# Dynamic Model of the Hot Strip Rolling Mill Vibration Resulting from Entry Thickness Deviation and Its Dynamic Characteristics

Authors: Xinxin Wang; Xiaoqiang Yan
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

The vibration demolding of the crystallizer leads to slab thickness deviation in continuous casting processes. Therefore, in this paper, the dynamic model of the hot strip rolling mill vibration resulting from entry thickness deviation is proposed and its dynamic characteristics are studied. First, the dynamic model of vertical vibration in the hot strip rolling mill is established based on Sims’ rolling force model. Then, the model validity is certified by comparing the simulation with field test data from a 1580 hot strip rolling mill. Finally, we investigate the influence of model parameters on the dynamic characteristics of hot strip rolling mill vibration resulting from entry thickness deviation. The result indicates that hot strip rolling mill vibration resulting from entry thickness deviation can be reduced by increasing the area of the hydraulic cylinder piston side and the entry thickness as well as by decreasing the deformation resistance, rolling speed, and equivalent stiffness.


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## Body

## 1. Introduction

It is well known that mill vibration has become a bottleneck in thin steel strip manufacturing and a global problem [1]. Rolling mill vibration can cause unacceptable thickness variation and severely damage mill equipment. Due to its complexity, the mechanism of rolling mill vibration is unclear and remains an open problem worldwide. Rolling mill vibration research aims to analyze dynamic characteristics and obtain vibration suppression measures based on mathematical modeling and data-driven methods.Regarding mathematical models, there are two common methods used to study the dynamic characteristics of rolling mill vibration. The first method is establishing a spring mass model or dividing the continuum into finite elements, with various degrees of simplification, to solve for the inherent frequency and the mode of vibration. Peng et al. [2] established a swing dynamic model of the roll system with the dynamic characteristics of the hot rolling process and the coupling relationship between the horizontal and vertical vibrations of the work roll. Kim et al. [3] proposed a mathematical model of a cold rolling mill by considering the friction forces between rolls and the contact stiffness caused by roller bearings and rolls. Zheng et al. [4] analyzed the spatial vibration of four-high rolling mills using the modified Riccati-transfer matrix method (Riccati-TMM) and the finite element method (FEM). Liu et al. [5] studied the vertical vibration of the strip mill using the piecewise nonlinear constraint arising from hydraulic cylinder. Gao et al. [6] simplified the rolling mill structure into an eight-degree-of-freedom (8-dof) spring mass model and analyzed the critical rolling speed using the stability criterion. The second method analyzes the rolling deformation zone to obtain its dynamic characteristics based on the Kalman equation or the Orowan equation. Yun [7] and Hu [8, 9] et al. proposed a classical dynamic model of cold rolling mill vibration, which considered the change in the roll spacing and the rate of change of the roll spacing. Pittner et al. [10–12] proposed a dynamic model of the hot rolling mill, which could be applied only to looper control analysis and not to vertical vibration analysis. Fan et al. [13] proposed a dynamic model of hot rolling mill vertical vibration to analyze the influence of rolling process parameters on the hot rolling mill vibration. Zeng et al. [14] formulated a dynamic rolling process model based on the Bland-Ford-Hill rolling force model and analyzed the influences of process parameters and structure parameters on system stability. Based on the Coulomb friction model and the Kalman differential equation, Liu [15, 16] established a dynamic rolling process model that considered the rolled piece hardening effect and the work roll flattening effect.Due to the vibration demolding of the crystallizer the slab presents the thickness deviation. It is necessary to study the effect of entry thickness deviation on hot rolling mill vibration. However, all above-mentioned models cannot be used to study the effect of entry thickness deviation on hot rolling mill vibration. Therefore, we propose a vertical vibration dynamic model of the hot strip rolling mill based on Sims’ rolling force model [17, 18] to study dynamic characteristics of hot strip rolling mill vibration resulting from entry thickness deviation. In Section 2, the detailed derivation process of the dynamic model is shown and the dynamic model is presented in a state-space form. In Section 3, a comparison between the numerical simulation and the field test data certifies the dynamic model validity. In Section 4, we analyze the influence of model parameters, such as deformation resistance, entry thickness, rolling speed, equivalent stiffness, and the area of the hydraulic cylinder piston side, on dynamic characteristics of the hot rolling mill vibration resulting from the entry thickness deviation. The conclusion is given in Section 5.


## 2. Dynamic Model

### 2.1. Mill Structure Dynamic Model

Generally the mill structure, which includes the rack, rollers, and the hydraulic cylinder, is described as a 5-dof lumped-mass model. For simplicity, the upper backup roll and the cylinder body are assumed to form a whole, and the piston rod and the mill rack are assumed to form a whole. As the servo valve adjusts only the pressure of the piston side and does not change the pressure of the rod side, the pressure of the rod side can be assumed to be constant. In addition, the lower part of the rack is assumed to be the ground. The schematic diagram of the mill structure is depicted in Figure1.Figure 1
Schematic diagram of the mill structure.To study vibration, only the dynamic parts of parameters are considered in our model. According to vibration and fluid theories, the kinetic equations of the mill structure are written as follows.(1)m1d2x1dt2+c1dx1dt+k1x1=-ppApApdx2dt-dx1dt+Vpβedppdt+chpp=0m2d2x2dt2+c2dx2dt-dx3dt+k2x2-x3=ppApm3d2x3dt2+c2dx3dt-dx2dt+k2x3-x2=-dFRm4d2x4dt2+c3dx4dt-dx5dt+k3x4-x5=dFRm5d2x5dt2+c3dx5dt-dx4dt+k3x5-x4+c4dx5dt+k4x5=0wherem1 and x1 are the equivalent mass and the displacement of the rack, respectively; m2 and x2 are the equivalent mass and the displacement of the upper backup roll, respectively; m3 and x3 are the equivalent mass and the displacement of the upper work roll, respectively; m4 and x4 are the equivalent mass and the displacement of the bottom work roll, respectively; m5 and x5 are the equivalent mass and the displacement of the bottom back roll, respectively; k1 and c1 are the equivalent stiffness and damping, respectively, between the rack and the ground; k2 and c2 are the equivalent stiffness and damping, respectively, between the upper work roll and the upper backup roll; k3 and c3 are the equivalent stiffness and damping, respectively, between the bottom work roll and the bottom back roll; k4 and c4 are the equivalent stiffness and damping, respectively, between the bottom back roll and the ground; Ap is the area of the piston side; pp is the pressure of the piston side; ch is the internal leakage coefficient of the cylinder; βe is the elastic modulus of the liquid; and dFR is the dynamic part of the vertical rolling force.


