UDK 62-462:621.774.35:519.673
Original scientific article/Izvirni znanstveni članek
ISSN 1580-2949
MTAEC9, 43(2)63(2009)
A MATHEMATICAL MODEL FOR THE STATIONARY PROCESS OF ROLLING OF TUBES ON A CONTINUOUS
MILL
MATEMATIČNI MODEL PROCESA KONTINUIRNEGA VALJANJA
CEVI
Yu. G. Gulyayev1, Ye. I. Shyfrin2, Ilija Mamuzic3
1National Metallurgical Academie of Ukraine, Dnipropetrovsk, Ukraine 2Tube MetallurgicCompany, Russia 3University of Zagreb, Faculty of Metallurgy Sisak, Croatia mamuzic@simet.hr
Prejem rokopisa — received: 2008-09-23; sprejem za objavo - accepted for publication: 2008-10-23
A mathematical model has been developed for the calculation of process parameters in continuous lengthwise plugless tube rolling. Examples of concrete calculations of rolling parameters, their comparison with experimental data and the results obtained with the application of other calculation procedures are given.
Key words: tubes, plugless rolling, mathematical model
Razvit je bil matematični model za izračun parametrov procesa neprekinjenega valjanja cevi brez notranjega trna. Dani so konkretni primeri izračunov parametrov valjanja, rezultati pa so primerjani z eksperimentalnimi podatki in z izračuni po drugih postopkih.
Ključne besede: cevi, valjanje brez trna, matematični model
1	INTRODUCTION
The prospects of enhancement of the production efficiency at numerous tube rolling units are closely linked with the possibility of a reliable prediction of the forming parameters at the final stage of plastic deformation in the plugless tube reducing or sizing processes. In this connection, the problem of development of a universal mathematical model applicable in studying the process of lengthwise plugless rolling in the tube rolling mills equipped with the roll drives of different types is of a high interest.
2	STATE OF THE ISSUE AND THE AIM OF INVESTIGATION
The analysis of the relevant references shows that the problem of determination of kinematical, deformational and power-and-force parameters of the continuous plugless lengthwise tube rolling process was solved up to now by consecutive analysis of forming in each individual stand. Solutions based on integration of the defor-mational parameters in all N stands of the continuous mill into a common system of equations are proposed, also 1,2. In the development of mathematical models of the continuous rolling process, e.g. in 1,2 two assumptions were made.
Firstly, the mean angle of the neutral section On (Figure 1) is defined for the condition of coincidence of the roll and the mother tube speeds within the section of
the deformation zone exit in the i-th stand, though it would be logical to choose some section between the entry and exit of the deformation zone.
Secondly, for the determination of the effective roll diameter Dki, the approximate formula is used: DM = Du - Di cosöni (where Dm-, D - are the ideal roll diameter and the mean tube diameter after rolling in the i-th stand respectively) that introduces an error because in reality Dki = Dui - 2rei (6ni) cosöni (where Du is the ideal roll diameter; rei (Oni) is the pass radius at 0 = Oni, see Figure 2).
Figure 1: Scheme for the determination of the value of the effective diameter Dk
Slika 1 : Shema za določitev efektivnega premera Dk
Materiali in tehnologije / Materials and technology 42 (2009) 2, 63-67
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YU. G. GULYAYEV ET AL.: A MATHEMATICAL MODEL FOR THE STATIONARY PROCESS ...
In accord with the model consisting of 2 N equations proposed1, the effective roll diameter DM cannot be greater than the ideal roll diameter Dui and smaller than the roll diameter at the swell Dbi. This distorts the real picture of the rolling process kinematics, namely, when the rolls slip on the mother tube surface two conditions are met: Dki < Dbi or Dk > Dui. The model proposed 2 is free from this shortcoming but it is a system of 3N equations that when being solved at N > 16 is connected with considerable difficulties because of the great number of unknowns to be determined.
This work is aimed at the verification and simplification of the mathematical models proposed12 and to the assessment of the verification results on the basis of comparison of calculated and experimental data and it is, for this reason, of scientific and practical interest.
