# Multiple-Modal-Coupling Modeling and Stability Analysis of Cold Rolling Mill Vibration

Authors: Lingqiang Zeng; Yong Zang; Zhiying Gao
Journal: Shock and Vibration
 (2016)

---

## Abstract

An effective dynamic model is the basis for studying rolling mill vibration. Through analyzing characteristics of different types of vibration, a coupling vibration structure model is established, in which vertical vibration, horizontal vibration, and torsional vibration can be well indicated. In addition, based on the Bland-Ford-Hill rolling force model, a dynamic rolling process model is formulated. On this basis, the rolling mill vertical-torsional-horizontal coupled dynamic model is constructed by coupling the rolling process model and the mill structure model. According to this mathematical model, the critical rolling speed is determined and the accuracy of calculated results is verified by experimental data. Then, the interactions between different subsystems are demonstrated by dynamic responses in both time and frequency domains. Finally, the influences of process parameters and structure parameters on system stability are analyzed. And a series of experiments are conducted to verify the correctness of these analysis conclusions. The results show that the vertical-torsional-horizontal coupled model can reasonably characterize the coupling relationship between the mill structure and the rolling process. These studies are helpful for formulating a reasonable technological procedure of the rolling process and determining a feasible dynamic modification strategy of the structure as well.


---

## Body

## 1. Introduction

Strips are the most widely used rolled products. With the increase of steel productions and the further optimization of the steel product mix, the industrial demand for strips has been growing, and the requirement for the product quality has become increasingly strict. However, the improvement of the production efficiency and the product quality is seriously restricted due to the presence of mill vibration.Generally, three kinds of typical vibration have been observed in rolling by scholars. They are identified as vertical vibration, torsional vibration, and horizontal vibration. And the vertical vibration is classified into two types: third-octave-mode chatter (120~250 Hz) and fifth-octave-mode chatter (500~700 Hz).Third-octave-mode chatter is considered to be the most harmful vibration phenomenon that occurred in high-speed cold tandem mills. It is characterized by a sudden occurrence, which tends to accumulate a lot of energy in a few seconds to make the vibration amplitude increase rapidly. A four-high mill structure model proposed by Yarita et al. [1] is the earliest model used to study the third-octave-mode chatter. Tamiya et al. [2] studied the third-octave-mode chatter on a cold rolling mill and reported that this type of vibration is a self-excited vibration due to the phase difference between the vertical movement of rolls and the strip tension at entry. On this basis, Yun et al. [3], Hu et al. [4], and Zhao and Ehmann [5], the scholars of Northwestern University, conducted a series of systematic studies on vibration of rolling mills. Yun et al. [3] reported that negative damping, mode coupling, and regeneration were the three possible causes of third-octave-mode chatter. Hu et al. [4] further developed the dynamic rolling process model based on the study of Yun and studied the correlation between different rolling parameters and the stability of the system. Zhao and Ehmann [5] researched the third-octave-mode chatter based on a multistand model and found that the dynamic characteristics of this system are more complex due to the regenerative effect. Meehan [6] studied the third-octave-mode chatter on a 4-high rolling mill and pointed that the roll stack damping and the tension stress have the most effect on the critical speed.Compared with third-octave-mode chatter, fifth-octave-mode chatter causes less damage to rolls and strips. But it affects the product appearance. Roberts [7] explained the mechanism of fifth-octave-mode chatter and its negative impacts in detail and put forward a series of measures to suppress fifth-octave-mode chatter. Hu and Ehmann [8] studied the critical condition in which the system would become unstable owing to fifth-octave-mode chatter.Torsional vibration may cause fatigue effect of transmission system components or even instantaneous damage. Gallenstein [9] studied the possible causes of the torsional vibration on a test cold mill and discussed the quantitative relationship between rolling process parameters and the rolling torque. Krot [10] created a drive transmission structure model of the rolling mill and presented a diagnostic method for torsional vibration impact strength and equipment wear.In addition, Paton and Critchley [11] found that rolls can vibrate not only in vertical direction, but also in horizontal direction. Shen and Li [12] pointed that the horizontal vibration would result in minor fluctuations of the roll gap, which would lead to the occurrence of abnormal excitation forces and break the equilibrium state of the system. Dwivedy et al. [13] also reported that horizontal vibration may do serious harm to rolling mills and quality of strips. It may cause light and dark marks on the work roll surface or even cause accidents such as the strip breakage and the piling-up of steel.In recent years, specialists and scholars have done a lot of researches on rolling mill vibration and have gained abundant achievements. But these studies mostly focused on the vibration of a single subsystem like torsional subsystem or vertical subsystem. Only a few scholars studied the coupling vibration of rolling mills. Swiatoniowski [14] proposed a typical vertical-torsional coupling structure model with a two-high rolling mill as an object. Yan et al. [15] studied the vertical-torsional coupling vibration using FEM and found that the torsional vibration may strengthen the vibration of vertical subsystem. Hou et al. [16] proposed a vertical-horizontal coupling model based on the interactions between the dynamic rolling force and the mill structure and found that the system parameters have significant effect on the rolling mill coupling vibration. Yun et al. [17] proposed a model which allows rolls to vibrate along two orthogonal directions to study the vertical-horizontal coupling vibration and suggested that mode coupling might also induce vibration. Since the vibration of a high-speed rolling mill always appears as the coupling of multiple vibration types, it becomes a hot topic to study multiple-modal-coupling vibration.As reported by Tlusty et al. [18], mill vibration is caused by the interaction between the rolling mill structure and the rolling process. Based on this interaction, [19] has constructed an overall coupled model of the rolling mill, but the calculation of the neutral point position is not sufficiently precise, and the stability analysis is not systematic. Therefore, a further research needs to be performed. In this paper, a vertical-torsional-horizontal coupling vibration model of the rolling mill is modeled according to the relationship shown in Figure 1. The structure model, in which vertical vibration, horizontal, vibration and torsional vibration can be well indicated, is established under the assumption that the vibration is symmetrical with respect to the center plane of the strip. The dynamic rolling process model, which can be used to calculate the strip velocities, tensions, and forces acting on the rolls, is constructed based on the Bland-Ford-Hill rolling force model. According to this mathematic model, the critical rolling speed of the system is calculated, the interaction between the different subsystems is discussed, and the influence of process parameters and structure parameters on the system stability is analyzed.Figure 1
Coupling relationship of structural model and rolling process model.


## 2. Structure Model

The three most typical strip rolling mills are two-high mill, four-high mill, and six-high mill. Each of them can be simplified to a mass-spring system. In order to facilitate the analysis, the mill structure is assumed to be symmetrical with respect to the center plane of the strip, and the vertical subsystem, horizontal subsystem, and torsional subsystem are all simplified as ones with single degree of freedom. The coupling structure model is illustrated in Figure2.Figure 2
Simplified vertical-torsional-horizontal coupling structural model.In this structural model, only the top work roll is allowed to vibrate in horizontal direction, and the impact of the top backup roll on horizontal vibration is reflected in the support stiffness. For vertical subsystem, the top work roll, the top backup roll, and the upper mill housing are treated as a whole, as their vibration in vertical direction is in phase. The model is effective when studying third-octave-mode chatter.As this structural model is symmetrical, it is only valid to study the symmetrical vibration of the rolling mill. For asymmetric vibration, an asymmetrical model is needed. In that case, the rolling process model and corresponding equations will be more complicated absolutely.As shown in Figure2, m
1 is the equivalent mass of the top work roll. m
2 is the lumped equivalent mass of the top work roll, top backup roll, and upper mill housing, which is simplified to the center of the top work roll. J
M is the equivalent rotational inertia of the upper rolls, including the top work roll and the top backup roll. k
1, c
1 are the horizontal equivalent stiffness and horizontal equivalent damping due to the support from unilateral pillars of the rolling mill and the support from the top backup roll to the top work roll because of the offset between them. k
2, c
2 are the vertical equivalent stiffness and vertical equivalent damping due to the support from the frame and the upper beam of the rolling mill. k
t, c
t are the torsional equivalent stiffness and torsional equivalent damping of the top main drive system. R
′ is the radius of the deformed top work roll. x
c is the displacement of horizontal vibration. y
c is the displacement of vertical vibration. θ
M is the angular displacement of torsional vibration.Thus the differential equations can be written as follows:(1)
m
1
x
¨
c
+
c
1
x
˙
c
+
k
1
x
c
=
F
x
v
a
r
,
m
2
y
¨
c
+
c
2
y
˙
c
+
k
2
y
c
=
F
y
v
a
r
,
J
M
θ
¨
M
+
c
t
θ
˙
M
+
k
t
θ
M
=
M
v
a
r
,where F
x
v
a
r and F
y
v
a
r are the fluctuations of forces acting on the rolls in x and y directions and M
v
a
r is the fluctuation of the rolling torque.


## 3. Dynamic Model of Rolling Process

### 3.1. Theoretical Model of Dynamic Rolling Process

#### 3.1.1. Roll Bite Geometry

The geometry of the roll bite is schematically presented in Figure3, where O
-
1 and O
-
2 are the original geometrical centers of the top and bottom rolls (before vibrations begin). O
1 and O
2 are the ever-changing centers of the rolls. x
c is the distance from the centerline O
1
O
2 to the centerline O
-
1
O
-
2. h
0 and h
1 are the thicknesses of the strip at entry and exit. h
c is the roll gap spacing measured along the centerline O
1
O
2. v
0 and v
1 are the velocities of the strip at entry and exit. v
r is the roll peripheral velocity. x
0, x
n, and x
1 are the entry position, neutral point position, and exit position measured from the centerline O
-
1
O
-
2. σ
0 and σ
1 are the tensions of the strip at entry and exit. α, γ, and δ are the inlet angle, the neutral angle, and the exit angle. As shown in Figure 3, the rolls of the top and the bottom can vibrate in x, y, and rotation directions symmetrically. In addition, it is assumed that the width of the strip always keeps as B during rolling.Figure 3
Roll bite geometry during vibration.As shown in Figure3, the dynamic roll gap spacing can be written as follows:(2)
h
c
=
h
-
c
+
2
y
c
,where the parameters with superscript like “i
-” represent the value of parameter “i” under steady state.In the rolling deformation zone, the roll surface was usually treated as a parabolic curve, so the strip thickness at any arbitrary position is written as follows:(3)
h
x
=
h
c
+
x
-
x
c
2
R
′
.During the vibration of rolls, the volume of the roll bite changes simultaneously. Considering the control volume of the material flow within the roll bite, Hu et al. [4] modified the metal flow equation and expressed it as follows:(4)
v
h
=
v
0
h
0
-
2
y
˙
c
x
-
x
0
-
x
˙
c
h
0
-
h
,where v and h are the strip velocity and thickness of any cross section in the roll bite. The second and third terms on the right hand side of (4) represent the material flow fluctuations of any cross section caused by the vibration of rolls in y and x directions.For the cold rolling, the rolling pressure is particularly high, so the work roll will be elastically flattened obviously. Here, the Hitchcock equation is used to calculate the effective flattened radius of the work roll:(5)
R
′
=
R
1
+
2
c
Δ
h
P
B
,where Δ
h is the reduction, Δ
h
=
h
0
-
h
1 and c
=
8
(
1
-
u
2
2
)
/
π
E
2. E
2 and u
2 are Young’s modulus and Poisson’s ratio of the roll.As the function relationship between the flattened work roll radius and the rolling force, it is very difficult to decouple them. In order to simplify the analysis, the original radius is replaced by the effective radius, which is calculated iteratively during computations of the static rolling process. In addition, it is assumed that the effective radius of the work roll keeps unchanged along the contact arc between the work roll and the strip. These measures and assumptions not only can help to improve the accuracy of calculations but also can obtain a solvable analytical model.


#### 3.1.2. Strip Entry Position and Exit Position

The entry position of the strip can be derived from (3) as follows:(6)
x
0
=
x
c
-
R
′
h
0
-
h
c
.From the geometric relationship shown in Figure3, the inlet angle is expressed as follows:(7)
α
=
arcsin
⁡
x
c
-
x
0
R
′
=
arcsin
⁡
h
0
-
h
c
R
′
.Substitutingh
(
x
) from (3) into (4), the horizontal velocity of the strip at any vertical cross-section plane within the roll bite can be expressed as follows:(8)
v
x
=
v
0
h
0
-
2
y
˙
c
x
-
x
0
-
x
˙
c
h
0
-
h
x
h
c
+
x
-
x
c
2
/
R
′
.The strain rate of the strip at any vertical cross-section plane within the roll bite can be derived from (3):(9)
d
h
d
t
=
∂
h
∂
t
+
∂
h
∂
x
∂
x
∂
t
+
∂
h
∂
x
c
∂
x
c
∂
t
=
∂
h
c
∂
t
+
2
x
-
x
c
R
′
v
x
-
x
˙
c
.The exit position of the strip can be determined by examining the condition of zero strain rates (Zhao and Ehmann [5]), so(10)
∂
h
c
∂
t
+
2
x
1
-
x
c
R
′
v
1
-
x
˙
c
=
0
.Substituting (2) and (8) into (10) and rewriting it,(11)
y
˙
c
x
1
-
x
c
2
-
v
0
h
0
-
2
y
˙
c
x
c
-
x
0
-
x
˙
c
h
0
x
1
-
x
c
-
R
′
h
c
y
˙
c
=
0
.Neglecting the quadratic termx
1
-
x
c
2, then the exit position of the strip can be represented as follows:(12)
x
1
=
x
c
-
R
′
h
c
y
˙
c
v
0
h
0
-
2
y
˙
c
x
c
-
x
0
-
x
˙
c
h
0
.It can be seen from the above equation that the exit plane no longer coincides with the centerline of the rolls as the occurrence of vibration, so the exit angle can be expressed as follows:(13)
δ
=
arcsin
⁡
x
c
-
x
1
R
′
.


#### 3.1.3. Friction Model

The change of the friction along the contact arc is very complicated. It neither obeys the dry friction law nor obeys the adhesive friction theory [20]. Since the resultant force on the rolling interface is the focus in the study of mill vibration, the selected friction model should be simple, accurate, and solvable.In cold rolling, vibration often happens on the last few stands, and friction coefficients for these stands are usually very small. In this case, the friction force is far lower than the shear strength of the strip material, which means the friction state is sliding friction and there is no sticking point in the deformation zone. Therefore, Coulomb friction model can be used in the study of vibration in cold rolling. In order to simplify the analysis, we assume that the friction coefficients along the contact arc keep unchanged and are represented asμ
m.


#### 3.1.4. Neutral Point Position

In general, there is a point on the contact arc of the rolling deformation zone, named neutral point, where the peripheral velocity of the work roll and the velocity of the strip are equal. The neutral point is one of the most important variables in the rolling process. Along the contact arc, the friction changes direction on different sides of this point.As shown in Figure4, the neutral angle can be determined by the equilibrium condition of the force system in the rolling deformation zone:(14)
-
∫
δ
α
p
x
sin
⁡
α
x
R
′
d
α
x
+
∫
γ
α
μ
m
p
x
cos
⁡
α
x
R
′
d
α
x
-
∫
δ
γ
μ
m
p
x
cos
⁡
α
x
R
′
d
α
x
+
T
1
-
T
0
2
B
=
0
,where T
1 and T
2 are tensile forces of the strip at entry and exit, T
1
=
σ
1
h
1
B, and T
0
=
σ
0
h
0
B. α
x is the central angle corresponding to any arbitrary vertical cross section. p
x is the radial unit pressure, and we assume that p
x along the contact arc is uniform distribution. So it can be approximated that p
x is equal to the mean unit pressure p
m, p
x
≈
p
m
=
P
/
B
l, and l is the horizontal projection length of the contact arc, l
=
x
1
-
x
0.Figure 4
Equilibrium of force system in the roll bite.From (14), the neutral angle can be formulated as follows:(15)
sin
⁡
γ
=
sin
⁡
α
+
sin
⁡
δ
2
+
cos
⁡
α
-
cos
⁡
δ
2
μ
m
+
T
1
-
T
0
l
4
μ
m
P
R
′
.From the geometric relationship shown in Figure3, the position of the neutral point is expressed as follows:(16)
x
n
=
x
c
-
R
′
sin
⁡
γ
=
x
c
-
R
′
2
sin
⁡
α
+
sin
⁡
δ
-
R
′
2
μ
m
cos
⁡
α
-
cos
⁡
δ
-
T
1
-
T
0
l
4
μ
m
P
.


#### 3.1.5. Strip Velocities at Entry and Exit

Once the position of the neutral point is obtained, the strip velocity at any arbitrary vertical cross section of the roll bite can be also determined. As the strip velocity at the neutral point equals the roll peripheral velocity, substitutingx
n from (16) into (8), the strip velocity at the neutral point can be expressed as follows:(17)
v
r
=
v
0
h
0
-
2
y
˙
c
x
n
-
x
0
-
x
˙
c
h
0
-
h
n
h
c
+
x
n
-
x
c
2
/
R
′
.Thus, the strip velocities at entry and exit are written as follows:(18)
v
0
=
1
h
0
v
r
-
x
˙
c
h
c
+
x
n
-
x
c
2
R
′
+
2
y
˙
c
x
n
-
x
0
+
h
0
x
˙
c
,
(19)
v
1
=
v
0
h
0
-
2
y
˙
c
x
1
-
x
0
-
x
˙
c
h
0
-
h
c
-
x
1
-
x
c
2
/
R
′
h
c
+
x
1
-
x
c
2
/
R
′
.


