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International Journal of Mechanical Sciences 52 (2010) 1193–1201

Contents lists available at ScienceDirect

International Journal of Mechanical Sciences

0020-74

doi:10.1

n Corr

E-m

(J. Math

(S. Desa

journal homepage: www.elsevier.com/locate/ijmecsci

A theoretical model to predict the effect of localized defect on vibrationsassociated with ball bearing

M.S. Patil a,n, Jose Mathew b, P.K. Rajendrakumar b, Sandeep Desai c

a Department of Mechanical Engineering, Gogte Institute of Technology, Belgaum 590008, Karnataka, Indiab Department of Mechanical Engineering, National Institute of Technology, Calicut 673601, Kerala, Indiac Application Engineering, Automotive Business Unit, SKF India Limited, Pune 411033, Maharashtra, India

a r t i c l e i n f o

Article history:

Received 10 May 2009

Received in revised form

4 March 2010

Accepted 17 May 2010Available online 27 May 2010

Keywords:

Contact deformation

Condition monitoring

Bearing defects

03/$ - see front matter & 2010 Elsevier Ltd. A

016/j.ijmecsci.2010.05.005

esponding author. Tel.: +91 831 2405500; fa

ail addresses: mspatil_git@rediffmail.com (M.

ew), pkrkumar@nitc.ac.in (P.K. Rajendrakum

i).

a b s t r a c t

The paper presents an analytical model for predicting the effect of a localized defect on the ball bearing

vibrations. In the analytical formulation, the contacts between the ball and the races are considered as

non-linear springs. The contact force is calculated using the Hertzian contact deformation theory.

A computer program is developed to simulate the defect on the raceways with the results presented in

the time domain and frequency domain. The model yields both the frequency and the acceleration of

vibration components of the bearing. The effect of the defect size and its location has been investigated.

Numerical results for 6305 deep groove ball bearing have been obtained and discussed. The results

obtained from the experiments have also been presented.

& 2010 Elsevier Ltd. All rights reserved.

1. Introduction

Rolling element bearings are essential parts of rotatingmachinery. A machine could be seriously jeopardized if faultsoccur in bearings during service. Early detection of the defects inbearings, therefore, is crucial for the prevention of damage to theother parts of a machine. Bearing defects may be categorized aslocalized and distributed. The localized defects include cracks, pitsand spalls caused by fatigue on rolling surfaces. The othercategory, i.e., distributed defects, includes surface roughness,waviness and misaligned races and off size rolling elements.These defects may be due to manufacturing errors and operatingconditions. Hence, condition monitoring of bearings has beenconsidered to be an essential and integral part of any modernmanufacturing facility. Adequate monitoring predicts the possi-bility of a breakdown before it actually occurs. Different methodsare used for detection and diagnosis of the bearing defects. Theymay be classified as vibration measurement, acoustic measure-ment, temperature measurement and wear analysis. Vibrationbased condition monitoring has been the most widely usedtechnique. Both time domain and frequency-domain methods areused for monitoring the health of bearings.

Theoretical models of vibration generation mechanism inbearings due to single and multiple defects, and the influence of

ll rights reserved.

x: +91 831 2441909.

S. Patil), josmat@nitc.ac.in

ar), sandeep.desai@skf.com

various parameters such as loading and the transmission pathhelp in enhancing our understanding of the vibration generated atthe onset of incipient failure. The first attempt for obtainingdynamic models of rolling element bearings occurred in the mid-1970s. An analytical formulation for the generalized ball, cage andrace motion in ball bearings has been presented by Gupta [1]. Themodel proposed by McFadden and Smith [2,3] describes thevibration produced by a single point defect and multiple defectsin the bearings. The initial model developed by McFaddenconsiders a series of impulses representing the transient force tomodel the vibration produced when the rolling elementsencounter the point defect. As the bearing rotates, the impulsesoccur periodically with a frequency which is dependent on thelocation of the defect. A mathematical model for ball bearingvibrations with distributed defects was proposed by Meyer et al.[4]. The distributed defects are simulated and the spectralcomponents resulting from these defects have been predicted.A model presented by Tandon and Choudhury [5,6] predictedfrequency spectrum having peaks at characteristic defect fre-quencies. Pulses of finite width have been used to model thedefect. The height of the pulse is used to represent the extent andthe severity of the damage. The results were obtained for differentshape of the pulses. A model to simulate the force variation andimpact formation when the rolling elements roll over a localdefect was proposed by Kiral and Karagulle [7]. The dynamic loadof the rolling element bearing was modeled using a Visual Basicprogramming language and the vibration spectrum was obtainedusing a finite element method (FEM) package. The proposedmethod could be used to determine the optimum sensor location.Su et al. [8] extended the original work by McFadden to

