Dynamic Model of a Continuous Cold Rolling Mill

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

St = 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 =

where, 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-

+ 1)

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

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

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

b31 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

° ¥

C 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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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 .

22. Durel li, A. J . , E. A. P h i l l i p s , and C. H, Tsao, Introduction to

the Theoretical, and Experimental Analysis of Stress and Strain, McGraw-Hill Book Co., New York, 1958.

23. Kokotovic, P. and S. Bingulac, "Model Feedback Concept in

Analog Computer Design of Control Systems", Proc. Analog. Comp. Conference, pp. 291-298, Belgrade, Yoguslavia.

24. Moorev, M. V., Multivariable Control Systems, ( tr ansl . Russian),

published for NASA and NSF Israel Program for Scientific Trans­ lation, Jerusalem, 1968.

25. Courcoulas, J. H„ and J. M. Ham, "Incremental Control Equations

26, Gourishankar, V. , Electromechanical Energy Conversion, Inter­ national Book Co., Scranten, Pennsylvania, 1967.

27, Coughanowr and Koppel, Process Systems Analysis and Control,

McGraw-Hill, New York, 1964.

28, Shinskey, M., Process Control Systems, McGraw-Hill, I960.

29, Perkins, W. R. and J. B. Cruz, J r . , Engineering of Dynamic

Systems, John Wiley and Sons Book Co., New York, 1969.

30, Johnson, F. G., J. H. Stoltz, and R. G. Chapman, J r . , "El ec tri cal

Features of Combinational Temper - Reduction M i l l s " , ISE Journal, October, 1965, pp. 125-134.

1

1

1

1

i

1

1

■

t. . the i nl et tension, t. belonging to

the jth stand.

1

|

1

i

I

1

■ (JO = angular velocity

1

oj = screw-down setting of jth stand r o l l s

I

m

R = radius of undeformed r o l l

i

i

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

J

(

₂

.

₁

)

e

(

2

.

2

)

2

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

Note 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