### 2.2. Rolling Process Dynamic Model

The slab method is one of the most widely used methods to describe the plastic deformation of rolled materials. Sims derived the specific roll force equation, which is often applied in hot rolling mills. In this section, based on the Sims’ rolling force model, a dynamic rolling process model with vibration is proposed.


#### 2.2.1. Strip Entry and Exit Position

The roll bite geometry with vibration is illustrated in Figure2. It is assumed that the exit plane does not coincide with the centerline of the work rolls once the rolls begin to vibrate.Figure 2
Roll bite geometry with vibration.Here,o-1 and o-2 are the original work roll centers; o1 and o2 are the work roll centers with vibration; h1, h2, and hn are the strip entry thickness, the strip exit thickness, and the strip thickness of the neutral plane, respectively; x1, x2, and xn are the strip entry position, the strip exit position, and the neutral point position, respectively; R′ is the deformed radius of the work roll; u1 and u2 are the strip speed at entry and exit, respectively; ϕ1, ϕ2, and ϕn are the strip entry angle, the strip exit angle, and the neutral angle, respectively; vr is the roll speed; hc is the roll gap spacing along the centerline of the work rolls; and h˙c is the rate of change of the roll gap spacing.The high rolling force causes the work roll deformation. According to the Hitchcock equation, the deformed radius is calculated as follows.(2)R′=R1+81-ν2FRπEh1-h2whereR is the undeformed radius of the work roll; ν is Poisson’s ratio; E is Young’s modulus; and FR is the total rolling force.In the roll bite, the roll surface is usually assumed to be a parabolic curve. From Figure2, the strip thickness at an arbitrary cross-section can be calculated by(3)h=hc+x2R′By combining (3), x1=R′sin⁡ϕ1, and sin⁡ϕ1≈ϕ1, the strip entry angle can be derived from(4)ϕ1=h1-hcR′By applying the principle of mass conservation to the roll bite, the flow of metal through an arbitrary cross-section can be written as(5)uh=u1h1-x1-xh˙cBy applying (5) to the neutral point position and the exit position, the corresponding mass conservation equations are listed as(6)u2h2=u1h1-x1-x2h˙cunhn=u1h1-x1-xnh˙cFrom Figure2, the exit angle can be calculated by(7)ϕ2≈tan⁡ϕ2=h˙c2u2By substituting both (3) and (6) into (7), the strip exit angle can be obtained.(8)ϕ2=hch˙c2vrhc+vrxn2/R′-xnh˙c


#### 2.2.2. Neutral Point Position

Hot rolling mills roll with low tensions which can be neglected. The stress distribution in a volume element in the roll gap is illustrated in Figure3, where the circumferential thickness of the slice is dϕ; the horizontal composite force acting on a cross-section is Q; the normal roll pressure is pϕ; the specific shear stress is τϕ; and the surface angularity is ϕ.Figure 3
Stresses distribution acting on a volume element in the roll gap.According to Sims’ method, the horizontal force equilibrium equation can be derived as(9)ddϕpϕk-π4=R′πϕ2hc+R′ϕ2±R′hc+R′ϕ2where the constant k is the deformation resistance of the arc of contact.Equation (9) is solved by starting at the plane of exit and extending toward the plane of entry. Thus, the normal roll pressure between the exit position and the neutral point position is obtained.(10)pϕ=kπ4ln⁡hϕh2+π4+R′hctan-1⁡R′hcϕ-R′hctan-1⁡R′hcϕ2Equation (9) is solved by starting at the plane of entry and extending toward the plane of exit. Thus, the normal roll pressure between the entry position and the neutral point position is obtained.(11)pϕ=kπ4ln⁡hϕh1+π4+R′hctan-1⁡R′hcϕ1-R′hctan-1⁡R′hcϕThe neutral angle can be calculated by the equalization of the normal roll pressure obtained from (10) and (11) at the neutral point; the neutral angle can be calculated by(12)ϕn=hcR′tan⁡π8hcR′ln⁡h2h1+12tan-1⁡R′hcϕ1+12tan-1⁡R′hcϕ2


#### 2.2.3. Calculating Specific Roll Force

Notably, due to the small angular coordinateϕ the differences between the normal roll pressure and the vertical pressure are negligible and the strip width is defined as unit width. When plane deformation occurs, the specific roll force can be calculated by(13)FR=R′∫ϕ2ϕ1pϕdϕBy substituting (10) and (11) into (13), the specific roll force can be derived as(14)FR=R′kπ4ϕ2-ϕ1+π2hcR′tan-1⁡R′hcϕ1-π2hcR′tan-1⁡R′hcϕ2+12ln⁡h1h2+ln⁡h2hn


