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Procedia Engineering 64 (2013) 1582 – 1591 Available online at www.sciencedirect.com 1877-7058 © 2013 The Authors. Published by Elsevier Ltd. Open access under CC BY-NC-ND license. Selection and peer-review under responsibility of the organizing and review committee of IConDM 2013 doi:10.1016/j.proeng.2013.09.240 ScienceDirect International Conference on DESIGN AND MANUFACTURING, IConDM 2013 Vibration Studies of Dynamically Loaded Deep Groove Ball Bearings in Presence of Local Defects on Races V. N. Patel a *, N. Tandon b , R. K. Pandey c a G H Patel College of Engineering and Technology, Vallabh Vidyanagar-388120, Gujarat, India b Industrial Tribology, Machine Dynamic and Maintenance Engineering Centre (ITMMEC), I.I.T. Delhi, New Delhi – 110 016, India c Department of Mechanical Engineering, I.I.T. Delhi, New Delhi – 110 016, India Abstract Theoretical and experimental vibration studies of dynamically loaded deep groove ball bearings having local circular shape defects on either race are reported in this paper. The shaft, housing, raceways and ball masses are incorporated in the proposed mathematical model. Coupled solutions of governing equations of motion have been achieved using Runge-Kutta method. The model provides the vibrations response for the shaft, balls, and housing in time and frequency domains. The results achieved based on the proposed mathematical model have been validated with the experimental results. In experiments, the test bearings were radially loaded using an electro-mechanical shaker. The radial load excitation frequency ranged 10 - 1000 Hz. The characteristic defect frequencies and related harmonics are broadly investigated and presented herein. Keywords : vibration, local defects, deep groove ball bearing, dynamic loading, defect frequency 1. Introduction Rolling bearings are widely used in various mechanical systems viz. mechanisms, equipment and machines in industries. Reliable functioning of such mechanical systems depends on the good health of the employed rolling bearings. The rolling bearings are manufactured using high precision machine tools and pass through the strict quality checks. In spite of these, rolling element bearings may develop early defects on their mating components during their usage depending upon the operating parameters and environments of operation. The existence or development of even tiny local defects on the mating surfaces of bearing components in a mechanical system may lead to its catastrophic failure due to progressively increase in defect size through passes of time. Such failures may cause both substantial monetary and human life losses. Therefore, detection of local defects in their early stages through observations of vibration signals of rolling bearings is essentially a vital issue. *Corresponding author: Tel: +91-2692-231651, Fax: +91-2692-236896 E-mail: [email protected] © 2013 The Authors. Published by Elsevier Ltd. Open access under CC BY-NC-ND license. Selection and peer-review under responsibility of the organizing and review committee of IConDM 2013

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Page 1: 1-s2.0-S1877705813017542-main

Procedia Engineering 64 ( 2013 ) 1582 – 1591

Available online at www.sciencedirect.com

1877-7058 © 2013 The Authors. Published by Elsevier Ltd. Open access under CC BY-NC-ND license.Selection and peer-review under responsibility of the organizing and review committee of IConDM 2013doi: 10.1016/j.proeng.2013.09.240

ScienceDirect

International Conference on DESIGN AND MANUFACTURING, IConDM 2013

Vibration Studies of Dynamically Loaded Deep Groove Ball Bearings in Presence of Local Defects on Races

V. N. Patela*, N. Tandonb, R. K. Pandeyc

aG H Patel College of Engineering and Technology, Vallabh Vidyanagar-388120, Gujarat, India bIndustrial Tribology, Machine Dynamic and Maintenance Engineering Centre (ITMMEC), I.I.T. Delhi, New Delhi – 110 016, India

cDepartment of Mechanical Engineering, I.I.T. Delhi, New Delhi – 110 016, India

Abstract

Theoretical and experimental vibration studies of dynamically loaded deep groove ball bearings having local circular shape defects on either race are reported in this paper. The shaft, housing, raceways and ball masses are incorporated in the proposed mathematical model. Coupled solutions of governing equations of motion have been achieved using Runge-Kutta method. The model provides the vibrations response for the shaft, balls, and housing in time and frequency domains. The results achieved based on the proposed mathematical model have been validated with the experimental results. In experiments, the test bearings were radially loaded using an electro-mechanical shaker. The radial load excitation frequency ranged 10 - 1000 Hz. The characteristic defect frequencies and related harmonics are broadly investigated and presented herein.

