optimum design methodology for gear design

7/28/2019 optimum design methodology for gear design

1/12

Mohsen Nakhaeinejade-mail: [email protected]

Michael D. Bryante-mail: [email protected]

Department of Mechanical Engineering,University of Texas at Austin,

Austin, TX 78712

Dynamic Modeling of RollingElement Bearings With SurfaceContact Defects Using BondGraphs Multibody dynamics of healthy and faulty rolling element bearings were modeled usingvector bond graphs. A 33 degree of freedom (DOF) model was constructed for a bearingwith nine balls and two rings (11 elements). The developed model can be extended to arolling element bearing with n elements and 3 n DOF in planar and 6 n DOF inthree dimensional motions. The model incorporates the gyroscopic and centrifugal effects, contact elastic deections and forces, contact slip, contact separations, and local-ized faults. Dents and pits on inner race and outer race and balls were modeled throughsurface prole changes. Bearing load zones under various radial loads and clearanceswere simulated. The effects of type, size, and shape of faults on the vibration response inrolling element bearings and dynamics of contacts in the presence of localized faults werestudied. Experiments with healthy and faulty bearings were conducted to validate themodel. The proposed model clearly mimics healthy and faulty rolling element bearings. DOI: 10.1115/1.4003088

Keywords: rolling element bearings, modeling, multibody dynamics, bond graphs, roll-ing contacts, faults, dents, pits

1 IntroductionUnder high load rates and speeds, bearings are susceptible to

faults and failures. Bearing faults comprise 41% of failures ininduction motors 1 . Early detection and precise isolation of bear-ing faults can decrease maintenance costs and increase machinesafety 2 . Diagnostic techniques for bearings can be either signalbased or model based. Signal-based methods key on specic fea-tures of waveforms or spectra; however, since designs, manufac-ture, process dynamics, and operating conditions vary with ma-

chines and change over time, features used in signal-basedtechniques can vary markedly, making these techniques unreliable 3 . Model-based methods, which gather more information andsort via a model, can overcome disadvantages of signal-basedmethods. Studies on diagnostics of rolling element bearings androtating machinery has shown model-based diagnostic techniquesare superior to signal-based methods for early detection and moreaccurate isolation of faults. However, these techniques are notadequately developed for real applications and require the supportof detailed modeling.

Several modeling software and virtual design tools have beendeveloped 4 . However, at present, there is no universal and de-tailed platform for the whole system. Many available models rep-resent dynamics of rolling element bearings, but a generic modu-lar model, which can be adjusted based on the complexity of thesystem to represent dynamics of both normal and defective bear-ings, is not available yet.

Despite the long history of dynamic modeling of rolling ele-ment bearings, relatively few models consider faults. Most sim-plify models or neglect the details of faults. In 1960, Jones 5introduced bearing element stiffness, damping, constraint forces,and moments. In 1979, Gupta 6 introduced a six degree of free-dom DOF model of a bearing and validated the model experi-

mentally. In the early 1980s, McFadden and Smith 7 incorpo-rated localized defects of bearings using impulse functions.Assuming a rotor-bearing system with a stationary outer ring, Af-shari and Loparo 8 introduced a one dimensional linear timeinvariant state space model. Adams 9 presented an analytical29DOF model of a shaft supported by two rolling element bear-ings using Lagranges approach, assuming only the radial motionof the balls with no contact slip. Harsha 10 developed an ana-lytical model using Lagranges approach, assuming no slip and nofriction between balls and races. Incorporating gyroscopic effectsand shaft bending, El-Saeidy et al. 11 formulated dynamics of rotor-bearing systems using nite elements and Lagranges equa-tions. In 2002, Liew et al. 12 obtained a 5DOF analytical modelof angular contact ball bearing, including bearing centrifugalloads and axial and tilting stiffness effects. In 2003, Sopanen andMikkolas model of a deep-groove ball bearing included localizedand distributed defects, Hertzian contact deformations, and elas-tohydrodynamic EHD effects 13,14 , but neglected centrifugaleffects and slip between components. Analytical models by Jangand Jeong 15 and a 5DOF model by Changqing and Qingyu 16considered the waviness faults neglecting contact slip and contactseparations. Patil et al. 17 studied the dynamics of defects inbearings using a planar model neglecting slip, centrifugal force,and gyroscopic moments. Wensing 18 developed a nite elementmodel of rolling element bearings. Sassi and Badri

19 developed

a simple 3DOF model of ball bearings and introduced a numericalmodel to predict damaged bearing vibrations. Karkkainen et al. 20 studied dynamics of rotor systems with misaligned retainerbearings. Ashtekar et al. 21 used a dry contact elastic model tomodify the Hertzian relationships for contacts with small surfacefaults. Also, they used superposition to include the effects of dentsor bumps smaller than the Hertzian ellipse on bearing dynamics 22 , assuming the contact stiffness K is unaffected by the pres-ence of faults.

Little has been done in bond graph modeling of rotating sys-tems 2,23,24 , including rolling element bearings, multibody dy-namics, tribological aspects, and localized defects. Bond graphs

Contributed by the Tribology Division of ASME for publication in the J OURNAL OFTRIBOLOGY . Manuscript received June 24, 2010; nal manuscript received November17, 2010; published online December 15, 2010. Assoc. Editor: Ilya I. Kudish.

Journal of Tribology JANUARY 2011, Vol. 133 / 011102-1Copyright 2011 by ASME

wnloaded From: http://tribology.asmedigitalcollection.asme.org/ on 05/12/2013 Terms of Use: http://asme.org/terms


2/12

are energy-based modeling techniques, with modularity that per-mits system growth. Different submodels can be integrated into alarge system. Despite the apparent simplicity of rolling elementbearings, the presence of faults complicates dynamics. Most ex-isting models cannot fully explain the dynamic behavior of bear-ings with faults since simplied models do not fully contain de-tails of the fault. Here, the tribological aspects of rolling elementbearings, nonlinear contacts, and surface defects are modeled indetail separately, and integrated into a unied bond graph model.

