laminacion_frio.pdf

7/29/2019 LAMINACION_FRIO.pdf

1/10

j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 9 ( 2 0 0 9 ) 24362445

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / j m a t p r o t e c

Numerical and experimental analysis of the dynamic effects

in compact cluster mills for cold rolling

Eugenio Brusa a,, Luca Lemma b

a Universit degli Studi di Udine, DIEGM, Via delle Scienze 208, 33100 Udine, Italyb SKF, Reliability Systems, Service division, Via Pinerolo, 54 10020 Airasca (Turin), Italy

a r t i c l e i n f o

Article history:

Received 19 April 2007

Received in revised form

21 May 2008

Accepted 23 May 2008

Keywords:

Cluster rolling mill

Cold rolling

Nonlinear dynamics

Multi-body dynamics

Vibration monitoring

a b s t r a c t

Prevention of dynamic instability caused by chatter phenomenon may be difficult in case of

the so-called cluster mill because of the number of back-up rolls used to avoid bending of

thework roll. A numericalsimulation of thedynamics of thecluster mill may help to prevent

the strip defects and to predict the life of bearings and rolls. Analytical models proposed

in the literature for the two- and four-high mills are unfortunately inapplicable. This paper

investigates whether the multi-body dynamics approach can be effective in analysing the

so-called Z-Mill. Service monitoring detected in a real case some clear and dark bands

on the strip surface, which disappeared after a calibration of rolls position. A numerical

model was built by assuming that the mill stand and the strip could be analysed separately.

Validation allowed concluding that modelling is effective, if relevant parameters are tuned

on the experimental evidences. In particular damping and friction coefficientslook the most

critical to predict the actual value of rolling force. Mechanical nonlinearities introduced by

the contact among rolls and by bearings increase the computational effort, but their role is

somewhere overestimated in the literature. In practice, the multi-body dynamics approach

still shows some limits, but they should be overcome, if a deeper experimental validation

will be performed on a dedicated test rig, suitably designed to be largely instrumented.

2008 Elsevier B.V. All rights reserved.

1. Introduction

Compact layouts are currently preferred to design the cold

rolling mills used for producing very thin metal strips. Some

examples of these systems are shown by Roberts (1978) in

his textbook about the cold rolling technology. A cluster ofback-up rolls is applied to assure an effective position con-

trol on the work roll, being usually very slender and prone

to bend (Siemens, 2008). To decrease the complexity of this

assembly, in some configurations like in the Z-mill few back-

up rolls areintroduced(Fig. 1) (Sendizimir, 2008). To effectively

design these plants some relevant phenomena occurring in

flat rolling process and affecting the dynamic behaviour of

Corresponding author. Tel.: +39 0432558299; fax: +39 0432558251.E-mail address: [email protected] (E. Brusa).

these mills have to be considered, as it was discussed by

Ginzburg and Ballas (2000). In addition, some peculiarities of

the thin foils are important from the point of view of the

rolling force computation, as evidenced by Zhang (1995). All

these aspects will be included in thefollowing sections,where

modelling activity will be described.Industrial experience demonstrates that compact layouts

may suffer vibration effects. If the mill installation is inac-

curate some defects appear on the strip surface as clear and

dark bands (Fig. 2). In case of steel production the quality is

compromised, because these marks cause the rejection of the

strip even when its thickness is uniform and the rupture is

averted. Uniform thickness, regular flatness and homogene-

0924-0136/$ see front matter 2008 Elsevier B.V. All rights reserved.

doi:10.1016/j.jmatprotec.2008.05.044
mailto:[email protected]://localhost/var/www/apps/conversion/tmp/scratch_1/dx.doi.org/10.1016/j.jmatprotec.2008.05.044http://localhost/var/www/apps/conversion/tmp/scratch_1/dx.doi.org/10.1016/j.jmatprotec.2008.05.044mailto:[email protected]


2/10

j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 9 ( 2 0 0 9 ) 24362445 2437

Fig. 1 Description of the compact Z-mill.

ity are highly required, if the thin strip is used to manufacture

coveringand protecting surfaces, as is in applicationslike cars,washing machines and masks for lithography. Investigating

the dynamic behaviour of rolls and strip is therefore crucial to

predict the performance of the cold rolling process.

Work roll vibration depends on the contact with strip and

back-up roll. In the cluster mill layout vibration is transmitted

from the work roll to the saddles through the back-up rolls.

It affects the strip quality (Nizio and Swiatoniowski, 2005).

Sometimes dynamic instability and chatter phenomenon

occur. Chatter typologies have been classified by Lin et al.

(2003), while a comprehensive review of models was proposed

by Yun et al. (1998). Chatter is self-excited and arises in rolling

operation as a consequence of the interaction between the

structural dynamics of the mill stand and the dynamics ofthe rolling operation itself. The dynamic response of the cold

rolling mill is affected by the irregularities occurring in the

strip tension and speed, as well as Drzymala and Bar demon-

strated by finding a correlation between the strip and the roll

vibrations (Drzymala et al., 2003), which was then validated in

Fig. 2 Clear and dark bands detected on the surface of the

strip.