#### 2.2.4. Linearized Dynamic Model of the Rolling Process

Since the vibration is dynamic rather than static, the variations in the system parameters are important. Linearization through first-order Taylor approximation can be used to simplify the dynamic model.The variation in strip entry thicknessdh1, the variation in roll gap spacing dhc, and the rate of change of roll gap spacing dh˙c are defined as inputs, and the variation in specific roll force dFR is defined as an output.From (14), the variation in specific roll force dFR can be expressed as(15)dFR=dFRdh1dh1+dFRdhcdhc+dFRdh˙cdh˙cwhere(16)dFRdh1=∂FR∂h1+∂FR∂ϕ1∂ϕ1∂h1+∂FR∂hn∂hn∂ϕn∂ϕn∂h1+∂ϕn∂ϕ1∂ϕ1∂h1(17)dFRdhc=∂FR∂hc+∂FR∂ϕ1∂ϕ1∂hc+∂FR∂h2∂h2∂hc+∂FR∂hn∂hn∂hc+∂hn∂ϕn∂ϕn∂ϕ1∂ϕ1∂hc+∂ϕn∂h2∂h2∂hc+∂ϕn∂hc(18)dFRdh˙c=∂FR∂ϕ2+∂FR∂h2∂h2∂ϕ2+∂FR∂hn∂hn∂ϕn∂ϕn∂ϕ2+∂ϕn∂h2∂h2∂ϕ2∂ϕ2∂h˙cAll partial derivatives in (16), (17), and (18) are listed in Appendix A.


### 2.3. State-Space Form of the Dynamic Model

The dynamic roll force from the rolling process acts on the work rolls and causes them to vibrate. The vibration of the work roll reacts to the rolling process. Clearly, there is coupling between the rolling process and the structure of the rolling mills. The coupling relationship between the work roll vibration and the dynamic roll gap spacing is shown as follows.(19)hc=x4-x3h˙c=x˙4-x˙3By combining (1), (15), and (19), the state space equation of the dynamic model whose system input is the entry thickness deviation can be written as(20)X˙=M1X+M2dh1where(21)X=x1x˙1ppx2x˙2x3x˙3x4x˙4x5x˙5TM1=01000000000-k1m1-c1m1-Apm1000000000βeApVp-βechVp0-βeApVp0000000000100000000Apm2-k2m2-c2m2k2m2c2m2000000000010000000k2m3c2m31m3dFRdhc-k21m3dFRdh˙c-c2-1m3dFRdhc-1m3dFRdh˙c000000000010000000-1m4dFRdhc-1m4dFRdh˙c1m4dFRdhc-k31m4dFRdh˙c-c3k3m4c3m4000000000010000000k3m5c3m5-k3-k4m5-c3-c4m5M2=000000-1m3dFRdh101m4dFRdh100T


## 2.1. Mill Structure Dynamic Model

Generally the mill structure, which includes the rack, rollers, and the hydraulic cylinder, is described as a 5-dof lumped-mass model. For simplicity, the upper backup roll and the cylinder body are assumed to form a whole, and the piston rod and the mill rack are assumed to form a whole. As the servo valve adjusts only the pressure of the piston side and does not change the pressure of the rod side, the pressure of the rod side can be assumed to be constant. In addition, the lower part of the rack is assumed to be the ground. The schematic diagram of the mill structure is depicted in Figure1.Figure 1
Schematic diagram of the mill structure.To study vibration, only the dynamic parts of parameters are considered in our model. According to vibration and fluid theories, the kinetic equations of the mill structure are written as follows.(1)m1d2x1dt2+c1dx1dt+k1x1=-ppApApdx2dt-dx1dt+Vpβedppdt+chpp=0m2d2x2dt2+c2dx2dt-dx3dt+k2x2-x3=ppApm3d2x3dt2+c2dx3dt-dx2dt+k2x3-x2=-dFRm4d2x4dt2+c3dx4dt-dx5dt+k3x4-x5=dFRm5d2x5dt2+c3dx5dt-dx4dt+k3x5-x4+c4dx5dt+k4x5=0wherem1 and x1 are the equivalent mass and the displacement of the rack, respectively; m2 and x2 are the equivalent mass and the displacement of the upper backup roll, respectively; m3 and x3 are the equivalent mass and the displacement of the upper work roll, respectively; m4 and x4 are the equivalent mass and the displacement of the bottom work roll, respectively; m5 and x5 are the equivalent mass and the displacement of the bottom back roll, respectively; k1 and c1 are the equivalent stiffness and damping, respectively, between the rack and the ground; k2 and c2 are the equivalent stiffness and damping, respectively, between the upper work roll and the upper backup roll; k3 and c3 are the equivalent stiffness and damping, respectively, between the bottom work roll and the bottom back roll; k4 and c4 are the equivalent stiffness and damping, respectively, between the bottom back roll and the ground; Ap is the area of the piston side; pp is the pressure of the piston side; ch is the internal leakage coefficient of the cylinder; βe is the elastic modulus of the liquid; and dFR is the dynamic part of the vertical rolling force.


## 2.2. Rolling Process Dynamic Model

The slab method is one of the most widely used methods to describe the plastic deformation of rolled materials. Sims derived the specific roll force equation, which is often applied in hot rolling mills. In this section, based on the Sims’ rolling force model, a dynamic rolling process model with vibration is proposed.


### 2.2.1. Strip Entry and Exit Position

The roll bite geometry with vibration is illustrated in Figure2. It is assumed that the exit plane does not coincide with the centerline of the work rolls once the rolls begin to vibrate.Figure 2
Roll bite geometry with vibration.Here,o-1 and o-2 are the original work roll centers; o1 and o2 are the work roll centers with vibration; h1, h2, and hn are the strip entry thickness, the strip exit thickness, and the strip thickness of the neutral plane, respectively; x1, x2, and xn are the strip entry position, the strip exit position, and the neutral point position, respectively; R′ is the deformed radius of the work roll; u1 and u2 are the strip speed at entry and exit, respectively; ϕ1, ϕ2, and ϕn are the strip entry angle, the strip exit angle, and the neutral angle, respectively; vr is the roll speed; hc is the roll gap spacing along the centerline of the work rolls; and h˙c is the rate of change of the roll gap spacing.The high rolling force causes the work roll deformation. According to the Hitchcock equation, the deformed radius is calculated as follows.(2)R′=R1+81-ν2FRπEh1-h2whereR is the undeformed radius of the work roll; ν is Poisson’s ratio; E is Young’s modulus; and FR is the total rolling force.In the roll bite, the roll surface is usually assumed to be a parabolic curve. From Figure2, the strip thickness at an arbitrary cross-section can be calculated by(3)h=hc+x2R′By combining (3), x1=R′sin⁡ϕ1, and sin⁡ϕ1≈ϕ1, the strip entry angle can be derived from(4)ϕ1=h1-hcR′By applying the principle of mass conservation to the roll bite, the flow of metal through an arbitrary cross-section can be written as(5)uh=u1h1-x1-xh˙cBy applying (5) to the neutral point position and the exit position, the corresponding mass conservation equations are listed as(6)u2h2=u1h1-x1-x2h˙cunhn=u1h1-x1-xnh˙cFrom Figure2, the exit angle can be calculated by(7)ϕ2≈tan⁡ϕ2=h˙c2u2By substituting both (3) and (6) into (7), the strip exit angle can be obtained.(8)ϕ2=hch˙c2vrhc+vrxn2/R′-xnh˙c


