending
Journal of Engineerin
X
for
Industry, Vol. 101, No. 3, August 1979, pp.304-310.
16.
Cook, G., Hansen, N.E., and Trostmann, E., "General Scheme
for Automatirc Control
of Continuous
Bending
Trans. of ASMEj Journal of Dynamic Systems
and Control,
17.
18.
Hardt,
-
Vol. 104, No. 2, June 1982, pp. 173-179.
Foster, GB.,
Forming
of Beams",
Measurement
"Springback Compensated Continuous Roll
Machines",
U.S. Patent No. 3955,389,
D.E., Roberts,
M.A., and Stelson,
May 11,1976.
K.A., "Closed-
Loop Shape Control of a Roll-Bending Process", Trans.
of ASME L Journal of Dnamic SstemsL Measurement and
Control,9 December
19.
Gossard,
1982, Vol. 104, No. 4
D.C., and Stelson,
pp. 37-322.
K.A., "An Adaptive
Pressbrake
Control Using an Elastic-Plastic Material Model", Trans.
of ASME. Journal of En ineering for Industry,
lo. 4, November 1982 , pp. 389-393.
20.
Hale, MB.,
and Hardt,
D.E., "Closed
Vol. 104,
Loop Control
of a
Roll Straightening Process", Annals of the CIRP,
Vol. 33, No. 1, August
21.
Lee, M., and Stelson,
1984, pp. 137-140.
K.A.,
"Adaptive
Control
for a
Straightening Process", Sensors and Controls for
Automated Manufacturing and Robotics, ASME, New York,
1984, pp. 153-166.
22,
Roberts,
MA.,
"Experimental Investigation of Material
Adaptive Springback Compensation in Roller Bending", M\S
Thesi:, Massachusetts
23.
Institute
of Technology,
1981.
Crandall, S.H., Dahl, N.C., and Lardner, T.J., editors,
An Introduction to the Mechanics of Solids, 2ed,
McGraw-Hill, New York, 1959, pp. 455-477.
123
24.
25.
Cook, N.H., Mechanics and Materials for Design,
McGraw-Hill, New York, 1984, pp. 388-396.
Franklin,
G.F., and Powell, J.D., Digital
Control of
Dynamic Systems, Addison-Wesley, Reading, Mass., 1980.
26.
Ogata, K., Modern Control Engineering, Prentice-Hall,
Englewood
Cliffs,
N.J.,
1970.
124
Appendix
EXPERIMENTAL BENDING APPARATUS
The experimental three-roll bending apparatus shown in
Figure 10 consists of three roll-pairs mounted on a Bridgeport milling machine.
The roll-pairs contain instrumentation
needed to implement the closed-loop control scheme outlined
in Chapter
2.
connected
sition
The
instruments
to a digital
and
servo
implementation
and
computer
control.
details
the milling
are
that is used for data acqui-
Descriptions
are
machine
given
of the hardware
below.
The
and
equations
for
calculating moment and curvature are also developed.
The Bridgeport milling machine is used as a drive mechanism to feed the workpiece
forming
center
mechanism.
roll
The
in the
through
feed
milling
is
the rolls
achieved
spindle,
milling machine spindle motor.
and also as the
by
which
mounting
is driven
the
by the
The forming action in any
three-roll pyramid bending machine is achieved by moving the
center
roll relative
case, the center
to the outer
rolls.
roll is fixed and the outer
result is identical in either case.
is changed
are
through
brushless
the
by moving
attached.
a
5
The
to
1
resolver
milling
In this
bed.
the milling
milling
ballscrew.
bed to which
is
driven
The
DC
motor
is used
The
position
125
rolls move.
The
The outer-roll position
bed
which
particular
the outer rolls
by
is
to measure
measurement
a
DC
fitted
the
motor
with
position
system
has
a
of
a
resolution
of 0.00032
Electric
General
in.
The
DC motor
is controlled
in
servo-controller which
commands
from
a
DEC LSI-11
equipped
with
A/D
and
computer.
D/A
The
converters,
turn
DEC
reads
by a
receives
computer,
stores
and
These measurements are
measurements from all transducers.
used to generate a control signal according to some algorithm
(see
Chapter
controller
4).
which
The
control
directs
the
signal
DC
is
motor
passed
to
move
to
the
GE
milling
the
This movement causes a change in unloaded curvature of
bed.
the workpiece.
Figure 62 shows the hardware
relationships
in
the roll bending apparatus.
The
three
roll-pairs
all
adjustable "pinch" roller.
consist
of
one fixed
curvature)
swivel
This
allows
scheme
forming.
about
allows
the
the
that no moment
piece
cocking
bidirectional
three
The
centerline
roll
one
The pinch rollers are adjustable
by means of bolts which clamp the rolls together.
roll
and
pairs
(positive
roll-pairs
of the
to seek
fixed
are
roll
a "neutral"
is applied
to the workpiece
in the rolls.
The rotation
pair is also used to find the location
This pinch
and
negative
all
free
(Figure
to
63).
position
so
due to the workof the center
of the maximum
roll
moment
and maximum curvature along the workpiece as shown in Figure
71.
The center-roll housing, shown in Figure 64, contains
instrumentation for measuring the curvature of the workpiece
and the
rotation
of the
roll-pair.
126
The
shaft
of the
fixed
Position
Figure 62.
Roll Bender Hardu
127
" Relationships
F
0
4C
1
a
^
a
co
H
Z
p d __-- 0
41a(
r^^
~~~~~~~~~~~~~~
1.
.ade
p
C)
.1
=3
,4
L;c
IH
pj
I
C)
IE
.- J.
C)
1-
128
I
-z
i
r
.
0
C)
C2)
-
I
C)
e
129
center
drive
roll is mounted
system
in the milling
is used
to drive
the center
workpiece through the apparatus.
by two linear variable
with
a measurement
The
roll
and
spindle
feed
the
The curvature is measured
differential
range
spindle.
transformers
of ±O.1
in.
The
(LVDT), each
LVDT measurement
resolution is infinite since LVDT's are analog transducers.
Digitizing
the
resolution.
a
For
resolution
apparatus,
roll.
signal
the
computer
this particular
of
about
one LVDT
Appendix
in
2
0.00005
is placed
contains
introduces
application
in.
On
the
side
discussion
of
various LVDT positioning schemes.
finite
the LVDT's
on either
a
a
have
experimental
of the center
the
merits
of
The position of the LVDT's
is adjustable
from 0.5 to 1.75 in from the centerline
of the
center
This
of the
roll.
transducers
to be used
stiffnesses.
but measure
allows
the full measurement
with workpieces
of different
the
linear
displacement
below.
of
the
can be used to estimate
any one of several methods.
The
LVDT
shafts
are
spring
roll pair is measured
with a potentiometer,
filtered
LVDT
before
67 are plots
and
potentiometer
being
of the
sent
The
by
Specific curvature equations are
as shown in Figure 65.
