Experimental and Numerical Investigation on the Roll System Swing Vibration Characteristics of a Hot Rolling Mill

Abstract

The model of the roll system vibration is usually limited to a one-dimensional or two-dimensional translation. In reality, however, the roll system also has a rotation behavior due to the uneven force distribution on the roll body. So the vibration, especially in rolling the wide steel strip, should perform a swing behavior, which consists of rotation and translation. In this paper, the experiment phenomena of plant vibration are analyzed, and the influences of roll system vibration on the rolling process are studied. A swing dynamic model of roll system is established considering the dynamic characteristics of the hot rolling process, the uneven force distribution on the roll body, and the coupling relationship between the horizontal and vertical vibration of the work roll. Through the numerical simulation, the natural characteristics of roll system swing are analyzed, the vibration responses of roll system swing are simulated with the arc tooth gear meshing impact force, and the stability of roll system is studied.

1. Introduction

Vibration phenomena in rolling mill are common and complicated. The severe vibration can seriously damage mill equipments, restrict rolling speed and cause variations in strip thickness and surface quality. The studies on the rolling mill vibration are mainly to establish the dynamic rolling process model and rolling mill structure model. In the dynamic rolling process, the research on dynamic rolling force is more comprehensive. Firstly, Tlusty et al.¹⁾ established a dynamic rolling force model in view of the roll gap fluctuation. Yun et al.²⁾ established a new dynamic rolling force model by modifying metal mass flow theorem on the basis of Tlusty’s model. Hou et al.³⁾ established a dynamic rolling force model with the impact of asymmetric hysteresis deformation. However, the research on the dynamic rolling torque or friction force of the rolling interface is less and imperfect. Zhong et al.⁴⁾ and Vladimir et al.⁵⁾ established the dynamic rolling torque model or friction force model considering the Stribeck effect only, and it ignored the dynamic transformation between the back slip and forward slip zone caused by the work roll vibration, which also has a significant effect on the rolling torque. So, the established dynamic rolling torque models are imperfect. In this paper, the vibration makes little effect on the rolling force (as shown in Fig. 3), but makes the rolling torque of the single work roll change greatly (as shown in Fig. 1(b)). And the change of the rolling torque also reacts to the vibration. So, the rolling torque or the friction force is the key factor affecting the stability rather than the rolling force. To study the vibration mechanism of the roll system exactly, it is necessary to establish an accurate dynamic rolling torque model or friction force model that considers not only the Stribeck effect but also the dynamic transformation between the back slip and forward slip zone.

Fig. 3.

Plant rolling force data.

Fig. 1.

Time domain and frequency domain vibration signal of the work roll. (a) time domain signal in vertical direction; (b) time domain signal in horizontal direction; (c) frequency domain signal in vertical direction; (d) frequency domain signal in horizontal direction.

In the rolling mill structure, it mainly includes torsional vibration analysis in horizontal plane studied by Vladimir et al.,⁵⁾ vertical vibration analysis in the vertical plane studied by Kimura et al.,⁶⁾ vertical-torsional coupling vibration analysis studied by Niroomand et al.,⁷⁾ and vertical-torsional-horizontal coupling vibration analysis studied by Zeng et al.⁸⁾ In addition, with the development of nonlinear dynamics theory, the nonlinear dynamic models of the rolling mill were studied.^9,10,11,12) The structural dynamic model became more and more comprehensively from the single-dimensional linear vibration model to multi-dimensional nonlinear coupled vibration model. These models were built by using the centralized mass method which neglected the axis asymmetrical factors of roll body. It was applicable for the narrow steel strip rolling process, and the vibration behavior was the translation. As the hot rolling mill develops towards the large-size, the steel strip width is growing. The axis asymmetrical factors can’t be neglected at all, such as the uneven force distribution on the roll body, different stiffness of the operation side and driving side, periodic interference force from driving spindles, and structural gap. These asymmetrical factors make the roll system derive a rigid rotation of roll body. So, the roll system vibration of a wide strip will perform a swing behavior, which consists of rotation and translation. The swing behavior also was illustrated by Shen et al.¹³⁾ At present, the study on vibration mechanism neglects the rotation of the roll system, which can’t fully reflect the dynamic characteristics of the roll system. In order to understand the dynamic characteristics of the roll system more comprehensively, a swing dynamic model of the roll system is established based on the experimental results.

2. Experiment

An intensive vibration happened at the second finisher (F2) of 2160 mm hot tandem rolling mill system, and a comprehensive vibration test was carried out by using ICP accelerometer. One of a typical vibration product was the steel of SPHC-FX with the finished thickness of 3.0 mm and the width of 1650 mm. And the entry thickness was 15.21 mm, the exit thickness was 6.1 mm, the rolling speed was 2.9 m/s at F2. The test data of the work roll vibration are shown in Fig. 1. It illustrates that the horizontal vibration is the most severe; the vertical vibration frequency consists of 55 Hz, 115 Hz and 215 Hz; the amplitude of 115 Hz is the largest. The horizontal vibration frequency consists of 55 Hz, 115 Hz and 215 Hz too, but the amplitude of 55 Hz is the largest. The frequency of 55 Hz is consistent with the frequency of arc tooth gear (ATG) periodical meshing impact force. The frequencies of 115 Hz and 215 Hz are multiple-frequency components caused by the nonlinear structure. The vibration results from the ATG periodical meshing impact force.