© 2013 The Authors. Published by Elsevier Ltd. Selection and peer-review under responsibility of the organizing and review committee of IConDM 2013.

Keywords : vibration, local defects, deep groove ball bearing, dynamic loading, defect frequency

1. Introduction

Rolling bearings are widely used in various mechanical systems viz. mechanisms, equipment and machines in industries. Reliable functioning of such mechanical systems depends on the good health of the employed rolling bearings. The rolling bearings are manufactured using high precision machine tools and pass through the strict quality checks. In spite of these, rolling element bearings may develop early defects on their mating components during their usage depending upon the operating parameters and environments of operation. The existence or development of even tiny local defects on the mating surfaces of bearing components in a mechanical system may lead to its catastrophic failure due to progressively increase in defect size through passes of time. Such failures may cause both substantial monetary and human life losses. Therefore, detection of local defects in their early stages through observations of vibration signals of rolling bearings is essentially a vital issue.

*Corresponding author: Tel: +91-2692-231651, Fax: +91-2692-236896 E-mail: [email protected]

© 2013 The Authors. Published by Elsevier Ltd. Open access under CC BY-NC-ND license. Selection and peer-review under responsibility of the organizing and review committee of IConDM 2013

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1583 V.N. Patel et al. / Procedia Engineering 64 ( 2013 ) 1582 – 1591

Nomenclature C equivalent viscous damping of housing, N-s/mm D pitch diameter of bearing, mm d ball diameter, mm ddefect diameter of defect, mm din diameter of inner race, mm dout diameter of outer race, mm ds diameter of shaft, mm E Young’s modulus, N/mm2 fs shaft rotation frequency, Hz K stiffness, N/mm

klin linear stiffness of the bearing (N/mm) Mbi mass of ith

ball, g Mh combined mass of housing and outer race, kg Ms combined mass of shaft and inner race, kg Nb number of balls in bearing P dynamic radial load applied to bearing, N Pd diametric clearance, mm X,Y deflection along the axes, mm

defect angle with respect to X axis additional displacementof ball centre in defect zone, mm contact deflection in radial direction, mm * dimensionless contact deflection angular velocity, rad/s angular position of the ball, rad 0 angle between two balls, rad

Subscript b ball c cage h housing i ball number j defect number in inner race out outer race s shaft

Some of the relevant papers dealing with the local defect detections in rolling bearings using vibration signals are

reviewed herein. Papers [1, 2] have thoroughly surveyed various methods/techniques for defect detections in rolling bearings using vibration signatures. McFadden and Smith [3, 4] have presented simple model to describe the vibration of rolling element bearings in presence of single and multiple point defects on the inner races. A comparison of predicted and measured vibration spectra is provided by the authors with relevant discussions. Later on, Su and Lin [5] have extended the model proposed by [3, 4] for describing the bearing vibration under various loadings. Sopanen and Mikkola [6, 7] proposed a dynamic model for vibration study of a deep groove ball bearing having localized and distributed defects. In their model, the authors have incorporated influence of lubrication and non-linear deformation at the contacts formed between the balls and races. In a relatively more realistic and simplified model [8], the authors have considered lumped masses of the shaft and housing on the vibration response of the locally defective rolling element bearings.

