Impact Analysis of Roller System Stability for Four-High Mill Horizontal Vibration

In order to study the hot Compact Strip Production (CSP), four-high mill vibration characteristics, and vibration suppression method, the roller system structure stability was analyzed and calculated at first in the paper. And then, the mill stand gap was measured at field and its influence on roll transverse vibration was analyzed. The drum gear coupling effect on the roller system stability and the automatic balance conditions of the coupling transmission torque were studied; the influence of axial force caused by the roller cross on the system stability was analyzed. Finally, the roller transverse friction chatter vibrationmechanics model was established; the simulation analysis was carried out with eliminating mill house-bearing clearance and adding floating support for coupling, respectively. And the characteristics of the roller “jump vibration” were studied. We applied copper gaskets to eliminate or reduce mill house-bearing clearance for suppressing the rolling mill vibration on the spot; the test results show that the roller transverse vibration was suppressed after eliminating clearance.


Introduction
For the convenience of installation work roller, there is clearance between work roller bearing seat and mill house in fourhigh mill, which makes work roller in an unstable state when there is no fixed side force on it.So it should have an offset  between work roller axis center and backup roller, and its value certainty principle is to make the lateral reaction force of the frame acting on work roller bearing always greater than zero, and the force direction is constant.Modern four-high mills all set corresponding offset value [1,2], such as 2050 CVC four-high rolling mill and 4300-medium plate mill of the German SMS, four-roll rolling mill of the British DAVY, 1580-pair cross four-high rolling mill of Japan Mitsubishi heavy industries.But these modern four-high mills with offset all have roller parallel positioning instability, and with the increasing of rolling load and speed, their disadvantages will become more serious.For these CSP mills, the offset value is 1.0∼1.2mm, which also facilitates replacement roller installation and disassembly and serves as a compensation of bearing thermal expansion space.
Although scholars have conducted a lot of research on rolling mill vibration, because the problem is too complicated, it has not been solved well.A. Bar and O. Bar presented the numerical analysis of vibrations with moderate frequencies excited during the milling process taking into account the transportation motion of the strip and its inertial properties [3].Bar and Świątoniowski built the nonlinear mathematical model of oscillated system: the continuous group of the rolling stands coupled by transferring strip [4].Panjković et al. found that the frictional conditions in the roll gap are the principal cause of chatter in this mill and the frictional conditions appear to be associated with the thickness and properties of oxide formed on rolls [5].Furumoto et al. designed a chamber in mill stabilizing device and optimized its size [6].Kim et al. modeled a rolling mill that includes the driving system by multibody dynamics to investigate the cause and characteristics of the chatter vibration.The chatter frequency was 1190 Hz and was caused by the rolling force.The amplitude of chatter vibration could be reduced by controlling the speed of the roll and the static and dynamic components of the rolling force [7].They also proposed a mathematical model of a cold rolling mill including the driving system and a novel combination of the direct integration method and quasistatic analysis to solve the model efficiently and found the horizontal chatter vibration had a strong effect on the dynamic characteristics [8].Swiatoniowski and Gregorczyk presented a probabilistic model of the friction phenomena on the work-backup rolls contact surface and fount that such character of the disturbance in distribution of zones with static and kinetic friction can be regarded as one of the sources of self-excited vibrations appearing in the system consisting of a rolling mill and a strip [9].Amer et al. studied the torsional vibration reduction for rolling mill's main drive system via negative velocity feedback under parametric excitation and found the resonance condition is the first natural frequency vibration as one of the worst resonance cases [10].
Usually, rolling mill vibration research does not distinguish whether rolling mill roll system is stable, and mechanics model is set up setting the mill under stabilization.A single vibration as the coupling mode with roll vertical, horizontal, axial, torsional, cross, and swing space vibration behavior is analyzed.Actually, roller vibration will damage roll system stability and cause bearing horizontal or vertical vibration.Therefore, the CSP four-high mill vibration characteristics were studied from the mill roll system stability and lateral dynamic behavior of the roll system in this paper.