### 2.2.2. Neutral Point Position

Hot rolling mills roll with low tensions which can be neglected. The stress distribution in a volume element in the roll gap is illustrated in Figure3, where the circumferential thickness of the slice is dϕ; the horizontal composite force acting on a cross-section is Q; the normal roll pressure is pϕ; the specific shear stress is τϕ; and the surface angularity is ϕ.Figure 3
Stresses distribution acting on a volume element in the roll gap.According to Sims’ method, the horizontal force equilibrium equation can be derived as(9)ddϕpϕk-π4=R′πϕ2hc+R′ϕ2±R′hc+R′ϕ2where the constant k is the deformation resistance of the arc of contact.Equation (9) is solved by starting at the plane of exit and extending toward the plane of entry. Thus, the normal roll pressure between the exit position and the neutral point position is obtained.(10)pϕ=kπ4ln⁡hϕh2+π4+R′hctan-1⁡R′hcϕ-R′hctan-1⁡R′hcϕ2Equation (9) is solved by starting at the plane of entry and extending toward the plane of exit. Thus, the normal roll pressure between the entry position and the neutral point position is obtained.(11)pϕ=kπ4ln⁡hϕh1+π4+R′hctan-1⁡R′hcϕ1-R′hctan-1⁡R′hcϕThe neutral angle can be calculated by the equalization of the normal roll pressure obtained from (10) and (11) at the neutral point; the neutral angle can be calculated by(12)ϕn=hcR′tan⁡π8hcR′ln⁡h2h1+12tan-1⁡R′hcϕ1+12tan-1⁡R′hcϕ2


### 2.2.3. Calculating Specific Roll Force

Notably, due to the small angular coordinateϕ the differences between the normal roll pressure and the vertical pressure are negligible and the strip width is defined as unit width. When plane deformation occurs, the specific roll force can be calculated by(13)FR=R′∫ϕ2ϕ1pϕdϕBy substituting (10) and (11) into (13), the specific roll force can be derived as(14)FR=R′kπ4ϕ2-ϕ1+π2hcR′tan-1⁡R′hcϕ1-π2hcR′tan-1⁡R′hcϕ2+12ln⁡h1h2+ln⁡h2hn


### 2.2.4. Linearized Dynamic Model of the Rolling Process

Since the vibration is dynamic rather than static, the variations in the system parameters are important. Linearization through first-order Taylor approximation can be used to simplify the dynamic model.The variation in strip entry thicknessdh1, the variation in roll gap spacing dhc, and the rate of change of roll gap spacing dh˙c are defined as inputs, and the variation in specific roll force dFR is defined as an output.From (14), the variation in specific roll force dFR can be expressed as(15)dFR=dFRdh1dh1+dFRdhcdhc+dFRdh˙cdh˙cwhere(16)dFRdh1=∂FR∂h1+∂FR∂ϕ1∂ϕ1∂h1+∂FR∂hn∂hn∂ϕn∂ϕn∂h1+∂ϕn∂ϕ1∂ϕ1∂h1(17)dFRdhc=∂FR∂hc+∂FR∂ϕ1∂ϕ1∂hc+∂FR∂h2∂h2∂hc+∂FR∂hn∂hn∂hc+∂hn∂ϕn∂ϕn∂ϕ1∂ϕ1∂hc+∂ϕn∂h2∂h2∂hc+∂ϕn∂hc(18)dFRdh˙c=∂FR∂ϕ2+∂FR∂h2∂h2∂ϕ2+∂FR∂hn∂hn∂ϕn∂ϕn∂ϕ2+∂ϕn∂h2∂h2∂ϕ2∂ϕ2∂h˙cAll partial derivatives in (16), (17), and (18) are listed in Appendix A.


## 2.2.1. Strip Entry and Exit Position

The roll bite geometry with vibration is illustrated in Figure2. It is assumed that the exit plane does not coincide with the centerline of the work rolls once the rolls begin to vibrate.Figure 2
Roll bite geometry with vibration.Here,o-1 and o-2 are the original work roll centers; o1 and o2 are the work roll centers with vibration; h1, h2, and hn are the strip entry thickness, the strip exit thickness, and the strip thickness of the neutral plane, respectively; x1, x2, and xn are the strip entry position, the strip exit position, and the neutral point position, respectively; R′ is the deformed radius of the work roll; u1 and u2 are the strip speed at entry and exit, respectively; ϕ1, ϕ2, and ϕn are the strip entry angle, the strip exit angle, and the neutral angle, respectively; vr is the roll speed; hc is the roll gap spacing along the centerline of the work rolls; and h˙c is the rate of change of the roll gap spacing.The high rolling force causes the work roll deformation. According to the Hitchcock equation, the deformed radius is calculated as follows.(2)R′=R1+81-ν2FRπEh1-h2whereR is the undeformed radius of the work roll; ν is Poisson’s ratio; E is Young’s modulus; and FR is the total rolling force.In the roll bite, the roll surface is usually assumed to be a parabolic curve. From Figure2, the strip thickness at an arbitrary cross-section can be calculated by(3)h=hc+x2R′By combining (3), x1=R′sin⁡ϕ1, and sin⁡ϕ1≈ϕ1, the strip entry angle can be derived from(4)ϕ1=h1-hcR′By applying the principle of mass conservation to the roll bite, the flow of metal through an arbitrary cross-section can be written as(5)uh=u1h1-x1-xh˙cBy applying (5) to the neutral point position and the exit position, the corresponding mass conservation equations are listed as(6)u2h2=u1h1-x1-x2h˙cunhn=u1h1-x1-xnh˙cFrom Figure2, the exit angle can be calculated by(7)ϕ2≈tan⁡ϕ2=h˙c2u2By substituting both (3) and (6) into (7), the strip exit angle can be obtained.(8)ϕ2=hch˙c2vrhc+vrxn2/R′-xnh˙c