Bar and Swiatoniowski (2004). Several sources of vibration are

superimposed and affect the rolling process (Geropp, 2003),

thus requiring a continuous monitoring activity on the mill

plant to prevent faults (Mackel, 2003). Lubrication and friction

are even important since they define the amplitude and the

direction of the rolling force applied by the strip on the work

roll and of the contact forces exerted among the back-up rolls.

Some typical values of friction coefficients in cold rolling wereidentified by Jeswiet (1998).

All the above mentioned aspects make hard an effective

modelling of the cold rolling process, according to the indus-

trial daily experience. For the cluster mill layout back-up rolls

make this task more difficult. Several authors dealt with the

roll bite dynamics. Classical methods were summarized by

Roberts (1978) and more deeply by Ginzburg and Ballas (2000)

in their well-known textbooks. Some analytical methods were

proposed to predict the dynamic component of the rolling

force and its effects on the frequency response of the mill.

A group of scientists developed over the time a preliminary

model of coupling between strip and work roll. A first contri-

bution was proposed by Wu and Duan (2002), then developedin three papers by Bar and Swiatoniowski, first in Drzymala

et al. (2003), then in Bar and Swiatoniowski (2004), where

some comments on experimental validation were added and

finally refined and completed in Nizio and Swiatoniowski

(2005). A complete dynamic analysis of the cluster mill layout

was never performed. Current industrial demand for larger

rolling pressures and faster cold rolling process motivates

the implementation of numerical models aimed to predict

the dynamic actions applied to work and back-up rolls, sad-

dles and bearings (Montmitonnet, 2006). Recent tests of active

vibration control on the cluster mills demonstrated that an

accurate modelling of the contact among rolls and bearings is

required to set properly the control system parameters and toavoid the effects of the sensors non-collocation. Mechatronic

approaches were even proposedby Bates et al.(1997), whopro-

posed a robustcontrol strategy forrolling mills, andby Knospe

(2002), who applied the active magnetic bearings technology

to provide a continuous and adaptive control of vibration.

2. Research goals

Modelling the dynamic behaviour of the Z-mill is the goal of

this paper. It was selected as test case because of its compact-

ness and the availability of some preliminary experimental

results. This research activity is aimed to assess an approachsuitable to model the cold rolling cluster mill. Performance of

themonitored plant will be considered as a secondary issue of

the analysis. Three main targets were defined by the industrial

partner, i.e. SKF Industries. The first was investigating fea-

sibility, limits and effectiveness of the multi-body dynamics

approachas it wasformulatedby Shabana (1999). In particular,

the ADAMS commercial code was selected, although some

aspects were modelled by means of Matlab subroutines. Pre-

dicting the dynamic response of work and back-up rolls and

the forces applied to saddles and bearings was a second issue.

A sensitivity analysis was performed to identify the design

parameters of the cluster mill to be accurately set or experi-

mentally measured. The architecture of the Z-mill is briefly


3/10

2438 j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 9 ( 2 0 0 9 ) 24362445

presented. It includes two back-up rolls (intermediate and

external) along the vertical axis and a couple of control rolls

(Fig. 1). This configuration can be referred to as cluster mill. In

fact, it is more compact than Sendzimir mill, where a cluster

of 20 rolls is usually assembled (Sendizimir, 2008). The exper-

imental evidences of a preliminary testing performed on the

Z-mill installation are presented. Tests detected a sequence of

clear and dark bands on the strip surface in correspondenceof the highest speed. Control rolls position appeared some-

how inaccurately calibrated and the surface quality of some

rolls was inadequate. Clear and dark bands were parallel to

the work roll axis and their surface roughness was different.

Thickness and hardness of the strip were constant all over the

surface (Fig. 2). These defects disappeared as soon as the con-

trol rolls position was calibrated. To decrease both the time

and the cost of the installation procedure a numerical model

of the Z-mill was developed and a correlation between the

defects and the dynamic behaviour of the system was found.

A coupled analysis of the dynamic behaviour of strip and rolls

is expensive in terms of the computational time. Therefore

it was investigated whether strip and rolls could be analysedseparately. Vibration of work and back-up rolls was computed

by means of the ADAMS code. Rolling pressure and force

were computed according to some available analytical mod-

els. Some classical approaches were described in Freshwater

(1996), while an original approach, which uncouples the strip

from the work roll, was proposed by Bar and Swiatoniowski

(2004). These models include the static and dynamic com-

ponents of the rolling force. Elastic contact among rolls was

introduced according to Lin et al. (2003) and conformingly

to the theory of contact mechanics applied to the numeri-

cal methods by Wriggers (2006). The mechanical behaviour of

rollers was studied and modelled according to Harris (2000)

and to some preliminary experiences performed by SKF anddocumented in SKF (A). A numerical investigation was exten-

sively performed. Experimental testing was used to validate

each step of theproposed model andto tune the model param-

eters.