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Nomenclature

d ball diameter (m)D pitch diameter of bearing (m)c damping factor (N s/m)Cr radial clearance (m)F Hertzian contact force (N)FX, FY, FXD, FYD components of restoring force in X and Y

directions (N)HD height of defect (m)K, Kp, Ki, Ko load–deformation factor (N/m3/2)M mass of the rotor (kg)n load–deflection exponentNs shaft speed (rpm)

W radial load (N)x, y deflections along X and Y axes (m)Z number of ballsz1, z2, z3, z4 state space variabledr radial deflection (m)dn dimensionless contact deformationSr curvature sumf length of defect in degrees.yi initial position of the defect (deg)o shaft speed (rad/s)oc cage speed (rad/s)BPFI ball pass frequency inner (Hz)BPFO ball pass frequency outer (Hz)

M.S. Patil et al. / International Journal of Mechanical Sciences 52 (2010) 1193–12011194

characterize the vibrations measured from bearings subjected tovarious loading conditions and with defects located on any of thebearing components. The main development of the work was thedetermination of the periodic characteristics of various loadingand transmission path effects and their influence on the vibration.These effects are generally associated with the misalignment ordynamic unbalance of the shaft, the axial or radial loading,preload and manufacturing imperfections. Akturk [9] presented amathematical model consisting of inner race, outer race and ballwaviness. The effect of number of waves on the amplitude ofvibration and the frequency was studied and concluded that forouter race waviness, most severe vibrations occur when the ballpassage frequency (BPF) and its harmonics coincide with thenatural frequency.

This work is focused towards the development of a theoreticalmodel to study the effect of defect size on bearing vibration.Instead of using periodically repeated impulse function for theimpulse due to defect, the defect itself is modeled as a part ofsinusoidal wave. The model makes it possible to detect thefrequency spectrum having peaks at the bearing defect frequen-cies. The amplitudes at these frequencies are also predicted.Comparison with the experimental values shows that the modelhelps to study the effect of defect size and predicts the spectralcomponents. For the experimental work, artificial defects areinduced separately on the outer race and inner race of single rowdeep groove ball bearings using electric discharge machine. Thevibration signals from the bearing are picked up using anaccelerometer. The data acquisition system has relevant softwareto acquire and store the data in the computer. The data are furtherprocessed using MATLAB software.

2. System modeling

To study the rolling element bearing structural vibrationcharacteristics, the rolling element–raceway contact can be

Fig. 1. Rolling elements replaced by spring and dash-pot.

considered as a spring mass system, in which the outer race isfixed in a rigid support and the inner race is fixed rigidly with themotor shaft. Elastic deformation between raceways and rollingelements produces a non-linear phenomenon between force anddeformation, which is obtained by the Hertzian theory. The rollingelement bearing is considered as non-linear contact spring asshown in Fig. 1.

In the model, the outer race of the bearing is fixed in a rigidsupport and inner race is held rigidly on the shaft. A constantradial load acts on the bearing. The contact force is calculatedusing the Hertzian contact deformation theory.

2.1. Calculating the contact force

According to the Hertzian contact deformation theory, thenon-linear relation load–deformation is given by [10]

F ¼ Kdnr ð1Þ

where K is the load–deflection factor or constant for Hertziancontact elastic deformation, dr the radial deflection or contactdeformation and n the load–deflection exponent; n¼3/2 for ballbearing and 10/9 for roller bearing.

The load–deflection factor K depends on the contact geometry.The ball and the raceway contact are as shown in Fig. 2(a) and (b).

Fig. 2. (a) Contact in the plane normal to plane of rotation and (b) contact in the

plane of rotation.

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Total deflection between two raceways is the sum of theapproaches between the rolling elements and each raceway.Using this we get

K ¼1

1=Ki

� �1=nþ 1=Ko

� �1=n

" #n

ð2Þ

Ki and Ko inner and outer raceways to ball contact stiffness,respectively, which are obtained using

Kp ¼ 2:15� 105X

r�1=2ðd�Þ�3=2ð3Þ

Fig. 3. Schematic diagram of a ball bearing.

Fig. 4. (a) Defect on outer race and (b) defect on inner race.

whereP

r is the curvature sum which is calculated using theradii of curvature in a pair of principal planes passing throughthe point contact. dn is the dimensionless contact deformationobtained using curvature difference (Refer Table 6.1, EssentialConcepts of Rolling Bearing Technology , Tedric Harris, Taylor andFrancis Group, 2007). The value of K for 6305 bearing is8.37536�109 N/m3/2.