## 2.2.2. Neutral Point Position

Hot rolling mills roll with low tensions which can be neglected. The stress distribution in a volume element in the roll gap is illustrated in Figure3, where the circumferential thickness of the slice is dϕ; the horizontal composite force acting on a cross-section is Q; the normal roll pressure is pϕ; the specific shear stress is τϕ; and the surface angularity is ϕ.Figure 3
Stresses distribution acting on a volume element in the roll gap.According to Sims’ method, the horizontal force equilibrium equation can be derived as(9)ddϕpϕk-π4=R′πϕ2hc+R′ϕ2±R′hc+R′ϕ2where the constant k is the deformation resistance of the arc of contact.Equation (9) is solved by starting at the plane of exit and extending toward the plane of entry. Thus, the normal roll pressure between the exit position and the neutral point position is obtained.(10)pϕ=kπ4ln⁡hϕh2+π4+R′hctan-1⁡R′hcϕ-R′hctan-1⁡R′hcϕ2Equation (9) is solved by starting at the plane of entry and extending toward the plane of exit. Thus, the normal roll pressure between the entry position and the neutral point position is obtained.(11)pϕ=kπ4ln⁡hϕh1+π4+R′hctan-1⁡R′hcϕ1-R′hctan-1⁡R′hcϕThe neutral angle can be calculated by the equalization of the normal roll pressure obtained from (10) and (11) at the neutral point; the neutral angle can be calculated by(12)ϕn=hcR′tan⁡π8hcR′ln⁡h2h1+12tan-1⁡R′hcϕ1+12tan-1⁡R′hcϕ2


## 2.2.3. Calculating Specific Roll Force

Notably, due to the small angular coordinateϕ the differences between the normal roll pressure and the vertical pressure are negligible and the strip width is defined as unit width. When plane deformation occurs, the specific roll force can be calculated by(13)FR=R′∫ϕ2ϕ1pϕdϕBy substituting (10) and (11) into (13), the specific roll force can be derived as(14)FR=R′kπ4ϕ2-ϕ1+π2hcR′tan-1⁡R′hcϕ1-π2hcR′tan-1⁡R′hcϕ2+12ln⁡h1h2+ln⁡h2hn


## 2.2.4. Linearized Dynamic Model of the Rolling Process

Since the vibration is dynamic rather than static, the variations in the system parameters are important. Linearization through first-order Taylor approximation can be used to simplify the dynamic model.The variation in strip entry thicknessdh1, the variation in roll gap spacing dhc, and the rate of change of roll gap spacing dh˙c are defined as inputs, and the variation in specific roll force dFR is defined as an output.From (14), the variation in specific roll force dFR can be expressed as(15)dFR=dFRdh1dh1+dFRdhcdhc+dFRdh˙cdh˙cwhere(16)dFRdh1=∂FR∂h1+∂FR∂ϕ1∂ϕ1∂h1+∂FR∂hn∂hn∂ϕn∂ϕn∂h1+∂ϕn∂ϕ1∂ϕ1∂h1(17)dFRdhc=∂FR∂hc+∂FR∂ϕ1∂ϕ1∂hc+∂FR∂h2∂h2∂hc+∂FR∂hn∂hn∂hc+∂hn∂ϕn∂ϕn∂ϕ1∂ϕ1∂hc+∂ϕn∂h2∂h2∂hc+∂ϕn∂hc(18)dFRdh˙c=∂FR∂ϕ2+∂FR∂h2∂h2∂ϕ2+∂FR∂hn∂hn∂ϕn∂ϕn∂ϕ2+∂ϕn∂h2∂h2∂ϕ2∂ϕ2∂h˙cAll partial derivatives in (16), (17), and (18) are listed in Appendix A.


## 2.3. State-Space Form of the Dynamic Model

The dynamic roll force from the rolling process acts on the work rolls and causes them to vibrate. The vibration of the work roll reacts to the rolling process. Clearly, there is coupling between the rolling process and the structure of the rolling mills. The coupling relationship between the work roll vibration and the dynamic roll gap spacing is shown as follows.(19)hc=x4-x3h˙c=x˙4-x˙3By combining (1), (15), and (19), the state space equation of the dynamic model whose system input is the entry thickness deviation can be written as(20)X˙=M1X+M2dh1where(21)X=x1x˙1ppx2x˙2x3x˙3x4x˙4x5x˙5TM1=01000000000-k1m1-c1m1-Apm1000000000βeApVp-βechVp0-βeApVp0000000000100000000Apm2-k2m2-c2m2k2m2c2m2000000000010000000k2m3c2m31m3dFRdhc-k21m3dFRdh˙c-c2-1m3dFRdhc-1m3dFRdh˙c000000000010000000-1m4dFRdhc-1m4dFRdh˙c1m4dFRdhc-k31m4dFRdh˙c-c3k3m4c3m4000000000010000000k3m5c3m5-k3-k4m5-c3-c4m5M2=000000-1m3dFRdh101m4dFRdh100T