Vibration of angular contact ball bearings in presence of local defects have been investigated by Arslan and Aktürk [9]

and Ashtekar et al. [10]. The authors [9] have developed a dynamic model of shaft-bearing system to study the components of defect frequency in vibration spectra of angular contact ball bearings. However, Ashtekar et al. [10] have studied the effect of surface defects/dents on the bearing motion. A paper of Patel et al. [11] presents vibration studies of shaft, balls, and housing in presence of local defects on either races by applying steadily radial load (constant magnitude and direction) on the deep groove ball bearings. The authors have compared theoretical and experimental vibrations of bearing housings for the cases of healthy and locally defective bearings.

Based on the literature review, it is observed that in past researchers have studied vibrations of locally defective rolling bearings by considering only steady loads (i.e. constant magnitude and direction). However, it is worth mentioning here that in majority of practical situations in mechanical systems the frequency of applied loads on the rolling bearings varies. Therefore,

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the objective of this paper is to study theoretically and experimentally the vibration responses of deep groove ball bearings in presence of local defects on either of race by applying varying frequency of applied loads. 2. Mathematical model

The mathematical model reported by authors [11] has been modified and used herein for study of vibrations by

incorporating the frequency dependent loading. Figure 1 shows schematic description of bearing system and its simplified physical model used in this numerical study of vibrations.

Fig. 1 Schematic description of bearing system and its mathematical model

2.1 Simulation of local circular defect

The role of the defect in simulation has been considered by adding or subtracting the additional displacement in the total deflection at the contact of ball and locally defective races. When centre of a ball approaches to the defect, the additional displacement of the ball remains minimum. However, when it passes through the defect the deflection ‘ ’ increases. It reaches its maximum value when ball reaches at the centre of the defect. After this the deflection decreases from maximum to zero as the ball passes from centre of the defect to other end of the defect. The locus (with dotted line) of the centre (C) of the ball during interaction with defect is illustrated in Fig. 2(a). The additional displacement increases from centre position (C1) to the centre position (C2) and thereafter it starts reducing and finally becomes zero as the ball comes out from the defect. Moreover, Figs. 2 (b) and 2(c) show the top view and side view of ball and defect interaction when a ball passes through the defect.

(a) (b) (c)

Fig.2. Schematic representation of ball interactions with defect lying on surface of outer race (a) Locus of ball centre while passing through the defect (Front view) (b) Contacts of ball through the defect (Top view)

(c) Contacts of ball in defect (Side view)

Test Bearing

Dynamic Load

Enlarged view of bearing in housing Shaft bearing

system

P

O

Mb

Cs

M

innerKi

outerKi Mb

Mb

Mb

Mb

Mb

Mb

Mb

outerKi

innerKi

Kh Ch

Free body diagram of shaft bearing system

Additional deflection ( )

C1 C2 C3

C C

ddefect

d4 d2

d1

f

d6

C e

d3 d5

d2 2* ball

C d1

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2.2 Dynamic loading on bearing

Dynamic loading with randomly varying frequency (10 - 1000 Hz) has been considered in this proposed investigation. For numerical simulation, the dynamic load ‘P’ has been considered by the following relation:

100

ext jj= 1

P = F sin 2 j f + (1)

Where, Fext and f are amplitude and frequency of applied force, respectively, and varies randomly between 0 and 2 .

2.3 Equations of motion The equations of the motion for housing in X and Y directions with dynamic loading are as follows:

( )( )

( )

3/2Nb bi h iouterh h h h h h i i

i=1 bi h i ij j

X - X cosM X + K X + C X - K cos = P

+ Y - Y sin - * (2)

( )( )

( )

3 /2N b b i h iou terh h h h h h i i

i= 1 bi h i ij j

X - X co sM Y + K Y + C Y - K s in = 0

+ Y - Y sin - * (3)

Where, ij =1, for ith ball, when it passes through jth defect, ij=0 for all other cases. Distance d1e (refer Fig. 2b) is written as:

2 21 1 2 defectd e = d d /2 = d /2 - e f (4)

Where, o u t ie f = d /2 s in ( ) The angle, ball, subtended by the ball with defect is expressed as (refer Fig. 2(c)):

-1ball 1 2= sin d d /d (5)

The additional displacement ( j) when ith ball passes through jth circular defect is expressed by the following relation: j ba llD = (d/2) ± (d/2) cos( ) (6)

In Eq. (6), sign ‘+’ is applicable for locally defective outer race and sign ‘ ’ is used for locally defective inner race.

