Stability of the coupled vibrations of work roll and strip in cold rolling process

Abstract

Undesirable vibrations that occurred in cold rolling mills, widely known as chatter, are studied in this article by considering the interaction of three types of vibrations, namely, the longitudinal vibration of the rolled strip and the torsional and vertical vibrations of the upper work roll. The dynamic component of rolling force is determined using the quasi-static model under the assumption that the changes in roll gap and strip tension produce the variation of rolling force. The coupled vibrations of the work roll and rolled strip are mathematically governed by a set of 3-degree-of-freedom non-linear equations. Under chatter conditions, a new variable is introduced to represent the motion of the quasi-neutral point. A stability criterion for the motion of the quasi-neutral point is developed by studying the eigenvalues of the corresponding characteristic equation of the linearized parts of the non-linear equations. The chatter stability can then be examined by evaluating the determinants of five matrices. Numerical examples are given to show the stable and unstable vibrations in the cold rolling process. The unstable vibration would lead to skidding phenomenon and even break the rolled strip. The results presented in this article provide new insights into the dynamic interaction of the coupled vibrations and the dynamics of the rolling process.

Keywords

Cold rolling process chatter coupled vibrations stability neutral plane

Introduction

Undesirable severe vibrations that occurred in cold rolling mills, generally referred to as chatter, have been one of the main concerns to mill operators. Excessive chatter may not only result in unacceptable gauge variations in strip thickness and chatter marks on strip surface but may also lead to a damage to mill components (such as backup rolls) and to undesirable operating conditions (such as high noise levels) for mill crews. For the vibrations of the work and backup rolls, basically there are three predominant types of oscillations in cold rolling mills, which are known as the torsional, third-octave-mode and fifth-octave-mode chatter in the literature. The corresponding resonant frequencies of these oscillations are approximately between 5 and 15 Hz for the torsional mode, 125 and 240 Hz for the third-octave mode, and 550 and 650 Hz for the fifth-octave mode. Roberts¹ noted that the actual resonant frequencies pertaining to a mill stand are directly related to the design of the stand such as roll size, screwdown design dimensions of the mill housing and so on.

Chatter has been an active topic in the literature of cold rolling process. Yun et al.² conducted a comprehensive review of the modelling of stand structure and rolling process for cold rolling. In studying the vertical vibration (which is regarded as the third-octave-mode and fifth-octave-mode chatter in the literature), the mill stand is typically modelled by a linear lumped parameter system with all the masses vibrating in the same direction and often assumed to be symmetrical with respect to the strip surface, which leads to a reduction in the number of degree of freedom by a factor of 2. In developing the roll gap model, Yun et al.², and Ginzburg and Ballas³ noticed that the resultant rolling force obtained from the rigorous modelling of the rolling process is strongly non-linear in nature and expressed approximately by the first-order terms in the Taylor series expansion when carrying out numerical simulations. To overcome the analytical intractability of the rolling force, a quasi-static model has been introduced to simplify the dynamics of the rolling process. The quasi-static model assumes that the variations in roll gap and vibrations engaged in the rolling process produce variations in rolling force, which are similar to those occurring under steady-state rolling conditions. It is widely believed that chatter is an action–reaction dynamic phenomenon in the sense that vibrations can lead to variations in roll gap and roll speed, which, in turn, result in further variations in the dynamic rolling force.

The existing research on the rolling chatter has focused on understanding the chatter phenomena and determining the stability conditions. For example, Gregorczyk⁴ developed a mathematical model of a continuous rolling mill by considering the vertical vibration of roll stand, torsional vibration of rolling mill drives and roll gap dynamics. Hu and Ehmann⁵ combined a linearized rolling process model and a structural dynamic model to study the fifth-octave-mode chatter. Meehan⁶ utilized a discrete spring–mass–damper system to study the vertical vibration of a four-high stand and examined the instability of the third-octave-mode chatter. Swiatoniowski and Bar⁷ simplified the transverse vibration of the strip linking two stands to a 1-degree-of-freedom non-linear equation. Kimura et al.⁸ studied the effects of rolling speed and frictional coefficient on the stability of mill vibration and showed that the mill vibration tends to be self-excited with an increase in rolling speed. Lin et al.⁹ presented a non-linear model describing the dynamic interaction between work rolls and metal sheets and studied the instability of strip rolling. Bar and Bar¹⁰ derived 2-degree-of-freedom non-linear equations to study the vertical vibration of a rolling stand and the transverse vibration of the strip. Nizoil and Swiatoniowski¹¹ studied the vertical vibrations of rolling mills and their negative effect on the sheet quality. Hu et al.¹² examined the third-octave-mode chatter and its stability by combining a structural dynamics model and a linearized rolling process model.

Later, Brusa et al.¹³ investigated the dynamic behaviour of a Sendzimir mill and examined the effects of roll unbalance, motor dynamic irregularities and bearing faults. Kim et al.¹⁴ developed a mathematical model of a cold rolling mill by considering the driving system, the frictional forces between the rolls and the stiffness caused by the roller bearings and the contact between rolls. Niroomand et al.¹⁵ used the Arbitrary Lagrangian–Eulerian (ALE) finite element method to simulate the chatter vibrations and compared two main chattering with those of the experimental measurements. Swiatoniowski and Bar¹⁶ developed a model for the self-excited high-frequency vibrations and studied the effects of rolling stand and process parameters on vibrations. Kim et al.¹⁷ considered the stiffness of roller bearings and contact between rolls in their model and studied the vertical and horizontal vibrations of work rolls. Zhao and Ehmann^18,19 formulated the state-space models of single- and multi-stand chatter and investigated the negative damping and regenerative effects on the stability of tandem mills. Heidari et al.²⁰ developed a chatter model of the cold strip rolling by considering unsteady lubrication and studied the effect of rolling lubricant on the chatter critical speed.

The chatter phenomenon in a cold rolling mill can be considered as a complicated coupling between the roll stand chatter and the strip vibration, following the understanding of the vertical vibrations of the work and backup rolls. As Johnson and Qi²¹ suggested, the interaction between the work roll and the rolled strip must be considered in order to develop an in-depth understanding of rolling chatter instability. The dynamic interaction between the work rolls and the metal sheet was studied by considering the transverse vibration of the strip coupled with the vertical vibration of the roll stand⁷ and incorporating a work roll sub-model and a metal sheet roll-bite sub-model.⁹ From a perspective different from the existing studies, this article considers the dynamic interaction of three types of vibrations, namely, the longitudinal vibration of the rolled strip and the torsional and vertical vibrations of the work roll. It should be noted that chatter vibrations can also occur in other manufacturing process including milling and grinding operations.^22–27

In strip rolling, the location of neutral plane is very important in that it governs the condition of the cold rolling process. Ideally, the neutral plane does not move along the arc of the roll contact if no vibration is involved in the rolling process. In the presence of rolling chatter, the position of neutral plane may vary along the arc, either on or outside the arc of the roll contact. Thus, an important technical issue needed to be addressed is under what conditions the neutral plane remains on the arc of the roll contact. This research issue has prominent practical relevance to ensuring a stable rolling operation.