## 3. Model Validity

To verify model validity, numerical analysis results are compared with the field test data from F2 of a 1580 hot strip rolling mill. The experiment equipment is shown in Figure4. It is a hot strip mill composed of seven stand 4-high rolling mills. The standard parameter settings are listed in Table 1. Equivalent mass and equivalent stiffness are calculated using the principle of conservation of energy [19]. Equivalent damping is assumed to be structural, and structural damping is proportional to equivalent stiffness [20]. The hydraulic cylinder parameters are found in the corresponding specification.Table 1
Standard parameter settings.
parameter Value parameter value R 0.37 5 m β e 1.4 × 10 9 P a v r 2.8 9 m / s V p 3.46 × 10 - 2 m 3 E 2.06 × 10 11 N / m 2 c h 5 × 10 - 14 m 3 / ( s . P a ) ν 0.3 m 1 1 × 10 4 k g h 1 16.6 2 m m m 2 3.6 × 10 4 k g h 2 8.1 7 m m m 3 2 × 10 4 k g k 201 × 10 6 N / m 2 m 4 2 × 10 4 k g k 1 4 × 10 9 N / m m 5 3.5 × 10 4 k g k 2 2 × 10 10 N / m c 1 5 × 10 6 N / ( m / s ) k 3 2 × 10 10 N / m c 2 2 × 10 6 N / ( m / s ) k 4 4 × 10 9 N / m c 3 2 × 10 6 N / ( m / s ) A p 0.8659 m 2 c 4 5 × 10 6 N / ( m / s )Figure 4
Experiment equipment.As the dynamic model of rolling mill vibration is formulated by coupling the rolling process model with the mill structure model, the validity of the rolling process model is first shown and then the validity of the dynamic model of rolling mill vibration is shown.


### 3.1. Validity of the Rolling Process Model

In the rolling process model, the specific rolling force is the most important parameter and can be measured by the load cells, which are installed between the bottom backup roll bearing and the lower housing. The specific roll force of F2 is treated as a criterion to evaluate the rolling process model. For four different types of steel products, the experimental data are compared with the numerical results, as listed in Table2.Table 2
Comparison between the experimental data and the numerical results.
Type of steel SPA-H SPA-H SPHC FB60-P F7 exit thickness (mm) 1.52 1.87 2.6 1.87 F2 entry thickness (mm) 16.62 16.62 20.72 18.09 F2 exit thickness (mm) 8.17 8.67 11.13 9.05 Deformation resistance (MPa) 201.11 190.47 150.28 181.96 Rolling speed(m/s) 2.89 2.5 2.34 2.56 Reduction ratio (%) 50.8 47.8 46.3 50 Calculated roll force (kN) 22373 19769 15636 20051 Experimental roll force (kN) 23813 20390 14294 21332 Error of roll force (%) -6.05 -3.05 8.58 -6From Table2, all errors between the experimental roll force and calculated roll force are less than 10% for different type of steel products, which shows that the rolling process model is effective.


### 3.2. Validity of the Rolling Mill Vibration Dynamic Model

The entry incoming thickness harmonic deviation is assumed to be excitation, its amplitude is 200μm, and its frequency is 41 Hz. In field manufacturing, the vibration velocity sensor on the up work roll is used to measure the work roll vibration. The up work roll vibration velocity is treated as a criterion to evaluate the rolling mill vibration intensity. It is known that many rolling process parameters, such as deformation resistance, reduction ratio, and entry thickness, play an important role in rolling vibration. However, it is impossible to change a single-factor variable in the field experiment. As a result, we choose three types of steel products, i.e., SPA-H (1.52 mm), SPA-H (1.87 mm), and FB60-P (1.87 mm) in Table 2, to qualitatively certify the validity of the rolling mill vibration dynamic model. For these three specifications, simulation results of the upper work roll vibration velocity are shown in Figure 5, and the field test data of the upper work roll vibration velocity are shown in Figure 6.Figure 5
Simulation results of the up work roll vibration velocity.
(a)
SAP-H (1.52 mm) (b)
SPA-H (1.87 mm) (c)
FB60-P (1.87 mm)Figure 6
Field test data of the upper work roll vibration velocity.
(a)
SAP-H (1.52 mm) (b)
SPA-H (1.87 mm) (c)
FB60-P (1.87 mm)From Figure5, we can find that the vibration velocity amplitudes of SPA-H (1.52 mm), SPA-H (1.87 mm), and FB60-P (1.87 mm) are 4.153 mm/s, 3.825 mm/s, and 3.61 mm/s, respectively, in the numerical simulation. From Figure 6, we can find that the vibration velocity amplitudes of SPA-H (1.52 mm), SPA-H (1.87 mm), and FB60-P (1.87 mm) are 4.681 mm/s, 4.129 mm/s, and 3.795 mm/s, respectively, in the field test data. The comparison between SPA-H (1.87 mm) and FB60-P (1.87 mm), which have the same F7 exit thickness and different steel grade, shows that the vibration velocity amplitude of FB60-P (1.87 mm) is 5.6% and 10.4% less than that of SPA-H (1.87 mm) in the numerical simulation and in the field test data, respectively. Similarly, the comparison between SPA-H (1.87 mm) and SPA-H (1.52 mm), which have different F7 exit thickness and the same steel grade, shows that the vibration velocity amplitude of SPA-H (1.87 mm) is 8% and 12% less than that of SPA-H (1.52 mm) in the numerical simulation and in the field test data, respectively. The vibration velocity amplitudes of SPA-H (1.52 mm), SPA-H (1.87 mm), and FB60-P (1.87 mm) in the numerical simulation and in the field test data are counted in Figure 7, which shows that, although there is a quantitative error, the numerical simulation qualitatively agrees with the field test data.Figure 7
The count of all vibration velocity amplitudes.