The cross-correlations of the upper and lower work rolls are analyzed on the vertical and horizontal directions, as shown in Fig. 2. The cross-correlation value is maximum at t=0 in Fig. 2(a), which can explain that the vibration phases of the upper and lower work rolls are same in the vertical direction. The cross-correlation value is minimum at t=0 in Fig. 2(b), which can explain that the vibration phases of the upper and lower work rolls are reverse in the horizontal direction. Due to the same vibration phases in the vertical direction, the vertical vibration has little impact on the rolling process, and the rolling force varies tinily. Due to the reverse vibration phases in the horizontal direction, the horizontal vibration has a significant impact on the rolling process, and the rolling torque varies intensively. In addition, the plant rolling force data of the vibration steel strips and stable steel strips are contrasted, as shown in Fig. 3. The rolling forces of both the vibration and stable steel strips vary tinily, and it illustrates that the vibration has no effect on the rolling force. Therefore, when the vibration model of rolling mill is established, the effect of vertical vibration on the rolling process can be ignored, but the effect of horizontal vibration on the rolling process must be considered.

Fig. 2.

Cross-correlation analysis of the upper and lower work rolls. (a) vertical direction; (b) horizontal direction.

Figure 4 is the dynamic displacement difference between the driving side and the operation side of the rolling mill. The displacement difference fluctuation of the vibration steel strip is more significant than the stable steel strip, which confirms that the swing vibration did exist when the roll system vibrates.

Fig. 4.

Displacement difference between driving side and the operation side of rolling mill. (a) vibration strip; (b) stable strip.

3. Mathematical Model

3.1. Mathematical Model of the Dynamic Friction Force

The horizontal vibration is the main factor that affects the rolling process, which will make the friction force and rolling toque vary greatly. So, the main research of this section is establishing the dynamic friction force model and rolling torque model in view of the horizontal vibration.

In this paper, the research object is F2, the relative reduction rate is 63%, the rolling speed is 2.9 m/s. Due to the heavy reduction, the metal volume flowing through the deformation zone in unit time is large, and the rolling interface is made up of the slipping zone mainly. In the slipping zone, the friction coefficient will decrease with the increasing of the relative velocity between the work roll and steel strip, the expression of friction coefficient is assumed to be linear and can be described as Eq. (1) based on Vladimir et al.⁵⁾   

$$\mu ={\mu }_{\text{s}}+\alpha \cdot v_{\text{ref}}$$(1)

Where μ is the friction coefficient. μ_s is the static friction coefficient, α is the friction negative coefficient which is less than 0, v_ref is the relative velocity between the work roll and steel strip.

Considering that the relative velocity of the work roll and steel strip is different at different points, it needs integrate the friction force of every point along the contact arc to establish the total friction force.   

$$f=p\cdot \left({\int }_{{\phi }_{\text{n}}}^{{\phi }_1}{\mu }_{\text{b}}\text{d}R\phi -{\int }_0^{{\phi }_{\text{n}}}{\mu }_{\text{f}}\text{d}R\phi \right)$$(2)

Where f is the total friction force between work roll and steel strip, μ_b and μ_f are the back slip and forward slip friction coefficient respectively, φ₁ is the bite angle, φ_n is the neutral angle, p is the rolling pressure in view of the slipping zone,¹⁴⁾ which is expressed as Eq. (3):   

$$P=2k\left(0.75\text{+}\frac{\mu l}{H+h}\text{+}\frac{H+h}{8l}\right)$$(3)

Where k is the material shear yield stress, H is the entry thickness, h is the exit thickness, l is the contact arc length.   

$$\begin{array}{l}{\mu }_{\text{b}}={\mu }_{\text{s}}+\alpha \cdot \left(v_{\text{r}}cos(\phi )-v_{\phi }\right)\\ {\mu }_{\text{f}}={\mu }_{\text{s}}+\alpha \cdot \left(v_{\phi }-v_{\text{r}}cos(\phi )\right)\end{array}\}$$(4)

Where v_r is the line velocity of the work roll, v_φ is the velocity of the steel strip on the contact arc.

Take Eqs. (4) to (2), and the unit width friction force can be deduced as:   

$$\begin{array}{l}f=bp{R}_{\text{w}}\cdot \\ \left({\mu }_{\text{s}}\left({\phi }_1-2{\phi }_{\text{n}}\right)+\alpha \cdot \left(v_{\text{r}}sin\left({\phi }_1\right)-\frac{v_{\text{r}}h\cdot arctan\left(R_{\text{w}}{\phi }_1/\sqrt{Rh}\right)}{\sqrt{R_{\text{w}}h}}\right)\right)\end{array}$$(5)

Where b is the steel strip width, R_w is the work roll radius.

