X. MA et al.: THREE-DIMENSIONAL MODEL OF EQUIVALENT ROLL GAP OF CVC MILL UNDER CONDITIONS ...
193–202
THREE-DIMENSIONAL MODEL OF EQUIVALENT ROLL GAP OF CVC
MILL UNDER CONDITIONS OF ROLL CROSSING AND NO LOAD
TRIDIMENZIONALNI MODEL EKVIVALENTNE VALJARSKE
RE@E CVC VALJARNE V POGOJIH BREZ OBREMENITVE IN
KRI@ANJA VALJEV
Xiaoxin Ma
1*
, Peisen Yuan
2
, Fuyi Li
2
, Jiang Ji
3
1
Faculty of Mechanical Engineering, Hebei University of Architecture, Zhangjiakou, Hebei, China
2
College of Mechanical and Vehicle Engineering, Taiyuan University of Technology, Taiyuan, Shanxi, China
3
China National Heavy Machinery Research Institute Co., Ltd, Xi’an, Shaanxi, 710032, China
Prejem rokopisa – received: 2023-11-14; sprejem za objavo – accepted for publication: 2024-02-07
doi:10.17222/mit.2023.1047
Traditional roll technologies focus solely on the contour of the roll gap at the outlet under ideal conditions. However, upon roll
crossing, the three-dimensional distribution of the deformation zone becomes asymmetrical, leading to an adverse impact on the
rolling pressure in the deformation zone. To study the screw-down load deviation of plate mills, it is necessary to accurately cal-
culate the equivalent three-dimensional roll gap in the deformation zone under roll-crossing conditions. In this paper a model for
calculating the equivalent three-dimensional roll gap under the conditions of roll crossing and no load was established based on
the method of the coordinate nonlinear transformation of the spatial rectangular coordinate system. Based on this model, the in-
fluences of roll-crossing angles and translation parameters on the symmetry of the equivalent three-dimensional roll gap were
analyzed. The distribution of the three-dimensional equivalent roll gap was calculated with different roll lengths, roll radii, and
the roll gap’s nominal thickness. The intuitive contour map reveals that under the same condition of crossing angle and transla-
tion parameters, the larger the roll length and roll radii and the smaller the roll gap’s nominal thickness are, the greater the influ-
ence of the roll crossing on the asymmetrical distribution of roll gap is. It put forward a quantitative calculation method for eval-
uating the symmetrical distribution of the roll gap, and the calculation results can provide the equivalent roll profile for the
roll-deformation model.
Keywords: roll crossing, nonlinear transformation, equivalent three-dimensional roll gap, asymmetrical distribution
Tradicionalne valjarni{ke tehnologije se osredoto~ajo predvsem na konturo re`e v idealnih pogojih na izhodu med valjema.
Vendar pa na kri`i{~u valjev nastane nesimetri~na tridimenzionalna deformacija, kar posledi~no vodi do ne`eljenega oziroma
neugodnega vpliva v coni deformacije. Zato, da bi lahko {tudirali nihanje obremenitve zategnitve plo{~ valjev so avtorji
natan~no izra~unali ekvivalentno tridimenzionalno re`o med valji vconi deformacije pri pogojih kri`anja valjev. V tem ~lanku
avtorji opisujejo model za izra~un ekvivalentne tridimenzionalne valjarske re`ev pogojih kri`anja valjev brez obremenitve.
Metoda temelji na koordinatni nelinearni transformaciji prostorskega pravokotnega koordinatnega sistema. Na osnovi tega
modela so avtorji analizirali vpliv kotov kri`anja valjev in translacijskih parametrov na simetrijo ekvivalentne tridimenzionalne
valjarske re`e. Izra~unali so porazdelitev tridimenzionalne ekvivalentne valjarske re`e za razli~ne dol`ine valjev, polmera valjev
in nominalno debelino re`e. Ugotovljena mapa konture je pokazala da je, pri enakih pogojih kota kri`anja in enakih
translacijskih parametrih, vpliv kri`anja ve~ji ~im ve~jaje dol`ina, ~im ve~ji je premer valjev in ~im manj{a je nominalna
debelina re`e med valjema. To je v nadaljevanju omogo~ilo izdelati kvantitativno ra~unsko metodo za ovrednotenje simetri~ne
porazdelitve valjarske re`e. Rezultati izra~una pa so lahko podlaga za ekvivalentni profil valjev v modelu za deformacijo valjev.
Klju~ne besede: kri`anje valjev, nelinearna transformacija, ekvivalentna tridimenzionalna re`a med valjema, nesimetri~na
porazdelitev
1 INTRODUCTION
Roll technology is the most active and effective
shape-control technology. Various advanced techniques
have been designed for different purposes, such as the
paired crossed (PC) rolling mill,
1
the continuously vari-
able crown (CVC) roll profile technology,
2
the asymme-
try self-compensating work roll (ASR) technology,
3
the
large concave roll (LCR) technology,
4
the edge variable
crown (EVC) roll technology,
5
the SmartCrown roll tech-
nology,
6
the SVT roll technology,
7
and linearly variable
crown (LVC) roll technology.
8
The core of new roll tech-
nologies is to change the roll gap by crossing or shifting
the upper and lower work rolls (WRs). The roll gap is an
important factor that affects both the strip profile and the
distribution of the rolling pressure.
Ideally, the strip profile is mainly affected by the
two-dimensional (2D) profile of the outlet at the defor-
mation zone. To realize precise control of the strip pro-
file, many studies are about the calculation of the equiva-
lent 2D roll gap of the outlet of the deformation zone.
The equivalent roll crown of the cubic CVC roll curve
was discussed in depth.
9
It showed that the equivalent
roll crown, considering the width of the strip, is a linear
function of both the amount of roll shafting and the roll
length, and is also a quadratic function of the strip width.
Li Hongbo et al.
10
designed a quintic CVC roll profile,
which can equalize the quadratic crown control capabil-
Materiali in tehnologije / Materials and technology 58 (2024) 2, 193–202 193
UDK 629.5.017.22:7.021.5 ISSN 1580-2949
Original scientific article/Izvirni znanstveni ~lanek MTAEC9, 58(2)193(2024)
*Corresponding author's e-mail:
mxxwzgz@163.com (Xiaoxin Ma)

