Research Article Impact Analysis of Roller System ...downloads.hindawi.com/journals/sv/2016/5693584.pdf · Research Article Impact Analysis of Roller System Stability for Four-High

Research ArticleImpact Analysis of Roller System Stability for Four-High MillHorizontal Vibration

Xiao-bin Fan1 Yong Zang2 Yuan-kui Sun1 and Ping-an Wang1

1School of Mechanical and Power Engineering Henan Polytechnic University Jiaozuo 454003 China2Mechanical Engineering Institute University of Science and Technology Beijing Beijing 100083 China

Correspondence should be addressed to Xiao-bin Fan fanxiaobinhpueducn

Received 26 March 2016 Revised 18 June 2016 Accepted 26 June 2016

Academic Editor Chao Tao

Copyright copy 2016 Xiao-bin Fan et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In order to study the hot Compact Strip Production (CSP) four-high mill vibration characteristics and vibration suppressionmethod the roller system structure stability was analyzed and calculated at first in the paper And then the mill stand gap wasmeasured at field and its influence on roll transverse vibration was analyzed The drum gear coupling effect on the roller systemstability and the automatic balance conditions of the coupling transmission torque were studied the influence of axial force causedby the roller cross on the system stability was analyzed Finally the roller transverse friction chatter vibrationmechanics model wasestablished the simulation analysis was carried out with eliminating mill house-bearing clearance and adding floating support forcoupling respectively And the characteristics of the roller ldquojump vibrationrdquo were studied We applied copper gaskets to eliminateor reduce mill house-bearing clearance for suppressing the rolling mill vibration on the spot the test results show that the rollertransverse vibration was suppressed after eliminating clearance

1 Introduction

For the convenience of installation work roller there is clear-ance between work roller bearing seat andmill house in four-high mill which makes work roller in an unstable state whenthere is no fixed side force on it So it should have an offset119890 between work roller axis center and backup roller and itsvalue certainty principle is to make the lateral reaction forceof the frame acting onwork roller bearing always greater thanzero and the force direction is constant Modern four-highmills all set corresponding offset value [1 2] such as 2050CVC four-high rolling mill and 4300-medium plate mill ofthe German SMS four-roll rolling mill of the British DAVY1580-pair cross four-high rolling mill of Japan Mitsubishiheavy industries But these modern four-high mills withoffset all have roller parallel positioning instability and withthe increasing of rolling load and speed their disadvantageswill become more serious For these CSP mills the offsetvalue is 10sim12mm which also facilitates replacement rollerinstallation and disassembly and serves as a compensation ofbearing thermal expansion space

Although scholars have conducted a lot of research onrolling mill vibration because the problem is too compli-cated it has not been solved well A Bar andO Bar presentedthe numerical analysis of vibrations with moderate frequen-cies excited during themilling process taking into account thetransportation motion of the strip and its inertial properties[3] Bar and Swiątoniowski built the nonlinear mathematicalmodel of oscillated system the continuous group of therolling stands coupled by transferring strip [4] Panjkovic etal found that the frictional conditions in the roll gap arethe principal cause of chatter in this mill and the frictionalconditions appear to be associated with the thickness andproperties of oxide formed on rolls [5] Furumoto et aldesigned a chamber in mill stabilizing device and optimizedits size [6] Kim et al modeled a rolling mill that includesthe driving system by multibody dynamics to investigatethe cause and characteristics of the chatter vibration Thechatter frequency was 1190Hz and was caused by the rollingforceThe amplitude of chatter vibration could be reduced bycontrolling the speed of the roll and the static and dynamic

Hindawi Publishing CorporationShock and VibrationVolume 2016 Article ID 5693584 10 pageshttpdxdoiorg10115520165693584

2 Shock and Vibration

120579 120574

1205882

R2pm

QS 120593

R1

1205881 FS

T1T0e

Figure 1 Upper roll system force diagram

components of the rolling force [7] They also proposed amathematical model of a cold rolling mill including the driv-ing system and a novel combination of the direct integrationmethod and quasistatic analysis to solve the model efficientlyand found the horizontal chatter vibration had a strongeffect on the dynamic characteristics [8] Swiatoniowski andGregorczyk presented a probabilistic model of the frictionphenomena on the work-backup rolls contact surface andfount that such character of the disturbance in distributionof zones with static and kinetic friction can be regarded asone of the sources of self-excited vibrations appearing inthe system consisting of a rolling mill and a strip [9] Ameret al studied the torsional vibration reduction for rollingmillrsquos main drive system via negative velocity feedback underparametric excitation and found the resonance conditionis the first natural frequency vibration as one of the worstresonance cases [10]

Usually rolling mill vibration research does not distin-guishwhether rollingmill roll system is stable andmechanicsmodel is set up setting the mill under stabilization A singlevibration as the coupling mode with roll vertical horizontalaxial torsional cross and swing space vibration behavior isanalyzed Actually roller vibration will damage roll systemstability and cause bearing horizontal or vertical vibrationTherefore the CSP four-high mill vibration characteristics

were studied from the mill roll system stability and lateraldynamic behavior of the roll system in this paper

2 Roller System StructureStability Calculation

Work roller force analysis is shown in Figure 1 where 119876119904is

the force onwork roller by backup roller119875 is the rolling force119865119904is the work roll bearing reaction force by mill house 119898 is

the friction arm distance (02mm) between work roller andbackup roller119879

1and1198790are the front and back tension119877

1and

1198772are the work roll and backup roll radius 120588

1and 1205882are work

roll and backup roll bearing friction circle radius 120579 is theincluded angle between the work roll and backup roll defiledline and there is sin 120579 = 119890(119877

1+ 1198772) 120593 is deviation angle of

front and back tension impact on rolling force direction andthere has been sin120593 = (119879

1minus 1198790)2119875 120574 is the included angle

between roller defiled line and reaction force 119876119904 and there

has been

sin 120574 =(1205882+ 119898)

1198772

(1)

Due to the included angle between the roller and drumgear spindle axis the spindle will produce additionalmoment119872119867on the roller namely in the horizontal direction it will

make the roller produces additional horizontal force 119865119867and

119865119867= 119872119867119871The force on the side is pointing in the direction

of work roll shift direction and the force is departure fromthe direction of work roll shift on the other side and maycause two rollersrsquo crossing where 119871 is the distance betweenthe center lines of the both work rollers bearing seat and119872119867= 119872119860sdot tan120572 120572 is included angle between the spindle

and roller and 119872119860is the sum of the rolling torque 119872

119877+

support roller rotation torque 119872119861+ work roller bearing

friction consumption torque119872119882 namely

119872119860= 119875 sdot 119886 + 119876

119878sdot 119888 + 119865

119878sdot 1205881 (2)

where 119886 is rolling force armWhen1198791gt 1198790 119886 = 119877

1sdotsin(120573minus120593)

120573 is the roller center angle corresponding to rolling forcepoint without considering tension and 120573 asymp (12)radicΔℎ1198771015840(1minus(12120583)radicΔℎ119877

1015840) where Δℎ is reduction and 119877

1015840 is the rollflattening radius The force of backup roller acting on workroller is 119876

