`Advances in Mechanical EngineeringVolume 2012 (2012), Article ID 734510, 7 pageshttp://dx.doi.org/10.1155/2012/734510`
Research Article

## Vertical Vibration Model for Unsteady Lubrication in Rolls-Strip Interface of Cold Rolling Mills

1School of Automation and Electrical Engineering, University of Science and Technology Beijing, Beijing 100083, China
2Key Laboratory of Advanced Control of Iron and Steel Process (Ministry of Education), University of Science and Technology Beijing, Beijing 100083, China

Received 6 October 2012; Accepted 23 November 2012

Copyright © 2012 Xu Yang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

According to the vertical vibration phenomena existing in cold rolling mills, the unsteady lubrication mechanism in roll gap and its influence to rolling stability was chosen as the case for analysis. On the basis of rolling theory, lubrication and friction theory, and mechanic vibration theory, the vertical vibration model for unsteady lubrication in rolls-strip interface was presented. The Geometry model of roll gap, the unsteady lubrication model of roller-strip working interface, the distribution model of normal rolling stress and friction stress, and the rolling vertical structure model were taken into account. Based on the rolling equipment and process parameters of aluminium mill, the rolling force curve and dynamic response of working roll displacement variation was simulated on Matlab/Simulink platform. A comparison with actual production data shows the validity of this vibration model.

#### 1. Introduction

The cold rolling process is a high speed, transient, and time-related metal machining molding process [1]. As cold rolling mill is a multimass and rotary motion object system, the vibration as an implicit factor commonly exists in the course of its operation, but it usually does not apparently present due to the conditional constraints of factors such as various rolling processes, equipment, and control. In fact, once the mill running state exceeds the accuracy requirements of the product or equipment bearing capacity, mill vibration phenomena will emerge. Of torsional vibration, vertical vibration, and vertical torsion-coupled vibration of transmission system in the rolling process [2], the third-octave vibration of vertical system causes the greatest harm [3, 4], which will have adverse effects on quality indicators of rolled product in thickness and shape and even cause a great deal of production loss such as serious equipment damage. Therefore, the modeling, analysis, and control for cold rolling mill with vibration factors have been the focus of research domestically and abroad all the time [5, 6], and also a number of meaningful research results have appeared [7, 8].

The rolling process is the process of roll contacting strip with each other, and the frictional force is the tangential resistance in the interface between these two objects. As we can see from [9, 10], the unstable lubrication state of two degrees of freedom can lead to a stick-slip phenomenon and therefore cause self-excited vibration. In order to reduce the adverse effects caused by the outer friction, it is essential to have the industrial lubricant mechanism in roll gap deformation zone [11]. In the process of high speed rolling, vertical vibration will cause the lubrication system in an unsteady state and may further cause the mill system oscillation instability. Therefore, studying this unsteady lubrication phenomenon on working interface in forming process of large deformation metal is especially of important significance, which is good for establishing a dynamic friction model that is more suitable to the actual working conditions, so as to derive a more accurate and comprehensive kinetic model of the rolling process and vertical vibration model of rolling mill system. However, the existing rolling friction models are mostly steady-state models [12, 13] that cannot effectively reflect lubricated friction state changes in the deformation zone caused by the parameters change in the rolling dynamic process. To this end, this paper, based on the roll gap geometry model in dynamic rolling process, comprehensively uses the unsteady lubrication friction model on rolls-strip contact interface and on this basis derives the normal rolling stress distribution model and friction stress distribution model in roll gap deformation zone. It ultimately combines the mill vertical system structure model, and vertical vibration model of single-stand cold rolling mill is established with coupling those models above. It contributes to the theoretical basis for the study on vibration mechanism analysis and control strategy in the high-speed rolling process.

#### 2. Vibration Model of Vertical System of Cold Rolling Mill

##### 2.1. Roll Gap Geometry Model

The roll gap (deformation zone) is an elastic-to-plastic deformation area produced by strip directly contacting roll in the rolling process. Set the coordinates and onto the center line of roll and the rolling centerline; the intersection point as the origin of coordinates and the rolling direction is directly reversed with the positive -axis. In steady rolling condition, usually assume that the strip outlet position is coincided with the center lines of the upper and lower work rolls, whereas this assumption is not valid in the dynamic rolling process. Figure 1 shows the roll gap geometry model in unsteady rolling process of the cold rolling mill, where are the front and back tension stresses of rolling strip; , and are strip entry velocity, exit velocity, and rolling speed, respectively; is the distance from strip entrance to centerline of two rolls; is the distance from strip at exit to the centerline of two rolls; is the distance from neutral plane to centerline of two rolls; are strip thicknesses at entrance and exit, respectively; is the variation rate of roll gap displacement at vertical vibration state; is work roll radius in consideration of roll elastic flattening and strip elastic recovery.