ity for both wide and thin strips and show excellent con-
trol capability for both the quadratic and the quartic
crowns for the ultra-wide strip. F. Schausberger et al.
11
put forward a continuum-mechanics approach to system-
atically derive the longitudinal contour evolution of the
plate based on the input and output thickness. The PC
technology can achieve large adjustment domains for the
strip crown. The equivalent roll-gap crown model after
roll crossing is very important for strip crown control,
and the calculation models after roll symmetrically and
asymmetrically crossing were established based on the
assumption that the cross sections of the rolls were circu-
lar or ellipse.
12–14
The thermal and wear contours of the
online work roll are also the main factors that influence
the strip profile. Therefore, online prediction models for
the 2D roll gap are also research cores. A computer
model based on the finite-difference method was applied
to predict the model for the work-roll thermal profile.
15
R. Servin Castañeda et al.
16
put forward a roll-wear pre-
diction model of a hot strip mill. It mainly considered the
influence of the Hertzian pressure distribution and the
mechanical contact between two cylinders with parallel
axes.
17
X. D. Wang et al.
18
put forward a predictive calcu-
lation model for the work roll comprehensive contour,
which is the superposition of the initial, thermal, and
wear contours of work rolls.
The above studies on the strip profile only concerned
the 2D roll gap at the outlet of the deformation zone. The
shape change of the roll gap has little effect on the load
distribution under the symmetry condition, thus, the in-
fluence is neglected. However, in the case of asymmetry,
the impact of the roll gap on the rolling load is severe.
The applications of the PC rolling mill and the CVC roll
contour have led to many load problems, especially the
thrust load and the screw-down load deviation. Both the
great deviation of the screw-down load and the roll dam-
age frequently occurred in the 1580-mm PC mill of
Baosteel. The main reason for the dynamic screw-down
load deviation was identified as the axial force resulting
from roll crossing.
19
In addition, the problems of thrust
bearing burning and short service life which frequently
arose in a 2350 aluminum foil mill were also considered
to be caused by the great thrust force due to the roll
crossing.
20
The reasons for those problems definitely
point to the stability of the rolls or roll crossing. Aiming
at roll crossing, Sheng et al.
21–23
firstly put forward new
theories of the roll spiral precession, the dynamic roll
crossing, and the microscale dimension behavior. They
pointed out that the mechanism of setting eccentricity
between rolls was adverse to the roll stability. It was
prone to cause dynamic roll crossing behavior and even-
tually led to the great axial force, which is the main rea-
son for the bearing shortened longevity and burn dam-
age. To this end, the mechanical statically determinate
structure of rolling mills was proposed to solve the prob-
lem. However, those investigations not only did not es-
tablish the effect evaluation of the roll crossing on the
rolling load, but also did not give an explanation for the
deviation of the screw-down load. The asymmetrial de-
formation zone is one of the characteristics of asyn-
chronous rolling.
24–26
However, the roll gap of asyn-
chronous rolling is asymmetric regarding up and down
rather than about left and right along the roll barrel. The
asynchronous rolling process is mainly affected by dif-
ferent circumferential velocities or different work-roll ra-
dii.
According to the above analyses, the present studies
aim only at the equivalent 2D roll gap at the outlet of the
deformation zone. There is little research on the equiva-
lent three-dimensional (3D) roll gap under roll-crossing
conditions, which has a vital influence on the rolling
load distribution along the deformation zone. The great
screw-down load deviation has occurred in a 5000-mm
heavy plate mill, which can reach 200 t to 400 t, account-
ing for 3–6 % of the total rolling force. It seriously
harms the mill equipment and leads to significant bent
defects for heavy plates. The axial force of the backup
roll (BUR), which would generate an additional moment
on the screw-down, is considered one of the main rea-
sons for the screw-down load deviation.
27
Furthermore,
another important factor for the screw-down load devia-
tion is the distribution of the rolling pressure, especially
under roll-crossing conditions. The asymmetrical distri-
bution of the deformation zone after roll crossing is a di-
rect reason for the asymmetrical distribution of the roll-
ing pressure in the deformation zone. It is obviously
one-sided to apply the 2-D equivalent roll gap at the out-
let of the deformation zone to evaluate the whole roll gap
in the deformation zone after roll crossing. In particular,
the work roll is the CVC profile, and the distribution of
the deformation zone is nonlinear after roll crossing,
which means that the equivalent roll contours from the
inlet to the outlet of the deformation zone are not paral-
lel. Therefore, one of the key issues for the research on
screw-down load deviation is to establish the equivalent
3D roll gap in the deformation zone. In this paper, the
3D distribution of both the upper and lower roll contours
after roll crossing is established based on the coordinate
nonlinear transformation of the spatial rectilinear coordi-
nate system. Finally, the 3D roll gap is reconstructed
without load conditions, and intuitive contour maps are
used to reveal the asymmetrical distribution of the defor-
mation zone.
2 NONLINEAR TRANSFORMATION MODEL OF
ROLL CROSSING
2.1 Roll crossing modeling principle
Reconstruction of the equivalent 3D roll gap under
the conditions of roll crossing and no load needs to es-
tablish the 3D roll model. Figure 1 is a sketch map of
the CVC WR under roll-crossing conditions. The Y
0
-axis
of the spatial rectangular coordinate system O
0
–X
0
Y
0
Z
0
is
the rotatory central line of the lower CVC WR, and the
X. MA et al.: THREE-DIMENSIONAL MODEL OF EQUIVALENT ROLL GAP OF CVC MILL UNDER CONDITIONS ...
194 Materiali in tehnologije / Materials and technology 58 (2024) 2, 193–202

