Vibration Analysis of Defected Ball Bearing Using

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such as finite element analysis or modal analysis, thenvibration measurements during its in service operation candefine the dynamic characteristics of the forces acting on themachines, moreover it can be known also whether the bearinghave defect or not.

Numerical techniques to simulate the vibration responseof structures have become popular in recent years. One of the

powerful numerical techniques for solving complexmechanical and structural vibration problem is the finiteelement method. Some researchers have applied this methodto study defect detection in rolling element bearings. Wensing

(1998) investigated the dynamic behavior of ball bearingsusing finite element model simulation, then vibration caused

by imperfections like surface waviness was studied. Holm-Hansen and Gao (2000) used finite element method tocalculate the changes in the dynamic loading and speedvariations associated with an outer ring fault. Kral andKaragulle (2003) developed the dynamic loading models for

rolling element bearing structures using finite element modeland performed the finite element vibration analysis to detectthe outer ring defect for several bearing geometries andloading conditions. In another study, Kral and Karagulle(2006) investigated the loading mechanism model in abearing structure which houses a deep groove ball bearinghaving different localized defects and carrying an unbalanced

force rotating with the shaft.In this study, finite element model simulation is

developed to analyze vibration response of ball bearing usinga commercial software ABAQUS. The transfer load fromrolling element to the outer race way is simulated with thedynamic loading model that represent reaction force on the

outer raceway due to the rotating of rolling elements into it.Then the vibration signature responses of healthy and

defected bearing are compared. Time domain parameter isused for the vibration analysis. RMS and peak to peak valueare used as time signal descriptors for condition monitoringpurpose. The effect of varying shafts rotational speed andload are investigated.

2. LOADING MECHANISM

The loads applied to rolling bearings are transmittedthrough the rolling elements from the inner ring to the outerring. The magnitude of the loading carried by the individual

ball or roller depends on the internal geometry of the bearingand the type of load applied to it.

In most bearing applications, only applied radial, axial,or a combination of radial and axial loadings are considered.However, under very heavy applied loading or if shafting ishollow, the shaft where the bearing is mounted may bend,

causing a significant moment load on the bearing. Also, thebearing housing may be nonrigid due to design targeted atminimizing both size and weight, causing it to bend whileaccommodating moment loading. This combined radial, axial,and moment loadings result in distorted distribution of load

among the bearings rolling element complement. This maycause significant changes in bearing deflections, contactstresses, and fatigue endurance compared to the operatingparameters which have the simpler load distributions (Harris

and Kotzalas, 2007). To simplify the analytical process, load

applied is assumed pure radial load in this study.The load distribution around the circumference of a

rolling element bearing under radial load (as shown in Figure1) is defined approximately by the Stribeck equation (Harris,2001):

( ) ( )[ ]noqq cos11 21 = (1)Where qo is the maximum load intensity at = 0

0, is theload distribution factor and n denotes the load deflectionexponent. For ball bearing n is 1.5 while for roller bearing n

is 1.11.

Figure 1: The load distribution in a bearing under radial load

The maximum load intensity, qo, for ball bearing havingzero clearance and subjected to a simple radial load can beapproximated by,

cos37.4

ZFq ro = (2)

whereFr is the radial load and Z is the number of balls and is contact angle.

The load distribution factor, , is defined by

=

r

dP

21

2

1(3)

wherePd denotes the diameter clearance, while r is the ringradial shift. If the diameter clearance is assumed as 0, hence

the value of is 0.5.The rolling elements transfer the radial load to the outer

ring during their rotation with the cage frequency, fc,

expressed as

= cos1

2 m

bsc

d

dff (4)

withfs is the shaft frequency, db and dm is ball diameter andpitch diameter respectively. Since in this study assumed thatthe ball bearing is subjected to a pure radial load, hence the

contact angle will be 00.

APIEMS 2008 Proceedings of the 9th Asia Pasific Industrial Engineering & Management Systems Conference

December 3rd 5th, 2008Nusa Dua, Bali INDONESIA

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3. FINITE ELEMENT MODEL SIMULATION

The bearing type that will be used in this study is a singlerow

Figure 2: SKF bearing 6205 geometry

The dim ring areas fo

ter diameter,D = 52 mm

is adapted from the NTNpillo

ed to thehou

ratio (v) of 0.3.