#### 3.1.6. Calculation of Rolling Force

The rolling force is one of the most important energetic parameters of the rolling mill. Specialists and scholars have done a lot of research on the rolling force and also achieved a variety of rolling force models for different mills or working conditions. Reference [21] indicated that most of the cold rolling mills can meet the conditions of Bland-Ford-Hill rolling force model. The form of Hill simplified formula is as follows:(20)
P
=
K
-
σ
m
B
Q
p
l
,where σ
m is the mean tension, usually taking σ
m
=
0.7
σ
0
+
0.3
σ
1. Q
p is the stress state factor. K is the mean deformation resistance.The stress state factor of Hill simplified formula is expressed as follows:(21)
Q
p
=
1.08
+
1.79
ε
μ
R
′
h
0
-
1.02
ε
,where ε is the reduction ratio, expressed as ε
=
h
0
-
h
1
/
h
0. μ is the friction coefficient along the contact arc, taking μ
=
μ
m.In the rolling mill, the strip tensions at entry and exit have a significant effect on the stability of the system. And the variations of strip tensions have a direct relationship with variations of strip velocities. As the vibration characteristic of a single stand mill is the focus of this paper, supposing both the variation of the exit velocity in the upstream stand and the variation of the entry velocity in the downstream stand are zero, tension variations at entry and exit can be expressed according to Hooke’s law as follows:(22)
σ
0
v
a
r
=
E
1
L
0
Δ
L
0
=
E
1
L
0
∫
0
t
v
0
-
v
-
0
d
t
,
σ
1
v
a
r
=
E
1
L
1
Δ
L
1
=
-
E
1
L
1
∫
0
t
v
1
-
v
-
1
d
t
,where E
1 is Young’s modulus of the strip. L
0 and L
1 are the distances from this stand to the upstream stand and the downstream stand.Hence, the dynamic tensions of the strip at entry and exit can be written as follows:(23)
σ
0
=
σ
-
0
+
σ
0
v
a
r
;
σ
1
=
σ
-
1
+
σ
1
v
a
r
.In addition, the strain hardening effect must be considered when calculating the deformation resistance of the cold-rolled strip. Because of the strain hardening effect, the deformation resistance of each rolling pass is not only related to the deformation extent of this rolling pass but also associated with the total deformation extent of rolling passes before. Furthermore, the deformation extent is also changing along the contact arc. As a consequence, the mean total deformation extent is used to calculate the mean deformation resistance (Zou [22]). Consider (24)
ε
∑
m
=
b
0
ε
0
+
b
1
ε
1
,where ε
0 is the total deformation extent at the entry of this mill stand, ε
0
=
H
-
h
0
/
H. ε
1 is the total deformation extent at the exit of this mill stand, ε
1
=
H
-
h
1
/
H. H is the initial strip thickness entering the first mill stand in a tandem mill configuration. b
0 and b
1 are coefficients, usually taking b
0
=
0.4 and b
1
=
0.6.For cold rolling, the following equation can be used to calculate the mean deformation resistance:(25)
K
=
A
·
ε
∑
m
n
,where A and n are the coefficients associated with the carbon content of the strip.


#### 3.1.7. Forces Acting on the Rolls and the Rolling Torque

As shown in Figure4, forces acting on the rolls in x and y directions can be expressed as follows:(26)
F
x
=
B
∫
δ
α
p
x
sin
⁡
α
x
R
′
d
α
x
+
B
∫
δ
γ
μ
m
p
x
cos
⁡
α
x
R
′
d
α
x
-
∫
γ
α
μ
m
p
x
cos
⁡
α
x
R
′
d
α
x
=
T
1
-
T
0
2
=
σ
1
h
1
B
-
σ
0
h
0
B
2
,
(27)
F
y
=
B
∫
δ
α
p
x
cos
⁡
α
x
R
′
d
α
x
-
B
∫
δ
γ
μ
m
p
x
sin
⁡
α
x
R
′
d
α
x
-
∫
γ
α
μ
m
p
x
sin
⁡
α
x
R
′
d
α
x
=
B
R
′
p
x
sin
⁡
α
-
sin
⁡
δ
-
B
R
′
μ
m
p
x
cos
⁡
α
+
cos
⁡
δ
-
2
cos
⁡
γ
=
P
-
B
R
′
μ
m
p
x
cos
⁡
α
+
cos
⁡
δ
-
2
cos
⁡
γ
.As the anglesα, γ, and δ are very small, the second term on the right hand side of (27) can be negligible; that is,(28)
F
y
=
P
=
K
-
σ
m
B
Q
p
l
.The rolling torque is expressed as follows:(29)
M
=
B
∫
δ
γ
μ
m
p
x
R
R
′
d
α
x
-
∫
γ
α
μ
m
p
x
R
R
′
d
α
x
=
μ
m
p
x
B
R
R
′
2
γ
-
α
-
δ
=
-
α
+
δ
-
2
γ
α
-
δ
μ
m
P
R
.


### 3.2. Linearization of the Dynamic Rolling Process Model

Although the theoretical model of the dynamic rolling process has been established in the previous section, it is not easy to study the rolling vibration analytically because of its nonlinear nature. Therefore, a linearization process is done by using a first-order Taylor series approximation for all the factors of interest, and then the resulting variations are used to formulate a linearized dynamic rolling process model.In this model, the variables likeσ
0, σ
1, h
0, y
c, y
˙
c, x
c, x
˙
c, and θ
˙
M are defined as inputs; therefore, the variation of any linearized model output can then be expressed as follows:(30)
Q
i
v
a
r
=
∂
Q
i
∂
σ
0
s
σ
0
v
a
r
+
∂
Q
i
∂
σ
1
s
σ
1
v
a
r
+
∂
Q
i
∂
h
0
s
h
0
v
a
r
+
∂
Q
i
∂
y
c
s
y
c
+
∂
Q
i
∂
y
˙
c
s
y
˙
c
+
∂
Q
i
∂
θ
˙
M
s
θ
˙
M
+
∂
Q
i
∂
x
c
s
x
c
+
∂
Q
i
∂
x
˙
c
s
x
˙
c
,where the subscript “s” designates that the corresponding derivative is evaluated at steady-state conditions.


#### 3.2.1. Variations of Strip Entry and Exit Positions

From (6), the variation of strip entry position can be expressed as follows:(31)
x
0
v
a
r
=
∂
x
0
∂
h
0
s
h
0
v
a
r
+
∂
x
0
∂
h
c
s
h
c
v
a
r
+
∂
x
0
∂
x
c
s
x
c
=
∂
x
0
∂
h
0
s
h
0
v
a
r
+
2
∂
x
0
∂
h
c
s
y
c
+
∂
x
0
∂
x
c
s
x
c
.Substitutingv
0 from (18) into (12), the strip exit position is written as follows:(32)
x
1
=
x
c
-
R
′
h
c
y
˙
c
v
r
-
x
˙
c
h
c
+
x
n
-
x
c
2
/
R
′
+
2
y
˙
c
x
n
-
x
c
.Therefore, the variation of strip exit position is expressed as follows:(33)
x
1
v
a
r
=
2
∂
x
1
∂
h
c
s
y
c
+
∂
x
1
∂
y
˙
c
s
y
˙
c
+
∂
x
1
∂
v
r
s
R
′
θ
˙
M
+
∂
x
1
∂
x
n
s
x
n
v
a
r
+
∂
x
1
∂
x
c
s
x
c
+
δ
x
1
δ
x
˙
c
s
x
˙
c
.


#### 3.2.2. Variation of Neutral Point Position

SubstitutingP from (20) into (16), the variation of neutral point position can then be calculated as follows:(34)
x
n
v
a
r
=
∂
x
n
∂
σ
0
s
σ
0
v
a
r
+
∂
x
n
∂
σ
1
s
σ
1
v
a
r
+
∂
x
n
∂
h
0
s
h
0
v
a
r
+
∂
x
n
∂
h
1
s
h
1
v
a
r
+
∂
x
n
∂
x
c
s
x
c
+
∂
x
n
∂
α
s
α
v
a
r
+
∂
x
n
∂
δ
s
δ
v
a
r
+
∂
x
n
∂
K
s
K
v
a
r
+
∂
x
n
∂
σ
m
s
σ
m
v
a
r
+
∂
x
n
∂
Q
p
s
Q
p
v
a
r
.It can be seen from (34) that some of the items are implicit expressions of independent variables, and they must be converted to explicit expressions. Substituting x
1 from (32) into (3), the strip thickness at exit is written as follows:(35)
h
1
=
h
c
+
R
′
h
c
2
y
˙
c
2
v
r
-
x
˙
c
h
c
+
x
n
-
x
c
2
/
R
′
+
2
y
˙
c
x
n
-
x
c
2
.Then the variation of strip thickness at exit is expressed as follows:(36)
h
1
v
a
r
=
2
∂
h
1
∂
h
c
s
y
c
+
∂
h
1
∂
y
˙
c
s
y
˙
c
+
∂
h
1
∂
v
r
s
R
′
θ
˙
M
+
∂
h
1
∂
x
n
s
x
n
v
a
r
+
∂
h
1
∂
x
c
s
x
c
+
δ
h
1
δ
x
˙
c
s
x
˙
c
.As the inlet angle and the exit angle are small in cold rolling, approximately takingα
≈
sin
⁡
α
=
h
0
-
h
c
/
R
′ and δ
≈
sin
⁡
δ
=
x
c
-
x
1
/
R
′, hence their first-order Taylor series approximations are expressed as follows:(37)
α
v
a
r
=
∂
α
∂
h
0
s
h
0
v
a
r
+
2
∂
α
∂
h
c
s
y
c
,
(38)
δ
v
a
r
=
∂
δ
∂
x
c
s
x
c
+
∂
δ
∂
x
1
s
x
1
v
a
r
.And the variations of the mean deformation resistance, the mean tension, and the stress state factor are formulated as follows:(39)
K
v
a
r
=
∂
K
∂
h
0
s
h
0
v
a
r
+
∂
K
∂
h
1
s
h
1
v
a
r
,
(40)
σ
m
v
a
r
=
∂
σ
m
∂
σ
0
s
σ
0
v
a
r
+
∂
σ
m
∂
σ
1
s
σ
1
v
a
r
=
0.7
σ
0
v
a
r
+
0.3
σ
1
v
a
r
,
(41)
Q
p
v
a
r
=
∂
Q
p
∂
h
0
s
h
0
v
a
r
+
∂
Q
p
∂
h
1
s
h
1
v
a
r
.Substituting (36)–(41) into (34), the resulting equation for the variation of neutral point position becomes(42)
x
n
v
a
r
=
a
x
n
,
σ
0
σ
0
v
a
r
+
a
x
n
,
σ
1
σ
1
v
a
r
+
a
x
n
,
h
0
h
0
v
a
r
+
a
x
n
,
y
c
y
c
+
a
x
n
,
y
˙
c
y
˙
c
+
a
x
n
,
x
c
x
c
.


#### 3.2.3. Variations of Strip Velocities at Entry and Exit

From (18) and (19), the variations of strip velocities at entry and exit are expressed as follows:(43)
v
0
v
a
r
=
∂
v
0
∂
h
0
s
h
0
v
a
r
+
2
∂
v
0
∂
h
c
s
y
c
+
∂
v
0
∂
y
˙
c
s
y
˙
c
+
∂
v
0
∂
x
c
s
x
c
+
∂
v
0
∂
x
˙
c
s
x
˙
c
+
∂
v
0
∂
v
r
s
R
′
θ
˙
M
+
∂
v
0
∂
x
n
s
x
n
v
a
r
+
∂
v
0
∂
x
0
s
x
0
v
a
r
,
(44)
v
1
v
a
r
=
∂
v
1
∂
h
0
s
h
0
v
a
r
+
2
∂
v
1
∂
h
c
s
y
c
+
∂
v
1
∂
y
˙
c
s
y
˙
c
+
∂
v
1
∂
x
c
s
x
c
+
∂
v
1
∂
x
˙
c
s
x
˙
c
+
∂
v
1
∂
v
0
s
v
0
v
a
r
+
∂
v
1
∂
x
1
s
x
1
v
a
r
+
∂
v
1
∂
x
0
s
x
0
v
a
r
.Therefore, the explicit expressions of (43) and (44) can be written as follows:(45)
v
0
v
a
r
=
a
v
0
,
σ
0
σ
0
v
a
r
+
a
v
0
,
σ
1
σ
1
v
a
r
+
a
v
0
,
h
0
h
0
v
a
r
+
a
v
0
,
y
c
y
c
+
a
v
0
,
y
˙
c
y
˙
c
+
a
v
0
,
x
c
x
c
+
a
v
0
,
x
˙
c
x
˙
c
+
a
v
0
,
θ
˙
M
θ
˙
M
,
(46)
v
1
v
a
r
=
a
v
1
,
σ
0
σ
0
v
a
r
+
a
v
1
,
σ
1
σ
1
v
a
r
+
a
v
1
,
h
0
h
0
v
a
r
+
a
v
1
,
y
c
y
c
+
a
v
1
,
y
˙
c
y
˙
c
+
a
v
1
,
x
c
x
c
+
a
v
1
,
x
˙
c
x
˙
c
+
a
v
1
,
θ
˙
M
θ
˙
M
.


#### 3.2.4. Variations of Forces Acting on the Rolls and the Rolling Torque

Applying the first-order Taylor series approximation to (26), the variation of the force acting on the rolls in x direction is expressed as follows:(47)
F
x
v
a
r
=
∂
F
x
∂
σ
0
s
σ
0
v
a
r
+
∂
F
x
∂
σ
1
s
σ
1
v
a
r
+
∂
F
x
∂
h
0
s
h
0
v
a
r
+
∂
F
x
∂
h
1
s
h
1
v
a
r
=
a
F
x
,
σ
0
σ
0
v
a
r
+
a
F
x
,
σ
1
σ
1
v
a
r
+
a
F
x
,
h
0
h
0
v
a
r
+
a
F
x
,
y
c
y
c
.Likewise, as the variation of horizontal projection length of the contact arc isl
v
a
r
=
x
1
v
a
r
-
x
0
v
a
r and considering (33) and (39)–(42), the variation of the force acting on the rolls in y direction can be derived from (28):(48)
F
y
v
a
r
=
∂
F
y
∂
K
s
K
v
a
r
+
∂
F
y
∂
σ
m
s
σ
m
v
a
r
+
∂
F
y
∂
Q
p
s
Q
p
v
a
r
+
∂
F
y
∂
l
s
l
v
a
r
=
a
F
y
,
σ
0
σ
0
v
a
r
+
a
F
y
,
σ
1
σ
1
v
a
r
+
a
F
y
,
h
0
h
0
v
a
r
+
a
F
y
,
y
c
y
c
+
a
F
y
,
y
˙
c
y
˙
c
+
a
F
y
,
x
c
x
c
.As the neutral angle is very small, approximately takingγ
≈
sin
⁡
γ
=
x
c
-
x
n
/
R
′, hence its first-order Taylor series approximation is expressed as follows:(49)
γ
v
a
r
=
∂
γ
∂
x
c
s
x
c
+
∂
γ
∂
x
n
s
x
n
v
a
r
.Therefore, the variation of the rolling torque can then be calculated from (29):(50)
M
v
a
r
=
∂
M
∂
α
s
α
v
a
r
+
∂
M
∂
γ
s
γ
v
a
r
+
∂
M
∂
δ
s
δ
v
a
r
+
∂
M
∂
K
s
K
v
a
r
+
∂
M
∂
σ
m
s
σ
m
v
a
r
+
∂
M
∂
Q
p
s
Q
p
v
a
r
+
∂
M
∂
l
s
l
v
a
r
=
a
M
,
σ
0
σ
0
v
a
r
+
a
M
,
σ
1
σ
1
v
a
r
+
a
M
,
h
0
h
0
v
a
r
+
a
M
,
y
c
y
c
+
a
M
,
y
˙
c
y
˙
c
+
a
M
,
x
c
x
c
.All the coefficients of the linearized parameters in this section are given in AppendixA.