vertical direction is the Z
0
-axis, the X
0
-axis is perpendic-
ular to the plane O
0
–Y
0
Z
0
. Under the ideal condition, the
upper and lower WR axes are parallel. When the upper
and lower WRs are crossing, the rotation axes of the up-
per and lower WR occur with a spatial intersection, re-
sulting in the rotation of the local spatial rectangular co-
ordinate system of the upper roll O–XYZ. The coordinate
transformation formula of two different spatial coordi-
nate systems is established to solve the equivalent 3-D
roll gap under the conditions of roll crossing and no
load.
2.2 Coordinate nonlinear transformation model of
space rectangular coordinate system
Figure 2 is a schematic diagram of the transforma-
tion of a spatial rectangular coordinate system. O
0
P is a
vector in the absolute spatial rectangular coordinate sys-
tem O
0
–X
0
Y
0
Z
0
. O
0
P is a vector in the local spatial rectan-
gular coordinate system O–XYZ. O
0
O is the vector from
the origin of the absolute spatial rectangular coordinate
system O
0
–X
0
Y
0
Z
0
to the origin of the local spatial rectan-
gular coordinate system O–XYZ. The coordinates of O in
the coordinate system O
0
–X
0
Y
0
Z
0
are (X
0
T
, Y
0
T
, Z
0
T
), i.e.,
they are the translation parameters of the rectangular co-
ordinate system O
0
–X
0
Y
0
Z
0
to the rectangular coordinate
system O–XYZ. As a result of the rotation of the coordi-
nate system, there are three rotation-angle parameters x
,
 y
,  z
, which the axes of the local coordinate system
O–XYZ are parallel to the axes of the absolute coordinate
system O
0
–X
0
Y
0
Z
0
after rotation. In addition, the unit
vector of two coordinate systems can be different, and
the unit dimension of the original absolute rectangular
coordinate system O
0
–X
0
Y
0
Z
0
is 1, while the scale of the
unit of the local rectangular coordinate system O–XYZ
increases K
1
.
The coordinates of a point P in the coordinate system
O
0
–X
0
Y
0
Z
0
are (X
i
0
,Y
j
0
, Z
k
0
), and in the coordinate system
O–XYZ are (X
i
, Y
j
,Z
k
). The relationship between them is
as follows:
28
X
Y
Z
X
Y
Z
K
X
i
j
k
i
0
0
0
0
0
0
1
1
⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ =
⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ++
T
T
T
() R Y
Z
j
k
⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ (1)
X. MA et al.: THREE-DIMENSIONAL MODEL OF EQUIVALENT ROLL GAP OF CVC MILL UNDER CONDITIONS ...
Materiali in tehnologije / Materials and technology 58 (2024) 2, 193–202 195
Figure 1: Sketch map of upper and lower CVC work rolls under the
condition of roll crossing
Figure 2: Schematic of the transformation of the spatial rectangular
coordinate system
R
xx x x
xx
() c o s s i n
sin cos


=−
⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ 10 0
0
0
, R
yy
yy
yy
()
cos sin
sin cos
 