Figure 3: Sim lified bearing housing (dimensio in mm).

opt enum

on the load zone, andthen

DEVELOPMENT

70

52

deep groove ball bearing. They are the most popular of

all rolling bearings because it is simple in design, non-separable, capable of operating at high even very high speeds,and require little attention or maintenance in service. Inaddition they have a price advantage (SKF general catalogue,

1989). The bearing model 6205 from SKF is used in thisstudy. This bearing has a bore diameter of 25 mm and widely

used for many applications. The geometry for this bearingtype is shown in Figure 2.

36,5

67 25

p n

ensions and parameters for the 6205 beallows:

- Ou- Bore diameter, d = 25 mm- Pitch diameter, dm = 39 mm- Ball diameter, db = 8 mm- Raceway width,B = 15 mm- Contact angle, = 00-Number of balls,Z = 9

The housing structure usedw type bearing unit. The bearing unit number F-

UCPM205/LP03 having a 25 mm shaft diameter is chosen.Modifications are carried out to simplify the modeling andanalytical process for this structure. This bearing unitstructure consists of housing and the bearing. Since the

bearing has been specified previously, so only the housingstructure is adapted. The modifications done include dispo-

sing the mounting part and making the width of the housinguniform. The width is chosen as 25 mm. The simplification ofthe housing structure is shown in Figure 3. Proper boundarycondition is applied to the bottom surface of the housing toreplace the mounting part of the original structure.

The outer ring is assumed perfectly attach

sing structure; hence the tie constraint is used as aninteraction type applied between the outer surface of the outerring and inner surface of the bearing housing. The materialused for both parts is steel with a density () of 7.8 E-6kg/mm

3, young modulus (E) of 209 E3 N/mm

2and Poissons

Mesh convergence test is performed to determine thei um number of elements on the model. In this study, th

mber of elements lying on the outer raceway part will

become the main parameter for performing this test, since the

load from the ball is transferred to this part. In this test, the

number of elements lying on the circumference of the outerrace will be varied from 40 until 88 elements with anincrement of 4 elements in each test.

Static analysis is performed for this test. The pressureload of 1000 Pa is applied uniformly

the displacement of the point P1 which is located on the

middle top of bearing housing structure as shown in Figure 4,will be investigated for each model. This point is chosenbecause this point is used as the location of the sensor foranalyzing the vibration of the bearing structure.

Figure 4: Geometry model used for mesh convergence test.

r

of element along the circumference of the bearings outerrace

The result of mesh convergence test showing the numbe

way with the corresponding displacement of point P1 isin Figure 5. It can be observed that the displacement of pointP1 give small differences (less than 0.2%) with the increasing

number element after the number of element alongcircumference of bearings outer race is 64, it indicates that

the model already converged. Hence, the model used foranalyzing vibration of bearing structure will have 64 elementslying along the circumference of bearings outer raceway.

Point P1



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Mesh Convergence Test

-0.13

-0.129

-0.128

-0.127

-0.126

-0.125

-0.124

-0.123

-0.122

-0.121

-0.12

32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92

Number of Element

Displacementof

pointP1(mm)

Figure 5: Mesh convergence test result

In this st e, Pd iszero, hence the load distribution actor, is 0.5, meaning the

load

of contact betwee ere and a flat surfaceis us

ith the indenter loadP, the indenter radiusR, and

the

udy it is assumed that diameter clearanc

fing zone on the outer race is between -90

0 90

0,

where is zero in the direction of the radial load Fr. So, the

number of element lying on the loading zone will be half of64, i.e. 32 elements, which resulting 33 nodes from -900until 900 with an increment of 5.625 degree. Therefore, 33dynamic loading is developed for each node in the loading

zone as the excitation force resulting from the contact withthe ball.

To determine the time period of contact on each node,the theory n a rigid sph

ed. Once the radius of the contact is determined, it can betransformed to the time period by dividing it with the angular

velocity .Hertz found that the relation of the radius of the circle of

contact a, w

elastic properties of the materials is given by (Fischer-Crips, 2007):

*3

43

PRa = (5)

E

whe E* is the cospecimen defined by

re mbined modulus of the indenter and the:

( ) ( )'E

+ (6)

E' and v', and E and , denotPoissons ratio of the indenter and the

'11

*

122 v

E

v

E

=

es the elastic modulus andspecimen respectively.