### 3.3. Mathematical Model of the Rolling Vibration

As equations in (22) have the integral terms of time, we take them as a derivative with respect to time. Thus they can also be written as follows:(51)
d
σ
0
v
a
r
d
t
=
E
1
L
0
v
0
-
v
-
0
=
E
1
L
0
v
0
v
a
r
,
d
σ
1
v
a
r
d
t
=
-
E
1
L
1
v
1
-
v
-
1
=
-
E
1
L
1
v
1
v
a
r
.In Section3.1, F
x
v
a
r
=
F
x
-
F
-
x; F
y
v
a
r
=
F
y
-
F
-
y; M
v
a
r
=
M
-
M
-; therefore, the mathematical model of rolling vibration based on the theoretical model of the dynamic rolling process is expressed as follows:(52)
m
1
x
¨
c
+
c
1
x
˙
c
+
k
1
x
c
=
F
x
-
F
-
x
,
m
2
y
¨
c
+
c
2
y
˙
c
+
k
2
y
c
=
F
y
-
F
-
y
,
J
M
θ
¨
M
+
c
t
θ
˙
M
+
k
t
θ
M
=
M
-
M
-
,
σ
˙
0
v
a
r
=
E
1
v
0
-
v
-
0
L
0
,
σ
˙
1
v
a
r
=
-
E
1
v
1
-
v
-
1
L
1
.And the mathematical model of rolling vibration based on the linearized dynamic rolling process model is expressed as a first-order differential equation in matrix form:(53)
d
X
d
t
=
D
1
X
+
D
2
h
0
v
a
r
,where(54)
X
=
x
c
x
˙
c
y
c
y
˙
c
θ
M
θ
˙
M
σ
0
v
a
r
σ
1
v
a
r
T
,
(55)
D
1
=
0
1
0
0
0
0
0
0
-
k
1
m
1
-
c
1
m
1
a
F
x
,
y
c
m
1
0
0
0
a
F
x
,
σ
0
m
1
a
F
x
,
σ
1
m
1
0
0
0
1
0
0
0
0
a
F
y
,
x
c
m
2
0
a
F
y
,
y
c
-
k
2
m
2
a
F
y
,
y
˙
c
-
c
2
m
2
0
0
a
F
y
,
σ
0
m
2
a
F
y
,
σ
1
m
2
0
0
0
0
0
1
0
0
a
M
,
x
c
J
M
0
a
M
,
y
c
J
M
a
M
,
y
˙
c
J
M
-
k
t
J
M
-
c
t
J
M
a
M
,
σ
0
J
M
a
M
,
σ
1
J
M
E
1
a
v
0
,
x
c
L
0
E
1
a
v
0
,
x
˙
c
L
0
E
1
a
v
0
,
y
c
L
0
E
1
a
v
0
,
y
˙
c
L
0
0
E
1
a
v
0
,
θ
˙
M
L
0
E
1
a
v
0
,
σ
0
L
0
E
1
a
v
0
,
σ
1
L
0
-
E
1
a
v
1
,
x
c
L
1
-
E
1
a
v
1
,
x
˙
c
L
1
-
E
1
a
v
1
,
y
c
L
1
-
E
1
a
v
1
,
y
˙
c
L
1
0
-
E
1
a
v
1
,
θ
˙
M
L
1
-
E
1
a
v
1
,
σ
0
L
1
-
E
1
a
v
1
,
σ
1
L
1
,
(56)
D
2
=
0
a
F
x
,
h
0
m
1
0
a
F
y
,
h
0
m
2
0
a
M
,
h
0
J
M
E
1
a
v
0
,
h
0
L
0
-
E
1
a
v
1
,
h
0
L
1
T
.


## 3.1. Theoretical Model of Dynamic Rolling Process

### 3.1.1. Roll Bite Geometry

The geometry of the roll bite is schematically presented in Figure3, where O
-
1 and O
-
2 are the original geometrical centers of the top and bottom rolls (before vibrations begin). O
1 and O
2 are the ever-changing centers of the rolls. x
c is the distance from the centerline O
1
O
2 to the centerline O
-
1
O
-
2. h
0 and h
1 are the thicknesses of the strip at entry and exit. h
c is the roll gap spacing measured along the centerline O
1
O
2. v
0 and v
1 are the velocities of the strip at entry and exit. v
r is the roll peripheral velocity. x
0, x
n, and x
1 are the entry position, neutral point position, and exit position measured from the centerline O
-
1
O
-
2. σ
0 and σ
1 are the tensions of the strip at entry and exit. α, γ, and δ are the inlet angle, the neutral angle, and the exit angle. As shown in Figure 3, the rolls of the top and the bottom can vibrate in x, y, and rotation directions symmetrically. In addition, it is assumed that the width of the strip always keeps as B during rolling.Figure 3
Roll bite geometry during vibration.As shown in Figure3, the dynamic roll gap spacing can be written as follows:(2)
h
c
=
h
-
c
+
2
y
c
,where the parameters with superscript like “i
-” represent the value of parameter “i” under steady state.In the rolling deformation zone, the roll surface was usually treated as a parabolic curve, so the strip thickness at any arbitrary position is written as follows:(3)
h
x
=
h
c
+
x
-
x
c
2
R
′
.During the vibration of rolls, the volume of the roll bite changes simultaneously. Considering the control volume of the material flow within the roll bite, Hu et al. [4] modified the metal flow equation and expressed it as follows:(4)
v
h
=
v
0
h
0
-
2
y
˙
c
x
-
x
0
-
x
˙
c
h
0
-
h
,where v and h are the strip velocity and thickness of any cross section in the roll bite. The second and third terms on the right hand side of (4) represent the material flow fluctuations of any cross section caused by the vibration of rolls in y and x directions.For the cold rolling, the rolling pressure is particularly high, so the work roll will be elastically flattened obviously. Here, the Hitchcock equation is used to calculate the effective flattened radius of the work roll:(5)
R
′
=
R
1
+
2
c
Δ
h
P
B
,where Δ
h is the reduction, Δ
h
=
h
0
-
h
1 and c
=
8
(
1
-
u
2
2
)
/
π
E
2. E
2 and u
2 are Young’s modulus and Poisson’s ratio of the roll.As the function relationship between the flattened work roll radius and the rolling force, it is very difficult to decouple them. In order to simplify the analysis, the original radius is replaced by the effective radius, which is calculated iteratively during computations of the static rolling process. In addition, it is assumed that the effective radius of the work roll keeps unchanged along the contact arc between the work roll and the strip. These measures and assumptions not only can help to improve the accuracy of calculations but also can obtain a solvable analytical model.


### 3.1.2. Strip Entry Position and Exit Position

The entry position of the strip can be derived from (3) as follows:(6)
x
0
=
x
c
-
R
′
h
0
-
h
c
.From the geometric relationship shown in Figure3, the inlet angle is expressed as follows:(7)
α
=
arcsin
⁡
x
c
-
x
0
R
′
=
arcsin
⁡
h
0
-
h
c
R
′
.Substitutingh
(
x
) from (3) into (4), the horizontal velocity of the strip at any vertical cross-section plane within the roll bite can be expressed as follows:(8)
v
x
=
v
0
h
0
-
2
y
˙
c
x
-
x
0
-
x
˙
c
h
0
-
h
x
h
c
+
x
-
x
c
2
/
R
′
.The strain rate of the strip at any vertical cross-section plane within the roll bite can be derived from (3):(9)
d
h
d
t
=
∂
h
∂
t
+
∂
h
∂
x
∂
x
∂
t
+
∂
h
∂
x
c
∂
x
c
∂
t
=
∂
h
c
∂
t
+
2
x
-
x
c
R
′
v
x
-
x
˙
c
.The exit position of the strip can be determined by examining the condition of zero strain rates (Zhao and Ehmann [5]), so(10)
∂
h
c
∂
t
+
2
x
1
-
x
c
R
′
v
1
-
x
˙
c
=
0
.Substituting (2) and (8) into (10) and rewriting it,(11)
y
˙
c
x
1
-
x
c
2
-
v
0
h
0
-
2
y
˙
c
x
c
-
x
0
-
x
˙
c
h
0
x
1
-
x
c
-
R
′
h
c
y
˙
c
=
0
.Neglecting the quadratic termx
1
-
x
c
2, then the exit position of the strip can be represented as follows:(12)
x
1
=
x
c
-
R
′
h
c
y
˙
c
v
0
h
0
-
2
y
˙
c
x
c
-
x
0
-
x
˙
c
h
0
.It can be seen from the above equation that the exit plane no longer coincides with the centerline of the rolls as the occurrence of vibration, so the exit angle can be expressed as follows:(13)
δ
=
arcsin
⁡
x
c
-
x
1
R
′
.


### 3.1.3. Friction Model

The change of the friction along the contact arc is very complicated. It neither obeys the dry friction law nor obeys the adhesive friction theory [20]. Since the resultant force on the rolling interface is the focus in the study of mill vibration, the selected friction model should be simple, accurate, and solvable.In cold rolling, vibration often happens on the last few stands, and friction coefficients for these stands are usually very small. In this case, the friction force is far lower than the shear strength of the strip material, which means the friction state is sliding friction and there is no sticking point in the deformation zone. Therefore, Coulomb friction model can be used in the study of vibration in cold rolling. In order to simplify the analysis, we assume that the friction coefficients along the contact arc keep unchanged and are represented asμ
m.


### 3.1.4. Neutral Point Position

In general, there is a point on the contact arc of the rolling deformation zone, named neutral point, where the peripheral velocity of the work roll and the velocity of the strip are equal. The neutral point is one of the most important variables in the rolling process. Along the contact arc, the friction changes direction on different sides of this point.As shown in Figure4, the neutral angle can be determined by the equilibrium condition of the force system in the rolling deformation zone:(14)
-
∫
δ
α
p
x
sin
⁡
α
x
R
′
d
α
x
+
∫
γ
α
μ
m
p
x
cos
⁡
α
x
R
′
d
α
x
-
∫
δ
γ
μ
m
p
x
cos
⁡
α
x
R
′
d
α
x
+
T
1
-
T
0
2
B
=
0
,where T
1 and T
2 are tensile forces of the strip at entry and exit, T
1
=
σ
1
h
1
B, and T
0
=
σ
0
h
0
B. α
x is the central angle corresponding to any arbitrary vertical cross section. p
x is the radial unit pressure, and we assume that p
x along the contact arc is uniform distribution. So it can be approximated that p
x is equal to the mean unit pressure p
m, p
x
≈
p
m
=
P
/
B
l, and l is the horizontal projection length of the contact arc, l
=
x
1
-
x
0.Figure 4
Equilibrium of force system in the roll bite.From (14), the neutral angle can be formulated as follows:(15)
sin
⁡
γ
=
sin
⁡
α
+
sin
⁡
δ
2
+
cos
⁡
α
-
cos
⁡
δ
2
μ
m
+
T
1
-
T
0
l
4
μ
m
P
R
′
.From the geometric relationship shown in Figure3, the position of the neutral point is expressed as follows:(16)
x
n
=
x
c
-
R
′
sin
⁡
γ
=
x
c
-
R
′
2
sin
⁡
α
+
sin
⁡
δ
-
R
′
2
μ
m
cos
⁡
α
-
cos
⁡
δ
-
T
1
-
T
0
l
4
μ
m
P
.


### 3.1.5. Strip Velocities at Entry and Exit

Once the position of the neutral point is obtained, the strip velocity at any arbitrary vertical cross section of the roll bite can be also determined. As the strip velocity at the neutral point equals the roll peripheral velocity, substitutingx
n from (16) into (8), the strip velocity at the neutral point can be expressed as follows:(17)
v
r
=
v
0
h
0
-
2
y
˙
c
x
n
-
x
0
-
x
˙
c
h
0
-
h
n
h
c
+
x
n
-
x
c
2
/
R
′
.Thus, the strip velocities at entry and exit are written as follows:(18)
v
0
=
1
h
0
v
r
-
x
˙
c
h
c
+
x
n
-
x
c
2
R
′
+
2
y
˙
c
x
n
-
x
0
+
h
0
x
˙
c
,
(19)
v
1
=
v
0
h
0
-
2
y
˙
c
x
1
-
x
0
-
x
˙
c
h
0
-
h
c
-
x
1
-
x
c
2
/
R
′
h
c
+
x
1
-
x
c
2
/
R
′
.


### 3.1.6. Calculation of Rolling Force

The rolling force is one of the most important energetic parameters of the rolling mill. Specialists and scholars have done a lot of research on the rolling force and also achieved a variety of rolling force models for different mills or working conditions. Reference [21] indicated that most of the cold rolling mills can meet the conditions of Bland-Ford-Hill rolling force model. The form of Hill simplified formula is as follows:(20)
P
=
K
-
σ
m
B
Q
p
l
,where σ
m is the mean tension, usually taking σ
m
=
0.7
σ
0
+
0.3
σ
1. Q
p is the stress state factor. K is the mean deformation resistance.The stress state factor of Hill simplified formula is expressed as follows:(21)
Q
p
=
1.08
+
1.79
ε
μ
R
′
h
0
-
1.02
ε
,where ε is the reduction ratio, expressed as ε
=
h
0
-
h
1
/
h
0. μ is the friction coefficient along the contact arc, taking μ
=
μ
m.In the rolling mill, the strip tensions at entry and exit have a significant effect on the stability of the system. And the variations of strip tensions have a direct relationship with variations of strip velocities. As the vibration characteristic of a single stand mill is the focus of this paper, supposing both the variation of the exit velocity in the upstream stand and the variation of the entry velocity in the downstream stand are zero, tension variations at entry and exit can be expressed according to Hooke’s law as follows:(22)
σ
0
v
a
r
=
E
1
L
0
Δ
L
0
=
E
1
L
0
∫
0
t
v
0
-
v
-
0
d
t
,
σ
1
v
a
r
=
E
1
L
1
Δ
L
1
=
-
E
1
L
1
∫
0
t
v
1
-
v
-
1
d
t
,where E
1 is Young’s modulus of the strip. L
0 and L
1 are the distances from this stand to the upstream stand and the downstream stand.Hence, the dynamic tensions of the strip at entry and exit can be written as follows:(23)
σ
0
=
σ
-
0
+
σ
0
v
a
r
;
σ
1
=
σ
-
1
+
σ
1
v
a
r
.In addition, the strain hardening effect must be considered when calculating the deformation resistance of the cold-rolled strip. Because of the strain hardening effect, the deformation resistance of each rolling pass is not only related to the deformation extent of this rolling pass but also associated with the total deformation extent of rolling passes before. Furthermore, the deformation extent is also changing along the contact arc. As a consequence, the mean total deformation extent is used to calculate the mean deformation resistance (Zou [22]). Consider (24)
ε
∑
m
=
b
0
ε
0
+
b
1
ε
1
,where ε
0 is the total deformation extent at the entry of this mill stand, ε
0
=
H
-
h
0
/
H. ε
1 is the total deformation extent at the exit of this mill stand, ε
1
=
H
-
h
1
/
H. H is the initial strip thickness entering the first mill stand in a tandem mill configuration. b
0 and b
1 are coefficients, usually taking b
0
=
0.4 and b
1
=
0.6.For cold rolling, the following equation can be used to calculate the mean deformation resistance:(25)
K
=
A
·
ε
∑
m
n
,where A and n are the coefficients associated with the carbon content of the strip.


### 3.1.7. Forces Acting on the Rolls and the Rolling Torque

As shown in Figure4, forces acting on the rolls in x and y directions can be expressed as follows:(26)
F
x
=
B
∫
δ
α
p
x
sin
⁡
α
x
R
′
d
α
x
+
B
∫
δ
γ
μ
m
p
x
cos
⁡
α
x
R
′
d
α
x
-
∫
γ
α
μ
m
p
x
cos
⁡
α
x
R
′
d
α
x
=
T
1
-
T
0
2
=
σ
1
h
1
B
-
σ
0
h
0
B
2
,
(27)
F
y
=
B
∫
δ
α
p
x
cos
⁡
α
x
R
′
d
α
x
-
B
∫
δ
γ
μ
m
p
x
sin
⁡
α
x
R
′
d
α
x
-
∫
γ
α
μ
m
p
x
sin
⁡
α
x
R
′
d
α
x
=
B
R
′
p
x
sin
⁡
α
-
sin
⁡
δ
-
B
R
′
μ
m
p
x
cos
⁡
α
+
cos
⁡
δ
-
2
cos
⁡
γ
=
P
-
B
R
′
μ
m
p
x
cos
⁡
α
+
cos
⁡
δ
-
2
cos
⁡
γ
.As the anglesα, γ, and δ are very small, the second term on the right hand side of (27) can be negligible; that is,(28)
F
y
=
P
=
K
-
σ
m
B
Q
p
l
.The rolling torque is expressed as follows:(29)
M
=
B
∫
δ
γ
μ
m
p
x
R
R
′
d
α
x
-
∫
γ
α
μ
m
p
x
R
R
′
d
α
x
=
μ
m
p
x
B
R
R
′
2
γ
-
α
-
δ
=
-
α
+
δ
-
2
γ
α
-
δ
μ
m
P
R
.


