4
ISSN 09670912, Steel in Translation, 2013, Vol. 43, No. 11, pp. 758–761. © Allerton Press, Inc., 2013. Original Russian Text © Yu.G. Gulyaev, E.I. Shifrin, I.I. Lube, D.Yu. Garmashev, Yu.N. Nikolaenko, 2013, published in “Stal’,” 2013, No. 11, pp. 53–55. 758 Precise determination of the shortest distance from the billet axis to the surface of the rollers in a piercing mill is very difficult. It entails solving fourth order equations, which may be found in various forms in [1–3]. At the same time, if determining the true geometric parameters of the deforming region in a rotaryrolling mill is only one component of a math ematical model—associated with determining the energy parameters of the process, say—it is more expedient to use approximate formulas, as computa tional experience shows [4, 5]. Existing approximate approaches to determining the true geometric param eters of the deforming region in a piercing mill [6–8] are inaccurate and relatively cumbersome, as noted in [3]. It is quite possible to obtain relatively simple and precise relationships between the geometric parame ters of the deforming region in piercing, according to [9]. In the present work, we propose a practical approach. The initial roller position (setup position) is assumed to be the position in which they are turned by the rolling angle β in the horizontal plane, with supply angle α = 0 (Fig. 1), as in [10]. (Here β > 0 for fungi form rollers and β < 0 for lenticular rollers.) To sim plify the calculations, we assume that the displace ment q of the rolling axis relative to the plane through the roller axes is zero, when α = β = 0. In the initial position, the distance between the rollers in the gorge is 2Δr. In the basic coordinate system, axis OX is aligned with the rolling axis. The origin is the coordinate of the point O 1 of roller rotation. When α = β = 0, the gorge of the rollers is displaced by a distance k (measured from the roller axis O 1 X 1 ) relative to their axes of rota tion in terms of the supply and rolling angles. (Here k < 0 with displacement of the gorge in the opposite direction to rolling, for mills with fungiform rollers; k > 0 with displacement of the gorge in the direction of rolling, for mills with lenticular rollers.) MATHEMATICAL MODEL The coordinate X A of point A 1 at the roller surface (Fig. 1) corresponding to a position of the roller gorge turned by the angles α and β is (1) where D p is the roller diameter in the gorge. The coordinates of the ends of the roller are as fol lows: for the piercing cone (2.1) X A 0.5 D p β sin k β cos + ( ) α, cos = X p l l p γ 1 cos γ bl cos X A α, cos = Geometry of the Deforming Region in RotaryRolling Mills Yu. G. Gulyaev a , E. I. Shifrin b , I. I. Lube b , D. Yu. Garmashev c , and Yu. N. Nikolaenko a a Ukrainian National Metallurgical Academy, Dnepropetrovsk, Ukraine b OAO Trubnaya Metallurgicheskaya Kompaniya, Moscow, Russia c OAO Interpipe Nizhnedneprovskii Truboprokatnyi Zavod, Dnepropetrovsk, Ukraine Abstract—A method is proposed for determining the geometric parameters of the deforming region in rotaryrolling mills. In this method, the change in size of the deforming region due to roller rotation and the supply is described by relatively simple formulas. Results of calculations by the proposed and familiar meth ods are compared for piercing and rolling mills. The proposed method may be used to correct the calibration parameters of the working rollers and for industrial adjustment of rotaryrolling mills. Keywords: pipe, rotary rolling, deformingregion geometry DOI: 10.3103/S0967091213110089 X M sinα M II Z r X 1 Y br X 2 B II X 0 M X X p X A X r X p X p l X M l X r l M I X r O 2 α β l r k γ 1 γ b1 A 1 X M Y Vr γ b2 γ 2 l p O 1 O Δr Fig. 1. Determining the geometric parameters of the deforming region in a rotaryrolling mill.

Geometry of the deforming region in rotary-rolling mills

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Page 1: Geometry of the deforming region in rotary-rolling mills

ISSN 0967�0912, Steel in Translation, 2013, Vol. 43, No. 11, pp. 758–761. © Allerton Press, Inc., 2013.Original Russian Text © Yu.G. Gulyaev, E.I. Shifrin, I.I. Lube, D.Yu. Garmashev, Yu.N. Nikolaenko, 2013, published in “Stal’,” 2013, No. 11, pp. 53–55.