This article presents a detailed model of rolling element bear-ings REBs with a direct physical correspondence between pa-

rameters of the model and physical system components and faults.The goal is a generic model of rolling element bearings, whichcan be simple or detailed, based on the complexity of the system,needed to support model-based diagnostics. First, multibody dy-namics of bearings, including elastic deections with nonlinearHertzian contacts, tractions with contact slips, and surface separa-tions, are modeled in a generic bond graph. Localized faults, suchas dents and pits, are modeled as surface prole changes andincorporated into the bond graph. The model is validated withexperiments, and then the effects of parameters, such as geometryof faults, clearance, and radial loads on vibration responses, areinvestigated. Finally, dynamics of contacts with faults are studiedin detail.

2 Modeling

A deep-groove ball bearing consisting of balls, inner race, andouter race was modeled as a multibody system using vector bondgraphs. Nonlinear contact stiffness, contact damping effects, con-tact slip, contact separations, bearing clearance, traction betweenelements, rotational friction torques, and localized faults were in-cluded. The axial motion, lubrication effects, structural exibilityexcept at contacts, and dynamics of the cage were neglected. A33DOF bond graph model was constructed for a bearing with nineballs and two rings 11 bodies . The model can be extended to anyn-body rolling element bearing with 3 n -DOF in planar mo-tions and 6 n -DOF in 3D motions.

2.1 Geometry Considerations. Assuming planar motions,each element of the rolling element bearing has 3DOFs. Figure 1shows a three-element rolling system representing inner race a ,ball b , and outer race c in contact, with global xed coordinatesystem XYZ and rotating body coordinates x i yi zi dened for eachmoving element, and coincident with the principal axes of movingelements.

Misalignments and eccentricity can be included by relocation of inner race and outer race centers. Position and orientation of eachelement can be described in the global coordinate system by a

position vector R i = X i Y i Z i T and i, respectively. For eachcontact between two elements i and j, a moving contact frame x ij yij zij with origin at the contact is dened with orientation ijdescribed in the global coordinate as

ij = arctanY i Y jX i X j

1

Any vector X dened in the global frame can be transformed to avector X

in the moving frame via

X = TX 2

where transformation matrix T for the body coordinate systems is

T i =cos i sin i 0 sin i cos i 00 0 1 3 and for the contact coordinate systems is

T ij =cos ij sin ij 0 sin ij cos ij 00 0 1 4 2.2 Contact Modeling

2.2.1 Contact Stiffness and Damping . According to Hertziantheory 25 , the load-deection relation in the x direction for thecontact between the two bodies with different radii of curvature ina pair of principal planes, as shown in Fig. 2, is

F = K 3 / 2 5

where the contact force F is in N, the deformation is in mm, andthe contact stiffness K is in N / mm 2 / 3.

The contact stiffness K is dened as

K =2 E 2

3 3 / 2 1 / 2 1 2

6

where E is the modulus of elasticity in MPa, is the curvature in1/mm dened as = 1 / r , and the curvature sum is

= 1r a 1+ 1

r a 2+ 1

r b1+ 1

r b2 7

Subscripts a and b refer to bodies a and b and subscripts 1 and 2refer to the bodies principal planes, in which curvatures are de-ned for each body. In rolling element bearings, includes maincurvatures of ball and raceway, as well as side groove curvatures,

Y

X

Zo

yaxab yb x

b

yc xc

c

a b

a

c

ab

yab

xa

b

A

B

C

Fig. 1 Geometry and coordinates of a three-body rolling ele-ment system

Body b

F

1br 2br

F

1ar 2a

r

xab

z ab yabBody a

Plane 1

Plane 1

P l a n e

2

P l a n e

2

Fig. 2 Geometry of two curved bodies a and b in contactunder the normal force F

011102-2 / Vol. 133, JANUARY 2011 Transactions of the ASME



3/12

considering the sign convention for convex and concave surfaces.In Eq. 6 , is a dimensionless deection factor graphed byHarris 25 as a function of curvature difference F dened as

F =

1

r a 1

1

r a 2+

1

r b1

1

r b2

8

Note that the order of planes 1 and 2 should be selected in such away that F is positive. For computational applications, the cur-vature function F

is formulated as

26

= 327.61 + 1883.34 F 3798.11 F 2 + 3269.62 F

3

1026.96 F 4

Since contact damping in ball bearings is relatively small, a con-stant damping value, based on the average load, can be used in themodel. Kramer 27 estimated bearing damping by frequencyanalysis of the equivalent linear spring-mass-damper system as

b = k 0.25 2.5 102 N s / mm 9

where k is the equivalent linear stiffness of the bearing in N/mm.Here, b is selected as b = 10 2k N s/mm . Therefore, the normalcontact force including the damping force along x ab is

F x = K 3 / 2 + b 10

where is the rst derivative of the contact deformation in mm/s.Note that when faults are smaller than the Hertzian contact ellipse,the Hertzian contact relationships should be modied for moreaccurate results 21,22 . Here, it is assumed that Hertzian contactrelationship is valid for contacts with faults.