## 3.1.1. Roll Bite Geometry

The geometry of the roll bite is schematically presented in Figure3, where O
-
1 and O
-
2 are the original geometrical centers of the top and bottom rolls (before vibrations begin). O
1 and O
2 are the ever-changing centers of the rolls. x
c is the distance from the centerline O
1
O
2 to the centerline O
-
1
O
-
2. h
0 and h
1 are the thicknesses of the strip at entry and exit. h
c is the roll gap spacing measured along the centerline O
1
O
2. v
0 and v
1 are the velocities of the strip at entry and exit. v
r is the roll peripheral velocity. x
0, x
n, and x
1 are the entry position, neutral point position, and exit position measured from the centerline O
-
1
O
-
2. σ
0 and σ
1 are the tensions of the strip at entry and exit. α, γ, and δ are the inlet angle, the neutral angle, and the exit angle. As shown in Figure 3, the rolls of the top and the bottom can vibrate in x, y, and rotation directions symmetrically. In addition, it is assumed that the width of the strip always keeps as B during rolling.Figure 3
Roll bite geometry during vibration.As shown in Figure3, the dynamic roll gap spacing can be written as follows:(2)
h
c
=
h
-
c
+
2
y
c
,where the parameters with superscript like “i
-” represent the value of parameter “i” under steady state.In the rolling deformation zone, the roll surface was usually treated as a parabolic curve, so the strip thickness at any arbitrary position is written as follows:(3)
h
x
=
h
c
+
x
-
x
c
2
R
′
.During the vibration of rolls, the volume of the roll bite changes simultaneously. Considering the control volume of the material flow within the roll bite, Hu et al. [4] modified the metal flow equation and expressed it as follows:(4)
v
h
=
v
0
h
0
-
2
y
˙
c
x
-
x
0
-
x
˙
c
h
0
-
h
,where v and h are the strip velocity and thickness of any cross section in the roll bite. The second and third terms on the right hand side of (4) represent the material flow fluctuations of any cross section caused by the vibration of rolls in y and x directions.For the cold rolling, the rolling pressure is particularly high, so the work roll will be elastically flattened obviously. Here, the Hitchcock equation is used to calculate the effective flattened radius of the work roll:(5)
R
′
=
R
1
+
2
c
Δ
h
P
B
,where Δ
h is the reduction, Δ
h
=
h
0
-
h
1 and c
=
8
(
1
-
u
2
2
)
/
π
E
2. E
2 and u
2 are Young’s modulus and Poisson’s ratio of the roll.As the function relationship between the flattened work roll radius and the rolling force, it is very difficult to decouple them. In order to simplify the analysis, the original radius is replaced by the effective radius, which is calculated iteratively during computations of the static rolling process. In addition, it is assumed that the effective radius of the work roll keeps unchanged along the contact arc between the work roll and the strip. These measures and assumptions not only can help to improve the accuracy of calculations but also can obtain a solvable analytical model.


## 3.1.2. Strip Entry Position and Exit Position

The entry position of the strip can be derived from (3) as follows:(6)
x
0
=
x
c
-
R
′
h
0
-
h
c
.From the geometric relationship shown in Figure3, the inlet angle is expressed as follows:(7)
α
=
arcsin
⁡
x
c
-
x
0
R
′
=
arcsin
⁡
h
0
-
h
c
R
′
.Substitutingh
(
x
) from (3) into (4), the horizontal velocity of the strip at any vertical cross-section plane within the roll bite can be expressed as follows:(8)
v
x
=
v
0
h
0
-
2
y
˙
c
x
-
x
0
-
x
˙
c
h
0
-
h
x
h
c
+
x
-
x
c
2
/
R
′
.The strain rate of the strip at any vertical cross-section plane within the roll bite can be derived from (3):(9)
d
h
d
t
=
∂
h
∂
t
+
∂
h
∂
x
∂
x
∂
t
+
∂
h
∂
x
c
∂
x
c
∂
t
=
∂
h
c
∂
t
+
2
x
-
x
c
R
′
v
x
-
x
˙
c
.The exit position of the strip can be determined by examining the condition of zero strain rates (Zhao and Ehmann [5]), so(10)
∂
h
c
∂
t
+
2
x
1
-
x
c
R
′
v
1
-
x
˙
c
=
0
.Substituting (2) and (8) into (10) and rewriting it,(11)
y
˙
c
x
1
-
x
c
2
-
v
0
h
0
-
2
y
˙
c
x
c
-
x
0
-
x
˙
c
h
0
x
1
-
x
c
-
R
′
h
c
y
˙
c
=
0
.Neglecting the quadratic termx
1
-
x
c
2, then the exit position of the strip can be represented as follows:(12)
x
1
=
x
c
-
R
′
h
c
y
˙
c
v
0
h
0
-
2
y
˙
c
x
c
-
x
0
-
x
˙
c
h
0
.It can be seen from the above equation that the exit plane no longer coincides with the centerline of the rolls as the occurrence of vibration, so the exit angle can be expressed as follows:(13)
δ
=
arcsin
⁡
x
c
-
x
1
R
′
.


## 3.1.3. Friction Model

The change of the friction along the contact arc is very complicated. It neither obeys the dry friction law nor obeys the adhesive friction theory [20]. Since the resultant force on the rolling interface is the focus in the study of mill vibration, the selected friction model should be simple, accurate, and solvable.In cold rolling, vibration often happens on the last few stands, and friction coefficients for these stands are usually very small. In this case, the friction force is far lower than the shear strength of the strip material, which means the friction state is sliding friction and there is no sticking point in the deformation zone. Therefore, Coulomb friction model can be used in the study of vibration in cold rolling. In order to simplify the analysis, we assume that the friction coefficients along the contact arc keep unchanged and are represented asμ
m.


## 3.1.4. Neutral Point Position

In general, there is a point on the contact arc of the rolling deformation zone, named neutral point, where the peripheral velocity of the work roll and the velocity of the strip are equal. The neutral point is one of the most important variables in the rolling process. Along the contact arc, the friction changes direction on different sides of this point.As shown in Figure4, the neutral angle can be determined by the equilibrium condition of the force system in the rolling deformation zone:(14)
-
∫
δ
α
p
x
sin
⁡
α
x
R
′
d
α
x
+
∫
γ
α
μ
m
p
x
cos
⁡
α
x
R
′
d
α
x
-
∫
δ
γ
μ
m
p
x
cos
⁡
α
x
R
′
d
α
x
+
T
1
-
T
0
2
B
=
0
,where T
1 and T
2 are tensile forces of the strip at entry and exit, T
1
=
σ
1
h
1
B, and T
0
=
σ
0
h
0
B. α
x is the central angle corresponding to any arbitrary vertical cross section. p
x is the radial unit pressure, and we assume that p
x along the contact arc is uniform distribution. So it can be approximated that p
x is equal to the mean unit pressure p
m, p
x
≈
p
m
=
P
/
B
l, and l is the horizontal projection length of the contact arc, l
=
x
1
-
x
0.Figure 4
Equilibrium of force system in the roll bite.From (14), the neutral angle can be formulated as follows:(15)
sin
⁡
γ
=
sin
⁡
α
+
sin
⁡
δ
2
+
cos
⁡
α
-
cos
⁡
δ
2
μ
m
+
T
1
-
T
0
l
4
μ
m
P
R
′
.From the geometric relationship shown in Figure3, the position of the neutral point is expressed as follows:(16)
x
n
=
x
c
-
R
′
sin
⁡
γ
=
x
c
-
R
′
2
sin
⁡
α
+
sin
⁡
δ
-
R
′
2
μ
m
cos
⁡
α
-
cos
⁡
δ
-
T
1
-
T
0
l
4
μ
m
P
.


## 3.1.5. Strip Velocities at Entry and Exit

Once the position of the neutral point is obtained, the strip velocity at any arbitrary vertical cross section of the roll bite can be also determined. As the strip velocity at the neutral point equals the roll peripheral velocity, substitutingx
n from (16) into (8), the strip velocity at the neutral point can be expressed as follows:(17)
v
r
=
v
0
h
0
-
2
y
˙
c
x
n
-
x
0
-
x
˙
c
h
0
-
h
n
h
c
+
x
n
-
x
c
2
/
R
′
.Thus, the strip velocities at entry and exit are written as follows:(18)
v
0
=
1
h
0
v
r
-
x
˙
c
h
c
+
x
n
-
x
c
2
R
′
+
2
y
˙
c
x
n
-
x
0
+
h
0
x
˙
c
,
(19)
v
1
=
v
0
h
0
-
2
y
˙
c
x
1
-
x
0
-
x
˙
c
h
0
-
h
c
-
x
1
-
x
c
2
/
R
′
h
c
+
x
1
-
x
c
2
/
R
′
.


## 3.1.6. Calculation of Rolling Force

The rolling force is one of the most important energetic parameters of the rolling mill. Specialists and scholars have done a lot of research on the rolling force and also achieved a variety of rolling force models for different mills or working conditions. Reference [21] indicated that most of the cold rolling mills can meet the conditions of Bland-Ford-Hill rolling force model. The form of Hill simplified formula is as follows:(20)
P
=
K
-
σ
m
B
Q
p
l
,where σ
m is the mean tension, usually taking σ
m
=
0.7
σ
0
+
0.3
σ
1. Q
p is the stress state factor. K is the mean deformation resistance.The stress state factor of Hill simplified formula is expressed as follows:(21)
Q
p
=
1.08
+
1.79
ε
μ
R
′
h
0
-
1.02
ε
,where ε is the reduction ratio, expressed as ε
=
h
0
-
h
1
/
h
0. μ is the friction coefficient along the contact arc, taking μ
=
μ
m.In the rolling mill, the strip tensions at entry and exit have a significant effect on the stability of the system. And the variations of strip tensions have a direct relationship with variations of strip velocities. As the vibration characteristic of a single stand mill is the focus of this paper, supposing both the variation of the exit velocity in the upstream stand and the variation of the entry velocity in the downstream stand are zero, tension variations at entry and exit can be expressed according to Hooke’s law as follows:(22)
σ
0
v
a
r
=
E
1
L
0
Δ
L
0
=
E
1
L
0
∫
0
t
v
0
-
v
-
0
d
t
,
σ
1
v
a
r
=
E
1
L
1
Δ
L
1
=
-
E
1
L
1
∫
0
t
v
1
-
v
-
1
d
t
,where E
1 is Young’s modulus of the strip. L
0 and L
1 are the distances from this stand to the upstream stand and the downstream stand.Hence, the dynamic tensions of the strip at entry and exit can be written as follows:(23)
σ
0
=
σ
-
0
+
σ
0
v
a
r
;
σ
1
=
σ
-
1
+
σ
1
v
a
r
.In addition, the strain hardening effect must be considered when calculating the deformation resistance of the cold-rolled strip. Because of the strain hardening effect, the deformation resistance of each rolling pass is not only related to the deformation extent of this rolling pass but also associated with the total deformation extent of rolling passes before. Furthermore, the deformation extent is also changing along the contact arc. As a consequence, the mean total deformation extent is used to calculate the mean deformation resistance (Zou [22]). Consider (24)
ε
∑
m
=
b
0
ε
0
+
b
1
ε
1
,where ε
0 is the total deformation extent at the entry of this mill stand, ε
0
=
H
-
h
0
/
H. ε
1 is the total deformation extent at the exit of this mill stand, ε
1
=
H
-
h
1
/
H. H is the initial strip thickness entering the first mill stand in a tandem mill configuration. b
0 and b
1 are coefficients, usually taking b
0
=
0.4 and b
1
=
0.6.For cold rolling, the following equation can be used to calculate the mean deformation resistance:(25)
K
=
A
·
ε
∑
m
n
,where A and n are the coefficients associated with the carbon content of the strip.


## 3.1.7. Forces Acting on the Rolls and the Rolling Torque

As shown in Figure4, forces acting on the rolls in x and y directions can be expressed as follows:(26)
F
x
=
B
∫
δ
α
p
x
sin
⁡
α
x
R
′
d
α
x
+
B
∫
δ
γ
μ
m
p
x
cos
⁡
α
x
R
′
d
α
x
-
∫
γ
α
μ
m
p
x
cos
⁡
α
x
R
′
d
α
x
=
T
1
-
T
0
2
=
σ
1
h
1
B
-
σ
0
h
0
B
2
,
(27)
F
y
=
B
∫
δ
α
p
x
cos
⁡
α
x
R
′
d
α
x
-
B
∫
δ
γ
μ
m
p
x
sin
⁡
α
x
R
′
d
α
x
-
∫
γ
α
μ
m
p
x
sin
⁡
α
x
R
′
d
α
x
=
B
R
′
p
x
sin
⁡
α
-
sin
⁡
δ
-
B
R
′
μ
m
p
x
cos
⁡
α
+
cos
⁡
δ
-
2
cos
⁡
γ
=
P
-
B
R
′
μ
m
p
x
cos
⁡
α
+
cos
⁡
δ
-
2
cos
⁡
γ
.As the anglesα, γ, and δ are very small, the second term on the right hand side of (27) can be negligible; that is,(28)
F
y
=
P
=
K
-
σ
m
B
Q
p
l
.The rolling torque is expressed as follows:(29)
M
=
B
∫
δ
γ
μ
m
p
x
R
R
′
d
α
x
-
∫
γ
α
μ
m
p
x
R
R
′
d
α
x
=
μ
m
p
x
B
R
R
′
2
γ
-
α
-
δ
=
-
α
+
δ
-
2
γ
α
-
δ
μ
m
P
R
.


## 3.2. Linearization of the Dynamic Rolling Process Model

Although the theoretical model of the dynamic rolling process has been established in the previous section, it is not easy to study the rolling vibration analytically because of its nonlinear nature. Therefore, a linearization process is done by using a first-order Taylor series approximation for all the factors of interest, and then the resulting variations are used to formulate a linearized dynamic rolling process model.In this model, the variables likeσ
0, σ
1, h
0, y
c, y
˙
c, x
c, x
˙
c, and θ
˙
M are defined as inputs; therefore, the variation of any linearized model output can then be expressed as follows:(30)
Q
i
v
a
r
=
∂
Q
i
∂
σ
0
s
σ
0
v
a
r
+
∂
Q
i
∂
σ
1
s
σ
1
v
a
r
+
∂
Q
i
∂
h
0
s
h
0
v
a
r
+
∂
Q
i
∂
y
c
s
y
c
+
∂
Q
i
∂
y
˙
c
s
y
˙
c
+
∂
Q
i
∂
θ
˙
M
s
θ
˙
M
+
∂
Q
i
∂
x
c
s
x
c
+
∂
Q
i
∂
x
˙
c
s
x
˙
c
,where the subscript “s” designates that the corresponding derivative is evaluated at steady-state conditions.


### 3.2.1. Variations of Strip Entry and Exit Positions

From (6), the variation of strip entry position can be expressed as follows:(31)
x
0
v
a
r
=
∂
x
0
∂
h
0
s
h
0
v
a
r
+
∂
x
0
∂
h
c
s
h
c
v
a
r
+
∂
x
0
∂
x
c
s
x
c
=
∂
x
0
∂
h
0
s
h
0
v
a
r
+
2
∂
x
0
∂
h
c
s
y
c
+
∂
x
0
∂
x
c
s
x
c
.Substitutingv
0 from (18) into (12), the strip exit position is written as follows:(32)
x
1
=
x
c
-
R
′
h
c
y
˙
c
v
r
-
x
˙
c
h
c
+
x
n
-
x
c
2
/
R
′
+
2
y
˙
c
x
n
-
x
c
.Therefore, the variation of strip exit position is expressed as follows:(33)
x
1
v
a
r
=
2
∂
x
1
∂
h
c
s
y
c
+
∂
x
1
∂
y
˙
c
s
y
˙
c
+
∂
x
1
∂
v
r
s
R
′
θ
˙
M
+
∂
x
1
∂
x
n
s
x
n
v
a
r
+
∂
x
1
∂
x
c
s
x
c
+
δ
x
1
δ
x
˙
c
s
x
˙
c
.