This article aims to study the stability of the coupled vibrations. This article is organized into six sections. The details on the modelling of the coupled vibrations are developed in section ‘The modelling of the coupled vibrations’. The dynamic component of the rolling force is analytically obtained in section ‘Dynamic component of the rolling force’. Section ‘Stability of the coupled vibrations’ presents the stability analysis for the coupled vibrations by introducing a new variable. The numerical results are discussed in section ‘Numerical examples’, and finally section ‘Conclusion’ is given.

Modelling of the coupled vibrations

In order to study the dynamic interaction of the coupled vibrations, the mathematical models to be developed in this article include the vertical and torsional vibrations of the upper work roll, the longitudinal vibration of the rolled strip and the dynamic component of the rolling force. Assumptions will be made to simplify the modelling of vibrations in this section, and the equations of motion governing the coupled vibrations will be derived by applying Newton’s second law of motion.

Assumptions and simplifications

Chatter vibrations in cold rolling mills are a complex dynamic problem and remain not fully understood, to the author’s best knowledge. In order to overcome the theoretical intractability of chatter phenomenon, the assumptions and simplifications to be made in this article are given below:

It is assumed that the work roll experiences vertical and torsional vibrations only and its horizontal vibration is ignored.

The lower and upper parts of a roll stand are assumed to be symmetrical with reference to the roll gap. The dynamic conditions of both parts of a roll stand are considered to be identical. Consequently, only the upper work roll will be considered in subsequent analysis. Then, the roll stand can be approximately simplified to be a two-high stand, as shown in Figure 1(a).

The backup roll is lumped with the stand and bearings. Their collective effect on the work roll is represented by linear stiffness $k_{2}$ and damping coefficient $c_{2}$ , as shown in Figure 1(b). The rolled strip is considered to act as a spring–mass–damper system that undergoes longitudinal vibration only. The transverse vibration of the strip is assumed to be negligible.

The resultant frictional force and rolling force acting in the roll bite are assumed to apply in the middle of the arc of roll contact. As shown in Ginzburg and Ballas,³ the resultant frictional force can be considered to be proportional to the rolling force that is perpendicular to the strip surface.

Figure 1.

Dynamic modelling of the work roll and rolled strip: (a) simplified schematic diagram of a single two-high stand, (b) modelling of the work roll and strip segment as a lumped mass–spring–damper system, (c) free-body diagram of the strip segment and (d) free-body diagram of the upper work roll.

Equations of motion governing the coupled vibrations

Under the assumptions made in section ‘Assumptions and simplifications’, the structural dynamics of the mill stand can be modelled as a linear lumped parameter system with the upper work roll vibrating in vertical and torsional directions, and the dynamics of the rolled strip modelled as a mass–spring–damper system with the lumped mass vibrating in the horizontal direction, as shown in Figure 1(b). For simplification, the strip back and forward tensions are expressed by the corresponding equivalent back and front stiffness $k_{b}$ and $k_{f}$ . The strip segment in the roll gap and between two adjacent stands is treated as a block of lumped mass $m_{1}$ . The horizontal forces acting on the strip segment are the spring forces and the frictional force. The free-body diagram of the strip segment is shown in Figure 1(c), where x is the position of the equivalent lumped mass which is chosen in such a way that the origin $x = 0$ corresponds to the neutral point. The variables $\overset{\cdot}{x}$ and $\overset{\cdot\cdot}{x}$ are the first and second derivatives of displacement $x$ with respect to time t, which can be regarded as the velocity and acceleration of the motion of the rolled strip. Without loss of generality, a coordinate system is established using the Lagrangian inertial reference frame with respect to the steady-state velocity field. The origin is at the neutral plane on the strip surface when there is no vibration involved in the rolling process. The displacement $x$ is the location of the particle (i.e. the lumped mass of the strip segment) measured from the origin. During the oscillatory rolling process, this particle oscillates longitudinally along the strip surface.

Under steady-state normal rolling operation without vibrations engaged, the whole system is in the equilibrium state of static balance conditions including the normal rolling force and the frictional force. This is an ideal situation that may be rarely observed in industry. Once vibrations begin, the main source of excitation for vibrations comes from the variable part of the normal rolling force, while the corresponding static part is on the balance with its counterpart relevant to the initial conditions. Accordingly, it is the variable part of the frictional force that induces the torsional vibration of the upper work roll and the longitudinal vibration of the rolled strip. It should be noted that the vibrations can in turn change the variable component of the normal rolling force and then the variable part of the frictional force. For the horizontal vibration of the rolled strip, an application of Newton’s second law of motion leads to a second-order differential equation of the form

m_{1} \overset{\cdot\cdot}{x} = k_{f} x - k_{b} x + c_{f} \overset{\cdot}{x} - c_{b} \overset{\cdot}{x} + F

(1)

where $c_{f}$ and $c_{b}$ are the coefficients of viscous damping of the front and back parts of the rolled strip, and F represents the dynamic component of the frictional force between the work roll and strip and is given by equation (4).

The torsional and vertical vibrations of the work roll are considered while the horizontal vibration is ignored in this article but will be pursued in the future. The symmetry assumption of the mill stand made in section ‘Assumptions and simplifications’ renders the vibration analysis of the two-high stand by considering one work roll only. The free-body diagram of the upper work roll is shown in Figure 1(d), where $y$ , $\overset{\cdot}{y}$ and $\overset{\cdot\cdot}{y}$ are the displacement, velocity and acceleration of the upper work roll in the vertical direction. The parameters $ϕ$ , $\overset{\cdot}{ϕ}$ and $\overset{\cdot\cdot}{ϕ}$ are the angular displacement, velocity and acceleration of the work roll. The origin of angular displacement $ϕ$ is the location of the particle at the neutral plane on the peripheral surface of the work roll for the rolling process without vibrations occurring. The equation of motion governing the vertical vibration of the upper work roll can be obtained by applying Newton’s second law of motion as

m \overset{\cdot\cdot}{y} = - k_{2} y - c_{2} \overset{\cdot}{y} + P_{var}

(2)

where m is the lumped mass of the upper work roll, k ₂ represents the stiffness resulting from the contact with backup roll and bearings, $c_{2}$ denotes the coefficient of viscous damping and $P_{var}$ is the dynamic component of the rolling force which will be determined in the next section.

For the torsional vibration of the work roll, the equation of motion can be written as

J \overset{\cdot\cdot}{φ} = - k_{t} φ - c_{t} \overset{\cdot}{φ} - rF

(3)

where J is the mass moment of inertia of the work roll, $k_{t}$ is the torsional stiffness, $c_{t}$ is the damping coefficient of the torsional friction and r is the radius of the work roll. The frictional force F is assumed to be proportional to the dynamic component of the rolling force as

F = μ P_{var}

(4)

where $μ$ is the frictional coefficient of the contact surface between the work roll and strip and depends on the roll surface roughness, lubrication, rolled material and rolling conditions (such as workpiece temperature and roll speed).