758

Precise determination of the shortest distancefrom the billet axis to the surface of the rollers in apiercing mill is very difficult. It entails solving fourth�order equations, which may be found in variousforms in [1–3]. At the same time, if determining thetrue geometric parameters of the deforming region ina rotary�rolling mill is only one component of a math�ematical model—associated with determining theenergy parameters of the process, say—it is moreexpedient to use approximate formulas, as computa�tional experience shows [4, 5]. Existing approximateapproaches to determining the true geometric param�eters of the deforming region in a piercing mill [6–8]are inaccurate and relatively cumbersome, as noted in[3]. It is quite possible to obtain relatively simple andprecise relationships between the geometric parame�ters of the deforming region in piercing, according to[9]. In the present work, we propose a practicalapproach.

The initial roller position (setup position) isassumed to be the position in which they are turned bythe rolling angle β in the horizontal plane, with supplyangle α = 0 (Fig. 1), as in [10]. (Here β > 0 for fungi�form rollers and β < 0 for lenticular rollers.) To sim�plify the calculations, we assume that the displace�ment q of the rolling axis relative to the plane throughthe roller axes is zero, when α = β = 0. In the initialposition, the distance between the rollers in the gorgeis 2Δr.

In the basic coordinate system, axis OX is alignedwith the rolling axis. The origin is the coordinate of thepoint O1 of roller rotation. When α = β = 0, the gorgeof the rollers is displaced by a distance k (measuredfrom the roller axis O1X1) relative to their axes of rota�tion in terms of the supply and rolling angles. (Herek < 0 with displacement of the gorge in the opposite

direction to rolling, for mills with fungiform rollers;k > 0 with displacement of the gorge in the direction ofrolling, for mills with lenticular rollers.)

MATHEMATICAL MODEL

The coordinate XA of point A1 at the roller surface(Fig. 1) corresponding to a position of the roller gorgeturned by the angles α and β is

(1)

where Dp is the roller diameter in the gorge.

The coordinates of the ends of the roller are as fol�lows: for the piercing cone

(2.1)

XA 0.5Dp βsin k βcos+( ) α,cos=

Xpl lp γ1cos

γblcos�������������� XA–⎝ ⎠⎛ ⎞ α,cos–=

Geometry of the Deforming Region in Rotary�Rolling MillsYu. G. Gulyaeva, E. I. Shifrinb, I. I. Lubeb, D. Yu. Garmashevc, and Yu. N. Nikolaenkoa

aUkrainian National Metallurgical Academy, Dnepropetrovsk, UkrainebOAO Trubnaya Metallurgicheskaya Kompaniya, Moscow, Russia

cOAO Interpipe Nizhnedneprovskii Truboprokatnyi Zavod, Dnepropetrovsk, Ukraine

Abstract—A method is proposed for determining the geometric parameters of the deforming region inrotary�rolling mills. In this method, the change in size of the deforming region due to roller rotation and thesupply is described by relatively simple formulas. Results of calculations by the proposed and familiar meth�ods are compared for piercing and rolling mills. The proposed method may be used to correct the calibrationparameters of the working rollers and for industrial adjustment of rotary�rolling mills.

Keywords: pipe, rotary rolling, deforming�region geometry

DOI: 10.3103/S0967091213110089

XMsinα

MII

Zr

X1

Ybr

X2

BII

X 0

M

X

Xp XA Xr

Xp Xpl

XMl Xr

l

MI

XrO2

α

β

l r

k

γ1

γb1

A1 XM YV

r

γb2 γ2

l p

O1

OΔr

Fig. 1. Determining the geometric parameters of thedeforming region in a rotary�rolling mill.

Page 2: Geometry of the deforming region in rotary-rolling mills

STEEL IN TRANSLATION Vol. 43 No. 11 2013

GEOMETRY OF THE DEFORMING REGION IN ROTARY�ROLLING MILLS 759

and for the rolling cone

(2.2)

where lp and lr are, respectively, the lengths of thepiercing and rolling cones along the roller axes; γ1 =⎯(β � γb1) and γ2 = –(β � γb2) are the inclinations ofthe roller surface to the rolling axes in the piercing androlling cones of the deforming region, respectively(the upper sign corresponds to fungiform rollers andthe lower sign to lenticular rollers); γb1 and γb2 are thetaper angles for the generatrices of the roller surface inthe piercing and rolling cones of the deforming region,respectively. (For all types of mills, γb1 and γb2 areassumed positive.)

The distance from the rolling axes to the roller sur�face in the initial position is as follows: for the piercingcone

(3.1)

and for the rolling cone

(3.2)

where r is half the initial distance between the rollers inthe gorge; X is the coordinate along the rolling axis OX.