Roller System Structure Stability Calculation
Work roller force analysis is shown in Figure 1, where Due to the included angle between the roller and drum gear spindle axis, the spindle will produce additional moment   on the roller; namely, in the horizontal direction, it will make the roller produces additional horizontal force   and   =   /.The force on the side is pointing in the direction of work roll shift direction and the force is departure from the direction of work roll shift on the other side and may cause two rollers' crossing, where  is the distance between the center lines of the both work rollers bearing seat and   =   ⋅ tan ,  is included angle between the spindle and roller and   is the sum of the rolling torque   + support roller rotation torque   + work roller bearing friction consumption torque   ; namely, where  is rolling force arm.When  1 >  0 ,  =   If cos  ≈ 1, cos( + ) ≈ 1, and tan( + ) ≈ sin  + sin  = /( 1 +  2 ) + ( 2 + )/ 2 , tan  ≈ sin  = ( 1 −  0 )/2, and then (3) can become as follows: At this time, the work roller offset should be satisfied as follows: By (5), if one considers the included angle between the spindle and the roller influence, the work roller offset should be increased Minimum offset change scope with the rolling force change under work roller diameter within a certain range is shown in Figure 2. According to the experience in the production, the offset values are usually larger than the theoretical calculation (3∼5 mm).The offset  of F3 rolling mill is 10 mm; when the work roller diameter is small, it probably meets the requirements of the static stability.

Mill House Clearance and Drum Gear Coupling Effect on the Roll System Stability
The CSP mill house and bearing plate never change from production in recent 4 years; their wear and tear may be large.So the F3 mill upper and lower work roller bearing seat width of drive side and operation side were measured.In order to improve the measurement accuracy, 4 points of bearing plate were selected and shown in Figure 3 (1∼4 points on operating side, 6∼9 points on drive side), and then the average was got.Five points were selected on mill house plate (1∼5 points on operation side, 6∼10 points on drive side), and then the average was got.The measurement results of the bearing seat and mill house and the original design values are shown in Table 1.
From Table 1, bearing width has negative deviation; the negative deviation is around 0.5 mm.Mill house width has positive deviation; the deviation value is around 1.8 mm.The upper work roller clearance averages of drive side and operation side are 2.343 mm and 2.442 mm, respectively.The lower work roller clearance averages of drive side and operation side are 2.194 mm and 2.319 mm, respectively.Obviously, the F3 mill house and bearing seat clearance are much larger original design requirement (transition fit); namely, the gap provides necessary conditions to roller transverse vibration.
According to the mechanical transmission theory, the condition of spindle can be normally operated as follows: gear shaft axis, roller axis, and drum gear spindle axis must be located on the same plane, and we must ensure that both the included angles with the spindle axis are equal.But with roll bearing seat and mill house plate wear (in the outlet side mainly), roll connected end to the spindle will shift toward the exit side.Then, the spindle will produce angle between the vertical planes; namely, the spindle axis, gear axis, and roller axis are not in a plane.This will make the spindle have additional torque on the roller, and the additional torque will make the work roller produce deflection movement in the horizontal direction, which will make the bearing seat acting force uneven (as shown in Figure 4) and the bearing seat tilt (Figure 5) and roller axis cross (Figure 6) and other adverse condition.The additional horizontal force directions of the operation side and transmission side of the roller are opposite, under the combined action of the clearance and the additional bending moment, which will make the roller bearing seat have crash tendency on mill house (this is confirmed from the scene that bearing seat entrance side plate also has certain wear and tear).In addition, because of the bearing seat plate force and uneven wear (Figure 4), it  will destroy the roller system stability under roller insufficient constraints.
Work roller axial force caused by backup roller cross and rolled strip cross is divided into roller axial force and deformation zone axial force.And the deformation zone axial force is dominant, which mainly comes from axial component of the friction force, and its direction depends on the metal particle longitudinal and transverse sliding velocity direction relative to the roller surface; its value is related to rolling pressure and the friction condition, as follows:  where  is the rolled strip width,  is the deformation zone length,  is the contact angle and tan  = −(1/2)ℎ   = ((ℎ 0 − ℎ 1 )/)(1 − /), ℎ  is rolled strip thickness at the  coordinate deformation zone, and ℎ 0 and ℎ 1 are the rolled strip thickness at entrance and export.  is axis component of deformation area friction force.According to the axial force vector diagram in Figure 7, there has been   = |sin( + )|, where  is the deformation area friction,  is sliding angle between metal and the rollers,  is roller crossing angle, V is the arbitrary point metal flow rate in deformation area, and V  is rotational speed of roller surface.Thus, according to the   calculation equation, the relationship curves between axial force and reduction rate and bearing-mill house clearance are shown in Figure 8.It can be seen that the axial force caused by roller cross increases with reduction rate and clearance increase; its effect on the system cannot be ignored.
The F3 mill upper work roller vibration acceleration testing time-domain parameters on field is shown in Table 2.It can be seen that although axial force is less than 1% of the rolling force, roller axial vibration acceleration amplitude is close to vertical acceleration and lateral acceleration, so the axial force (caused by rollers cross) destruction on the system stability cannot be ignored; we should try to eliminate the gap between roller bearing seat and the mill house; at the same time, it also shows the gap has important influence on roller vibration.
In addition, because the length of mill drum gear coupling is about 5 m and its weight is about 7 tons and it has no  support, in the presence of the spindle imbalance, due to the internal or external gear eccentricity and the gravity center offset, these will bring large unbalanced force and affect the roll system dynamic stability.In the presence of drum gear tooth tip clearance, centrifugal force   will be generated:  where  is the outer gear shaft spindle weight, kg;  is the coupling working speed, r/min; Δ is the relative radial displacement, mm, as shown in Figure 9.As a result of the existence of the centrifugal force, in order to make the interior and exterior of tooth coupling automatically, work roller needs to transmit the minimum torque  min ; namely, where  is the weight of the intermediate shaft or internal gear ring, kg;  is inside and outside of the teeth relative eccentricity, mm;  is the coupling speed, r/min. is tooth reference circle diameter, mm.Substituting parameters into (8), we get  min = 4.52×10 7 N⋅m, and the measured torque is 4×10 5 N⋅m; that is, it cannot meet self-centering requirement; there always is centrifugal force or unbalanced force.The unbalanced force on bearing seat diagram is shown in Figure 10; there, point 1 shows unbalanced force upward, point 2 shows the unbalanced force toward left which makes  transverse vibration natural frequency was calculated with finite element method, and within the roller diameter available range, the first three-order natural frequency of the roll system horizontal vibration is 4.80∼4.99Hz, 50.9∼55.5Hz, and 55.5∼82.4Hz, respectively [11].
It can be seen from Figure 13 that the measured vibration main frequency has almost linear relationship with speed, and the 20 Hz torsional vibration is mainly around concentrated in the gear mesh frequency of distribution box and the first-order natural frequency of torsional vibration system intersection.The 50 Hz vibration mainly concentrated in the drum gear meshing frequency and the first-order natural frequency of the vertical vibration system intersection.The F3 rolling mill main vibration mode is horizontal and vertical vibration; vibration main frequency is 50 Hz or so and has approximate linear relationship with the rolling speed.The frequency is consistent with drum gear shaft gear meshing frequency, first-order natural frequency of vertical vibration system, and the-third order natural frequency of horizontal vibration, which indicates that the drum gear meshing impact excites the stand vertical vibration system first-order and the horizontal vibration third-order resonance.It also can be seen from the spot photos (Figure 14) that the spindle drum gear removed from mill has wear and tear; both sides of the drum gear tooth and top all produced excessive wear phenomenon.So it will be beneficial to eliminate vibration mill if drum tooth meshing clearance is eliminated.