### 3.2.2. Variation of Neutral Point Position

SubstitutingP from (20) into (16), the variation of neutral point position can then be calculated as follows:(34)
x
n
v
a
r
=
∂
x
n
∂
σ
0
s
σ
0
v
a
r
+
∂
x
n
∂
σ
1
s
σ
1
v
a
r
+
∂
x
n
∂
h
0
s
h
0
v
a
r
+
∂
x
n
∂
h
1
s
h
1
v
a
r
+
∂
x
n
∂
x
c
s
x
c
+
∂
x
n
∂
α
s
α
v
a
r
+
∂
x
n
∂
δ
s
δ
v
a
r
+
∂
x
n
∂
K
s
K
v
a
r
+
∂
x
n
∂
σ
m
s
σ
m
v
a
r
+
∂
x
n
∂
Q
p
s
Q
p
v
a
r
.It can be seen from (34) that some of the items are implicit expressions of independent variables, and they must be converted to explicit expressions. Substituting x
1 from (32) into (3), the strip thickness at exit is written as follows:(35)
h
1
=
h
c
+
R
′
h
c
2
y
˙
c
2
v
r
-
x
˙
c
h
c
+
x
n
-
x
c
2
/
R
′
+
2
y
˙
c
x
n
-
x
c
2
.Then the variation of strip thickness at exit is expressed as follows:(36)
h
1
v
a
r
=
2
∂
h
1
∂
h
c
s
y
c
+
∂
h
1
∂
y
˙
c
s
y
˙
c
+
∂
h
1
∂
v
r
s
R
′
θ
˙
M
+
∂
h
1
∂
x
n
s
x
n
v
a
r
+
∂
h
1
∂
x
c
s
x
c
+
δ
h
1
δ
x
˙
c
s
x
˙
c
.As the inlet angle and the exit angle are small in cold rolling, approximately takingα
≈
sin
⁡
α
=
h
0
-
h
c
/
R
′ and δ
≈
sin
⁡
δ
=
x
c
-
x
1
/
R
′, hence their first-order Taylor series approximations are expressed as follows:(37)
α
v
a
r
=
∂
α
∂
h
0
s
h
0
v
a
r
+
2
∂
α
∂
h
c
s
y
c
,
(38)
δ
v
a
r
=
∂
δ
∂
x
c
s
x
c
+
∂
δ
∂
x
1
s
x
1
v
a
r
.And the variations of the mean deformation resistance, the mean tension, and the stress state factor are formulated as follows:(39)
K
v
a
r
=
∂
K
∂
h
0
s
h
0
v
a
r
+
∂
K
∂
h
1
s
h
1
v
a
r
,
(40)
σ
m
v
a
r
=
∂
σ
m
∂
σ
0
s
σ
0
v
a
r
+
∂
σ
m
∂
σ
1
s
σ
1
v
a
r
=
0.7
σ
0
v
a
r
+
0.3
σ
1
v
a
r
,
(41)
Q
p
v
a
r
=
∂
Q
p
∂
h
0
s
h
0
v
a
r
+
∂
Q
p
∂
h
1
s
h
1
v
a
r
.Substituting (36)–(41) into (34), the resulting equation for the variation of neutral point position becomes(42)
x
n
v
a
r
=
a
x
n
,
σ
0
σ
0
v
a
r
+
a
x
n
,
σ
1
σ
1
v
a
r
+
a
x
n
,
h
0
h
0
v
a
r
+
a
x
n
,
y
c
y
c
+
a
x
n
,
y
˙
c
y
˙
c
+
a
x
n
,
x
c
x
c
.


### 3.2.3. Variations of Strip Velocities at Entry and Exit

From (18) and (19), the variations of strip velocities at entry and exit are expressed as follows:(43)
v
0
v
a
r
=
∂
v
0
∂
h
0
s
h
0
v
a
r
+
2
∂
v
0
∂
h
c
s
y
c
+
∂
v
0
∂
y
˙
c
s
y
˙
c
+
∂
v
0
∂
x
c
s
x
c
+
∂
v
0
∂
x
˙
c
s
x
˙
c
+
∂
v
0
∂
v
r
s
R
′
θ
˙
M
+
∂
v
0
∂
x
n
s
x
n
v
a
r
+
∂
v
0
∂
x
0
s
x
0
v
a
r
,
(44)
v
1
v
a
r
=
∂
v
1
∂
h
0
s
h
0
v
a
r
+
2
∂
v
1
∂
h
c
s
y
c
+
∂
v
1
∂
y
˙
c
s
y
˙
c
+
∂
v
1
∂
x
c
s
x
c
+
∂
v
1
∂
x
˙
c
s
x
˙
c
+
∂
v
1
∂
v
0
s
v
0
v
a
r
+
∂
v
1
∂
x
1
s
x
1
v
a
r
+
∂
v
1
∂
x
0
s
x
0
v
a
r
.Therefore, the explicit expressions of (43) and (44) can be written as follows:(45)
v
0
v
a
r
=
a
v
0
,
σ
0
σ
0
v
a
r
+
a
v
0
,
σ
1
σ
1
v
a
r
+
a
v
0
,
h
0
h
0
v
a
r
+
a
v
0
,
y
c
y
c
+
a
v
0
,
y
˙
c
y
˙
c
+
a
v
0
,
x
c
x
c
+
a
v
0
,
x
˙
c
x
˙
c
+
a
v
0
,
θ
˙
M
θ
˙
M
,
(46)
v
1
v
a
r
=
a
v
1
,
σ
0
σ
0
v
a
r
+
a
v
1
,
σ
1
σ
1
v
a
r
+
a
v
1
,
h
0
h
0
v
a
r
+
a
v
1
,
y
c
y
c
+
a
v
1
,
y
˙
c
y
˙
c
+
a
v
1
,
x
c
x
c
+
a
v
1
,
x
˙
c
x
˙
c
+
a
v
1
,
θ
˙
M
θ
˙
M
.


### 3.2.4. Variations of Forces Acting on the Rolls and the Rolling Torque

Applying the first-order Taylor series approximation to (26), the variation of the force acting on the rolls in x direction is expressed as follows:(47)
F
x
v
a
r
=
∂
F
x
∂
σ
0
s
σ
0
v
a
r
+
∂
F
x
∂
σ
1
s
σ
1
v
a
r
+
∂
F
x
∂
h
0
s
h
0
v
a
r
+
∂
F
x
∂
h
1
s
h
1
v
a
r
=
a
F
x
,
σ
0
σ
0
v
a
r
+
a
F
x
,
σ
1
σ
1
v
a
r
+
a
F
x
,
h
0
h
0
v
a
r
+
a
F
x
,
y
c
y
c
.Likewise, as the variation of horizontal projection length of the contact arc isl
v
a
r
=
x
1
v
a
r
-
x
0
v
a
r and considering (33) and (39)–(42), the variation of the force acting on the rolls in y direction can be derived from (28):(48)
F
y
v
a
r
=
∂
F
y
∂
K
s
K
v
a
r
+
∂
F
y
∂
σ
m
s
σ
m
v
a
r
+
∂
F
y
∂
Q
p
s
Q
p
v
a
r
+
∂
F
y
∂
l
s
l
v
a
r
=
a
F
y
,
σ
0
σ
0
v
a
r
+
a
F
y
,
σ
1
σ
1
v
a
r
+
a
F
y
,
h
0
h
0
v
a
r
+
a
F
y
,
y
c
y
c
+
a
F
y
,
y
˙
c
y
˙
c
+
a
F
y
,
x
c
x
c
.As the neutral angle is very small, approximately takingγ
≈
sin
⁡
γ
=
x
c
-
x
n
/
R
′, hence its first-order Taylor series approximation is expressed as follows:(49)
γ
v
a
r
=
∂
γ
∂
x
c
s
x
c
+
∂
γ
∂
x
n
s
x
n
v
a
r
.Therefore, the variation of the rolling torque can then be calculated from (29):(50)
M
v
a
r
=
∂
M
∂
α
s
α
v
a
r
+
∂
M
∂
γ
s
γ
v
a
r
+
∂
M
∂
δ
s
δ
v
a
r
+
∂
M
∂
K
s
K
v
a
r
+
∂
M
∂
σ
m
s
σ
m
v
a
r
+
∂
M
∂
Q
p
s
Q
p
v
a
r
+
∂
M
∂
l
s
l
v
a
r
=
a
M
,
σ
0
σ
0
v
a
r
+
a
M
,
σ
1
σ
1
v
a
r
+
a
M
,
h
0
h
0
v
a
r
+
a
M
,
y
c
y
c
+
a
M
,
y
˙
c
y
˙
c
+
a
M
,
x
c
x
c
.All the coefficients of the linearized parameters in this section are given in AppendixA.


## 3.2.1. Variations of Strip Entry and Exit Positions

From (6), the variation of strip entry position can be expressed as follows:(31)
x
0
v
a
r
=
∂
x
0
∂
h
0
s
h
0
v
a
r
+
∂
x
0
∂
h
c
s
h
c
v
a
r
+
∂
x
0
∂
x
c
s
x
c
=
∂
x
0
∂
h
0
s
h
0
v
a
r
+
2
∂
x
0
∂
h
c
s
y
c
+
∂
x
0
∂
x
c
s
x
c
.Substitutingv
0 from (18) into (12), the strip exit position is written as follows:(32)
x
1
=
x
c
-
R
′
h
c
y
˙
c
v
r
-
x
˙
c
h
c
+
x
n
-
x
c
2
/
R
′
+
2
y
˙
c
x
n
-
x
c
.Therefore, the variation of strip exit position is expressed as follows:(33)
x
1
v
a
r
=
2
∂
x
1
∂
h
c
s
y
c
+
∂
x
1
∂
y
˙
c
s
y
˙
c
+
∂
x
1
∂
v
r
s
R
′
θ
˙
M
+
∂
x
1
∂
x
n
s
x
n
v
a
r
+
∂
x
1
∂
x
c
s
x
c
+
δ
x
1
δ
x
˙
c
s
x
˙
c
.


## 3.2.2. Variation of Neutral Point Position

SubstitutingP from (20) into (16), the variation of neutral point position can then be calculated as follows:(34)
x
n
v
a
r
=
∂
x
n
∂
σ
0
s
σ
0
v
a
r
+
∂
x
n
∂
σ
1
s
σ
1
v
a
r
+
∂
x
n
∂
h
0
s
h
0
v
a
r
+
∂
x
n
∂
h
1
s
h
1
v
a
r
+
∂
x
n
∂
x
c
s
x
c
+
∂
x
n
∂
α
s
α
v
a
r
+
∂
x
n
∂
δ
s
δ
v
a
r
+
∂
x
n
∂
K
s
K
v
a
r
+
∂
x
n
∂
σ
m
s
σ
m
v
a
r
+
∂
x
n
∂
Q
p
s
Q
p
v
a
r
.It can be seen from (34) that some of the items are implicit expressions of independent variables, and they must be converted to explicit expressions. Substituting x
1 from (32) into (3), the strip thickness at exit is written as follows:(35)
h
1
=
h
c
+
R
′
h
c
2
y
˙
c
2
v
r
-
x
˙
c
h
c
+
x
n
-
x
c
2
/
R
′
+
2
y
˙
c
x
n
-
x
c
2
.Then the variation of strip thickness at exit is expressed as follows:(36)
h
1
v
a
r
=
2
∂
h
1
∂
h
c
s
y
c
+
∂
h
1
∂
y
˙
c
s
y
˙
c
+
∂
h
1
∂
v
r
s
R
′
θ
˙
M
+
∂
h
1
∂
x
n
s
x
n
v
a
r
+
∂
h
1
∂
x
c
s
x
c
+
δ
h
1
δ
x
˙
c
s
x
˙
c
.As the inlet angle and the exit angle are small in cold rolling, approximately takingα
≈
sin
⁡
α
=
h
0
-
h
c
/
R
′ and δ
≈
sin
⁡
δ
=
x
c
-
x
1
/
R
′, hence their first-order Taylor series approximations are expressed as follows:(37)
α
v
a
r
=
∂
α
∂
h
0
s
h
0
v
a
r
+
2
∂
α
∂
h
c
s
y
c
,
(38)
δ
v
a
r
=
∂
δ
∂
x
c
s
x
c
+
∂
δ
∂
x
1
s
x
1
v
a
r
.And the variations of the mean deformation resistance, the mean tension, and the stress state factor are formulated as follows:(39)
K
v
a
r
=
∂
K
∂
h
0
s
h
0
v
a
r
+
∂
K
∂
h
1
s
h
1
v
a
r
,
(40)
σ
m
v
a
r
=
∂
σ
m
∂
σ
0
s
σ
0
v
a
r
+
∂
σ
m
∂
σ
1
s
σ
1
v
a
r
=
0.7
σ
0
v
a
r
+
0.3
σ
1
v
a
r
,
(41)
Q
p
v
a
r
=
∂
Q
p
∂
h
0
s
h
0
v
a
r
+
∂
Q
p
∂
h
1
s
h
1
v
a
r
.Substituting (36)–(41) into (34), the resulting equation for the variation of neutral point position becomes(42)
x
n
v
a
r
=
a
x
n
,
σ
0
σ
0
v
a
r
+
a
x
n
,
σ
1
σ
1
v
a
r
+
a
x
n
,
h
0
h
0
v
a
r
+
a
x
n
,
y
c
y
c
+
a
x
n
,
y
˙
c
y
˙
c
+
a
x
n
,
x
c
x
c
.


## 3.2.3. Variations of Strip Velocities at Entry and Exit

From (18) and (19), the variations of strip velocities at entry and exit are expressed as follows:(43)
v
0
v
a
r
=
∂
v
0
∂
h
0
s
h
0
v
a
r
+
2
∂
v
0
∂
h
c
s
y
c
+
∂
v
0
∂
y
˙
c
s
y
˙
c
+
∂
v
0
∂
x
c
s
x
c
+
∂
v
0
∂
x
˙
c
s
x
˙
c
+
∂
v
0
∂
v
r
s
R
′
θ
˙
M
+
∂
v
0
∂
x
n
s
x
n
v
a
r
+
∂
v
0
∂
x
0
s
x
0
v
a
r
,
(44)
v
1
v
a
r
=
∂
v
1
∂
h
0
s
h
0
v
a
r
+
2
∂
v
1
∂
h
c
s
y
c
+
∂
v
1
∂
y
˙
c
s
y
˙
c
+
∂
v
1
∂
x
c
s
x
c
+
∂
v
1
∂
x
˙
c
s
x
˙
c
+
∂
v
1
∂
v
0
s
v
0
v
a
r
+
∂
v
1
∂
x
1
s
x
1
v
a
r
+
∂
v
1
∂
x
0
s
x
0
v
a
r
.Therefore, the explicit expressions of (43) and (44) can be written as follows:(45)
v
0
v
a
r
=
a
v
0
,
σ
0
σ
0
v
a
r
+
a
v
0
,
σ
1
σ
1
v
a
r
+
a
v
0
,
h
0
h
0
v
a
r
+
a
v
0
,
y
c
y
c
+
a
v
0
,
y
˙
c
y
˙
c
+
a
v
0
,
x
c
x
c
+
a
v
0
,
x
˙
c
x
˙
c
+
a
v
0
,
θ
˙
M
θ
˙
M
,
(46)
v
1
v
a
r
=
a
v
1
,
σ
0
σ
0
v
a
r
+
a
v
1
,
σ
1
σ
1
v
a
r
+
a
v
1
,
h
0
h
0
v
a
r
+
a
v
1
,
y
c
y
c
+
a
v
1
,
y
˙
c
y
˙
c
+
a
v
1
,
x
c
x
c
+
a
v
1
,
x
˙
c
x
˙
c
+
a
v
1
,
θ
˙
M
θ
˙
M
.


## 3.2.4. Variations of Forces Acting on the Rolls and the Rolling Torque

Applying the first-order Taylor series approximation to (26), the variation of the force acting on the rolls in x direction is expressed as follows:(47)
F
x
v
a
r
=
∂
F
x
∂
σ
0
s
σ
0
v
a
r
+
∂
F
x
∂
σ
1
s
σ
1
v
a
r
+
∂
F
x
∂
h
0
s
h
0
v
a
r
+
∂
F
x
∂
h
1
s
h
1
v
a
r
=
a
F
x
,
σ
0
σ
0
v
a
r
+
a
F
x
,
σ
1
σ
1
v
a
r
+
a
F
x
,
h
0
h
0
v
a
r
+
a
F
x
,
y
c
y
c
.Likewise, as the variation of horizontal projection length of the contact arc isl
v
a
r
=
x
1
v
a
r
-
x
0
v
a
r and considering (33) and (39)–(42), the variation of the force acting on the rolls in y direction can be derived from (28):(48)
F
y
v
a
r
=
∂
F
y
∂
K
s
K
v
a
r
+
∂
F
y
∂
σ
m
s
σ
m
v
a
r
+
∂
F
y
∂
Q
p
s
Q
p
v
a
r
+
∂
F
y
∂
l
s
l
v
a
r
=
a
F
y
,
σ
0
σ
0
v
a
r
+
a
F
y
,
σ
1
σ
1
v
a
r
+
a
F
y
,
h
0
h
0
v
a
r
+
a
F
y
,
y
c
y
c
+
a
F
y
,
y
˙
c
y
˙
c
+
a
F
y
,
x
c
x
c
.As the neutral angle is very small, approximately takingγ
≈
sin
⁡
γ
=
x
c
-
x
n
/
R
′, hence its first-order Taylor series approximation is expressed as follows:(49)
γ
v
a
r
=
∂
γ
∂
x
c
s
x
c
+
∂
γ
∂
x
n
s
x
n
v
a
r
.Therefore, the variation of the rolling torque can then be calculated from (29):(50)
M
v
a
r
=
∂
M
∂
α
s
α
v
a
r
+
∂
M
∂
γ
s
γ
v
a
r
+
∂
M
∂
δ
s
δ
v
a
r
+
∂
M
∂
K
s
K
v
a
r
+
∂
M
∂
σ
m
s
σ
m
v
a
r
+
∂
M
∂
Q
p
s
Q
p
v
a
r
+
∂
M
∂
l
s
l
v
a
r
=
a
M
,
σ
0
σ
0
v
a
r
+
a
M
,
σ
1
σ
1
v
a
r
+
a
M
,
h
0
h
0
v
a
r
+
a
M
,
y
c
y
c
+
a
M
,
y
˙
c
y
˙
c
+
a
M
,
x
c
x
c
.All the coefficients of the linearized parameters in this section are given in AppendixA.