The
workpiece.
the curvature
against the workpiece
68.
bendin S
The LVDT's do not measure curvature directly,
linear displacement
derived
range
signals
130
shown
bear
of the
in Figure
amplified
Figures
characteristics
and potentiometer signals, respectively.
to
Rotation
are
to the computer.
static noise
loaded
and
66 and
of the LVDT
'J.;;
I','
q
I
0
C)
ZC
a)
E
C)
c
)
4%
7,
C)
.-i
12
I
131
E-04
8
4
I
I
0
-4
-0
-12
0
1
2
3
4
5
TIME (SEC)
Figure 66.
LVDT
6
E.O0
Noise
E-02
O
8
4
2
-
a
I-
-2
-4
-6
-8
-10
0
1
2
3
4
TIME (SEC)
Figure 67.
Potentiometer Noise
132
5
6
E+00
The
infeed
roll-pair,
which
has
no instrumentation,
mounted directly on the milling bed.
is
The outfeed roll-pair
is attached to a strain-gage force transducer (Figure 68)
which
has a measurement
force
transducer
Advanced
force
and
lb
0.028
force
of
model
Inc.
The
is 0.087
measurement
in the
250 lb on each axis.
SRMC3-2-250
Technology
y direction.
to an L-bracket
bolted
The
is
Mechanical
digitized
range
transducer
which
the
acting on the outfeed roll-pair.
by
of
the
resolution
force
x direction
transducer
to the milling
is clamped
measures
manufactured
lb in the
The
The
two
is
bed.
forces
orthogonal
The relationship between
the forces acting
on the roll-pair
and the maximum
moment
is
developed below.
The signals from the force transducer are
amplified and filtered before being sent to the computer.
Figures
69 and
70 are
plots
of the
static
noise
character-
istics of the force transducer.
Figure
71
is
a
schematic
view
of
the
experimental
bending apparatus.
The equations needed to calculate moment
and
developed
curvature
Figures
equation
are
71 and 72.
are
below
contact
very
small
small
compared
to the
that
the
neutral
to radius
then an adjustment
and
can
be
(see Appendix 2).
assumes
point
using
the
notation
of
Note that many of the terms in the moment
bending conditions
shown
below
distance
axis
133
for
certain
The moment equation
from
of the
of the rolls.
can be made to r1
neglected
the
center
workpiece
If this
roll
is very
is not true
and r 2 to reflect
this.
01)
Ojo
134
E-01
'u
a
8
4
2
&
-jn
I
U.
0
-2
-4
-1
-8
-10
0
1
2
3
4
5
TINE (SEC)
Figure
69.
6
E+00
F x Noise
E-O1
10
8
8
4
2
U
-2
-4
-6
-8
-10
O0
1
2
3
4
T I ME (SEC)
Figure
70.
F
Y
135
Noise
5
Ea
EO0
a
-
_- C
I
C =
H (ntI
\
/
I-j
0
I
0
a)
a)
1~0
0)
'-4
N
-J
.
CO
X~co
I
LL
I
'C
136
\
00
*64
-
-
d2
I
x
-
w lI
z
z
R-LY
R
R
Figure 72.
Curvature Measurement Scheme
137
__
The equation for calculating the maximum moment is:
M
= F D
max
x
+ F D
y
y
(34)
x
where
Dy = z -
Dx
There
loaded
=
d2 + r
are
curvature
is constant.
(see Figure
1
sine1 - r2cose
22
2
several
curvature,
workpiece
r1 (1 - cose1 ) - r 2 (1 - sine 2 )
KL.
possible
One
option
in the region
A curvature
can
then
schemes
is
to
of the
for
calculating
assume
that
the maximum
be calculated
the
moment
as follows
72):
s2 + (R - L
)
2
= 2
Rearranging this equation yields:
K
Equation
L
= 1/R = 2L2 /(s 22 + L
35 can be applied
2
2.
2
2
)
(35)
to each LVDT measurement
results averaged as discussed in Appendix 2.
and the
An alternate
method of calculating curvature can be found from linear beam
138
beam shown
Consider the cantilever
theory.
in
Figure 72
which represents the portion of the workpiece between the
center
and
deflection
outer
The
rolls.
point x along
of the beam at any
assuming
loaded curvature,
maximum
relationship
the
between
the beam and the
small
is
deflections,
developed below.
2
ay
F(d 2 -=K
a2
KL
ax
where
EI
maximum
is the
curvature
bending
-
x)
~~~~~~~~(36)
2
EL
EI
(36)
of
stiffness
the
workpiece.
The
occurs at x = 0 which means that:
KL
= Fd 2 /EI
(37)
Integrating Equation 36 twice with
ay
-(0)
and
= O
y(O)
= 0
ax
yields:
F
d x2
_
Yx
EI
where
x3
Yx is the deflection
2
=
6
of the workpiece
139
(38)
at any
point
x
along the workpiece.
37
gives
the
Substituting Equation 38 into Equation
between
relationship
curvature and the deflection
the
maximum
loaded
of the beam.
(39)
KL
This
the
equation
shows
displacement
limiting
case,
is
the
just
curvature
of
that the loaded
the
workpiece
is a linear
at any
for x = d 2 , the displacement
displacement
of
the
center
point
function
x.
of
In the
of the workpiece
roll.
The
loaded
is then:
KL = 3z/(d2 )
where z is the center-roll
displacement.
140
(40)
Appendix
2
MEASUREMENT ALTERNATIVES
The
closed-loop
developed
control
in Chapter
of
2 depends
a
roll
entirely
bending
process
as
on
ability
to
the
measure the unloaded curvature, Ku, while the workpiece is
still
in the
loaded
state.
As
shown
in Equation
4, K
is
u
actually a combination of three separate measurements.
These
are loaded curvature, KL; moment, M; and bending stiffness,
dM/dK.
The bending stiffness is considered constant for a
given workpiece and can be calculated rather than measured if
enough information is known about the workpiece.
there
are advantages
to measuring
it will be considered
The details
are
given
a measurement
of several
in Appendix
methods.
This
the bending
1.
appendix
contains
stiffness,
so
for the general case.
measurement
These
In practice
are
a
methods
not
for K
L
the only
discussion
of
and M
available
all
the
measurement methods used on the experimental roll bending
apparatus.
depend
on
required
Because
the
type
performance,
advantages
to accuracy
of
an
bending
to
attempt
is made
and disadvantages
be
performed
to
point
and
the
out
the
of each of the methods in regard
and type of bending.
It is important
loaded
the choice of measurement method will
curvature
and
to notice
moment
that absolute
are
not
measurements
required.
As shown
of
in
the control program listing in Appendix 4, the analog instru-
141
ments
on the experimental
workpiece.
each
using
initialized
necessary
to
curvature.
curvature
Point
But,
that
extreme
any
point
for each workpiece
absolute
be
zero
moment
and
The unloaded
of whether
zero moment and loaded curv-
along
the
the origin.
in setting
must
for
it is not
will be the same regardless
be considered
care
instruments
of the workpiece,
at
B is considered
In fact
may
them
the
be initialized
this consider Figure 73.
measurement
line
though
a flat section
To se
A or Point
ature.
even
initialize
must
apparatus
linear
elastic
Practically,
up the
roll
loading
this means
bending
apparatus
This is a nice feature of
is unnecessary.
the unloaded curvature measurement scheme since less setup
time results
in higher
productivity.
