- Email: [email protected]
Closed Loop Control of a Roll Straightening Process
Closed Loop Control of a Roll Straightening Process David E. Hardt, Michael Hale
-
Submitted by N. H. Cook ( l ) , Massachusetts Institute of Technology
SUMt4ARY
Manufacture of basic metal stock forms by hot o r cold rolling o r by extrusion often leads to a product that is long, thin and far from straight. This lack of straightness is exaggerated by subsequent heat treating in the case of aluminum products. A necessary step in manufacture of this stock is, therefore, selective bending of the material to improve straightness. However, this is a labor intensive, time consuming step since each piece has a unique shape, which must be accounted for individually. In this paper a novel strategy for controlling a three roll bending process is described that employs closed-loop regulation of the desired material shape. With this approach, lack of straightness in the in-feed material is treated as an output ( o r shape) disturbance, and by proper design the control system can be made to totally eliminate these variations. Considerations of speed, accuracy, and the material adaptive qualiti-es of this system are presented along with experimental results from a lab-scale machine. INTRODUCTION
The production of metal shapes by extrusion, forging and hot rolling often leads to materials that are long relative to the cross section. In addition, because of thermally induced stresses and heat treatment the finished pieces are far from straight along the long axis. Consequently, a major step in the production of these items is a straightening operation. For materials with large cross sections, such as plate and large forgings, the preferred method is sequential brakebending o r bumpforming. Although a slow and inherently difficult process to control, a skilled operator can eventually achieve the desired results since an iterative procedure is employed. Recent attempts to apply closed-loop control techniques to the bumpforming process [ l ] could significantly improve the efficiency of this straightening method, but it will still be a slow process relative to continuous r o l l bending. If the material to be straightened is long with a small cross section (i.e. included in a 40 mm diameter circle) then r o l l bending is commonly used for straightening. This affords a much shorter processing time, but still involves considerable iteration on the part of the operator. This is because a part will typically have a random distribution of curvature along any one axis, which will manifest itself in the form of dominant elevations along the piece when laid on a flat reference surface. Thus the manual straightening process involves application of selective "unbending" moments that vary according to the location on the part. Compounding this problem is the very low domain of curvature involved (typically <0.03 l / m ) which implies that very small ratios of plastic to elastic strains are required. To significantly impact this process, a means for single-pass straightening of arbitrary cross sections must be developed. A s will be discussed below this mandates that the straightening process employ closed-loop control to become insensitve to material property variations and to respond to the arbitrary curvatures that must be removed. This paper describes such a method that is based on in-process measurement of the bending operation. By extending a method previously developed by Hardt et al. [2] it will be shown that straightening can be automatically performed when a closed-loop strategy is employed. BACKGROUND 2 Closed-Loop Control of a Roll Bending Process A typical three r o l l o r pyramid roll
machine
is
shown
schematically L
d
in
Fig.
bending 1. The
SHEET
CENTER DRIVE
'&''FIXED" BED ROLLER
Figure 1.
A
Typical Three Roll Bending Geometry
Annals of the ClRP Vol. 33/1/1984
material is driven through by one of the r o l l s and the center r o l l is moved transverse to the outer rolls to vary the bending moment on the sheet. As a point on the sheet traverses the rolls it sees an increasing bending moment until, at the point of tangency with the center r o l l , it sees the maximum moment. Beyond that point the moment decreases until the material point leaves the machine. If we follow that same point on a moment-curvature diagram appropriate for that material and cross section shape, (see Fig. 2 . ) we can see that the final curvature imparted to the point depends upon how far up the curve the maximum moment occurs and upon the slope of the elastic loading (and unloading) line. It can be seen that the resulting unloaded curvature at that point is given by : KU = K1 - M/(dM/dK) (1) where dM/dK is the elastic slope. If conventional predictive o r open-loop control is used on this process the relationship between moment and curvature for the workpiece material as well as a relationship between the center r o l l position and the maximum bending moment and loaded
Figure 2.
A General Moment Curvature Relationship With a Roll Bending Cycle Indicated
curvature must be precisely known. As demonstrated by Hansen and Jannerup [31 the lager is extremely complex and must be found by an iterative calculation. On the other hand if one directly measures K on the outlet side of the machine, as was done by Foster 141 , the data is available too late to correct the error at that point and can only be used to eventually maintain constant final curvature. Even the novel use of this post-forming measurement for property indentification as proposed by Hansen et al. [51 has severe limitations when considering arbitrary curvatures. In addition, the highly variable plastic properties of all materials nakes precise a D r i ori determination of the moment-curvature relationship virtually impossible. By contrast, elastic bending properties of metals are well behaved and can be exploited to effect a highly accurate closed-loop control scheme. From Eqn 1 it is apparent that if we can measure the loaded curvature ( K ) and the corresponding maximum moment (M), and if the effective bending stiffness (dM/dK) of the material is known, we can then indirectly measure the expected unloaded curvature simply by applying Eqn 1. Such a scheme was employed by Hardt et al. (21 and it results in the control system block diagram
137
RA
- ACTUAL
RD-
* - ,090"
.parts were formed to within 3% of the desired radius on the first pass. (It should be noted that the measurement errors of the machine used in these experi-ments and the final part radius measurement were on the order of 3%).
