vibration analysis of bearings chapter 2

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Mechanical Systems

and

Signal ProcessingMechanical Systems and Signal Processing 20 (2006) 19671991

Vibration analysis of rolling element bearings with various

defects under the action of an unbalanced force

Zeki Kral, Hira Karagu lle

Department of Mechanical Engineering, Dokuz Eylul University, Engineering Faculty, 35100 Bornova, Izmir, Turkey

Received 2 December 2004; received in revised form 16 May 2005; accepted 17 May 2005

Available online 5 July 2005

Abstract

In this paper, a method based on the finite element vibration analysis is presented for defect detection in

rolling element bearings with single or multiple defects on different components of the bearing structure

using the time and frequency domain parameters. A dynamic loading model is proposed in order to create

the nodal excitation functions used in the finite element vibration analysis as external loading. A computer

code written in Visual Basic programming language with a graphical user interface is developed to create

the nodal excitations for different cases including the outer ring, inner ring or rolling element defects.Forced vibration analysis of a bearing structure is performed using the commercial finite element package I-

DEAS under the action of an unbalanced force transferred to the structure via a ball bearing. Time and

frequency domain parameters such as rms, crest factor, kurtosis and band energy ratio for the frequency

spectrum of the enveloped signals are used to analyse the effect of the defect location and the number of

defects on the time and frequency domain parameters. The role of the receiving point for vibration

measurements is also investigated. The vibration data for various defect cases including the housing

structure effect can be obtained using the finite element vibration analysis in order to develop an optimum

monitoring method in condition monitoring studies.

r 2005 Elsevier Ltd. All rights reserved.

Keywords: Rolling element bearings; Unbalanced force; Localised defects; Finite element vibration analysis

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www.elsevier.com/locate/jnlabr/ymssp

0888-3270/$ - see front matter r 2005 Elsevier Ltd. All rights reserved.

doi:10.1016/j.ymssp.2005.05.001

Corresponding author. Tel.: +90 232 388 3138 130; fax: +90 232 388 7864.

E-mail addresses: [email protected] (Z. Kral), [email protected] (H. Karagu lle).
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1. Introduction

Among the other mechanical components, researchers pay great attention to the rolling element

bearings due to their unquestionable industrial importance. Rolling element bearings arefrequently encountered in rotating machinery due to their carrying capacity and low-friction

characteristics. Rolling element bearings work under different conditions and frequently under

heavy loadings generated in the machinery and they are subjected to time and space varying

dynamic loads. The complexity of the loading mechanism in the bearings shows its effect in the

form of local defects. It is important to detect a defect at its incipient stage in order to prevent

long-term breakdowns or in some cases possible catastrophic failures. Different monitoring

techniques are used in industry to prevent machinery failures caused by the rolling element

bearings and new techniques are being developed. Vibration analysis is among the most common

method used in the monitoring applications since a local defect produces successive impulses at

every contact of defect and the rolling element, and the housing structure is forced to vibrate at itsnatural modes. The vibration pattern of a damaged bearing includes the low-frequency

components related to the impacts and the high-frequency components in which the structural

information of the bearing structure or the machine is stored.

Different techniques are employed in the studies related to the rolling element bearings. A

detailed review of these methods is given in Ref. [1]. The vibration signal containing the

information about the anomaly in the bearing structure is created artificially using the theoretical

considerations or collected from the experimental measurements. Basically, the time domain, the

frequency domain and recently the timefrequency domain analyses are used to extract useful

information about the existence of a localised defect. A well-established model including the load

distribution around the circumference of the rolling element bearings having localised defect and

the impulse response of the bearing structure are proposed in Ref. [2]. Multi-point defect case witharbitrary locations is analysed in Ref. [3]. Wang and Harrap presented a method which combines

the time-synchronous averaging and envelope spectral analysis techniques for diagnosing multiple

element defects of rolling bearings [4]. Natural modes are used to obtain the dynamic response of

the rings in Refs. [5,6] and the vibration signal is modelled as the combination of different

components such as fault, modulation due to non-uniform loading, inner or outer ring modes,

structural vibrations and noise in Ref. [7].