3	PROBLEM STATEMENT
The following values have to be calculated:
-	angular roll rotation velocity mi in each i-th mill stand (for the mill with individual roll drives);
-	angular velocities of rotation of the main (Nr) and auxiliary (Nb) motors (for the mill with differential-group roll drives);
-	ideal roll diameters Dui (for the mill with group roll drives).
These values ensure that tubes of required size (Dt-Su mm) are rolled from the mother tube of given size ((Do x So) mm) at a specified rolling speed W(mjs) in the first stand of the multiple-stand mill.
Initial data for the calculation are as follows:
-	the total diameter and wall reduction (or just diameter reduction), i.e. initial mother tube dimensions D0 x S0 (or just D0) and final tube dimensions Dt x St;
-	the distribution of partial mother tube diameter reductions mi (%) among the mill stands of total number of N;
-	the value of external friction f;
-	the mother tube rolling speed V)j(mjs) in the first mill stand (the problem can also be stated for Vo as the value to be determined);
-	the gear ratios nri, nBi from the motors to the rolls in the lines of the main and auxiliary drives (for the mills with differential-group roll drives);
-	the absence of backward pull in the first mill stand (Z31 = 0) and of front pull in the last mill stand (Znv = 0);
-	the number of rolls Nb forming passes in the mill stands.
4	PHYSICAL MODEL OF THE PROCESS
No mother tube forming occurs in interstand spaces and the wall thickness SJ at the exit from the stand of ordinal number J = i - 1 is equal to the wall thickness S01
at the entry to the stand of ordinal number i. The deformation resistance Kfj of the mother tube material at the exit from the stand of ordinal number J is equal to the deformation resistance Kf0i of the mother tube material at the entry to the stand of ordinal number i. It follows that the coefficient of front plastic pull ZnJ for the stand of ordinal number J is equal to coefficient of backward plastic push Z3i for the stand of ordinal number i. The area Fki of the contact surface of the mother tube with one roll in the stand of ordinal number i is equal to the area of a rectangle with sides
L =
ßi • D, • £ i • (Dui - D,)
with
2 sin ß i
= ß i • DJ
(1)
(2)
where Dj and Di are the mean mother tube diameters at the entry to and at the exit from the deformation zone in the stand of ordinal number i;
£ i j% = ß i i 100 ' l N bi
The area F? of the zone of forward creep at the surface of contact between one roll and the mother tube in the deformation zone of the stand of ordinal number i is defined as the surface of a rectangle with sides
and
L + = L
L + = e D
(3)
(4)
where eni is the neutral section angle characterizing the position of the neutral line differentiating the zone of forward creep and the zone of backward creep on the surface of contact between the mother tube and the roll in the deformation zone (Figure 1).
In a real process, the magnitude of angle eni is a function of the angle a characterizing the position of a concrete diametrical section of the deformation zone relative to the diametrical section of the mother tube exit from the reduction zone. In accord with the assumption4, the magnitude of angle eni is assumed to be equal to some quantity averaged over the contact surface length. It will be regarded that eni is the value of the neutral angle in the "neutral" diametrical section of the deformation zone where the extension is equal to the mean extension in the i-th stand. The axial velocity VMn of metal and axial component of the roll surface velocity VBn in the "neutral" diametrical section are given with
VMni = V0 ft j V =nn bA- Dee
' Tir.;
I
(5)
(6)
where ft jp = -
2 S 0 (D 0 - S 0)
is the total elongation
S, ( D, - S, ) + S, ( Di - Si )
from the mill entry to the "neutral" diametrical section of the i-th stand;
64
Materiali in tehnologije j Materials and technology 43 (2009) 2, 63-67
YU. G. GULYAYEV ET AL.: A MATHEMATICAL MODEL FOR THE STATIONARY PROCESS
r = O k A = R .
Ol	k	or
1 -
f \2 e„
v oI J
(1 - cos2 O)-
e . cos O
r
is
the varying across the pass perimeter value of the pass diameter;
Roi = OA = radius;
e
7TT h(12i +1 - 21 kl sinp.
= O k A =
2(1-1 k sin p)
( 12i -1)
2(1-1 kl sin p)
is the pass generatrix eccen-
tricity;
b.
1 = — is the pass ovality;
p =
h.
(N b - 2) .
