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Satonin A.V., Korenko M.G., Nastoyashchaya S.S., Perekhodchenko V.A.
THE DEVELOPMENT OF ENGINEERING METHODS FOR CALCULATING ENERGY-POWER PARAMETERS OF RELATIVELY THIN STRIPS HOT ROLLING PROCESS
Abstract. Based on the analytical solutions the engineering mathematical model of the relatively thin strips hot rolling process energy-power parameters has been and expanded.Criterion evaluation has confirmed a sufficient degree of accuracy of the engineering techniques obtained. It was done on the basis of the skidding lines fields' method which allows us to take into account two-dimensional character of plastic metal forming absolutely.
Keywords: power parameters, relatively thin strips, roll gap, engineering mathematical model.
Further development of ferrous and non- ferrous metallurgy is inextricably linked with the increasing automation of relatively thin sheet and strip hot rolling process. Alongside with the numeral solutions some engineering mathematical models of the technological scheme energy-power parameters are also of interest as the ones that are able to ensure the rational combination of complexity, performance, and the reliability of the results.
As for engineering methods for calculating the rolling of relatively thin sheets and strips one of the most frequently used technique is the A.I Tselikov's one, which is based on the analytical description of the external contact friction conditions expressed as the Amontons-Coulomb law. [1]. From the point of view of the conditions of the hot-rolling process realization it is more appropriate to use the Siebel's law as the one providing condition of non-exceeding tangential contact stresses r shear resistance of
■V
the rolled metal Kx to the entire range of possible relative reduction. In this case it is necessary as well to take into account availability of an elastic recovery zone, as well as the influence of the stress de-viator tangential components that may reach sufficiently large relative values.
The purpose of the work is to refine and expand the engineering power parameters of the mathematical model process of relatively thin strips hot rolling.
Engineering mathematical model of power parameters of relatively thin strips hot rolling process having been developed as similar to the methods of [1, 2] is based on the following basic assumptions adoption:
• rolled strips plastic changes are two-dimensional and temporally steady;
• deformation zone (Fig. 1a) includes the area of plastic forming with the length of Lm and area of the elastic recovery of the rolled strip with the length of LJK ;the boundaries of plastic and elastic forming zones, and the
cross section area of conjugation lag with the length of L and timing zone, with the length of L , are
om o ' o on '
vertical, and parallel to the plane which is passing through the axis of the work rolls rotation;
the current value of doubled 2Kx shear resistance and coefficient of friction of the plastic zone along the length of nx plastic forming is constant and equal to their average integral estimates 2KC and ¡j.c ;
forming surface work rolls in zone plastic metal forming approximated by chords, whereupon the current values of the angle ax are constant (see Fig. 1); normal axial stress ax on height cross-section plastic
forming zone not changed, and tangent components
y
b c
Fig. 1. Settlement schemes of the deformation integrated center (a), and also elementary volumes of metal in relation to engineering mathematical modeling of power parameters of relatively thin strips hot rolling process allocated in a lag zone (6) and in an advancing zone (b)
stress deviatorT change in law close to linear and
their average integral evaluation corresponds to the expression:
V =TX/2 = 2KC^C/2.
(1)
Taking into account character of the accepted assumptions the equation of balance of metal elementary volume allocated in a zone of the lag (see fig. 1, a, b), looks like:
ox hx-(ox +dox )(hx+dhx)+ +2pX0T tgadx- 2ixdx = 0,
(2)
ax =a = arctg (0,5AA / Lnn );
K = \ + Ah( x / 4,)-
(3)
Proceeding from the law of external contact friction used and neglecting infinitesimal of the second and more orders, the expression (2) is transformed into a kind:
-a* dhx - d°x hx +Pxot dhx -
-2Kc2jucdhx / (2tqa) = 0.
(4)
Following a full form of a plasticity condition record [1], the quantity of normal axial tension ox can be presented as:
ct = v - /4K2 - 4r2 = v -
X Jr xom V c xy r xom
-J4K2 - 4K2u2 = p - 2Ka,
V c c r c ± xom c k
(5)
where ak = yj 1 — /r - is the auxiliary variable providing
a form record simplification.
After substitution of the condition (5) in the expression (2), we will receive:
2KcakdK - dPxoxhx - 2KcSndK = 0
(6)
Where = Lru / Ah - is the auxiliary quantity
characterizing influence of geometrical parameters and conditions of external contact friction in the deformation center.