Time-domain parameters such as skewness and kurtosis are used to analyse the vibration

signals [811]. The vibration signal of a faulty bearing is dominated by the high-frequency ringings

related with the natural modes of the bearing structure and the impact frequencies are suppressed

in the frequency spectrum. To overcome this problem, the envelope method in which the resonantringings are subtracted from their envelopes and the vibration signal is represented by the

exponential decay functions with the frequency of related defect, is employed. The basic

understandings of the high-frequency resonance technique are presented by McFadden and Smith

[12]. McFadden and Toozhy combined the high-frequency resonance technique with the

synchronous averaging method for vibration monitoring of rolling element bearings [13].

The classical approaches in the frequency domain methods such as the short-time

Fourier analysis has not been widely used due to the non-stationary characteristics of the

bearing vibration signals. Recently, the wavelet method which provides variable time and

frequency resolution becomes very popular. This method is used by researchers frequently

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[1418]. Recently, a new method called as basis pursuit which provides fine resolution and sparsity

is used [19,20].

In addition to the use of the vibration signals in condition monitoring studies, acoustic signals

having ultrasonic frequencies are used for monitoring the condition of the bearings [2124]. Artificialneural networks are used by researchers entering different statistical indices obtained from the time

and frequency domain form of the vibration signals for training purpose of the network and then the

network reaches a decision for the condition of the examined bearing [25,26].

The finite element method is also used in the studies for defect detection in rolling element

bearings. Holm-Hansen and Gao [27] used the finite element method to calculate the changes in

the dynamic loading and speed variations associated with an outer ring fault. Kral and Karagu lle

[28] used the finite element model of a bearing structure to model the dynamic loading in a rolling

element bearing and performed the finite element vibration analysis to detect the outer ring fault

for different bearing geometries and loading conditions.

The aim of this study is to model the loading mechanism in a bearing structure which houses adeep groove ball bearing having different localised defects and carrying an unbalanced force

rotating with the shaft. This study differs from the others due to the load carried by the bearing.

In most studies related to the dynamics of the rolling element bearings, the load carried is

considered as a point load that means the direction and the magnitude of the load remain

unchanged during the operation. In practice, the case of an unbalanced force exists in the bearings

of a washing machine due to the non-uniform distribution of the drum content. Dynamic loading

model of the bearing structure is obtained considering the bearing kinematics and the loading

type, and the vibration response of the structure is calculated by means of the finite element

vibration analysis. The effects of different parameters such as the rotational speed, sensor

location, angular position and number of the outer ring defects, defect type (inner ring defect and

rolling element defect) on the vibration monitoring methods are examined by using the time andfrequency domain parameters.

2. Bearing model

In the studies related to the rolling element bearing vibrations, the vibration signal xt from arolling element bearing with a single-point fault is expressed as [7]

xt xftxqtxbst xst nt, (1)

where xft is the impulses produced by the fault, xqt is the modulation effect due to the non-uniform load distribution and xbst is the bearing-induced vibrations determined by the bearingstructure, xst is the machinery-induced vibrations determined by the machine structure or othercomponents, and n(t) is the noise which is encountered in any measurement system. Bearing and

machinery-induced vibrations are composed of damped oscillations having the structural

resonant frequencies excited by the impulse train whose amplitude and periodicity are determined

by the defect location, bearing properties and load distribution. The reader can find detailed

description of the signal components in Ref. [7]. The vibration response xt is determined mainlyby the transfer function of the bearing structure. The transfer function is estimated from the

impulse response of the bearing structure and the vibration signals with various faults are