2.2.2 Traction Model . Based on the rolling or slip regions, thetraction model follows different behaviors with different coef-cients of friction. Kragelsky et al. 28 described the dry contactcoefcient of the friction as a function of sliding speed as

= C 1 + C 2s2 e s

2sign s + C 2s

where C 1, C 2, and are constants obtained experimentally basedon the materials and surfaces. Here the traction force is modeledusing traction-slip behavior in elastohydrodynamic contacts 29 .Traction forces in directions yab and zab are functions of the nor-mal force and the friction coefcient as

F y = F x 11

The friction coefcient can be calculated based on the contactlubrication modes, determined by the oil lm parameter a ratioof oil lm thickness and contact surface roughness 30 as

=

bd , 0.01

bd hd 0.01 1.6 6

1.5 6 + hd , 0.01 1.5

hd

, 1.5

12 Friction coefcient hd under hydrodynamic lubrication mode is afunction of EHD parameters and temperature. Under boundarylubrication mode, friction coefcient bd is a function of slide-rollratio s as

bd = 0.1 + 22.28 s e 181.46 s + 0.1 13

Therefore, assuming planar motion F z= 0 , the contact force vec-tor, consisting of stiffness, damping, and traction forces as a func-tion of contact , , and slip s, is given by

F =F x F yF z=K 3 /2 + b

bd K 3/ 2 + b

0= f , , s 14

2.2.3 Bearing Friction Torques . In rolling element bearings,contact deformation, viscous lubrication, and seal rubbing gener-ate rotational resistive torque,

T f = T load + T lub + T seal 15

where T load is the torque caused by bearing radial loads and con-tact deformations. For a ball bearing under radial load, T load hasbeen formulated 25 as

T load = 0.0003F sC s

0.55

F d p 16

where C s is the bearing static load rating dened by the bearingsmanufacturer, F s is the static equivalent load dened as

F s =0.6 F r + 0.7 F a F r F a

F r F r F a 17

and F depends on the magnitude and direction of the appliedloads, which equals to F r in radial ball bearings under pure radialloads.

The lubricant friction torque, T lub , is dened by empirical equa-

tions 30 as

T lub =10 7 f o o f r

2/ 3d p3, o f r 2000

16 106 f od p3, o f r 2000

18

where f r is the rotational speed in rpm, o is the kinematic viscos-ity of the lubricant in cSt, and coefcient f o, which depends on thebearing type and lubrication, equals unity for deep-groove ballbearings lubricated with oil mist and is given in Ref. 25 for othertypes of bearings with different lubrications. The rubbing sealfriction torque, T seal , which depends on the geometry and designof the seal, is assumed constant.

2.3 Fault Modeling. Bearings fail by numerous fault modes:corrosion, wear, plastic deformation, fatigue, lubrication failure,electrical damage, fracture, and incorrect design, among others.Most common spalling fatigue leaves pits on races or rollers, as inFig. 3 a , because of periodic contact stress 31 . Most models forrolling element bearing faults have introduced mathematical im-pulse functions based on fault frequencies. Here, a kinematics-based fault modeling is introduced, where the faults are dened inthe model based on the surface prole change.

For each fault, the width w, depth h, and location of the fault f in the body coordinate system are dened, as shown in Fig. 3 c .Parameters T f , w, h, r f , and f in the model dene the type, size,shape, and location of localized faults, as shown in the block diagram of Fig. 4. T f denes the type of the fault, w and h repre-sent size, r f determines the shape of the fault, and the location of the fault on each element is determined by f in the model.

As a fault passes through the contact, the prole changes inducedeections that result in dynamic interactions between elements,which generate force impulses that induce fault vibrations. Mul-tiple faults of different sizes and shapes, at any location of thebearing, can be modeled via the proposed scheme.

f

h

(b )(a ) (c )

Fig. 3 a Subsurface-initiated fatigue spall 31 , b outer racefault model, and c cross section of an outer race fault

Journal of Tribology JANUARY 2011, Vol. 133 / 011102-3



4/12

3 Bond Graph ModelingBond graphs, introduced in the 1950s by Paynter 32 , map

energy storage, energy dissipation, and power ow between com-ponents in a system through energy bonds and elements 33 .Associated with each bond are power conjugates, effort, and ow,the product of which is the instantaneous power owing to orfrom physical elements. Efforts and ows in mechanical systemsare forces and velocities. Energy dissipation is modeled by theresistive element R , energy storage devices such as springs andinertia are modeled by elements C and I , sources of efforts andows are represented by Se and Sf , and power transformations aremodeled by transformers TF or gyrators GY . Power bonds join at0 junctions summing ows to zero with equal efforts or at 1 junctions summing efforts to zero with equal ows, which imple-ments Newtons second law. With constitutive laws for each ele-ment such as Eqs. 5 and 10 , state equations are extracted frombond graphs. Bond graphs have many advantages. Multiphysicsdynamic systems, such as electrical, mechanical, magnetic, uid,chemical, and thermodynamic systems, can be modeled andlinked together. Also, nite element model FEM can be embed-ded in model 11 . Furthermore, the modularity characteristic of bond graphs permits system growth.

3.1 Multibody Dynamics With Bond Graphs. The dynam-ics of a 6DOF rigid body system with the global coordinate XYZand the body coordinate xyz, as shown in Fig. 5 a , can be repre-sented with NewtonEulers equations,

Mx = 1

n

F i 19

J

=

+ 1

n

r

i F

i J

20

expressed in the global and the body coordinate systems,respectively.

The body is subjected to external forces F i and moment , alsoexpressed in the global and the body coordinates, respectively.The second term on the right side of Eulers equation 20 repre-sents moments due to external forces. The last term is the Eulerian

junction term representing gyroscopic forces.A cross product C B can be represented in matrix form using

the cross product operator A as

C B =C x C yC z B x B yB z=0 C z C yC z 0 C x C y C x 0

B x B yB y= C B

21

Applying this to the second term, Eulers equation becomes

J = + 1

n

r i F i J 22

External forces in the global coordinates can be transformed to thebody coordinates as

F i = T GBF i 23 where T GB is the transformation matrix obtained from Eulerangles or direction cosines 34 .