3. Experimental set-up

The cold rolling process is operated by Z-mill at the room tem-

perature by cylindrical and smooth work rolls. They usually

have a quite small diameter, since it allows bearing higher

rolling pressures. Slender roll is prone to bend. To avoid this

bending a cluster of back-up rolls supports the work roll.Stiffness is assured by the intermediate roll, being simul-

taneously in contact with the work roll and the external

back-up roll. Theexternal roll is connectedto the motor, which

applies the required torque. To control the lateral motion

of the work roll a couple of so-called control rolls regulate

its position. Each control roll is in contact with two small

rollers. These have the inner ring fixed and the outer ring

rotating. Rollers are connected to the saddles. Position of

rollers may be either fixed, after a calibration of the Z-mill, or

actively controlled by a hydraulic actuator. The Z-mill selected

as test case is equipped with a monitoring system which

measured the strip thickness before and after the roll bite

(Fig. 3).

Fig. 3 Vibration monitoring system and Non Destructive

Testing equipment.

A closed loop control regulates the tension of the strip.

This is a reversible cluster mill. Work rolls rotate alter-nately clockwise and counter-clockwise. Strip thickness is

gradually reduced by the cold rolling process after five

steps. To allow both the frontal and backward motions

strip is stored by two coilers. High strength steels are usu-

ally rolled to produce very thin sheets. The standard AISI

340 Steel was rolled in the experimental tests. Strip had

a constant width of 505 mm. Thickness was reduced from

0.26mm to 0.095 mm. Roll diameters were 69.6 mm (work),

139.95 mm (intermediate), 450.31 mm (external) and 51 mm

(control roll). Table 1 summarizes the relevantinformationcol-

lected during the experimental tests and some geometrical

parameters.

The monitoring system consists of several components.X-rays measure the strip thickness at the in-gauge and out-

gauge respectively (Fig. 3). A torque control is applied to the

coilers and allows monitoring speed and tension of the rolled

strip. Rolling force is measured by a dynamometer applied on

the work roll. Vibration monitoring was performed by means

of few accelerometers mounted at the saddles behind rollers

and on the mill stand (Fig. 3). Vibration signals were detected

by SKF Microlog CMXA50 analyser and data collector, which

was operated in connection with SKF Machine Analyst soft-

ware. A dynamic analyser acquired the acceleration signals

along three axes. The SKF CMSS2200-M8 accelerometers were

used (100mV/g, 80 g peak, 15000 Hz, linearity 1%). Because of

theinaccessibility of theother parts of theZ-mill no additionalsensors were mounted. This experimental set-up allowed

drawing somepreliminary conclusions, but it demonstratedto

be unsuitable for a complete experimental validation. A spe-

cial test rig able to simulate the cold rolling process and more

accessible for measurement equipments has to be designed

and built.

4. Experimental evidences

Strip faults caused by cold rolling mill vibration are basically

chattermarks, flatness faults and flat rupture. They are exten-

sively classified in Ginzburg and Ballas (2000) and shown in


4/10


Table 1 Cold rolling parameters monitored on the Z-mill test case

Step Thickness (mm) Tension (kN) Speed Force (kN) (minimummaximum) Angular speeds (rad/s)

In Out In Out M (min) m (s) WR CNT INT EXT

1 0.260 0.180 38 35 178 2.966 11501200 85.23 116.31 42.39 13.17

2 0.180 0.145 31 34 235 3.196 19001950 112.56 153.61 55.98 17.40

3 0.145 0.120 31 28 248 4.133 15001550 118.79 162.12 59.08 18.36

4 0.120 0.105 23 25 265 4.416 14501500 126.93 173.22 63.12 19.62

5 0.105 0.095 22 20 500 8.333 13001350 239.45 326.78 119.09 37.01

WR = work roll; CNT= control roll; INT = intermediate roll; EXT = external roll.

Mackel (2003). Experiments neither show flatness nor rupture

problems. Few chattermarks appeared on the stripsurface, but

without modifying its thickness. According to the literature

vibration monitoring was aimed to detect the effects of the

so-called 3rd octave, 5th octave and torsional chatters as

they were defined by the literature (Yun et al., 1998). No mark

of torsional chatter was found. This kind of chatter usually

occurs at the lowest frequencies and depends on the irregu-lar torque applied by the motors. Analysis was focused on the

3rd and 5th octave chatters. These are due to some technolog-

ical and dynamic problems. Material pre-damage, rotundity

and balance errors upon rolls, roll bearings errors and drive

irregularities cause a speed-dependent excitation. A speed-

independent dynamic excitation is due to the self-excited or

roll stand vibration, front tension fluctuation, lack of material

homogeneity, stick-slip or to the control system. No mate-

rial pre-damage and problems with homogeneity were found.

Surface irregularities, rotundity or severe balance errors on

rolls were absent. Bearings did not show failures or unex-

pected clearances. A continuous monitoring of the electric

current flowing in the motor circuits is required to detectmotor dynamics irregularities. It was unavailable, although

unbalance magnetic pull of the electric motor is a typical

source of dynamic instability as it was found by Amati and

Brusa (2001).