BEND RADIUS
DESIRED BEND RADIUS ALUMINUM 2024-16
060" 3 0 2 STAINLESS STEEL
UNDERBEND t 30
-
+ 20 -
.. . . .-- _ _ _ _ ----_ _ _ _ : -
+loRA -RD (INCHES)
7
9
4
a
- - - - - --.
o----
a
. B
1
--2
.
,
-101
1 6
,
-.20
-.30
5"
OVERBEND
6"
7"
8"
9"
10"
R D (INCHES)
I
Figure 3 .
I
I
The Closed Loop Roll Bending Control System
CLOSED-LOOP STRAIGHTENING
To apply the above method to straightening,it is necessary to consider the effect of inital curvature of the in-feed material. If we again focus attention on the point moment-curvature properties of the workpiece, the effect of initial curvature is simply to shift the origin of the curve from zero to the initial curvature K (see Fig. 5). Assuming the same closed-loop controy scheme as above it is clear that when zero moment is measured a non-zero loaded curvature will be sensed. Thus th: system as it exists will be aware of a "disturbance in the form of a shift in the moment-curvature relationship. If
1
1
I Figure 5.
SPRINGBACK
~
4
- K (CURVATURE)
The Straightening Operation on a Moment Curvature Diagram. The inital curvature is assumed negative and has the value K O
the desire is to have a final unloaded curvature of zero, it can be seen from Fig. 5 that a positive bending moment must be applied to produce a positive loaded curvature sufficient to springback to zero when unloaded. The loading and unloading cycle presented by the 5 is, however, no straightening operation in Fig. 2. It is different that that illustrated in Fig. apparent than the existing control scheme will automatically compensate for lack of straightness in the material since it continuously measures the loaded state(M and K ) of the sheet. Also, the block diagram of Fig. 3 'implies that the system is an absolute curvature regulator, which carries with it the property of implicit disturbance rejection. Another way to view this property is to consider the fact that the only material property measured by the machine is a single point on the moment-curvature relationship (the point of maximum moment and curvature) and the elastic unloading slope. If the system has the ability to correctly form a desired radius when variations of plastic properties occur, then it must just as well succeed when a moment-curvature offset occurs since both phenomena manifest themselves to the controller in the same set of variables.
138
Figure 4.
Curvature E r r o r for Constant Radius Bends (from Hardt et al. [ 2 ] )
The performance of the closed-loop r o l l bending device will thus depend upon the ability to respond quickly to what could be rapid variations in the moment-curvature relationship. F o r low speed rolling this is not a problem, but when higher speeds are required the occurtence of a non-flat region on the in-feed will be a transient phenomenon and the control system will need either extremely fast response or will have to take special account of the straightness disturbance. EXPERIMENTS
The above closed-loop straightening method was investigated experimentally using a 1ab.scale machine. The purpose of these experiments was to verify the disturbance rejection characteristics of the process and to define more clearly the effect of measurement errors on the ultimate accuracy of the closed-loop system, To that end, a series of metal strips with initial curvature were run through the system and automatic straightening was attempted. The apparatus comprised all of the components indicated in the block diagram of Fig. 3 . The basic machine was mounted on a Bridgeport milling machine with the table position under servo control. The machine layout and instrumentation are shown in Fig. 6. while retaining the basic pyramid roll arrangement, the control system requires measurement of the bending moment and the loaded curvature at the center roll. The moment is measured using a two-axis force transducer on which one of the outside rolls is mounted. As shown in Fig. 6 this allows the force normal to the sheet at the r o l l contact point to be measured. The moment arm between that point and the center r o l l is then calculated using the known displacement between the center and outer roll and the measured contact point between the center roll and the sheet. Since the maximum moment point will not, in general, be directly below the center r o l l , the pinch roll was allowed to articulate about the center r o l l axis and thereby settles at the maximum moment point. The angle at which this occurs, denoted the sheet tangent position in Fig. 6, is then used along with the known original roll spacing distances to calculate the moment arms. FORCE MEASUREMENT ROLLER
f
CENTER
"FIXED" BED ROLLER
' ROLLER DISPLACEMENT CURVATURE MEASURE
/
v
SHEET TANGENT POSITION
Figure 6 .
The Instrumented Roll Bendinq Machine The distances dl and d2 were measured using two LVDT s.