2N
is the pass shape index;
VMm — VR
0
(7)
O=
arc cos Qi if cos ß i < Qi < 1 0 if 1 < Q.
ß i if cos ß i > Q.
(8)
The quantity Q. in (8) is defined as the root of equation
nn B. AC
2 S o Vo( Do - S o) __
Sj (Dj - Sj ) + St (D t - St ) |
Dui - 2QiR0
1 -
2
e
Ro
(1 - Q.2 )-
eoQ
R„
= 0 (9)
Figure 2: Scheme for the determination of the value of the variable
across the pass perimeter of the roll diameter Do
Slika 2: Shema za določitev spremembe premera valjev Do na obodu
vtika
Atcp is the mean value of the guiding cosine of the contact friction stresses
Dor = Du. - 2roi cos O is the varying of the roll pass diameter across the pass perimeter (Figure 2);
Note that as distinct from the conditions used12, the condition (8) reflects the relation of the neutral angle On value with the roll design parameters (Ro., ew, Ik, b , h).
Taking into consideration relationships (5) and (6), the effective roll diameter Dk can be defined by the following equation:
2 S o Vo( Do - S o)|
D k =
n[ Sj (Dj - Sj ) + S, (Dt - St )]n r Abc
(1o)
is the pass generatrix
For Db. < Du < Du, on the contact surface of each roll appears a forward creep zone with the area equal of F+ = On DjL in accord with (3), (4) and the backward creep zone with the area equal to Fr = (ß. - On) DjLu For Dw > Dk , the backward creep zone extends over the entire contact surface area and the "forward roll slippage" takes place on the metal surface. If Du < DM, the forward creep zone extends over the entire contact surface area and "backward roll slippage" takes place on the metal surface.
The magnitude of neutral angle On must meet the condition of force equilibrium of the metal volume in the geometrical deformation zone of the .-th stand 1 that can
be expressed as:
O n = ß,
X. if o < X t < 1
0	if o > X.
1	if 1 < X;
(11)
bu h. are the pass width and the pass height correspondingly;
| = 61o4 is coefficient of quantity dimension reduction
(smm—r—m) min
The angle On is defined as root of the transcendental equation
where
x =-
zn S , (d.
• s. ) - znj s j (dj - sj )
f.A? (2L + j)n, S D [1 -( Zcp )]
Taking in account that in a physical sense o < On < ß , the condition for the determination of the neutral angle assumes the following form (in symbols of MathCAD programming language)
is the coefficient of forward creep in the .-th mill stand calculated for the equilibrium of forces in the volume of the metal in the geometrical deformation zone of the .-th stand;
Acp is the mean, over the contact surface, value of the
guiding cosine for normal contact stresses;
f is coefficient of external friction;
nt. = 1 + o,36 f is coefficient accounting for the effect of
the contact friction stresses upon normal contact
stresses3;
Zn is coefficient of forward plastic pull;
Z 2Z n.
(Zcp ) = + °1 is the mean value of the plastic pull coefficient in the . -th stand.
cj

+
F
Materiali in tehnologije / Materials and technology 43 (2009) 2, 63-67
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YU. G. GULYAYEV ET AL.: A MATHEMATICAL MODEL FOR THE STATIONARY PROCESS ...