Also high
precision
is
sometimes difficult to obtain on a factory floor.
Loaded Curvature Measurement
According to the bending control algorithm developed in
Chapter
at
a
2, it is necessary
specific
center
roll.
point,
But
to know, or measure,
namely
the point
a point
measuring
nearly
lated
constant
methods,
then,
in the region
by finding,
contact
curvature
is
with
the
extremely
The assumption behind the fol-
difficult, if not impossible.
.lowing measurement
of
the curvature
is that
of interest
or estimating,
the shape
the
curvature
is
and can
be calcu-
of the
workpiece
in this region.
Figure
74
shows
two
of
the
142
methods
used
to
measure
c
of
S
E
a
N
A
L
K
Figure 73.
Curvature
Ju
Translated Moment-Curvature Origin
I
Method
152
Method
S2
Figure 74.
w
2
Sl
LVDT Configurations
143
loaded
curvature.
contacts
These
the sheet.
to measure
methods
A linear displacement
the displacement
non-contacting
noise
is more
generally
methods
of
In both methods
ments
were
greater
used.
decreased
linear
such
due
Because
optical
because
of
implement
4),
than
measurements,
surface
measurements
increased
but
irregulari-
two displacement
measure-
might
provide
complexity
computation
side of the center
unloading
(Figure
is used
as it deflects
to
relationship between moment
the
that
and
in
the
In the first method, both measurements
the moment is
position
as
to
on the unloading
in
simpler
shown, only
performance
because the
transducer
but at the cost of greater
discrete controller.
are made
much
Additional
precision,
roller
Contacting methods of curvature
a problem
ties.
a
of the workpiece
during the bending process.
measurement are
employ
it
region
as
is
and
seen
nearly a linear
is easy
to see
roll.
This
is
curvature
is
in
Figure
3.
function of sheet
that
the
curvature
should vary linearly with distance along the workpiece on the
unloading
behaved
side,
side.
function
Thus
on
which
includes
measurements
of each
estimate curvature.
the loaded
curvature
the unloading
side
the nonlinear
method
than
plastic
can be combined
The three known
is actually
mated
by
the
circular,
inverse
of
then
the
the
radius
144
on the
region.
better
loading
The
two
in many ways
to
points are enough to
define a unique circle through the points.
shape
is a much
If the workpiece
curvature
of
can be esti-
curvature
for
this
circle.
are
Or
assumed
known
then
curvature,
to be
points
average
of
symmetric
for
the curvatures
is the basis
a circle
option
and
workpiece.
the
found
point
then
the
between
a
curvature
above
options
curvature
is
and
two
to
circles
curvature.
can
This
Another
measurements
of
the
location
curvature.
assume
that
displacement
to define
curvature
it is possible
to the point of maximum
five
A weighted
displacements.
change
from
the
35, which is the equation
of the
of
backward
these
for the loaded
by one
rate
1,
two circles.
from
With this information
transducer
all
displacements,
about
for Equation
defined
therefore
is to use the two displacement
curvature
late
and
can be used to define
be used as an estimate
approach
a
if the
is
along
to extrapoof the
first
A variation
the
as
the
on
relationship
defined
for
an
elastic cantilever beam (see Equation 39) rather than assuming a circular shape.
options
is that many
One problem with all these measurement
of the assumptions
near the point of interest.
displacement
measurements
Greater
made
are
only
error is expected
are made farther
valid
as the
from the point of
maximum curvature under the center roll.
A more practical consideration eliminated all of these
options
as
useful
experimental
shown
measurement
apparatus
in Figure
64,
used.
methods,
The
has a tendency
center
roll
is
the
drive
145
roll,
least
center-roll
to vibrate
piece is rolled through the rollers.
the
at
for
the
apparatus,
as the work-
In addition, because
a
moment
is
generated
between
the
center
roll
and
the workpiece
which
causes
the
center-roll apparatus to rotate slightly when the drive motor
Because the linear displacement transducers
is turned on.
are
attached
to the
transducers.
rotation
The
the
of
any
rotation
of
on the displacement
in a false reading
results
the apparatus
apparatus,
center-roll
apparatus
center-roll
causes an offset in the curvature reading while the vibration
causes large amplitude noise at about the natural frequency
to runout-of
the
Another source of error is due
apparatus.
of the center-roll
center
roll.
roll
If the center
has
any
runout, the workpiece will translate perpendicular to the
longitudinal
detected
of
axis
the
This
workpiece.
transducers
by the displacement
translation
is
as a
and interpreted
change in curvature.
the errors due to center roll rotation,
To eliminate
of the displacement
of the center
so
roll.
relationship
ment
this
method
first method.
the errors
was moved to the loading
transducers
As noted
above,
is not as well
would
appear
not
one
side
the curvature-displace-
behaved
to
on the
be as
input
accurate
side,
as
the
However, the new configuration greatly reduces
mentioned
above.
Any rotation
of the center
roll
will cause one of the transducer displacement measurements to
increase,
but at the same time
site side will decrease.
the same distance
ments
will
If the transducers
from the contact
be the same
magnitude.
146
~~~~~~~~.
. . ...
the measurement
point,
on the oppo-
are both located
then the displace-
Otherwise,
the
displace-
ments will only be proportional, with the proportion depending on the relative
Taking
measurement.
these
Figure
75
the
the contact
average
of
shows
the
the
point for each
curvature
based
curvature
the errors
should eliminate
displacements
tion.
from
distance
due
to rotafor
measurements
on
each
transducer and the average curvature measured for a disturbance test with the transducers
the center
roll.
easily
several
This
be
indicates
Notice
times
that
on opposite
mounted
due to rotation
that the errors
as
large
the
measuring
as
sides
the
actual
can
curvature.
on both
displacement
of
the
loading and unloading sides is necessary despite the shortcomings
mentioned
above.
The
errors
due
to
runout
of
but careful
center roll are not eliminated with this method,
E-3
im
10
0
-5
-10
-15
-.2n
0
1
3
2
4
5
a
E..00
TIME CSEmn
Figure 75.
Opposing LVDT Curvature Measurement
147
srzyr6L..`iYWb.Y;-3irteiLILkb-
the
of
machining
the
roll
center
reduced
This reduced error due to runout
to
noise
below
quite
most
in.
measurement
used
was
This
experiments.
large
curvatures,
displacements.
in large
result
curvatures
0.001
method
measurement
for bending
successful
the large
because
in the curvature
in many of the bending
successfully
method was
This
levels.
to
runout
For straightening applications, the signal-to-noise ratio was
low and good readings
to obtain.
were difficult
A final method of measuring curvature, which was also
used
is
successfully,
above.
discussed
a
For
case
specialized
bending
some
the
of
method
such
operations,
straightening, the curvature magnitudes are very small.
means
that the displacements
very near
the
center
necessary
farther
be
the
roll,
will
the
resolution,
from the center
greater.
measured,
In
the
roll
be very
moved outward as far as the outer
can
be
moved
displacements
transducers
the
rolls.
are
To achieve
small.
transducers
case
This
if the transducers
so that the
limiting
as
will
can
Measurements
be
from
this limiting case result in exactly the same displacement as
measuring
the displacement
outer rolls.
mated
of the center
roll relative
The loaded curvature, therefore, can be esti-
using only a single measurement of
displacement.
measurements
The
and
advantage
of
calculations
the center-roll
this
method
are
needed,
increased speed in the discrete controller.
this
to the
measurement
is very
easy
to make.
