K 3 * COORDINATED SCIEN CE LABORATORY
DYNAMIC MODEL
OF A CONTINUOUS
COLD ROLLING MILL
M. J A M S H I D I R K 0 K 0 T 0 V I C '
U N IVERSITY OF ILLIN O IS - URBANA, ILLIN O IS
by
M. Jamshidi and P. Kokotovic
This work was supported in part by the Joint Services Electronics Program (U.S. Army, U.S. Navy, U.S. Air Force) under Contract DAAB-07- 67-C-0199; also in part by the U. S. Air Force under Grant AFOSR 68-1579A; and also in part the Alcoa Foundation.
Reproduction in whole or in part is permitted for any purpose of the United States Government.
This document has been approved for public release and sale; its distribution is unlimited
by
M. Jarnshidi and P. Kokotovic
Abstract
The dynamic model of a continuous cold reduction mill is developed. The model makes use of the force=torque equation expansion technique developed in an e a r lie r report and presents the generalized (7N+8)th order state equation
for an N-stand ro llin g m ill. The equations of inter stand tensions are found
to be highly nonlinear with the arguments delayed due to the transition of strip
between the stands. The model developed can be used in the design of a
The authors are thankful to Mrs. Sandra Bowles, Mrs. Barbara Champagne, Mrs. Sherry Kallembach, Mrs. Rose Lane, and Mrs. Nicola S t illin g s for their
I. INTRODUCTION
In this report a nonlinear model of an N-stand ro llin g m ill is
developed. The model represents an improvement on previously proposed
models [1 ,2 ].
Figure 1 shows an N-stand m ill presented with c o ile r , decoiler, r o l l screw-down and drive motors.
As seen, the decoiler feeds the strip into the f i r s t stand where i t s thickness is reduced by a combination of interstand tension
and r o l l separating force. The r o lls of each stand are driven by a dc-
motor which in turn is connected to i t s generator. The generators, the
c ircu it breakers and emergency stopping circuits are not shown for
simplicity of the diagram [3 ]. The drive motors can be controlled
by both armature and f i e l d , while the screwdown motors are only armature controlled.
The model developed in this report w i l l be used in the design
of a controller for the r o llin g process. Each stand, c o ile r and decoiler
are considered as dynamic subsystems coupled by an e la stic s trip . In
general the coupled model is nonlinear and it s order is high. Conse
quently, straight forward applications of optimal control theory would be v ir t u a lly impossible and some approximation methods are needed. The development of this npdel takes into consideration properties of a
class of near optimum design methods. In p articu lar, i t is adapted for
an application of singular perturbation [4 - 6 ], decoupling methods [7 ], three time scale design method [8] and design of systems with time
Figure 1, remains general in it s mechanical subsystems and formulation of
state equations for the m ill in interstand variables. The N-stand m ill
system of Figure 1 is divided into two major subsystems: mechanical
and e le c t r ic a l.
2. MECHANICAL SUBSYSTEM
A b r i e f review of ro llin g theory w i l l explain the deformation process, interstand relations and other r o llin g equations relatin g m ill process variables such as thickness, tension, torque, force, and
coefficients of fric tio n . Appendix 1 presents the notation and nomen
clature for the entire report. 2.1 Deformation Process
In general, there is no "exact" r o llin g theory presented in
the lite ra tu re . The "exactness" of a r o llin g theory depends on assump
tions involved. Thus fa r , the theory of Orowan [13] is most "exact".
I t permits variation of both yield stress and coefficient of fric tio n .
Xt is furthermore useful for both cold and hot strip r o llin g . Orowan’ s
theory has been extended by Finne, et a l [14]. However, due to it s
complexity, this theory is approximated by Bland and Ford [15].
The theory of Bland and Ford is used throughout our analysis. Using this theory several authors have developed graphical techniques and it e r a t iv e methods for computations of force and torque to be
applied to the r o lls during the process [1,2,16-20]. In an e a r lie r
report we developed a computational procedure for determining the coefficients of a three-term Taylor series expansion of ro llin g
force and torque [21].
The deformation of strip takes place at each stand's r o l l gap. A r o l l gap configuration is shown in Figure 2.