The loaded curvature is more difficult to measure since it is a point property and must instead be measured indirectly. The assumption is made that the curvature very near the center roll is nearly the same as the curvature under the roll itself. The extent to which the moment has decreased as one moves away from the contact point will determine how much this assumption is in error, but to minimize this discrepancy the curvature is measured on the unloading side of the center r o l l . Examination of the moment curvature relation of Fig. 2 illustrates that the curvature is much less sensitive to the moment in the elastic case than in the plastic case. The actual measurement is a pair of sheet displacements relative to a line perpendicular to the center and pinch rolls, as shown in Fig. 6. These measurements were made with linear displacement transducers (LVDT s ) mounted 15 and 30 mm from the By assuming that the sheet exits center roll axis. the center roll perpendicular to the line between the drive and pinch roll and then forms a constant radius, each displacement measurement was considered to lie on that circle. The curvature was found for each LVDT measurement according to: Kl
=
2 d/ (d2+12)
(2)
where d is the displacement measurement and 1 is the distance from the center roll axis to the transducer. The two separate measurements (one from each LVDT) were then averaged to improve the signal-to-noise ratio of the measurement. Two observations concerning this measurement method can be made. The first is that the curvature is measured at a finite distance away from the center roll and will introduce a time delay into the control system. This has not been considered for this study but will be of concern when maximum rolling speed is considered in the future. The second is that the curvature measured by this method will indeed be less than the actual curvature under the center r o l l . It is possible to extrapolate from this curvature back to the center roll position by assum-ing that the bending moment varies linearly with distance from the center roll. Then, given the slope of the elastic unloading line and the true maximum bending moment, the curvature can be updated according to: * dx/D (3i where K is the actual loaded curvature, K is the measureA curvature, and dx and D are the Pransducer and outer roll distance from the center r o l l axis. For the apparatus used in these experiments the term dx/D has a value of 0.1, and since the moments required for straightening are nearly equal to the yield moment, it is expected that the correction term is negligible. In addition, the calibration method used here will compensate for the lack of this term by applying a proportional correction factor to the transducer estimate of K so as to minimize the error in the unloaded curvaturh. The signals from these transducers were sent to a microcomputer (a CROMEMCO Z2-D) and sampled by a 1 2 bit A-D converter. The control system of Fig. 3 was implemented with the control block, Gc(s) containing the term K/s, i.e. an integral controller. This controller, combined with first order filters on the feedback transducers gives a dominant low-frequency ( ( 1 5 Hz.) bandwidth second-order system characteristic with critical damping. The output from the integral controller calculation was then sent as a digital command to a highly accurate, high bandwidth (>60 Hz.) servosystem that controlled the roll positions. The center r o l l was driven by the spindle motor of the milling machine and yielded a rolling speed of approximately 7 cm/sec. The cycle time o f the controller program was 25 msec. The calibration of the transducers is critical to the steady-state accuracy of this system and presented considerable difficulties. A s discussed above, errors in determining the loaded moment, curvature o r elastic slope will directly affect the resulting unloaded (desired zero) curvature. For these experiments the procedure was to calibrate the force and displacement transducers, measure the critical machine dimensions and then perform a straightening test. The overall error in the system will then be reflected in the curvature of the final piece as compared to the desired zero curvature. A detailed procedure for using such tests to fine-tune the calibration of all three quantities: force, curvature and slope can be employed; a simple procedure was used here where it was assumed that the errors were all caused by errors in curvature measurement, since it is the most difficult sensor to calibrate.
The initial estimate for the elastic slope is simply the bending stiffness for an elastic beam. However, since the test specimens were wide and thin, a plane-strain correction must be made, thus the actual calculation was made according to : dM/dK = EI / ( 1 - Y 2 , where E is the elastic modulus, I is the area moment of inertia and Y is poisson's ratio. The actual tests were performed on 1 m long strips of 2024-T4 aluminum and 304 stainless steel with a 25 x 3 mm rectangular cross section. In the first test a specimen was prepared by intentionally bending three regions of increasing curvature into the piece. It was then rolled through the closed-loop apparatus and the shape of the inital and final part measured. A plot of one if these shapes with the 7. The vertical axis exaggerated is shown in Fig. initial shape of the workpiece is seen to have regions of high curvature and a maximum elevation of 2.5 mm over 30 cm. The straightened part has a maximum elevation of 0.6 mm, but more importantly can be seen to have regiomof very low curvature with small "bumps" between. These correspond to transients in the control system caused by the curvature changes in the initial part. To completely remove them will require that either the bandwidth of the control system be increased or the rolling speed decreased. The loaded moment and curvature and the calculated unloaded curvature for this test is plotted in Fig. 8. The variations on the loaded curvature correspond to those observed in Fig. 7, A l s o shown is the calculated unloaded curvature that was used to drive the control system. The extent to which the predicted unloaded curvature is non-zero in the final piece is, therefore, a reflection of errors in calibration of the transducers and of the finite bandwidth of the control system as presented here.