The total rolling torque of two work rolls can be expressed as:   

$$M=2bf{R}_{\text{w}}$$(6)

  

$$\begin{array}{l}M=4kb{R}_{\text{w}}^2\left(0.75\text{+}\frac{\mu l}{H+h}+\frac{H+h}{8l}\right)\cdot \\ \left({\mu }_{\text{s}}\left({\phi }_1-2{\phi }_{\text{n}}\right)+\alpha \cdot \left(v_{\text{r}}sin\left({\phi }_1\right)-\frac{v_{\text{r}}h\cdot arctan\left(R_{\text{w}}{\phi }_1/\sqrt{R_{\text{w}}h}\right)}{\sqrt{R_{\text{w}}h}}\right)\right)\end{array}$$(7)

The total rolling torque is calculated based on Eq. (7) and the plant PDA data (inlet thickness, exit thickness and rolling speed). To verify the validity of Eq. (7), the comparison between the simulation rolling torque and experiment rolling torque extracted from the plant PDA data is carried on, as shown in Fig. 5. The relative error is less than 5%. So, the rolling torque model (Eq. (7)) is precise, and the friction force model (Eq. (5)) is precise too.

Fig. 5.

Comparison between rolling torques of simulation and experiment. (a) rolling torque; (b) relative error.

Because the horizontal vibration affects the rolling process greatly, the dynamic friction force model should be established considering the horizontal vibration. When the horizontal vibration acts on the work roll, the horizontal line velocity of work roll will be fluctuant, which makes the friction coefficient vary as well, as shown in Eq. (8).   

$$\begin{array}{l}{\mu }_{\text{b}}={\mu }_{\text{s}}+\alpha \cdot \left(v_{\text{r}}cos(\phi )+\dot{x}-v_{\phi }\right)\\ {\mu }_{\text{f}}={\mu }_{\text{s}}+\alpha \cdot \left(v_{\phi }-v_{\text{r}}cos(\phi )-\dot{x}\right)\end{array}\}$$(8)

Where $\dot{x}$ is the horizontal vibration velocity of the work roll.

Take Eqs. (8) to (2), and the unit width dynamic friction force of the steel strip can be deduced as:   

$$\begin{array}{l}f(\dot{x})=p{R}_{\text{w}}\cdot \\ \left({\mu }_{\text{s}}\left({\phi }_1-2{\phi }_{\text{n}}\right)+\alpha \cdot \left(\left(v_{\text{r}}+\dot{x}\right)\cdot {\phi }_1-\frac{v_{\text{r}}h\cdot arctan\left(R_{\text{w}}{\phi }_1/\sqrt{Rh}\right)}{\sqrt{R_{\text{w}}h}}\right)\right)\end{array}$$(9)

The netural angle also varies due to the work roll vibration, as shown in Eq. (10).   

$${\phi }_{\text{n}}\text{=}\sqrt{s_1/\left(R/h-0.5\right)}$$(10)

Where, $s_1=\left(v_1-v_{\text{r}}-\dot{x}\right)/\left(v_{\text{r}}+\dot{x}\right)$ it is the forward slip coefficient; v₁ is the exit velocity of steel strip.

Equation (9) consists of two parts. The first part shows that the vibration of work roll absolute velocity affects the friction force by means of changing the neutral angle. The change of the neutral angle makes the back slip and forward slip zones transform each other, which can restrain the vibration. So, this part plays a positive role to keep the roll system stability just as a positive damping. The other part shows that the vibration of the relative velocity between the work roll and steel strip affects the friction force, which can aggravate the vibration. So, this part plays a negative role to keep the roll system stability just as a negative damping.

3.2. Mathematical Model of the Roll System Swing

F2 is taken as an object for analysis. Assume that the roll system vibration is a rigid motion, the upper and lower roll systems along the rolling line are symmetrical. Figure 6 shows the mathematical model of the roll system swing which consists of translation and rotation. The rigid swing of the backup roll is considered in the z–y plane, which can be decomposed into the translation of the center C along the z axis and the rotation of the center C around the x axis. While the rigid swing of the work roll is considered in the z–y plane and x–y plane, which can be decomposed into the translation of the center C along the z axis and the rotation of the center C around the x axis, and the translation of the center C along the x axis and the rotation of the center C around the z axis.

Fig. 6.

Swing vibration mechanical model of roll system.

Where m_b and m_w are the backup roll and work roll equivalent translational inertial masses, respectively; J_b and J_w are the backup roll and work roll equivalent rotational inertia masses, respectively; k_bz is the equivalent stiffness of the square column, beam, screwdown and backup roll bearing block; k_wb is the elastic contact stiffness between the back roll and work roll; k_wxA and k_wxB are the horizontal equivalent stiffnesses between the work roll block and square column of A point and B point, respectively. z_bc and θ_bc are the backup roll translation and rotation displacements of C, respectively; z_wc and θ_wcz are the work roll translation and rotation displacements of C in the z–y plane, respectively; x_wc and θ_wcx are the work roll translation and rotation displacements of C in the x–y plane , respectively; A, B and C are the center of the operation side bearing block, the center of the driving side bearing block, and the center of the roll body; the origin of y axis is at C; y_A, y_B and y_D are the coordinate values, respectively; y_D is the force point of ATG meshing impact force; e is the offset of the backup roll and work roll; ee is the deformation of the work roll block and square column in the rolling process; P_x and P_z are the forces from the steel strip acting on the work rolls along the x axis and z axis, respectively; ΔF_x and ΔF_z are the periodic interference forces from the ATG driving system acting on the work roll along the x axis and z axis, respectively; T₀ and T₁ are the back tension and forward tension, respectively.