=
−
⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ 0
010
0
, R
zz
zz
zz
()
cos sin
sin cos  
 =
−
⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ 0
0
001
R =
− cos( )cos( ) cos( )sin( )sin( ) cos( )sin( 
yzzyxxzxyzxz
y
) cos( )sin( )cos( ) sin( )sin( )
cos( )sin(


+
zxyzxzx
) sin( )sin( )sin( ) cos( )cos( ) cos( )sin(  +
yzzx
yy x
)sin( ) cos( )sin( )
sin( ) cos( )sin( ) cos(

 
−
− 
xy
)cos( )
⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ (2)

Where R = R
z
( z
)R
y
( y
)R
x
( x
),
As a result of the roll flattening between the BUR
and the WR, the setting of the offset distance becomes
invalid, leading to a failure of the horizontal force.
18
Ad-
ditionally, the initial gap and the extent of the wear be-
tween the roll bearing chocks and the mill housings are
inevitable. In particular, the screw-down stroke of the
heavy plate rolling mill is large and the AGC frequently
and dynamically adjusts so that the amount of wear and
the gap are relatively large. All of those will lead to the
roll crossing. However, the offset distance, the initial gap
and the wear between the bearing chock and housing are
relatively small compared with the roll length and the ra-
dius of the heavy-plate rolling mill. The crossing angle
of the upper and lower WRs is generally less than 0.5°.
Therefore, the values of x
, y
, and z
can be treated as
minute quantities. According to the Taylor formula, the
sin( ) and cos( ) can be expanded to a power series of ,
which can be expressed as follows:
sin( ) ...
()
() !
...  
      
=++ +
−
−
+
−
1
21
121 nn
n
(3)
cos( ) ...
()
() !
... 
     
=++ +
−
+ 1
1
2
2 nn
n
(4)
From Equations (3) and (4), as well as the derivative
relationship of Equation (2), and ignoring the higher-or-
der infinitesimal that is more than the quadratic term of
the , the following equations can be obtained:
R =
−
+
−+
−
+
−
−
1
2
1
2

   
 
 



yz
xy z xz y
z
xz
yz x
yx
1
2
−
+
⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ 

xy
(5)
From Equations (1) and (5), the relationship between
the two coordinate systems of the upper and lower WR
can be established:
X
Y
Z
XK X
i
j
k
yz i
0
0
0
01
11
2
⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ≈
++ −
+ ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ T
()


+−++
⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ++ +−
()()
()
   
  xy z
j
xz y
k
z
i
YZ
YKX
01
11
T xz j
yz x
k
y
YZ
ZK

  

+ ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ +−
⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ++
2
1
01
()
()
T
XY Z
i
x
j xyk
+++
+ ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥  

1
2
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ (6)
The position and direction of the local coordinate
system of the upper and lower WRs change before and
after the roll crossing, while the scale factor of the local
coordinate system will not change, i.e., K
1
=0 .T h e
translation parameters from the absolute spatial rectan-
gular coordinate system of the lower roll O
0
–X
0
Y
0
Z
0
to
the local spatial rectangular coordinate system of the up-
per roll O–XYZ can be determined by the offset distance
of the roll and the crossing angle. If the rotation angle of
the three directions is given, the equivalent roll gap in the
deformation zone can be obtained.
3 MODEL OF EQUIVALENT 3D ROLL GAP
UNDER ROLL-CROSSING CONDITIONS
In the spatial rectangular coordinate system
O
0
–X
0
Y
0
Z
0
, the equation of the lower roll is expressed as:
xzr
222
+=
rRal ys e
al ys e al ys
=+ −−++
+− −++− −
www
ww ww
1
2
2
3
(())
(())(()) + e
3
(7)
where y s
w
, s
w
+ l
w
 ; s
w
is the distance from the
center line of the housing to the end of the WRs; l
w
is the
length of the WRs; e is the shifting displacement of the
WRs; R
w
is the nominal radius of the WRs; a
1
, a
2
, and a
3
are the coefficients of the cubic curve of the contour of
the CVC WR.
According to Equations (1) and (6), the transforma-
tion relationship between the spatial rectangular coordi-
nate system O–XYZ and the spatial rectangular coordi-
nate system O
0
–X
0
Y
0
Z
0
is obtained, and the upper-roll
equation in the spatial rectangular coordinate system
O
0
–X
0
Y
0
Z
0
is:
[] {
[]
() ( ) ZAB z CARa lse B
al s eB
yx 0
22
1
2
T
www
ww
+−++=+++−+
+++ −

[]
}
2
3
3
2
+++ − al s eB
ww
(8)
AX x y z
T
yz
zxy yxz
=−
+
−
⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟−−++
0
22
2
1

  () ()
BY X x y
z
z
yz
zxy
=+ −
+
−
⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎧ ⎨ ⎪ ⎩ ⎪ −−+
+
00
22
2
1
TT
 