148
Most
is
that
which
In
fewer
means
addition,
servo systems
incorporate the position measurement in the servo controller,
so little additional hardware is needed.
is that this measurement
sense
and therefore
Another advantage
is a non-contacting
measurement
does not have the noise
problems
ated with the contacting methods described above.
in a
associ-
The major
disadvantage is that there is no simple, precise relationship
between
center-roll displacement
approximation
is given by Equation
and curvature.
40.
A linear
But the errors due to
the estimates and assumptions outlined above increase as the
measurements
curvature
method
made
farther
from
the
and as the speed of response
is
certain
are
less
accurate
than
some
bending applications,
point
of
increases.
other
such as
methods,
maximum
Thus
but
this
for
straightening stiff
workpieces, the advantages might outweigh the disadvantages.
Moment Measurement
Figure 71 shows the measurements which are necessary to
calculate
the moment
curvature.
sary
at the point of maximum
In the most general case, it might also be neces-
to include
particularly
in the workpiece
an applied
if the outfeed
moment
at the
roll-pair
does
outfeed
roll,
not rotate.
Mb,
For
this most general one-dimensional bending case there are two
force measurements, each with a respective moment arm, and
one
moment
measurement.
Thus
it is necessary
quantities to calculate the maximum moment.
to know
five
The relationship
between each of these quantities and the maximum moment is
149
-
given by:
M
max
= Fx
D
y
+x
Mb
y
(41)
where
z -
Dy=
D
d
Dx
r (1 - cosOl) - r 2 (1 - sinG 2 )
+ r
2
1
sine
-
1
r cosO
2
2
2
Careful study of this equation along with intuition should
indicate
that
important
some
than
process under
adjustment
for
quantities will
depending
that would
straightening
very
D
x
be incurred
to be constant
stiff
very small
for this particular
curvatures
91
can
Note, however,
to r,
even if 91 is very
design
This example
one
satisfactory
type
moment
for
all
of
more
bending
angle
the
81
to the moment
the
will be negligible
workpiece.
even
E1
since
will
be
For bending
large
for
stiff
fairly
that if d2 is very large compared
large,
indicates
measurement
bending
various bending processes.
much
by eliminating
workpieces,
substantial,
be
workpieces.
be small.
the
be
71 is used to make adjustments
and assuming
then
on
For instance,
consideration.
error
x
the
others,
shown in Figure
arm D . The
of
the effect
on D
x
will
that it is not possible
scheme
machines
and
that
for
will
all
of
to
be
the
The following observations can be
150
used as general guidelines.
in Equation
values of all variables
only if typical
obtained
More specific information can be
41 are available.
Mb
is almost
always
negligible
free
to rotate
or
there
The moment
is
roll-pair
if
only
is
outer
roll, as there might be in unidirectional
might
also be negligible
if
the
outer
of
levels
is
are
low.
it is
where
necessary
eliminate
the
outer
moment
applied
a
If
It
process,
small
even
because
rotate
of curvature
the levels
to allow
the
to
single
the
can
error
be
certainly simplifies the hardware.
tolerated, neglecting Mb
For bending,
free
not
a
bending.
for the straightening
roll-pair
curvature
if the outer
or
are
to rotate
roll-pair
to
much higher,
measure
the
to
applied
moment.
The
force
F
the
is generally
largest
force
and
when
Y
combined with D , which is generally the largest moment arm,
is the most dominant
it is easy to see that F
measurement
in
Y
of the maximum
the calculation
moment.
F
be ignored
cannot
Y
in any of the bending
for most straightening
will be small
F X, however,
processes.
operations.
is also very
And since D
Y
small for straightening,
the combined
due
to F
x
might
also
of F
x
and D y
is
The component of the maximum
negligible for straightening.
moment
effect
be negligible
for many
bending
operations if high precision is not a concern.
The moment
measured
using
described
in the experiments
both the F y and F
151
x
in Chapter
measurements,
5 was
but with
no
A first-order approximation
adjustment to the moment arm D .
Y
to
the
sine
moment
term
in D
measurement
was
x
The
retained.
approximation
was
result
less
of
precision
this
but
improved dynamic response because the decreased computation
Nevertheless,
resulted in a much faster discrete controller.
the
of
precision
experiments
this
in Chapter
bending
apparatus
resulted
in a very high
was
approximation
5
because
the
and
the
good
geometry
stiffness
signal-to-noise
very
of
the
of
the
ratio
for
the
roll
workpieces
from the force
transducer.
Bending Stiffness Measurement
As mentioned earlier, the bending stiffness can usually
be
considered
value
constant
formed,
constant
is
for
known
a
for
given
workpiece.
workpiece
each
general
will
that
this
be
into the control
then the value can simply be entered
program.
If
Measuring the bending stiffness, though, is a more
procedure
which
properties to be formed.
allows
workpieces
with
unknown
The procedure employed for measur-
ing the bending stiffness with the experimental apparatus is
as
follows.
apparatus
workpiece
and
The
the
workpiece
is
first
instruments
is loaded throughout
are
loaded
in
initialized.
the linear
elastic
the
bending
Then
the
range.
At
specific intervals in this loading cycle the computer calculates loaded curvature and maximum moment.
After the loading
cycle is completed, a least squares curve fitting routine is
152
used
to
This
straight
line
shown
bending
just
fit
a
straight
line
in
by
is,
Figure
the
the
of
slope
the
line
shown
elastic
both
negative
and
positive,
elastic
this
in
loading
loading
line
is
Figure
the
3
is
which
line
The full elastic
continues into the negative loading region.
region,
data.
moment-curvature
definition,
loading
of
half
positive
to
The
3.
The
stiffness.
the
line
is used
to measure
the
bending stiffness.
Measuring the bending stiffness in this manner increases
the roll bending system flexibility because no prior knowledge of the workpiece
properties
But there is an
is needed.
even more compelling reason for measuring the bending stiffThe curvature and moment measure-
ness of each workpiece.
ments
are
error
in
both
calibrated
either
calibration
measurement
stiffness
will
then
calibration uses
the
tend to reduce
unloaded curvature calculation.
stiffness
but
separately,
if
there
calibrated
the errors
is an
bending
in the
This is because the bending
the measured curvature
and the
measured moment to establish the relationship between the
elastic
moment
calibration
two.
and curvature.
for the
loaded
curvature
instance,
is off
suppose
by a factor
the
of
In other words, suppose the actual loaded curvature is
half the measured value.