As seen the arc of contact is divided into three segments: A is the e la stic arc at the entry.
B is the p la s t ic arc.
C is the e la s tic recovery arc at the e xit.
In the two e la s tic regions the fr ic t io n a l force exerted by the r o l l s to the strip is commonly assumed to [13-15] be proportional to r o l l pressure with
proportionality being the coefficient of f r ic t io n . I t is in these two
regions where the r o llin g metal is said to be "s lip p in g ” [13-20]. In
region B, however, the fr ic t io n a l force increases up to the magnitude of the y ield stress in shear and a p la stic shear occurs in the metal. The surface of the r o llin g metal is then said to be "sticking" to the
r o lls [13]. One of the basic assumptions made in [13] is that the
deformation in the p la s tic region is a "homogeneous compression", i . e . the strip is divided into a number of v e r t ic a l segments which remain
plane and perpendicular to the direction of r o llin g . Three of these
planes are shown in Figure 2. As seen from this figu re, in the neighbor
hood of the entry plane these v e rtic a l segments (Planes) are "squeezed backward" while close to the exit plane they are "squeezed forward". In the middle of the gap there is a segment where the strip is being
pulled out to i t s right and pushed back to it s l e f t . Such a segment
is called the "neutral plane" [13,20].
The r o llin g theory of Bland and Ford calculates r o l l force and torque items of the "m ill v a ria b le s ", h ., h , t . , t , and y,
1 o 1 o
F = F(h. , h , t . , t , y)
t = t(h ± , hQ, t± , t Q, y) . (2.2) As i t w i l l become clear later a third equation for the neutral point thickness, h^ (See Figure 2) is also required.
h = h ( h . , h , t . , t , y)
n n i o ’ l* o ’ (2.3)
Analytic expressions for (2.1, 2 and 3) developed by Bland and Ford are given in Appendix 2.
These expressions are complicated nonlinear functions of m ill
variables and a simplification of their calculation is needed. In [18]
a graphical method was proposed. Some computer ite ra tiv e methods have
been suggested [1 ,2 ], where small (1%) incremental changes of different m ill variables are made to perform linearization of the force-torque
equations. A computationally simpler and in the same time more accurate
approximation is given in [21] where 2.1, 2 and 3 are expanded using
e x p lic it d ifferen tiation . The truncated series are
F = (k ,q ) + ( q ,Kq) (2.4)
f = (4,q> + ( q , Lq> (2.5)
‘n = <m,q> + ( q , Mq ) (2.6)
where q is a vector with the mill variables as its components
Tj r- ^ /v a ^ n
q = Lh. h t . t mJ ,
k is a 5X1 normalized vector, k. = ^—
* l F oq, i = 1 , . . . 5 and K is a 5X5
symmetric matrix whose elements are, k „ = —
q* q^*
i 1 ô2F
F* dq. dq.
i J
Similar
definitions apply for i , L, m, and M for (2.5) and (2 .6 ). The computational
scheme developed in [ 2 l ] consists of two main subroutines which allows a l l the mill coefficients to be obtained in less than 20 seconds on a CDC-1604 computer.
A common assumption in a l l r o llin g theories is that the volume of the material per unit time passing through each v e r t ic a l segment is constant, i . e .
v. h. = . . . . = v h = . . . . = v h
i l n n o o (2.8)
2,2. Stand Relations
The output tension of the jth stand is assumed to be equal to the input tension of the ( j+1) th stand, i . e .
t . - (t) = t . (t) .
i , J + l oj (2.9)
The output tension is related to the strip velocity by the e la s t ic it y principle (Hook's law) [22-24],
‘ oj ■ T <v i , j + l - vo ; j ) (2.10)
where it is assumed that the delay time T (tran sit time between stands j
and j+1) is constant [ l 4 , 1 8 ] . The input thickens at the (j+1)th stand is the
= h . (t -1 ) .
o,J (2.11)
As noted before when force is applied to the r o l l s an e la stic deformation occurs to the "m ill housing" and the r o l l s , i . e . the entire structure (stand and its four highs) is considered to be a s t i f f spring with a
modulus of e la s t ic it y (spring constant) M [1 ,2 ,2 5 ]. Thus,
F.