Dynamic component of the rolling force

A number of different models for rolling force and roll torque have been developed over the past decades. Roberts¹, Yun et al.,² and Ginzburg and Ballas³ summarized several models developed in their text books and literature review article. Two of the widely used modelling methods are the slab analysis and the quasi-static model of the rolling process. The slab method is applicable to the steady-state rolling process only as it does not consider the dynamics of the rolling process induced by the vibrations of work rolls. A quasi-static model can incorporate the dynamics of the rolling process into the rolling models. Essentially, the quasi-static model method assumes that the dynamic component of the rolling force depends on the dynamic variations in roll gap and strip tension. Tlusty et al.²⁸ developed the dynamic model of the rolling force that consists of three variable components due to the variations of both strip tension and contact length. However, they assumed that the vertical vibration of sinusoidal function form is known beforehand in formulating the variable rolling force. Then they studied the vibrations of the work roll excited by the dynamic component of the rolling force. On the contrary, in this article, the coupled vibrations of the work roll and strip are assumed to be unknown before determining the variable rolling force using a quasi-static model. This assumption is more rational for studying the chatter problem because the vibrations engaged in the work rolls and strip will affect the rolling force exerting on the work roll and strip, and vice visa. Figure 2 shows a schematic of a rolled strip in a simple two-stand cold rolling mill for developing the quasi-static model. It is assumed that a strip of width $w$ leaves the gap between two work rolls of Stand 1 with velocity $v_{1}$ and thickness $h_{1}$ and exits the gap between two work rolls of Stand 2 with velocity $v_{2}$ and thickness $h_{2}$ . The fixed distance between two stands is L. The work rolls of Stand 2 have a radius r. Stand 1 does not engage vibrations, and only Stand 2 experiences vibrations, as assumed by Tlusty et al.²⁸ The neutral point in Stand 2 is on the exit plane passing through the roll axes. Figure 2 is a simplified case but contains all the pertinent feature of the phenomenon.

Figure 2.

Schematic of a two-stand quasi-static model of the rolling process.

The dynamic component of the rolling force is assumed to have four parts

P_{var} = P_{1 var} + P_{2 var} + P_{3 var} + P_{4 var}

(5)

where $P_{1 var}$ is the dynamic component due to the vertical vibration of the work roll. $P_{2 var}$ and $P_{3 var}$ are the dynamic components due to the variation of interstand tension, which is induced by the torsional vibration of the work roll and the horizontal vibration of the strip, respectively. $P_{4 var}$ is the variable component due to the variation of the contact length which is caused by the vertical vibration of the work roll and the longitudinal vibration of the rolled strip. Here, the basic ideas of determining the dynamic component of the rolling force are based on the fact that the coupled vibrations produce the variations of interstand tension and contact length, which will in turn generate variable components of the rolling force.

As suggested by Tlusty et al.,²⁸ and Swiatoniowski and Bar,⁷ the dynamic component of the rolling force due to the vertical vibration of the work roll can be expressed as

P_{1 var} = (S - σ_{m}) w [\sqrt{r (h_{d} - 2 y)} - \sqrt{{rh}_{d}}]

(6)

where $S$ is the yield stress of the rolled material, $σ_{m}$ is the average stress of the front and back tension in the strip, $w$ is the strip width, $r$ is the roll radius, $h_{d} = h_{1} - h_{2}$ is the reduction in strip thickness between the entry and exit of Stand 2 and $y$ is the displacement of vertical vibration of the work roll.

It should be noted that even in the presence of vibrations, the amplitude of the vertical vibration of the work roll is in practice far less than the thickness reduction of the strip, that is, $h_{d} >> y$ . It is thus reasonable to use Taylor series expansion to approximate the variable force $P_{1 var}$ . Expanding equation (6) in Taylor series about equilibrium $y = 0$ to the third-order terms yields

P_{1 var} = (S - σ_{m}) w \sqrt{{rh}_{d}} (- \frac{y}{h_{d}} - \frac{y^{2}}{2 h_{d}^{2}} - \frac{y^{3}}{2 h_{d}^{3}})

(7)

Equation (7) indicates that the first dynamic component of the rolling force is a non-linear function of the displacement of the vertical vibration of the work roll.

The contributions of the other two types of vibrations to the dynamic component of the rolling force will be studied by adopting some ideas introduced by Tlusty et al.²⁸ and Yun et al.² As shown in Figure 2, both the entry thickness $h_{1}$ and the exit thickness $h_{2}$ are assumed to be constant in the sense that the torsional vibration of the work roll and the longitudinal vibration of the rolled strip do not affect the reduction in strip thickness. By applying the principle of mass conservation, the flow of the strip through the roll gap of Stand 2 is given as

v_{1} h_{1} = v_{2} h_{2}

(8)

The exit speed $v_{2}$ is equal to the circumferential speed of the work roll and is changeable due to both the torsional vibration of the work roll and the longitudinal vibration of the rolled strip. At the entry side, the roll surface speed is lower than the peripheral speed of the work roll, while at the exit side the roll speed is higher. In terms of the pressure distributions, the shear stress between the work roll and strip is tangent to the roll surface and variable in magnitude, and changes its sense at the neutral point as demonstrated in the text book of Talbert and Avitzur.²⁹ In the presence of torsional vibration only, the exit speed $v_{2}$ can be written as

v_{2} = r (ω_{0} + \overset{\cdot}{φ})

(9)

Here, the longitudinal vibration of the rolled strip is ignored at this stage, and its contribution to the interstand tension will be discussed in the component $P_{3 var}$ , where the torsional vibration of the work roll will be ignored.

If no torsional vibration happens, the nominal exit speed, denoted by $v_{20}$ , is given by $v_{20} = r ω_{0}$ , where $ω_{0}$ is the steady-state angular velocity of the work roll under steady-state normal operating conditions. Substitution of equation (9) into equation (8) yields

v_{1} = \frac{(v_{20} + r \overset{\cdot}{φ}) h_{2}}{h_{1}}

(10)

It is shown that the entry speed of the strip has a mean and variable component. It follows correspondingly that the variable entry speed is given by

v_{1 var} = \frac{h_{2} r \overset{\cdot}{φ}}{h_{1}}

(11)

where $v_{1 var}$ represents the variation of strip entry speed.

The variable entry speed $v_{1 var}$ produces a changeable elongation of the rolled strip between Stands 1 and 2, which is the integral of the variable speed

L_{var} = \int_{0}^{t} v_{1 var} dt = \int_{0}^{t} (\frac{h_{2} r \overset{\cdot}{φ}}{h_{1}}) dt = \frac{h_{2} r φ}{h_{1}}

(12)

where $L_{var}$ is the elongation of the rolled strip. Here, the initial angular displacement of the torsional vibration of the work roll is assumed to be $φ_{0} = 0$ .