Rewriting the NewtonEuler equations 19 and 22 ,

Mx = 1

n

F i

J = + 1

n

A iF i J 24

where

A i = r i T GB 25 In terms of vector bond graphs, as shown in Figs. 5 b and 5 c ,Newtons and Eulers equations can be represented as 1 junctionsthat embed conservation of linear momentum and conservation of angular momentum. Transformers TF with modulus of A i trans-form coordinates between Eulers and Newtons equations andalso convert external forces to moments acting on the center of mass. External moments i are represented by sources of effortsSe . The gyroscopic term J is incorporated as a modulated R Eelement. The overall bond graph structure of the system in Fig.5 a with dynamic equations 24 is the diamond-shaped bondgraph structure of Figs. 5 b and 5 c . A multibody system con-sisting of n rigid bodies can be modeled by connecting ndiamond-shape bond graph structures together through constraintmodels.

3.2 Bond Graph Model of Rolling Element Bearings. Adeep-groove ball bearing consisting of nine balls, inner race, and

T f =1 IRF2 BF3 ORF

wh[mm 2]

r f = w/h0 < f


5/12

outer race is modeled as a multibody system using vector bondgraphs. The outer race is xed in a housing characterized by stiff-ness and damping in the vertical and horizontal directions. Theinner race moves and rotates under external forces and torques Fig. 6 a . Weights are applied as external loads on each body.Each element is modeled using the diamond-shaped vector bondgraph structure of Fig. 5. Contacts are modeled as nonlinear Celements representing nonlinear stiffness, damping, and tractionforces inherent in Eq. 14 .

The bond graph model of a bearing with races and one ball, inFig. 6 d , has three of the diamond-shape bond graph structureswith two contact models in between. Each diamond structure ap-plies NewtonEulers equations 24 to inner race, balls, and outerrace via the 1 junction. Each has rotational inertia J , translationalinertia M , transformer TF with modulus matrix A transformingbody coordinate system to the global coordinate system Eqs. 23 25 , and Eulerian junction elements R E representing gyro-scopic moments. External torque and external force vector F areapplied to the 1 junctions through Se elements. Bearing rotationalfriction torques are modeled as resistive elements R in the innerrace model, with the constitutive law described by Eqs. 15 18 .Housing stiffness and damping effects are modeled through C andR elements with structural stiffness and damping matrices K v ,hand R v ,h. Each contact model consists of a transformer and a C celement. Transformers TF transform the contact coordinate sys-tems to the global coordinate system, as expressed in Eq. 2 .Nonlinear elements C c with the constitutive law of Eq. 14 rep-resent contact and traction forces. Here, the bond graph model of the bearing with nine balls and two races consists of 11 diamond-shape models combined through 18 contact models. Faults aremodeled by surface prole changes, as shown in Fig. 3 c . Duringsimulations, the geometry of elements changes based on locationsand growth of faults. Fault impulses and forces are generated viakinematics and dynamics.

4 Simulations and Experiment SetupA bond graph model of a deep-groove ball bearing with speci-

cations in Table 1 was built and numerically simulated with thesoftware 20-SIM 4.0 . The RungeKutta Dormand Prince integrationmethod with variable step size was used in simulations. Initialvelocities and accelerations were zero. A radial load was appliedto the inner race vertically downward. Running torque and exter-nal rotational loads were applied to the inner race.

The effects of the clearance and radial loads on the load zonedistribution were studied. Three cases of clearance and two casesof radial loads were simulated. Healthy and faulty bearings withlocalized faults on inner race, outer race, and balls were simu-lated. For each type of fault, 16 cases with different fault sizes Table 2 were simulated to study the effects of fault type, size,and shape on the vibration response. For each case, the verticalvibration signal of the housing in Y direction of Fig. 6 c will bepresented. Also, normal contact force and displacements in thepresence of ball faults are presented.

For a bearing with stationary outer race and rotating inner raceunder downward vertical load F , the loading zone is stationary. Inthe load zone, reaction forces transfer load F between housingsand shaft. Based on the angular position of the outer race fault ORF , the vibration response can change. However, in the case of inner race fault IRF and ball fault BF , fault locations do not

Table 1 Bearing specications and simulation parameters

Parameter Value

Bearing model MB ER-16KNumber of balls 9Material density g / cm 3 7.75Elastic modulus MPa 210Ball/inner race/outer race diameter mm 7.94/31.38/47.26Pitch diameter mm 39.32Race groove radius mm 4.1Poissons ratio 0.25Radial load on inner race N 110Inner race rotational speed Hz 35

Housing vertical/horizontal stiffness N/mm 4 107

Table 2 Studied faults

faults size: ][mmhw 2

5hw = 2hw = hw = 0.5hw =

Faul t Level I 2 .250 .45 1. 00 0.50 0. 71 0.71 0 .2 20 .44Fau lt Level I I 3 .150.63 1 .420.71 1 .001.00 0 .320.64Fau lt Level I II 6 .301.26 2 .821.41 1 .411.41 0 .450.90

C

SeF

CcR E

R

Se

TF TF : T b o

1

00

1

0

1

T F

Sf 0

0 0

1

1

I: Jo

0

1

inner race ba lls outer racecontact i/b contact b /o

:K v,h

R

CcE ulers E qs .

in rotat ingf ra m e

N e wtonsE qs . in

g loba l f ra m e

: A o

I:M o

:R v,h

R ER

E

I:M b

I: J b T F : A

b

T F

: A

b : T i b

I:M i

I: J iT F

: A

i

T f

i &

b & o

&

(d)

Rh

K h

K v Rv

F

F

F t_ib

F k_ib

F b _ib

F t_ib

F k_oh

F b _oh

F t_ b o F k_ b o

F b _ b o F t_ b o

F k_ b o

F b _ b o

F k_ov F b _ov

f T

y

x

i

y

x

b

y

x

o

inner race ba ll outer race

(a) (b) (c)

Fig. 6 Rolling element bearing with single roller: a bearing with housing, b system with external loads and constraints, c free body diagram, and d bondgraph models of each element and contacts




6/12

affect the response since faults are not stationary with respect tothe load zone. Here, the model is simulated with ORF locateddown at the load zone