The schematic diagram for determining the radial deflection isshown in Fig. 3.

If x and y are the deflections along X- and Y-axis and Cr is theinternal radial clearance, the radial deflection at the ith ball, at anyangle yi is given by [(x cos yi+y sin yi)�Cr].

Substituting in Eq. (1),

F ¼ K ðxcosyiþysinyiÞ�Cr

� �3=2ð4Þ

Since the Hertzian forces arise only when there is contactdeformation, the springs are required to act only in compression.In other words the respective spring force comes into play whenthe instantaneous spring length is shorter than its unstressedlength (the term in the bracket should be positive); otherwise theseparation between ball and race takes place and the resultingforce is set to zero. The total restoring force is the sum of therestoring forces from each of the rolling elements. Resolving thetotal restoring force along the X- and Y-axis we obtain

FX ¼XZ

i ¼ 1

K ðxcosyiþysinyiÞ�Cr

� �3=2cosyi ð5Þ

FY ¼XZ

i ¼ 1

K ðxcosyiþysinyiÞ�Cr

� �3=2sinyi ð6Þ

Table 1Inputs for the model.

1. Geometric properties of the bearing 6305Inner race diameter (Di) 32.1 mm

Outer race diameter (D0) 54.67 mm

Pitch diameter (D) 43.385 mm

Ball diameter (d) 11.274 mm

Number of balls (Z) 7

Contact angle (a) 01 (assumed)

Radial clearance (Cr) 11.285 mm

2. Other inputsMass of rotor (M) 3 kg

Damping factor (c) 200 N s/m

Fig. 5. Flow chart of the program.

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M.S. Patil et al. / International Journal of Mechanical Sciences 52 (2010) 1193–12011196

Eqs. (5) and (6) are modified to consider the effect of the defecton the bearing surface. The defect is modeled as a circumferentialhalf sinusoidal wave. Fig. 4(a) and (b) shows the defect on theouter raceway and inner raceway, respectively.

The total deflection of the paths of bearing is the sum of thecharacteristic of the defect and that of the static deflectionconsidered. Radial displacement is obtained by considering theresulting distortion. The restoring force for the presence of defecton the bearing race is given as

FXD ¼XZ

i ¼ 1

K ½ðxcosyiþysinyiÞ�ðCrþHD sinðp=jðyt�yiÞÞÞ�3=2 cosyi

ð7Þ

Fig. 6. (a) Time domain and power spectrum of vibrations due to ou

FXD ¼XZ

i ¼ 1

K½ðxcosyiþysinyiÞ�ðCrþHD sinðp=jðyt�yiÞÞÞ�3=2 sinyi

ð8Þ

j¼ Defect size

Raceway radius

If the defect is on the outer race

yt ¼octþ2p=ZðZ�iÞ ð9Þ

where i¼Z to 1.The inner race is moving at the shaft speed (o) and the ball

center at the speed of the cage (oc). If a point on the inner raceand a point at the ball center are considered at the same distance

ter race defect and (b) power spectrum due to outer race defect.

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Fig. 6. (Continued)

M.S. Patil et al. / International Journal of Mechanical Sciences 52 (2010) 1193–1201 1197

from the X-axis at the initial time, it is seen that after a time t, thepoint at the ball center will lag the point on the inner race by anangle of �(o�oc)t. Eqs. (7) and (8) can be used to calculate therestoring force due to defect on inner race by substituting‘yt’ byEq. (10).

For the defect on the inner race

yt ¼ ðoc�oÞtþ2p=ZðZ�iÞ ð10Þ

2.2. Equation of motion

The equations of motion for a two degree of freedom systemcan be written as follows:

M €xþc _xþFXD ¼W ð11Þ

M €yþc _yþFYD ¼ 0 ð12Þ

Eqs. (11) and (12) are second order non-linear differentialequations. The solution to these equations is obtained byconverting these into first order differential equations using statespace variable method.

Let z1 ¼ x, z2 ¼ _x, z3 ¼ y, z4 ¼ _y

_z1 ¼ _x ¼ z2, _z2 ¼ €x, _z3 ¼ _y ¼ z4, _z4 ¼ €y

Therefore,

z¼

z1

z2

z3

z4

8>>><>>>:

9>>>=>>>;¼

x

_x

y

_y

8>>>><>>>>:

9>>>>=>>>>;

and _z ¼

_z1

_z2

_z3

_z4

8>>><>>>:

9>>>=>>>;¼

z2

_z2

z4

_z4

8>>><>>>:

9>>>=>>>;¼

_x

€x

_y

€y

8>>>><>>>>:

9>>>>=>>>>;

A computer program is developed to obtain the solution andplot the results.