119878= 119875 sdot cos120593 cos(120579 + 120574) 119888 is the arm of force

of 119876119878acting on work roller and 119888 = 119898 cos 120574 + 119877

1sin 120574

When 1198791gt 1198790 119865119878= 119876119878sin(120579 + 120574) + 119875 sin120593 At this time

the work roller system stability condition is that the reactionforce of mill house acting on work roller meets 1198651015840

119904gt 0 or

1198651015840

119878= 119865119878+ 119865119867gt 0 namely

1198651015840

119878= 119865119878+ 119865119867

= 119875 cos120593 [tan120593 + tan (120579 + 120574)]

+ 119875 tan1205721198771sin (120573 minus 120593) + (cos120593 cos (120579 + 120574)) sdot (119898 cos 120574 + 119877

1sin 120574) + cos120593 [tan120593 + tan (120579 + 120574)] 120588

1

119871

(3)

Shock and Vibration 3O

ffset

e(m

m)

Rolling force P (MN)20 21 22 23 24 25 26 27 28 29 30

R1 = 400mm

R1 = 390mm

R1 = 380mm

R1 = 370mm

R1 = 360mm535

54

545

55

555

56

565

Figure 2 Minimum offset calculation

If cos120593 asymp 1 cos(120579 + 120574) asymp 1 and tan(120579 + 120574) asymp sin 120579 + sin 120574 =119890(1198771+ 1198772) + (1205882+ 119898)119877

2 tan120593 asymp sin120593 = (119879

1minus 1198790)2119875 and

then (3) can become as follows

1198791minus 1198790

2119875

+

119890

1198771+ 1198772

+

119898 + 1205882

1198772

+

2 tan120572119871

[1198771sin (120573 minus 120593) + 119898 cos 120574 + 119877

1sin 120574]

gt 0

(4)

At this time the work roller offset should be satisfied asfollows

119890 gt 1198900= (1198771+ 1198772)

1198791minus 1198790

2119875

+

119898 + 1205882

1198772

+

2 tan120572119871

[1198771sin (120573 minus 120593) + 119898 cos 120574 + 119877

1sin 120574]

(5)

By (5) if one considers the included angle between thespindle and the roller influence the work roller offset shouldbe increased (2 tan120572119871)(119877

1+ 1198772)[1198771sin(120573 minus 120593) + 119898 cos 120574 +

1198771sin 120574] based on the originalMinimum offset change scope with the rolling force

change under work roller diameter within a certain range isshown in Figure 2 According to the experience in the pro-duction the offset values are usually larger than the theoret-ical calculation (3sim5mm) The offset 119890 of F3 rolling mill is10mm when the work roller diameter is small it probablymeets the requirements of the static stability

3 Mill House Clearance and Drum GearCoupling Effect on the Roll System Stability

The CSP mill house and bearing plate never change fromproduction in recent 4 years their wear and tearmay be largeSo the F3mill upper and lower work roller bearing seat widthof drive side and operation side were measured In order toimprove the measurement accuracy 4 points of bearing plate

1 2

3 4

5

Operatingside

Bearing seat

Work roller

Mill house

6 7

8 9

10

Workside

Figure 3 Clearance measurement

were selected and shown in Figure 3 (1sim4 points on operatingside 6sim9 points on drive side) and then the average wasgot Five points were selected on mill house plate (1sim5 pointson operation side 6sim10 points on drive side) and then theaverage was got The measurement results of the bearing seatand mill house and the original design values are shown inTable 1

From Table 1 bearing width has negative deviation thenegative deviation is around 05mm Mill house width haspositive deviation the deviation value is around 18mmTheupper work roller clearance averages of drive side and opera-tion side are 2343mm and 2442mm respectivelyThe lowerwork roller clearance averages of drive side and operation sideare 2194mm and 2319mm respectively Obviously the F3mill house and bearing seat clearance aremuch larger originaldesign requirement (transition fit) namely the gap providesnecessary conditions to roller transverse vibration

According to the mechanical transmission theory thecondition of spindle can be normally operated as followsgear shaft axis roller axis and drum gear spindle axis mustbe located on the same plane and we must ensure that boththe included angles with the spindle axis are equal But withroll bearing seat and mill house plate wear (in the outlet sidemainly) roll connected end to the spindle will shift towardthe exit side Then the spindle will produce angle betweenthe vertical planes namely the spindle axis gear axis androller axis are not in a plane This will make the spindle haveadditional torque on the roller and the additional torquewill make the work roller produce deflection movement inthe horizontal direction which will make the bearing seatacting force uneven (as shown in Figure 4) and the bearingseat tilt (Figure 5) and roller axis cross (Figure 6) and otheradverse condition The additional horizontal force directionsof the operation side and transmission side of the rollerare opposite under the combined action of the clearanceand the additional bending moment which will make theroller bearing seat have crash tendency on mill house (thisis confirmed from the scene that bearing seat entrance sideplate also has certain wear and tear) In addition because ofthe bearing seat plate force and uneven wear (Figure 4) it


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

Bearing seat width Mill house widthDrive side Operating side Drive side Operating side

Design size 8900

minus0169000

minus0168900

minus0169000

minus016

MeasurementUpper roller 890 minus 0490 900 minus 0540 890 + 1853 900 + 1902

Lower roller 890 minus 0445 900 minus 0495 890 + 1749 900 + 1824

F

Figure 4 Bearing seat and mill house force diagram

Δ

Figure 5 Bearing seat tilt

will destroy the roller system stability under roller insufficientconstraints

Work roller axial force caused by backup roller crossand rolled strip cross is divided into roller axial force anddeformation zone axial force And the deformation zoneaxial force is dominant which mainly comes from axialcomponent of the friction force and its direction depends onthe metal particle longitudinal and transverse sliding velocitydirection relative to the roller surface its value is related torolling pressure and the friction condition as follows

119865119886= int

1198872

minus1198872

int

119897

0

120591119860

cos 120579119889119909 119889119910 (6)

120593

Figure 6 Roller cross

where 119887 is the rolled strip width 119897 is the deformation zonelength 120579 is the contact angle and tan 120579 = minus(12)ℎ

1015840

119909=

((ℎ0minus ℎ1)119897)(1 minus 119909119897) ℎ

119909is rolled strip thickness at the 119909

coordinate deformation zone and ℎ0and ℎ

1are the rolled

strip thickness at entrance and export 120591119860is axis component of

deformation area friction force According to the axial forcevector diagram in Figure 7 there has been 120591

119860= 120591|sin(120595+120593)|

where 120591 is the deformation area friction 120595 is sliding anglebetween metal and the rollers 120593 is roller crossing angle V isthe arbitrary point metal flow rate in deformation area andV119877is rotational speed of roller surface Thus according to the

119865119886calculation equation the relationship curves between axial

force and reduction rate and bearing-mill house clearance areshown in Figure 8 It can be seen that the axial force causedby roller cross increases with reduction rate and clearanceincrease its effect on the system cannot be ignored

The F3 mill upper work roller vibration accelerationtesting time-domain parameters on field is shown in Table 2It can be seen that although axial force is less than 1 of therolling force roller axial vibration acceleration amplitude isclose to vertical acceleration and lateral acceleration so theaxial force (caused by rollers cross) destruction on the systemstability cannot be ignored we should try to eliminate the gapbetween roller bearing seat and the mill house at the sametime it also shows the gap has important influence on rollervibration

In addition because the length ofmill drumgear couplingis about 5m and its weight is about 7 tons and it has no