Considering the coupling relation of the work roll horizontal and vertical movements caused by the offset between the work roll and backup roll,¹⁵⁾ the dynamic equilibrium equation of the roll system swing is established according to the quality dynamic balance condition.   

$$\begin{array}{l}{m}_{\text{b}}{\ddot{z}}_{\text{bc}}+k_{\text{bz}}\left(z_{\text{bA}}+z_{\text{bB}}\right)+{\int }_{-B/2}^{B/2}{k}_{\text{wbz}}\left(z_{by}-z_{wy}\right)\text{d}y+C_1{\dot{z}}_{\text{bc}}=0\\ J_{\text{b}}{\ddot{\theta }}_{\text{bc}}+k_{\text{bz}}{z}_{\text{bA}}{y}_{\text{A}}+k_{\text{bz}}{z}_{\text{bB}}{y}_{\text{B}}+{\int }_{-B/2}^{B/2}{k}_{\text{wbz}}\left(z_{by}-z_{wy}\right)y\text{d}y+C_2{\dot{\theta }}_{\text{bc}}=0\\ m_{\text{w}}{\ddot{z}}_{\text{wc}}-m_{\text{w}}sin\varphi cos\varphi {\ddot{x}}_{\text{wc}}+{\int }_{-B/2}^{B/2}{k}_{\text{wb}}cos\varphi cos\varphi z_{wy}\text{d}y\\ -{\int }_{-B/2}^{B/2}{k}_{\text{wb}}{z}_{by}\text{d}y-{\int }_{-B/2}^{B/2}{k}_{\text{wb}}cos\varphi sin\varphi x_{wy}\text{d}y+C_3{\dot{z}}_{\text{wc}}=\mathrm{\Delta }{P}_{\text{z}}+\mathrm{\Delta }{F}_{\text{z}}\\ J_{\text{w}}{\ddot{\theta }}_{\text{wcz}}-J_{\text{w}}sin\varphi cos\varphi {\ddot{\theta }}_{\text{wcx}}+{\int }_{-B/2}^{B/2}{k}_{\text{wb}}cos\varphi cos\varphi z_{wy}y\text{d}y\\ -{\int }_{-B/2}^{B/2}{k}_{\text{wb}}{z}_{by}y\text{d}y-{\int }_{-B/2}^{B/2}{k}_{\text{wb}}cos\varphi sin\varphi x_{wy}y\text{d}y+C_4{\dot{\theta }}_{\text{wcz}}\\ =\mathrm{\Delta }{M}_{\text{z}}+\mathrm{\Delta }{F}_{\text{z}}{y}_{\text{D}}\\ m_{\text{w}}{\ddot{x}}_{\text{wc}}-m_{\text{w}}sin\varphi cos\varphi {\ddot{z}}_{\text{wc}}+k_{\text{wxA}}{x}_{\text{wA}}+k_{\text{wxB}}{x}_{\text{wB}}\\ +{\int }_{-B/2}^{B/2}{k}_{\text{wb}}sin\varphi sin\varphi x_{wy}\text{d}y-{\int }_{-B/2}^{B/2}{k}_{\text{wb}}{z}_{by}\text{d}y\\ -{\int }_{-B/2}^{B/2}{k}_{\text{wb}}cos\varphi sin\varphi z_{wy}\text{d}y+C_5{\dot{x}}_{\text{wc}}=\mathrm{\Delta }{P}_{\text{x}}+\mathrm{\Delta }{F}_{\text{x}}\\ J_{\text{w}}{\ddot{\theta }}_{\text{wcx}}-J_{\text{w}}sin\varphi cos\varphi {\ddot{\theta }}_{\text{wcz}}+k_{wxA}{x}_{wA}{y}_{\text{A}}+k_{wxB}{x}_{wB}{y}_{\text{B}}\\ +{\int }_{-B/2}^{B/2}{k}_{\text{wb}}sin\varphi sin\varphi x_{wy}y\text{d}y-{\int }_{-B/2}^{B/2}{k}_{\text{wb}}cos\varphi sin\varphi z_{wy}y\text{d}y\\ +C_6{\dot{\theta }}_{wcx}=\mathrm{\Delta }{M}_{\text{x}}+\mathrm{\Delta }{F}_{\text{x}}{y}_{\text{D}}\end{array}$$(11)

Where ϕ is the angle between z axis and the line of the work roll and backup roll centers, sinϕ=e/(R_w+R_b).   

$$\begin{array}{l}{z}_{\text{bA}}=z_{\text{bc}}+{\theta }_{\text{bc}}{y}_{\text{A}}\\ z_{\text{bB}}=z_{\text{bc}}+{\theta }_{\text{bc}}{y}_{\text{B}}\\ x_{\text{wA}}=x_{\text{wc}}+{\theta }_{\text{wcx}}{y}_{\text{A}}\\ x_{\text{wB}}=x_{\text{wc}}+{\theta }_{\text{wcx}}{y}_{\text{B}}\end{array}\}$$(12)