## 3.3. Mathematical Model of the Rolling Vibration

As equations in (22) have the integral terms of time, we take them as a derivative with respect to time. Thus they can also be written as follows:(51)
d
σ
0
v
a
r
d
t
=
E
1
L
0
v
0
-
v
-
0
=
E
1
L
0
v
0
v
a
r
,
d
σ
1
v
a
r
d
t
=
-
E
1
L
1
v
1
-
v
-
1
=
-
E
1
L
1
v
1
v
a
r
.In Section3.1, F
x
v
a
r
=
F
x
-
F
-
x; F
y
v
a
r
=
F
y
-
F
-
y; M
v
a
r
=
M
-
M
-; therefore, the mathematical model of rolling vibration based on the theoretical model of the dynamic rolling process is expressed as follows:(52)
m
1
x
¨
c
+
c
1
x
˙
c
+
k
1
x
c
=
F
x
-
F
-
x
,
m
2
y
¨
c
+
c
2
y
˙
c
+
k
2
y
c
=
F
y
-
F
-
y
,
J
M
θ
¨
M
+
c
t
θ
˙
M
+
k
t
θ
M
=
M
-
M
-
,
σ
˙
0
v
a
r
=
E
1
v
0
-
v
-
0
L
0
,
σ
˙
1
v
a
r
=
-
E
1
v
1
-
v
-
1
L
1
.And the mathematical model of rolling vibration based on the linearized dynamic rolling process model is expressed as a first-order differential equation in matrix form:(53)
d
X
d
t
=
D
1
X
+
D
2
h
0
v
a
r
,where(54)
X
=
x
c
x
˙
c
y
c
y
˙
c
θ
M
θ
˙
M
σ
0
v
a
r
σ
1
v
a
r
T
,
(55)
D
1
=
0
1
0
0
0
0
0
0
-
k
1
m
1
-
c
1
m
1
a
F
x
,
y
c
m
1
0
0
0
a
F
x
,
σ
0
m
1
a
F
x
,
σ
1
m
1
0
0
0
1
0
0
0
0
a
F
y
,
x
c
m
2
0
a
F
y
,
y
c
-
k
2
m
2
a
F
y
,
y
˙
c
-
c
2
m
2
0
0
a
F
y
,
σ
0
m
2
a
F
y
,
σ
1
m
2
0
0
0
0
0
1
0
0
a
M
,
x
c
J
M
0
a
M
,
y
c
J
M
a
M
,
y
˙
c
J
M
-
k
t
J
M
-
c
t
J
M
a
M
,
σ
0
J
M
a
M
,
σ
1
J
M
E
1
a
v
0
,
x
c
L
0
E
1
a
v
0
,
x
˙
c
L
0
E
1
a
v
0
,
y
c
L
0
E
1
a
v
0
,
y
˙
c
L
0
0
E
1
a
v
0
,
θ
˙
M
L
0
E
1
a
v
0
,
σ
0
L
0
E
1
a
v
0
,
σ
1
L
0
-
E
1
a
v
1
,
x
c
L
1
-
E
1
a
v
1
,
x
˙
c
L
1
-
E
1
a
v
1
,
y
c
L
1
-
E
1
a
v
1
,
y
˙
c
L
1
0
-
E
1
a
v
1
,
θ
˙
M
L
1
-
E
1
a
v
1
,
σ
0
L
1
-
E
1
a
v
1
,
σ
1
L
1
,
(56)
D
2
=
0
a
F
x
,
h
0
m
1
0
a
F
y
,
h
0
m
2
0
a
M
,
h
0
J
M
E
1
a
v
0
,
h
0
L
0
-
E
1
a
v
1
,
h
0
L
1
T
.


## 4. Model Verification

As the model of rolling mill vibration is formulated by coupling the dynamic rolling process model with the mill structure model, it is necessary to verify the effectiveness of the rolling process model by test data. In addition, in order to study the stability of the mill system, a linearization process for the theoretical rolling process model is done by using a first-order Taylor series approximation; thus it is also necessary to verify the accuracy of the linearized rolling process model.


### 4.1. Effectiveness of Rolling Process Model

The experimental data are selected from a series of actual rolling parameters in a certain five-stand cold tandem mill. Here the fourth stand is particularly investigated. The setting steady-state values of rolling process parameters are listed in AppendixB, and the invariable parameters are R
=
0.28 m, μ
m
=
0.035, n
=
0.29, E
2
=
210 GPa, and u
2
=
0.3. The rolling force and the rolling torque can be obtained by using equations of Section 3.1. The calculated results are compared with the setting values in Figure 5, where all the relative errors of the rolling force and the rolling torque are less than 10%. Therefore, the proposed rolling process model can be used to construct the vibration model effectively.Figure 5
Model validation through comparing calculated results with experimental data.


### 4.2. Accuracy of Linearized Rolling Process Model

Since the linearization process listed in Section3.2 is a first-order approximation, the variations should not be large, or the results may include an unacceptable amount of error. For this problem, the simulation analysis is carried out on the two mathematical vibration models of Section 3.3. The simulation parameters are listed in Table 1, which are taken from the 4th stand of a 2030 five-stand tandem cold mill.Table 1
Parameters of mill structure, rolling process, and strip material.
m
1 (kg)
m
2 (kg)
J
M (kg⋅m2)
k
1 (N⋅m−1)
k
2 (N⋅m−1)
k
t (N⋅m⋅rad−1)
c
1 (N⋅s⋅m−1)
c
2 (N⋅s⋅m−1)
9200
203200
1381
1 × 109
6.9 × 1010
7.9 × 106
5 × 104
1.64 × 107
c
t (N⋅m⋅s⋅rad−1)
R (m)
L
0 (m)
L
1 (m)
E
1,2 (GPa)
u
2
h
0 (mm)
h
1 (mm)
4178
0.3
4.75
4.75
210
0.3
0.789
0.577
B (mm)
σ
0 (MPa)
σ
1 (MPa)
μ
m
v
-
r (m⋅s)
H (mm)
A (MPa)
n
1206
180
189
0.04
19
2
810
0.29For the initial conditiony
c
0
=
10
-
6 m, the system dynamic responses based on the theoretical model and the linearized model are shown in Figure 6. It is observed that the theoretical model can be well approximated by the linearized model. The mean absolute errors of following three subgraphs are 6.1
×
10
-
9 m, 2.6
×
10
-
9 m, and 6.2
×
10
-
8 m, respectively. And all of them are much smaller than the amplitudes of each subsystem. Therefore, using the linearized rolling process model to establish the mathematical model of mill vibration not only can ensure the accuracy but also can facilitate the stability analysis of the system.Figure 6
Accuracy verification of linearized dynamic rolling process model.


## 4.1. Effectiveness of Rolling Process Model

The experimental data are selected from a series of actual rolling parameters in a certain five-stand cold tandem mill. Here the fourth stand is particularly investigated. The setting steady-state values of rolling process parameters are listed in AppendixB, and the invariable parameters are R
=
0.28 m, μ
m
=
0.035, n
=
0.29, E
2
=
210 GPa, and u
2
=
0.3. The rolling force and the rolling torque can be obtained by using equations of Section 3.1. The calculated results are compared with the setting values in Figure 5, where all the relative errors of the rolling force and the rolling torque are less than 10%. Therefore, the proposed rolling process model can be used to construct the vibration model effectively.Figure 5
Model validation through comparing calculated results with experimental data.


## 4.2. Accuracy of Linearized Rolling Process Model

Since the linearization process listed in Section3.2 is a first-order approximation, the variations should not be large, or the results may include an unacceptable amount of error. For this problem, the simulation analysis is carried out on the two mathematical vibration models of Section 3.3. The simulation parameters are listed in Table 1, which are taken from the 4th stand of a 2030 five-stand tandem cold mill.Table 1
Parameters of mill structure, rolling process, and strip material.
m
1 (kg)
m
2 (kg)
J
M (kg⋅m2)
k
1 (N⋅m−1)
k
2 (N⋅m−1)
k
t (N⋅m⋅rad−1)
c
1 (N⋅s⋅m−1)
c
2 (N⋅s⋅m−1)
9200
203200
1381
1 × 109
6.9 × 1010
7.9 × 106
5 × 104
1.64 × 107
c
t (N⋅m⋅s⋅rad−1)
R (m)
L
0 (m)
L
1 (m)
E
1,2 (GPa)
u
2
h
0 (mm)
h
1 (mm)
4178
0.3
4.75
4.75
210
0.3
0.789
0.577
B (mm)
σ
0 (MPa)
σ
1 (MPa)
μ
m
v
-
r (m⋅s)
H (mm)
A (MPa)
n
1206
180
189
0.04
19
2
810
0.29For the initial conditiony
c
0
=
10
-
6 m, the system dynamic responses based on the theoretical model and the linearized model are shown in Figure 6. It is observed that the theoretical model can be well approximated by the linearized model. The mean absolute errors of following three subgraphs are 6.1
×
10
-
9 m, 2.6
×
10
-
9 m, and 6.2
×
10
-
8 m, respectively. And all of them are much smaller than the amplitudes of each subsystem. Therefore, using the linearized rolling process model to establish the mathematical model of mill vibration not only can ensure the accuracy but also can facilitate the stability analysis of the system.Figure 6
Accuracy verification of linearized dynamic rolling process model.


## 5. Stability Analysis and Dynamic Response

### 5.1. System Critical Rolling Speed

#### 5.1.1. Calculation of Critical Rolling Speed

The necessary and sufficient condition for a linear time-invariant system to be stable is that all the eigenvalues of the system or the roots of the characteristic equation possess negative real parts. As shown in (55), the characteristic matrix of the mathematical model is an eight-order square, and the elements of the characteristic matrix are generally represented by complex mathematical relationships of rolling process parameters. Thus it is difficult to achieve the eigenvalues of the system in a symbolic or analytical form. However, it is rather easy to calculate the eigenvalues by using a numerical method and then to determine the system stability. The calculation flow chart is shown in Figure 7.Figure 7
Calculation flow chart of critical rolling speed.With parameters provided in Table1, the critical rolling speed can be calculated as v
-
r
c
=
20.675 m·s−1. For v
-
r
=
20 m·s−1 and v
-
r
=
21 m·s−1, the eigenvalues of the system are calculated, respectively, and compared in Table 2. Evidently, when the rolling speed is higher than the critical value, one pair of characteristic roots have positive real parts, and the system lost stability accordingly.Table 2
Characteristic roots for the neighbor parameters of the instability threshold.
v
-
r (m⋅s−1)
λ
1,2
λ
3,4
λ
5
λ
6,7
λ
8
20
-
4.38
±
839.64
i
-
3.19
±
329.87
i
-
275.30
-
2.45
±
75.39
i
-
120.86
21
2.09
±
841.36
i
-
3.18
±
329.85
i
-
288.59
-
2.39
±
75.39
i
-
126.61


#### 5.1.2. Comparison of Calculated Results with Test Data

In order to verify the correctness of the proposed model, the comparisons of calculated critical rolling speeds with test data are carried out. The test data are also taken from the 4th stand of a 2030 five-stand tandem cold mill. In the vibration test, there were thirteen vibration phenomena have been observed. Here, selecting ten sets of data as examples, which are corresponding to relatively strong vibration phenomena, the comparisons are listed in Table3. It can be seen that the calculated critical rolling speeds are very close to test values, where all relative errors are less than 14%. Therefore, the proposed rolling mill multiple-modal-coupling model is accurate. Moreover, through further observation of Table 3, it can be found that all the calculated values are greater than the test values. It is because the content of this study is the vibration characteristic of a single stand mill. But the interaction between adjacent stands in tandem rolling mills can also worsen the stability of the system. Therefore, it is reasonable for the calculated value to be relatively larger.Table 3
Comparisons of calculated critical rolling speeds with test data.
Number
H (mm)
h
0 (mm)
h
1 (mm)
B (mm)
σ
0 (MPa)
σ
1 (MPa)
A (MPa)
R (m)
Critical rolling speed (m/min)
Test
Calculated
Relative error
1
2.000
0.789
0.577
1206
180
189
810
0.30
1090
1240
13.76%
2
1.974
0.719
0.512
1018
180
190
845
0.30
1229
1263
2.77%
3
1.962
0.658
0.472
1006
150
160
810
0.30
1271
1287
1.26%
4
1.845
0.645
0.468
1006
150
160
810
0.30
1274
1300
2.04%
5
2.022
0.659
0.468
1006
150
160
810
0.30
1274
1277
0.24%
6
1.966
0.720
0.513
1018
180
190
840
0.30
1156
1271
9.95%
7
1.966
0.720
0.513
1018
180
190
810
0.30
1059
1206
13.89%
8
2.000
0.755
0.568
1206
156
197
810
0.30
1220
1315
7.79%
9
1.998
0.754
0.575
1206
160
200
820
0.29
1188
1329
11.87%
10
1.964
0.718
0.517
1006
176
119
830
0.30
1300
1305
0.38%


### 5.2. Dynamic Response of System Coupling Vibration

#### 5.2.1. Dynamic Response to Initial Disturbance of Roll Rotational Angle

For an initial disturbanceθ
M
0
=
10
-
5 rad, the simulations on the conditions of v
-
r
=
20
m
/
s and v
-
r
=
21
m
/
s are carried out, and the results are shown in Figures 8 and 9.Figure 8
Dynamic response to the variation in roll rotational angle forv
-
r
=
20 m⋅s−1.Figure 9
Dynamic response to the variation in roll rotational angle forv
-
r
=
21 m⋅s−1.As shown in Figures8 and 9, although the initial disturbance is only acting on the torsional subsystem, the vibration occurs on the whole mill system. In the time-history diagrams, it is found that the system is stable when v
-
r
<
v
-
r
c. Conversely, the system is unstable when v
-
r
>
v
-
r
c. From the frequency spectrums, it is found that the initial roll rotational angle not only causes the vibration of each subsystem containing the frequency component of torsional vibration modal (12.21 Hz) but also motivates the free vibration of vertical subsystem and horizontal subsystem with their own modal frequency (133.5 Hz and 52.64 Hz, resp.). And the frequency of 133.5 Hz is among the frequency domain of third-octave-mode chatter. Once the system is unstable, the energy of this frequency component will increase and extend sharply in the mill system.


#### 5.2.2. Dynamic Response to Periodic Disturbance of Strip Entry Thickness

Assuming that the variation of strip entry thickness ish
0
v
a
r
=
1
0
-
6
×
s
i
n
(
200
·
t
) and without any other initial disturbance, the dynamic responses on the conditions of v
-
r
=
20
m
/
s and v
-
r
=
21
m
/
s are analyzed and shown in Figures 10 and 11.Figure 10
Dynamic response to periodic disturbance of strip entry thickness forv
-
r
=
20 m/s.Figure 11
Dynamic response to periodic disturbance of strip entry thickness forv
-
r
=
21 m/s.Likewise, the system is stable whenv
-
r
<
v
-
r
c and unstable when v
-
r
>
v
-
r
c. Since the periodic fluctuation of the strip entry thickness is equivalent to introducing a periodic excitation into the system, the vibration frequencies of each subsystem are all close to 32 Hz (the frequency of excitation) when the system is stable. However, when the system lost stability, the energy of the excitation frequency component is almost unchanged, but the frequency component, which belongs to the frequency domain of third-octave-mode chatter, would rise and spread rapidly in the system.