Shock and Vibration 5

120593

120591

R

Figure 7 Axial force vector diagram

Reduction rate04 045 05 055 06

110

115

120

125

130

Axi

al fo

rceF

a(k

N)

Δ = 28mmΔ = 20mm

Δ = 12mmΔ = 04mm

Figure 8 Axial force theory diagram

Table 2 Roller vibration measurement (unit msdotsminus2)

Peak value Averageamplitude

Effectivevalue

Vertical acceleration 16383 04096 05074Lateral acceleration 21308 04341 05499Axial acceleration 12711 02919 03599

support in the presence of the spindle imbalance due to theinternal or external gear eccentricity and the gravity centeroffset these will bring large unbalanced force and affect theroll system dynamic stability In the presence of drum geartooth tip clearance centrifugal force 119865

119888will be generated

119865119888=

119898Δ1199101198992

182 times 105N (7)

Fc2

Fr

Tmin

Δy

O1

O

Fr

Fc2

Figure 9 Gear coupling centrifugal force

1

4

3

2

Figure 10 Unbalanced force diagram

where 119898 is the outer gear shaft spindle weight kg 119899 isthe coupling working speed rmin Δ119910 is the relative radialdisplacement mm as shown in Figure 9 As a result ofthe existence of the centrifugal force in order to make theinterior and exterior of tooth coupling automatically workroller needs to transmit the minimum torque 119879min namely

119879min =1198981198901198992

119889

364 times 105(N sdotmm) (8)

where 119898 is the weight of the intermediate shaft or internalgear ring kg 119890 is inside and outside of the teeth relativeeccentricity mm 119899 is the coupling speed rmin 119889 is toothreference circle diameter mm Substituting parameters into(8) we get119879min = 452times10

7N sdotm and themeasured torque is4times105Nsdotm that is it cannotmeet self-centering requirement

there always is centrifugal force or unbalanced forceThe unbalanced force on bearing seat diagram is shown

in Figure 10 there point 1 shows unbalanced force upwardpoint 2 shows the unbalanced force toward left which makes


(1) (2) (3) (4)

Figure 11 Drum tooth coupling mesh diagram

29500 30000 30500 31000 31500 32000 32500

432 Torque1

Lateral acceleration

Bending moment

Time (ms)

Vertical acceleration

Am

plitu

de (m

V)

29500 30000 30500 31000 31500 32000 32500

29500 30000 30500 31000 31500 32000 32500

29500 30000 30500 31000 31500 32000 32500minus5000

05000

minus5000

05000

010002000

minus1500minus1000minus500

0500

(a)

70000 70500 71000 71500 72000 72500 73000

70000 70500 71000 71500 72000 72500 73000

70000 70500 71000 71500 72000 72500 73000

70000 70500 71000 71500 72000 72500 73000

42 Torque


Bending moment

Time (ms)


Am

plitu

de (m

V)

minus1200minus1000minus800minus600

010002000

minus3000minus1500

015003000

minus3000minus1500

015003000

(b)

Figure 12 Roller system vibration measurement results

the bearing seat on left from mill house point 3 showsunbalanced force downward and point 4 shows unbalancedforce to the right and has a tendency to make the bearingseat pressure on the mill house For each point correspondsto 1sim4 drum gear meshing clearance is shown in Figure 11accordingly

Gear spindle shaft torque bending moment roller trans-verse vibration and vertical vibration test curve are shownin Figure 12 Figure 12(a) shows the strip bite stage and Fig-ure 12(b) shows strip rolling intermediate stage In the figureplumb line 1 is corresponding to spindle top point plumbline 2 is corresponding to spindle left point plumb line 3 iscorresponding to spindle bottom point and plumb line 4 iscorresponding to spindle right point and they are corre-sponding to Figures 10 and 11 From Figure 12(a) in the initialstage of strip entrance when drum teeth are in upper andlower vertices namely perpendicular lines 1 and 3 meshingimpact has greater influence on roller vibration At stablerolling stage in the left vertex (at perpendicular line 2) the

vibration ismost violent and in the right vertex the vibrationnearly disappeared This is because unbalanced force makesroller depart from the mill house and be in an unstable stateat the left point and makes meshing impact increase sharplyOn the contrary the imbalance force makes roller close tomill house and will not produce unstable state in right vertexThus this makes the waveform produce ldquogourdrdquo shape thisis also because some mills rarely have this vibration shapeequipped with slider or cross type spindle

In order to determine the vibration nature and find outthe vibration source considering the characteristics of thecollective of upper and lower roll drive speed frequencymap values contained the F3 first-order natural frequency oftorsional vibration system the first-order natural frequencyof stand vertical vibration system and the mill stand third-order natural frequency of transverse vibration system Con-centrated mass-spring model was adopted for the main drivesystem and the first four-order natural frequency is 217Hz73Hz 873Hz and 229Hz respectively The rolling mill

Shock and Vibration 7Fr

eque

ncyf

(Hz)

V-mode 1 56HzH-mode 3 51Hz

T-mode 1 21Hz

1

2

3

4

05 10 1500

Work roller speed 120596 (Hz)

0

20

40

60

1ndash4 meshing frequency of middle shaft gear reducerarc tooth spindle and pinion stand

work roller measured vertical and horizontal accelerationroll torque dominant frequency

V T-mode 1 first natural frequency of vertical and torsionalvibration system

H-mode 3 the third natural frequency of horizontal vibrationsystem

Figure 13 F3 rolling mill speed map

Figure 14 Spindle arc tooth wear photos

transverse vibration natural frequency was calculated withfinite element method and within the roller diameter avail-able range the first three-order natural frequency of the rollsystem horizontal vibration is 480sim499Hz 509sim555Hzand 555sim824Hz respectively [11]

It can be seen from Figure 13 that the measured vibrationmain frequency has almost linear relationship with speedand the 20Hz torsional vibration is mainly around concen-trated in the gear mesh frequency of distribution box andthe first-order natural frequency of torsional vibration systemintersection The 50Hz vibration mainly concentrated in thedrum gear meshing frequency and the first-order naturalfrequency of the vertical vibration system intersection TheF3 rollingmill main vibrationmode is horizontal and verticalvibration vibration main frequency is 50Hz or so and hasapproximate linear relationship with the rolling speed Thefrequency is consistent with drum gear shaft gear meshingfrequency first-order natural frequency of vertical vibrationsystem and the-third order natural frequency of horizontalvibration which indicates that the drumgearmeshing impactexcites the stand vertical vibration system first-order and thehorizontal vibration third-order resonance It also can be seenfrom the spot photos (Figure 14) that the spindle drum gearremoved frommill has wear and tear both sides of the drumgear tooth and top all produced excessive wear phenomenonSo it will be beneficial to eliminate vibration mill if drumtooth meshing clearance is eliminated

4 Roller Horizontal Friction-ImpactVibration Characteristic Analysis

Combined with structural characteristics of the roll systemroll horizontal vibration dynamic model is set up Accordingto the roll loading characteristics when the extraneous forceis more than offset drift force 119865

119890and has a gap between roll

bearing seat and the mill house the work roller bearing seatmay jump from themill house and there is bearing impact onmill house conditions the system stability is damaged at thistime Considering the mill housing after elasticity we makeit as an elastic collision process in this paper Based on thisthe roller horizontal vibration system ldquojumprdquo model was setup in Figure 15 and the diagrams (a) (b) and (c) show workroll bearing pressing on the right side of the housing pillarand bearing seat in the middle and bearing pressing on theleft side of the housing pillar in sequence In the figure 119865