Dynamic loading model

at 0 deg node

0

100

200

300

400

500

600

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

time (s)

q

(N)

Figure 6: Dynamic loading models for node at 0 degree

In this case, the indenter load is defined by q(), which isdetermined using Eq. (1) for each node, while the indenterradius is the radius of the ball, that is 4 mm. Material propertysuch as elastic modulus and Poissons ratio of the indenterand the specimen is similar, that is steel. A sample of

dynamic loading model for node at 00

is shown in Figure 6,with the applied radial load is 1000 N and the shaft speed is

1000 RPM.

4. VIBRATION ANALYSIS OF THE BEARING

n

anal

is shown in Fig 7, including the location ofs

ingbear

Dynamic explicit step is used to perform vibratio

ysis of the bearing model in Abaqus. The time period ofthis analysis is taken for one second. The history output is

requested as 2500 point during interval of analysis, meaningthe result data is written every 4 E-4 second of simulationtime.

Vibration analysis is performed by plotting the

acceleration, velocity, and displacement response as afunction of time at the point P1. Finite element model used in

this analyzed urepoint P1 and its axis direction. The housing structure idiscretized into 1920 finite elements and the outer r

ing part discretized into 1920 finite elements also.

Figure 7: Finite element model used for vibration analysis.

The simulation is carried out by applying 1000 N radialload to the model with a shaft speed of 1000 RPM. In thisstudy, the vibration analysis is performed for healthy anddefected bearing model, and then the response of both modelsis compared. Moreover, in order to validate the resultobtained from the finite element model simulation, the result

is compared with the experimental study.

4.1 Analysis of Healthy Bearing Model

ponse of 1000 N radial load, to the finite element

Analyzing the vibration response of the healthy bearingperformed by applying the dynamic loading model due tois

the res

P1



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model. Then the response of displacement, velocity, andacceleration at point P1 in three axis direction, that isx,y, andzdirection, is analyzed. Figure 8 shows the dynamic responseof displacement, velocity, and acceleration inx direction.

ux response

0.001

-0.001

-0.0008

-0.0006

-0.0004

-0.0002

0

0.0002

0 0.2 0.4 0.6 0.8 1

time (s)

u(mm)

0.0004

0.0008

0.0006

(a)

vx response

-12

-8

-4

0

4

8

12

0 0.2 0.4 0.6 0.8 1

time (s)

v(mm/s)

(b)

ax response

-500000

-400000

-300000

-200000

-100000

0

100000

200000

300000

400000

500000

0 0.2 0.4 0.6 0.8 1

time (s)

a(mm/s2)

(c)

Figure 8: The dynamic response of (a) displacement,(b) velocity, and (c) acceleration inx direction at point P1 for

healthy bearing.

Time domain analysis is used to analyze the vibrationresponse of this model. RMS (Root Mean Square) and peak

to peak value are used as time signal descriptors. Peak topeak value is determined as the difference of the maximumand minimum peak value. The RMS is defined as:

(7)

where Ns is the number of and xi is the amplitude ofvibr

shown in Table 1. T gnificant in x and z

dire

onse ofhealthy b

=

=sN

i

is

xN

RMS1

21

dataation.RMS and peak to peak value for displacement, velocity,

and acceleration response at point P1 of the healthy bearing ishe response is si

ction, since the radial load is transformed into these axisdirection. Response in y direction is just the response of the

structure from the deformation due to applied load, hence themagnitude of vibration is not significant.

Table 1: Time signal parameter for vibration respearing

Type of Axis Peak to

Response DirectionRMS

Peak Value

u 0.000274 0.0014454xuy 2.08E-06 1.454E-05

Displacement

(mm)uz 0.000287 0.0014559

vx 2.675483 16.32028vy 0.102216 0.750409

Velocity

(mm/s)vz 2.359384 19.50575

ax 127135.2 824651

ay 5725.132 40960.3Acceleration

2(mm/s )az 105861.2 686747

4.2 Analysis of Defected Bearing Model

is a ated on the no ema tensit o, at a hedirect load his po en n

of the de because this areainten plied al loa arin

A local defect on t ng ce i by

ampli magnitudes of t on henod e defected area of tioncon n sim as 6, ed andKarag 03). Th dth o s de hewid of the contact area on the con d for

m. The first

e local defecthile t

The defect ssumed loc de lying in thximum load in

ion of radial

fect

y q = 00

or p rallel with tFr. T int is chos as the locatio

experiences the largest loadsity due to ap

fying the

radi d on the be g.he beari s outer ra s modeled

he excitati force on tes lying in th

stant is chose

The value amplifica

ply as propos by Kiraluelle (20 e wi f defect i

stitutive nofined as te, henceth

this simulation the defect width is 0.506 m

contact point acts as the leading edges of thw he last contact point acts as the trailing edges.