J (2.12)
where is the screw-down setting (separation of r o l l s during r o llin g ,
positive upward) of the jth stand and S . is the screw-down setting when
°> J
Fj = 0. I f the e lastic recovery of the metal is neglected as in the case in
Bland and Ford's theory [1,2,15,25] then the output thickness h . = S. and
°> J J
(2.12) in incremental form becomes [
2
] ,3. ELECTRICAL SUBSYSTEM
The e le c tric a l subsystem consists of the control r e c t i f i e r motor groups for c o ile r, decoiler, stands, drives, and screw-down servomechanisms. In order to preserve generolity, our model is derived with the assumption
that the armature voltages of the motors are control variables. In a
solution of an optimum control problem the dynamics of the generators or
rectifiers w i l l have to be included, and the motor armature voltages w i l l
become state variables.
In this section the dynamic equations of motors are given. A
block diagram of both mechanical and e le c t r ic a l subsystems shows the in te r connection between process variables.
3.1 Coiler and Decoiler
The coiler and decoiler are driven by a dc motor which is con trolled by both armature and f ie ld voltages.
strip
c
The torque equation at the c o i l e r ’ s shaft, Figure 3, is [23,26]
where the term J c
(oo
cSt = t = Jd) + J ( u ) - 0) ) + B u) H— ■ r t __ (3 ,1 )
c c c c s c c c n c oN
U)^) accounts for the fact that the inertia of
the c o ile r, Jc , is a time-varying function [27-29], The term u>c -
is the magnitude of the angular velocity of incoming strip re la tiv e to
that of the c o ile r. The remaining relationships for the system in
Fig, 3 are T C = Kl c 0c iac = R i + L ■ ac ac ac a 1 f c = Rfc i „f c + Nfc r c h oN 2n 00c di d0( + K ao DC C fc Cl 0c + C30c J ( r ) = J + ^ JT = J + npW r 4 c c m 2 L m 0 2 r c n Zn and B = B + ~ BT , c m 2 ^ n
A l l the quantities are defined in the nomenclature (Appendix 1), A similar set of equations can be written for decoiler by
interchanging subscripts c (c o i l e r ) to d (decoiler) and N to 1, For
example, as an equivalent to 3,1 one obtains,
St = Td = J i d + Jd <«»]_ - u>d) + BdU)d - l r t (3.2) (3.3) (3.4) (3.5) (3.6) (3.7) (3.8) where xd = Kld0di ad, etc
Note that a nonlinear magnitization relation, (3.6) is assumed between the f i e ld current and the flux of the motors.
3.2 Main Drive
The torque, armature and fie ld c ircu it ^equations for the jth stand
are presented. The stand's drive motor must provide torque su ffic ie n t to
balance frictions ro llin g torque, torque caused by inlet tension and the
acceleration torque. The torque caused by outlet tension is helping the
r o lls to rotate in the prescribed directions. Thus,
where t^j o, f , i , j indicate the torque of motor, outlet tension, f r ic t io n ,
inlet tension, and jth r o l l s , respectively. Equation (3.9) can be written
for the jth stand as
Et = t + t - J - t , - t . - t. = 0 m o tu f i j (3.9) t . = Jd). + Bcjd. + ~ Rt. . = — Rt . + t. mj J j n i j n o j j (3.10) where, (3.11) The armature and fie ld currents are governed by,
(3.12)
(3.13) The fie ld current-flux are related by the familiar magnitization charac
t e r is t ic approximated by third order polynomial,
(3.14) Thus three variables, u)., i . , and 0; are sufficient for a state space
J aJ v
3,3 Screw-Down
The screw-down motors are assumed to be of the armature-
controlled direct current type. The task of these motors is the
adjustment of the vertical position of r o l l s (settings or screw
pitch) [3 0 ], The equations for the jth stand screw-down motor are,
(3.15)
(3.16)
(3.17)
and the screw speed is assumed to be directly proportional to the angular velocity of the motor,
t . - J cb . + B a) . sj s sj s sj t . = K. 0 . i . Sj 1 sj a s j d ( i .) e . — R i . + L l~" ~l "3' * + K, cu asj a asj d t b sj S . ° J (3,18)
where S . is the screw pitch of the ith stand r o l l s ,
oj J
3,4 M ill Block Diagram
The mechanical and e le c tric a l subsystems for the jth
stand can be shown in a general block diagram as in Fig, 4, Note
that for j = 1 and N, the diagram involves the decoiler or coiler motors, respectively and are considered as special cases of this diagram.