Mathematically, the position of ith ball in inner race defect zone is expressed as: s defect in i s defect inw t - d /d w t + d /d (7)

The local defects on outer race are normally found in the loaded region of the bearing. Moreover, for the case of stationary outer race, the position of defect does not change with shaft rotation. However, the deflection of ith ball varies when it passes through the defect of the outer race. Mathematically, the angular position of ith ball in defect zone is written as follows: out defect out i out defect out- d /d + d /d (8)

2.4 Computational procedure

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The coupled solution of governing equations of motion by satisfying miscellaneous relations as outlined in subsections

(2.1 to 2.3) is achieved using Runge Kutta method. Displacements (in X and Y directions) and velocities ( X andY ) at time (t+ t) are calculated using Runge-Kutta method. The time step ( t) of 10-7 sec has been considered in the present computation. The system equations are iteratively solved for specified time period. 3. Experimentation

For validating the proposed mathematical model, experiments were performed using the experimental setup as shown in Fig.3. The rotating shaft is supported by two deep groove ball bearings (SKF 6311) named as supporting bearings. However, experiments for vibration responses were conducted on test bearings as indicated in Fig.3. Test bearing is a deep groove ball bearing (Designation: SKF BB1B420205), which was mounted at the projected end of the shaft towards right hand side. The left end of the shaft is coupled to an electric motor through V- belt. Faults were introduced on the races of the test bearings through electric discharge machining (EDM). The photographic view of a typical circular defect on inner race can be seen in Fig. 4. Dynamic loading in form of random vibrations varying in the frequency range 10-1000 Hz were added to the defective test bearing through an electro-mechanical shaker. The vibration signals from the test bearing housing were captured by the accelerometer (B&K - 4370), which was mounted on the top of the test bearing housing. The captured signal was amplified by the charge amplifier and transferred to Fast Fourier Transform analyser (ONO SOKKI) for post processing in MATLAB.

Fig.3. Photographic view of experimental set up

Fig. 4.Image of a circular defect on inner race [12] 4. Results and discussions

Vibrations of test bearings having circular defect of 600 μm diameter on either of the race have been investigated theoretically and experimentally considering dynamic load (frequency ranged 10-1000 Hz). The input data of bearing system is listed in Table 1. The amplitude of the dynamic load has been measured by a load sensor attached between the housing of test bearing and electro-mechanical shaker. Velocity of vibration is captured on top of test bearing housing using an accelerometer.

Test bearing

Support bearing

FFT analyser Shaker

Accelerometer

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1587 V.N. Patel et al. / Procedia Engineering 64 ( 2013 ) 1582 – 1591

Table 1. Input data for bearing system

Bore of bearing, mm 25 Inner race diameter (di), mm 29.2 Outer race diameter (do), mm 46.6 Pitch diameter (D), mm 37.9 Ball diameter (d), mm 8.7 Diametric clearance (Pd), m 10 Grooves radii, mm 5 Number of balls (Nb) 8 Mass of inner race, g 25.25 Mass of outer race, g 49.40 Mass of ball, g 2.70 Mass of shaft, kg 1.50 Mass of housing, kg 1.0 Overhung length of shaft, mm 140.5

It is worth mentioning here that simulated results shown in subsections (4.1- 4.2) have different sample size (i.e. much

less) than experimental ones. This makes appearance of experimental spectra different to simulated in these figures. It is necessary to mention here that sample size of experimental data could not be provided in simulation due to memory scarcity in computing system. Moreover, dynamic radial load is applied by an electro-mechanical shaker, which is also noisy. Many frequencies appearing in experimental spectra are due to excitation of other components of the experimental setup. Such frequencies (except fs, BPFI, BPFI ± fs) cannot appear in simulated results. Therefore, comparison between simulated and experimental spectra should be seen for presence of fs, BPFI, and BPFI ± fs.