4-
-300mrn Figure 7.
Profile for Test A 1 2. The upper line is the initial part. The vertical scale is multiplied bya factor of 6 and the horizontal scale by 1/12.
r-
E+02 30
~~
E
2
20-
Y 3
ca
KI
15 -
I'
Y A
J
10
-5
-
1
0
I
L -
2
I
I
I
I
1
I
4
6
€3
10
12
14
E+01 DISTANCE (am)
Figure 8.
The Measured Loaded Moment, Curvature and Calculated Unloaded Curvature f o r Test A 1 12.
-
I
1Figure 9.
300rnm--------------$( hOrl&ffM
Profile for Test SS 4. The scale is identical to Fig. 7 (the vertr'ca( scale k not)
.
139
In Fig. 9 a similar test performed on the stainless steel specimens is displayed and it is apparent that identical straightening characteristics exist for this material. The result is a more nearly flat part (0.6/5.2 mm) but this appears to be caused entirely by the less severe changes in curvature presented the initial piece. Indeed the only difference brn the experiments for this material is a change in the value calculated f o r the elastic slope of the moment-curvature relationship because of different values for E and y The results for a series of testson both aluminum and stainless steel are presented in table 1. The measure of straightness is the maximum deviation from a horizontal datum using a gauge section of 30 cm.
References
1.
2.
Hardt, D.E. Roberts, M.A. and Stelson, K.A., "Closed-Loop Shape Control of a Roll Bending Process," ASME. Journal of Dy.&mic S y . s k a s . Measurema& u 4 , no. 4 , Dec. 1982, pp. 317-321. Con-,
3.
Hansen, N.E. and Jannerup, O., "Modelling of ElasticPlastic Bending of Beams Using a Roller Bending Machine," ASME Paper 78-WA/Prod-6.
4.
Foster, G.B., "Springback Compensated Continuous Roll Forming Machines," U.S. Patent No. 3,955,389, May TI, 1976.
5.
Trotsmann, E., Hansen, N.E., and Cook, G., "General Scheme for Automatic Control of Continuous Bending of Dvnamic S y s t e L of Beams," T Measurement and Contra;L, U, no. 2, 1982, pp. 173-179.
.
TABLE 1
TEST A1 0 A1 2 A1 3 A1 7 A1 8 ss 2
ss ss
4
5
Initial 'max 1.6 mm 2.5 2.6 2.0 1.9 3.9 5.2 6.5
Final Ymax 0.6 mm 0.6 0.9 0.7 0.9 0.2 0.6 0.6
%
Reduction 63 % 76 65 65 53 95 88 91
From these data it can be seen that the steel appears to straighten to smaller deflections than the aluminum. This can be attributed to the lesser springback for the steel and the higher moments involved, which improved the signal to noise ratio on the force transducer. However, the profiles f o r all of these specimens were similar to thoseco~y~UmFigs. 7 and 9 thus the major cause of residualvapbears to be the relatively slow settling time of the control system. CONCLUSIONS
The closed-loop roll bending control scheme presented here has the inherent property of disturbance rejection that all feedback systems possess. In this case the disturbance is initial curvature in the workpiece. Accordingly the control system will automatically accorffadate non-straight workpieces and if the specified final curvature is zero, a flat piece will be produced. It should be noticed that the straightening problem is in many ways more severe than the bending problem since arbitrary initial curvatures must be dealt with. Accordingly, a closed-loop system such as that demonstrated in this paper is a basic requirement for automated straighten ing The evaluation of the worth of this system as compared to a manual system requires an assessment of the importance of production speed. Manual operation is iterative and with non-uniform initial curvatures will require several bending passes simply to identify and respond to the various sections o f differing curvature on the workpiece. However, with a closed-loop control strategy the problems of arbitrary initial curvature and unknown plastic properties of the material are explicitly considered within the control loop and are, therefore, transparent to the operator. This implies then that single pass straightening will be possible. Given the prospect of single pass straightening, the importance of rolling speed can be seen. The next phase of this research will consider in greater detail the dynamics of this control system in an effort to determine the upper bound on rolling speed. As discussed in 121 the dynamics will be a function of the bending machine, the transducers and the workpiece itself. Since the latter will be of variable net stiffness and mass, the best control system will probably have to consider explicitly the changing dynamic properties of the system being controlled.
.
Acknowledgements
The authors gratefully acknowledge the assistance of Mr. Lee Mallet working under the Undergraduate Research Opportunities Program (UROP).
140
Hardt, D.E., and Chen, B., "Control of a Sequential Brakeforming Process," Control of Manufacturinu Processes qnd Robotic SY stems, ASME Pub. # H00279, 1983, pp. 171-182.