5. Model set-up

5.1. Approach and layout

To perform a complete dynamic analysis of the cluster mill

a model of the whole system is required. The most com-

plete analysis should be performed by modelling strip androlls within the cage of the cold rolling mill. Both the elastic

mechanical behaviour and the rigid body motions should be

considered (Shabana, 1999). This approach requires to resort

to a 3D model of all the structural components and needs for a

virtual prototyping activity based on the FEM and multi-body

dynamics methods. Computational effort is huge because of

the time required to solve the problem and the amount of

numerical results to be stored for a significant number of

rotations of the work roll. Therefore this approach appears

unpractical as it was remarked by Zhang (1995). When only

the vibration of the rolls composing the cluster has to be anal-

ysed the multi-body dynamics approach can be applied. This

paper investigates the performance of the proposed method

in the context of the cold rolling mill design. Symmetry prop-

erties allowed studying the upper part of the Z-mill (Fig. 3).

Work and back-up rolls and bearings were modelled. Inter-

action between the work roll and the strip was given by the

rolling force distribution. It was computed first according to

Bland and Ford (1948), then to the refined model of Yun et

al. (1998) and finally implemented by following the approach

developed by Bar and Swiatoniowski (2004). A nonlinear rela-tion between the contact force and the displacement was

assumed for bearings according to Harris (2000). The role of

this nonlinearity on the system dynamic response was dis-

cussed. Numerical data of bearings were provided by SKF

(2007).

5.2. Static component of the rolling force

Rolling force is composed of two components, static and

dynamic respectively. To compute the static contribution the

analytical models proposed by Bland and Ford (1948) and by

Yun et al. (1998) can be implemented. The first above men-

tioned approach looks the simplest and did not include the

vertical speed of the work roll, while the formulation of Yun,

Hu and Ehmann did take into account a variable position of

the work roll and its instantaneous speed. Bland and Fords

model was implemented. Few parameters are enclosed and

described in Fig. 4.

Thickness values at in-gauge and out-gauge are respec-

tively hin and hout. Work roll radius is R, friction coefficient

between strip and roll is named , rolling pressure is p, rolling

angle is called andthe angular position withinthe rolling arc

Fig. 4 Relevant parameters of the rolling force model.


5/10


is referred to as . These parameters are related as follows:

P = wR

N0

poutd+

N

pind

pin =hk

hin 1

inkin

exp[(Hi H)]

pout = hkhout

1 out

kout

exp(H)

Hin =

2

R

hout

arctan

R

hout

;

H =

2

R

hout

arctan

R

hout

. (1)

Rolling force P is computed as a result of the numerical

integration of the rolling pressure applied at in-gauge (pin)

and out-gauge (pout) arcs. Roll width is w. Force P depends

on several geometric parameters, related to the strip reduc-

tion (hout/hin), but much more on the friction coefficient

and the compression resistance of the strip material, k. This

model assumes that Coulomb friction forces, Von Mises fail-

ure criterion and plane strain hypotheses are all applicable

(Timoshenko, 1983). According to Von Mises criterion k is pro-

portional to the yielding stress measured by a unidirectional

test. Values ofk can be distinguished at in-gauge (kin) and out-

gauge (kout) in presence of stress (Ginzburg and Ballas, 2000).

Pressure is an exponential function of the local thickness h

and of the angle called . It is computed as a function of the

back and front stresses in and out respectively. In above Eq.

(1) pin describes the local value of pressure for each section of

the rolling arc from the entry to the neutral section, while poutis valid from the neutral section to the exit. Neutral section

can be identified as follows:

Hn =1

2

Hin

1

ln

hin

hout

1 (out/kout)

1 (in/kin)

;

n =

hout

Rtan

1

2

hout

RHN

. (2)

In Eq. (2) Hn and n are expressed for the neutral section,

where in is equal to out. Resultant P is then convention-

ally applied at the section corresponding to =0.4, like Fig. 4

shows. Rolling force of Eq. (1) depends on a unique value of

friction coefficient . This is an approximation, since it wasdemonstrated that at the in-gauge and out-gauge these coef-

ficients are different. Stick-slip phenomenon often occurs, but

in this case it is neglected. Hu and Ehmann take into account

two friction factors, min and mout respectively and the actual

length of the rolling arc, corresponding to a given position of

the work roll (Yun et al., 1998). Friction factor does include the

possibility of sticking, since its value corresponds to 0 in case

of no friction andto 1 in stickingcondition,respectively. These

authors implemented the following formulation:

P = (2y + in)(xout xin)

+

y

R [mout(x2out x

2n)+min(x

2in x

2n)]+ yRmout ln

houthn

+yRmin ln

hinhn

+ 2yxout ln

hin

hout

+2y(min +mout)xout

R

houtarctan

xnRhout

+4yRhout 2ymin

xoutRhout

arctan xinRhout

4y

Rhout + 2ymoutxout

R

hout

arctan

xoutRhout

.

(3)

In Eq. (3) y corresponds to the yielding tangential stress.

This model actually allows describing even the dynamic inter-

action between the strip and the work roll, if the vertical

position described by h varies over the time as a consequence

of the fluctuation of the tensile stresses in and out (Yun et

al., 1998). Nevertheless, in this case the dynamic behaviour ofthe strip was introduced according to Bar and Swiatoniowski

(2004) to avoid including the strip as a structural component

within the numerical model.