Table 1 shows a possible selection of state, control and output
variables of the system of Figures 1,2. For a single stand r o l l i n g m ill,
i . e . , j = N = 1 there may be as many as 7 + 8 = 15 state variables, 3 + 4 = 7 control variables and 2 output variables.
Let x. . = (JD . J x 0 . = t 2j Oj x 0 . = i . 3j aj x. . = 0 . 4j x r . = U) . 5j s j x . . = S 6j Oj x_ . = i 7j as j u. . = e . ! j a J u0 . = e 2j f J u_ . = e 3 j as j (4.1)
Considering (2.4) and rewrite it as:
f j = <k>qj ) + ( q j- K q?
= k00 iin . + (k0 + k10 ii. + k0 t . . + k0 p. .) ii .
22 Oj 2 12 lg 23 i j 25 y aj
+ ( k ,q j) + <qy K q (4.2)
where
k = Ik, K = IK, I = diag [ l 0 1 1 l ] ,
Table 1
System Variables in an N-Stand Rolling M ill Stand j Total no. of Variables Type of Variable Roll
Gap Drive Motor Screw-Down
Decoiler Coiler
State
*0 j (JO. J j i a J . ’ 6 .J (10 s j’ .S-*, i Oj aSj U)di o’d0dr d (jo? i c ac c c> (t>> r 7N + 8
Control e .. e , . f j eas . J ead',ef d e 9 0. . ac f c 3N + 4 Output fl , OJ t . OJ - - - - 2N
the subscript has been omitted on them for Simplicity-
Eliminating ft from (4.2) and (2 .1 3 ),
P j k22^0j + ^ j (k2 + k12^ih + k23^ij + k24^0j + k2 ^ V
-1] + (<*jS0j + p . <is>qj) + 3 j ( q j ^ q ^ ) ) =
where O'. = ( ^ —) , 3. = (F ./Mb-.) as defined in (2.13).
J n0 j J A J J
Solving for h^ from the above and considering (2
notations of (4 .1 ),
R0j =
a. .o . x 3 J 2, j-1 . . + a. .x 4j 2j - + 7] 2. (t - T ) + a_ . 5jwhere, a _ , i = 3,4,5 and Tj^(t-T) are defined in Appendix 3. Similarly torque and thickness equations of (2.5)
after the substitution of . from (4.3) can be rewritten as
Oj
t . = b x . + b x + 0 . (t - T ) + b
J 3 j 2, j - 1 4j 2 j 5 j
b ^ , i = 3,4,5 and 0 ^ (t-T ) are expressed in Appendix 3 Similarly 2.6 can be reduced to,
fi . = C x . - + C. .x + i|r . (t - T ) + C,. .
nJ 3j 2 , j-1 4j 2j Yj v 5j
where C ., n = 3,4,5 and \lf . have expressions identical with
nj TJ
and 0^ when replacing & and L by m and M, respectively.
9) and (4.3) and (2.6) (4.4) (4.5) b ., n = 3,4,5 nj
To derive the f i r s t state equation let us write the torque
equation at the shaft of the stand drive motor,
J cl). = -Bud. + K. i .0 . - — A . . t . . + — , t .
J J 1 a J J n i j i j n o j
1 * * 1 * -k *
- T . T . - —( R ( t . . - tA .) + X .)