4.1 Inner race defect in bearing

Figure 5 illustrates variation of additional displacement during interaction of a ball with local defect of inner race while

passing through it. In Fig. 5, it can be seen that when a ball approaches the defect how the displacement of the ball changes from zero to maximum. The additional displacement reaches zero from its maximum value when ball reaches from the centre of the defect to the other end of the defect. Vibration response captured at the housing of the bearing is presented in Fig. 6 at the shaft rotational speed of 1500 rpm. It is difficult to identify the impulses generated during the interaction of defect and balls in time domain. The impulses are merged into the noise generated due to the dynamic load. Thus, it is necessary to analyse the time domain vibration results in to the frequency domain for meaningful interpretation. At the shaft rotational speed of 1500 rpm (25 Hz), the ball pass frequency for inner race (BPFI) is computed 123 Hz b sB P FI = (N /2 )* (N /60)* (1 + d/D ) .

Fig. 5. Illustration of additional displacement of a ball when it passes through the inner race defect (Ns = 1500 rpm, defect diameter = 600 μm)

In case of inner race defect, the defect rotates at the shaft’s speed, so the BPFI is amplitude modulated due to shaft

0.00780.0079

0.008

7272.57373.574-0.015

-0.01

-0.005

0

Time (sec)Ball position (Deg)

Bal

l dis

plac

emen

t (m

m)

Ball approaching to defect

Ball at centre ofdefect

Ball leaving defect

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1588 V.N. Patel et al. / Procedia Engineering 64 ( 2013 ) 1582 – 1591

rotational frequency. Therefore, side bands are expected at two frequencies i.e. BPFI ± fs = 99 Hz, 148 Hz. Figures 7(a) and 7(b) show, respectively, the numerical and experimental vibration velocity spectra of bearing housing having inner race defective bearing. The shaft rotational frequency (fs), ball pass frequency of inner race (BPFI) and side bands (BPFI fs) are present in both numerically simulated and experimental spectra. The additional frequencies, other than BPFI are observed in the spectra in Fig. 7(b). It is due to the noise added by the dynamic loading. The additional frequencies observed in the experimental spectra in Fig. 7(b) are due to the excitation of the other components of the experimental setup caused by the electromechanical shaker.

Fig. 6. Vibration response of bearing housing having circular defect on inner race of bearing

(Ns = 1500 rpm, defect diameter = 600 μm)

(a) (b)

Fig.7. Vibration velocity spectra of housing with circular defected inner race of bearing (Ns = 1500 rpm, defect diameter = 600 μm) (a) Numerically simulated (b) Experimental

4.2 Outer race defect in bearing

Variations in additional displacement of a ball when it passes through the circular defect (diameter = 600 m) on the outer race of bearing are shown in the Fig.8. . In Fig. 8, it can be seen that when a ball approaches the defect how the deflection of the ball changes from zero to maximum. The additional displacement reaches zero from its maximum value when ball reaches from the centre of the defect to the other end of the defect.

The vibration response of bearing’s housing for the defective outer race of bearing is shown in Fig. 9 in time domain. The impulses generated due to interaction of defect and balls are not clearly visible in Fig.9 due to presence of noise. Therefore, vibration response achieved through numerical and experimental simulations have been plotted in frequency domain in Fig.10 (a) and Fig.10 (b), respectively. The BPFO and its second harmonics are visible in both the spectra. Normally, the defect on stationary outer race is expected in the loaded zone. Therefore, when moving ball approaches to the defect same magnitude of impulse is expected for every time of contact between ball and defect. For present case, the BPFO is computed 77.045 Hz

* *b sB P F O = (N /2) (N /60) (1 - d /D ) .