(slope
For
of
the
actual value.
elastic
Then the measured bending stiffness
loading
line)
will
also
be half
the
The unloaded curvature based on the incorrect
loaded curvature measurement, the correct moment measurement,
153
and the calibrated bending stiffness will be exactly correct
in the linear
elastic
plastic region
than
bending stiffness.
bending
stiffness
region,
and more nearly
a measurement
based
on
In fact the error with
will,
for
this
example,
correct
in the
precalculated
the calibrated
be
equal
to
the
actual unloaded curvature, while the error with a precalculated
bending
curvature.
between
the
operations,
stiffness
would
be equal
to the
actual
loaded
The reduction in error is equal to the difference
loaded and
this reduction
unloaded curvatures.
For
may not be significant
bending
because for
large curvatures and stiff workpieces, the difference between
the
loaded
and
unloaded
curvatures
straightening operations, where
zero,
measuring
stantial
the
reduction
bending
be
the unloaded
stiffness
of error.
will
can
In this
small.
For
curvature is
result
in a sub-
particular
bending
operation the bending stiffness measurement is essentially a
"system" calibration
which
encompasses
bending system.
154
the
complete
roll
Appendix
3
ERROR ANALYSIS
The experiments described in Chapter 5 were performed
strictly
to evaluate
the
new control
Chapter 4. Because the purpose was
scheme
proposed in
to attain the fastest
system response possible, the system precision was compromised to increase speed.
For this reason, system precision
was ignored and no attempt
of the final workpiece
was made to measure
shape.
The assumption
the precision
is that if the
closed-loop control system has the required response, then
the system
cision.
precision
is only limited by the measurement
It is instructive,
curvature
equation
though,
to examine
to see the effect of errors
pre-
the unloaded
in the indivi-
dual measurements on the final workpiece shape.
The follow-
ing analysis presents some typical values which illustrate
the precision required for a straightening operation, which
generally
has
the
bending processes.
bending
operations
most
stringent
requirements
of
all
the
The analysis can be generalized for other
to
determine what
instrumentation and
measurement methods are most appropriate.
The equation used to calculate unloaded curvature is:
K u = KL -M/S
where
K
is
curvature, M
curvature,
M
unloaded
is
is
maximum
maximum
curvature,
KL
moment
moment
S
S
155
si
_
_
AL
_Ee._kLUi-U*L~--
(42)
and
and
is
is
is
maximum
the
the
slope
slope
loaded
of
of
the
the
elastic
error
loading
in
unloaded
the
line
shown
three
curvature
in Figure
measurements,
can be expressed
3.
KL,
If
M
there
or
S,
is any
then
the
by:
M + dM
+ dK
K
u
where
dK ,
terms.
dKL,
u
= K
+ dK L
(43)
KL~dLd
S + dS
dM, and dS are the error
Equation
43
can
be
combined
for the respective
with
Equation
36
and
rearranged to obtain the following error equation.
dK
M dS
2(44)
S2 + SdS
= dK L +
u
dM
S + dS
Figure 76 shows how each of the measurement errors affects
the unloaded
curvature
maximum acceptable
examination
1.
If M and
S are
If
KL
and
u
is the
44 and Figure 76 reveals:
known
acceptable
measurement,
If dK (max)
error for the unloaded curvature, then
of Equation
maximum
2.
of the workpiece.
error
dKL(max),
S
are
exactly
(dM = dS =
for
is exactly
known
exactly
the
)
loaded
then
the
curvature
dK (max).
(dK
=
S
=
)
then
dM
=
)
then
dM(max) = -dK (max)S.
U
3.
If KL
and
M
are
known
exactly
(dKL
dS(max) = dK (max)S2/(M-dK (max)S).
U
U
156
=
/:
4-
C
I:T
E
a
/iT
| : dM
/1
dKL
(
mI
I
I
I
I
I
of a
H
dKu
dKu
Figure 76.
precision
S
II
I
dKu
can
S+dS
I
I
analysis
I
I
I
I
II
I
This
/
I
/I
I
I
I
be
Curvature
Measurement Errors
used
to
find
given measurement.
the
minimum
Consider
two
required
specific
examples:
1) 1/8"
X 1" Aluminum
S = 1880
lb-in
Strip
2)
2
L
= 0.1
in
X 1!' Aluminum
S = 29000
Strip
.2
lb-in
M = 1400 in-lb
M = 180 in-lb
K
1/4"
1
K
L
= 0.05
in
For the straightening operation, specifications for maximum
acceptable curvature error are generally provided in terms of
deviation from a flat surface.
157
A typical maximum acceptable
__
deviation
Assuming
ness
for
the
a constant
specification
the
two
cases
curvature
can
above
is
0.0125
in/ft.
over a 5 ft span, the straight-
be changed
to
curvature
by applying
Equation 35.
The maximum acceptable unloaded curvature error
is
be
found
to
0.0007
in
1
.
Using
these
numbers
it
is
possible to establish typical values for the maximum acceptable error for each of the measurements.
Case
1:
dKL(max)
=
0.0007 in - 1 or
dM(max)
=
1.3 in-lb
or ± 0.7%
dS(max)
=
14 lb-in 2
or ± 0.7%
dKL(max)
=
dM(max)
= + 20 in-lb
dS(max)
=
L
Case
+
0.7%
2:
L
0.0007
in- 1 or ± 1.4%
or ± 1.4%
420 lb-in2
or
1.4%
These numbers show that required precision drops rapidly for
the moment and bending stiffness measurements as the bending
stiffness of the workpiece increases.
curvature
measurement
workpiece properties.
precision
The required loaded
is independent
of any of the
Of course the maximum allowable errors
given above only define the minimum required precision since
the errors may all occur
increase
at once.
The errors
the total error or to cancel
158
can combine
one another.
to
Appendix
4
COMPUTER PROGRAMS
This appendix contains a listing of all the programs
used
in modeling,
analysis,
bending apparatus.
Many
and control
of the
variations of these
experimental
programs are
used, depending on the instrumentation and control algorithm
being
employed
programs
all
for
a particular
contain
the
same
experiment,
structure
and
but
the various
differ
only
in
minor areas not important to the essential logic.
Bending Control Program
The program BEND is used to control the experimental
bending apparatus during the closed-loop forming operation.
A flow chart
of the program
is presented
in Figure 77 and the
program listing is given on the following pages.
tion of the program and its operation
is provided
A descripbelow along
with a definition of the major variables and subroutines.
Variables:
K1 - Reading
from the x direction
of the force
transducer.
K2 - Reading
from the y direction
of the force
transducer.
K3 - Reading from the center-roll rotation
potentiometer.
K4 - Reading from the servo tachometer.
K5 - Reading from the servo resolver.
159
C1 - Calibration
factor to convert K1 to an equivalent t
curvature.
C2 - Calibration
factor to convert K2 to an equivalent
curvature.
C3 - Calibration
factor
to convert K3 to an equivalent
curvature.
C4 - Calibration factor to convert K4 into center-rollt
velocity.