Experiments were conducted on the test rig shown in Fig. 7with three types of rolling element bearing faults IRF, BF, andORF. The test rig consists of a rotating shaft supported by tworolling element bearings Table 1 . A three-phase induction motorrotates the shaft through a beam coupling. Bearing loaders applystatic radial loads to bearings, and a magnetic loader applies rota-tional loads to the shaft through a gearbox and belts. Accelerom-eters measure the vertical and horizontal vibrations of the out-board bearing housing. Eddy current proximity probes measurethe vertical and horizontal displacements of the shaft close to thebearing. A tachometer and an encoder measure shaft speed. Also,voltages and currents of the induction motor can be measured.Localized faults were created on inner race, outer race, and ballsby spot grindings. A vertical load was applied to the inner race viathe bearing loader. Since excessive vibrations can invalidate re-sults, waterfall plots and hammer tests found machine criticalspeeds and natural frequencies, which were avoided during tests.At the shaft speed of 35 Hz, bearing vibration signals and rotorspeed were recorded at a sampling rate of 76,000 Hz and com-puter analyzed.

5 Results and Discussions

5.1 Load Zone Tests. Clearance in bearings changes the pro-le of contact tractions in the load zone. Defects within the loadzone generate much bigger impulses than at other locations in abearing. This behavior is observed as modulations of the vibrationresponses for IRF and BF, wherein faults are not stationary. Outerrace faults, when located in the load zone, create a much largerresponse than at other locations. Vibration responses in faulty

bearings can change signicantly with clearance of the bearing.The model should be detailed enough to include this behavior.Figure 8 shows the simulation results for a rolling element bearingwith 0.08 mm clearance, 0 clearance, and 0.02 mm preloads. Foreach case, the load zone proles for radial load conditions of 100N and 60 N were calculated.

For a given clearance, the load zone widens and lengthens withincreased radial load. For zero clearance, the load zone covers half the ring regardless of the radial load and expands with load. In thecase of preloads usually demanded by the design of the bearing,the load zone extends completely around the ring with zero radialloads and shifts to one side when radial load is applied. The simu-lated results of the load zone show the ability of the detailedmodel to incorporate load zone effects in vibration results.

5.2 Model Verication. To validate the model, experimentswere conducted and vibration signals of the bearing in the verticaldirection, in time and frequency domains, were measured andcompared with simulations. Results are presented in Fig. 9 forIRFs, in Fig. 10 for BFs, and in Fig. 11 for ORFs. For each fault,the upper plots are simulations and the bottom plots are experi-mental results. The left plots present vibration waveforms with the

magnied portions in middle plots. The right plots are the powerspectra of these vibration signals. To isolate fault signals, bearingfundamental frequencies, which depend on the bearing geometryand rotor speed, are as follows:

inner race fault frequency

f IRF =n

2 f r 1 +

d bd p

cos 26

acce lero meters

stud ied bear ing

acce lero metersba lance d is ks

bear ing loader

Data acqu is ition board

vibrat ions & speed

Signa ls are ana lyzed in Vibra Quest andMATLAB

Fig. 7 Test bed for bearing experiments

Preloa d : 0.02 mm

(c )

0 clearance

(b )

clearance: 0.08 mm

(a )

Ra d ial Loa d : 100 N

Ra dial Loa d : 60 NRa dial Loa d : 100 N

Ra d ial Loa d : 60 N

Ra d ial Loa d : 100 NRa dial Loa d : 60 N

Fig. 8 Simulation results: effects of clearance and radial loads on loaddistribution in rolling element bearings. a 0.08 mm clearance, b 0clearance, and c 0.02 mm preload.




7/12

outer race fault frequency

f ORF =n2

f r 1 d bd p

cos 27

ball fault frequency

f BF =d pd b

f r 1 d b

2

d p2 cos

2 28

Si m ul a tion

Exp er imen t

Time sig na l Po wer sp ec tr u m o f vibra tion

Vibra tion re sp on s e o f th e bear in g with I nner Race Fa ults (I RF)

Time sig na l (ma g n ifie d)

(a ) (b ) (c )

(d) (e ) (f)

1 /f IRF

f IRF

Fault Width1 /f r

Fig. 9 Simulation versus experiment: Vertical vibration response of the bearing with inner race fault fault size: w h =3 1 mm 2. Fault is located at the load zone down . a and d Time response, b and e time response magni-ed , and c and f vibration power spectrum shaft speed: 35 Hz .

Fig. 10 Simulation versus experiment: vibration response of bearing with ball faults fault size: w h =1.0 0.8 mm 2.a and d Time response, b and e time response magnied , and c and f vibration power spectrum shaftspeed: 35 Hz .




8/12

case frequency

f cage = f r 2

1 d bd p

cos 29

Here n is the number of balls, f r is the rotor frequency, d b is theball diameter, d p is the bearing pitch diameter, and is the contactangle, which is zero for radial bearings. Given f r =35 Hz, for thebearing of Table 1, the bearing frequencies are

f IRF = 189.28 Hz, f ORF = 125.72 Hz, f BF

= 166.32 Hz, f cage = 13.96 Hz

Figure 9 a contains simulated vibration signals of accelerationversus time for IRF. Here, fault impact signals are modulated withrotor frequency f r because the inner race fault, xed on the innerrace, passes through the load zone at a rate of f r as the inner racerotates. Similar behavior with the same frequency is observed inthe corresponding experimental signals shown in Fig. 9 d . Themagnied portion of vibration signal plotted in Fig. 9 b showsthe fundamental fault frequencies of the bearing, which is con-rmed by the experiments in Fig. 9 e . As the IRF passes throughthe contact, two impact responses appear in the vibration signal.The fault model of Fig. 3 suggests that the leading and trailingedges of the surface prole dent would cause impacts with theballs in the load zone. The distance between these two impactsignals contains information on the size faults. Figures 9 c and9 f present power spectrum of the vibration signals. Dependingon the frequency range, harmonics of the fault frequencies indi-cated by dashed lines might appear in the power spectrum.