n J J i j Oj7 y (4.6)
Substituting for % from (4»7)
B 1 * x, . = " T x - — (b X . + RA. .) x. . . l j J J n 3 j j i j ' 2 , j - 1 — (b . r . - RAn .)x 0 . — 0 . - dn . Jn 4j j 0 j 7 2j Jn l j (4.7) where d. . = — (R (t. 4 - t n .) + (b .+1) t .) l j Jn i j Oj7 v 5j 7 j 7 (4.8)
To drive the state quantum for t ^ , consider equations ( 2 . 8 ), (2 .10), and (2.11), Vi>j+1 - 1 H----V3? i* j+1 = -1 + V n, 1+1 1 h n , j + l1 h. . Vi, j + 1 “ i , j+1 Rh* ... U).,.(fi + 1) n, .1+1 1+1 n, i+l v* - L h ( t L . ( t - T ) + 1) i , J + l i , J + l 0 j v -1 + (4.9)
Sim ilarly, + Rh .h J 9T~ oj CJU . (ft .+1)
• jrf-
oj+ 1)
(4.10) then by (2.10) *2 j = ^2j ( —*—( t_T) »—( t_2T) ) (4.11)where is defined in Appendix 3. Note that by x in (4.11) it is meant a
I
vector containing a l l state variables as defined by (4 .1 ).
The remaining state equations are simple and are written accord ing to (3.12-18). Ra Kb , 1 LaX3.j L Xl j a J L Ulja J V i Rf e3 3 . 1 Nf v4j Nf x. . + — 4j Nf B K. • s . i s 5J " “ Js X5j Js X7j (4.12) (4.13) (4.14) -K x s 5j (4.15) R as L as as 5j (4.16)
Thus 4.7, 4.11, 4.12 to 4.16 are Severn s t a t e equations d e s c r i b i n g the
th
dynamics of the j stand. In addition to the above, eight other equations
let
M = 7N, N is the number of stands in the r o lli n g m ill in s t a l lation, and .1er : C o i le r : XM+1 = “ a XM+5 = (JOc XM+2 Lad XM+6 Lac XM+3 XM+7 K XM+4 r d XM+8 = rc i
U2N+1 = ®ad U3N+3 = eac
U3N+2 CD
II
U3N+4 = e, f c
(4.17)
The corresponding state equations are,
M+l * XM+4X11 *M+1 Bd Jd(x) 2WPhi j Jd(x) 2 3 , XM+lXM+4 , i l XM+4XM+5 + 2Wph . . ■_ ~ t---- + i l Jd (x) n Jd (x) + K XM+2XW-3 Id Jd (x ) (4.18) where J(x) Jm + 2 TTpWxM+4 2n R , K, , . _ ad bd , 1 ^ + 2 "L , XM+2 L , XM+1 L , U3N+1 ad ad ad (4.19)
Rf dl Rfdd3 3 1 XM+3 ” N , , XM+3 " N£j XM+3 + N£J U3N+2 rd rd fd i l ^ + 4 2tt XM+1 ^ + 5 3 „ CM+5 , * XM+8XN . = - B x + 2Wph „ —i—r~r + c J (x) c oN J (x) c 2 3 „„ , * XM+5XM+8 ON XM+8X2N , TT XM+8XM+7 2Wph .. ■_ r~ — - — ---mi 1 j \ + K, oN J (x) c n J (x) c lc J (x)c when 1 4 J (x) = J H---x npWx..,0 c v ' m _ 2 M+8 2n R ac K M+6 be 1 X_, + T---- u. L M+6 L M+5 L 3N+3 ac ac ac Rr C fc 1 Rfc C3 XM+7 Nr aM+7 f c Nfc U3N+3 Rfc Cl R fc 3r C 3 XM+7 = N, XM+7 fc N, f c XM+7 ' 'fc = _oN XM+8 2tt XM+5 (4.20) (4.21) (4.22) (4.23) (4.23) (4.24) (4.25)
The above equations as w e ll as those for the j stand can now
be written in a vector form for an N- stand r o llin g mill as follows, Let
x X1
xn
ui U1 x2 X21 u2 0 S tand 1 0 u3 « X71 • U3 O • • • • 0 • • * . • • • = x . J u = • St u . J • X2j Stand j • U2j • K7j • _ *3 j _ • %f • » » # o o « X1N • "n ° X2N Stand N • U2N • X7N • _ ^3N _ • X7N+1 Coiler • U3N+1 0 * and j • Decoiler X7N+8 X7N+8 U3N+4 U3N+4 _ _ _ — — —1 (4.26)be 7N + 8th and 2N + 4th state and control vectors for the N- stand
x = Ax + Bu + f ( x ) + g ( x , x ( t - T ) , x (t -2 T )) + d
where
A 7N + 8 X 3N + 8 constant matrix
B 7N + 8 X 2N + 4 constant matrix (4.27)
f , g, d, and x are 7N + 8 vector
u is 7N + 4 vector
f and g are nonlinear vector functions.