Having transformed the differential equation (6) to a kind:
2KC(ak-SM)(dhx /hx) = dpx0T, (7)
As a result of its integration we shall receive:
2 Kc ( ak-<5„)ln hx = pX0T + C0T,
(8)
where C0T — is the constant of integration defined for a lag zone which is preceded from known values of normal contact tension in section to the deformation center entrance p
= 2Kcak
Cor = 2Kc (ak "5Jln K - 2Kcak + CT0
(9)
where px, ox, xx — are the current values of normal contact tension, normal axial and tangent contact tension, and compression tension is accepted as positive values of tension p and a ; a , h —are the current quantities of
IX X ' x' X ^
contact corners and thickness of the rolled strip corresponding to certain dependences of a kind (we take into account accepted assumptions):
where ao — is tension quantity of a rolled strip back tension.
As a result of the substitution (9) in the condition (8) finally for normal contact tension pxom in a lag zone is we have the following:
PX0T = 2KC(^ -ak)ln(ho /hx) + 2Kcak(10)
By analogy with (2) for an advancing zone (see Fig. 1 a, c) the differential equation of a condition of static balance of the allocated elementary volume looks like:
hx- ( °x + d°x )(h* + dhx )+2Pxondxt^ax +
+2t dx = 0,
(11)
and following that we will get the in accordance with above mentioned (3)-(8):
2 Kc (ak +Su)ln K = Pxon + Con,
(12)
where Con - is integrations permanent that is determined for the zone of passing, and based on the well-known quantities of normal pin tensions in a section on the plastic form destructing zone exit p
XOn| hx =h
Con = 2KC(ak + 5)lnh, -2Kcak +
= 2Kcak
(13)
where o^ — is tension quantity of a rolled strip front tension.
Finally in relation to the tension quantities which are current on the length of advancing zone we have pxon :
Pxon = 2KC(^ + ak)ln(hx /h) + 2Kcak-ov (14)
On the principle that the value of normal pin tensions in a neutral section in thick hH (see Fig. 1 a) are equivalent both for the lag zone and for the one of advancing
Pi = pi in accordance with (10) and (14), it is
XOT| hx =hi XOn| hx=*H
possible to write down:
2Kc(8^~ak)ln(ho /hH)+ = (15)
= 2Kc(5^+ak )ln(hH/h )+2Kca ^
From where after some mathematical transformations the thickness of stripe in a neutral section hH, and also extent of advancing zone Lon (see Fig. 1 a) can be defined as:
h = V^Â expx
x{aklnft/h0)+CTl/2Kc-oJIK )/(2<5j}; L = (h -h)L /Ah.
on ^ H l s nn
(16)
(17)
and issued from that the quantity of average integrals on the length of plastic deformation zone of normal pin tensions pc may correspond:
p = 2K n
± C C G
(21)
having integrated the distributions of normal pin tensions Pxot (10) and pxon (14) it is possible to define the quantity of the tense state coefficient nCT as:
Taking into account the well-known values 2KC, , Lm, of radiuses R of working rollers and widths B of the rolled stripe the value of rolling total moment can be determined as:
n = (1/L )
a v nn '
J Pxondx + J Pxotdx
(18)
Ms= 2 x 2Kc^cRB(Lm - 2Lon) = = 2x2KcMcRBLJ1 -2(hH -h,)/Ah].
(22)
from where with an account of (10), (14):
n„ =-
L
bn
S
<A+at)ln
( h+Ahx / L ^
i n
K
■J ak )ln(
h + Ahx / L
i ni
h
j-,
)]dx+J (ak
dx + J
2K
a —
2K
As a result of integration (19) with the use of auxiliary variables (h1+Ahx/Ltm)/h = U1, (h1 + Ahx/Lm)/h0 = U2, we will get finally:
n.= ak) + ak )
Â_ ln K. J Â - h1
Ah ^ \ Ah
K - h h , h
0 h__^ln_0_
2 K
Criterion estimation of authenticity degree of the decisions received (10) and (14) was exeA cuted with the use of the fields of skid-dx + ding lines method, allowing to take into account two-dimensional character of
(19) plastic metal deformation. Construction and subsequent analysis of the fields of descriptions in physical plane and plane of hodograph of speeds (Fig. 2) were produced numerically in accordance with
methodologies of works [2, 3].
As an example of numeral realization results of solutions offered are presented on a Figure 3 the comparisons of calculation distributions of current n values of the metal
ax
(20) tense state coefficient which were received by analytical decisions using (10), (14) and on basis of the skidding lines fields method [2, 3].