Figure 10 shows the vibration signals of a bearing with a singleBF. Fault impact signals spaced by the fault frequency timing 1 / f BF are clearly observed in both simulation and experimentaldata. Also, simulated signals are modulated by the cage frequency f cage . However, in real bearings, three dimensional rotations of balls can prevent engaging of ball faults with races, causing ran-domness in measured signals. This phenomenon generates random

modulations of vibration signals see Fig. 10 d , which can beused to separate ball faults from other types of bearing faults.Another vibration signal feature that agrees with experiments isshape of the impact signals, as shown in Figs. 10 b and 10 e . Asshown in the magnied plots, vibration signals due to BF consistof a single impact response signal for each fault, compared withthe IRF and ORF that generate two impacts. The power spectrumof simulated signals Fig. 10 c reveals harmonics of the faultfrequency, which are not very clear in measured signals becauseof three dimensional motions of balls.

Simulated vibration signals of the bearing with ORF are com-pared with experimental results in Fig. 11. Simulated vibrationsignals shown in Fig. 11 a reveal a regular pattern of fault im-pacts spaced at the fault frequency timing, which appears in theexperimental data of Fig. 11 d . Two impact signals generated inORF by the two edges of fault are observed in simulations. Simi-lar to the IRF case, the ORF impact signals contain information onthe fault size, which indicate the fault severity. In ORF, since thefault is stationary not moving , the vibration signal has a regularpattern, and fault frequency harmonics can be clearly observed inthe power spectrum.

Comparing simulation signals and measured signals in Figs.911, the natural frequencies of the model and the machine seem

to be slightly different. To investigate this issue, critical speedsand natural frequencies of the machine were obtained usingstartup/coast down and hammer tests. First and second criticalspeeds were about 70 Hz and 85 Hz. Two main natural frequen-cies obtained from hammer tests were about 480 Hz and 2100 Hz.The most dominant natural frequencies observed in the measure-ment signals during bearing experiments are about 2000 Hz, veryclose to the main natural frequency of the machine. The structureof the machine, including rotor and other components, is not in-cluded in the model. Therefore, there are natural frequencies of the real machine in measured signals that do not appear in simu-lations. These unmodeled frequencies likely from the machine

Fig. 11 Simulation versus experiment: vibration response of bearing with outer race faults fault size: w h =2.7 1.0 mm 2. a and d Time response, b and e time response magnied , and c and f vibration powerspectrum shaft speed: 35 Hz




9/12

frame, bearing housing, rotating shaft, couplings, etc., can be eas-ily considered in the model by appended submodels of machinecomponents to the bond graph model.

Simulations suggest that information on type and size of faultsexist in the vibration signals from the bearing. However, the realsystem has numerous natural frequencies not in the bearing modelthat might mask the original spikes from faults. Tools such as highfrequency lters might remove masking signals and retrieve faultinformation in the real system. Also, factors such as imbalanceand misalignments can alter the regular patterns of the faults in areal machine. Interpreting the pattern of damage in real faultsneeds more than one snapshot of the signal and some statisticalanalysis to determine the condence level of the results. This needis more critical in the case of ball faults and inner race faults, inwhich faults are moving.

The model includes centrifugal forces on balls. To illustrate thisfacet, a roller bearing with zero clearance under radial load wassimulated for different shaft speeds. Bearing radial deections al-most overlay data of Harris 35 in Fig. 12. The quadratic relationbetween radial deections and shaft speed in Fig. 12 representsthe centrifugal effects in the bearing. The centrifugal force in-creases quadratically with the orbital speed of balls, which in-

creases the contact force and bearing radial deections. Centrifu-gal effects mainly appear at high speeds and are not signicant atlow speeds.

5.3 Fault Severity Test. A model-based diagnostic employs amodel to detect and assess faults, by estimating parameters in themodel associated with faults. Therefore, a parametric study isneeded. Here the effects of type IRF, BF, and ORF , size w h , and shape of the fault on vibration responses are studied.Referring to Table 2, faults in three levels corresponding to dif-ferent sizes of w h are simulated and shown in Figs. 1315 for

IRF, BF, and ORF, respectively. In time waveform and frequencysignals, fundamental fault frequencies f IRF , F BF , F ORF for alltypes of faults, cage frequency f cage for BF, and rotor frequency f r for IRF are observed. More severe faults with bigger defect sizesgenerate impact with higher amplitudes, for all types of faults. InIRF and ORF cases, vibration signals have a fault-element impactwaveform characterized by two distinct peaks, with distance be-tween these peaks proportional to the fault width, as shown inFigs. 13 b , 13 e , and 13 h and Figs. 15 b , 15 e , and 15 h .However, in the case of BF as shown in Fig. 14, the fault size onlyaffects impact amplitudes rather than the distance between twoimpact peaks.

To investigate the effect of fault shapes on the vibration re-sponse, faults with different size ratios w / h =5 ,2 ,1 were simu-lated and results are shown in Fig. 16. For IRF and ORF, increas-ing the fault size ratio w / h enlarges the distance between the twoimpact peaks but does not change the impact amplitude much. ForBF, the vibration response is relatively insensitive to w / h.

To understand dynamics of the contact with faults, detailedsimulations of contacts in the presence of BFs are shown in Fig.17. The contact is within the load zone, but once the fault passesthrough the contact, nonlinear dynamic interactions can inducetransient surface separations. Figures 17 a and 17 b show con-tact displacements solid line and forces dashed line for thecontacts between inner race balls and outer race balls, for BFbetween inner race and ball. Negative force and displacementsdenote compression. Positive displacements denote surface sepa-rations, leading to zero forces. As shown in Fig. 17 a , the innerrace does not touch the bottom surface of the fault, and ball-raceseparation occurs until the inner race surface hits the trailing cor-ner of the fault, causing the main impact. During this time, asshown in Fig. 17 b at about 45.1 ms, the contact between ball andouter race unloads, which momentarily separate the surfaces. Justbefore 45.2 ms, a fault impact shock reloads the contact. These

Fig. 12 Radial deection versus speed for a roller bearing.The centrifugal force on rollers increases contact loads andradial deections when shaft speed increases. Simulations arecompared with the results of Harris 35 .