## 3.1. Validity of the Rolling Process Model

In the rolling process model, the specific rolling force is the most important parameter and can be measured by the load cells, which are installed between the bottom backup roll bearing and the lower housing. The specific roll force of F2 is treated as a criterion to evaluate the rolling process model. For four different types of steel products, the experimental data are compared with the numerical results, as listed in Table2.Table 2
Comparison between the experimental data and the numerical results.
Type of steel SPA-H SPA-H SPHC FB60-P F7 exit thickness (mm) 1.52 1.87 2.6 1.87 F2 entry thickness (mm) 16.62 16.62 20.72 18.09 F2 exit thickness (mm) 8.17 8.67 11.13 9.05 Deformation resistance (MPa) 201.11 190.47 150.28 181.96 Rolling speed(m/s) 2.89 2.5 2.34 2.56 Reduction ratio (%) 50.8 47.8 46.3 50 Calculated roll force (kN) 22373 19769 15636 20051 Experimental roll force (kN) 23813 20390 14294 21332 Error of roll force (%) -6.05 -3.05 8.58 -6From Table2, all errors between the experimental roll force and calculated roll force are less than 10% for different type of steel products, which shows that the rolling process model is effective.


## 3.2. Validity of the Rolling Mill Vibration Dynamic Model

The entry incoming thickness harmonic deviation is assumed to be excitation, its amplitude is 200μm, and its frequency is 41 Hz. In field manufacturing, the vibration velocity sensor on the up work roll is used to measure the work roll vibration. The up work roll vibration velocity is treated as a criterion to evaluate the rolling mill vibration intensity. It is known that many rolling process parameters, such as deformation resistance, reduction ratio, and entry thickness, play an important role in rolling vibration. However, it is impossible to change a single-factor variable in the field experiment. As a result, we choose three types of steel products, i.e., SPA-H (1.52 mm), SPA-H (1.87 mm), and FB60-P (1.87 mm) in Table 2, to qualitatively certify the validity of the rolling mill vibration dynamic model. For these three specifications, simulation results of the upper work roll vibration velocity are shown in Figure 5, and the field test data of the upper work roll vibration velocity are shown in Figure 6.Figure 5
Simulation results of the up work roll vibration velocity.
(a)
SAP-H (1.52 mm) (b)
SPA-H (1.87 mm) (c)
FB60-P (1.87 mm)Figure 6
Field test data of the upper work roll vibration velocity.
(a)
SAP-H (1.52 mm) (b)
SPA-H (1.87 mm) (c)
FB60-P (1.87 mm)From Figure5, we can find that the vibration velocity amplitudes of SPA-H (1.52 mm), SPA-H (1.87 mm), and FB60-P (1.87 mm) are 4.153 mm/s, 3.825 mm/s, and 3.61 mm/s, respectively, in the numerical simulation. From Figure 6, we can find that the vibration velocity amplitudes of SPA-H (1.52 mm), SPA-H (1.87 mm), and FB60-P (1.87 mm) are 4.681 mm/s, 4.129 mm/s, and 3.795 mm/s, respectively, in the field test data. The comparison between SPA-H (1.87 mm) and FB60-P (1.87 mm), which have the same F7 exit thickness and different steel grade, shows that the vibration velocity amplitude of FB60-P (1.87 mm) is 5.6% and 10.4% less than that of SPA-H (1.87 mm) in the numerical simulation and in the field test data, respectively. Similarly, the comparison between SPA-H (1.87 mm) and SPA-H (1.52 mm), which have different F7 exit thickness and the same steel grade, shows that the vibration velocity amplitude of SPA-H (1.87 mm) is 8% and 12% less than that of SPA-H (1.52 mm) in the numerical simulation and in the field test data, respectively. The vibration velocity amplitudes of SPA-H (1.52 mm), SPA-H (1.87 mm), and FB60-P (1.87 mm) in the numerical simulation and in the field test data are counted in Figure 7, which shows that, although there is a quantitative error, the numerical simulation qualitatively agrees with the field test data.Figure 7
The count of all vibration velocity amplitudes.


## 4. The Influence of Model Parameters on Vibration Resulting from Entry Incoming Thickness Deviation

Since the validity of the dynamic model is qualitatively certified, it can be used to qualitatively analyze the influence of model parameters on the vibration resulting from the entry thickness deviation. The maximum and area of [20 Hz, 70 Hz], i.e., the general range of the vibration frequency in hot rolling, in the magnitude-frequency plot serve as the criterion for evaluating the vibration dynamic characteristics. The lower maximum and the lower area result in the better vibration suppression. In this section, we take the model parameters of SPA-H (1.52 mm) as an example to qualitatively analyze the influence of model parameters, such as deformation resistance, entry thickness, rolling speed, equivalent stiffness, and the area of the piston side, on the vibration resulting from the entry thickness deviation.


### 4.1. Deformation Resistance

In the model parameters of SPA-H (1.52 mm) the deformation resistance is 201.11 MPa. Therefore, the range of the deformation resistance is set as[150 MPa, 250 MPa] and other parameters remain unchanged. The relationships between the deformation resistance and the maximum and area are shown in Figure 8, from which we find that the maximum and area increase with deformation resistance; i.e., as deformation resistance increases, vibration suppression decreases. With the same entry thickness deviation the bigger the deformation resistance, the bigger the dynamic rolling force.Figure 8
The influence of deformation resistance on vibration suppression.


### 4.2. Entry Thickness

In the model parameters of SPA-H (1.52 mm) the entry thickness is 16.62 mm, and the range of the entry thickness is set as [14 mm, 19 mm]. The relationship between the entry thickness and the maximum and area is shown in Figure9, from which we find that the maximum and the area decrease with increasing entry thickness; i.e., as entry thickness increases vibration suppression increases. The reduction ratio increases, which leads to the increase of the steady rolling force, but with the same entry thickness deviation, the bigger the entry thickness, the smaller the dynamic rolling force.Figure 9
The influence of entry thickness on vibration suppression.