Identifying the actual values of friction factors min and moutis very hard. Some empiric approaches have been proposed

and described by Ginzburg and Ballas (2000) and tested by

SKF. Friction coefficient depends on the roll surface rough-

ness, temperature, steel composition, pressure, strip speed

and lubrication condition. For a known tension of the strip,

friction coefficient in Eq. (1) can be roughly predicted by SKF

(A):

= a1[a2 + a3 ln(v)] (4)

where v isthe strip speed (ms1), and coefficients a are related

to the surface condition (a1), lubrication method (a2) and

material (a3). In the test case these coefficients were evalu-

ated for mirror surface, oil lubrication and AISI 340 as a1 = 1,

a2 =0.066723 and a3 =0.00662. A first comparison with the

experimental results measured on the Z-mill was performed,

for each step of the strip reduction. As Table 1 shows values

of thickness, speed, stress and measured rolling force were

known. A good agreement was found only in case of the first

rolling step. Speed was about 178m/min, i.e. it washigher than

50 m/min suggested by Bland and Ford for implementing their

model. Numerical computation and measures disagreed forthe fourth and fifth rolling steps. Actually, the highest value

of speed was 500 m/min. Larke (1963) demonstrated that it

is possible to compensate for the inefficiency of the rolling

force model, by modifying the friction coefficient and increas-

ing the numerical value of parameter k. Friction coefficient

depends on the lubrication condition and varies along the arc

of contact between the strip and the work roll. Resistance to

a plane deformation is a function of the percentage reduction

of thickness. According to Larke (1963) to improve the accu-

racy of the above mentioned models in predicting the static

component of rolling force an equivalent friction coefficient

has to be extracted from experiments at each rolling step. For

the faster rolling speeds k has to be increased. In practice,


6/10


only an experimental tuning of k and for each rolling step

allows introducingcoherent valuesof therolling force into the

numerical model.

5.3. Dynamic component of the rolling force

In principle rolling force should be constant for a given speed

of the steel strip. Actually a dynamic component superim-poses to the static value previously computed because of the

mill stand and strip vibration. To predict this contribution

the model proposed by Bar and Swiatoniowski (2004) was

implemented.It avoids modellingthe strip dynamicbehaviour

and assumes that a certain correlation between the vibration

modes of the strip and of the mill stand can be found.

If the total action is P:

P = Ps + Pd (5)

static contribution Ps is computed according to Eq. (1), while

dynamic component Pd can be found as follows:

Pd = (K m)w

R(h+ 2y)

R(h) (6)

where K is the yielding stress for unidirectional compressive

test, m is the average value of stress applied on the strip, w is

strip width, R roll radius, h thickness reduction andy vertical

speed of the work roll. This speed is assumed to be periodic

and can be evaluated as:

y =A sin(kt + 0)

A =(n)2ka

20hout

16LVout

(7)

where n is the number of the vibration mode of the strip con-sidered, k is the critical angular speed corresponding to a

frequency of resonance of the mill stand, t is time, while 0 is

the phase angle, L is thedistancebetweentwo mills in tandem

configuration, Vout is the strip speed at out-gauge. Amplitude

a0 is computed in Bar and Swiatoniowski (2004) as function of

some parameters of the dynamic excitation of the strip vibra-

tion. This coefficient is easily found on a diagram for a given

value of the frequency ratio , being defined as:

=k

2n(8)

where k is the frequency of resonance of the mill stand and

n the resonance of the strip. For each frequency of reso-nance of the mill stand, being indicated by counter k, the

excited frequency of resonance of the strip is n = k 1. In this

case, the dynamic component computed according to Bar and

Swiatoniowski (2004) appeared compatible with the dynamic

rolling force monitored during vibration monitoring.

5.4. Rolls modelling

Rolls have been modelled in the Adams code, as rigid bodies,

although they are flexible structures (Fig. 5).

A preliminary numerical investigation was performed by

means of FEM rotordynamic code DYNROT. It demonstrated

that for each value of strip speed angular speeds of rolls

Fig. 5 Multi-body dynamics model of the Z-mill.

are sub-critical with respect to the critical speeds of flexu-

ral behaviour (Genta, 2005). Rolls were free for rotating andtranslating,within a range of alloweddisplacements. Supports

were introduced as bearing forces, which were described by

means of look-up table functions. These were built accord-

ing to the nonlinear dependence of force on displacement as

it was provided by SKF laboratories, as result of the exper-

imental characterization (SKF, A). Gravitational effect was

applied along the vertical direction of Fig. 5. The ADAMS

code describesthe contact among therolls bymeansof impact

forces, suitable to transmit momentum between two bodies.

These forces are active only when the distance between two

rolls is less than the sum of radii of the rolls. Impact force is

found as follows:

f = max(fu fV); fu = kC(z0 z)p

fV = cCz (z0, j0) z z0 1 p 1 (9)

In above Eq. (9) z is the relative position of the two bodies,

while z0 is a reference position,being usually the roll diameter.