Fig. 13 Vibration response of a bearing with IRF for different fault severities: w h =1 0.5 level I , 1.42 0.71 level II , 2.82 1.41 level I mm 2, and f r =35 Hz




10/12

Fig. 14 Vibration response of a bearing with BFs for different fault severities: w h =1 0.5 level I , 1.42 0.71 level II , 2.82 1.41 level I mm 2, and f r =35 Hz

Fig. 15 Vibration response of a bearing with ORF for different fault severities: w h =1 0.5 level I ,1.42 0.71 level II , 2.82 1.41 level I mm 2, and f r =35 Hz

N o r m a l

i z e d

A c c e

l e r a t i o n

Fig. 16 Vibration responses of faulty bearings IRF, BF, and ORF withdifferent fault shapes, f r =35 Hz




11/12

nonlinear contact dynamics generate fault impulses, which trans-ferred through elements, and generate vibrations, as observed inFig. 17 c .

6 Summary and ConclusionsA detailed model of rolling element bearings developed in vec-

tor bond graphs incorporated multibody dynamics of elements,centrifugal effects, dynamics of contacts, and surface defects.NewtonEulers equations for each element were encoded intobond graphs, with dynamics of contacts, traction forces, and rota-tional frictions formulated as constitutive laws of elements. Akinematics-based fault model was introduced. Tribological faultswere modeled as surface prole changes, which generate impulsesvia dynamic interactions of faults and bearing elements. Faultparameters dene type, size, shape, and locations of faults. Simu-lations for different clearances and radial loads show ability of themodel to incorporate load zone effects in vibration signals. Ex-periments for healthy and faulty rolling element bearings vali-dated the model. A parametric study investigated the effects of type, size, and shape of faults on vibration responses. Dynamicsof contacts under faults demonstrated how fault impulses arephysically generated.

The modular and generic rolling element bearing bond graphmodel represent complex dynamics of both normal and defectivebearings, for rolling element bearings with different geometry and

specications. With physical and kinematic parameters assignedto faults, the model can predict bearing response for fault condi-tions, and thus be used in model-based diagnostics of rolling ele-ment bearings, for information processing necessary for predictivemaintenance of machinery.

NomenclatureC S bearing static load rating E modulus of elasticityF i external force vector acting on the bodyF S static equivalent load

F x , F y , F r , F a forces in x , y, radial, and axial directions,respectively

F curvature differenceJ body rotational inertia matrix

M body mass matrixT coordinate transformation matrix

T GB transformation matrix from the global to thebody coordinate

T f rotational resistive torqueT f fault type indicators 1: IRF, 2: BF, 3:ORF

T load , T lub , T seal resistive torques due to contact deformations,lubrications, and rubbing seal

XYZ global xed coordinateb contact damping factor

d b balls diameterd p bearing pitch diameter

f r bearing rotational speed in rpmr f fault size ratio

v 0 kinematic viscosity of the lubricantK contact stiffnessk equivalent linear stiffness of the bearingn number of balls in the rolling element bearingr radius of curvaturer position vector with respect to the body

coordinates contact slide-roll ratiox body position vector with respect to the global

coordinate x i yi zi moving coordinate associated with the body i

x ij yij zij moving contact coordinate between bodies iand j

v Poissons ratiow , h width and depth of localized faults

oil lm parameter which is a function of oillm thickness

ij orientation of the contact coordinate describedin the global coordinate

f angular position of the fault dened in bodycoordinate

dimensionless deection factor body angular position vector i orientation of the body coordinate described in

the global coordinate friction coefcient

bd , hd under boundary lubrication and hydrody-namic lubrication modes

curvature external torque vector acting on the body bearing contact angle

bar accent assigned to vectors described in the

body coordinate system

References 1 Raison, B., Rostaing, G., Butscher, O., and Maroni, C. S., 2002, Investiga-

tions of Algorithms for Bearing Fault Detection in Induction Drives,IECON02, pp. 16961701.

2 Nakhaeinejad, M., and Bryant, M. D., 2010, Multibody Dynamics of RollingElement Bearings With Faults, Proceedings of the ASME/STLE InternationalJoint Tribology Conference 2009, pp. 249251.

3 Isermann, R., and Balle, P., 1997, Trends in the Application of Model-BasedFault Detection and Diagnosis of Technical Processes, Control Eng. Pract.,5 5 , pp. 709719.

4 Purdue University, 2003, Virtual Machine Design WorkshopDening theRequirements for the Next Generation of Analytical Software for the Design of Rotating Components, Technical Report.

5 Jones, A. B., 1960, A General Theory for Elastically Constrained Ball andRoller Bearing Under Arbitrary Load and Speed Conditions, ASME J. BasicEng., 82 21 , pp. 309320.

6 Gupta, P. K., 1979, Dynamics of Rolling-Element Bearings.1. CylindricalRoller Bearing Analysis, Mech. Eng. Am. Soc. Mech. Eng. , 101 1 , pp.9596.

Fig. 17 Faults in rolling element generate impacts, which can cause contact separations. a

Force and displacement at inner race-ball contact, b force and displacement at ball-outer racecontact, and c outer race vibration ball fault: w h =1.42 0.71 mm 2 engaged at inner race-ball contact, f r =35 Hz .




12/12

7 McFadden, P. D., and Smith, J. D., 1984, Vibration Monitoring of RollingElement Bearings by the High-Frequency Resonance TechniqueA Review,Tribol. Int., 17 1 , pp. 310.