0 20 40 60 80 100 120 140 160 180 2000

0.2

0.4

0.6

0.8

1 x 10-3

Frequency (Hz)

Vel

ocity

(m/s

)

BPFI-

fs

fsBPFI

fsBPFI+

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2-0.1

-0.05

0

0.05

0.1

Time (sec)

Vel

ocity

(m/s

)

0 20 40 60 80 100 120 140 160 180 2000

1

2

3

4

5

6 x 10-4

Frequency (Hz)

Vel

ocity

(m/s

)

BPFIfs

fsBPFI-

BPFI+fs

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1589 V.N. Patel et al. / Procedia Engineering 64 ( 2013 ) 1582 – 1591

Fig.8. Illustration of additional displacement of a ball when it passes through defect on outer race of the bearing (Ns =1500 rpm, defect diameter = 600 m)

Fig. 9. Vibration response of bearing housing having circular defect on outer race of bearing (Ns =1500 rpm, defect diameter = 600 m)

(a) (b)

Fig. 10. Vibration velocity spectra of housing with circular defect on outer race of bearing

(Ns =1500 rpm, defect diameter = 600 m) (a) Numerical results (b) Experimental results Table 2 shows comparison between the frequency and vibration amplitudes for the cases of inner and outer races defects

of a bearing. Good correlations between theoretical and experimental extracted frequencies can be seen in this table. Moreover, reasonably fair correlations are also visible for vibration amplitudes. Overall good matching of theoretical and experimental results develops good confidence in the proposed dynamic model. It is necessary to mention here, that the simulated and experimental velocity spectra (Fig. 7 and Fig. 10) are not exactly similar due to different sample size in both cases. This variation in sample size is because of the limitation in handling the data in computer due to memory constraints.

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2-0.06

-0.04

-0.02

0

0.02

0.04

0.06

Time (sec)

Vel

ocity

(m/s

)

0 20 40 60 80 100 120 140 160 180 2000

0.2

0.4

0.6

0.8

1x 10-3

Frequency (Hz)

Vel

ocity

(m/s

) fsBPFO

2*BPFO

0.012 0.013 0.014359.5360360.53610

0.005

0.01

0.015

Time (sec)Ball position (deg)B

all d

ispl

acem

ent (

mm

)

Ball approachingto defect

Ball leavingdefect

Ball at centre ofdefect

0 20 40 60 80 100 120 140 160 180 2000

0.5

1

1.5

2 x 10-3

Frequency (Hz)

Vel

ocity

(m/s

)

2*BPFOBPFO

fs

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1590 V.N. Patel et al. / Procedia Engineering 64 ( 2013 ) 1582 – 1591

Table 2. Comparison of vibration amplitudes and characteristic defective frequencies (Ns =1500 rpm, Defect diameter = 600 m, Dynamic loading)

Defect location

Simulated results

Experimental results

Defect on inner race

BPFI =122.9 Hz, Amplitude= 0.00019 m/s

BPFI =124.9 Hz, Amplitude= 0.00016 m/s

Defect on outer race

BPFO =76.68 Hz, Amplitude= 0.00055 m/s

BPFO =78.13 Hz, Amplitude= 0.0004 m/s

5. Conclusions

A dynamic model for vibration study of deep groove ball bearings having local defects on either race is presented in this paper considering dynamic loading. The vibration amplitudes (velocities) and frequencies are numerically computed by solving the coupled governing equations of motion. Simulated and experimental results pertaining to vibration of bearing housings are compared and discussed. Based on this study the following conclusions are drawn:

When a ball approaches to the inner race defect, the additional displacement of the ball changes from zero to

maximum, while, it reaches to zero from its maximum value when ball reaches from the centre of the defect to the other end of the defect.