3. Flow chart of the program

The program consists of a function and the main program.The function is used to compute the state derivatives. Themain program uses the function as the input and solves theequations.

The flow chart of both the parts is shown in Fig. 5.

4. Results

The non-linear equations are solved to obtain the radialdisplacements. In order to obtain the results the inputs used areshown in Table 1.

4.1. Initial conditions

The time step for the investigation is assumed as thetime required for 0.11 of rotation. For the shaft speedof 1200 rpm, the time increment is 37.543 ms. The initialdisplacements set to the following values: x0 ¼ 10�6 m andy0 ¼ 10�6 m. The initial velocities are assumed to be zero: _x0 ¼ 0and _y0 ¼ 0.

For a shaft speed, Ns¼1200 rpm.Shaft speed, o¼ 2pNs=60¼125.66 rad/s.Shaft frequency, Fs ¼Ns=60¼20 Hz.Cage speed, oc ¼ ðo=2Þð1�ðd=DÞcosaÞ¼46.5 rad/s.Ball pass frequency outer (BPFO)¼ ZNs

2�60 1� dD cosa

� �¼51.8 Hz.

Ball pass frequency inner (BPFI)¼ZNs=2� 60ð1þd=DcosaÞ¼88.19 Hz.

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4.2. Effect of defect size

The defect is modeled as half sine wave. The defect sizecorresponds to the length of the defect in the direction of themotion. The depth of the defect is taken as 0.1 mm. The resultsare obtained for a radial load of 100 N and shaft speed of1200 rpm. Fig. 6(a) and (b) shows the time domain and thepower spectrum of the vibrations due to outer race defect ofdifferent sizes. Peaks at BPFO and its harmonics are seen inthe power spectrum. The dominant frequency is 103.6 Hz

Fig. 7. Effect of de

(2BPFO). There is significant increase in the amplitude at thisfrequency with the increase in the size.

4.3. Effect of position of the defect

The rolling elements present in the load zone [10] support theexternal load. Fig. 7 shows the variation in the amplitude level ofvibration with respect to the position of the defect on the outer racefor the bearing. It is clear from the figure that maximum amplitude of

fect location.

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vibration occurs when the defect is at zero degree. In this position thedefect is directly below the load (i.e., present in the load zone). As thedefect moves away from the load zone, the amplitude decreases.

4.4. Inner race defect

The results for the effect of inner race defect size are presentedin Fig. 8(a) and (b). BPFI for a shaft speed of 1200 rpm is 88.19 Hz.Peaks at BPFI and the combination of the BPFI and inner ring

Fig. 8. (a) Results of defect on inner race a

(shaft frequency) are found. The frequencies corresponding to thepeaks are 88.79 Hz (1BPFI), 308.9 Hz (1BPFI+11Fs), 353.3 Hz(4BPFI) and 397.7 (E2BPFI+11Fs).

5. Experimentation

The experimental set-up used for this study is shown in Fig. 9.It consists of a shaft supported on two bearings and driven by a

nd (b) results of defect on inner race.

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Fig. 9. Experimental set-up.

Fig. 10. Defect on the bearing race.

M.S. Patil et al. / International Journal of Mechanical Sciences 52 (2010) 1193–12011200

variable speed motor. The test bearing, a single-row deep grooveball bearing (SKF 6305) is placed on the non-drive end of the shaftand a double-row self aligning ball bearing is placed on the driveend side. A hydraulic loading arrangement which is used to applyload on the bearing is placed between these two bearings. Apiezo-electric accelerometer with a sensitivity of 50 mV/g is usedto measure the vibrations. It is mounted on the housing of the testbearing. The accelerometer is connected to the charge amplifier,the output of which is connected to a computer. The relevanthardware and the software required to acquire the data, store it,and display the time domain signal are installed in the computerused for this work. The signals are sampled at 9 kHz with asampling size of 4096 (212) samples. Low pass filter of 1 kHz isused to remove the unwanted high frequency noise. Theexperiments have been performed on separate test bearingshaving defects of size 0.5, 1 and 1.5 mm artificially induced onouter race and inner race separately. Electric discharge machinewas used to create the defect on the bearing surface. Fig. 10 shows

the bearing outer race with 0.5 mm defect. The experiment wasperformed at a shaft speed of 1200 rpm and radial load of 1000 N.