Roller Horizontal Friction-Impact Vibration Characteristic Analysis
Combined with structural characteristics of the roll system, roll horizontal vibration dynamic model is set up.According to the roll loading characteristics, when the extraneous force is more than offset drift force   and has a gap between roll bearing seat and the mill house, the work roller bearing seat may jump from the mill house, and there is bearing impact on mill house conditions; the system stability is damaged at this time.Considering the mill housing after elasticity, we make it as an elastic collision process in this paper.Based on this, the roller horizontal vibration system "jump" model was set up in Figure 15, and the diagrams (a), (b), and (c) show work roll bearing pressing on the right side of the housing pillar and bearing seat in the middle and bearing pressing on the left side of the housing pillar in sequence.In the figure,   is the interface friction damping force.So its motion differential equation is as follows: where  is the weight of work roller,  is extraneous force, Ω is the roll rotation frequency, and (, ẋ ) expression is as follows: where  1 is the linear stiffness coefficient between work roller and mill house left side,  2 is linear stiffness coefficient of mill house pillar,  is the linear damping coefficient,  is the nonlinear damping coefficient, and Δ max is the gap between bearing seat and mill house considering the roller crossover and bearing seat tilt.Then, the numerical simulation system The oversize clearance between mill house and bearing seat can cause roller crossover and roll system transverse vibration; in order to verify the clearance influence on the roll system dynamic characteristics, it is necessary to carry out simulation and analysis.Due to large weight of drum gear spindle, when it has slight eccentric or installation error, which would cause arc tooth meshing impact, the spindle support with arc tooth will play an important role on vibration suppression.But limited by various factors, it is difficult to be implemented at the scene, but these achievements have certain inspiration to rolling mill manufacturers.In order to eliminate the gap and eliminate the phenomenon of roll bearing seat collision on mill house, free expansion hydraulic plate can be used; in order to reduce or eliminate the drum gear shaft imbalance force, the drum gear spindle floating support can be used.In order to test and verify the effect of these measures, the numerical simulation was carried out, respectively, in this paper.In Figure 16, the simulation curve is the roller displacement response diagram, phase diagram, and displacement power spectrum diagram, respectively.
According to the roll horizontal vibration mechanics model (Figure 15), with the gap between roll bearing seat and mill house eliminated, roll horizontal vibration differential equations can be expressed as follows: The values of the parameters are the same as the mentioned above.When adding drum spindle floating support, the spindle unbalance excitation disappears; roller horizontal vibration differential equation is expressed as follows: When the two measures of eliminating the gap between the roller bearing seat and mill house and adding the drum gear shaft floating support were adopted at the same time, roll horizontal vibration differential equations can be expressed as follows: The simulation results are shown in Figure 17 of eliminating the gap between the roller bearing seat and mill house and adding the drum gear shaft floating support, adopted at the same time.From the figure, we can see that after these two measures are taken at the same time, the vibration amplitude reduced to one-tenth of the original, and by the phase diagram and the spectrum diagram, system property has been improved significantly and the roller "jump vibration" phenomenon disappeared.
We adopted vibration suppression measures as adding copper gaskets within the bearing liner to eliminate or reduce the bearing seat and mill house clearance at site.The roller horizontal vibration results after eliminating the clearance are shown in Figure 18, and it can be seen after eliminating the clearance that chopped or touch vibration phenomenon disappears; vibration amplitude is declined greatly and the energy spread.So the measure of gap eliminating can reduce mill vibration (horizontal).