Amplification constant of 6 is applied on the trailing edges ofthe defects, while 3 is applied on the leading edges of the

defects.Table 2 shows the amplified dynamic loading modelon the defected node, which is the node at = 0

0, and

compare with the original/healthy dynamic loading model.The acceleration response in thex direction at point P1 of

defected bearing is given in Figure 9(a), and compared to theresponse of healthy bearing that is shown Figure 9(b). This

response direction is observed because most of theexperiment uses single axis accelerometer as the sensor;hence the direction of the accelerometers sensitive axis isperpendicular with the mounted accelerometer.



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Table 2: Defected and healthy dynamic loading model

Defected Loading

Model

Original/Healthy

Loading Model

time (s) q (N) time (s) q (N)

0 2185 0 485.55560.000259 2913.332 0.000259 485.5553

0.000269 0 0.000269 0

0.016505 0 0.016505 0

0.016515 1456.666 0.016515 485.5553

0.017033 2913.332 0.017033 485.5553

0.017043 0 0.017043 0

0.033279 0 0.033279 0

0.033289 1456.666 0.033289 485.5553

0.033807 2913.332 0.033807 485.5553

0 0.033817 0 .033817 0

0.0500 0 0.54 050054 0

0.050064 1 4456.666 0.050064 85.5553

0.050582 29 2 0.05058213.33 485.5553

ax r es po ns e o f ec t

0. 0 8

tim

d ef ed bear in g

-1000000

-800000

-600000

-400000

-2000000

a(

0mm/s2

200000

400000

)

600000

800000

1000000

0.1 0.2 0.3 4 .5 0.6 0.7 0. 0.9 1

e (s)

(a)ax response of t

-1000000

-800000

-600000

-400000

-200000

0

200000

400000

600000

800000

1000000

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

time (s)

a(mm/s2)

heal hy bearing

(b)

Figure 9: The acceleratio of (a) defected and(b) healthy bearing inx direction at point P1.

As observed from Figure 9, the acceleration response ofthe defected bearing model has larger magnitude and randomspiky characteristic. This simulated vibration pattern hassimilar characteristics with the experimental result given inTao, et al. study (2007), which is shown in Figure 10.

n response

(a)

(b)

Figure 10: The vibration response of (a) defected and(b) healthy bearing in Tao, et al. experiment (2007).

Statistical parameter of RMS and peak to peak value foracceleration response at point P1 in all of three axis direction

of defected bearing which compare with the healthy bearingresponse is shown in Table 3. The response for defectedbearing is most significant in the x direction, wi themagnitude about two times e healthy bearing response,since the defect is locate in the direction of x directionloading in node at 0 degree. The response in zdirection is notsignificant, that is below 7 %, while in y direction even the

difference is between 33 44 %, but the magnitude is verysmall, compare to response inx andzdirection.

Table 3: Time signal parameter for accelerometer responseof defected bearing

th

of th

AccelerationDirection Type of Model RMS Peak toPeak Value

Healthy 127135.194 824651

Defected 228011.174 1664960ax

Difference (%) 79.35 101.90

Healthy 5725.132 40960.3Defected 8275.828 54483.7ay

Difference (%) 44.55 33.02

Healthy 105861.156 686747

Defected 105205.376 731820az

Difference (%) -0.62 6.56



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5. EFFECT OF SHAFT ROTATIONAL SPEED

The effect of various shaft rotational speeds for the

simulation is investigated in this section. The shaft speed willbe varied from 1000, 2000, 3000, and 4000 RPM. Radial load

applied in this test is 1000 N.Since the vibration response at point P1 is most sensitive

n the x axis direction, hence in this test vibration parameterx axis direction only

iwill be analyzed in . RMS and peak to

placement, velocity, and acceleo hy a ected

in haft speed. The RMS of divelocity, and acceleration respon haf sshow Figure 11 e pe lueFigure 12.