  

$$\begin{array}{l}{z}_{by}=z_{\text{bc}}+{\theta }_{\text{bc}}y\\ z_{wy}=z_{\text{wc}}+{\theta }_{\text{wcz}}y\\ x_{wy}=x_{\text{wc}}+{\theta }_{\text{wcx}}y\end{array}\}$$(13)

A horizontal force acts on the work roll due to the offset e, which makes the exit side of the work roll press on the square column with a deformation of ee, as shown in Fig. 6. In addition, there is a clearance between the entrance side of the work roll and square column, which makes the work roll get a certain motion space. When the horizontal displacement of work roll is less than −ee, the work roll is separated from the square column, and the horizontal equivalent stiffness between the work roll block and square column is 0, as shown in Eqs. (14) and (15).   

$$k_{\text{wxA}}=\{\begin{array}{c}\begin{array}{cc}{k}_{\text{ws}}& x_{\text{wA}}>-ee\end{array}\\ \begin{array}{cc}0& x_{\text{wA}}<-ee\end{array}\end{array}$$(14)

  

$$k_{\text{wxB}}=\{\begin{array}{c}\begin{array}{cc}{k}_{\text{ws}}& x_{\text{wB}}>-ee\end{array}\\ \begin{array}{cc}0& x_{\text{wB}}<-ee\end{array}\end{array}$$(15)

  

$$ee=\left(P_{\text{z}}sin\varphi +\left(T_1-T_0\right)/2\right)/k_{\text{ws}}$$(16)

Where k_ws is the horizontal equivalent stiffness of the square column.

ΔP_z and ΔM_z are the rolling force increment along the z axis and rotational torque increment around the x axis, respectively, which are caused by the uneven distribution of the rolling force along the roll body. The plant experiment results points out that the rolling force fluctuation is small and can be ignored. So,   

$$\mathrm{\Delta }{P}_z=0,\mathrm{\Delta }{M}_z=0.$$

ΔP_x and ΔM_x are the horizontal force increment along the x axis and rotational torque increment around the z axis , respectively, which are caused by the uneven distribution of the horizontal force along the roll body. And the horizontal force can be assumed as the friction force of the work roll because the bite angle is small. So, the horizontal force increment can be expressed as:   

$$\mathrm{\Delta }{p}_{xy}=p{R}_{\text{w}}\cdot \left({\mu }_{\text{s}}\frac{v_1\sqrt{h}}{v_{\text{r}}^{\text{2}}\sqrt{s_{1}R}}+\alpha \sqrt{\frac{\mathrm{\Delta }h}{R}}\right){\dot{x}}_{wy}$$(17)

  

$$\mathrm{\Delta }{P}_{\text{x}}={\int }_{-b/2}^{b/2}\mathrm{\Delta }{p}_{xy}\text{d}y$$(18)

  

$$\mathrm{\Delta }{M}_{\text{x}}={\int }_{-b/2}^{b/2}\mathrm{\Delta }{p}_{xy}y\text{d}y$$(19)

ΔF_x and ΔF_z are the interference force caused by the ATG meshing along the x axis and z axis, respectively. The experiment illustrates that the main vibration frequencies are the meshing frequency and its second harmonic frequency. In addition, taking the symmetry of the ATG driving system along the x axis and z axis into consideration, it is assumed that the meshing impact forces are equal in value along the x axis and z axis. So, ΔF_x and ΔF_z can be expressed as:   

$$\mathrm{\Delta }{F}_{\text{x}}=\mathrm{\Delta }{F}_{\text{z}}=A_s\left(sin\left(2\pi ft\right)+\gamma sin\left(2\pi (2f)t\right)\right)$$(20)

Where A_s is the amplitude of impact force, which is relative to the ATG wear degree and the rolling speed; γ is the coefficient of second harmonic frequency caused by nonlinear structure, this paper takes γ = 0.1 ; f is the ATG meshing frequency, which is taken as f = 52v_r/2πR; 52 is the number of arc teeth.

C₁…C₆ are the structural damping, which can be solved according to the damping ratio. And the value of the damping ratio is 0.02 in the general mechanical system.

4. Numerical Analysis and Discussion

4.1. Analysis of the Natural Characteristics of the Roll System Swing

The swing dynamic model of the roll system is established in the hot rolling process. The main structure and process parameters are shown as the following:   