Conclusions
In order to deeply study the rolling mill vibration source, the roll system structure stability was analyzed and calculated firstly and it had been found that when the F3 rolling mill work roller diameter is small, the offset basically conforms to the requirements of the static stability.Then, the field measurement of the clearance between mill stand and bearing seat has been carried out, and it was found that F3 clearance is much larger than the original design (transition fit) requirement; namely, the gap provides necessary conditions for roller horizontal vibration.The drum gear coupling effect on the roll system stability is analyzed, and it was found that the coupling transmission torque cannot meet self-centering requirement, roller driving system always has unbalanced force or centrifugal force, and this is verified by the field vibration test.At the same time, the axial force by rollers cross caused by roller system structure defect has great destruction on the system stability and it cannot be allowed to be ignored.Finally, the roll horizontal friction chatter vibration mechanics model was established and the simulation analysis was carried out; it showed that when eliminating the gap between the roller bearing seat and mill house and adding the drum gear shaft floating support at the same time, roll vibration amplitude was reduced to one over ten of the original, and by the phase diagram and the spectrum diagram, system dynamics have been improved thoroughly and the roller "jump vibration" phenomenon disappeared.At the same time, adding copper gaskets to eliminate or reduce the clearance between the bearing seat and mill house for vibration suppression at the scene, the results showed that after the clearance is eliminated, chopped or touch vibration phenomenon disappeared and vibration amplitude value has fallen obviously, and the energy has spread; that is, it is helpful to suppress roller horizontal vibration.

Figure 18 :
Figure 18: Roller horizontal vibration measured results after gap elimination.
is the force on work roller by backup roller,  is the rolling force,   is the work roll bearing reaction force by mill house,  is the friction arm distance (0.2 mm) between work roller and backup roller,  1 and  0 are the front and back tension,  1 and  2 are the work roll and backup roll radius,  1 and ), where Δℎ is reduction and   is the roll flattening radius.The force of backup roller acting on work roller is   =  ⋅ cos / cos( + ). is the arm of force of   acting on work roller and  = cos  +  1 sin .When  1 >  0 ,   =   sin( + ) +  sin .

Table 1 :
Mill house and bearing seat clearance design value and measurements (unit: mm).