## 5.1. System Critical Rolling Speed

### 5.1.1. Calculation of Critical Rolling Speed

The necessary and sufficient condition for a linear time-invariant system to be stable is that all the eigenvalues of the system or the roots of the characteristic equation possess negative real parts. As shown in (55), the characteristic matrix of the mathematical model is an eight-order square, and the elements of the characteristic matrix are generally represented by complex mathematical relationships of rolling process parameters. Thus it is difficult to achieve the eigenvalues of the system in a symbolic or analytical form. However, it is rather easy to calculate the eigenvalues by using a numerical method and then to determine the system stability. The calculation flow chart is shown in Figure 7.Figure 7
Calculation flow chart of critical rolling speed.With parameters provided in Table1, the critical rolling speed can be calculated as v
-
r
c
=
20.675 m·s−1. For v
-
r
=
20 m·s−1 and v
-
r
=
21 m·s−1, the eigenvalues of the system are calculated, respectively, and compared in Table 2. Evidently, when the rolling speed is higher than the critical value, one pair of characteristic roots have positive real parts, and the system lost stability accordingly.Table 2
Characteristic roots for the neighbor parameters of the instability threshold.
v
-
r (m⋅s−1)
λ
1,2
λ
3,4
λ
5
λ
6,7
λ
8
20
-
4.38
±
839.64
i
-
3.19
±
329.87
i
-
275.30
-
2.45
±
75.39
i
-
120.86
21
2.09
±
841.36
i
-
3.18
±
329.85
i
-
288.59
-
2.39
±
75.39
i
-
126.61


### 5.1.2. Comparison of Calculated Results with Test Data

In order to verify the correctness of the proposed model, the comparisons of calculated critical rolling speeds with test data are carried out. The test data are also taken from the 4th stand of a 2030 five-stand tandem cold mill. In the vibration test, there were thirteen vibration phenomena have been observed. Here, selecting ten sets of data as examples, which are corresponding to relatively strong vibration phenomena, the comparisons are listed in Table3. It can be seen that the calculated critical rolling speeds are very close to test values, where all relative errors are less than 14%. Therefore, the proposed rolling mill multiple-modal-coupling model is accurate. Moreover, through further observation of Table 3, it can be found that all the calculated values are greater than the test values. It is because the content of this study is the vibration characteristic of a single stand mill. But the interaction between adjacent stands in tandem rolling mills can also worsen the stability of the system. Therefore, it is reasonable for the calculated value to be relatively larger.Table 3
Comparisons of calculated critical rolling speeds with test data.
Number
H (mm)
h
0 (mm)
h
1 (mm)
B (mm)
σ
0 (MPa)
σ
1 (MPa)
A (MPa)
R (m)
Critical rolling speed (m/min)
Test
Calculated
Relative error
1
2.000
0.789
0.577
1206
180
189
810
0.30
1090
1240
13.76%
2
1.974
0.719
0.512
1018
180
190
845
0.30
1229
1263
2.77%
3
1.962
0.658
0.472
1006
150
160
810
0.30
1271
1287
1.26%
4
1.845
0.645
0.468
1006
150
160
810
0.30
1274
1300
2.04%
5
2.022
0.659
0.468
1006
150
160
810
0.30
1274
1277
0.24%
6
1.966
0.720
0.513
1018
180
190
840
0.30
1156
1271
9.95%
7
1.966
0.720
0.513
1018
180
190
810
0.30
1059
1206
13.89%
8
2.000
0.755
0.568
1206
156
197
810
0.30
1220
1315
7.79%
9
1.998
0.754
0.575
1206
160
200
820
0.29
1188
1329
11.87%
10
1.964
0.718
0.517
1006
176
119
830
0.30
1300
1305
0.38%


## 5.1.1. Calculation of Critical Rolling Speed

The necessary and sufficient condition for a linear time-invariant system to be stable is that all the eigenvalues of the system or the roots of the characteristic equation possess negative real parts. As shown in (55), the characteristic matrix of the mathematical model is an eight-order square, and the elements of the characteristic matrix are generally represented by complex mathematical relationships of rolling process parameters. Thus it is difficult to achieve the eigenvalues of the system in a symbolic or analytical form. However, it is rather easy to calculate the eigenvalues by using a numerical method and then to determine the system stability. The calculation flow chart is shown in Figure 7.Figure 7
Calculation flow chart of critical rolling speed.With parameters provided in Table1, the critical rolling speed can be calculated as v
-
r
c
=
20.675 m·s−1. For v
-
r
=
20 m·s−1 and v
-
r
=
21 m·s−1, the eigenvalues of the system are calculated, respectively, and compared in Table 2. Evidently, when the rolling speed is higher than the critical value, one pair of characteristic roots have positive real parts, and the system lost stability accordingly.Table 2
Characteristic roots for the neighbor parameters of the instability threshold.
v
-
r (m⋅s−1)
λ
1,2
λ
3,4
λ
5
λ
6,7
λ
8
20
-
4.38
±
839.64
i
-
3.19
±
329.87
i
-
275.30
-
2.45
±
75.39
i
-
120.86
21
2.09
±
841.36
i
-
3.18
±
329.85
i
-
288.59
-
2.39
±
75.39
i
-
126.61


## 5.1.2. Comparison of Calculated Results with Test Data

In order to verify the correctness of the proposed model, the comparisons of calculated critical rolling speeds with test data are carried out. The test data are also taken from the 4th stand of a 2030 five-stand tandem cold mill. In the vibration test, there were thirteen vibration phenomena have been observed. Here, selecting ten sets of data as examples, which are corresponding to relatively strong vibration phenomena, the comparisons are listed in Table3. It can be seen that the calculated critical rolling speeds are very close to test values, where all relative errors are less than 14%. Therefore, the proposed rolling mill multiple-modal-coupling model is accurate. Moreover, through further observation of Table 3, it can be found that all the calculated values are greater than the test values. It is because the content of this study is the vibration characteristic of a single stand mill. But the interaction between adjacent stands in tandem rolling mills can also worsen the stability of the system. Therefore, it is reasonable for the calculated value to be relatively larger.Table 3
Comparisons of calculated critical rolling speeds with test data.
Number
H (mm)
h
0 (mm)
h
1 (mm)
B (mm)
σ
0 (MPa)
σ
1 (MPa)
A (MPa)
R (m)
Critical rolling speed (m/min)
Test
Calculated
Relative error
1
2.000
0.789
0.577
1206
180
189
810
0.30
1090
1240
13.76%
2
1.974
0.719
0.512
1018
180
190
845
0.30
1229
1263
2.77%
3
1.962
0.658
0.472
1006
150
160
810
0.30
1271
1287
1.26%
4
1.845
0.645
0.468
1006
150
160
810
0.30
1274
1300
2.04%
5
2.022
0.659
0.468
1006
150
160
810
0.30
1274
1277
0.24%
6
1.966
0.720
0.513
1018
180
190
840
0.30
1156
1271
9.95%
7
1.966
0.720
0.513
1018
180
190
810
0.30
1059
1206
13.89%
8
2.000
0.755
0.568
1206
156
197
810
0.30
1220
1315
7.79%
9
1.998
0.754
0.575
1206
160
200
820
0.29
1188
1329
11.87%
10
1.964
0.718
0.517
1006
176
119
830
0.30
1300
1305
0.38%


## 5.2. Dynamic Response of System Coupling Vibration

### 5.2.1. Dynamic Response to Initial Disturbance of Roll Rotational Angle

For an initial disturbanceθ
M
0
=
10
-
5 rad, the simulations on the conditions of v
-
r
=
20
m
/
s and v
-
r
=
21
m
/
s are carried out, and the results are shown in Figures 8 and 9.Figure 8
Dynamic response to the variation in roll rotational angle forv
-
r
=
20 m⋅s−1.Figure 9
Dynamic response to the variation in roll rotational angle forv
-
r
=
21 m⋅s−1.As shown in Figures8 and 9, although the initial disturbance is only acting on the torsional subsystem, the vibration occurs on the whole mill system. In the time-history diagrams, it is found that the system is stable when v
-
r
<
v
-
r
c. Conversely, the system is unstable when v
-
r
>
v
-
r
c. From the frequency spectrums, it is found that the initial roll rotational angle not only causes the vibration of each subsystem containing the frequency component of torsional vibration modal (12.21 Hz) but also motivates the free vibration of vertical subsystem and horizontal subsystem with their own modal frequency (133.5 Hz and 52.64 Hz, resp.). And the frequency of 133.5 Hz is among the frequency domain of third-octave-mode chatter. Once the system is unstable, the energy of this frequency component will increase and extend sharply in the mill system.


### 5.2.2. Dynamic Response to Periodic Disturbance of Strip Entry Thickness

Assuming that the variation of strip entry thickness ish
0
v
a
r
=
1
0
-
6
×
s
i
n
(
200
·
t
) and without any other initial disturbance, the dynamic responses on the conditions of v
-
r
=
20
m
/
s and v
-
r
=
21
m
/
s are analyzed and shown in Figures 10 and 11.Figure 10
Dynamic response to periodic disturbance of strip entry thickness forv
-
r
=
20 m/s.Figure 11
Dynamic response to periodic disturbance of strip entry thickness forv
-
r
=
21 m/s.Likewise, the system is stable whenv
-
r
<
v
-
r
c and unstable when v
-
r
>
v
-
r
c. Since the periodic fluctuation of the strip entry thickness is equivalent to introducing a periodic excitation into the system, the vibration frequencies of each subsystem are all close to 32 Hz (the frequency of excitation) when the system is stable. However, when the system lost stability, the energy of the excitation frequency component is almost unchanged, but the frequency component, which belongs to the frequency domain of third-octave-mode chatter, would rise and spread rapidly in the system.


## 5.2.1. Dynamic Response to Initial Disturbance of Roll Rotational Angle

For an initial disturbanceθ
M
0
=
10
-
5 rad, the simulations on the conditions of v
-
r
=
20
m
/
s and v
-
r
=
21
m
/
s are carried out, and the results are shown in Figures 8 and 9.Figure 8
Dynamic response to the variation in roll rotational angle forv
-
r
=
20 m⋅s−1.Figure 9
Dynamic response to the variation in roll rotational angle forv
-
r
=
21 m⋅s−1.As shown in Figures8 and 9, although the initial disturbance is only acting on the torsional subsystem, the vibration occurs on the whole mill system. In the time-history diagrams, it is found that the system is stable when v
-
r
<
v
-
r
c. Conversely, the system is unstable when v
-
r
>
v
-
r
c. From the frequency spectrums, it is found that the initial roll rotational angle not only causes the vibration of each subsystem containing the frequency component of torsional vibration modal (12.21 Hz) but also motivates the free vibration of vertical subsystem and horizontal subsystem with their own modal frequency (133.5 Hz and 52.64 Hz, resp.). And the frequency of 133.5 Hz is among the frequency domain of third-octave-mode chatter. Once the system is unstable, the energy of this frequency component will increase and extend sharply in the mill system.


## 5.2.2. Dynamic Response to Periodic Disturbance of Strip Entry Thickness

Assuming that the variation of strip entry thickness ish
0
v
a
r
=
1
0
-
6
×
s
i
n
(
200
·
t
) and without any other initial disturbance, the dynamic responses on the conditions of v
-
r
=
20
m
/
s and v
-
r
=
21
m
/
s are analyzed and shown in Figures 10 and 11.Figure 10
Dynamic response to periodic disturbance of strip entry thickness forv
-
r
=
20 m/s.Figure 11
Dynamic response to periodic disturbance of strip entry thickness forv
-
r
=
21 m/s.Likewise, the system is stable whenv
-
r
<
v
-
r
c and unstable when v
-
r
>
v
-
r
c. Since the periodic fluctuation of the strip entry thickness is equivalent to introducing a periodic excitation into the system, the vibration frequencies of each subsystem are all close to 32 Hz (the frequency of excitation) when the system is stable. However, when the system lost stability, the energy of the excitation frequency component is almost unchanged, but the frequency component, which belongs to the frequency domain of third-octave-mode chatter, would rise and spread rapidly in the system.


## 6. Influence of Parameters on System Stability

### 6.1. Rolling Process Parameters

#### 6.1.1. Tension at Entry and Exit

With parameters provided in Table1, the influences of tensions at entry and exit on the system stability are depicted in Figure 12. As shown in Figure 12(a), with the increase of tensions at entry and exit, the critical rolling speed decreases gradually. The reason is that larger tensions correspond to the smaller rolling stiffness (a
F
y
,
y
c). Thus the stability of the system is weakened. Through further comparison, it can be seen that the influence of the entry tension is greater than the exit tension. Figure 12(b) is the change of the critical rolling speed under the coupling effect of entry tension and exit tension. Figure 12(c) is the critical rolling speed contour in the coupling parameters plane (σ
1, σ
0). It can be seen from Figures 12(b) and 12(c) that the influences of tensions at entry and exit on the system stability are approximately linear.Figure 12
Influences of tensions at entry and exit on system stability.
(a)
(b)
(c)


#### 6.1.2. Reduction Ratio

Similarly, with parameters provided in Table1, the change of the critical rolling speed with different reduction ratios is plotted in Figure 13. In this section, the conditions of invariable entry thickness and invariable exit thickness are considered, respectively. In Figure 13, the two curves almost overlap. And both conditions show that the increasing reduction ratio can result in lower system stability. The reason is that the larger reduction ratio means greater rolling force. So, the fluctuation of the rolling force will increase under the same disturbance, and the emergence of vibration happens more easily.Figure 13
Influence of reduction ratio on system stability.


#### 6.1.3. Friction Coefficient and Strip Width

Figure14 represents the influences of friction coefficient and strip width on system stability. From Figure 14(a), it can be seen that the critical rolling speed increases significantly with the increase of the friction coefficient. The reason is that the friction in the roll gap acts as a positive damping in the system. As Figure 14(b) displays, the wider the strip is, the more stable the system will be. It is because the wider strip corresponds to relatively larger rolling contact stiffness and rolling damping, which can strengthen the system stability.Figure 14
Influences of friction coefficient and strip width on system stability.
(a)
(b)


#### 6.1.4. Strip Material Parameters

Figures15(a) and 15(b) depict that the critical rolling speed increases with the increase of material parameter A and increases with the decrease of material parameter n. Equation (25) indicates that the mean deformation resistance K is an increasing function of A and is a decreasing function of n. Therefore, with the increase of the mean deformation resistance K, the critical rolling speed increases monotonically. The reason is that the larger mean deformation resistance signifies the greater stiffness of the system. Naturally, the stability of the system will be strengthened.Figure 15
Influences of strip material parameters on system stability.
(a)
(b)
(c)From Figure15(c), increasing Young’s modulus of the strip leads to destabilization of the system. It is because the increasing Young modulus means larger variations of tensions at entry and exit. In addition, the steep slope of the curve indicates a high sensitivity of the system stability to Young’s modulus of the strip.


### 6.2. Mill Structure Parameters

As the vertical-torsional-horizontal coupled dynamic model is established by associating the rolling process model with the structure model, so mill structure parameters also influence the system stability. The impact trend of each structure parameter on the system stability is shown in Figures16–19. And the impact extents of process parameters and structure parameters will be compared in Section 6.4.Figure 16
Influences of horizontal structural parameters on system stability.
(a)
(b)
(c)Figure 17
Influences of vertical structural parameters on system stability.
(a)
(b)
(c)Figure 18
Influences of torsional structural parameters on system stability.
(a)
(b)
(c)Figure 19
Influences of work roll radius and distances between adjacent stands on system stability.
(a)
(b)
(c)As shown in Figures16 and 18, the structure parameters of horizontal subsystem and torsional subsystem almost have no effect on the critical rolling speed. However, the vertical subsystem structure parameters have a significant impact on the stability of the system (Figure 17). It is because the third-octave-mode chatter mainly reflects in the vertical subsystem.From Figure19(a), it can be seen that the critical rolling speed will increase with the increase of the work roll radius. It is because the increase of the work roll radius means increasing the contact arc length, which will greaten the friction effect of the roll gap and then enhance the system stability. In Figures 19(b) and 19(c), the critical rolling speed increases with the increase of L
0 and L
1. The reason is that the larger distances L
0 and L
1 will reduce the variations of tensions and then strengthen the system stability. Since the effect of the entry tension is greater than the exit tension, the effect of L
0 is relatively larger.


### 6.3. Experimental Verification

As rolling mill vibration will affect the normal production and even lead to equipment failure, therefore, it is not easy to conduct an all-round rolling mill vibration test to verify the influences of all aforementioned parameters on the system stability. In this section, a series of experiments, which are corresponding to the friction coefficient, the reduction ratio, and the exit tension, respectively, are used to verify the above relevant conclusions. Here, the experiment used to verify the influence of the friction coefficient is taken from the field vibration test of the 4th stand of a 2030 five-stand tandem cold mill. The experiments used to verify the influences of the reduction ratio and the exit tension are performed on a laboratory test rolling mill.In order to observe the influence of the friction coefficient on the system stability, rolling lubricants of different types and different concentrations are applied to the 2030 five-stand tandem cold mill system in the field test, as shown in Table4. For this rolling mill system, the original rolling lubricant type is N54. In that case, the system would lose its stability when the rolling speed was raised to 1250 m/min. Then, the rolling lubricant was changed to N428, where lubricating property is better than N54. As a result, the system was unstable when the rolling speed was only raised to 1000 m/min. Thus, the system is more prone to vibration under the condition of the smaller friction coefficient. The same conclusion can also be obtained by the experiment of rolling lubricants with different concentrations. As shown in Table 4, for the same type of rolling lubricant, the lower the concentration of the emulsion is (scilicet, the friction coefficient is larger), the more stable the system will be. Obviously, these experimental results are consistent with the analysis conclusion mentioned in Section 6.1.3.Table 4
Influence of rolling lubricants on system stability.
Rolling specification: 2→ 0.5 (mm)
Rolling lubricants
⁢Emulsion concentration (%)
Rolling speed (m/min)
System stability
Tank #3
Tank #5
N54
2.2
1.2
1200
Stable
2.2
1.2
1250
Unstable
1.5
0.6
1480
Stable
N428
2.2
1.2
988
Stable
2.2
1.2
1000
Unstable
1.6
0.7
1413
StableAs the 2030 five-stand tandem cold mill system is highly automatic, it is difficult for artificial participation. Therefore, in the field test, the influences of other parameters on the system stability are not investigated. In the following, the laboratory test rolling mill is used to verify some of other parameters’ influence. Since it is not possible to generate vibration on the laboratory-scale rolling mill, hence, in the experiments, these influences can only be verified indirectly through comparing the vibration intensities of the system under different parameters.Figure20 shows the dynamic rolling experiment under different reduction ratios. In this experiment, the rolling speed is 0.17 m/s, and the tensile forces of the strip at entry and exit are 8000 N and 4000 N, respectively. As shown in Figure 20, the reduction ratio in Region II is larger than that in Region I, and the system vibration intensity in Region II is also relatively strong. Therefore, it can be inferred that the system is more prone to vibration with larger reduction ratio. This conclusion is consistent with the analysis in Section 6.1.2.Figure 20
Dynamic rolling experiment under different reduction ratios.Figure21 shows the dynamic rolling experiment under different exit tensions and different rolling speeds. In this experiment, the entry tensile force is 10000 N, and the reduction ratio is kept unchanged. As the rolling speed in Region II is larger than that in Region I, the system vibration intensity is also relatively strong, while, in Region IV, even though the rolling speed is smaller than that in Region II, the system vibration intensity is stronger. It is because the exit tension in that region is larger than Region II. This experiment result confirmed the analysis conclusion in Section 6.1.1. Scilicet, the larger tensions will weaken the system stability. In Region III, as both the rolling speed and the exit tension are larger than other regions, the system vibration intensity became strong significantly.Figure 21
Dynamic rolling experiment under different exit tensions and different rolling speeds.