$$\begin{array}{l}{m}_{\text{b}}\text{=61}\mathrm{\hspace{0.17em} }\text{811}\mathrm{\hspace{0.17em} }\text{kg,}\hspace{0.17em}\hspace{0.17em}{m}_{\text{w}}\text{=19}\mathrm{\hspace{0.17em} }\text{511}\mathrm{\hspace{0.17em} }\text{kg,}\hspace{0.17em}\hspace{0.17em}{J}_{\text{b}}\text{=105}\mathrm{\hspace{0.17em} }\text{931}\mathrm{\hspace{0.17em} }\text{kg}\times {\text{m}}^{\text{2}}\text{,}\\ J_{\text{w}}\text{=30}\mathrm{\hspace{0.17em} }\text{238}\mathrm{\hspace{0.17em} }\text{kg}\times {\text{m}}^{\text{2}}\text{,}\hspace{0.17em}\mathrm{\hspace{0.17em} }{k}_{\text{bz}}\text{=1}\text{.2}\times {\text{10}}^{\text{10}}\hspace{0.17em}\text{N/m,}\hspace{0.17em}\hspace{0.17em}{k}_{\text{wb}}\text{=3}\text{.47}\times {\text{10}}^{\text{10}}\hspace{0.17em}\text{N/m,}\mathrm{\hspace{0.17em} }\\ k_{\text{ws}}\text{=1}\text{.3}\times {\text{10}}^{\text{9}}\hspace{0.17em}\text{N/m}\mathrm{\hspace{0.17em} }{y}_{\text{A}}\text{=}-\text{1}\text{.75}\mathrm{\hspace{0.17em} }\text{m,}\mathrm{\hspace{0.17em} }{y}_{\text{B}}\text{=1}\text{.75}\mathrm{\hspace{0.17em} }\text{m,}\mathrm{\hspace{0.17em} }{y}_{\text{D}}\text{=2}\text{.945}\mathrm{\hspace{0.17em} }\text{m,}\mathrm{\hspace{0.17em} }\\ H\text{=15}\text{.27}\mathrm{\hspace{0.17em} }\text{mm,}\mathrm{\hspace{0.17em} }h\text{=6}\mathrm{\hspace{0.17em} }\text{mm,}\mathrm{\hspace{0.17em} }{v}_r\text{=2}\text{.9}\mathrm{\hspace{0.17em} }\text{m/s,}\mathrm{\hspace{0.17em} }{R}_w\text{=470}\hspace{0.17em}\text{mm,}\mathrm{\hspace{0.17em} }\\ R_D\text{=800}\mathrm{\hspace{0.17em} }\text{mm,}\mathrm{\hspace{0.17em} }B\text{=2}\text{.26}\mathrm{\hspace{0.17em} }\text{m,}\mathrm{\hspace{0.17em} }b\text{=1}\text{.7}\mathrm{\hspace{0.17em} }\text{m,}\mathrm{\hspace{0.17em} }e\text{=8}\mathrm{\hspace{0.17em} }\text{mm}\text{.}\end{array}$$

Assume that the driving side structure is same with the operation side. According to these parameters, the natural frequencies and vibration modes of the rolling mill are calculated, as shown in Table 1.

Table 1. Natural frequencies and vibration modes of roll system.

Frequency/Hz		f1	f2	f3	f4	f5	f6
		57.9	81.7	85.9	108.6	204.4	369.6
Vibration mode	zbc	0.039	0	−0.676	0	0	0.322
	zwc	0.047	0	−0.728	0	0	−0.947
	xwc	0.998	0	0.113	0	0	−0.002
	θbc	0	0.006	0	−0.501	0.442	0
	θwcz	0	0.015	0	−0.865	−0.897	0
	θwcx	0	0.999	0	0.018	−0.002	0

The translation and rotation of the roll system are independent of each other, which is because of the assumption that the structure of the driving side is same with the operation side. The translational frequencies are 57.9 Hz, 85.9 Hz and 369.6 Hz. The vibration mode is mainly the horizontal translation when the frequency is 57.9 Hz; the vibration mode is mainly the vertical translation of the work roll and backup roll with the same phase when the frequency is 85.9 Hz; the vibration mode is mainly the vertical translation of work roll and backup roll with the reverse phase when the frequency is 369.6 Hz. The rotational frequencies are 81.7 Hz, 108.6 Hz and 204.4 Hz. The vibration mode is mainly the horizontal rotation when the frequency is 81.7 Hz; the vibration mode is mainly the vertical rotation of the work roll and backup roll with the same phase when the frequency is 108.6 Hz; the vibration mode is mainly the vertical rotation of the work roll and backup roll with the reverse phase when the frequency is 204.4 Hz. The translation of 57.9 Hz and the rotation of 108.6 Hz are close to the arc ATG meshing frequencies (55 Hz, 110 Hz), which makes the resonance happen. In practice, the rolling speeds of different steel strips are different. The rolling speed is in the range of 2.0 m/s to 3.2 m/s at F2. So, when serious wear or assembly error of the ATG occurs, the ATG meshing impact force will happen with the frequency of 39.4 Hz to 63.4 Hz and the second harmonic frequency of 78.8 Hz to 126.8 Hz. The first four order natural frequencies of the roll system are in these frequencies range, which may cause severe vibration. In summary, the three new natural frequencies and vibration modes are derived with an eye to the rotation of the roll system, and it enriches the natural characteristics of the roll system. The new model is in line with the experiment phenomena, and the vibration mechanism is also more accurate.