Since the neutral point has been assumed to be at the exit of the roll bite, the contribution from the variable strip tension at the exit of Stand 2 to the dynamic component of the rolling force is negligible and thus ignored here. The stretch of the rolled strip produces a variable strip tension at the entry of Stand 2. The variation of the tension stress is given by

σ_{var} = E \frac{L_{var}}{L} = \frac{{Eh}_{2} r φ}{{Lh}_{1}}

(13)

where $E$ is Young’s modulus of the strip material.

The dynamic component of the rolling force due to the variable interstand tension can be obtained by borrowing the formulation given by Tlusty et al.²⁸ and expressed as

P_{2 var} = - σ_{var} w \sqrt{{rh}_{d}} = - \frac{{Eh}_{2} r φ w \sqrt{{rh}_{d}}}{{Lh}_{1}}

(14)

The dynamic component $P_{3 var}$ due to the longitudinal vibration of the rolled strip can be studied in the same manner as $P_{2 var}$ and is found to be

P_{3 var} = - \frac{{Eh}_{2} xw \sqrt{{rh}_{d}}}{{Lh}_{1}}

(15)

The presence of the vertical vibration of the work roll and the longitudinal vibration of the rolled strip produces the variation of the contact length at the exit of the strip. During an instant $Δ t$ , the work roll moves upward by the amount $Δ y$ and the rolled strip moves to the right by $Δ x$ . The contact length change $Δ l$ due to the two incremental motions is

Δ l = - \frac{r Δ y}{Δ x} = - \frac{r \overset{\cdot}{y}}{v_{20} + \overset{\cdot}{x}}

(16)

where the negative sign indicates the shortness of the contact length.

The dynamic component $P_{4 var}$ can be expressed using the formula proposed by Tlusty et al.²⁸ as

P_{4 var} = - \frac{(S - σ_{m}) wr \overset{\cdot}{y}}{v_{20} + \overset{\cdot}{x}}

(17)

Applying Taylor series expansion about the nominal speed $v_{20}$ by considering the fact that $\overset{\cdot}{x} << v_{20}$ gives rise to

P_{4 var} = - (S - σ_{m}) wr (\frac{\overset{\cdot}{y}}{v_{20}} - \frac{\overset{\cdot}{x} \overset{\cdot}{y}}{v_{20}^{2}} + \frac{{\overset{\cdot}{x}}^{2} \overset{\cdot}{y}}{v_{20}^{3}})

(18)

Here, only the first three terms are kept in the series.

Substituting equations (7), (14), (15) and (18) into equation (5) leads to the resultant dynamic component of the rolling force

\begin{matrix} P_{var} = & p_{1} y + p_{2} y^{2} + p_{3} y^{3} + p_{4} φ \\ + p_{5} x + p_{6} \overset{\cdot}{y} + p_{7} \overset{\cdot}{x} \overset{\cdot}{y} + p_{8} {\overset{\cdot}{x}}^{2} \overset{\cdot}{y} + α_{2} {\overset{\cdot}{y}}^{3} \end{matrix}

(19)

where

\begin{matrix} p_{1} = - \frac{(S - σ_{m}) w \sqrt{{rh}_{d}}}{h_{d}}, p_{2} = - \frac{(S - σ_{m}) w \sqrt{{rh}_{d}}}{2 h_{d}^{2}}, \\ p_{3} = - \frac{(S - σ_{m}) w \sqrt{{rh}_{d}}}{2 h_{d}^{3}}, p_{4} = - \frac{{Eh}_{2} wr \sqrt{{rh}_{d}}}{{Lh}_{1}}, \\ p_{5} = - \frac{{Eh}_{2} w \sqrt{{rh}_{d}}}{{Lh}_{1}}, p_{6} = - \frac{(S - σ_{m}) wr}{v_{20}}, \\ p_{7} = \frac{(S - σ_{m}) wr}{v_{20}^{2}}, p_{8} = - \frac{(S - σ_{m}) wr}{v_{20}^{3}} . \end{matrix}

The term $α_{2} {\overset{\cdot}{y}}^{3}$ has been introduced to account for the non-linear dissipative property of the dynamic rolling force as given by Bar and Bar,¹⁰ and $α_{2}$ denotes the non-linear damping coefficient. It is easy to notice that the dynamic component of the rolling force is non-linear in nature and depends on the coupled vibrations of the work roll and strip.

Stability of the coupled vibrations

The dynamic behaviour of rolling chatter can be examined by studying equations (1)–(3). These three equations will be rewritten by introducing a new variable in section ‘Introduction of a new variable’. The stability of the coupled vibrations will then be examined in section ‘Stability analysis’.

It is known that the neutral plane (point) plays an important role in strip rolling. As discussed by Yuen et al.,³⁰ when the neutral plane is at or beyond the roll-bite exit, there is not enough traction to pull the strip through the roll bite. This phenomenon is known as skidding. When the neutral plane is at or ahead of the roll-bite entry, the strip is drawn through the roll bite and moves faster than the roll throughout the entire contact region. This phenomenon, although rare, may occur under certain cold rolling conditions and is known as slipping. During the steady-state rolling process, the neutral plane is at a certain point on the arc of the roll contact. Ideally, the neutral plane does not move along the arc of contact if no vibration is involved. Practically, in the presence of rolling chatter, the neutral plane may either stay at or move forward and backward about its original position. A larger vibrational velocity of either work roll or strip has the potential to move the neutral plane out of the arc of contact.

Introduction of a new variable

In section ‘The modelling of the coupled vibrations’, the origin of the longitudinal displacement of the rolled strip and the torsional displacement of the upper work roll is assumed to be at the location of the neutral plane for the rolling operation without vibrations occurring. This origin will be referred to here as the original neutral plane (point). For the purpose of comparison, the neutral plane for the rolling operation involving vibrations will be referred to here as the quasi-neutral point. The vibrations of the work roll and strip can cause the quasi-neutral point moving forward and backward about the position of the original neutral plane, towards either the entry side or the exit side of the roll bite. The new position of the quasi-neutral plane is at best indicated by the displacement of motion from its origin. It is thus meaningful to introduce a new variable to represent the movement of the quasi-neutral plane in the rolling process. This new parameter can easily track the motion of the quasi-neutral point along the arc of roll contact.

When vibrations occur in the rolling process, the peripheral velocity of the work roll is given by

v_{p} = ω_{0} r + r \overset{\cdot}{φ}

(20)

where $ω_{0}$ represents the angular velocity of the work roll under non-vibrational rolling operation, r is the radius of the work roll and $\overset{\cdot}{ϕ}$ denotes the angular velocity of the torsional vibration of the work roll.

The surface velocity of the rolled strip in the presence of longitudinal vibration is

v_{s} = v_{0} + \overset{\cdot}{x}

(21)

where $v_{0}$ is the strip velocity under normal operation condition and $\overset{\cdot}{x}$ represents the strip velocity of the longitudinal vibration.