Table 1: Parameters of rolling a (57 X 11.6) mm tube from a (117 X 14.8) mm mother tube Tabela 1: Parametri valjanja cevi (57 X 11.6) mm iz cevi (117 X 14.8) mm
i	D./mm	m./%	^ki	Si/mm			Zni		ej°		Pi/kN		Mi/(kN-m)	
				E	A	B	A	B	A	B	A	B	A	B
1	115.25	1.50	1.037	14.77	14.86	14.85	0.327	0.361	53.7	56.8	93	92	-14.5	-16.6
2	112.59	2.30	1.024	14.82	14.82	14.81	0.568	0.606	51.3	53.2	74	69	-10.0	-10.6
3	109.67	2.60	1.032	15.55	14.68	14.64	0.721	0.756	46.9	48.2	49	44	-5.3	-5.2
4	106.49	2.90	1.028	14.35	14.47	14.40	0.729	0.766	19.9	19.1	37	32	3.0	2.7
5	102.97	3.30	1.038	14.07	14.22	14.13	0.730	0.761	17.7	14.0	38	32	3.8	3.9
6	99.47	3.40	1.034	13.75	13.97	13.86	0.729	0.755	17.00	14.4	37	33	3.8	3.9
7	96.09	3.40	1.036	13.50	13.73	13.60	0.724	0.742	16.6	13.5	37	33	3.8	4.2
8	92.82	3.40	1.035	13.28	13.49	13.35	0.715	0.743	16.3	17.2	37	33	3.9	3.4
9	89.57	3.50	1.037	13.00	13.26	13.10	0.705	0.721	16.7	14.1	37	34	4.0	4.2
10	86.44	3.50	1.036	12.70	13.03	12.87	0.693	0.711	17.2	15.4	38	35	3.9	4.0
11	83.41	3.50	1.037	12.49	12.82	12.65	0.679	0.693	17.5	16.1	38	36	3.9	4.0
12	80.49	3.50	1.036	12.41	12.62	12.44	0.665	0.676	18.2	17.3	39	37	3.8	3.9
13	77.68	3.50	1.036	12.31	12.42	12.25	0.650	0.659	18.8	18.2	40	38	3.7	3.7
14	74.96	3.50	1.036	12.18	12.24	12.06	0.635	0.642	19.4	18.9	41	39	3.7	3.7
15	72.33	3.50	1.036	11.95	12.07	11.89	0.618	0.623	19.9	19.4	42	41	3.6	3.6
16	69.80	3.50	1.036	11.78	11.91	11.74	0.595	0.597	19.7	19.2	43	42	3.8	3.8
17	67.36	3.50	1.036	11.69	11.77	11.59	0.568	0.564	20.1	19.6	45	44	3.8	3.9
18	65.00	3.50	1.036	11.58	11.64	11.47	0.532	0.518	20.1	19.2	47	47	4.0	4.4
19	62.73	3.50	1.036	11.53	11.53	11.39	0.459	0.414	18.2	16.1	52	54	5.3	6.3
20	60.53	3.50	1.036	11.40	11.49	11.41	0.267	0.078	12.6	6.5	63	72	9.2	13.8
21	59.02	2.50	1.020	11.50	11.53	11.44	0.036	-0.140	14.3	17.4	71	87	9.4	9.3
22	57.96	1.80	1.018	11.58	11.58	11.58	0.001	-0.105	25.7	29.0	74	85	3.0	1.3
23	57.60	0.61	1.000	11.60	16.00	11.60	0	0	28.4	34.3	51	54	1.0	-1.6
NOTES:
A = calculation by the procedure proposed in 1 B = calculation by the procedure proposed in this work E = experimental data
5 MATHEMATICAL MODEL
Equate right parts of equations (8) and (11) and use the equation of relation between the change of the mean wall thickness and force conditions of the mother tube deformation in each i-th mill stand 4 5 to obtain the mathematical model of the continuous mother tube rolling process in N stands of the mill as a system of 2N equations:
arc cos Qi if cos ß i < Qi < 1
Q = 0 if 1 < Q.	=
i i
ß i if cos ß i > Qt
Xi if 0 < X. < 1 = ß. • 0 if 0 > X. 1 if 1 < X;
(12)
! 2(Z ). (Tt -1) + (1 - 2Tt )
st - s. n+tpi • p . .——-- *
(Zcp)i(1 - T)-(2- T)
1
+ -
2
Vi ■-
2( Z cp) i (Ti -1) + (1 - 2 T. )
( Z cp) i (1 - T ) - (2 - T )
(13)
where
Vi = ln
Di - si:
T=
Sj Si — + —
DD
\K
1
(Z cp) i = 2(Z nj
+Z „i ); 2; K:
1.20 for Nb = 3; i = 1, 2,
K =1.57 for Nb N-1, N
Distinct from the known solution2, the mathematical model includes 2N and not 3N equations that simplifies the search of solution and makes it possible to analyze the rolling process in stretch-reducing mills with N < 25 stands.