4.2. Analysis of the Swing Characteristics of the Roll System

Because of the inaccurate installation, unreasonable lubrication, not timely maintenance to the ATG driving system, and violent friction force caused by its own work, etc., the faults of the ATG driving system often happened, such as the wear and spalling of the gear surface. These faults will result in some dynamic meshing impact force acting on the roll system, which reduces the stability of the roll system seriously. Then the dynamic characteristics of the roll system are analyzed in view of the meshing impact force of ATG. Based on the plant vibration characteristics, the ATG meshing impact force can be expressed as:   

$$\mathrm{\Delta }{F}_{\mathrm{x}}=\mathrm{\Delta }{F}_{\mathrm{z}}=30\mathrm{\hspace{0.17em} }000\mathrm{s}\mathrm{i}\mathrm{n}(2\pi \cdot 55\cdot t)+3\mathrm{\hspace{0.17em} }000\mathrm{s}\mathrm{i}\mathrm{n}(2\pi \cdot 110\cdot t)$$

Figures 7 and 8 show the results of the translation and rotation simulations at the C point. In the time-domain diagram, it shows that the horizontal vibration of the work roll is the most violent, followed by the vertical vibration, which is because of the horizontal stiffness of work roll being the smallest. In the frequency-domain diagram, the results of the translational vibration show that the main frequency is 55 Hz, the forced vibration occurs in the vertical direction, and the violent resonance occurs in the horizontal direction. While, the results of the rotational vibration show that the vertical vibration frequencies are 55 Hz and 110 Hz, the amplitude of 110 Hz close to its natural frequency is the maximum; the horizontal vibration frequency is 55 Hz, and the violent forced vibration occurs in the horizontal direction. From the above analysis, it can be seen that if only the translational characteristics are taken into account, the effect of 110 Hz vibration on the roll system stability will be ignored and the true vibration state can’t be reflected. Therefore, it is necessary to consider the rotation to establish a more comprehensive dynamical model of the roll system.

Fig. 7.

Displacement response of roll system in time domain. (a) vertical translation of backup roll; (b) vertical translation of work roll; (c) horizontal translation of work roll; (d) vertical rotation of backup roll; (e) vertical rotation of work roll; (f) horizontal rotation of work roll.

Fig. 8.

Displacement response of roll system in frequency domain. (a) vertical translation of backup roll; (b) vertical translation of work roll; (c) horizontal translation of work roll; (d) vertical rotation of backup roll; (e) vertical rotation of work roll; (f) horizontal rotation of work roll.

The vibration mechanism of the roll system is the forced vibration, so the vibration phase of roll system is determined by the external force, namely the ATG meshing impact force. When the phases of the external force acting on the upper and lower work rolls are same in the vertical direction and are reverse in the horizontal direction, the vibration phase characteristics will be consistent with the experimental, as shown in Fig. 9.

Fig. 9.

Vibration response of the upper and lower work rolls. (a) in vertical direction; (b) in horizontal direction.

Based on the relationship that the rolling toque approximately equals to the friction force times roll radius, the rolling torque of single work roll is simulated, as shown in Fig. 10(a). Figure 10(b) is the actual dynamic rolling toque measured by wireless telemetry devices. The simulation result is consistent with the measured result, which can confirm the validity of the dynamic model established.

Fig. 10.

Rolling torque of single work roll. (a) simulation result; (b) measured result.

Figure 11 shows the dynamic swing response of the work roll with two different interference forces in the horizontal direction. In Fig. 11(a), the vibration amplitudes of both ends of the roll body are larger than that of the middle portion at beginning, and then the vibration amplitude of the operation side becomes the largest. In Fig. 11(b), the vibration amplitude of the driving side is the largest. It can be seen that the vibration amplitudes at one end or both ends of the roll body are larger than that of the middle portion due to the rotation. So, both ends of the roll body are more prone to generate the vibration mark. And it is coincided with the vibration mark distribution on the roll body in the plant, as shown in Fig. 12. The plant vibration mark is mainly distributed on the operation side or driving side of the work roll.

Fig. 11.

Horizontal swing response of work roll. (a) A₀=30000 N; (b) A₀=10000 N.

Fig. 12.

Vibration marks on the roll body in the plant. (a) operation side; (b) driving side.

4.3. Stability Analysis of the Roll System

The Eq. (11) illustrates that the influence of the horizontal force increment ΔP_x and the periodic meshing impact ΔF_x, ΔF_z generated by the ATG are the main factors which affect the roll system stability.

4.3.1. Effect of ΔP_x on the Stability

In the plant, some steel strips vibrated and some steel strips ran stably with the same rolling mill structure. It indicates that the rolling process parameters have a significant impact on the stability of the roll system. ΔP_x is the key link affected by the rolling process parameters and is a damping term of the roll system. Equation (17) declares that the damping term includes both positive damping and negative damping. So, the system stability can be improved by increasing the positive damping and reducing the negative damping, such as increasing the exit thickness h, reducing the reduction Δh, reducing the rolling speed v_r, and improving the rolling friction coefficient μ_s.

The parameters (inlet thickness H, exit thickness h and rolling speed v_r) of the vibration steel strips and stable steel strips are extracted from the plant PDA data, then the influence of Δh/h and v_r on the system stability are studied, as shown in Fig. 13. It can be seen that the greater Δh/h and v_r are, the worse the stability is. And the critical curve is also drawn based on Eq. (17). The damping on the left side of the curve is larger than the right side, so the process parameters on the left side of the curve are more stable than the right side. Contrasting the critical curve and the plant data, all of vibration steel strips data are on the right side of the curve, and most of stable steel strips data are on the left side of the curve except for some data near the critical curve. The reason that some data near the critical curve don’t meet the curve is because of the critical curve only considering the rolling speed, inlet thickness and exit thickness parameters and ignoring other factors. In reality, there are also other factors deciding the stability of the roll system such as the structure assembly precision of rolling mill, rolling mileage, steel strip surface quality, which cause a little difference between the theoretical critical curve and the plant data. In general, the curve is basically consistent with the plant data and both of them can indicate the effect of the rolling process parameters on the roll system stability effectively. It will provide a theoretical basis for the rolling schedule design in view of the roll system stability.