The above matrices and vectors are defined on the following pages.
a31 0 a33 0 0 0 0 1 0 0 0 a44 0 0 0 11 0 0 0 0 a55 0 a 57 i 0 0 0 0 a65 0 0 0 0 0 0 a_ 7 5 0 a77 1 0 a7N-6,7N-12 ° ° l a 7N-6,7N-6 a 7N-6,7N-5 1 0 0 0 ia I /N-7, 7N-6 0 a7N 0 0 0 ; o 0 0 1 0 0 0 0 0 0
J
NO U>0 0 0 0 0 0 í-3,7N-3 0 0 a7N-2,7N-2 0 a7N-l,7N-2 0 a7N,7N-2
o
o o o l7N-2,7N O L7N,7N a7N+2,7N+l a7N+2,7N+2 O l7N+4,7N+1 O O O o o o l7N+3, 7N+3 O O o o o 0 0 o 0 0 o 0 0 o 0 0 o 0 0 o 0 a7N+6,7N+5 a7N+6,7N+6 0 0 0 a 0 a 7N+8, 7N+5 0 0 0 o o o o 7N+7, 7N+7 ° 0 0 C o il e r a n d Deco il er S ta n d N .. . S tab31 0 0 0 b42 0 0 0 0 0 0 0 0 0 b • • • B= 1I 0 0 0 1 0 0 0 0 ! b7N-4,3N-2 0 0 ! ° b7N-3,3N-l 0 i ° 0 0 i ° 0 0 1 0 0 b. 7N,3N 0 0 0 0 b7N+2,3N+l 0 0 0 0 b7N+3,3N+2 0 0 0 0 0 0 0 0 0 0 0 0 b 7N+6 j 3N+3 0 0 0 0 b 0 0 0 0 7N+7, 3N+4 C o il e r a n d Deco il er l S ta n d N r . . . 1 S ta n d 1
o o o o o o o o I I Q* I " J I O O O O O O I I I CT> I I I CL O O O O O O (-> I l-h —J o 2¡ o a Ln Mi Mi •^J
° ¥
03 o ¡z¡ + -J O a I 0 3 M> II I I Mi O O O 4> O O O I OQ II r O O O O o o o o i OQ ! o o o o o s; OQ a O © • • « 1 » OQ OQ • <* O O O O O N3 h- 1 1 1 Ul CT\ 1 » 1C o i l e r and D e c o i l e r Stand N • • • Stand 1
has been developed. A generalized state equation of (7N + 8)th order was derived taking into account the dynamics of c o ile r , stands, drive,
screw~down motors and decoiler. The equations turned out to be highly
nonlinear and in general involve time lags due to the transition of strip from one stand to the next.
This model is expected to be a more accurate description of a r o llin g mill than the models proposed thus far since i t uses less approximations such as linearizations of force-torque as well as the
conservation of the strip volume (eq. 2 .8 ). The main reasons for avoid
ing these approximations is that with the existence of d i g i t a l computer and e ffic ie n t numerical methods more r e a l i s t i c models can be handled.
The state equations can be modified for r o llin g mills with different e le c t r ic a l equipment and introduction of new variables cause no d i f f i c u l t y in formulating the new set of equations.
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a System with Time Varying Transport Lags", IEEE Trans. AC, October, 1966.
13. Orowan, E., "The Calculation of Roll Pressure in Hot and Cold
14. Finne, R., J. D. Jacob, and J„ Rissanen, "A Mathematical Model of Strip Rolling M i l l s " , IBM Nordiska Laboratorier TP18.0 94,
January, 1963, Nordiska, Sweden.
15. Bland, D. P. and H. Ford, "The Calculation of Roll Force and
Torque in Cold Strip Rolling With Tensions", Proc. Inst. Mech. Engrs. , V o l . 159, pp. 144-63, 1948.