-
Submitted by N. H. Cook ( l ) , Massachusetts Institute of Technology
SUMt4ARY
Manufacture of basic metal stock forms by hot o r cold rolling o r by extrusion often leads to a product that is long, thin and far from straight. This lack of straightness is exaggerated by subsequent heat treating in the case of aluminum products. A necessary step in manufacture of this stock is, therefore, selective bending of the material to improve straightness. However, this is a labor intensive, time consuming step since each piece has a unique shape, which must be accounted for individually. In this paper a novel strategy for controlling a three roll bending process is described that employs closed-loop regulation of the desired material shape. With this approach, lack of straightness in the in-feed material is treated as an output ( o r shape) disturbance, and by proper design the control system can be made to totally eliminate these variations. Considerations of speed, accuracy, and the material adaptive qualiti-es of this system are presented along with experimental results from a lab-scale machine. INTRODUCTION
The production of metal shapes by extrusion, forging and hot rolling often leads to materials that are long relative to the cross section. In addition, because of thermally induced stresses and heat treatment the finished pieces are far from straight along the long axis. Consequently, a major step in the production of these items is a straightening operation. For materials with large cross sections, such as plate and large forgings, the preferred method is sequential brakebending o r bumpforming. Although a slow and inherently difficult process to control, a skilled operator can eventually achieve the desired results since an iterative procedure is employed. Recent attempts to apply closed-loop control techniques to the bumpforming process [ l ] could significantly improve the efficiency of this straightening method, but it will still be a slow process relative to continuous r o l l bending. If the material to be straightened is long with a small cross section (i.e. included in a 40 mm diameter circle) then r o l l bending is commonly used for straightening. This affords a much shorter processing time, but still involves considerable iteration on the part of the operator. This is because a part will typically have a random distribution of curvature along any one axis, which will manifest itself in the form of dominant elevations along the piece when laid on a flat reference surface. Thus the manual straightening process involves application of selective "unbending" moments that vary according to the location on the part. Compounding this problem is the very low domain of curvature involved (typically <0.03 l / m ) which implies that very small ratios of plastic to elastic strains are required. To significantly impact this process, a means for single-pass straightening of arbitrary cross sections must be developed. A s will be discussed below this mandates that the straightening process employ closed-loop control to become insensitve to material property variations and to respond to the arbitrary curvatures that must be removed. This paper describes such a method that is based on in-process measurement of the bending operation. By extending a method previously developed by Hardt et al. [2] it will be shown that straightening can be automatically performed when a closed-loop strategy is employed. BACKGROUND 2 Closed-Loop Control of a Roll Bending Process A typical three r o l l o r pyramid roll
machine
is
shown
schematically L
d
in
Fig.
bending 1. The
SHEET
CENTER DRIVE
'&''FIXED" BED ROLLER
Figure 1.
A
Typical Three Roll Bending Geometry
Annals of the ClRP Vol. 33/1/1984
material is driven through by one of the r o l l s and the center r o l l is moved transverse to the outer rolls to vary the bending moment on the sheet. As a point on the sheet traverses the rolls it sees an increasing bending moment until, at the point of tangency with the center r o l l , it sees the maximum moment. Beyond that point the moment decreases until the material point leaves the machine. If we follow that same point on a moment-curvature diagram appropriate for that material and cross section shape, (see Fig. 2 . ) we can see that the final curvature imparted to the point depends upon how far up the curve the maximum moment occurs and upon the slope of the elastic loading (and unloading) line. It can be seen that the resulting unloaded curvature at that point is given by : KU = K1 - M/(dM/dK) (1) where dM/dK is the elastic slope. If conventional predictive o r open-loop control is used on this process the relationship between moment and curvature for the workpiece material as well as a relationship between the center r o l l position and the maximum bending moment and loaded
Figure 2.
A General Moment Curvature Relationship With a Roll Bending Cycle Indicated
curvature must be precisely known. As demonstrated by Hansen and Jannerup [31 the lager is extremely complex and must be found by an iterative calculation. On the other hand if one directly measures K on the outlet side of the machine, as was done by Foster 141 , the data is available too late to correct the error at that point and can only be used to eventually maintain constant final curvature. Even the novel use of this post-forming measurement for property indentification as proposed by Hansen et al. [51 has severe limitations when considering arbitrary curvatures. In addition, the highly variable plastic properties of all materials nakes precise a D r i ori determination of the moment-curvature relationship virtually impossible. By contrast, elastic bending properties of metals are well behaved and can be exploited to effect a highly accurate closed-loop control scheme. From Eqn 1 it is apparent that if we can measure the loaded curvature ( K ) and the corresponding maximum moment (M), and if the effective bending stiffness (dM/dK) of the material is known, we can then indirectly measure the expected unloaded curvature simply by applying Eqn 1. Such a scheme was employed by Hardt et al. (21 and it results in the control system block diagram
137
RA
- ACTUAL
RD-
* - ,090"
.parts were formed to within 3% of the desired radius on the first pass. (It should be noted that the measurement errors of the machine used in these experi-ments and the final part radius measurement were on the order of 3%).