119889is

the interface friction damping force So itsmotion differentialequation is as follows

119898 + 119891 (119909 ) = 119865 cosΩ119905 cos 51Ω119905 (9)

where 119898 is the weight of work roller 119865 is extraneous forceΩ is the roll rotation frequency and 119891(119909 ) expression is asfollows

119891 (119909 ) =

minus (1198961+ 1198962) 119909 + 119860 minus 119861

3

119909 ge 0 or 119909 le minusΔmax

minus1198961119909 + 119860 minus 119861

3

minusΔmax lt 119909 lt 0 contact pressure is zero(10)

where 1198961is the linear stiffness coefficient between work roller

and mill house left side 1198962is linear stiffness coefficient of

mill house pillar 119860 is the linear damping coefficient 119861 is the

nonlinear damping coefficient and Δmax is the gap betweenbearing seat and mill house considering the roller crossoverand bearing seat tilt Then the numerical simulation system


k1

c1

Fd

k2

k2

Δmaxx

m

F(t)

(a)

k1

c1

Fd

k2

k2

Δ1 Δ2x

m

F(t)

(b)

k1

c1

Fd

k2

k2

Δmaxx

m

F(t)

(c)

Figure 15 Roller-mill impact vibration mechanics model

Disp

lace

men

t(120583

m) 10

0minus10minus20

Time t (s)08 09 1 11 12

Velo

city

(mm

s) 10

0minus10

Displacement (120583m)minus20 0 20

Pow

ersp

ectr

um

4

2

0

times10minus3

Frequency f (Hz)0 100 200

Figure 16 F3 horizontal response without measures

was carried out with the following set of basic parameters ora certain fluctuation119898 = 1times10

5 kg 1198961= 1times 10

10Nm 1198962=

1times1011NmΩ = 2120587119860 = 4times10

6Nsdotsm119861 = 4times1010Nsdotsm119865 = 1MN and Δmax = 02mm

The oversize clearance between mill house and bearingseat can cause roller crossover and roll system transversevibration in order to verify the clearance influence on theroll system dynamic characteristics it is necessary to carryout simulation and analysis Due to large weight of drumgear spindle when it has slight eccentric or installation errorwhich would cause arc tooth meshing impact the spindlesupport with arc tooth will play an important role on vibra-tion suppression But limited by various factors it is difficultto be implemented at the scene but these achievements havecertain inspiration to rolling mill manufacturers In orderto eliminate the gap and eliminate the phenomenon of rollbearing seat collision onmill house free expansion hydraulicplate can be used in order to reduce or eliminate the drumgear shaft imbalance force the drum gear spindle floatingsupport can be used In order to test and verify the effectof these measures the numerical simulation was carried outrespectively in this paper In Figure 16 the simulation curveis the roller displacement response diagram phase diagramand displacement power spectrum diagram respectively

According to the roll horizontal vibration mechanicsmodel (Figure 15) with the gap between roll bearing seat andmill house eliminated roll horizontal vibration differentialequations can be expressed as follows

119898 = minus (1198961+ 21198962) 119909 + 119860 minus 119861

3

+ 119865 cosΩ119905 cos 51Ω119905 (11)

1

0

minus1

Time t (s)1 11 12 13 14

020

minus02


5

0

times10minus4


Disp

lace

men

t(120583

m)

Velo

city

(mm

s)

Pow

ersp

ectr

um

Figure 17 F3 horizontal response with measures

The values of the parameters are the same as the mentionedabove When adding drum spindle floating support thespindle unbalance excitation disappears roller horizontalvibration differential equation is expressed as follows

119898 + 119891 (119909 ) = 119865 cos 51Ω119905

119891 (119909 )

=

minus (1198961+ 1198962) 119909 + 119860 minus 119861

3


minus1198961119909 + 119860 minus 119861

3

minusΔmax lt 119909 lt 0

(12)

When the two measures of eliminating the gap betweenthe roller bearing seat and mill house and adding the drumgear shaft floating support were adopted at the same time rollhorizontal vibration differential equations can be expressed asfollows

119898 = minus (1198961+ 21198962) 119909 + 119860 minus 119861

3

+ 119865 cos 51Ω119905 (13)

The simulation results are shown in Figure 17 of elim-inating the gap between the roller bearing seat and millhouse and adding the drum gear shaft floating supportadopted at the same time From the figure we can see thatafter these two measures are taken at the same time thevibration amplitude reduced to one-tenth of the original andby the phase diagram and the spectrum diagram systemproperty has been improved significantly and the roller ldquojumpvibrationrdquo phenomenon disappeared

We adopted vibration suppression measures as addingcopper gaskets within the bearing liner to eliminate or reduce

Shock and Vibration 9A

mpl

itude

2000

1000

0

minus1000

minus2000

minus3000

(ms)0 150 300 450

Am

plitu

de

100

80

60

40

20

0

(Hz)0 40 80 120 160 200 240

Figure 18 Roller horizontal vibration measured results after gapelimination

the bearing seat and mill house clearance at site The rollerhorizontal vibration results after eliminating the clearanceare shown in Figure 18 and it can be seen after eliminatingthe clearance that chopped or touch vibration phenomenondisappears vibration amplitude is declined greatly and theenergy spread So the measure of gap eliminating can reducemill vibration (horizontal)

5 Conclusions

In order to deeply study the rolling mill vibration source theroll system structure stability was analyzed and calculatedfirstly and it had been found that when the F3 rolling millwork roller diameter is small the offset basically conformsto the requirements of the static stability Then the fieldmeasurement of the clearance betweenmill stand and bearingseat has been carried out and it was found that F3 clear-ance is much larger than the original design (transition fit)requirement namely the gap provides necessary conditionsfor roller horizontal vibrationThe drum gear coupling effecton the roll system stability is analyzed and it was found thatthe coupling transmission torque cannot meet self-centeringrequirement roller driving system always has unbalancedforce or centrifugal force and this is verified by the fieldvibration test At the same time the axial force by rollers crosscaused by roller system structure defect has great destructionon the system stability and it cannot be allowed to beignored Finally the roll horizontal friction chatter vibrationmechanics model was established and the simulation analysiswas carried out it showed that when eliminating the gapbetween the roller bearing seat and mill house and addingthe drum gear shaft floating support at the same timeroll vibration amplitude was reduced to one over ten ofthe original and by the phase diagram and the spectrumdiagram system dynamics have been improved thoroughlyand the roller ldquojump vibrationrdquo phenomenon disappeared Atthe same time adding copper gaskets to eliminate or reduce

the clearance between the bearing seat and mill house forvibration suppression at the scene the results showed thatafter the clearance is eliminated chopped or touch vibrationphenomenon disappeared and vibration amplitude value hasfallen obviously and the energy has spread that is it is helpfulto suppress roller horizontal vibration

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

Acknowledgments

This research was supported by National UndergraduateTraining Programs for Innovation and Entrepreneurship(no 201510460001) Henan Polytechnic University EducationTeaching Reform Research Projects (no 2015JG034) andColleges and Universities Focus on Soft Science ResearchProject Plan (no 16A630049)