Fig. 13.

Statistics of vibration strips and stable strips parameters.

4.3.2. Effect of ΔF_x, ΔF_z on the Stability

ΔF_x, ΔF_z are related to the degree of ATG wear and the rolling speed. In the gear dynamics, the meshing impact force will be strengthened by the increasing of the tooth side clearance and the rolling speed. Assume that the meshing impact force is proportional to the square of the rolling speed, and is proportional to the degree of gear wear. And then, take the above section’s amplitude (30000 N) as a basis value, and the amplitude of the meshing impact force can be expressed as:   

$$A_s=A_1{A}_2{A}_0$$(21)

Where A₁ is the influence coefficient of ATG wear degree on the impact force; A₂ is the influence coefficient of the rolling speed on the impact force, A₂=(v_r/2.9)²; A₀ is the basis value, A₀=30000 N.

(1) Influence of ATG Wear Degree on the Stability of the Roll System

The bifurcation diagram of the roll system with the change of A₁ is shown in Fig. 14. The system is periodic-one motion when A₁=[0, 0.616]. A jumping phenomenon appears with the vibration amplitude changing at A₁=0.616. And then the system enters chaotic motion. At A₁=0.96, the system enters periodic-one motion again. Then a transient chaotic motion occurs at A₁=1.08, and the system enters periodic-one again at A₁=1.13. Then period-doubling bifurcation occurs at A₁=1.58, from periodic-two motion to periodic-four motion to periodic-two motion, and it finally enters chaotic motion at A₁=2.11. The dynamic behavior of the system is rich with the change of A₁. In overall, the smaller the A₁ is, the more stable the roll system will be. So, the high installation accuracy, full lubrication system and timely maintenance to the ATG can improve the stability of the rolling mill system. In the plant, the vibration was under control by replacing the excessive wear ATG shaft.

Fig. 14.

Bifurcation diagram of roll system with the change of A₁.

(2) Influence of the Rolling Speed on the Stability of the Roll System

The change of rolling speed v_r affects the meshing impact force amplitude and frequency and the horizontal force ΔP_x. The bifurcation diagram of the roll system with the change of v_r is shown in Fig. 15. The system is periodic-one motion with the smaller amplitude when v_r=[1.5, 2.67]. Among this range, a local maximum appears at v_r =2.08, and then the system enters chaotic motion at v_r=2.67. At v_r=2.8, the system enters periodic-one motion again. Then, a period-doubling bifurcation occurs at v_r=3.02, from periodic-one motion to periodic-three motion. At v_r=3.17, a jumping phenomenon appears which makes the system enter periodic-one motion. And then, a period-doubling bifurcation occurs again at v_r=3.59, from periodic-one motion to periodic-two motion to periodic-four motion. Finally, it enters chaotic motion at v_r=4.3. Through the above analysis, the speed region and the critical value of the chaotic motion and multi-period motion are obtained, which can artificially control and avoid the uncontrollable and unpredictable movement.

Fig. 15.

Bifurcation diagram of roll system with the change of V_r.

5. Conclusion

(1) The vibration modes of the upper and lower work rolls are the same phase in vertical direction and the reverse phase in the horizontal direction; the vibration has little effect on the rolling force and has a significant influence on the rolling torque.

(2) A dynamic friction force model is built taking the horizontal vibration of the work roll into consideration, which characterizes the positive and negative damping terms. It provides an accurate theoretical basis for the dynamic model analysis.

(3) A swing model of the roll system is established. And the natural characteristics are analyzed. The results show that the translation of 57.9 Hz and the rotation of 108.6 Hz are close to the arc tooth gear meshing frequencies (55 Hz, 110 Hz), which makes the severe resonance occur.

(4) The influence of the periodic meshing impact force on the roll system stability is analyzed. The results show that if only the translational characteristics are taken into account, it will weaken the influence of the external disturbance force on the stability of roll system; the vibration amplitude at the end of the roll body is larger than that of the middle portion, both ends are more prone to generate vibration marks.

(5) The stability of the roll system is analyzed. The results show that the greater Δh/h and v_r are, the worse the stability is. And the critical curve is basically consistent with the plant data. In addition, the dynamic behavior of roll system will transform from periodic-one motion to chaotic state with the increasing of the wear degree of ATG and rolling speed.

(6) The complex vibration mode of the driving system also has a significant impact on the stability of the roll system. A perfect rolling mill dynamic model will be researched consisting of the dynamic characteristics of the driving system and the roll system, which is contribute to researching the true “temper” of the rolling mill.

Acknowledgements

This research is supported by National Natural Science Foundation of China (Grant No. 51375424). The experiments were made in Shougang Qian’an Iron and Steel Co., Ltd. We gratefully acknowledge the technical support.

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