16. Ford, H. "Researches into the Deformation of Metals by Cold Rolling",
Proc. Inst. Mech. Engrs., Vol. 159, p. 115, 1948.
17. Ford, H., F. E l l i s , and D. R. Bland, "Cold Rolling with Strip
Tensions", J. Iron and Steel I n s t r .. Vol. 168, pp. 57-68, London, England.
18. Hessenberg, W. C. F . , and R. B. Sims, "Principles of Continuous
Gauge Control in Sheet and Strip Rol ling", I n s t . Mech. Engrs. J ., Vol. 155, 1952, pp. 75-81, London, England.
19. Ford, H., F. E l l i s , "Comparison of Calculated and Experimental
Results", J„ Iron and Steel I n s t ., Vol. 171, pp. 239-245, 1951.
20. Bland, D. R. and H. Ford, "An Approximate Treatment of the
Elastic Compression of the Strip in Cold Rolling", J. Iron and Steel I n s t . , Vol. 171, October, 1951, pp. 245-249.
21. Jamshidi, M. and P. Kokotovic, "An Approximation of Force-Torque
Equation in Rolling M i l l s " , Coordinated Science Laboratory Report R-452, University of I l l i n o i s , Urbana, I l l i n o i s .
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the Theoretical, and Experimental Analysis of Stress and Strain, McGraw-Hill Book Co., New York, 1958.
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published for NASA and NSF Israel Program for Scientific Trans lation, Jerusalem, 1968.
25. Courcoulas, J. H„ and J. M. Ham, "Incremental Control Equations
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McGraw-Hill, New York, 1964.
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Features of Combinational Temper - Reduction M i l l s " , ISE Journal, October, 1965, pp. 125-134.
1
311
Appendix 1. Notations and Nomenclature1
Notations :1
i = "entry" or i nl et plane m o = " e x i t " , or outlet planei
n = neutral plane a j = number of stand j = 1 , . . . , N1
N = maximum number of stands in the mi ll1
A = F = f - F*F , denotes the operating value■
t. . the i nl et tension, t. belonging to
the jth stand.
1
s = screw down motors|
c = coil er d = decoiler1
m = motor ( c o i l e r , decoiler)i
L = loadI
Nomenclature: • ef = f i e l d voltage1
e a = armature voltage■ (JO = angular velocity
1
Soj = screw-down setting of jth stand r o l l s
I
9 = arc of contact ( r o l l s and s tr ip )m
R = radius of undeformed r o l l
i
R' = radius of deformed r o l li
i
tension per unit area area of the strip
r o l l torque per unit width r o l l force per unit width coefficient of f r i c t i o n strip velocity
modulus of e l a s t i c i t y (Young's modulus) time lag ( s t r i p transit period)
modulus of e l a s t i c i t y (spring constant) of "mill housing" moment of inertia
f r i c ti o n loss gear ratio
coiler or decoiler radius torque constant
magnetic flux armature current armature resistance armature inductance back emf constant f i e l d current back emf constant f i e l d inductance f i e l d resistance
f i e l d winding number of turns coefficients of f i e l d current polynomial
w = width of the strip K
s = screw-down position gears constant
d = disturbance vector
X = state vector
Appendix 2.
where
and
Roll Force and Torque Equations [Bland and Ford]
F = F(h.,h , t . , t , r ) = R ' f f n ^ (1 - ^ ) ' i * o i * o 7 L*Jo h v k 7 M<H(h , 0) d0 rei k h n V - H( ho>e0 l
+ Je
Ï T (1 " k 7 ) eJ
n i(
2
.
1
)
e
T = = RR' J1^ jp- (1 - J2 ) ee»1 + 1ae + O to ^O + 1 0. i,ui kh Jo" if1 C1 “ t t) + t. h . t. - h t lv . i t o o 0 h. H 1 k.i 2R '(
2
.