References

[1] X Shu and G Shen ldquoAnalysis of roll system stability about alu-minum foil millsrdquo Chinese Journal of Mechanical Engineeringvol 16 no 2 pp 181ndash189 2003

[2] G Shen and M Li ldquoStatically determinate characteristics ofmicrodisplacement in a four-high millrdquo Journal of MaterialsProcessing Technology vol 209 no 11 pp 5002ndash5007 2009

[3] A Bar and O Bar ldquoTypes of mid-frequency vibrations appear-ing during the rolling mill operationrdquo Journal of MaterialsProcessing Technology vol 162-163 pp 461ndash464 2005

[4] A Bar and A Swiątoniowski ldquoInterdependence between therolling speed and non-linear vibrations of the mill systemrdquoJournal of Materials Processing Technology vol 155-156 no 11pp 2116ndash2121 2004

[5] V Panjkovic R Gloss J Steward S Dilks R Steward andG Fraser ldquoCauses of chatter in a hot strip mill observationsqualitative analyses and mathematical modellingrdquo Journal ofMaterials Processing Technology vol 212 no 4 pp 954ndash9612012

[6] H Furumoto S Kanemori K Hayashi and A Sako ldquoEnhanc-ing technologies of stabilization of mill vibration by millstabilizing device in hot rollingrdquo Procedia Engineering vol 81no 12 pp 102ndash107 2014

[7] Y Kim C-W Kim S-J Lee and H Park ldquoExperimental andnumerical investigation of the vibration characteristics in a coldrolling mill using multibody dynamicsrdquo ISIJ International vol52 no 11 pp 2042ndash2047 2012

[8] Y Kim H Park S S Lee and C-W Kim ldquoDevelopment of amathematical model for the prediction of vibration in a coldrollingmill including the driving systemrdquo ISIJ International vol52 no 6 pp 1135ndash1144 2012

[9] A Swiatoniowski and R Gregorczyk ldquoSelf-excited vibrationsin four-high rolling mills caused by stochastic disturbance offriction conditions on the roll-roll contact surfacerdquo Mechanicsand Control vol 29 no 3 pp 158ndash162 2010

[10] Y A Amer A T EL-Sayed and F T El-Bahrawy ldquoTorsionalvibration reduction for rolling millrsquos main drive system vianegative velocity feedback under parametric excitationrdquo Journal


of Mechanical Science and Technology vol 29 no 4 pp 1581ndash1589 2015

[11] F Xiaobin Vibration problem research for CSP mill stand [PhDthesis] University of Science and Technology Beijing BeijingChina 2007

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



120579 120574

1205882

R2pm

QS 120593

R1

1205881 FS

T1T0e

Figure 1 Upper roll system force diagram

components of the rolling force [7] They also proposed amathematical model of a cold rolling mill including the driv-ing system and a novel combination of the direct integrationmethod and quasistatic analysis to solve the model efficientlyand found the horizontal chatter vibration had a strongeffect on the dynamic characteristics [8] Swiatoniowski andGregorczyk presented a probabilistic model of the frictionphenomena on the work-backup rolls contact surface andfount that such character of the disturbance in distributionof zones with static and kinetic friction can be regarded asone of the sources of self-excited vibrations appearing inthe system consisting of a rolling mill and a strip [9] Ameret al studied the torsional vibration reduction for rollingmillrsquos main drive system via negative velocity feedback underparametric excitation and found the resonance conditionis the first natural frequency vibration as one of the worstresonance cases [10]

Usually rolling mill vibration research does not distin-guishwhether rollingmill roll system is stable andmechanicsmodel is set up setting the mill under stabilization A singlevibration as the coupling mode with roll vertical horizontalaxial torsional cross and swing space vibration behavior isanalyzed Actually roller vibration will damage roll systemstability and cause bearing horizontal or vertical vibrationTherefore the CSP four-high mill vibration characteristics

were studied from the mill roll system stability and lateraldynamic behavior of the roll system in this paper

2 Roller System StructureStability Calculation

Work roller force analysis is shown in Figure 1 where 119876119904is

the force onwork roller by backup roller119875 is the rolling force119865119904is the work roll bearing reaction force by mill house 119898 is

the friction arm distance (02mm) between work roller andbackup roller119879

1and1198790are the front and back tension119877

1and

1198772are the work roll and backup roll radius 120588

1and 1205882are work

roll and backup roll bearing friction circle radius 120579 is theincluded angle between the work roll and backup roll defiledline and there is sin 120579 = 119890(119877

1+ 1198772) 120593 is deviation angle of

front and back tension impact on rolling force direction andthere has been sin120593 = (119879

1minus 1198790)2119875 120574 is the included angle

between roller defiled line and reaction force 119876119904 and there

has been

sin 120574 =(1205882+ 119898)

1198772

(1)

Due to the included angle between the roller and drumgear spindle axis the spindle will produce additionalmoment119872119867on the roller namely in the horizontal direction it will

make the roller produces additional horizontal force 119865119867and

119865119867= 119872119867119871The force on the side is pointing in the direction

of work roll shift direction and the force is departure fromthe direction of work roll shift on the other side and maycause two rollersrsquo crossing where 119871 is the distance betweenthe center lines of the both work rollers bearing seat and119872119867= 119872119860sdot tan120572 120572 is included angle between the spindle

and roller and 119872119860is the sum of the rolling torque 119872

119877+

support roller rotation torque 119872119861+ work roller bearing

friction consumption torque119872119882 namely

119872119860= 119875 sdot 119886 + 119876

119878sdot 119888 + 119865

119878sdot 1205881 (2)

where 119886 is rolling force armWhen1198791gt 1198790 119886 = 119877

1sdotsin(120573minus120593)

120573 is the roller center angle corresponding to rolling forcepoint without considering tension and 120573 asymp (12)radicΔℎ1198771015840(1minus(12120583)radicΔℎ119877

1015840) where Δℎ is reduction and 119877

1015840 is the rollflattening radius The force of backup roller acting on workroller is 119876

119878= 119875 sdot cos120593 cos(120579 + 120574) 119888 is the arm of force

of 119876119878acting on work roller and 119888 = 119898 cos 120574 + 119877

1sin 120574

When 1198791gt 1198790 119865119878= 119876119878sin(120579 + 120574) + 119875 sin120593 At this time

the work roller system stability condition is that the reactionforce of mill house acting on work roller meets 1198651015840

119904gt 0 or

1198651015840

119878= 119865119878+ 119865119867gt 0 namely

1198651015840

119878= 119865119878+ 119865119867

= 119875 cos120593 [tan120593 + tan (120579 + 120574)]

+ 119875 tan1205721198771sin (120573 minus 120593) + (cos120593 cos (120579 + 120574)) sdot (119898 cos 120574 + 119877

1sin 120574) + cos120593 [tan120593 + tan (120579 + 120574)] 120588

1

119871

(3)


ffset

e(m

m)


R1 = 400mm

R1 = 390mm

R1 = 380mm

R1 = 370mm

R1 = 360mm535

54

545

55

555

56

565



2 tan120593 asymp sin120593 = (119879

1minus 1198790)2119875 and


1198791minus 1198790

2119875

+

119890

1198771+ 1198772

+

119898 + 1205882

1198772

+

2 tan120572119871

[1198771sin (120573 minus 120593) + 119898 cos 120574 + 119877

1sin 120574]

gt 0

(4)