Symbol indicates a step function whose height is propor-

tional to cC and width is given by parameterj0. In this casethis

parameter was suggested by the code itself for each material.

Nevertheless, it can be tuned according to some indentation

tests (SKF, A). Damping coefficient of the roll material is cC,

while contact stiffness is described by kC. It was computed

according to Yun et al. (1998). In case of harmonic dynamic


7/10


Table 2 Rolling force analysis

Step Measured rolling force (kN) Computed static component (kN) Computed dynamic component (kN)

1 11501200 1148 0.40

2 19001950 1878 0.70

3 15001550 1507 0.70

4 14501500 1459 1.00

5 13001350 1302 0.85

response, equivalent viscous damping ceq may be computed

by introducing the material loss factor and the frequency

at which hysteresis occurs (Genta, 2005):

ceq =kC

=

kCn

. (10)

Since frequency is unknown it is usually approximated by

the frequency of resonance of the mechanical system, being

n. Values ofkC and cC are crucial to find out a good agreement

between numerical and experimental results. In the ADAMS

code power transmission among rolls has to be expressively

activated. The power of therolling process foreachrolling step

was computed as:

Pw = kwwvhout ln

hinhout

(11)

kw being the local stiffness for the current temperature, pres-

sure and speed conditions. Torques applied to work and

back-up rolls were then computed, by knowing their angular

velocities and diameters. No slip effect was included.

5.5. Bearings modelling

A nonlinear relation between the radial force and the related

displacement was measured on the bearings installed on this

mill by SKF. In practice, bearing compliance and stiffness C

are computed according to Harris (2000) as:

= kFp C =F

=

F1p

k C =

dF

d=

F1p

kp=

1p/p

pkp(12)

F being the applied load, k a constant, p equal to 2/3 (ball

bearing) or 9/10 (roller bearing). This relation is nonlinear,

but a reference value for a given equilibrium condition can

be computed as tangent stiffness C as it was shown in Eq.

(12). This approach is used to describe radial and axial dis-

placements in ball androller bearings,with null clearanceand

to compute the stiffness coefficients. This computation was

straightforward for all the SKF bearings mounted on the mon-

itored Z-mill, but for rollers of control rolls. An estimation of

their stiffness was done as a result of a comparison with SKF

NA6902. Damping coefficient wasevaluated by SKF fromdirect

measurements on the installed bearing.

6. Numerical investigation

Severalnumerical analyseswere carried out to testthe numer-

ical model. Vertical motion was mainly considered, although

model included horizontal displacements too. Each rolling

step was analysed separately. Computation of rolling pressure

and static component of rolling force was performed accord-

ing to above Eq. (3). To fit the measured values of force it was

required tuning two parameters, i.e. friction coefficient and

strength coefficient Y. The latter was used to compute y as

Yun et al. (1998):

y =K

2= Ylog

h0

houtn

(13)

where h0 is the initial thickness of the strip and n the strain

hardening coefficient of the material (Dieter, 2000). Values of

friction coefficient which allowed fitting the measured force

disagreed with prediction of Eq. (4) at higher speeds, although

they were found compatible with the typical values identified

for the cold rolling process by Jeswiet (Jeswiet, 1998). Strength

coefficient Yhad to be increased from the nominal value cor-

respondingto the first rolling speed, step by step up to thefifth

one. Numerical results were included into the range indicated

by Larke (1963). Dynamic component of the rolling force was

then computed according to Bar and Swiatoniowski (2004).

Rolling step two was the most difficult to be predicted in

terms of both the static and the dynamic component. Stepfour showed a larger dynamic force. Comparison with exper-

iments was more difficult for the dynamic force than for the

static contribution, since the measuring equipment recorded

acceleration with a resolution of 50 kN. Table 2 summarizes

numerical and experimental results.

Dynamic force Pd did correspond to a few percent of the

total force P. Nevertheless, it was sufficient to appreciate and

monitor vibration. It is remarkable that nonlinear stiffness of

bearings described by Eq. (12) can be approximated by the

value computed about the static equilibrium condition cor-

responding to force Ps according to Friswell et al. (1995, 1996).

This approachallowedsimplifying the numerical analysis and

performing a preliminary computation of the resonance fre-quencies of the mill stand, being described in Table 3.

Table 3 Critical frequencies of the cluster of rolls

Step Vibration mode frequency (Hz)

1 621

2 647

3 648

4 667

5 2475

6 6123

7 7614

8 10160


8/10


Fig. 6 Experimental waterfall diagram measured at the

saddles.

Two frequency ranges appeared critical. One included

vibration modes from 600 Hz to 1000 Hz, while a second range

corresponded to vibration modes 6 and 7 (2.5kHz through

7kHz). This result agrees with the evidence of the waterfall

diagram experimentally built and depicted in Fig. 6. Monitor-

ing system did not allow detecting modes above 10 kHz.