peak value of disresponse at p

vestigated for each s

rationbearing is

splacement,

int P1 for healt nd def

se for each s t speed in in , while th ak to peak va shown in

RMS

0

0.0002

0.0004

0 1000 2000 3000 4000 5000

RPM

0.0006

0.0008

splacemen

0.001

0.0012

0.0014

0.0016

Di

t(mm)

healthy

defected

(a)

RMS

2

4

6

8

10

12

Velocity(mm/s)

0

0 1000 2000 3000 4000 5000

RPM

healthy

defected

(b)

RMS

0

50000

100000

150000

200000

250000

300000

350000

400000

450000

500000

0 1000 2000 3000 4000 5000

RPM

Acceleration(mm/s

2)

healthy

defected

Peak to peak value

0

0.0005

0.001

0.0015

0.002

0.0025

0.003

0.0035

0.004

0.0045

0 1000 2000 3000 4000 5000

RPM

Displacem

ent(mm)

healthy

defected

(a)

Peak to peak value

0

10

20

30

40

50

60

70

80

0 1000 2000 3000 4000 5000

RPM

Ve

locity(mm/s)

healthy

defected

(b)

Peak to Peak Value

3000000

3500000

0

500000

1000000

1500000

2000000

2500000

0 1000 2000 3000 4000 5000

RPM

Acceleration(mm/s2)

healthy

defected

(c)

Figure 12: The peak to peak value of vibration response for(a) displacement, (b) velocity, and (c) acceleration ith

various s speeds.

The RMS magnitude for displacement, velocity andacceleration responses for both healthy and defected bearingtend to be higher for faster shaft speed, only in 2000 RPM the

trend is not showing the consistency. For displacementresponse of the defected bearing, the magnitude is a bit lowerthan the response at 1000 RPM, while the velocity is a bithigher than the response at 3000 RPM. For the healthy

bearing the inconsistency is just seen for the velocityresponse. This is caused by the dynamic characteristic of thestructure; otherwise the magnitude response is sti theacceptable range.

The peak to peak value also tend to be higher for fastershaft speed, even at speed of 2000 RPM the trend shows a bitinconsistency. Hence it can be observed that the faster the

shaft speed, the larger the magnitude of vibration response.

w

haft

ll in

(c)

Figure 11: The RMS of vibration response for(a) displacement, (b) velocity, and (c) acceleration with

various shaft speeds.



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This is because part of the machine vibrations isproduced by the repeatedly changes of rolling elementsangular position with time, and this change causes the innerand outer ring to experience periodic relative motion.Therefore, the higher the rotational speed of the bearing, the

faster the periodic relative motion and the higher themagnitude of vibration.

RMS

0

0.0002

0 500 1000 1500 2000 2500

Radial Load (N)

0.0004

0.0012

0.0014

0.0016

D

m)

0.0006

0.0008

0.001

isplacement(m

healthy

defected

(a)

RMS

3

4

5

6

7

8

9

Velocity(mm/s)

0

1

2

0 500 1000 1500 2000 2500

Radial Load (N)

healthy

defected

(b)

RMS

150000

200000

250000

300000

350000

400000

cceleration(mm/s2)

0

50000

100000

0 500 1000 1500 2000 2500

Radial Load (N)

A

healthy

defected

(c)

Figure 13: The RMS of vibration response for

(a) displacement, (b) velocity, and (c) acceleration withvarious radial loads.

6. EFFECT OF RADIAL LOADING

The effect of various ra loadings for the simulationwill be investigated in this section. The radial load is variedfrom 500, 1000, 1500, and 2000 Newton. The shaft rotational

speed for this test is 1000 RPM. The vibration response at

point P1 in thex axis direction is analyzed, RMS and peak topeak value of displacement, velocity, and accelerationresponse for healthy and defected bearing is investigated foreach radial loading condition. The RMS of displacement,velocity, and acceleration response for each radial load is

shown in Figure 13, while the peak to peak value shown inFigure 14.

dial

Peak to peak value

0

0.001

0.002

0.003

0.004

0.005

0.006

0 500 1000 1500 2000 2500

Radial Load (N)

Displacement(mm)

healthy

defected

(a)

Peak to peak value

0

0 500 1000 1500 2000 25

10

20

50

60

70

00

Radial Load (N)

30

Velocity

40

(mm/s)

healthy

defected

(b)

Peak to Peak Value

500000

1000000

1500000

2000000

2500000

3000000

3500000

Acceleration(mm/s2)

0

0 500 1000 1500 2000 2500

Radial Load (N)

healthy

defected

(c)

Figure 14: The peak to peak value of vibration response for(a) displacement, (b) velocity, and (c) acceleration with

various radial loads.