C5 - Calibration factor to convert K5 into center-roll
position.
SLOPE - Bending stiffness of the workpiece.
GAIN1,GAIN2 - Controller gains.
RKD - Desired curvature.
RKU - Measured unloaded curvature.
ICOM - Controller command to the velocity servo.
Subroutines:
OUTPOS - sends a position command to the servo.
INPOS - reads the servo
position.
OUTVEL - sends a velocity command to the servo.
SCAL - measures the workpiece bending stiffness.
SZERO - returns the initial value of all analog
transducers.
ATOD - reads the analog transducers into the computer.
Lines
1-10 of the program
calibration
geometry,
factors
C-C5
are all
but they are scaled
varies with
(lines 36-91)
initialize
fixed
for a given
by the bending
the particular workpiece.
is used
all variables.
to measure
160
the
The
machine
stiffness
which
The subroutine SCAL
bending
stiffness
(see
2 for a discussion
Appendix
SCAL calculates
ing stiffness).
loading
the
workpiece
ways to find the bend-
of various
and
the bending stiffness
taking
measurements
of
by
loaded
curvature and moment at various points in the elastic loading
region.
These
by means
slope
f a simple
the
by definition,
and
defined
J,
are
44-62,
fixed
once.
a
The
entered
lines
piece
is set,
of
the average
five
The
the
This
for
each
does
technique
but M, N, and J are
and
must
listing
but it could
the subroutine
to read the initial
noise.
this
N,
lines
that
stiffness
be
shows
found
only
SZERO
(lines
values of the analog
of 20 measurements
value.
This
minimizes
the
desired
easily be read in as an
desired curvature is variable.
th e
initial
that
workpiece
11-19.
a s a constant,
if
the
in
to ensure
used
of the workpiece,
particular
array
takes
are
is,
M,
variables
a
controller gains and the desired curvature are
in
curvature
The slope
workpieces
of the bending
Notice
workpiece.
som e knowledge
for
The
calculate
loaded throughout the full elastic region.
particular
require
to
regression.
different
which
th e best estimate
provides
used
stiffness.
several
factors
then
linear
bending
for
scale
iS
workpiece
are
measurements
When the work-
92-120)
is called
transducers.
for each
any
SZERO
transducer
variance
due
as
to
The subroutine as written actually averages four sets
average
possibility
of
measurements.
overflow
This
because the
is
eliminate
computer has
capacity in representing integer numbers.
161
to
the
limited
The real-time bending control algorithm is contained in
21-27.
lines
because
For
data.
the
ignored,
22 and
could
memory
DO-loop
the experi-
to store
data acquisition
in an
be placed
a
reads
computer
all
is
loop
infinite
for the end of the workpiece.
to check
23 the
within
placed
runs, where
production
algorithm
with a statement
is
has limited
the computer
mental
lines
algorithm
The
In
measurements.
The
subroutine ATOD is a machine language procedure which reads
the analog
transducers
measurements
in the
which
and the initial
K-K4.
variables
is
in digital
values.
INPOS
form
for
The
returns
this
the
Ku =KL
This
is easily
tions
from
equation
loaded
Appendix
such
- (FxD
verified
the
in line 24 uses
measurement
computation
2 contains
compromises.
speed
at
a complete
Line
25
is
the transducer
the appropriate
given
center-roll
(see
the
expense
discussion
the
substituThe
above.
position
as the
1).
This
Appendix
of
precision.
of the
implementation
digital controller design (see Chapter 4).
162
Next,
is of the form:
definitions
scaled
servo.
+ F D )/SLOPE
by making
variable
curvature
increases
The equation
(line 24).
position,
servo
particular
the
is returned
result
the unloaded curvature is calculated using
measurements
between
takes the difference
and
effect
of
of
the
In this particu-
lar
case
the
difference
controller
is
just
a
gain
operating
on
between the desired curvature and the
measurement.
The
feedback
measurement
the
feedback
is a weighted
sum of
the unloaded curvature and the center-roll velocity, which is
related
to the rate of change
in Chapter
in
line
25
4.
is of
Thus
of unloaded
the velocity
the
command
curvature
to the
as shown
servo shown
form:
COMMAND = GAINI(K desired - (K measured + GAIN2(K )))
U
U
The velocity
command
is then
sent
U
to the
servo
through
subroutine OUTVEL.
This completes one control cycle.
computer
a new
then reads
new control
cycle.
set
of measurements
At the completion
of the control
the program writes all data to disk storage.
163
and
the
The
begins
a
DO-loop,
Initialize
Transducers
Yes
Figure 77.
Bending Program Flow Chart
164
C
BEND
C
C
C
BENDING CONTROL PROGRAM
4/04/85
C
C
C
DIMENSION Kl(1000l),K2(1000),K3(1000),K4(1000),K5(1000)
COMMON C1,C2,C3,C4,C5
DATA K,K2,K3,K4,K5,/1000*0,1000*O,1000*O,1000*0
& 1000*0/
0001
0002
0003
C
C
MOVE THE SERVO TO ALLOW WORKPIECE INSERTION
C
CALL OUTPOS(O)
0004
C
C
MEASURE THE BENDING STIFFNESS
C
CALL SCAL(SLOPE)
0005
C
C
INITIALIZE ALL CONSTANTS
C
C1
C2
C3
C4
C5
0006
0007
0008
0009
0010
=
=
=
=
=
0.00032/(35.86*SLOPE)
6.0/(11.5*SLOPE)
(0.875/2.O*0.000126)/(11.5*SLOPE)
0.00272
3.0*0.000032/36.