The distance from the roller axis O1X1 to its surfacein the initial position is as follows: for the piercingcone

(4.1)

and for the rolling cone

(4.2)

where Y0A = Dp/2cosβ.The distance between the rolling axis OX and the

roller axis O2X2 in roller rotation by the angles α and βis as follows: for the piercing cone

(5.1)

and for the rolling cone

(5.2)

The shortest distance from the rolling axis OX to theroller surface with roller rotation by the angles α and βis as follows: for the piercing cone

(6.1)

and for the rolling cone

(6.2)

where Rp and Rr are the shortest distances from theroller axis to its surface in the piercing and rollingcones, respectively.

Xrl lr γ2cos

γb2cos������������� XA+⎝ ⎠⎛ ⎞ α,cos=

YVp Δr XA X–( ) γ1,tan+=

YVr Δr X XA–( ) γ2tan ,+=

Ybp Y0A XA X–( ) γ1tan βtan+( ),–=

Ybr Y0A X XA–( ) γ2tan βtan–( ),–=

Zp X2 α2sin Ybp YVp+( )2+ ,=

Zr X2 α2sin YbrYVr( )2+ .=

rp Zp Rp,–=

rr Zr Rr,–=

We assume that, with roller rotation by the angle α,the shortest distance from the roller axis to its surface isrelatively unchanged. Accordingly, Rp ≈ Ybp and Rr ≈ Ybr,while

(7.1)

(7.2)

With rotation by the angle α, the change in theshortest distance from the rolling axis to the roller sur�face is as follows: for the piercing cone

(8.1)

and for the rolling cone

(8.2)

MODELING RESULTS

To assess the error due to our assumptions, wecompare the results for the variation in the shortestdistance from the rolling axis to the roller surface Δrpand Δrr in a two�roller piercing mill with fungiformrollers according to the proposed model and the modelin [3]; the latter results are taken from [10]. Theparameters adopted are as follows: Dp = 860 mm; Δr =60 mm; lp + lr = 500 mm; β = 17°; γ1 = γ2 = 2°. In thecalculations by the proposed model, for purposes ofcomparison, k is chosen so that the coordinate of theroller gorge along the OX axis is zero when the rollersturn by the angle β (in the initial position. In otherwords, we assume that k = –0.5Dptanβ. As we see inFig. 2, the discrepancy between the results of the pro�posed (engineering) method and the familiar (precise)method is small (no more than 10%) even with consid�erable values of β (up to 17°) and α (up to 24°).

rp Zp Ybp,–≈

rr Zr Ybr.–≈

Δrp rp YVp,–=

Δrr rr YVr.–=

14

12

10

6

4

2

025015050–50–150

–200–250

8

–100 0 100 200

24.7°

10.0° 15.0°

20.0°

Var

iati

on in

th

e sh

orte

st d

ista

nce

Distance from the plane of rotation X, mm

Δr p

an

d Δ

r r,

mm

Fig. 2. Variation in the shortest distance from the rollingaxis to the roller surface according to the proposed method(continuous curves) and the data of [10] (dashed curves).Values of the supply angle α are given on the curves.

Page 3: Geometry of the deforming region in rotary-rolling mills

760

STEEL IN TRANSLATION Vol. 43 No. 11 2013

GULYAEV et al.

In Fig. 3, we show the values of Δrr in the deformingregion of a three�roller rolling mill calculated by theproposed method and by the model in [1]. The param�eters adopted are as follows: Dp = 600 mm; Δr = 70 mm;lr = 300 mm; α = 12°; γ2 = 2°; k = 0. The calculationin [1] proceeds from the zero coordinate of the gorgecross section when α = β = 0. Therefore, to obtaincomparable results, we modify the coordinate system:in the corrected system, Xn = X – XA/cos α. As we seein Fig. 3, the results of the two methods are in rela�tively good agreement; the discrepancy is no morethan 15%).

In Fig. 4, we plot ΔRp = rp – Δr and ΔRr = rr– Δr forthe deforming region of a piercing mill with barrel�shaped rollers according to the proposed method andthe model in [11]. The parameters adopted are as fol�lows: Dp = 1000 mm; Δr = 75 mm; lp + lr = 600 mm;β = 0°; γ1 = 3.5°; γ2 = 4.5°; k = 0. The discrepancybetween the results of the two methods is no morethan 5%.

It follows from Figs. 2–4 that the proposed modelmay be used to correct the calibration parameters ofthe working rollers and to set up rotary�rolling mills inindustrial conditions, as confirmed in practice.

CONCLUSIONS

(1) A mathematical model has been proposed fordetermining the geometric parameters of the deform�ing region in a piercing mill. In this method, the cal�culation is greatly simplified by means of assumptionsthat do not significantly impair the computationalaccuracy.