The RMS magnitude for displacement, velocity and

acceleration responses for both healthy and defected bearingtend to be higher for larger radial load. The peak to peak

value also shows the same trend. Even the peak to peak valuefor the velocity and acceleration response for load 1500 N ishigher than 2000 N, but the displacement response sh s thatthe magnitude of vibration response for 2000 N is higher than

ow



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the response for 1500 N. Hence it can be clearly concludedthat the larger the given load, the larger the magnitude ofvibration response of the structure. Since the higher the loadwill result the higher excitation force subjected to the bearing,hence the magnitude of vibration also will increase.

7. CONCLUSIONS

A finite element model simulation for analyzingvibration response of a bearing has been develo d. Adynamic loading model simulates the distribution lo in theouter race due to transfer lo rom the ball. Moreover, the

model to simulate the impulse force due to impact betweenthe ball and the defect located in the outer race is proposed.Time domain analysis is performed to evaluate the outputresult of vibration analysis from the finite element software.RMS and peak to peak value is used as the time signaldescriptors and can be used as a parameter for conditionmonitoring purposes.

The vibration response of healthy and defected bearing iscompared. The simulated vibration pattern has similarcharacteristics with results from experimental study inliterature. The effect of shaft rotational speed and ra al loadis investigated. It can be ob d that the faster the shafts

oto

etermine the vibra for various shaftpee

. (2007) Introduction to Contact

Mec

Mechanical Systems and SignalPro

eatritain.

Tandon, N. and Choudhury, (1999) A. A review ofent methods for the detection

f defects in rolling element bearings, TribologyInte

Engi ng and System Safety92, pp. 660 670.ensing, J. A. (1998). On the Dynamics of Ball Bearing.

niversity of Twente, Enschede, The

Y

tudent in Department of

Tec

pead

ad f

diserve

peed and the larger the radial load, the larger the magnitudef vibration response.

The proposed simulation method can be usedtion signal responsed

s ds and loading conditions, which can be used as thecondition monitoring application for the bearing structure.

ACKNOWLEDGMENTS

The authors would like to thank for ASEAN UniversityNetwork / Southeast Asia Engineering Education Develop-ment Network (AUN/SEED-Net) and JICA for their financialsupport in this research.

EFERENCER

Fischer-Crips, A.C

ndhanics, 2 ed. Springer Science and Business Media,USA.

Harris, T. A. (2001) Rolling Bearing Analysis, 4th ed.,John Wiley & Sons, Inc., Canada.

Harris, T.A. and Kotzalas, M.N. (2007) AdvancedConcepts of Bearing Technology, Taylor & Francis Groups,USA.

Holm-Hansen, B.T. and Gao, R.X. (2000) Structural

design and analysis for a sensor-integrated ball bearing.Finite element in Analysis and Design34, pp. 257 270.

Kral, Z. and Karagulle, H. (2003) Simulation andanalysis of vibration signals generated by rolling elementbearing with defects, Tribology International 36, pp. 667678.

Kral, Z. and Karagulle, H. (2006) Vibration analysis of

rolling element bearings with various defects under the actionof an unbalanced force,

cessing20, pp. 19671991.Rao, S. S. (2005), Mechanical Vibration, Pearson

Education South Asia Pte Ltd., Singapore.

SKF General Catalogue, (1989). SKF Groups. Gr

B

vibration and acoustic measuremo

rnational32 (1999), pp. 46980.Tao, B., Zhu, L., Dang, H. and Xiong, Y. (2007) An

alternative time-domain index for condition monitoring of

rolling element bearings A comparison study. ReabilityneeriW

PhD Thesis, Uetherlands.N

BIOGRAPH

Purwo Kadarno is a Master s

Engineering Design and Manufacture, University of Malaya,Malaysia. He received a Bachelor of Science fromDepartment of Aerospace Engineering, Bandung Institute of

hnology, Indonesia in 2006. His research interests include

vibration and stress analysis using finite element method. Hisemail address is [email protected].



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Vibration Analysis of Defected Ball Bearing Using