BEND RADIUS
DESIRED BEND RADIUS ALUMINUM 2024-16
060" 3 0 2 STAINLESS STEEL
UNDERBEND t 30
-
+ 20 -
.. . . .-- _ _ _ _ ----_ _ _ _ : -
+loRA -RD (INCHES)
7
9
4
a
- - - - - --.
o----
a
. B
1
--2
.
,
-101
1 6
,
-.20
-.30
5"
OVERBEND
6"
7"
8"
9"
10"
R D (INCHES)
I
Figure 3 .
I
I
The Closed Loop Roll Bending Control System
CLOSED-LOOP STRAIGHTENING
To apply the above method to straightening,it is necessary to consider the effect of inital curvature of the in-feed material. If we again focus attention on the point moment-curvature properties of the workpiece, the effect of initial curvature is simply to shift the origin of the curve from zero to the initial curvature K (see Fig. 5). Assuming the same closed-loop controy scheme as above it is clear that when zero moment is measured a non-zero loaded curvature will be sensed. Thus th: system as it exists will be aware of a "disturbance in the form of a shift in the moment-curvature relationship. If
1
1
I Figure 5.
SPRINGBACK
~
4
- K (CURVATURE)
The Straightening Operation on a Moment Curvature Diagram. The inital curvature is assumed negative and has the value K O
the desire is to have a final unloaded curvature of zero, it can be seen from Fig. 5 that a positive bending moment must be applied to produce a positive loaded curvature sufficient to springback to zero when unloaded. The loading and unloading cycle presented by the 5 is, however, no straightening operation in Fig. 2. It is different that that illustrated in Fig. apparent than the existing control scheme will automatically compensate for lack of straightness in the material since it continuously measures the loaded state(M and K ) of the sheet. Also, the block diagram of Fig. 3 'implies that the system is an absolute curvature regulator, which carries with it the property of implicit disturbance rejection. Another way to view this property is to consider the fact that the only material property measured by the machine is a single point on the moment-curvature relationship (the point of maximum moment and curvature) and the elastic unloading slope. If the system has the ability to correctly form a desired radius when variations of plastic properties occur, then it must just as well succeed when a moment-curvature offset occurs since both phenomena manifest themselves to the controller in the same set of variables.
138
Figure 4.
Curvature E r r o r for Constant Radius Bends (from Hardt et al. [ 2 ] )
The performance of the closed-loop r o l l bending device will thus depend upon the ability to respond quickly to what could be rapid variations in the moment-curvature relationship. F o r low speed rolling this is not a problem, but when higher speeds are required the occurtence of a non-flat region on the in-feed will be a transient phenomenon and the control system will need either extremely fast response or will have to take special account of the straightness disturbance. EXPERIMENTS
The above closed-loop straightening method was investigated experimentally using a 1ab.scale machine. The purpose of these experiments was to verify the disturbance rejection characteristics of the process and to define more clearly the effect of measurement errors on the ultimate accuracy of the closed-loop system, To that end, a series of metal strips with initial curvature were run through the system and automatic straightening was attempted. The apparatus comprised all of the components indicated in the block diagram of Fig. 3 . The basic machine was mounted on a Bridgeport milling machine with the table position under servo control. The machine layout and instrumentation are shown in Fig. 6. while retaining the basic pyramid roll arrangement, the control system requires measurement of the bending moment and the loaded curvature at the center roll. The moment is measured using a two-axis force transducer on which one of the outside rolls is mounted. As shown in Fig. 6 this allows the force normal to the sheet at the r o l l contact point to be measured. The moment arm between that point and the center r o l l is then calculated using the known displacement between the center and outer roll and the measured contact point between the center roll and the sheet. Since the maximum moment point will not, in general, be directly below the center r o l l , the pinch roll was allowed to articulate about the center r o l l axis and thereby settles at the maximum moment point. The angle at which this occurs, denoted the sheet tangent position in Fig. 6, is then used along with the known original roll spacing distances to calculate the moment arms. FORCE MEASUREMENT ROLLER
f
CENTER
"FIXED" BED ROLLER
' ROLLER DISPLACEMENT CURVATURE MEASURE
/
v
SHEET TANGENT POSITION
Figure 6 .
The Instrumented Roll Bendinq Machine The distances dl and d2 were measured using two LVDT s.