119890 gt 1198900= (1198771+ 1198772)

1198791minus 1198790

2119875

+

119898 + 1205882

1198772

+

2 tan120572119871

[1198771sin (120573 minus 120593) + 119898 cos 120574 + 119877

1sin 120574]

(5)


1+ 1198772)[1198771sin(120573 minus 120593) + 119898 cos 120574 +





1 2

3 4

5

Operatingside

Bearing seat

Work roller

Mill house

6 7

8 9

10

Workside








Design size 8900

minus0169000

minus0168900

minus0169000

minus016



F


Δ




119865119886= int

1198872

minus1198872

int

119897

0

120591119860

cos 120579119889119909 119889119910 (6)

120593



1015840

119909=

((ℎ0minus ℎ1)119897)(1 minus 119909119897) ℎ



1are the rolled



119860= 120591|sin(120595+120593)|







120593

120591

R



110

115

120

125

130

Axi

al fo

rceF

a(k

N)

Δ = 28mmΔ = 20mm

Δ = 12mmΔ = 04mm




Effectivevalue




119865119888=

119898Δ1199101198992

182 times 105N (7)

Fc2

Fr

Tmin

Δy

O1

O

Fr

Fc2


1

4

3

2



119879min =1198981198901198992

119889







(1) (2) (3) (4)


29500 30000 30500 31000 31500 32000 32500

432 Torque1


Bending moment

Time (ms)


Am

plitu

de (m

V)

29500 30000 30500 31000 31500 32000 32500

29500 30000 30500 31000 31500 32000 32500

29500 30000 30500 31000 31500 32000 32500minus5000

05000

minus5000

05000

010002000


0500

(a)

70000 70500 71000 71500 72000 72500 73000

70000 70500 71000 71500 72000 72500 73000

70000 70500 71000 71500 72000 72500 73000

70000 70500 71000 71500 72000 72500 73000

42 Torque


Bending moment

Time (ms)


Am

plitu

de (m

V)


010002000

minus3000minus1500

015003000

minus3000minus1500

015003000

(b)







eque

ncyf

(Hz)


T-mode 1 21Hz

1

2

3

4

05 10 1500


0

20

40

60













119889is


119898 + 119891 (119909 ) = 119865 cosΩ119905 cos 51Ω119905 (9)


119891 (119909 ) =

minus (1198961+ 1198962) 119909 + 119860 minus 119861

3


minus1198961119909 + 119860 minus 119861

3







k1

c1

Fd

k2

k2

Δmaxx

m

F(t)

(a)

k1

c1

Fd

k2

k2

Δ1 Δ2x

m

F(t)

(b)

k1

c1

Fd

k2

k2

Δmaxx

m

F(t)

(c)


Disp

lace

men

t(120583

m) 10

0minus10minus20

Time t (s)08 09 1 11 12

Velo

city

(mm

s) 10

0minus10


Pow

ersp

ectr

um

4

2

0

times10minus3




5 kg 1198961= 1times 10

10Nm 1198962=

1times1011NmΩ = 2120587119860 = 4times10




119898 = minus (1198961+ 21198962) 119909 + 119860 minus 119861

3

+ 119865 cosΩ119905 cos 51Ω119905 (11)

1

0

minus1

Time t (s)1 11 12 13 14

020

minus02


5

0

times10minus4


Disp

lace

men

t(120583

m)

Velo

city

(mm

s)

Pow

ersp

ectr

um



119898 + 119891 (119909 ) = 119865 cos 51Ω119905

119891 (119909 )

=

minus (1198961+ 1198962) 119909 + 119860 minus 119861

3


minus1198961119909 + 119860 minus 119861

3


(12)


119898 = minus (1198961+ 21198962) 119909 + 119860 minus 119861

3

+ 119865 cos 51Ω119905 (13)




mpl

itude

2000

1000

0

minus1000

minus2000

minus3000

(ms)0 150 300 450

Am

plitu

de

100

80

60

40

20

0

(Hz)0 40 80 120 160 200 240



5 Conclusions



Competing Interests


Acknowledgments


References
















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










ffset

e(m

m)


R1 = 400mm

R1 = 390mm

R1 = 380mm

R1 = 370mm

R1 = 360mm535

54

545

55

555

56

565



2 tan120593 asymp sin120593 = (119879

1minus 1198790)2119875 and


1198791minus 1198790

2119875

+

119890

1198771+ 1198772

+

119898 + 1205882

1198772

+

2 tan120572119871

[1198771sin (120573 minus 120593) + 119898 cos 120574 + 119877

1sin 120574]

gt 0

(4)


119890 gt 1198900= (1198771+ 1198772)

1198791minus 1198790

2119875

+

119898 + 1205882

1198772

+

2 tan120572119871

[1198771sin (120573 minus 120593) + 119898 cos 120574 + 119877

1sin 120574]

(5)


1+ 1198772)[1198771sin(120573 minus 120593) + 119898 cos 120574 +





1 2

3 4

5

Operatingside

Bearing seat

Work roller

Mill house

6 7

8 9

10

Workside








Design size 8900

minus0169000

minus0168900

minus0169000

minus016



F


Δ




119865119886= int

1198872

minus1198872

int

119897

0

120591119860

cos 120579119889119909 119889119910 (6)

120593



1015840

119909=

((ℎ0minus ℎ1)119897)(1 minus 119909119897) ℎ



1are the rolled



119860= 120591|sin(120595+120593)|







120593

120591

R



110

115

120

125

130

Axi

al fo

rceF

a(k

N)

Δ = 28mmΔ = 20mm

Δ = 12mmΔ = 04mm




Effectivevalue




119865119888=

119898Δ1199101198992

182 times 105N (7)

Fc2

Fr

Tmin

Δy

O1

O

Fr

Fc2


1

4

3

2



119879min =1198981198901198992

119889







(1) (2) (3) (4)


29500 30000 30500 31000 31500 32000 32500

432 Torque1


Bending moment

Time (ms)


Am

plitu

de (m

V)

29500 30000 30500 31000 31500 32000 32500

29500 30000 30500 31000 31500 32000 32500

29500 30000 30500 31000 31500 32000 32500minus5000

05000

minus5000

05000

010002000


0500

(a)

70000 70500 71000 71500 72000 72500 73000

70000 70500 71000 71500 72000 72500 73000

70000 70500 71000 71500 72000 72500 73000

70000 70500 71000 71500 72000 72500 73000

42 Torque


Bending moment

Time (ms)


Am

plitu

de (m

V)


010002000

minus3000minus1500

015003000

minus3000minus1500

015003000

(b)







eque

ncyf

(Hz)


T-mode 1 21Hz

1

2

3

4

05 10 1500


0

20

40

60













119889is


119898 + 119891 (119909 ) = 119865 cosΩ119905 cos 51Ω119905 (9)


119891 (119909 ) =

minus (1198961+ 1198962) 119909 + 119860 minus 119861

3


minus1198961119909 + 119860 minus 119861

3







k1

c1

Fd

k2

k2

Δmaxx

m

F(t)

(a)

k1

c1

Fd

k2

k2

Δ1 Δ2x

m

F(t)

(b)

k1

c1

Fd

k2

k2

Δmaxx

m

F(t)

(c)