### 4.3. Rolling Speed

In the model parameters of SPA-H (1.52 mm), the rolling speed is 2.89 m/s, and the range of the rolling speed is set as [2 m/s, 4 m/s]. The relationships between the rolling speed and the maximum and area are shown in Figure10, from which the maximum and area increase with rolling speed; i.e., as rolling speed increases, vibration suppression decreases. Due to the increase of rolling speed, the equivalent damping in roll gap dFR/dh˙c decreases, which increases the dynamic rolling force.Figure 10
The influence of rolling speed on vibration suppression.


### 4.4. Area of the Piston Side

In the model parameters of SPA-H (1.52 mm) the area of the piston side is 0.8659 m2, and the range of the area of the piston side is set as [0.6 m2, 1 m2]. It should be noted that the volume of the piston side increases with the area of the piston side and the length of the piston side remains unchanged. The relationships between the area of the piston side and the maximum and area are shown in Figure 11, from which the maximum and area decrease with increasing area of the piston side. As the area of the piston side increases, hydraulic stiffness decreases. With the same entry thickness deviation, the smaller the hydraulic stiffness, the better the vibration suppression.Figure 11
The influence of the area of the piston side on vibration suppression.


### 4.5. Equivalent Stiffness

The equivalent stiffness is classified into two categories:k2 and k3, which are 2×1010N/m in the model parameters of SPA-H (1.52 mm); k1 and k4, which are 4×109N/m in the model parameters of SPA-H (1.52 mm). The range of k2 and k3 is set as [1×1010 N/m, 3×1010 N/m] and the range of k1 and k4 is set as [3×109 N/m, 5×109 N/m]. The relationships between the equivalent stiffness and the maximum and area are shown in Figure 12, from which the maximum and area increase with equivalent stiffness. With the same entry thickness deviation, the lower the equivalent stiffness, the smaller the impact force on the work roll, which leads to a smaller dynamic rolling force.Figure 12
The influence of equivalent stiffness on vibration suppression.
(a)
k 2 and k3 (b)
k 1 and k4


## 4.1. Deformation Resistance

In the model parameters of SPA-H (1.52 mm) the deformation resistance is 201.11 MPa. Therefore, the range of the deformation resistance is set as[150 MPa, 250 MPa] and other parameters remain unchanged. The relationships between the deformation resistance and the maximum and area are shown in Figure 8, from which we find that the maximum and area increase with deformation resistance; i.e., as deformation resistance increases, vibration suppression decreases. With the same entry thickness deviation the bigger the deformation resistance, the bigger the dynamic rolling force.Figure 8
The influence of deformation resistance on vibration suppression.


## 4.2. Entry Thickness

In the model parameters of SPA-H (1.52 mm) the entry thickness is 16.62 mm, and the range of the entry thickness is set as [14 mm, 19 mm]. The relationship between the entry thickness and the maximum and area is shown in Figure9, from which we find that the maximum and the area decrease with increasing entry thickness; i.e., as entry thickness increases vibration suppression increases. The reduction ratio increases, which leads to the increase of the steady rolling force, but with the same entry thickness deviation, the bigger the entry thickness, the smaller the dynamic rolling force.Figure 9
The influence of entry thickness on vibration suppression.


## 4.3. Rolling Speed

In the model parameters of SPA-H (1.52 mm), the rolling speed is 2.89 m/s, and the range of the rolling speed is set as [2 m/s, 4 m/s]. The relationships between the rolling speed and the maximum and area are shown in Figure10, from which the maximum and area increase with rolling speed; i.e., as rolling speed increases, vibration suppression decreases. Due to the increase of rolling speed, the equivalent damping in roll gap dFR/dh˙c decreases, which increases the dynamic rolling force.Figure 10
The influence of rolling speed on vibration suppression.


## 4.4. Area of the Piston Side

In the model parameters of SPA-H (1.52 mm) the area of the piston side is 0.8659 m2, and the range of the area of the piston side is set as [0.6 m2, 1 m2]. It should be noted that the volume of the piston side increases with the area of the piston side and the length of the piston side remains unchanged. The relationships between the area of the piston side and the maximum and area are shown in Figure 11, from which the maximum and area decrease with increasing area of the piston side. As the area of the piston side increases, hydraulic stiffness decreases. With the same entry thickness deviation, the smaller the hydraulic stiffness, the better the vibration suppression.Figure 11
The influence of the area of the piston side on vibration suppression.


## 4.5. Equivalent Stiffness

The equivalent stiffness is classified into two categories:k2 and k3, which are 2×1010N/m in the model parameters of SPA-H (1.52 mm); k1 and k4, which are 4×109N/m in the model parameters of SPA-H (1.52 mm). The range of k2 and k3 is set as [1×1010 N/m, 3×1010 N/m] and the range of k1 and k4 is set as [3×109 N/m, 5×109 N/m]. The relationships between the equivalent stiffness and the maximum and area are shown in Figure 12, from which the maximum and area increase with equivalent stiffness. With the same entry thickness deviation, the lower the equivalent stiffness, the smaller the impact force on the work roll, which leads to a smaller dynamic rolling force.Figure 12
The influence of equivalent stiffness on vibration suppression.
(a)
k 2 and k3 (b)
k 1 and k4


## 5. Conclusion

In this paper, the dynamic model of hot strip rolling mill vibration resulting from entry thickness deviation is proposed and its dynamic characteristic is studied. By means of mathematic modeling, numerical simulation, and field test data, the following conclusions are obtained.(1)
The proposed dynamic model, based on the Sims’ rolling force, is effective and can be used to study the dynamic characteristics of hot rolling mill vibration resulting from the entry thickness deviation;(2)
The hot rolling mill vibration resulting from the entry thickness deviation decreases with decreasing deformation resistance and rolling speed, as well as increasing entry thickness;(3)
The hot rolling mill vibration resulting from the entry thickness deviation decreases with increasing area of the hydraulic cylinder piston side and decreasing equivalent stiffness.



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*Source: 5868740-2019-02-25.xml*