C
C
ENTER THE NECESSARY CONTROL VARIABLES AND INPUTS
C
0011
0012 10
0013
0014 20
0015
0016 30
WRITE(5,10)
ENTER CONTROL GAINS (GAIN1,GAIN2):')
FORMAT(/2X,'
READ(5,20) GAIN1,GAIN2
FORMAT(2F13.0)
WRITE(5,30)
ENTER THE DESIRED CURVATURE: ')
FORMAT(/2X,'
0017
READ(5,40)
0018 40
FORMAT(F10.O)
RKD
C
C
INITIALIZE ALL TRANSDUCERS
C
0019
0020
CALL SZERO(I1,I2,I3,I4)
PAUSE ' PRESS RETURN TO BEGIN'
C
C
*******
REAL TIME CONTROL BEGINS HERE *******
C
C
K1 - FX
K2 - FY
K3 - ROTATION
C
C
C
K5 - POSITION
K4 - VELOCITY
*******************************************
C
165
DO 50 J=l,1000
0021
C
C
READ ALL TRANSDUCERS
C
0022
CALL ATOD(I1,Kl(J),I2,K2(J),I3,K3(J),I4,K4(J))
0023
CALL INPOS(K5(J))
C
C
CALCULATE THE UNLOADED CURVATURE
C
RKU = C5*K5(J)-Kl(J)*Cl*K5(J)-K2(J)*(C2+C3*K3(J))
0024
C
C
CALCULATE THE CONTROLLER COMMAND
C
ICOM = INT(GAIN1*(RKD-RKU-K4(J)*C4*GAIN2))
0025
C
C
SEND COMMAND TO THE SERVO
C
CALL OUTVEL(ICOM)
0026
C
C
REPEAT THE CONTROL LOOP
C
CONTINUE
0027 50
C
C
C
C
C
C
C
WRITE
DATA TO DISK
C
0028
0029
0030
CALL NAMEIN(NAME1)
OPEN (UNIT=i,NAME=NAME1,FORM='UNFORMATTED')
WRITE(1) SLOPE,C1,C2,C3,C4,C5,RKD
0031
DO 60 J = 1,1000
0032
WRITE(1) K(J),K2(J),K3(J),K4(J),K5(J)
0033 60
CONTINUE
0034
CLOSE (UNIT=1)
0035
END
C
C
C
C
C
SUBROUTINE SCAL(SLOPE)
DIMENSION RK(lOO),RM(100)
COMMON C,C2,C3,C4,C5
0036
0037
0038
C
C
SET SCALE FACTORS ACCORDING TO THE DESIRED WORKPIECE
C
WRITE(5,10)
FORMAT(/2X,' ENTER OPTION '/'
0039
0040 10
&
'
2) 1/4 X 1 ALUMINUM
166
'/'
1) 1/8 X 1 ALUMINUM '/
3) 1/8 X 1 STAINLESS
'
/
&
1/8
X
1/4
X
1
1/8 X
0056 60
0057
0058
0059
5)
1/2
X 3/4
TEE
IMAT
ALUMINUM
N =
J =
100
40
GO TO 80
1
ALUMINUM
M = 600
N = 100
J = 10
GO TO 80
1
0052 50
0053
0054
0055
C 1/4 X
X 1 STAINLESS'/'
M = 2000
0048 40
0049
0050
0051
C
1/4
FORMAT(I3)
GOTO(30,40,50,60,70),IMAT
0044 30
0045
0046
0047
C
4)
READ(5,20)
0041
0042 20
0043
C
'
STAINLESS
M = 200
N = 100
J = 4
GO TO 80
1
STAINLESS
M =
100
N =
J =
100
2
GO TO 80
C 1/2 X 3/4 TEE
M = 100
0060 70
N = 100
0061
0062
0063 80
0064
0065
0066
0067
J
=
2
CALL OUTPOS(0)
CALL SZERO(I1,I2,I3,I4)
CALL INPOS(I5)
CALL OUTPOS(M+10)
PAUSE 'PRESS RETURN TO CALIBRATE'
C
C
LOAD THE WORKPIECE THROUGHOUT THE ELASTIC REGION
C
CALL OUTPOS(M)
0068
0069
0070
0071
0072
0073
DO 90 I=I,N
CALL OUTPOS(M-J*I)
CALL INPOS(N5)
CALL ATOD(I1,Nl,I2,N2,I3,N3,I4,N4)
N5 = N5-I5
C
C
CALCULATE MOMENT AND CURVATURE AT VARIOUS POINTS
C
0074
0075
0076 90
0077
RK(I) = C5*N5
RM(I) = Nl*Cl*N5+N2*(C2+C3*N3)
CONTINUE
CALL OUTPOS(0)
C
C
PERFORM LINEAR REGRESSION TO CALCULATE THE BENDING
C
STIFFNESS
167
'/)
C
0078
0079
0080
0081
A1 = O.
A2 = O.
A3 = O.
A4 = O.
0082
0083
0084
0085
0086
0087 100
0088
0089
DO 100 I=1,N
A1 = A+RK(I)/FLOAT(N)
A2 = A2+RM(I)
A3 = A3+RK(I)*RK(I)
A4 = A4+RK(I)*RM(I)
CONTINUE
SLOPE = (A4-Al*A2)/(A3-Al*Al*N)
ROFF = A2/N-SLOPE*A1
0090
RETURN
0091
END
C
C
C
C
C
0092
0093
SUBROUTINE SZERO(I1,I2,I3,I4)
PAUSE 'PRESS RETURN TO ZERO TRANSDUCERS'
C
C
SEQUENCING START, ZERO ALL INPUTS
C
0094
I1
0095
0096
0097
0098
0099
0100
0101
0102
0103
0104
I2 = 0
I3 = 0
I4 = 0
DO 20 J=1,4
N1 = 0
N2 = 0
N3 = 0
N4 = 0
DO 10 I=1,5
CALL ATOD2(0,LI,O,L2,0,L3,0,L4)
= 0
0105
N1 = N1 + L1
0106
0107
0108
0109 10
N2 = N2 + L2
N3 = N3 + L3
N4 = N4 + L4
CONTINUE
0110
0111
+ N1/5
I
= I
I2 = I2 + N2/5
0112
0113
0114 20
I3 = I3 + N3/5
I4 = I4 + N4/5
CONTINUE
0115
0116
I
I2
0117
0118
I3 = I3/4
I4 = I4/4
= I1/4
= I2/4
0119
RETURN
0120
END
168
Modeling Program
The
roll
consists
of a
bending
main
system
program
modeling
and
two
program,
subroutines.
MODEL,
The
sub-
routine RK4 is a general fourth-order Runge-Kutta integration
routine.
The
continuous
modeling
subroutine
system
program
in
SYSTEM
standard
is used
is
used
state
to model
to
variable
the
nonlinear
define
the
form.
The
effects
of
the workpiece and also to implement the discrete controller.
A listing
of the program
brief description
34,
and
are
in lines
and command
used
to
simulate
in Chapter
4 and
implemented
in
25.
12
a
Line
moment-curvature
starting
is
the
The system
in lines
digital
controller
of
the
once per time
program
shift
due to a disturbance.
13 is completed
10, 11,
in the control
calculation
relationship
at line
A
below.
1-12.
equations
derived
line
pages.
in a file which is read into the program
The error
35
is presented
are initialized
is stored
in line 7.
on the following
of program logic
All variables
definition
is given
in
the
The loop
step.
First
the differential equations representing the continuous system
are
defined
integrated
in
the
at the
subroutine
current
time
SYSTEM.
step
in
The
the
equations
are
subroutine
RK4
using a standard Runge-Kutta integration routine.
Lines 20-
24 represent a velocity saturation nonlinearity.
Lines 25-
31 are
3).
used
to model
the
nonlinear
workpiece
(see
Chapter
The equations shown represent a unidirectional bending,
moving workpiece model.
The equations can be expanded to
169
include bidirectional bending and a stationary workpiece, but
this introduces considerable complexity without a compensating
increase
are
the
in information.
implementation
that the controller
of
command
The
the
discrete
time.