 ()
}
() () 


yx z
xz
xyz
yz +−
+
−
⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟−−
22
2
1
C
xy
=
+
−
⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ 
22
2
1
xl
x
= (9)
where l
x
 [0, l]; l is the projection length of the contact
arc of the deformation zone. Combining Equations (7),
(8), and (9), the following equivalent equation of the up-
per and lower roll curves at the position x = l
x
in the de-
formation zone can be obtained:
fyz
fyz
x
xl R zR
s
xlR zZ
x
x
(,)
(,)
()() =− ≤ ≤+
=< ≤
,
,
ww
w
T

0
(10)
where  is the difference between the maximum and
minimum CVC roll radii. Then the distribution of the
equivalent roll gap in the deformation zone is obtained:
hyz f yz f yz ( x = l)
xx
(,) (,) (,) =−
s
(11)
X. MA et al.: THREE-DIMENSIONAL MODEL OF EQUIVALENT ROLL GAP OF CVC MILL UNDER CONDITIONS ...
196 Materiali in tehnologije / Materials and technology 58 (2024) 2, 193–202

To facilitate the quantitative evaluation of the asym-
metry of the 3D roll gap in the deformation zone, the
asymmetry degree of the roll gap is defined as:
A
hh
hh
hh
h
ii
ii
o o
o
ag
wd
wd
w d
w
=+
1
2
max( , )
min( , )
max( , )
min( ,) h
od
⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ (12)
Further, the relative asymmetry of the roll gap can be
obtained:
A
h
h
A
rg ag
=
Δ
(13)
where h
iw
and h
id
, as shown in Figure 3, are the thick-
nesses at the working side and the driving side of the in-
let of the roll gap, respectively. h
ow
and h
od
are the thick-
nesses at the working side and the driving side of the
outlet of the roll gap, respectively. The thickness reduc-
tion and the average thickness of the roll gap can be ex-
pressed as:
Δh
hhhh
ii o o
=
+−+ () ()
wd w d
2
(14)
h
hhhh
iioo
=
+++ () ()
lrl r
4
(15)
Both values of A
ag
and A
rg
represent the asymmetry of
the roll gap. The closer the value of A
ag
to 1, the more
symmetrical the roll gap, and the less the value of A
rg
, the
smaller the impact of roll crossing.
4 EFFECT OF ROLL CROSSING ON THE
SYMMETRY OF THE EQUIVALENT 3D ROLL
GAP OF CVC ROLLING MILLS
Many factors will lead to roll crossing, such as the
gap between the bearing chocks of the WR and the BUR
and the mill stand housing, the symmetry of the mill
housing installation and manufacture, the oil-film thick-
ness of the BUR bearing, the additional moment from
the transmission shaft acting on the WR, the roll eccen-
tricity between the rolls, the roll contour, and so on. The
size of the intersection angle can only be determined by
main factors. Under the no-load condition, the influence
factors of the equivalent 3D roll gap shape include the
three rotation parameters, the three translation parame-
ters, the WR radius and the length, and the roll gap’s
nominal thickness. The equipment parameters and clear-
ance parameters of a 5000-mm-wide and heavy plate
rolling mill and a 1580-mm hot-rolling mill are shown in
Table 1. The parameters of the two CVC roll mills were
used in the model calculation, and the crossing angles of
the upper and lower WRs were estimated according to
those parameters.
Table 1: Parameters of 5000 mm and 1580 mm CVC rolling mill
Rolling mill type 5000-mm mill1580-mm mill
WR radius Rw
/mm 600 400
WR barrel length lw
/mm 5300 1580
Distance between both
screw-downs la/mm
7000 2760
Eccentricity between rolls be-
tween center lines of WR and
BUR /mm
10 10
Gap between WR chock and
housing w
/mm
1–4 1–4
Gap between BUR chock and
housing b
/mm
3–8 3–5
Equations (7) and (8) of the WR are binary cubic
functions about y and z, and the final derived Equation
X. MA et al.: THREE-DIMENSIONAL MODEL OF EQUIVALENT ROLL GAP OF CVC MILL UNDER CONDITIONS ...
Materiali in tehnologije / Materials and technology 58 (2024) 2, 193–202 197
Figure 3: Sketch map of the roll gap
Figure 4: 3D work rolls and equivalent roll gap: a) work rolls,
b) equivalent roll gap

(11) of the equivalent roll gap is an implicit function, it
will be extremely difficult to change the implicit function
into the explicit form. It is recommended to use some
mathematical calculation tools to solve this implicit
function. This study established a calculation program of
equivalent 3D roll gaps using MATLAB software.
Taking the parameter of 1580-mm CVC work rolls as an
example, the results of the 3D distribution of the work
rolls and the equivalent roll gap obtained are shown in
Figures 4a and 4b, respectively.
The crossing angle  z
means that the upper WR ro-
tates around the Z
0
axis in the horizontal plane O
0
–X
0
Y
0
.
It is mainly affected by the gaps w
, b
, between the bear-
ing blocks of both the WRs and the BURs and the mill
housing. The rolling distance of the BUR is large, and
the roll change period is so long that the gap between the
BUR bearing chocks and the mill housing is larger than
that between the WR bearing chocks and the mill hous-
ing. Hot-rolling mills, especially the heavy-plate rolling
mill, are characterized by the large opening degree of
roll gap, the large screw-down stroke, and the frequent
movement of hydraulic AGC, it is very likely to cause
roll crossing. In the case of serious wear, the gap be-
tween the BUR bearing chock and stand housing was 1.2
times the technical requirement. Furthermore, if the
function of the eccentricity between rolls is failure, the
crossing angle is about:
 