The additional displacement reaches to zero from its maximum value when ball reaches from the centre of the outer defect to the other end of the defect.

The vibration peaks at characteristic defect frequencies (BPFO and BPFI) with their harmonics are recorded for defective bearings.

In case of defective inner race, characteristic defective frequency along with the side bands at shaft rotation frequency is noticed.

The additional vibrations are also observed due to the noise added by the electro-mechanical shaker. Moreover, vibration enhances in presence of local defects on outer race in comparison to inner race.

Acknowledgements

Authors are grateful to staff members of Machine Dynamics Laboratory (ITMMEC) for assisting in the experiments. The authors acknowledge the financial support provided by Department of Science and Technology, India, via project SERC No. SR/S3/MERC/47/2006 dated 9/7/2007. Moreover, first author acknowledges and thanks the management of his parental institute (G H Patel College of Engg. & Tech., Vallabh Vidyanagar, Gujarat, India) for granting the permission to pursue his Ph.D. programme at ITMMEC, I.I.T. Delhi, India.

References

[1] Tandon, N., Choudhury, A., 1999. A Review of Vibration and Acoustic Measurement Methods for the Detection of Defects in Rolling Element Bearings, Tribology International 32, p. 469-480.

[2] Patil, M. S., Mathew J., Rajendrakumar, P. K., 2008. Bearing Signature Analysis as a Medium for Fault Detection: A Review, Journal of Tribology Transaction of ASME 130, p. 014001-014007.

[3] McFadden, P. D., Smith, J. D., 1984. Model for the Vibration Produced by a Single Point Defect in a Rolling Element Bearing, Journal of Sound and Vibration 96, p. 69-82.

[4] McFadden, P. D., Smith, J. D., 1985.The Vibration Produced by Multiple Point Defect in a Rolling Element Bearing, Journal of Sound and Vibration 98, p. 263-273.

[5] Su, Y. T., Lin, S. J., 1992. On Initial Fault Detection of a Tapered Roller Bearing: Frequency Domain Analysis, Journal of Sound and Vibration 155, p. 75–84.

[6] Sopanen, J., Mikkola, A., 2003. Dynamic Model of a Deep-Groove Ball Bearing Including Localized and Distributed Defects. Part 1: Theory, Proc. Instn. Mech. Engrs., Part K: Journal of Multi-Body Dynamics 217, p. 201-211.

[7] Sopanen, J., Mikkola, A., 2003. Dynamic Model of a Deep-Groove Ball Bearing Including Localized and Distributed Defects. Part 2: Implementation and Results, Proc. Instn. Mech. Engrs., Part K: Journal of Multi-Body Dynamics 217, p. 213- 223.

[8] Choudhury, A., Tandon, N., 2006. Vibration Response of Rolling Element Bearings in a Rotor Bearing System to a Local Defect under Radial Load, Journal of Tribology Transaction of ASME 128, p. 252-261.

[9] Arslan, H., Aktürk, N. 2008. An Investigation of Rolling Element Vibrations Caused by Local Defects, Journal of Tribology Transaction of ASME 130, p. 041101-1-041101-12.

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[10] Ashtekar, A., Sadeghi, F., Stacke, L., 2008. A New Approach to Modelling Surface Defects in Bearing Dynamics Simulations, Journal of Tribology Transaction of ASME 130, p. 041103-1-041103-8.

[11] Patel, V. N., Tandon N., Pandey, R. K., 2010. A Dynamic Model for Vibration Studies of Deep Groove Ball Bearings Considering Single and Multiple Defects in Races, Journal of Tribology Transaction of ASME 132, p. 041101-1-041101-10.

[12] Patel, V. N., Tandon N., Pandey, R. K., 2012. Defect detection in Deep Groove Ball Bearing in Presence of External Vibration using Envelope Analysis and Duffing Oscillator, Measurement 45, p. 960-970.