Figure 1: Geometry of roll gap in cold rolling.

Assume that roll gap shape appears as a quadratic parabola curve; a height of any position in deformation zone is.

Since the strip is thinner and harder during cold rolling so the contact arc gets larger unit pressure, which makes the roll produce flattening effect at a contact arc, thus increases the actual length of the contact arc and affects the variation rate of the rolling force. Therefore, based on the existing research results [14], this paper considers the roll elastic flattening and strip elastic recovery and derives a roll radius formula that is more suitable to actual working conditions as follows: where is working roll radius before flattening, is unit rolling force, is a downward distance variation by roll pressure, is strip elastic recovery amount, , is Poisson’s ratio for roll, and is roll’s elastic modulus.

##### 2.2. Unsteady Lubrication Friction Model on the Working Interface of Roll-Strip

In the mill vertical vibration accidents of recent years, people gradually noticed that lubrication conditions are important reasons that cause or influence vibration, especially to mill vertical self-excited vibration [5]. For example, the viscosity of the lubricant, the emulsion stability, and the instability of oil film strength and thickness may lead to the emergence of vibrations and also, in the cold rolling process, the state of lubrication of the roll-strip interface is mostly a mixed lubrication; that is, partial fluid film lubrication, adsorbed film lubrication, and asperity contact (dry friction) coexist [15]. In this state, one part of the load and the frictional force of the contact interface is borne by the rough contact surface, and the other part is borne by the pressure emulsion in notch on the contact surface. When the friction lubrication state of certain areas of the deformation zone is in the former, its friction stress equation can be expressed by the Coulomb friction model: where is friction shear stress in deformation zone, is coefficient of friction, and is rolling normal stress.

If in a state of fluid lubrication, the lubricating fluid shear stress equation on contact interface is where is dynamic viscosity of the lubricant, is tangent velocity of fluid parallel to the friction stress, and is -coordinate perpendicular to the rolling velocity.

Based on (2) and (3) and the above analysis, it was found that mixed friction state of dry and wet friction coexists in the cold rolling process [16]. Therefore, the unsteady friction stress model on roll-strip interface can be expressed as where is roll gap unsteady coefficient of friction with mixed lubrication.

##### 2.3. Stress Distribution Model in Deformation Zone

In the rolling process, the metal withstands the effect of rolling force by two rolls and the plastic deformation occurs. When rolling with the front and back tensions occurs, the metal in the middle of deformation zone presents the state of three side compressive stresses. Due to the tension, the metal withstands one-dimensional tensile stress and two-dimensional compressive stresses at the place near the entrance and exit. In this paper, based on classic Karman unit rolling force differential equations, it is assumed that the metal material in the deformation zone has homogeneous deformation with plane strain, meanwhile some of the assumptions of Karman unit rolling force equation are relaxed: considering the unsteady lubrication friction state variations, the coefficient of friction is no longer a constant considering the work roll elastic flattening and strip elastic recovery, the roll is no longer treated as absolutely rigid body inelastic deformation. Now take microunit from strip, the micro unit is (assume the unit width of strip as 1), and make the coordinate direction of the micro unit coincide with direction of main deformation as shown in Figure 2, where letting , as angle is very small, so approximately .

Figure 2: Slab analysis on a volume element of roll gap.

In direction, the force balance equation of front and back slip zones is (where “+” corresponding to back slip zone, “−” corresponding to front slip zone)

Expand the polynomial and ignoring the miniterm, the equation is

By the yield conditions of metal plastic deformation, the following equation can be obtained: where is unidirectional deformation yield limit of metal material and is deformation resistance in condition of plain strain (metal flow stress).

Make derivative of (8), (1) and substitute both derived equations and (5) into (7):

In order to study the normal rolling stress and friction stress distribution in deformation zone, now solving rolling normal pressure stress and friction stress for front slip zone and back slip zone, respectively. For front slip zone, the front slip zone formula in (9) is converted to the following form:

Seting , (10) can use solution of equation to obtain the following solution:

Assuming that there is no work-hardening effect in deformation zone and considering mill front and back tension effects, the boundary condition can be obtained.