K - Â
Ah
b
Lm / K = 10,0 ; «= 1,5° ; Vam = 0,4 ; = 0,42 ; aJ2Kc = 0,0 ; aJ2Kc = 0,175
Fig. 2. The fields of descriptions in the physical plane of XOY(a) and planes of hodograph of speeds XVO'YV (b) as it applies to the process of the hot rolling of relatively thin stripes [2, 3].
n<K<=p*/2Kc
r;
J A 2 \
i! t "V1
J \\
/
r
i.'XI
0,2
0,4
0,6
0,8
L =
^RAh
+ x2 +xT
R =L2 / Ah, (23)
n nnc ' ^ ^
where xL = 8Rpc(1- m^)/ ) is the auxiliary variable used for simplification of a form of record [1]; rns, Ee is Poisson's coefficient and a model of working rolls material elasticity.
The accounting of existence of a rolled strip elastic restoration zone in sections at the working rolls exit (see. Fig. 1, a) can be carried out on the basis of algorithmic sequence of a kind:
Pxi =2kx1 -Oi; Shi= pxl(1-©2)/En;
L yn =^/R:5h;,
(24)
where pxl, 2Kx1, 5h1 is the normal contact tension, doubled value of resistance to rolled metal shift and the
quantity of a strip elastic deformation in section at the plastic forming exit zone; , En is Poisson's coefficient and elasticity model of the rolled strip material, which take into consideration its temperature.
Proceeding from dependences structure (20), (21), (23) and (24) force value at hot rolling of relatively thin strips is defined as (and we take into account working rolls elastic flattening and existence of a elastic restoration zone):
P = (pc Lnnc + pxlL /2)B
(25)
Fig. 3 Settlement data, received according to the considered technique (1) and on the basis of the skidding lines fields method of (2) distributions of the metal tension coefficient current values.
The analysis of these results confirmed the sufficient degree of reliability of the received engineering calculation procedure.
It should be noted that one of current trends of calculation engineering methods of power parameters development is the accounting of working rolls elastic flattening and existence of a strip elastic restoration zone not only at cold, but at hot rolling as well [1, 2, 4]. In relation to working rolls elastic flattening this approach can be realized by iterative determination of plastic forming zone extent LnjIC and the subsequent determination of the elastic deformed working rolls radius Rfl:
Thus, owing to existence of functional interrelations
Pc = F(L imc) and Limc = F(pc) direct determination of
P force can be carried out iteratively using standard approaches [1, 2] at which in the calculations first cycle working rolls are accepted to be absolutely rigid, normal contact tension pc, the value of a plastic forming zone extent Ln^c are defined and after taking it into account calculation pc is repeated and so on. As convergence criteria assessment of the decision iterative procedure relative degree of discrepancy of extents Lnnc in this and in previous calculation cycles was used.
Conclusions
On the basis of analytical decisions the engineering mathematical model of hot rolling process power parameters of relatively thin strips is specified and expanded. This model is distinguished by the correct accounting of external contact friction conditions and also of tangents components of a tension deviator influence indicators, as well as accounting of rolled metal elastic restoration zone existence. The criterion sufficient degree of authenticity of the decision is confirmed by the results of comparison with the analogical quantitative estimations, which we got on the basis of method of the skidding lines fields that allow taking into account two-dimensional character of plastic form destruction throughout metal at the relatively thin stripes hot rolling.
References
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2. Fedorinov V.A., Satonin A.V., Gribkov E.P. Matematicheskoe modeliro-vanie napryazhenij, deformacj i osnovnyx pokazatelej kachestva pri pro-katke otnositel'no shirokix listov ipolos: monografiya. [Mathematical modeling of stresses, strains, and the main indicators of the quality of the rolling sheets and relatively broad bands: monoraph]. Kramatorsk : DGMA, 2010, 243 p.
3. Potapkin V.F. Metod polej linij sko'zheniya v teorii prokatki shirokix polos : monografiya. [The method of slip-line field theory rolling in broad bands: Monograph]. Kramatorsk : DGMA, 2005, 316 p.
4. Kozhevnikova I.A. Garber E.A. Proizvodstvo prokata. Razvitie teorii tonkolistovoj prokatki dlya povysheniya e'ffektivnosti raboty shirokopo-losnyx stanov. [Production of rolled products. The development of the theory of sheet rolling to improve the efficiency of broadband mills]. Moscow : Teplotexnik, 2010, vol. 1, kn. 2, 251 p.