Disp

lace

men

t(120583

m) 10

0minus10minus20

Time t (s)08 09 1 11 12

Velo

city

(mm

s) 10

0minus10


Pow

ersp

ectr

um

4

2

0

times10minus3




5 kg 1198961= 1times 10

10Nm 1198962=

1times1011NmΩ = 2120587119860 = 4times10




119898 = minus (1198961+ 21198962) 119909 + 119860 minus 119861

3

+ 119865 cosΩ119905 cos 51Ω119905 (11)

1

0

minus1

Time t (s)1 11 12 13 14

020

minus02


5

0

times10minus4


Disp

lace

men

t(120583

m)

Velo

city

(mm

s)

Pow

ersp

ectr

um



119898 + 119891 (119909 ) = 119865 cos 51Ω119905

119891 (119909 )

=

minus (1198961+ 1198962) 119909 + 119860 minus 119861

3


minus1198961119909 + 119860 minus 119861

3


(12)


119898 = minus (1198961+ 21198962) 119909 + 119860 minus 119861

3

+ 119865 cos 51Ω119905 (13)




mpl

itude

2000

1000

0

minus1000

minus2000

minus3000

(ms)0 150 300 450

Am

plitu

de

100

80

60

40

20

0

(Hz)0 40 80 120 160 200 240



5 Conclusions



Competing Interests


Acknowledgments


References
















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












Design size 8900

minus0169000

minus0168900

minus0169000

minus016



F


Δ




119865119886= int

1198872

minus1198872

int

119897

0

120591119860

cos 120579119889119909 119889119910 (6)

120593



1015840

119909=

((ℎ0minus ℎ1)119897)(1 minus 119909119897) ℎ



1are the rolled



119860= 120591|sin(120595+120593)|







120593

120591

R



110

115

120

125

130

Axi

al fo

rceF

a(k

N)

Δ = 28mmΔ = 20mm

Δ = 12mmΔ = 04mm




Effectivevalue




119865119888=

119898Δ1199101198992

182 times 105N (7)

Fc2

Fr

Tmin

Δy

O1

O

Fr

Fc2


1

4

3

2



119879min =1198981198901198992

119889







(1) (2) (3) (4)


29500 30000 30500 31000 31500 32000 32500

432 Torque1


Bending moment

Time (ms)


Am

plitu

de (m

V)

29500 30000 30500 31000 31500 32000 32500

29500 30000 30500 31000 31500 32000 32500

29500 30000 30500 31000 31500 32000 32500minus5000

05000

minus5000

05000

010002000


0500

(a)

70000 70500 71000 71500 72000 72500 73000

70000 70500 71000 71500 72000 72500 73000

70000 70500 71000 71500 72000 72500 73000

70000 70500 71000 71500 72000 72500 73000

42 Torque


Bending moment

Time (ms)


Am

plitu

de (m

V)


010002000

minus3000minus1500

015003000

minus3000minus1500

015003000

(b)







eque

ncyf

(Hz)


T-mode 1 21Hz

1

2

3

4

05 10 1500


0

20

40

60













119889is


119898 + 119891 (119909 ) = 119865 cosΩ119905 cos 51Ω119905 (9)


119891 (119909 ) =

minus (1198961+ 1198962) 119909 + 119860 minus 119861

3


minus1198961119909 + 119860 minus 119861

3







k1

c1

Fd

k2

k2

Δmaxx

m

F(t)

(a)

k1

c1

Fd

k2

k2

Δ1 Δ2x

m

F(t)

(b)

k1

c1

Fd

k2

k2

Δmaxx

m

F(t)

(c)


Disp

lace

men

t(120583

m) 10

0minus10minus20

Time t (s)08 09 1 11 12

Velo

city

(mm

s) 10

0minus10


Pow

ersp

ectr

um

4

2

0

times10minus3




5 kg 1198961= 1times 10

10Nm 1198962=

1times1011NmΩ = 2120587119860 = 4times10




119898 = minus (1198961+ 21198962) 119909 + 119860 minus 119861

3

+ 119865 cosΩ119905 cos 51Ω119905 (11)

1

0

minus1

Time t (s)1 11 12 13 14

020

minus02


5

0

times10minus4


Disp

lace

men

t(120583

m)

Velo

city

(mm

s)

Pow

ersp

ectr

um



119898 + 119891 (119909 ) = 119865 cos 51Ω119905

119891 (119909 )

=

minus (1198961+ 1198962) 119909 + 119860 minus 119861

3


minus1198961119909 + 119860 minus 119861

3


(12)


119898 = minus (1198961+ 21198962) 119909 + 119860 minus 119861

3

+ 119865 cos 51Ω119905 (13)




mpl

itude

2000

1000

0

minus1000

minus2000

minus3000

(ms)0 150 300 450

Am

plitu

de

100

80

60

40

20

0

(Hz)0 40 80 120 160 200 240



5 Conclusions



Competing Interests


Acknowledgments


References
















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










120593

120591

R



110

115

120

125

130

Axi

al fo

rceF

a(k

N)

Δ = 28mmΔ = 20mm

Δ = 12mmΔ = 04mm




Effectivevalue




119865119888=

119898Δ1199101198992

182 times 105N (7)

Fc2

Fr

Tmin

Δy

O1

O

Fr

Fc2


1

4

3

2



119879min =1198981198901198992

119889







(1) (2) (3) (4)


29500 30000 30500 31000 31500 32000 32500

432 Torque1


Bending moment

Time (ms)


Am

plitu

de (m

V)

29500 30000 30500 31000 31500 32000 32500

29500 30000 30500 31000 31500 32000 32500

29500 30000 30500 31000 31500 32000 32500minus5000

05000

minus5000

05000

010002000


0500

(a)

70000 70500 71000 71500 72000 72500 73000

70000 70500 71000 71500 72000 72500 73000

70000 70500 71000 71500 72000 72500 73000

70000 70500 71000 71500 72000 72500 73000

42 Torque


Bending moment

Time (ms)


Am

plitu

de (m

V)


010002000

minus3000minus1500

015003000

minus3000minus1500

015003000

(b)







eque

ncyf

(Hz)


T-mode 1 21Hz

1

2

3

4

05 10 1500


0

20

40

60













119889is


119898 + 119891 (119909 ) = 119865 cosΩ119905 cos 51Ω119905 (9)


119891 (119909 ) =

minus (1198961+ 1198962) 119909 + 119860 minus 119861

3


minus1198961119909 + 119860 minus 119861

3







k1

c1

Fd

k2

k2

Δmaxx

m

F(t)

(a)

k1

c1

Fd

k2

k2

Δ1 Δ2x

m

F(t)

(b)

k1

c1

Fd

k2

k2

Δmaxx

m

F(t)

(c)


Disp

lace

men

t(120583

m) 10

0minus10minus20

Time t (s)08 09 1 11 12

Velo

city

(mm

s) 10

0minus10


Pow

ersp

ectr

um

4

2

0

times10minus3




5 kg 1198961= 1times 10

10Nm 1198962=

1times1011NmΩ = 2120587119860 = 4times10




119898 = minus (1198961+ 21198962) 119909 + 119860 minus 119861

3

+ 119865 cosΩ119905 cos 51Ω119905 (11)

1

0

minus1

Time t (s)1 11 12 13 14

020

minus02


5

0

times10minus4


Disp

lace

men

t(120583

m)

Velo

city

(mm

s)