For friction stress distribution on front slip zone, converting equation (9) can obtain the following equation:

The metal flow velocity equation in front slip zone can be expressed as substituting it with (12) into (5), the friction stress equation can be derived as follow:

Similarly, by the above-mentioned derivation method, the rolling normal pressure stress distribution and friction stress distribution equations in back slip zone can be obtained: where and is back slip rate.

##### 2.4. Vertical Vibration Model on Unsteady Lubrication Condition in Cold Rolling

A set of mill vertical system includes the upper and lower work rolls, the upper and lower support rolls, hydraulic systems, and stand components. Usually depending on different research purposes and the accuracies, it is simplified to the multidegrees-of-freedom vibration system of “mass-damping-spring” using the mechanical model method. This paper as shown in Figure 3 takes a single stand with four rolls cold mill as a study object and assumes that physical contact between the work roll and support roll is considered as a spring [17] besides, two rolls always remain in contact status during the rolling process. Since mill is basically symmetric along the rolling line, the four-roll mill can be simplified as single degree of freedom vibration system, as shown in Figure 4.

Figure 3: Single-stand 4-high cold rolling mill.
Figure 4: Single-freedom vertical vibration system.

In Figure 4, is equivalent spring stiffness between the work rolls and support rolls, is equivalent damping coefficient of upper roll system, and is rolling pressure value passed through the bearing seat by the hydraulic system, in which the value can be calculated by the following equation: where is width of strip, is unit rolling force, and are rolling normal pressure values in front slip and back slip zones, respectively.

The differential equation of vertical vibration model of cold mill in condition of unsteady lubrication based on the model shown in Figure 4 can be obtained (where is work roll equivalent mass):

#### 3. Simulation Verification and Result Analysis

In order to verify the validity of the above model, a simulation verification analysis was carried out by using 1850 single-stand cold mill parameters and actual production data in an aluminum plant, using Matlab/Simulink platform to do simulation verification analysis in which the strip width is 760 mm and such specification can roll three kinds of passes such as 6.0 mm3.5 mm, 3.5 mm2.0 mm, and 2.0 mm1.1 mm, respectively, with metal flow stress , and other major material parameters as shown in Table 1.

Table 1: Material parameters.
##### 3.1. Simulation Analysis on Rolling Force and Normal Rolling Stress

Based on the former research [18], the rolling resistance and friction model were coupled in dynamic rolling process. So its worth noting that the approximants as well as the relationship between these two factors should be emphasized. Under this circumstance, by selecting 3102 aluminum alloy strip with 760 mm width in second pass (3.5 mm2.0 mm) to do rolling pressure simulation comparison verification, the rolling force distribution curve in roll seal deformation zone can be obtained. As shown in Figure 5, the -direction is the conveying direction of the strip, including front and back slip zones, and is the direction of the rolling pressure. The rolling force along the band steel plate width direction basically presents a uniform distribution with a slight rise of rolling force on the part of edge area, which is due to the relatively narrow strip, and both sides of the work rolls are too bonded with strip that causes the width spread deformation on the edge, which does not greatly affect the calculated value of the rolling force in the unsteady lubrication conditions.

Figure 5: Rolling force distribution in the roll gap.

Then, based on the established rolling force model calculation equation in this paper, the actual process, and device parameters, the rolling process for 760 mm width strip with three passes were calculated and compared with the actual rolling force, as shown in Table 2. The calculation results show that the errors between the simulation values and measured values did not exceed 5%, indicating that the simulation results are more in line with actual requirements.

Table 2: Rolling Force Comparison.

The normal rolling stress model and friction stress model derived and established by this paper are based on the conditions of unsteady lubrication in roll gap deformation zone, and calculation is carried out in the front and back slip areas. By simulation experiment, it is found that the normal rolling pressure stress in deformation area calculated based on this model exists a jump occurrence phenomena at the neutral plane, which is shown in Figure 6. According to the existing research results [1517], it is known that the friction lubrication state variation in the deformation zone, especially the negative damping effect in the deformation zone, would have direct impact on vertical system stability and may result in the mill system self-oscillation. Thus, the stress hopping phenomenon shown in the figure is also one of the reasons why self-excited vibration phenomena happen to the cold rolling mill in the high-speed rolling sheet.