2
)
H(h,0) = 2/ f tan" 1 0 h = h (h. ,h , t. ,t t) n n i o i o 1 + \ tan2
. h. (1 - to/ko) J — In 1 t. 2(1 h (1 - ~ ) J o k. i (2.3)In Equation ( 4 . 3) ;
“23 “24
a3j 2k22 ’ “4j 2k22 ’
^25^1
a5j - - 2k22 + 2P.k22 , a l l k' s are for jth stand,
+ 71. (t-T) ” J 12 2k
22
ko , j - l ^ t_T^ ± f ^ ( k2+ki 2ho , + k23X2 , j -+ k24X2,j -+ k25M' j ) “ 1-' " 4ßk22(aj x6j + ^ k ’ qj^ + ß .( q. , K q . ) ) P . J J JNote that here
qj Cho , j - l ( t _ I ) h0 j (t) X2 , j - 1 X2,j ^
0 j ( t - T ) ( X22a3j + 'e23a3 j ) x 2 , j - l + ^ 2 2 a4j + X24a4 j ^ X2 , j
+ (X23a4j + ^24a3j + 2'e22a3 j a4j) x 2 , j - l x2j
+ [ ( ^ 23 + 2'e22a3 j) x 2 , j - l + ( ^24 + 2 i 22a4 j ) x 2j + X12ho,
+ (¿2 + 2^22a5j + <e25M' j ^ Tij ( t “T) + ^ 1 2 (a3j X2 , j - l + a4 j X2,j
+ a 5 . ) + ^2^ho , j - l ( t " T^ + ^qj ’- ,qj^ + <e22Tij 2 ( t " 1^ *
Note that L = rL and is as in ( 4 . 2 ) . Equation (4.11);
A ER S2j T h* X. (c - .X . - + c, .X + Y. (t-T) + c,. .+1) _£J__ - L l 3 1 2 , i - l 4i 2i i x 5i V . h*. (a,,.x0 . - + a. .x0 . + Ti. (t-T) + ac .+l) °J °J 3j 2 , j - 1 4j 2j 3 5j h* , ‘n . i + l * 1 . i + l (c3 . 1+1*2.1 + C4 . 1+1*2 ■ 1+1 + vi , j + l hi , j + l <a3 jX2 , j - l ( t * ^ + a4Jx2J(t_l5 + 'i' i + l ( t ' ^ + C5.i+1 + ^ Tlj (t-2T) + a5j + 1)
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R E P O R T S E C U R I T Y C L A S S I F I C A T I O N
2 b . G R O U P
3 . R E P O R T T I T L E
DYNAMIC MODEL OF A CONTINUOUS COLD ROLLING MILL
4 . D E S C R I P T I V E N O T E S ( T y p e o t r e p o rt a n d i n d u s iv e d a te s ) 5- A U T H O R ( S ) ( F ir s t name, m id d le in it ia l, la s t name)
JAMSHIDI, M. & KOKOTOVIC, P.
6 W m 9D tf f 7 0 7a. T O T A L . N O . O F P A G E S
35
7 b . N O . O F R E F S 33
" d ® s ' w T- e r - v m w i a i so m part afosr 68-1579A» and in part Alcoa Foundation. 9 a . O R I G I N A T O R ' S R E P O R T N U M B E R ( S ) b. P R O J E C T N O . R-459 9b. O T H E R R E P O R T N O ( S ) (A n y o th e r n u m b e rs th a t m a y be a s s i g n e e / th i s re p o rt) UILU-ENG 70-204 1 0 . D I S T R I B U T I O N S T A T E M E N T
This document has been approved for public release and sale; its distribution is unlimited.
1 1 . S U P P L E M E N T A R Y N O T E S D N . S O . R I N G M I L I T A R Y A C T I V I T Y
Joint Services Electronics Program thru U.S. Army Electronics Command Fort Monmouth, New Jersey 07703 1 3 . A B S T R A C T
The dynamic model of a continuous cold reduction mill is developed. The model
makes use of the force-torque equation expansion technique developed in an e a r l i e r report and presents the generalized (7N+8)th order state equation for an N-stand
r o l l i n g mi ll . The equations of interstand tensions are found to be highly nonlinear
with the arguments delayed due to the transition of strip between the stands. The
model developed can used in the design of a controller for the entire r ol l i n g process.
DD ,F,r.,1473
System Modelling Rolling Mil ls Control Systems State-space Approach ! j * -S e c u r i t y C l a s s i f i c a t i o n