Rolls vibration was investigated. The most significant

results came from step first and five, which will be here dis-

cussed with reference to Table 4. Work roll is indicated as 1,

intermediate roll as 2 and control roll as 4 (Fig. 1). Dynamic

rolling action is fairly smaller than the static component. Ver-

tical displacement of work roll is fairly larger than the strip

thickness, but the magnitude is compatible with experiments.Since step five falls within the frequency range of the 5th

octave chatter, rolls speed and acceleration are higher than

those measured at the first rolling step. Dynamic force is so

small that numerical values are very low, for all the rolls

described in Table 4.

Last two rows include theforce applied bythe control roll to

the external roller and the acceleration imposed on the saddle.

Values are compatible with the experimental curve depicted

Fig. 7 FFT of the fifth rolling step measured at the saddles.

in Fig. 7. It describes theacceleration measured on thesaddles

at rolling step five.Actually, values found for rolling step one at 600 Hz are

fairly lower than the experimental results of Fig. 6. In prac-

tice, numerical model appears unable to detect peaks of the

acceleration monitored at steps one and five respectively. This

result motivated a further investigation on the mill plant.

Accelerometer mounted on the mill stand far from the clus-

ter of rolls (Fig. 3) measured a strong impulsive force in the

range 450600 Hz, thus showing a direct interference between

the mill cage structure and the cluster of the rolls (Fig. 8).

This result demonstrated that the proposed model was unable

to detect the mechanical coupling between the structural

dynamics of the cage and of the cluster, since the first was

not modelled.Position control system which regulates the action of lat-

eral control rolls on the work roll is calibrated for each rolling

step on the strip thickness. A direct measure on the sad-

dle support demonstrated that acceleration peaks found in

the range 46kHz were higher. In fact, even in this case, an

interference between the dynamic response of the position-

ing control actingon the saddles andof thecluster of rolls was

found. Chattermarks found on the strip surface were depen-

Table 4 Numerical results for rolling steps one and five

Step one Step five

Frequency 0.61 kHz, =0.08


Frequency 2.5 kHz, = 0


Frequency 6 kHz, = 0

Pd (kN) 0.40 0.60 0.88 0.88 0.33

P (kN) 1150 1150 1302 1302 1.300

y1 (mm) 0.120 0.127 0.145 0.143 0.143

y1 (m s1) 0.005 0.005 0.05 0.01 0.05

y1 (m s2) 200 400 2000 400 1800

y2 (m s1) 0.004 0.005 0.008 0.002 0.008

y2 (m s2) 50 10 400 50 350

y4 (m s1) 0.005 0.005 0.008 0.002 0.008

y4 (m s2) 40 20 400 50 300

Acceleration (g) 0.00015 0.00015 0.00033 0.00020 0.00024

F (kN) 4.5 4.5 10 6.5 9.5


9/10


Fig. 8 FFT of the first rolling step measured on the mill

cage.

dent on this phenomenon. Distance between two consecutive

peaks in Fig. 6 was about 49 Hz. This value corresponds to a

rotation of the control roll at the strip speed of 500 m/min and

is compatible with the distance between two marks equal to

2mm.

In practice, the numerical model implemented allowed

identifying the source vibration of the highest peaks and of

chattermarks detected during the preliminary tests. It some-

how demonstrated its limits in predicting the influence of the

fixed parts of the cold rolling mill on the dynamic behaviour.

An additional doubt remains unsolved. It concerns the role

of the strip stiffness and damping. In fact, model proposedby Bar and Swiatoniowski (2004) appears suitable to describe

the self-excitation of the system caused by the strip vibra-

tion resonance. Nevertheless, since the stripwas not explicitly

modelled its structural properties of stiffness and damping do

not appear in the proposed approach. This aspect looks lead-

ing to a lack of information in the numerical analysis of the

self-excitation out of the resonance frequencies of the mill

stand.

A final remark concerns the impact forces used to model

the contact among the rolls. Table 4 shows that contact damp-

ingcomputed by means of thematerialloss factor can havea

significant role in rolls vibration. Step five shows lower values

of speed and acceleration, if a typical value loss factor for thesteel is applied to compute the structural damping.

7. Conclusions

This preliminary study investigated the effectiveness of com-

mercial multi-body dynamics codes in predicting the dynamic

behaviour of cluster cold rolling mill. Compact Z-mill was

selected as test case for an experimental validation of numer-

ical models. Analytical formulations were applied to compute

first the rolling force, being the dynamic excitation. Models

revealed that an accurate prediction of friction is crucial to fit

the measured values of rolling force. Interaction among rolls

was then dealt with. Elastic contact wasmodelled by means of

the impact force exerted among the rolls. Dynamic behaviour

of the cluster mill appears affected more by the structural

damping of the rolls material than by the nonlinearity of the

contact itself. Even in the case of the bearings, where relation

between force and displacement is nonlinear, it was observed

that if the dynamic component of the rolling force is small,

bearing stiffness can be approximated with the value com-

puted for the static equilibrium condition. This aspect maylead to perform a simple linear dynamic analysis. Numeri-

cal results computed by ADAMS were compatible with the

experimental evidences. Nevertheless, some measured reso-

nances caused by the mill cage vibration and by the control

rolls positioning system were unpredictable. In fact, the vibra-

tion monitoring system equipping the Z-mill demonstrated to

be unsuitable to perform a complete experimental investiga-

tion. More motion sensors have to be applied to the rolls of

the cluster and a dedicated test rig has to be designed for this

purpose.