The velocity of the quasi-neutral point is the difference of the peripheral velocity and the strip velocity that are given by equations (20) and (21)

v_{var} = v_{p} - v_{s} = (r ω_{0} - v_{0}) + r \overset{\cdot}{φ} - \overset{\cdot}{x}

(22)

The first term vanishes under normal rolling conditions without vibrations engaged. The second term is related to the torsional vibration of the work roll and the longitudinal vibration of the rolled strip. The velocity of the quasi-neutral point is then expressed as

v_{var} = r \overset{\cdot}{φ} - \overset{\cdot}{x}

(23)

In the presence of vibrations, the speed of the quasi-neutral point may not be equal to 0. Only if the kinematic relation $\overset{\cdot}{x} = r \overset{\cdot}{φ}$ holds, the speed of the quasi-neutral point would be equal to 0, that is, $v_{var} = 0$ . This case indicates that as long as the kinematic relation holds, the neutral point will stay at its original position even if both work roll and strip engage vibrations.

A new variable $u$ is introduced here to denote the motion of the quasi-neutral point along the arc of the roll contact in terms of $u = r φ - x$ , $\overset{\cdot}{u} = r \overset{\cdot}{φ} - \overset{\cdot}{x}$ and $\overset{\cdot\cdot}{u} = r \overset{\cdot\cdot}{φ} - \overset{\cdot\cdot}{x}$ . Accordingly, the equation $\overset{\cdot}{u} = 0$ indicates that the neutral point will stay at its original position, whereas the expression $\overset{\cdot}{u} \neq 0$ implies that the quasi-neutral point will move about its original position. If the velocity $\overset{\cdot}{u}$ converges to 0 with time, the quasi-neutral point will return its original position within a limited period of time; otherwise, the quasi-neutral point will move away from its original position, leading to an unstable rolling process. Introducing the variable $u$ into equations (1)–(3) and re-organising the resultant equations give rise to the following equations

\begin{matrix} \overset{\cdot\cdot}{y} = - \frac{k_{2}}{m} y - \frac{c_{2}}{m} \overset{\cdot}{y} + \frac{P_{var}}{m} \\ \overset{\cdot\cdot}{φ} = - \frac{k_{t}}{J} φ - \frac{c_{t}}{J} \overset{\cdot}{φ} - \frac{{rF}_{var}}{J} \\ \overset{\cdot\cdot}{u} = - \frac{k_{1}}{m_{1}} u - \frac{c_{1}}{m_{1}} \overset{\cdot}{u} + (\frac{k_{1}}{m_{1}} - \frac{k_{t}}{J}) r φ \\ + (\frac{c_{1}}{m_{1}} - \frac{c_{t}}{J}) r \overset{\cdot}{φ} - (\frac{1}{m_{1}} + \frac{r^{2}}{J}) F_{var} \end{matrix}

(24)

where $k_{1} = k_{b} - k_{f}$ and $c_{1} = c_{b} - c_{f}$ . It is easy to notice that in addition to the implicit cross-terms in forces $P_{var}$ and $F_{var}$ , there are two cross-terms explicitly involving $φ$ and $\overset{\cdot}{φ}$ in the third equation, which indicates the interaction between the torsional vibration of the work roll and the longitudinal vibration of the rolled strip.

Substituting equations (4) and (19) into equation (24) and simplifying the resultant equations yield

\begin{matrix} \overset{\cdot\cdot}{y} = & a_{1} y - a_{2} \overset{\cdot}{y} + a_{3} φ + a_{4} u + a_{5} y^{2} + a_{6} y^{3} \\ + a_{7} {\overset{\cdot}{y}}^{3} + a_{8} \overset{\cdot}{y} \overset{\cdot}{φ} + a_{9} \overset{\cdot}{y} \overset{\cdot}{u} + a_{10} {\overset{\cdot}{φ}}^{2} \overset{\cdot}{y} + a_{11} \overset{\cdot}{y} \overset{\cdot}{u} \overset{\cdot}{φ} \\ + a_{12} \overset{\cdot}{y} {\overset{\cdot}{u}}^{2} \\ \overset{\cdot\cdot}{φ} = & b_{1} φ - b_{2} \overset{\cdot}{φ} + b_{3} y + b_{4} \overset{\cdot}{y} + b_{5} u + b_{6} y^{2} \\ + b_{7} y^{3} + b_{8} {\overset{\cdot}{y}}^{3} + b_{9} \overset{\cdot}{y} \overset{\cdot}{φ} + b_{10} \overset{\cdot}{y} \overset{\cdot}{u} + b_{11} \overset{\cdot}{y} {\overset{\cdot}{φ}}^{2} \\ + b_{12} \overset{\cdot}{y} \overset{\cdot}{u} \overset{\cdot}{φ} + b_{13} \overset{\cdot}{y} {\overset{\cdot}{u}}^{2} \\ \overset{\cdot\cdot}{u} = & g_{1} u - g_{2} \overset{\cdot}{u} + g_{3} y + g_{4} \overset{\cdot}{y} + g_{5} φ + g_{6} \overset{\cdot}{φ} \\ + g_{7} y^{2} + g_{8} y^{3} + g_{9} {\overset{\cdot}{y}}^{3} + g_{10} \overset{\cdot}{y} \overset{\cdot}{φ} \\ + g_{11} \overset{\cdot}{y} \overset{\cdot}{u} + g_{12} \overset{\cdot}{y} {\overset{\cdot}{φ}}^{2} + g_{13} \overset{\cdot}{y} \overset{\cdot}{u} \overset{\cdot}{φ} + g_{14} \overset{\cdot}{y} {\overset{\cdot}{u}}^{2} \end{matrix}

(25)