Fig. 11 depicts the results for the bearing with defect on itsouter race. The percentage variation between the theoretical andexperimental defect frequency is found to be 4.5–5.6%. This errormay be attributed to the slip existing between the rolling elementand the bearing races. There is difference between the amplitudesof vibration predicted by the model and the experiment because itis difficult to take into account the effects of the total rotorbearing system in the theoretical model.

6. Conclusion

1.

A mathematical model for the ball bearing vibrations due todefect on the bearing race has been developed. This model, inwhich the defect itself is modeled as half sinusoidal, helpsto simulate the effect of the defect size and its position andpredict the spectral components due to this. The modelpredicts the frequency spectrum having peaks at characteristicdefect frequencies and the amplitudes at these frequenciesemanating from the bearings. The frequency spectrum of thebearing vibrations due to the defects comprises mainly BPFOand its harmonics for outer race defect and, BPFI and thecombination of BPFI and the shaft frequency for the inner racedefect. The frequency components obtained from the proposedmodel are similar to those appearing in the frequency spectraof the experimental data for various defect conditions. Thisverifies the validity of the proposed model.

2.

It is found that the amplitude level of vibrations for the caseof outer race defect is more than that for the inner race defectand the ball defect. The defect present on the inner race movesin and out of the load zone during each revolution of theshaft. In this instance, the strong fault signatures producedwhile the defect is in the load zone are averaged with theweaker signatures acquired while the defect is outside theload zone. This has the effect of attenuating the magnitude ofthe inner-race characteristic fault frequency. It is predictedfrom the model that the amplitude of vibration increases withthe increase in the defect size. The same is observedexperimentally.

3.

The prediction of the actual amplitudes of vibration is notpossible by the model because it is difficult to incorporate intothe model, the effect of the rotor bearing system, whichincludes the shaft, motor, bearing and the supporting frame.However, it helps to predict the effect of the defect size and itsposition and the spectral components due to this.

4.

The present model is somewhat limited in treating the ballskidding effect. Ball slip is a function of lubrication, ball cageclearance, angular alignment, as well as speed and load.Lubrication traction has dominant effect on the ball skidding.Slip arises when the moment due to the drag forces on the ball,which is created by the viscous shearing resistance of thegrease, exceeds the traction moment at the raceway contacts.The present effort was aimed at a simpler approach to obtain atheoretical model to study the effect of defect size, load andspeed on the bearing vibration and predict the spectralcomponents. Including the ball raceway interaction and ballskidding phenomenon would make the model more rigorous.The authors believe that the moment load effects are not largeenough and therefore, may not lead to large error in theresults. However, this model will be upgraded at a future timeto incorporate the effect of ball skidding to predict the spectralcomponents of bearing with defects.

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Fig. 11. Power spectrum for outer race defect (experimental).

M.S. Patil et al. / International Journal of Mechanical Sciences 52 (2010) 1193–1201 1201

Acknowledgements

The authors would like to acknowledge the financial supportby KSCST (Karnataka State Council for Science and Technology)Bangalore, for fabricating the experimental set-up and SKF India,Pune for providing the bearings required for the tests.

References

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[3] McFadden PD, Smith JD. Model for the vibration produced by multiple pointdefects in a rolling element bearing. Journal of Sound and Vibration1985;98(2):263–73.

[4] Meyer LD, Ahlgren FF, Weichbrodt B. An analytic model for ballbearing vibrations to predict vibration response to distributeddefects. Journal of Mechanical Design—Transactions of the ASME 1980;102:205–10.

[5] Tandon N, Choudhury A. An analytical model for the prediction of thevibration response of rolling element bearings due to a localized defect.Journal of Sound and Vibration 1997;205(3):275–92.

[6] Tandon N, Choudhury A. A theoretical model to predict vibration response ofrolling bearings to distributed defects under radial load. Journal of Vibrationsand Acoustics 1998;l20:214–20.

[7] Kiral Z, Karagulle H. Simulation and analysis of vibration signals generated byrolling element bearing with defects. Tribology International 2003;36:667–78.

[8] Su YT, Lin MH, Lee MS. The effects of surface irregularities on rollerbearing vibrations. Journal of Sound and Vibration 1993;163(3):455–66.

[9] Akturk N. The effect of waviness on vibrations associated with ball bearings.Journal of Tribology 1999;121:667–77.

[10] Harris TedricA, Kotzalas MichealN. Rolling bearing analysis—essentialconcepts of bearing technology, 5th ed. Taylor and Francis; 2007.