z
l
max
(.)
=
++ 21 2
w b
a
Δ
(16)
The crossing angle y
means that the upper WR ro-
tates around the Y
0
axis in the vertical plane O
0
–X
0
Z
0
.It
is due to the rotation of the CVC roll contour, which has
no effect on the 3D roll gap. The crossing angle x
means
that the upper WR tilts around the X
0
axis in the vertical
plane O
0
–Y
0
Z
0
. It is mainly affected by the zero adjust-
ment of the roll gap, the vertical tilting of the WR, the
horizontal crossing angle, and the symmetry of the actual
grinding roll contour.
To analyze the influence of different crossing angles
on the symmetry of the equivalent 3D roll gap, the con-
tour maps of the equivalent roll gap in the deformation
zone are shown in the following figures. The longitudi-
nal coordinate is the axial position along the roll barrel,
and the horizontal coordinate is the position along the
contact arc. The values of the cross angle and translation
parameters in the simulation are shown in Table 2. The
parameters of the rolls used in the simulation adopt the
those of the 1580-mm and 5000-mm rolling mills in Ta-
ble 1.
Figure 5 shows contour maps of the equivalent roll
gap affected by rotational parameters, namely x
, y
, and
 z
. and the roll contours in the roll gap from the inlet to
the outlet of the deformation zone. Among the three rota-
tion parameters, rotation around the Y
0
-axis has no effect
on the roll gap, which is consistent with the theoretical
analysis and proves the correctness of the model. The ab-
solute asymmetry degrees of the roll gap, A
ag
,i nFig-
ures 5a to 5c are approximately 1.31, 1, and 1.06, re-
spectively. This indicates that rotation around the X
0
-axis
has the greatest impact on the asymmetry of the roll gap,
while rotation around the Z
0
-axis has a secondary impact
on the roll gap. In fact, a large number of simulation
analyses shows that the roll-gap asymmetry develops sig-
nificantly with the increase of x
. The larger the tilt angle
 x
, the greater the influence on the asymmetry degree of
roll gap. Even if the tilt angle x
is equal to 0.0001, the
absolute asymmetry degrees of the roll gap, A
ag
will
reach 1.05, which is roughly equal to that of example 3,
and the vertical height difference, h
= l
a
 x
, between the
working and driving sides of the 1580-mm rolling mill
will reach 0.276 mm at the same time. It is necessary to
strictly control the vertical leveling of the roll gap. In
Figure 5d, the roll contours do not keep parallel from
the inlet to the outlet of the deformation zone when the
roll axis rotates around the X
0
-axis and Z
0
-axis, simulta-
neously. Namely, it is inaccurate that the 2D equivalent
roll contour is used to evaluate the entire gap. It causes
an error if the rolling force is calculated by using the roll
contour at the outlet of the deformation. The thinner the
roll gap and the greater the steel deformation resistance,
the larger the error.
According to Equations (7), (8), (10), and (11), under
the conditions of roll rotation, the equivalent 3D roll gap
is a function of the offset position X
0
T
, Y
0
T
, Z
0
T
,x , y, and
z. X
0
T
and Z
0
T
will alter the relative position of the ends
of the upper and lower rolls in the radial direction, and
the offset position Y
0
T
can affect the symmetry of the up-
per and lower WR profiles after shifting, thereby chang-
ing the end position of roll crossing. In fact, the axial di-
rection of the rolls is constrained by the shifting
hydraulic cylinder, and the radial direction along the roll-
ing direction is constrained by the mill housing and bear-
ing chocks, thus the offset positions X
0
T
and Y
0
T
vary in a
limited range. The simulation results indicate that the
limited offset positions have little influence on the sym-
metry of the roll gap. Z
0
T
is the distance between two
work roll axes, which is influenced by both roll radius,
R
w
, and nominal thickness of roll gap, h. The x-coordi-
nate and z-coordinate are distributed along the radial di-
X. MA et al.: THREE-DIMENSIONAL MODEL OF EQUIVALENT ROLL GAP OF CVC MILL UNDER CONDITIONS ...
198 Materiali in tehnologije / Materials and technology 58 (2024) 2, 193–202
Table 2: Rotation parameters and translation parameters in simulated instances
 x  y  z X
0
T
/mm Y
0
T
/mm Z
0
T
/mm
Example 1 0.0005 0 0 10 1 2Rw
+4
Example 2 0 0.0005 0 10 1 2Rw
+4
Example 3 0 0 0.003 10 1 2Rw
+4