Figure 6: Rolling stress distribution.
##### 3.2. The Response Analysis of Vertical System in Dynamic Rolling Process

The vertical vibration of a cold rolling mill as a dynamic process can mainly be expressed by the amount of change of displacement of the work rolls and strip during the rolling process. If the change is too large, it means that the vertical vibration has affected the rolling thickness at exit and surface quality. Therefore, discussing the displacement dynamic response curve of cold rolling mill work rolls or strip under different working conditions through simulation can effectively simulate mill vertical vibration phenomenon, analyze vibration mechanism, and can thus propose further vibration avoidance strategy. The past traditional coefficient of friction of cold rolling deformation zone often used the experimental constant or a small variation range of absolute constant (coefficient of friction is usually between 0.01~0.3), which is not quite suitable in the analysis of mill vertical vibration, that is, parameter sensitive dynamic process, while the friction stress distribution calculation equation established by this paper uses the roll-strip interface unsteady friction model, which is very suitable for simulation analysis of the mixed lubrication friction state in cold deformation zone and the mill system instability phenomena caused by this state. Therefore, this paper selected the roll gap friction model with the constant coefficient of friction and compared it with the model proposed by this paper for comparative analysis, to discuss the amount of displacement of the work rolls under the influence of different parameters based on (16). Let us take the rolling speed  m/min, and 760 mm bandwidth with the third pass. Figures 7 and 8 are comparison charts of amount of displacement of work rolls under an ideal work conditions (other parameters remain the same), based on the different friction stress distribution model. It can be seen from Figures 7 that the mill vertical system is already in the critical steady state due to the current speed exceeding the rolling speed proposed for this pass specification, and if the presence of the slight interior and exterior interference of the system or the rolling speed continues to increase, it may lead to the mill system instability oscillation.

Figure 7: Dynamic response curve of work roll with constant friction coefficient.
Figure 8: Dynamic response curve of work roll under the influence by unsteady friction stress equation.

Figure 8 shows the dynamic response curve of work roll displacement calculated by unsteady lubrication friction model of roll-strip work interface established by this paper. As it can be seen, as rolling speed exceeds the safety range, unsteady-state friction model has predicted divergent work roll displacement response at this time, and if no action is taken, the vertical vibration equation will continuously add energy to system along with constant increase of back tension fluctuation and negative damping effect in deformation zone, which causes the system to increase the amplitude of the vibration with stronger vibration.

In addition, collecting the field measured data of thickness variations of strip of cold rolling mill under the conditions of this process, and processing offline spectrum analysis with the effect shown in Figure 9, it can be found that in such rolling speed the aluminum cold rolling mill has appeared frequency about 100 Hz. A third octave vertical vibration which is also consistent with the phenomenon shown in Figure 8 indicates the validity and applicability of the unsteady lubrication mill vertical vibration model established by this paper.

Figure 9: Frequency spectrum analysis of strip thickness under  m/min.

#### 4. Conclusion

(1)For frequently appeared self-excited vibration phenomenon in high-speed rolling process on cold strip, this paper comprehensively uses rolling theory, lubrication friction theory and mechanical vibration theory, to establish the vertical vibration system model of rolling Mill under the condition of unsteady lubrication. The model is composed of roll gap geometry model, unsteady lubrication friction model of roll-strip contact interface, normal rolling stress and friction stress distribution models in the roll gap deformation zone, and mill vertical mechanical structural model dynamic rolling process. (2)Based on the above models, actual rolling mill device and process parameters were used to simulate dynamic rolling process of single-stand aluminum cold rolling mill. By simulation and prediction of rolling force and normal rolling stress the comparison between measured value and calculated value was made to show its model validity. Besides, the dynamic response analysis of work roll displacement with constant friction coefficient and dynamic friction coefficient was simulated separately; meanwhile the frequency spectrum with field measured data shows third octave vertical vibration, which is also consistent with the phenomenon shown in Figure 8, indicating the validity and applicability of the unsteady lubrication mill vertical vibration model established by this paper.

#### Acknowledgments

The authors gratefully acknowledge the support by the National Natural Science Foundation of China (51205018), China Postdoctoral Science Foundation Funded Project (2012M510321), and Fundamental Research Funds for the Central Universities (FRF-TP-12-104A, FRF-AS-11-004B). Meanwhile, great thanks also go to former researchers for their excellent works, which give great help for our academic study.

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