Pow

ersp

ectr

um



119898 + 119891 (119909 ) = 119865 cos 51Ω119905

119891 (119909 )

=

minus (1198961+ 1198962) 119909 + 119860 minus 119861

3


minus1198961119909 + 119860 minus 119861

3


(12)


119898 = minus (1198961+ 21198962) 119909 + 119860 minus 119861

3

+ 119865 cos 51Ω119905 (13)




mpl

itude

2000

1000

0

minus1000

minus2000

minus3000

(ms)0 150 300 450

Am

plitu

de

100

80

60

40

20

0

(Hz)0 40 80 120 160 200 240



5 Conclusions



Competing Interests


Acknowledgments


References
















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










(1) (2) (3) (4)


29500 30000 30500 31000 31500 32000 32500

432 Torque1


Bending moment

Time (ms)


Am

plitu

de (m

V)

29500 30000 30500 31000 31500 32000 32500

29500 30000 30500 31000 31500 32000 32500

29500 30000 30500 31000 31500 32000 32500minus5000

05000

minus5000

05000

010002000


0500

(a)

70000 70500 71000 71500 72000 72500 73000

70000 70500 71000 71500 72000 72500 73000

70000 70500 71000 71500 72000 72500 73000

70000 70500 71000 71500 72000 72500 73000

42 Torque


Bending moment

Time (ms)


Am

plitu

de (m

V)


010002000

minus3000minus1500

015003000

minus3000minus1500

015003000

(b)







eque

ncyf

(Hz)


T-mode 1 21Hz

1

2

3

4

05 10 1500


0

20

40

60













119889is


119898 + 119891 (119909 ) = 119865 cosΩ119905 cos 51Ω119905 (9)


119891 (119909 ) =

minus (1198961+ 1198962) 119909 + 119860 minus 119861

3


minus1198961119909 + 119860 minus 119861

3







k1

c1

Fd

k2

k2

Δmaxx

m

F(t)

(a)

k1

c1

Fd

k2

k2

Δ1 Δ2x

m

F(t)

(b)

k1

c1

Fd

k2

k2

Δmaxx

m

F(t)

(c)


Disp

lace

men

t(120583

m) 10

0minus10minus20

Time t (s)08 09 1 11 12

Velo

city

(mm

s) 10

0minus10


Pow

ersp

ectr

um

4

2

0

times10minus3




5 kg 1198961= 1times 10

10Nm 1198962=

1times1011NmΩ = 2120587119860 = 4times10




119898 = minus (1198961+ 21198962) 119909 + 119860 minus 119861

3

+ 119865 cosΩ119905 cos 51Ω119905 (11)

1

0

minus1

Time t (s)1 11 12 13 14

020

minus02


5

0

times10minus4


Disp

lace

men

t(120583

m)

Velo

city

(mm

s)

Pow

ersp

ectr

um



119898 + 119891 (119909 ) = 119865 cos 51Ω119905

119891 (119909 )

=

minus (1198961+ 1198962) 119909 + 119860 minus 119861

3


minus1198961119909 + 119860 minus 119861

3


(12)


119898 = minus (1198961+ 21198962) 119909 + 119860 minus 119861

3

+ 119865 cos 51Ω119905 (13)




mpl

itude

2000

1000

0

minus1000

minus2000

minus3000

(ms)0 150 300 450

Am

plitu

de

100

80

60

40

20

0

(Hz)0 40 80 120 160 200 240



5 Conclusions



Competing Interests


Acknowledgments


References
















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










eque

ncyf

(Hz)


T-mode 1 21Hz

1

2

3

4

05 10 1500


0

20

40

60













119889is


119898 + 119891 (119909 ) = 119865 cosΩ119905 cos 51Ω119905 (9)


119891 (119909 ) =

minus (1198961+ 1198962) 119909 + 119860 minus 119861

3


minus1198961119909 + 119860 minus 119861

3







k1

c1

Fd

k2

k2

Δmaxx

m

F(t)

(a)

k1

c1

Fd

k2

k2

Δ1 Δ2x

m

F(t)

(b)

k1

c1

Fd

k2

k2

Δmaxx

m

F(t)

(c)


Disp

lace

men

t(120583

m) 10

0minus10minus20

Time t (s)08 09 1 11 12

Velo

city

(mm

s) 10

0minus10


Pow

ersp

ectr

um

4

2

0

times10minus3




5 kg 1198961= 1times 10

10Nm 1198962=

1times1011NmΩ = 2120587119860 = 4times10




119898 = minus (1198961+ 21198962) 119909 + 119860 minus 119861

3

+ 119865 cosΩ119905 cos 51Ω119905 (11)

1

0

minus1

Time t (s)1 11 12 13 14

020

minus02


5

0

times10minus4


Disp

lace

men

t(120583

m)

Velo

city

(mm

s)

Pow

ersp

ectr

um



119898 + 119891 (119909 ) = 119865 cos 51Ω119905

119891 (119909 )

=

minus (1198961+ 1198962) 119909 + 119860 minus 119861

3


minus1198961119909 + 119860 minus 119861

3


(12)


119898 = minus (1198961+ 21198962) 119909 + 119860 minus 119861

3

+ 119865 cos 51Ω119905 (13)




mpl

itude

2000

1000

0

minus1000

minus2000

minus3000

(ms)0 150 300 450

Am

plitu

de

100

80

60

40

20

0

(Hz)0 40 80 120 160 200 240



5 Conclusions



Competing Interests


Acknowledgments


References
















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










k1

c1

Fd

k2

k2

Δmaxx

m

F(t)

(a)

k1

c1

Fd

k2

k2

Δ1 Δ2x

m

F(t)

(b)

k1

c1

Fd

k2

k2

Δmaxx

m

F(t)

(c)


Disp

lace

men

t(120583

m) 10

0minus10minus20

Time t (s)08 09 1 11 12

Velo

city

(mm

s) 10

0minus10


Pow

ersp

ectr

um

4

2

0

times10minus3




5 kg 1198961= 1times 10

10Nm 1198962=

1times1011NmΩ = 2120587119860 = 4times10




119898 = minus (1198961+ 21198962) 119909 + 119860 minus 119861

3

+ 119865 cosΩ119905 cos 51Ω119905 (11)

1

0

minus1

Time t (s)1 11 12 13 14

020

minus02


5

0

times10minus4


Disp

lace

men

t(120583

m)

Velo

city

(mm

s)

Pow

ersp

ectr

um



119898 + 119891 (119909 ) = 119865 cos 51Ω119905

119891 (119909 )

=

minus (1198961+ 1198962) 119909 + 119860 minus 119861

3


minus1198961119909 + 119860 minus 119861

3


(12)


119898 = minus (1198961+ 21198962) 119909 + 119860 minus 119861

3

+ 119865 cos 51Ω119905 (13)




mpl

itude

2000

1000

0

minus1000

minus2000

minus3000

(ms)0 150 300 450

Am

plitu

de

100

80

60

40

20

0

(Hz)0 40 80 120 160 200 240



5 Conclusions



Competing Interests


Acknowledgments


References
















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










mpl

itude

2000

1000

0

minus1000

minus2000

minus3000

(ms)0 150 300 450

Am

plitu

de

100

80

60

40

20

0

(Hz)0 40 80 120 160 200 240



5 Conclusions



Competing Interests


Acknowledgments


References
















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









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