where

\begin{matrix} a_{1} = \frac{p_{1} - k_{2}}{m}, a_{2} = \frac{c_{2} - p_{6}}{m}, a_{3} = \frac{p_{4} + {rp}_{5}}{m}, \\ a_{4} = - \frac{p_{5}}{m}, a_{5} = \frac{p_{2}}{m}, a_{6} = \frac{p_{3}}{m}, a_{7} = \frac{α_{2}}{m}, \\ a_{8} = \frac{{rp}_{7}}{m}, a_{9} = - \frac{p_{7}}{m}, a_{10} = \frac{r^{2} p_{8}}{m}, a_{11} = - \frac{2 {rp}_{8}}{m}, \\ a_{12} = \frac{p_{8}}{m}, \\ b_{1} = - \frac{k_{t} + μ r (p_{4} + {rp}_{5})}{J}, b_{2} = \frac{c_{t}}{J}, b_{3} = - \frac{r μ p_{1}}{J}, \\ b_{4} = - \frac{r μ p_{6}}{J}, b_{5} = \frac{r μ p_{5}}{J}, b_{6} = - \frac{r μ p_{2}}{J}, \\ b_{7} = - \frac{r μ p_{3}}{J}, b_{8} = - \frac{r μ α_{2}}{J}, b_{9} = - \frac{r^{2} μ p_{7}}{J}, \\ b_{10} = \frac{r μ p_{7}}{J}, b_{11} = - \frac{r^{3} μ p_{8}}{J}, b_{12} = \frac{2 r^{2} μ p_{8}}{J}, \\ b_{13} = - \frac{r μ p_{8}}{J}, \\ g_{1} = \frac{μ p_{5} - k_{1}}{m_{1}} + \frac{r^{2} μ p_{5}}{J}, g_{2} = \frac{c_{1}}{m_{1}}, \\ g_{3} = - \frac{μ p_{1}}{m_{1}} - \frac{r^{2} μ p_{1}}{J}, g_{4} = - \frac{μ p_{6}}{m_{1}} - \frac{r^{2} μ p_{6}}{J}, \\ g_{5} = \frac{{rk}_{1} - μ (p_{4} + {rp}_{5})}{m_{1}} - \frac{{rk}_{t} + r^{2} μ (p_{4} + {rp}_{5})}{J}, \\ g_{6} = \frac{{rc}_{1}}{m_{1}} - \frac{{rc}_{t}}{J}, g_{7} = - \frac{μ p_{2}}{m_{1}} - \frac{r^{2} μ p_{2}}{J}, \\ g_{8} = - \frac{μ p_{3}}{m_{1}} - \frac{r^{2} μ p_{3}}{J}, g_{9} = - \frac{μ α_{2}}{m_{1}} - \frac{r^{2} μ α_{2}}{J}, \\ g_{10} = - \frac{r μ p_{7}}{m_{1}} - \frac{r^{3} μ p_{7}}{J}, g_{11} = \frac{μ p_{7}}{m_{1}} + \frac{r^{2} μ p_{7}}{J}, \\ g_{12} = - \frac{r^{2} μ p_{8}}{m_{1}} - \frac{r^{4} μ p_{8}}{J}, g_{13} = \frac{2 r μ p_{8}}{m_{1}} + \frac{2 r^{3} μ p_{8}}{J}, \\ g_{14} = - \frac{μ p_{8}}{m_{1}} - \frac{r^{2} μ p_{8}}{J} \end{matrix}

Here, the variable $x$ involved in forces $P_{var}$ and $F_{var}$ has been replaced by the new variable $u$ and variable $φ$ .

Stability analysis

In order to study the stability and later perform numerical integration, the three second-order coupled differential equations given by equation (25) should be rewritten as a system of the first-order coupled differential equations. This can be achieved by introducing new variables $z_{j}$ (with $j = 1, 2, \dots, 6$ ) in terms of $z_{1} = y$ , $z_{2} = \overset{\cdot}{y}$ , $z_{3} = φ$ , $z_{4} = \overset{\cdot}{φ}$ , $z_{5} = u$ and $z_{6} = \overset{\cdot}{u}$ into equation (25). Then, equation (25) becomes

\begin{matrix} {\overset{\cdot}{z}}_{1} = z_{2} \\ {\overset{\cdot}{z}}_{2} = a_{1} z_{1} - a_{2} z_{2} + a_{3} z_{3} + a_{4} z_{5} \\ + f_{1} (z_{1}, z_{2}, z_{3}, z_{4}, z_{5}, z_{6}) \\ {\overset{\cdot}{z}}_{3} = z_{4} \\ {\overset{\cdot}{z}}_{4} = b_{1} z_{3} - b_{2} z_{4} + b_{3} z_{1} + b_{4} z_{2} + b_{5} z_{5} \\ + f_{2} (z_{1}, z_{2}, z_{3}, z_{4}, z_{5}, z_{6}) \\ {\overset{\cdot}{z}}_{5} = z_{6} \\ {\overset{\cdot}{z}}_{6} = g_{1} z_{5} - g_{2} z_{6} + g_{3} z_{1} + g_{4} z_{2} + g_{5} z_{3} \\ + g_{6} z_{4} + f_{3} (z_{1}, z_{2}, z_{3}, z_{4}, z_{5}, z_{6}) \end{matrix}

(26)

where the non-linear functions $f_{i} (z_{1}, z_{2}, z_{3}, z_{4}, z_{5}, z_{6})$ are given by

\begin{matrix} f_{1} (z_{1}, z_{2}, z_{3}, z_{4}, z_{5}, z_{6}) = a_{5} z_{1}^{2} + a_{6} z_{1}^{3} + a_{7} z_{2}^{3} + a_{8} z_{2} z_{4} \\ + a_{9} z_{2} z_{6} + a_{10} z_{2} z_{4}^{2} + a_{11} z_{2} z_{4} z_{6} + a_{12} z_{2} z_{6}^{2}, \\ f_{2} (z_{1}, z_{2}, z_{3}, z_{4}, z_{5}, z_{6}) = b_{6} z_{1}^{2} + b_{7} z_{1}^{3} + b_{8} z_{2}^{3} \\ + b_{9} z_{2} z_{4} + b_{10} z_{2} z_{6} + b_{11} z_{2} z_{4}^{2} + b_{12} z_{2} z_{4} z_{6} + b_{13} z_{2} z_{6}^{2}, \\ f_{3} (z_{1}, z_{2}, z_{3}, z_{4}, z_{5}, z_{6}) = g_{7} z_{1}^{2} + g_{8} z_{1}^{3} + g_{9} z_{2}^{3} \\ + g_{10} z_{2} z_{4} + g_{11} z_{2} z_{6} + g_{12} z_{2} z_{4}^{2} + g_{13} z_{2} z_{4} z_{6} + g_{14} z_{2} z_{6}^{2} \end{matrix}

The non-linear system given by equation (26) admits the trivial (zero) solution, $z_{j} = 0$ . This trivial solution can be regarded as the equilibrium of the motion of the quasi-neutral point along the arc of contact. The zero solution of equation (26) is stable only if all the eigenvalues of the corresponding characteristic equation have negative real parts. The stability conditions of the trivial equilibrium of equation (26) can be studied in a way that is presented in Rao.³¹ It is assumed that equation (26) admits a set of solutions of the form in the neighbourhood of the trivial solution

z_{j} = A_{j} e^{st}, j = 1, 2, \dots, 6

(27)

where $s$ is a complex number to be determined and $A_{j}$ is the amplitude of the solution $z_{j}$ . Substituting equation (27) into equation (26) and letting the determinant of the coefficient matrix be equal to 0 lead to the characteristic equation

f (s) = s^{6} + q_{0} s^{5} + q_{1} s^{4} + q_{2} s^{3} + q_{3} s^{2} + q_{4} s + q_{5}