(2) The results of calculations by the proposed andfamiliar methods have been compared.

(3) The proposed method may be used to correctthe calibration parameters of the working rollers andalso permit industrial adjustment of rotary�rollingmills.

REFERENCES

1. Mogilevkin, F.D., Osadchii, V.Ya., Matveev, Yu.M.,et al., Geometry of the deforming region in a rotary�rolling mill, Proizvodstvo svarnykh i besshovnykh trub(Production of Welded and Seamless Pipe), Moscow:Metallurgiya, 1971, issue 12, pp. 45–53.

2. Teterin, P.K., Teoriya poperechnoi i vintovoi prokatki(Theory of Transverse and Helical Rolling), Moscow:Metallurgiya, 1983, 2nd ed.

3. Potapov, I.N. and Ol’khovoi, V.G., Computer calcula�tion of the geometric parameters in rotary�rolling mills,Plasticheskaya deformatsiya metallov i splavov (PlasticDeformation of Metals and Alloys), Moscow: Metal�lurgiya, 1972, issue 71, pp. 138–144.

4. Gulyaev, Yu.G., Shifrin, E.N., Garmashev, D.Yu.,et al., Developing a mathematical model for determiningthe energy parameters of piercing, Teor. Prakt. Metall.,2013, no. 1/2, pp. 86–90.

5. Gulyaev, Yu.G., Shifrin, E.I., Lube, I.I., andNikolaenko, Yu.N., Mathematical model for deter�mining the parameters of piercing in a rotary�rollingmill, Trudy IX kongressa prokatchikov (Proceedings ofthe Ninth Congress of Rolling Specialists), Chere�povets: MOO Ob’’edinenie Prokatchikov, 2013, vol. 1,pp. 338–344.

6. Fomichev, I.A., Kosaya prokatka (Skew Rolling), Mos�cow: Metallurgizdat, 1963.

7. Mironov, Yu.M., Geometric parameters of skew roll�ing, Proizvodstvo trub (Pipe Production), Moscow:Metallurgizdat, 1962, issue 6, pp. 37–46.

2.0

1.61.41.2

0.80.6

0.2

300250200150100500

0.4

1.0

1.8

β = 6°

β = 8°

β = 10°β = 12°

β = 14°

Var

iati

on in

th

e sh

orte

st d

ista

nce

Distance from the roller gorge Xn, mm

Δr,

mm

Fig. 3. Variation in the shortest distance from the rollingaxis to the roller surface over the length of the deformingregion’s output section, according to the proposed method(continuous curves) and the data of [1] (dashed curves).

30

20

15

10

5

3002001000–100–200–300

25

α = 6°

α = 12°

Gap

s Δ

r p a

nd

Δr r

, m

m

Distance from the roller gorge X, mm

Fig. 4. Variation in ΔRp and ΔRr over the length of thedeforming region, according to the proposed method(continuous curves) and the data of [11] (dashed curves).

Page 4: Geometry of the deforming region in rotary-rolling mills

STEEL IN TRANSLATION Vol. 43 No. 11 2013

GEOMETRY OF THE DEFORMING REGION IN ROTARY�ROLLING MILLS 761

8. Matveev, Yu.M., Polukhin, P.I., Golubchik, R.M., andTsodokova, N.S., Geometry of the deforming region ina rotary�rolling mill, Proizvodstvo svarnykh i bes�shovnykh trub (Production of Welded and SeamlessPipe), Moscow: Metallurgiya, 1968, issue 9, pp. 79–87.

9. Potapov, I.N. and Ol’khovoi, V.G., Engineeringmethod of determining the geometric parameters of thedeforming region in a skew�rolling mill, Plasticheskayadeformatsiya metallov i splavov (Plastic Deformation ofMetals and Alloys), Moscow: Metallurgiya, 1972,issue 71, pp. 144–147.

10. Potapov, I.N. and Ol’khovoi, V.G., Analysis of thedeforming�region distortion in a skew�rolling mill with

roller rotation by the supply angle, Plasticheskayadeformatsiya metallov i splavov (Plastic Deformation ofMetals and Alloys), Moscow: Metallurgiya, 1972,issue 66, pp. 125–130.

11. Chekmarev, A.P., Vatkin, Ya.L., and Umerenkov, V.N.,Determining the geometric parameters of the deform�ing region in a skew�rolling mill, Obrabotka metallovdavleniem (Pressure Treatment of Metals), Moscow:Metallurgiya, 1967, issue 52, pp. 124–141.

Translated by Bernard Gilbert