The loaded curvature is more difficult to measure since it is a point property and must instead be measured indirectly. The assumption is made that the curvature very near the center roll is nearly the same as the curvature under the roll itself. The extent to which the moment has decreased as one moves away from the contact point will determine how much this assumption is in error, but to minimize this discrepancy the curvature is measured on the unloading side of the center r o l l . Examination of the moment curvature relation of Fig. 2 illustrates that the curvature is much less sensitive to the moment in the elastic case than in the plastic case. The actual measurement is a pair of sheet displacements relative to a line perpendicular to the center and pinch rolls, as shown in Fig. 6. These measurements were made with linear displacement transducers (LVDT s ) mounted 15 and 30 mm from the By assuming that the sheet exits center roll axis. the center roll perpendicular to the line between the drive and pinch roll and then forms a constant radius, each displacement measurement was considered to lie on that circle. The curvature was found for each LVDT measurement according to: Kl
=
2 d/ (d2+12)
(2)
where d is the displacement measurement and 1 is the distance from the center roll axis to the transducer. The two separate measurements (one from each LVDT) were then averaged to improve the signal-to-noise ratio of the measurement. Two observations concerning this measurement method can be made. The first is that the curvature is measured at a finite distance away from the center roll and will introduce a time delay into the control system. This has not been considered for this study but will be of concern when maximum rolling speed is considered in the future. The second is that the curvature measured by this method will indeed be less than the actual curvature under the center r o l l . It is possible to extrapolate from this curvature back to the center roll position by assum-ing that the bending moment varies linearly with distance from the center roll. Then, given the slope of the elastic unloading line and the true maximum bending moment, the curvature can be updated according to: * dx/D (3i where K is the actual loaded curvature, K is the measureA curvature, and dx and D are the Pransducer and outer roll distance from the center r o l l axis. For the apparatus used in these experiments the term dx/D has a value of 0.1, and since the moments required for straightening are nearly equal to the yield moment, it is expected that the correction term is negligible. In addition, the calibration method used here will compensate for the lack of this term by applying a proportional correction factor to the transducer estimate of K so as to minimize the error in the unloaded curvaturh. The signals from these transducers were sent to a microcomputer (a CROMEMCO Z2-D) and sampled by a 1 2 bit A-D converter. The control system of Fig. 3 was implemented with the control block, Gc(s) containing the term K/s, i.e. an integral controller. This controller, combined with first order filters on the feedback transducers gives a dominant low-frequency ( ( 1 5 Hz.) bandwidth second-order system characteristic with critical damping. The output from the integral controller calculation was then sent as a digital command to a highly accurate, high bandwidth (>60 Hz.) servosystem that controlled the roll positions. The center r o l l was driven by the spindle motor of the milling machine and yielded a rolling speed of approximately 7 cm/sec. The cycle time o f the controller program was 25 msec. The calibration of the transducers is critical to the steady-state accuracy of this system and presented considerable difficulties. A s discussed above, errors in determining the loaded moment, curvature o r elastic slope will directly affect the resulting unloaded (desired zero) curvature. For these experiments the procedure was to calibrate the force and displacement transducers, measure the critical machine dimensions and then perform a straightening test. The overall error in the system will then be reflected in the curvature of the final piece as compared to the desired zero curvature. A detailed procedure for using such tests to fine-tune the calibration of all three quantities: force, curvature and slope can be employed; a simple procedure was used here where it was assumed that the errors were all caused by errors in curvature measurement, since it is the most difficult sensor to calibrate.
The initial estimate for the elastic slope is simply the bending stiffness for an elastic beam. However, since the test specimens were wide and thin, a plane-strain correction must be made, thus the actual calculation was made according to : dM/dK = EI / ( 1 - Y 2 , where E is the elastic modulus, I is the area moment of inertia and Y is poisson's ratio. The actual tests were performed on 1 m long strips of 2024-T4 aluminum and 304 stainless steel with a 25 x 3 mm rectangular cross section. In the first test a specimen was prepared by intentionally bending three regions of increasing curvature into the piece. It was then rolled through the closed-loop apparatus and the shape of the inital and final part measured. A plot of one if these shapes with the 7. The vertical axis exaggerated is shown in Fig. initial shape of the workpiece is seen to have regions of high curvature and a maximum elevation of 2.5 mm over 30 cm. The straightened part has a maximum elevation of 0.6 mm, but more importantly can be seen to have regiomof very low curvature with small "bumps" between. These correspond to transients in the control system caused by the curvature changes in the initial part. To completely remove them will require that either the bandwidth of the control system be increased or the rolling speed decreased. The loaded moment and curvature and the calculated unloaded curvature for this test is plotted in Fig. 8. The variations on the loaded curvature correspond to those observed in Fig. 7, A l s o shown is the calculated unloaded curvature that was used to drive the control system. The extent to which the predicted unloaded curvature is non-zero in the final piece is, therefore, a reflection of errors in calibration of the transducers and of the finite bandwidth of the control system as presented here.