8 Afshari, N., and Loparo, K. A., 1998, A Model-Based Technique for the FaultDetection of Rolling Element Bearings Using Detection Filter Design andSliding Mode Technique, Proceedings of the 37th IEEE Conference on De-cision and Control, Vols. 14, pp. 25932598.

9 Adams, M. L., 2010, Rotating Machinery Vibration: From Analysis toTroubleshooting , 2nd ed., CRC Press, Boca Raton, FL.

10 Harsha, S. P., 2006, Nonlinear Dynamic Analysis of Rolling Element Bear-ings Due to Cage Run-Out and Number of Balls, J. Sound Vib., 289 12 ,pp. 360381.

11 El-Saeidy, F. M. A., and Sticher, F., 2010, Dynamics of a Rigid Rotor Linear/ Nonlinear Bearings System Subject to Rotating Unbalance and Base Excita-

tions, J. Vib. Control, 16 3 , pp. 403438. 12 Liew, A., Feng, N., and Hahn, E. J., 2002, Transient Rotordynamic Modeling

of Rolling Element Bearing Systems, ASME J. Eng. Gas Turbines Power,124 4 , pp. 984991.

13 Sopanen, J., and Mikkola, A., 2003, Dynamic Model of a Deep-Groove BallBearing Including Localized and Distributed Defects. Part 1: Theory, Pro-ceedings of the Institution of Mechanical Engineers Part K-Journal of Multi-Body Dynamics, 217 3 , pp. 201211.

14 Sopanen, J., and Mikkola, A., 2003, Dynamic Model of a Deep-Groove BallBearing Including Localized and Distributed Defects. Part 2: Implementationand Results, Proceedings of the Institution of Mechanical Engineers PartK-Journal of Multi-Body Dynamics, 217 3 , pp. 213223.

15 Jang, G., and Jeong, S. W., 2004, Vibration Analysis of a Rotating SystemDue to the Effect of Ball Bearing Waviness, J. Sound Vib., 269 35 , pp.709726.

16 Changqing, B., and Qingyu, X., 2006, Dynamic Model of Ball Bearings WithInternal Clearance and Waviness, J. Sound Vib., 294 12 , pp. 2348.

17 Patil, M. S., Mathew, J., Rajendrakumar, P. K., and Desai, S., 2010, A The-oretical Model to Predict the Effect of Localized Defect on Vibrations Asso-

ciated With Ball Bearing, Int. J. Mech. Sci., 52 9 , pp. 11931201. 18 Wensing, J. A., 1998, Dynamic Behaviour of Ball Bearings on Vibration TestSpindles, IMAC, Proceedings of the 16th International Modal Analysis Con-ference, pp. 788794.

19 Sassi, S., Badri, B., and Thomas, M., 2007, A Numerical Model to PredictDamaged Bearing Vibrations, J. Vib. Control, 13 11 , pp. 16031628.

20 Karkkainen, A., Helfert, M., Aeschlimann, B., and Mikkola, A., 2008, Dy-

namic Analysis of Rotor System With Misaligned Retainer Bearings, ASMEJ. Tribol., 130 2 , p. 021102.

21 Ashtekar, A., Sadeghi, F., and Stacke, L. E., 2008, A New Approach toModeling Surface Defects in Bearing Dynamics Simulations, ASME J. Tri-bol., 130 4 , pp. 041103.

22 Ashtekar, A., Sadeghi, F., and Stacke, L. E., 2010, Surface Defects Effects onBearing Dynamics, Proc. Inst. Mech. Eng., Part J: J. Eng. Tribol., 224 , pp.2535.

23 Bhattacharyya, K., and Mukherjee, A., 2006, Modeling and Simulation of Centerless Grinding of Ball Bearings, Simul. Model. Pract. Theory, 14 7 ,pp. 971988.

24 Nakhaeinejad, M., Lee, S. H., and Bryant, M. D., 2010, Finite Element BondGraphs of Rotors, Proceeding of Ninth International Conference on BondGraph Modeling & Simulation ICBGM 2010 , Vol. 42, Issue No. 2, pp.164171.

25 Harris, T., and Kotzalas, M., 2007, Essential Concepts of Bearing Technology ,Taylor & Francis, Boca Raton, FL.

26 Aktrk, N., Uneeb, M., and Gohar, R., 1997, The Effects of Number of Ballsand Preload on Vibrations Associated With Ball Bearings, ASME J. Tribol.,119 4 , pp. 747753.

27 Kramer, E., 1993, Dynamics of Rotors and Foundations , Springer, Berlin. 28 Kragselsky, V., Dobychin, N., and Kombalov, S., 1982, Friction and Wear:

Calculated Methods , Pergamon, Elmsford, NY. 29 Gupta, P., 1984, Advanced Dynamic of Rolling Elements , Springer-Verlag,

New York. 30 Palmgren, A., 1959, Ball and Roller Bearing Engineering , SKF Industries,

Inc., Philadelphia. 31 Shin, H., 2004, Bearing Diagnosis Using Embedded Modeling Method and

System Identication Technique, Ph.D. thesis, Rensselaer Polytechnic Insti-tute, Troy, NY.

32 Paynter, H. M., 1961, Analysis and Design of Engineering Systems , MIT,Cambridge, MA.

33 Karnopp, D. C., Margolis, D. L., and Rosenberg, R. C., 2000, System Dynam-ics: Modeling and Simulation of Mechatronic System , Wiley, New York.

34 Sinokrot, T., Nakhaeinejad, M., and Shabana, A. A., 2008, A Velocity Trans-formation Method for the Nonlinear Dynamic Simulation of Railroad VehicleSystems, Nonlinear Dyn., 51 12 , pp. 289307.

35 Harris, T., and Kotzalas, M., 2007, Advanced Concepts of Bearing Technology ,Taylor & Francis, Boca Raton, FL.


optimum design methodology for gear design

Documents