The
remainder
in
170
32-35
Notice
every
time step,
to the discrete
controller
of the program stores
disk.
lines
controller.
is not computed
but at time steps which correspond
cycle
equations
the data on
-
C
C
ROLL BENDING MODELING PROGRAM
C
C
C
DIMENSION X(7,750)
COMMON /XYDATA/F(4),Y(4),SAVEY(4),PHI(4),M,RKU,COM,DT
0001
0002
/CONST/C1,C2,C3,C4,C5,C6,C7
&
0003
0004
DATA COM,ICOUNT,RKU,YO/O.,O,O.,O./
DATA Y,X/4*O.0,5250*O.0/
0005
0006
CALL NAMEIN(NAME1)
OPEN(UNIT=1,NAME=NAME1,TYPE='OLD')
READ(I1,*) RKIN,DT,NT,ITC,YB,VMAX,SLOPE,RL,GAIN,
0007
&
0008
C1,C2,C3,C4,C5,C6,C7,RKD
CLOSE
(UNIT=1)
C
C
INITIALIZE THE DISCRETE CONTROLLER
C
ERROR = RKIN-RKU-C7*Y(2)
COM = ERROR*GAIN
YO = (RKD*(RL**2))/3.0
0009
0010
0011
C
C
CALCULATE THE SYSTEM PARAMETERS FOR EACH TIME STEP
C
0012
0013
DO 110 IT=1,NT
ICOUNT = ICOUNT + 1
0014
0015
C
K =
N =
C
1
2
DEFINE SYSTEM
C
0016
DO 10 M=1,4
0017
CALL SYSTEM
C
C
INTEGRATE SYSTEM EQUATIONS
C
0018
CALL RK4(N,K)
0019 10
CONTINUE
C
C
CHECK FOR VELOCITY SATURATION
C
0020 20
0021
0022
0023 30
0024
0025
IF(Y(2).LT.VMAX) GO TO 30
Y(2) = VMAX
Y(1) = X(1,IT-1)+VMAX*DT
GO TO 40
IF(Y(2).GT.(-VMAX))
Y(2) = -VMAX
Y(1) = X(1,IT-1)-VMAX*DT
C
C
WORKPIECE MODEL
C
0026 40
RKL = 3.0*Y(1)/(RL**2)
171
-
0027
0028 50
0029
0030 60
-
IF((Y(1)-YO).GT.YB)
GO TO 60
RM = 3.0*(Y(1)-YO)*SLOPE/(RL**2)
GO TO 70
RM = ((9.0*YB*SLOPE)/(2.0*(RL**2)))*
& (1-(((YB/(Y(1)-YO))**2)/3.0))
0031 70
RKU = RKL-(RM/SLOPE)
C
C
CALCULATE DISCRETE CONTROLLER OUTPUT
C
0032
IF(ICOUNT.NE.ITC)
0033
ICOUNT
0034
0035
ERROR = RKIN-RKU-C7*Y(2)
COM = ERROR*GAIN
GO TO 90
= 0
C
C
STORE DATA
C
DO 100 I=1,N
0036 90
0037
0038 100
0039
0040
0041
0042
0043 110
0044
0045
0046
X(I,IT)=Y(I)
CONTINUE
X(N+1,IT)
X(N+2,IT)
X(N+3,IT)
= RKU
= RM
= ERROR
X(N+4,IT) = COM
CONTINUE
CALL NAMEIN(NAME1)
OPEN(UNIT=1,NAME=NAME1,TYPE='NEW' ,FORM='UNFORMATTED')
WRITE(1) RKIN,DT,NT,ITC,YB,VMAX,SLOPE,RL,GAIN,
&
0047
0048
0049 120
0050
0051
C1,C2,C3,C4,C5,C6,C7
DO 120 I=1,NT
WRITE(1) X(1,I),X(2,I) ,X(3,I),X(4,I),X(5,I),X(6,I)
CONTINUE
CLOSE(UNIT=1)
END
C
C
C
C
C
0052
0053
SUBROUTINE SYSTEM
COMMON /XYDATA/F(4),Y(4),SAVEY(4),PHI(4),M,RKU,COM,DT
&
0054
0055
0056
0057
/CONST/C1,C2,C3,C4,C5,C6,C7
F(1) = Y(2)
F(2) = C*Y(2)+C2*Y(1
)+C3*COM
RETURN
END
C
C
C
C
C
0058
0059
SUBROUTINE RK4(N,K)
COMMON /XYDATA/F(4),Y(4),SAVEY(4),PHI(4),M,RKU,COM,DT
172
0060
0061 10
0062
0063
0064
0065 20
0066
0067 30
0681
0069
GO TO (10,30,50,70),M
DO 20 J=K,N
SAVEY(J) = Y(J)
PHI(J) = F(J)
Y(J) = SAVEY(J)+0.5*DT*F(J)
CONTINUE
GO TO 90
DO 40 J=K,N
PHI(J) = PHI(J)+2.0*F(J)
Y(J) = SAVEY(J)+0.5*DT*F(J)
0070 40
CONTINUE
0071
0072 50
GO TO 90
DO 60'J=K,N
0073
PHI(J) = PHI(J)+2.0*F(J)
0074
0075 60
0076
0077 70
0078
0079 80
Y(J) = SAVEY(J)+DT*F(J)
CONTINUE
GO TO 90
DO 80 J=K,N
Y(J) = SAVEY(J)+(PHI(J)+F(J))*DT/6.0
CONTINUE
0080 90
RETURN
0081
END
173
-
L
-
Step and Frequency Response Program
Program
input
STEP
to the servo
is used
system
shown is for a velocity
servo by changing
step
input
remainder
to generate
and store
a
step
the output.
or
frequency
The
program
servo, but can be used for a position
lines 12 and 13 as shown in the listing.
is generated
of the program
by
specifying
zero
is self-explanatory.
174
frequency.
A
The
L_
I_
C
C
C
STEP AND FREQUENCY RESPONSE PROGRAM
C
C
0001
0002
DIMENSION K(3000),N(3000)
CALL OUTPOS(O)
C
C
INITIALIZE SERVO INPUT
C
0003
WRITE(5,10)
0004 10
0005
0006 20
0007
0008
0009 30
FORMAT(/2X,' ENTER
MAG. AND FREQ.
READ(5,20) A,W
FORMAT(2F10.O)
DO 30 I=1,3000
N(I) = INT(A*COS(W*I))
CONTINUE
0010
PAUSE 'PRESS RETURN WHEN READY'
0011
DO 40 I=1,3000
')
C
C
SEND VELOCITY COMMAND
C
0012
CALL OUTVEL(N(I))
C
C
READ SERVO VELOCITY
C
0013
CALL ATOD(O,Il,O,I2,0,I3,0,K(I))
0014 40
CONTINUE
************************************************
C
C
C
THE PROGRAM CAN BE USED FOR A POSITION
C
BY CHANGING EQUATIONS 12 AND 13 TO
SERVO
C
C
C
C
CALL OUTPOS(N(I))
CALL INPOS(K(I))
************************************************
C
C
STORE DATA
C
0015
0016
0017
CALL OUTPOS(O)
CALL NAMEIN(NAME1)
OPEN (UNIT=1,NAME=NAME1,FORM='UNFORMATTED')
0018
DO 50 I=1,3000
0019
0020 50
0021
WRITE(1) K(I),N(I)
CONTINUE
CLOSE (UNIT=i)
0022
END
175