(28)

where

q_{0} = a_{2} + b_{2} + g_{2}

q_{1} = - a_{1} - b_{1} - g_{1} + a_{2} b_{2} + a_{2} g_{2} + b_{2} g_{2}

\begin{matrix} q_{2} = & - a_{2} b_{1} - a_{1} b_{2} - a_{3} b_{4} - a_{2} g_{1} - b_{2} g_{1} \\ - a_{1} g_{2} - b_{1} g_{2} - b_{5} g_{6} - a_{4} g_{4} + a_{2} b_{2} g_{2} \end{matrix}

\begin{matrix} q_{3} = & a_{1} b_{1} + a_{1} g_{1} + b_{1} g_{1} - a_{3} b_{3} - a_{4} g_{3} - b_{5} g_{5} \\ - a_{2} b_{2} g_{1} - a_{2} b_{1} g_{2} - a_{1} b_{2} g_{2} - a_{2} b_{5} g_{6} - a_{3} b_{4} g_{2} \end{matrix}

- a_{4} b_{2} g_{4} - a_{4} b_{4} g_{6}

\begin{matrix} q_{4} = & a_{2} b_{1} g_{1} + a_{1} b_{2} g_{1} + a_{1} b_{1} g_{2} - a_{3} b_{3} g_{2} - a_{4} b_{2} g_{3} \\ - a_{2} b_{5} g_{5} - a_{4} b_{3} g_{6} + a_{1} b_{5} g_{6} + a_{3} b_{4} g_{1} \end{matrix}

+ a_{4} b_{1} g_{4} - a_{3} b_{5} g_{4} - a_{4} b_{4} g_{5}

\begin{matrix} q_{5} = & - a_{1} b_{1} g_{1} + a_{3} b_{3} g_{1} + a_{4} b_{1} g_{3} - a_{3} b_{5} g_{3} \\ - a_{4} b_{3} g_{5} + a_{1} b_{5} g_{5} \end{matrix}

According to the criterion of Routh-Hurwitz,³² all roots of the polynomial $f (s)$ have negative real parts if and only if the following determinantal inequalities hold

\begin{matrix} | q_{0} | > 0, | \begin{matrix} q_{0} & q_{2} \\ 1 & q_{1} \end{matrix} | > 0, | \begin{matrix} q_{0} & q_{2} & q_{4} \\ 1 & q_{1} & q_{3} \\ 0 & q_{0} & q_{2} \end{matrix} | > 0, \\ | \begin{matrix} q_{0} & q_{2} & q_{4} & 0 \\ 1 & q_{1} & q_{3} & q_{5} \\ 0 & q_{0} & q_{2} & q_{4} \\ 0 & 1 & q_{1} & q_{3} \end{matrix} | > 0 | \begin{matrix} q_{0} & q_{2} & q_{4} & 0 & 0 \\ 1 & q_{1} & q_{3} & q_{5} & 0 \\ 0 & q_{0} & q_{2} & q_{4} & 0 \\ 0 & 1 & q_{1} & q_{3} & q_{5} \\ 0 & 0 & q_{0} & q_{2} & q_{4} \end{matrix} | > 0, \\ | \begin{matrix} q_{0} & q_{2} & q_{4} & 0 & 0 & 0 \\ 1 & q_{1} & q_{3} & q_{5} & 0 & 0 \\ 0 & q_{0} & q_{2} & q_{4} & 0 & 0 \\ 0 & 1 & q_{1} & q_{3} & q_{5} & 0 \\ 0 & 0 & q_{0} & q_{2} & q_{4} & 0 \\ 0 & 0 & 1 & q_{1} & q_{3} & q_{5} \end{matrix} | > 0 \end{matrix}

(29)

The first determinant simply states that the coefficient $q_{0}$ should be positive, which is automatically satisfied as long as the three damping coefficients in equation (25) are positive. The necessary and sufficient condition for the stable trivial solution is that all the coefficients are positive and the last five determinants defined in equation (29) are positive. The Routh-Hurwitz conditions defined by equation (29) will be used to study the stability of the trivial solution and to obtain the stability boundary for certain combinations of system parameters. Stable vibrations indicate that the quasi-neutral plane initially moves around the original position of the neutral plane and returns to the original position within a limited period of time, which is desirable in rolling operation. Unstable vibrations imply that the vibration amplitude increases with time and the quasi-neutral plane moves away from its original position. Theoretically, if no action is taken, the amplitude of unstable vibrations will become very large within a certain period of time, eventually to such an extent that the non-linear effects become significant and the dynamic response of the rolling process will be strongly non-linear. Practically, due to the restrictions of roll stand to the vibration amplitude, unstable vibrations would produce large-amplitude dynamic rolling force and potentially damage the roll stand. Unstable vibrations, which are unwanted in the rolling process, have the tendency to move the neutral point out of the arc of roll contact, which may lead to skidding or slipping phenomena or even break the strip.

Numerical examples

Numerical simulations using MATLAB are carried out to obtain the dynamic behaviour and stability boundary of the coupled vibrations by integrating equation (26) and evaluating the inequality equation (29). The values of system parameters used are given in Appendix 1, unless otherwise specified (Tables 1 –3).

Table 1.

Parameters used in calculating the dynamic part of rolling force.

Entry thickness of the strip	$h_{1} = 0.0013 m$
Exit thickness of the strip	$h_{2} = 0.0010 m$
Reduction in strip thickness	$h_{d} = h_{1} - h_{2} = 0.0003 m$
Difference of yield and axial stresses	$(S - σ_{m}) = 550 MPa$
The modulus of elasticity (Young’s modulus)	$E = 200.0 GPa$
Width of strip	$w = 1.2 m$
Distance between two stands	$L = 3.2 m$
Diameter of work roll	$d = 0.43 m$
The coefficient of non-linear dissipative term	$α_{2} = 150.0 N {(s / m)}^{3}$

Table 2.

Parameters relevant to the rolled strip.

Front tension of the rolled strip	$80 MPa$
Back tension of the rolled strip	$160 MPa$
Damping coefficient of the rolled strip	$c_{s} = 188.59 N (s / m)$

Table 3.

Parameters related to the work roll.

Equivalent stiffness supporting the work roll	$k_{2} = 1.4 \times 10^{9} N / m$
Damping coefficient of the work roll	$c_{2} = 3.58 \times 10^{4} N (s / m)$
Mass moment of inertia of the work roll	$J = 36.94 kg m^{2}$
Torsional stiffness of the work roll	$k_{t} = 2.82 \times 10^{5} N / rad$
Coefficient of the torsional friction	$c_{t} = 77.44 N (s / rad)$

Figure 3(a) shows the variation of the strip front tension with the frictional coefficient $μ$ on the stability boundary. In this figure and Figure 3(b), the solid line connecting the circles indicates the stability boundary and the circles are the points obtained from stability condition (29) using numerical methods. The coupled vibrations are stable for the set of system parameters chosen out of the regime below the curve and unstable for the parameters above the curve. The coupled vibrations will decrease as time goes and eventually die out for the rolling operation in the stable regime. However, if the system parameters of the rolling operation fall within the unstable regime, the coupled vibrations will grow in amplitude as time goes and are likely to become severe in a short period of time. Unstable vibrations may move the quasi-neutral plane out of the arc of roll contact, thereby resulting in skidding or slipping phenomena. Actions must be taken before the unstable vibrations lead to damage of the roll stand and break the rolled strip. A decrease in the forward tension requires increasing the frictional coefficient to ensure a stable rolling operation. This is reasonable because a decrease in the forward tension will move the neutral plane towards the entry side. In order to ensure a stable rolling condition, a larger frictional force is required to pull the neutral plane forward along the arc of the roll contact. Figure 3(b) shows the variation of roll stiffness with the frictional coefficient. The coupled vibrations are unstable for the rolling operation in the regime above the curve and stable for the system parameters chosen below the curve. A decrease in roll stiffness requires an increase in the frictional coefficient to ensure the stability of the coupled motion. This is rational in that a smaller roll stiffness would generate a smaller normal rolling force, and thus a larger frictional coefficient is required for producing the frictional force necessary to pull the rolled strip into the roll bite. For high values of frictional coefficients, rolling operation requires soft roll stiffness to ensure a stable rolling condition. This prediction is qualitatively in good agreement with the measured results observed in industry.³⁰

Figure 3.