### 6.4. Sensitivity Analysis of Parameters on the System Stability

In order to compare the impact of parameters on the system stability, a dimensionless relative sensitivity factor is defined as follows:(57)
S
u
=
max
⁡
S
i
u
;
S
i
u
=
v
r
c
,
i
+
1
-
v
r
c
,
i
/
v
r
c
,
i
u
i
+
1
-
u
i
/
u
i
i
=
1,2
,
…
.The relative sensitivity factors for aforementioned parameters are given in Table5, in which the negative sign denotes the fact that the critical rolling speed decreases with the relevant parameter increasing.Table 5
Relative sensitivity factors for the following parameters.
S
(
σ
0)
S
(
ε)
S(B)
S
(
A)
S
(
m
1)
S
(
c
1)
S
(
k
1)
S
(
m
2)
S
(
c
2)
S
(
k
2)
−0.227
−0.458
0.294
1.231
1.502 × 10−3
1.469 × 10−5
−6.126 × 10−4
−0.649
0.479
0.374
S
(
σ
1)
S
(
μ
m)
S
(
E
1)
S
(
n)
S
(
J
M)
S
(
c
t)
S
(
k
t)
S
(
L
0)
S
(
L
1)
S
(
R)
-
0.139
0.504
-
0.583
-
0.136
0.003
−4.261× 10−6
−2.891× 10−5
0.527
0.085
0.395It can be seen from Table5 that the influences of process parameters on the system stability are generally larger than structure parameters. Among structure parameters, the relative sensitivity factors of the work roll radius R, the distance between this stand and the upstream stand L
0, and the structure parameters of vertical subsystem are relatively large. Clearing the impact trend and extent of these parameters on the system stability can help to formulate a reasonable rolling procedure specification and achieve a reasonable structural dynamic modification strategy as well.


## 6.1. Rolling Process Parameters

### 6.1.1. Tension at Entry and Exit

With parameters provided in Table1, the influences of tensions at entry and exit on the system stability are depicted in Figure 12. As shown in Figure 12(a), with the increase of tensions at entry and exit, the critical rolling speed decreases gradually. The reason is that larger tensions correspond to the smaller rolling stiffness (a
F
y
,
y
c). Thus the stability of the system is weakened. Through further comparison, it can be seen that the influence of the entry tension is greater than the exit tension. Figure 12(b) is the change of the critical rolling speed under the coupling effect of entry tension and exit tension. Figure 12(c) is the critical rolling speed contour in the coupling parameters plane (σ
1, σ
0). It can be seen from Figures 12(b) and 12(c) that the influences of tensions at entry and exit on the system stability are approximately linear.Figure 12
Influences of tensions at entry and exit on system stability.
(a)
(b)
(c)


### 6.1.2. Reduction Ratio

Similarly, with parameters provided in Table1, the change of the critical rolling speed with different reduction ratios is plotted in Figure 13. In this section, the conditions of invariable entry thickness and invariable exit thickness are considered, respectively. In Figure 13, the two curves almost overlap. And both conditions show that the increasing reduction ratio can result in lower system stability. The reason is that the larger reduction ratio means greater rolling force. So, the fluctuation of the rolling force will increase under the same disturbance, and the emergence of vibration happens more easily.Figure 13
Influence of reduction ratio on system stability.


### 6.1.3. Friction Coefficient and Strip Width

Figure14 represents the influences of friction coefficient and strip width on system stability. From Figure 14(a), it can be seen that the critical rolling speed increases significantly with the increase of the friction coefficient. The reason is that the friction in the roll gap acts as a positive damping in the system. As Figure 14(b) displays, the wider the strip is, the more stable the system will be. It is because the wider strip corresponds to relatively larger rolling contact stiffness and rolling damping, which can strengthen the system stability.Figure 14
Influences of friction coefficient and strip width on system stability.
(a)
(b)


### 6.1.4. Strip Material Parameters

Figures15(a) and 15(b) depict that the critical rolling speed increases with the increase of material parameter A and increases with the decrease of material parameter n. Equation (25) indicates that the mean deformation resistance K is an increasing function of A and is a decreasing function of n. Therefore, with the increase of the mean deformation resistance K, the critical rolling speed increases monotonically. The reason is that the larger mean deformation resistance signifies the greater stiffness of the system. Naturally, the stability of the system will be strengthened.Figure 15
Influences of strip material parameters on system stability.
(a)
(b)
(c)From Figure15(c), increasing Young’s modulus of the strip leads to destabilization of the system. It is because the increasing Young modulus means larger variations of tensions at entry and exit. In addition, the steep slope of the curve indicates a high sensitivity of the system stability to Young’s modulus of the strip.


## 6.1.1. Tension at Entry and Exit

With parameters provided in Table1, the influences of tensions at entry and exit on the system stability are depicted in Figure 12. As shown in Figure 12(a), with the increase of tensions at entry and exit, the critical rolling speed decreases gradually. The reason is that larger tensions correspond to the smaller rolling stiffness (a
F
y
,
y
c). Thus the stability of the system is weakened. Through further comparison, it can be seen that the influence of the entry tension is greater than the exit tension. Figure 12(b) is the change of the critical rolling speed under the coupling effect of entry tension and exit tension. Figure 12(c) is the critical rolling speed contour in the coupling parameters plane (σ
1, σ
0). It can be seen from Figures 12(b) and 12(c) that the influences of tensions at entry and exit on the system stability are approximately linear.Figure 12
Influences of tensions at entry and exit on system stability.
(a)
(b)
(c)


## 6.1.2. Reduction Ratio

Similarly, with parameters provided in Table1, the change of the critical rolling speed with different reduction ratios is plotted in Figure 13. In this section, the conditions of invariable entry thickness and invariable exit thickness are considered, respectively. In Figure 13, the two curves almost overlap. And both conditions show that the increasing reduction ratio can result in lower system stability. The reason is that the larger reduction ratio means greater rolling force. So, the fluctuation of the rolling force will increase under the same disturbance, and the emergence of vibration happens more easily.Figure 13
Influence of reduction ratio on system stability.


## 6.1.3. Friction Coefficient and Strip Width

Figure14 represents the influences of friction coefficient and strip width on system stability. From Figure 14(a), it can be seen that the critical rolling speed increases significantly with the increase of the friction coefficient. The reason is that the friction in the roll gap acts as a positive damping in the system. As Figure 14(b) displays, the wider the strip is, the more stable the system will be. It is because the wider strip corresponds to relatively larger rolling contact stiffness and rolling damping, which can strengthen the system stability.Figure 14
Influences of friction coefficient and strip width on system stability.
(a)
(b)


## 6.1.4. Strip Material Parameters

Figures15(a) and 15(b) depict that the critical rolling speed increases with the increase of material parameter A and increases with the decrease of material parameter n. Equation (25) indicates that the mean deformation resistance K is an increasing function of A and is a decreasing function of n. Therefore, with the increase of the mean deformation resistance K, the critical rolling speed increases monotonically. The reason is that the larger mean deformation resistance signifies the greater stiffness of the system. Naturally, the stability of the system will be strengthened.Figure 15
Influences of strip material parameters on system stability.
(a)
(b)
(c)From Figure15(c), increasing Young’s modulus of the strip leads to destabilization of the system. It is because the increasing Young modulus means larger variations of tensions at entry and exit. In addition, the steep slope of the curve indicates a high sensitivity of the system stability to Young’s modulus of the strip.


## 6.2. Mill Structure Parameters

As the vertical-torsional-horizontal coupled dynamic model is established by associating the rolling process model with the structure model, so mill structure parameters also influence the system stability. The impact trend of each structure parameter on the system stability is shown in Figures16–19. And the impact extents of process parameters and structure parameters will be compared in Section 6.4.Figure 16
Influences of horizontal structural parameters on system stability.
(a)
(b)
(c)Figure 17
Influences of vertical structural parameters on system stability.
(a)
(b)
(c)Figure 18
Influences of torsional structural parameters on system stability.
(a)
(b)
(c)Figure 19
Influences of work roll radius and distances between adjacent stands on system stability.
(a)
(b)
(c)As shown in Figures16 and 18, the structure parameters of horizontal subsystem and torsional subsystem almost have no effect on the critical rolling speed. However, the vertical subsystem structure parameters have a significant impact on the stability of the system (Figure 17). It is because the third-octave-mode chatter mainly reflects in the vertical subsystem.From Figure19(a), it can be seen that the critical rolling speed will increase with the increase of the work roll radius. It is because the increase of the work roll radius means increasing the contact arc length, which will greaten the friction effect of the roll gap and then enhance the system stability. In Figures 19(b) and 19(c), the critical rolling speed increases with the increase of L
0 and L
1. The reason is that the larger distances L
0 and L
1 will reduce the variations of tensions and then strengthen the system stability. Since the effect of the entry tension is greater than the exit tension, the effect of L
0 is relatively larger.


## 6.3. Experimental Verification

As rolling mill vibration will affect the normal production and even lead to equipment failure, therefore, it is not easy to conduct an all-round rolling mill vibration test to verify the influences of all aforementioned parameters on the system stability. In this section, a series of experiments, which are corresponding to the friction coefficient, the reduction ratio, and the exit tension, respectively, are used to verify the above relevant conclusions. Here, the experiment used to verify the influence of the friction coefficient is taken from the field vibration test of the 4th stand of a 2030 five-stand tandem cold mill. The experiments used to verify the influences of the reduction ratio and the exit tension are performed on a laboratory test rolling mill.In order to observe the influence of the friction coefficient on the system stability, rolling lubricants of different types and different concentrations are applied to the 2030 five-stand tandem cold mill system in the field test, as shown in Table4. For this rolling mill system, the original rolling lubricant type is N54. In that case, the system would lose its stability when the rolling speed was raised to 1250 m/min. Then, the rolling lubricant was changed to N428, where lubricating property is better than N54. As a result, the system was unstable when the rolling speed was only raised to 1000 m/min. Thus, the system is more prone to vibration under the condition of the smaller friction coefficient. The same conclusion can also be obtained by the experiment of rolling lubricants with different concentrations. As shown in Table 4, for the same type of rolling lubricant, the lower the concentration of the emulsion is (scilicet, the friction coefficient is larger), the more stable the system will be. Obviously, these experimental results are consistent with the analysis conclusion mentioned in Section 6.1.3.Table 4
Influence of rolling lubricants on system stability.
Rolling specification: 2→ 0.5 (mm)
Rolling lubricants
⁢Emulsion concentration (%)
Rolling speed (m/min)
System stability
Tank #3
Tank #5
N54
2.2
1.2
1200
Stable
2.2
1.2
1250
Unstable
1.5
0.6
1480
Stable
N428
2.2
1.2
988
Stable
2.2
1.2
1000
Unstable
1.6
0.7
1413
StableAs the 2030 five-stand tandem cold mill system is highly automatic, it is difficult for artificial participation. Therefore, in the field test, the influences of other parameters on the system stability are not investigated. In the following, the laboratory test rolling mill is used to verify some of other parameters’ influence. Since it is not possible to generate vibration on the laboratory-scale rolling mill, hence, in the experiments, these influences can only be verified indirectly through comparing the vibration intensities of the system under different parameters.Figure20 shows the dynamic rolling experiment under different reduction ratios. In this experiment, the rolling speed is 0.17 m/s, and the tensile forces of the strip at entry and exit are 8000 N and 4000 N, respectively. As shown in Figure 20, the reduction ratio in Region II is larger than that in Region I, and the system vibration intensity in Region II is also relatively strong. Therefore, it can be inferred that the system is more prone to vibration with larger reduction ratio. This conclusion is consistent with the analysis in Section 6.1.2.Figure 20
Dynamic rolling experiment under different reduction ratios.Figure21 shows the dynamic rolling experiment under different exit tensions and different rolling speeds. In this experiment, the entry tensile force is 10000 N, and the reduction ratio is kept unchanged. As the rolling speed in Region II is larger than that in Region I, the system vibration intensity is also relatively strong, while, in Region IV, even though the rolling speed is smaller than that in Region II, the system vibration intensity is stronger. It is because the exit tension in that region is larger than Region II. This experiment result confirmed the analysis conclusion in Section 6.1.1. Scilicet, the larger tensions will weaken the system stability. In Region III, as both the rolling speed and the exit tension are larger than other regions, the system vibration intensity became strong significantly.Figure 21
Dynamic rolling experiment under different exit tensions and different rolling speeds.


## 6.4. Sensitivity Analysis of Parameters on the System Stability

In order to compare the impact of parameters on the system stability, a dimensionless relative sensitivity factor is defined as follows:(57)
S
u
=
max
⁡
S
i
u
;
S
i
u
=
v
r
c
,
i
+
1
-
v
r
c
,
i
/
v
r
c
,
i
u
i
+
1
-
u
i
/
u
i
i
=
1,2
,
…
.The relative sensitivity factors for aforementioned parameters are given in Table5, in which the negative sign denotes the fact that the critical rolling speed decreases with the relevant parameter increasing.Table 5
Relative sensitivity factors for the following parameters.
S
(
σ
0)
S
(
ε)
S(B)
S
(
A)
S
(
m
1)
S
(
c
1)
S
(
k
1)
S
(
m
2)
S
(
c
2)
S
(
k
2)
−0.227
−0.458
0.294
1.231
1.502 × 10−3
1.469 × 10−5
−6.126 × 10−4
−0.649
0.479
0.374
S
(
σ
1)
S
(
μ
m)
S
(
E
1)
S
(
n)
S
(
J
M)
S
(
c
t)
S
(
k
t)
S
(
L
0)
S
(
L
1)
S
(
R)
-
0.139
0.504
-
0.583
-
0.136
0.003
−4.261× 10−6
−2.891× 10−5
0.527
0.085
0.395It can be seen from Table5 that the influences of process parameters on the system stability are generally larger than structure parameters. Among structure parameters, the relative sensitivity factors of the work roll radius R, the distance between this stand and the upstream stand L
0, and the structure parameters of vertical subsystem are relatively large. Clearing the impact trend and extent of these parameters on the system stability can help to formulate a reasonable rolling procedure specification and achieve a reasonable structural dynamic modification strategy as well.


## 7. Conclusions

For the wide existence of vibration in cold rolling mill, this paper has conducted a systematic study on multiple-modal-coupling modeling and stability analysis of rolling mill vibration. The following conclusions are drawn:(1)
A rolling mill multiple-modal-coupling vibration model is established by coupling the rolling mill structure model and the dynamic rolling process model. In the structure model, the rolls are allowed to vibrate in vertical, horizontal, and torsional directions. And the dynamic rolling process model, in which dynamic changes of the roll gap in the state of vibration are taken into consideration, is formulated based on the Hill simplified formula of Bland-Ford rolling force model.
(2)
The effectiveness of the theoretical rolling process model is verified by the test data of a cold rolling mill, and the accuracy of the linearized rolling process model is testified by the simulation analysis. This work can lay the foundation for further studying of dynamic characteristics of the mill system.
(3)
Based on the proposed vertical-torsional-horizontal coupling vibration model, the critical rolling speed for system to be stable is calculated by the method of nonlinear dynamics. And the accuracy of calculated critical rolling speed is verified by experimental data. On this basis, the system dynamic responses in both time and frequency domains are analyzed under different initial disturbances. And the results are used to discuss the interaction effect of different mill subsystems.
(4)
Finally, the influences of rolling process parameters and mill structure parameters on the system stability are investigated. And some experiments, which are corresponding to the friction coefficient, the reduction ratio, and the exit tension, respectively, are performed to verify the correctness of the relevant analysis conclusion. Then, the impact extents of these parameters on the system stability are compared by calculating their relative sensitivity factors. These conclusions not only can provide technical support for formulating a reasonable rolling process planning but also can lay a theoretical foundation for mill structure dynamics modification.



---

*Source: 2347386-2016-08-25.xml*