rection of the roll barrel, and the y-coordinate is distrib-
uted along the axial direction, thus the roll barrel length,
l
w
, and roll radius, R
w
, also influence the symmetry of the
roll gap under the condition of roll rotation.
To further analyze the influence of the roll barrel
length, roll radius, and nominal thickness of the roll gap
on the asymmetry of equivalent 3D roll gaps under
roll-crossing conditions, simulations were conducted
based on the parameters shown in Table 3. In examples 4
and 5, both the roll radius and roll gap’s nominal thick-
ness of the 5000-mm and 1580-mm rolling mills are
600 mm and 10 mm, the roll-barrel length is variable.
X. MA et al.: THREE-DIMENSIONAL MODEL OF EQUIVALENT ROLL GAP OF CVC MILL UNDER CONDITIONS ...
Materiali in tehnologije / Materials and technology 58 (2024) 2, 193–202 199
Figure 5: Contour maps of the equivalent roll gaps affected by the rotational parameters and roll contours: a) example 1, b) example 2, c) exam-
ple 3, d) roll contours in roll gap from inlet to outlet of the deformation zone
Figure 6: Contour maps of the equivalent roll gaps affected by the work roll-barrel length: a) example 4, b) example 5.

Figure 6 shows that the asymmetry degree of the whole
equivalent 3D roll gap increases with the roll-barrel
length increasing under the condition of the same rota-
tion parameters and translation parameters. The absolute
asymmetry degrees of roll gap, A
ag
,inFigure 6a and 6b
are about 1.51 and 1.09, respectively. The result indicates
that the longer the roll-barrel length, the greater the in-
fluence of the roll crossing on the asymmetry degree of
the roll gap. Specifically, when the roll barrel l
w
is equal
to 5300 mm, the whole equivalent 3D roll gap from the
inlet to the outlet exhibits pronounced asymmetry. This
means that the wide-plate mill is more susceptible to roll
crossing, in terms of the degree of roll-gap asymmetry.
In fact, it is precisely because of the roll length of the
heavy-plate mill, and the difference in the constraints of
both mill housings on roll bearing chocks is particularly
prominent. The roll bearing chocks at the operating side
bear strong fixed constraints of shifting hydraulic cylin-
ders or thrust plates, while those at the driving side be-
long to free ends. Furthermore, the roll driving end with-
stands the disturbance of the additional moment from the
main transmission system.
29
Thus, the longer the roll bar-
rel, the easier the rolls to cross, while the asymmetrical
roll gap, in turn, is more likely to make the roll crossing.
Moreover, Figure 6 shows that the asymmetry near the
outlet is greater than that near the inlet. This indicates
that the thinner the roll gap, the greater the influence of
roll crossing.
From examples 6 to 7, all the roll-barrel lengths and
the roll gap’s nominal thicknesses are 1580 mm and 10
mm, respectively, and the influential variable is the roll
radius. Figure 7 shows that the asymmetry degree of the
whole equivalent 3D roll gap increases with the roll ra-
dius increasing under the condition of the same rotation
parameters and translation parameters. However, the ab-
solute asymmetry degrees of roll gap, A
ag
,inFigure 7a
and 7b are about 1.0129 and 1.0178, respectively. Al-
though the asymmetry degree of the equivalent 3D roll
gap grows with roll radius increasing, the variation de-
gree is not significant. It can be inferred that the WR ra-
dius of the rolling mills has little effect on the asymme-
try of the 3D roll gap. For plate-rolling mills, under the
condition of constant outlet thickness, large roll radii
will cause an increase of the contact arc length in the de-
formation zone, and the influence of asymmetry three-di-
mensional roll gap on the distribution of rolling pressure
will still be amplified, resulting in screw-down load devi-
ation under the condition of roll crossing.
In examples 8 and 9, all the roll-barrel lengths and
the roll radii are 1580 mm and 400 mm, respectively, and
the influential variable is the roll gap’s nominal thick-
ness. The absolute asymmetry degrees of the roll gap,
A
ag
,inFigure 8a and 8b are about 1.0571 and 1.013, re-
spectively, while the relative asymmetry degrees of the
roll gap, A
rg
are about 0.42 and 0.12, respectively. The re-
sults indicate that the asymmetry degree of the whole
equivalent 3D roll gap decreases with the roll gap’s nom-
inal thickness increasing under the condition of the same
rotation parameters and translation parameters. That is,
the last pass with a thin gap or the downstream rolling
mill, in the matter of the asymmetry degree of roll gap, is
much more easily affected by the roll crossing. It indi-
cates that the screw-down load deviation is more likely
to occur in the last pass or the downstream rolling mill
X. MA et al.: THREE-DIMENSIONAL MODEL OF EQUIVALENT ROLL GAP OF CVC MILL UNDER CONDITIONS ...
200 Materiali in tehnologije / Materials and technology 58 (2024) 2, 193–202
Table 3: Examples of the influence of the work roll barrel length, work roll radius, and roll gap’s nominal thickness
Parameters Example 4 Example 5 Example 6 Example 7 Example 8 Example 9
WR radius R
w
/mm 600 600 400 800 400 400
WR barrel length l
w
/mm 5300 1580 1580 1580 1580 1580
Roll gap h/mm 10 10 10 10 4 15
Figure 7: Contour maps of the equivalent roll gaps affected by the work roll radii: a) example 6, b) example 7