Depending on the type of the mill drive, the problem of determination of the rolling parameters with the use of the system of equations (12)-(13) can be formulated in different ways. For the mills with individual drives, it is necessary to determine 2N values of (where i =1,2, ... , N) and S. (where i = 0, 1, 2, ... , N-1) for the specified values of St, V0 and Z„. (where i = 1, 2, ... , N-1). For the mills with differential-group drives, it is necessary to find 2(N-1) values of the quantities S. and Zni (where i =1,2,., N-1) and the values of Nr and Nb for the specified values of S0, St, V0. For the mills with individual drives, it is necessary to determine N-1 value
64
Materiali in tehnologije j Materials and technology 43 (2009) 2, 63-67
YU. G. GULYAYEV ET AL.: A MATHEMATICAL MODEL FOR THE STATIONARY PROCESS
of the quantities D„ (where i = 2, 3, ... , N), N values of the quantities Si (i = 0, 1, 2, ... , N-1) and the angular roll velocity nB that is constant for all stands at the specified values of Du1, V0 and Zn (where i = 1, 2, ... , N-1).
Solve the system of equations (12)-(13) using (8) or (11) to find values of neutral angles Oni and determine the values of the effective diameters in correspondence with expression (10).
6 RESULTS OF MODEL CALCULATIOINS
The model has been successfully used in the calculation of tube rolling parameters for mills with the roll drives of individual, group and differential-group types.
As an example, let us consider the results of the calculation of the nature of change in the mean wall thickness Si, rolling pressure Pi, rolling moments acting in the stand Mi and the values of Zni, Oni in rolling a Dt ■ St = (57 x 11.6) mm tube from a D0 ■ S0 = (117 x 14.8) mm mother tube in 23 stands of the tube rolling unit "30-102" reducing mill with differential-group roll drives (Nb = 3, V0 = 0.7 m/s with the mother tube material: Grade 45 steel). In this case, the mathematical model (12)-(13) is a system of 46 equations with 46 unknowns: 22 values of Si and Zni each and the values of Nr as well as Nb. The rolling parameters, results of calculation by the procedure given1, by the proposed model and experimental data are given in Table 1. For the calculation of Pi and Mi values, the procedure3 was used.
The processing of data in Table 1 shows that when the procedure1 was used, the standard deviation
A =
I(Sic - sa)2
N-1
of calculated values of the wall thickness SiC from the actual values of this parameter S,A was A = 0.150 mm. When the present mathematical model was used, the value of A was of 0.085 mm and for 1.76 times smaller. Hence, the rolling parameter calculation accuracy is improved when the proposed procedure is used.
7	CONCLUSION
The mathematical model of the lengthwise continuous plugless tube rolling process has been developed and successfully tested. It improves the accuracy of calculation of the process parameters in comparison with the earlier developed procedure.
ACKNOWLEDGEMENT
The authors are indebted to prof. F. Vodopivec for the revision of the manuscript.
8	REFERENCES
1	Gulyayev Yu. G., Shyfrin Ye. I., Kvitka N. Yu. The mathematical model of the continuous plugless lengthwise tube rolling process in the tube rolling mills with individual roll drives. Teoriya i Practika Metallurgii (2006) 3, 66-74
2	Gulyayev Yu. G., Shyfrin Ye. I., Kvitka N. Yu. The mathematical model of the continuous plugless lengthwise tube rolling. Teoriya i Praktika Metallugrii (2006) 6, 63-70
3	The procedure of determination of a maximum roll pressure in the continuous plugless tube rolling process / G. I. Gulyayev, Yu. G. Gulyayev, Ye. I. Shyfrin, N. Yu. Kvitka, C. V. Darragh. - Material International Conference on New Developments in Long and Forged Products. - Winter Park (Colorado, USA), (2006), 127-132
4	The technology of continuous plugless tube rolling: Edited by G. I. Gulyayev / G. I. Gulyayev, P. N. Ivshin, I. N. Yerokhin et al. -Moscow, Metallurgiya Publishers, 1975 - 264
5	Shevchenko A. A., Yurgelenas V. A. Continuous plugless tube rolling under conditions of limiting values of the pull force. Trudy UkrNTO ChM. Dnepropetrovsk: UkrNTO ChM, 13 (1958), 77-86'
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