Stability boundary of the coupled vibrations: (a) variation of the front tension with the frictional coefficient and (b) variation of the roll stiffness with the frictional coefficient.

To validate these analytical results, numerical integrations are performed to equation (26). In carrying out numerical integration, the values of system parameters are picked up out of the regions above and below the stability boundary curve as shown in Figure 3(a). The results are illustrated in Figure 4 for two values of the front tension stresses σ_f = 60 and 210 MPa, respectively. The trivial solution is asymptotically stable when the front tension is less than the critical value and unstable when the front tension is greater than the critical value. As shown in Figure 4(a), the three types of vibrations decrease their amplitudes as time goes and eventually die out in a short period of time. The motion of quasi-neutral point also gradually converges to 0 as time goes, indicating that the quasi-neutral plane remains along the arc of roll contact and rests on its original position. On the contrary, the unstable vibration eventually goes to infinity after some periods of oscillation. As shown in Figure 4(b), the vibrations starting from a very small value of initial conditions increase their amplitudes very quickly. Within t=1.4 s, the amplitudes of the torsional and vertical vibrations increase from the order of $10^{- 6} m$ to the order of $10^{- 5} m$ and will further increase exponentially thereafter. For the motion of the quasi-neutral point, the vibration amplitude increases from the order of $10^{- 6} m$ to the order of $10^{- 5} m$ . As the time further increases, the vibration amplitude will become larger and larger, which can lead to the quasi-neutral point moving out of the roll contact. The unstable vibration indicates the occurrence of skidding or slipping. In terms of the frequency components involved, a frequency spectrum analysis indicates that the vibrations contain three frequency components, 13.73, 45.78 and 222.8 Hz. The lower and higher frequencies can be considered as the frequencies of the torsional vibration and the third-octave mode of the work roll, while the middle frequency is the frequency of the motion of the quasi-neutral plane. Different values of the front and back tensions will result in different frequencies of the quasi-neutral point.

Figure 4.

Stable and unstable coupled vibrations: (a) stable vibrations for the front tension stress σ _f = 60 MPa and (b) unstable vibrations for the front tension stress σ _f = 120 MPa.

The numerical simulations of stability condition (29) also suggest that the stability boundary is strongly dependent on the system parameters. For some combinations of system parameters, the stability boundary curves are different in terms of the curve shapes. It is also found by numerical simulations that the stability of the trivial solution is sensitive to the initial conditions of integration. Different sets of initial values can lead to distinct dynamic behaviour of the system, namely, stable and unstable vibrations. The underlying mechanisms would be the sensitivity of the rolling process on the rolling conditions. From the perspective of dynamic analysis, the dynamic system given by equation (26) is non-linear in nature and may have fractal basin of attraction. This indicates that the stability of rolling operation depends on the initial conditions. For some sets of initial conditions, the rolling chatter is stable, while for others the rolling chatter is unstable even if under the similar set of system parameters. This may be the reason for the fact observed in practice that chatter occurs in the same stand under similar operating conditions on different dates. Specifically, no chatter is observed on one day and chatter happens on another day when rolling the same materials under the nearly same conditions.

Conclusion

A chatter model incorporating the structural dynamics and roll gap dynamics has been developed in order to study the dynamic interactions of the longitudinal vibration of rolled strip and the vertical and torsional vibrations of the work roll. It was shown that these three types of vibrations are strongly coupled in a non-linear manner.

For the rolling process under chatter conditions, a stability criterion was obtained for the motion of the quasi-neutral point. The effects of the frictional coefficient on the stability of the coupled vibrations were studied under variations of the front tension and roll stiffness. A decrease in the forward tension requires increasing the frictional coefficient to ensure a stable rolling operation. A decrease in roll stiffness also requires an increase in the frictional coefficient to ensure the stability of the coupled vibrations. Stable vibrations indicate that the quasi-neutral plane moves towards the original position of the neutral plane and eventually locates at the original position. As a result, the peripheral speed of the work roll and roll surface speed have a tendency to be identical again, although their values may be changed due to vibrations engaged. An unstable vibration grows its amplitude quickly and leads to excessive tension fluctuation if no action is taken. Unstable vibrations imply the occurrence of skidding or slipping phenomena and may eventually damage the roll stand or break the rolled strip. To avoid unstable vibrations, the rolling process should be operated in the stability region. Rolling chatter is a complex dynamic phenomenon. The stability boundary and dynamic behaviour of the coupled vibrations strongly depend on the system parameters and rolling conditions. For some combinations of system parameters, different sets of initial conditions (rolling conditions) can lead to different dynamic behaviour, either stable or unstable vibrations.

Footnotes

Appendix 1

The values of system parameters that are used in numerical simulations are based on a four-stand cold rolling process, which is currently operating in a strip production company. Due to the commercially-in-confidential nature, the values of certain parameters are slightly changed from the industry data. Some data are obtained by performing finite element analysis of the stand and rolling process using the design parameters and operation conditions. For the sake of brevity, the detailed procedure is not reproduced here. The SI units of the system parameters are used here instead of the units that are preferred by the engineers and workers in the production company.

The front and back stiffness can be obtained using formulae $k_{b} = σ_{b} A_{b} / L_{b var}$ and $k_{f} = σ_{f} A_{f} / L_{f var}$ , where $σ_{b}$ and $σ_{f}$ are the strip back and front tension stresses, $A_{b}$ and $A_{f}$ are the cross-sectional areas of the strip at entry and exit sides, and $L_{b var}$ and $L_{f var}$ are the variations of strip length at entry and exit due to the back and front tensions, respectively. The damping coefficient of the rolled strip is equivalent to the non-dimensional damping coefficient $ξ_{s} = 0.001$ .

The damping coefficient of the work roll is equivalent to the non-dimensional coefficient of damping $ξ_{2} = 0.002$ . The coefficient of the torsional friction is equivalent to the non-dimensional frictional coefficient $ξ_{t} = 0.016$ .

Declaration of conflicting interest

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

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