Acknowledgements

Authors thank SKF Industries for supporting this research

activity and are grateful to Dr. Denis Benasciutti (University

of Udine) and M.Sc. Eng. Vittorio Colavitti for their kind sug-

gestions and fruitful discussions about this topic.

r e f e r e n c e s

Amati, N., Brusa, E., 2001. Vibration condition monitoring ofrotors on AMB fed by induction motors. In: Proc. 2001

IEEE/ASME AIM01, Advanced Intelligent and MechatronicSystems, Como, 812 July 2001, pp. 750756.

Bar, A., Swiatoniowski, A., 2004. Interdependence between therolling speed and non-linear vibrations of the mill system. J.Mat. Proc. Tech. 155156, 21162121.

Bates, D.G., Ringwood, J.V., Holohan, A.M., 1997. Robust shapecontrol in a sendzimir cold-rolling steel mill. Control Eng.Pract. 5 (12), 16471652.

Bland, D.R., Ford, H., 1948. The calculation of roll force and torquein cold strip rolling with tensions. Proc. Inst. Mech. Eng. 159,144.

Dieter, J., 2000. Mechanical Metallurgy. Mc Graw Hill, New York.Drzymala, Z., Swiatoniowski, A., Bar, A., 2003. Non-linear

vibrations in cold rolling mills. Mcanique Industries 4,151158.

Freshwater, J., 1996. Simplified theories of flat rolling. Part I. Thecalculation of roll pressure, roll force and roll torque. Int. J.Mech. Sci. 38 (6), 633648.

Friswell, M.I., Penny, J.E.T., Garvey, S.D., 1995. Using linear modelreduction to investigate the dynamics of structures with localnon-linearities. Mech. Syst. Sign. Proc. 9 (3), 317328.

Friswell, M.I., Penny, J.E.T., Garvey, S.D., 1996. The application ofthe IRS and balanced realization methods to obtain reducedmodels of structures with local non-linearities. J. SoundVibration 196 (4), 453468.

Genta, G., 2005. Dynamics of Rotating Systems. Springer Verlag,New York.

Geropp, B., 2003. Vibration Problems in Aluminum and Steelmills, Industrial Handbook. ACIDA GmbH.

Ginzburg, V.B., Ballas, R., 2000. Flat Rolling Fundamentals.

Dekker, New York.


10/10


Harris, T.D., 2000. Rolling Bearing Analysis, fourth ed. WileyInterscience, New York.

Jeswiet, J., 1998. A comparison of friction coefficients in coldrolling. J. Mat. Proc. Tech. 8081, 239244.

Knospe, C.R., 2002. Control of Chatter Using Active MagneticBearings, Lecture Note. University of Virginia, Charlottesville(USA).

Larke, E.C., 1963. The Rolling of Strip, Sheet and Plate. Chapman

and Hall Ltd., London.Lin, Y.J., Suh, C.S., Langari, R., Noah, S.T., 2003. On the

characteristics and mechanism of rolling instability andchatter. J. Manuf. Sci. Eng., Trans. ASME 125, 778786.

Mackel, J., 2003. Condition Monitoring and DiagnosticEngineering for Rolling Mills, Industrial Handbook. ACIDA.

Montmitonnet, P., 2006. Hot and cold strip rolling processes.Comput. Methods Appl. Mech. Eng. 195, 66046625.

Nizio, J., Swiatoniowski, A., 2005. Numerical analysis of thevertical vibrations of rolling mills and their negative effect onthe sheet quality. J. Mat. Proc. Technol. 162163, 546550.

Roberts, W.L., 1978. Cold Rolling of Steel. Dekker, New York.Sendizimir, 2008. www.sendzimir.com.Shabana, A., 1999. Dynamics of Multi-body Systems. Cambridge

Press, Cambridge.Siemens, 2008. www.industry.siemens.com .SKF, 2007. General Catalogue, SKF.SKF Industries, ref. A. Computation of the rolling force in cold

rolling mills, Industrial Technical Report, RIV-SKF, Riv. Cusc.

no. 141142.Timoshenko, S., 1983. Strength of Materials, third ed. Krieger

Publishing Co.Wriggers, P., 2006. Computational Contact Mechanics. Springer

Verlag, Berlin and New York.Wu, Y.X., Duan, J.A., 2002. Frequency modulation of high speed

mill chatter. J. Mat. Proc. Tech. 129, 148151.Yun, I.S., Wilson, W.R.D., Ehmann, K.F., 1998. Review of chatter

studies in cold rolling. Int. J. Mach. Tools Manuf. 38, 14991530.Zhang, L.C., 1995. On the mechanism of cold rolling thin foils. Int.

J. Mach. Tools Manuf. 35 (3), 363372.
http://www.sendzimir.com/http://www.industry.siemens.com/http://www.industry.siemens.com/http://www.sendzimir.com/

laminacion_frio.pdf

Documents