under the condition of roll crossing. The problem be-
comes more significant, especially for the last pass of the
heavy-plate rolling mill as the deformation resistance of
the plate increases.
According to the influence rules of the roll-barrel
length, the roll radius, and the roll gap’s nominal thick-
ness on the asymmetry of the roll gap, it can be con-
cluded that the heavy-plate mills are more likely to be af-
fected by roll crossing than ordinary mills, and the great
screw-down load deviation is more likely to happen in
last pass compared with initial passes, which is consis-
tent with the production practice and reports.
27
For the
wide and heavy-plate rolling mill, the rolls would be lev-
eled before rolling, so the rotation of the work rolls
around the x-axis is controllable. When the rotation an-
gle z
of the work rolls around the z-axis is less than or
equal to 0.004, according to Equation (16), it can be ob-
tained that the sum of clearance and wear extent between
the liners of both bearing chocks and mill housing in
both the working and driving side should be less than 9
mm.
5 CONCLUSION
To analyze the influence of roll crossing on the asym-
metry of roll gap, this study established an equivalent 3D
roll-gap model and an evaluation method for asymmetry
degree. The influences of roll-crossing parameters on the
asymmetry of the 3D roll gap are analyzed. Furthermore,
the sensitivity of the work roll-barrel length, roll radius,
and nominal thickness of the roll gap to the asymmetry
degree of the equivalent 3D roll gap caused by roll cross-
ing is analyzed. Based on the above research, the follow-
ing conclusions can be drawn:
(1) The 3D model of the upper and lower WR and the
equivalent 3D roll-gap model are established based on
the coordinate nonlinear transformation model of the
spatial rectangular coordinate system. It can be widely
applicable to calculate the equivalent 3D roll gap under
small crossing angle conditions, and analyze the equiva-
lent roll contours from the inlet to the outlet of the defor-
mation zone for the roll-deformation model.
(2) The equivalent 3D roll gap is affected by the
translation parameters (X
0
T
, Y
0
T
, Z
0
T
) and the crossing an-
gle parameters ( x
, y
, z
) of the WR. The rotation param-
eter y
does not affect the roll gap. The influence of the
rotation parameter  x
, on the roll gap is much greater
than the rotation parameter  z
. The translation parame-
ters, X
0
T
and Y
0
T
, vary in a limited range and have little
influence on the symmetry of the roll gap.
(3) Under a certain crossing-angle condition, the
larger the roll barrel length, the larger the WR radius,
and the thinner the roll gap, the greater the effect of roll
crossing on the roll-gap asymmetry. Heavy mills should
give more attention to factors or phenomena that is
caused by roll crossing, such as the liner wear of bearing
chocks and mill housing, tremendous roll axial force,
etc.
(4) To prevent the occurrence of serious roll crossing
in a 5000-mm rolling mill, the following criteria are rec-
ommended: the sum of the clearance and wear extent be-
tween the liners of both bearing chocks and mill housing
in both the working and driving side is less than 9 mm.
Acknowledgments
The financial support of the Shanxi Province Natural
Science Foundation Research Program(Grant No.
202203021222121), the Shanxi "1331Project" Key Inno-
vative Research Team Fund, the Open Project of Re-
search Institute of Hai’an-Taiyuan University of Tech-
nology (Grant No. 2023HA-TYUTKFYF008), the
Chinese Postdoctoral Science Foundation (Grant No.
2021M702544), the School Fund of Taiyuan University
of Technology (Grant No. 2022QN007), and the Shanxi
Province Major Project of Science and Technol-
ogy(Grant No. 20181102016) are gratefully acknowl-
edged.
X. MA et al.: THREE-DIMENSIONAL MODEL OF EQUIVALENT ROLL GAP OF CVC MILL UNDER CONDITIONS ...
Materiali in tehnologije / Materials and technology 58 (2024) 2, 193–202 201
Figure 8: Contour maps of the equivalent roll gaps affected by the roll gap’s nominal thickness: a) example 8, b) example 9

Author contributions
Xiaoxin Ma: Investigation, Formal analysis, Original
draft writing, Programming; Peisen Yuan: Original draft
writing, Programming, Editing; Fuyi Li: Editing, Re-
view; Jiang Ji: Supervision, Review.
Declaration of competing interest
The authors declare that they have no known compet-
ing financial interests or personal relationships that could
have appeared to influence the work reported in this pa-
per.
Data availability statement
Almost all data generated or analyzed during this
study are included in this published article. The other in-
formation or parameters used and/or analyzed during the
current study are available from the corresponding au-
thor on reasonable request.
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X. MA et al.: THREE-DIMENSIONAL MODEL OF EQUIVALENT ROLL GAP OF CVC MILL UNDER CONDITIONS ...
202 Materiali in tehnologije / Materials and technology 58 (2024) 2, 193–202