4-
-300mrn Figure 7.
Profile for Test A 1 2. The upper line is the initial part. The vertical scale is multiplied bya factor of 6 and the horizontal scale by 1/12.
r-
E+02 30
~~
E
2
20-
Y 3
ca
KI
15 -
I'
Y A
J
10
-5
-
1
0
I
L -
2
I
I
I
I
1
I
4
6
€3
10
12
14
E+01 DISTANCE (am)
Figure 8.
The Measured Loaded Moment, Curvature and Calculated Unloaded Curvature f o r Test A 1 12.
-
I
1Figure 9.
300rnm--------------$( hOrl&ffM
Profile for Test SS 4. The scale is identical to Fig. 7 (the vertr'ca( scale k not)
.
139
In Fig. 9 a similar test performed on the stainless steel specimens is displayed and it is apparent that identical straightening characteristics exist for this material. The result is a more nearly flat part (0.6/5.2 mm) but this appears to be caused entirely by the less severe changes in curvature presented the initial piece. Indeed the only difference brn the experiments for this material is a change in the value calculated f o r the elastic slope of the moment-curvature relationship because of different values for E and y The results for a series of testson both aluminum and stainless steel are presented in table 1. The measure of straightness is the maximum deviation from a horizontal datum using a gauge section of 30 cm.
References
1.
2.
Hardt, D.E. Roberts, M.A. and Stelson, K.A., "Closed-Loop Shape Control of a Roll Bending Process," ASME. Journal of Dy.&mic S y . s k a s . Measurema& u 4 , no. 4 , Dec. 1982, pp. 317-321. Con-,
3.
Hansen, N.E. and Jannerup, O., "Modelling of ElasticPlastic Bending of Beams Using a Roller Bending Machine," ASME Paper 78-WA/Prod-6.
4.
Foster, G.B., "Springback Compensated Continuous Roll Forming Machines," U.S. Patent No. 3,955,389, May TI, 1976.
5.
Trotsmann, E., Hansen, N.E., and Cook, G., "General Scheme for Automatic Control of Continuous Bending of Dvnamic S y s t e L of Beams," T Measurement and Contra;L, U, no. 2, 1982, pp. 173-179.
.
TABLE 1
TEST A1 0 A1 2 A1 3 A1 7 A1 8 ss 2
ss ss
4
5
Initial 'max 1.6 mm 2.5 2.6 2.0 1.9 3.9 5.2 6.5
Final Ymax 0.6 mm 0.6 0.9 0.7 0.9 0.2 0.6 0.6
%
Reduction 63 % 76 65 65 53 95 88 91
From these data it can be seen that the steel appears to straighten to smaller deflections than the aluminum. This can be attributed to the lesser springback for the steel and the higher moments involved, which improved the signal to noise ratio on the force transducer. However, the profiles f o r all of these specimens were similar to thoseco~y~UmFigs. 7 and 9 thus the major cause of residualvapbears to be the relatively slow settling time of the control system. CONCLUSIONS
The closed-loop roll bending control scheme presented here has the inherent property of disturbance rejection that all feedback systems possess. In this case the disturbance is initial curvature in the workpiece. Accordingly the control system will automatically accorffadate non-straight workpieces and if the specified final curvature is zero, a flat piece will be produced. It should be noticed that the straightening problem is in many ways more severe than the bending problem since arbitrary initial curvatures must be dealt with. Accordingly, a closed-loop system such as that demonstrated in this paper is a basic requirement for automated straighten ing The evaluation of the worth of this system as compared to a manual system requires an assessment of the importance of production speed. Manual operation is iterative and with non-uniform initial curvatures will require several bending passes simply to identify and respond to the various sections o f differing curvature on the workpiece. However, with a closed-loop control strategy the problems of arbitrary initial curvature and unknown plastic properties of the material are explicitly considered within the control loop and are, therefore, transparent to the operator. This implies then that single pass straightening will be possible. Given the prospect of single pass straightening, the importance of rolling speed can be seen. The next phase of this research will consider in greater detail the dynamics of this control system in an effort to determine the upper bound on rolling speed. As discussed in 121 the dynamics will be a function of the bending machine, the transducers and the workpiece itself. Since the latter will be of variable net stiffness and mass, the best control system will probably have to consider explicitly the changing dynamic properties of the system being controlled.
.
Acknowledgements
The authors gratefully acknowledge the assistance of Mr. Lee Mallet working under the Undergraduate Research Opportunities Program (UROP).
140
Hardt, D.E., and Chen, B., "Control of a Sequential Brakeforming Process," Control of Manufacturinu Processes qnd Robotic SY stems, ASME Pub. # H00279, 1983, pp. 171-182.