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generated using this function. A model for bearing vibration is given by Ocak and Loparo [29] by

using the structural transfer function. In addition, the structural model for the bearing geometry

can be obtained by using the finite element method. The vibration response of the bearing

structure is calculated with the proper loading model.In this study, a bearing structure housing of a 6205 model deep groove ball bearing is

considered. The finite element method is employed in order to calculate the dynamic response of

the structure. The finite element model of the structure is shown in Fig. 1. The housing structure is

discretised into 23 964 10-node parabolic tetrahedron elements having 3 dof (ux, uy and uz) at

each node. The total number of nodes is 37 894. The material of the housing structure is isotropic

steel, E 206GPa, r 7860 kg=m3, and the material properties of the outer ring areE 200 GPa, r 7860 kg/m3. The housing structure is assumed to be subjected to an unbalancedforce. The magnitude of the rotational radial load is taken as 500 N. The unbalanced force rotates

with the shaft and excites the whole structure unlike the radial load having constant direction. The

governing equations of the system are written as

M xf g C _xf g K xf g ft , (2)

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Fig. 1. Rolling element bearing structure (dimensions in mm).



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where M, C, K are, respectively, the mass, damping and stiffness matrices. x, _x and x are,

respectively, the nodal displacement, velocity and acceleration vectors. The right-hand side of the

equation consists of time-dependent excitation functions acting on the each dof of the finite

element nodes. In this study, the bearing structure is excited by a moving distributed radial loadon the inner surface of the outer ring. The right-hand side of Eq. (2) is formed considering the

bearing kinematics and radial load distribution, and the vibration response of the bearing

structure is examined under the action of a set of excitation functions modelling the rotating

unbalanced force. Dimensions of the ball bearing used in this study are given in Fig. 2.

The load distribution should be considered while forming the dynamic loading model for the

finite element vibration analysis. In a rolling element bearing, the form of the load distribution is

given in Fig. 3.

The form of the distributed radial load is given as [2]

qf q0 1 1

2 1 cos f n0

( ) for fzofofz;elsewhere;

(3)

where e denotes the load distribution factor and is given as 0:51 Cd=2dmax in Ref. [6]. Cddenotes the diametral clearance, dmax denotes the maximum deflection in the direction of the

radial load and n 32

for ball bearings. In ball bearings e has the value between 0 and 0.5. q0 is the

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Fig. 2. Dimensions of the ball bearing.

Fig. 3. The form of the load distribution.



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maximum load intensity at f 01, and can be approximated by q0 5Fr=Z cos a [30], where Fris the radial load and Z is the number of balls. fz is the angular extent of the load zone. The

rolling elements transfer the radial load to the outer ring during their rotation with the cage

frequency expressed as fc fs=21 db=dm cos a where fs is the shaft frequency. The contactangle a is taken as 01 assuming that the ball bearing subjected to a pure radial load.

Having obtained the mass and stiffness matrices for the bearing structure, the natural

frequencies and the mode shapes of the structure are calculated by solving the following

K o2 M f 0, (4)where o is the angular natural frequency vector of the bearing structure and f is the mode shape

vector.

The governing equations of the system given in Eq. (2) are expressed in the modal form by using

the modal transformation

xf g P q , (5)where q is called as the principal coordinates and becomes the primary unknowns and [P] is the

modal matrix composed of the normalised mode shape vectors. The substitution of Eq. (5) into

Eq. (2) yields

M P q C P _q K P q ft . (6)If Eq. (6) is premultiplied by PT and applying the orthogonal properties of the normal modes, the

following equation can be obtained:

q

P

T C

P

_q X

2 q P T f

t

ut , (7)where u(t) is called as the modal force vector and [O] is the natural frequency matrix in diagonal

form.

Assuming the Rayleigh damping C aM M bK K ; where aM and bK are, respectively, themass and stiffness damping coefficients, Eq. (7) is written as

q aM I bK X 2

_q X 2 q ut , (8)where [I ] is the identity matrix. Eq. (7) can be written in the scalar form for each individual

equation

qi

2xioi_qi

o2qi

ui, (9)

where xi is the modal damping ratio for the ith mode and is taken as 0.005 for all the modes

considered in the calculation of the vibration response of the bearing structure.

The modal responses of the bearing structure are determined by the time step integration

methods. The time integration is applied to the Eq. (9), and the solution for the uncoupled

differential equations is obtained from Duhamel integral

qit 1

o0i

Zt0

uitexioitt sin o0it t dt, (10)

where o0i oiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 x2iq

:

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Using the modal displacements q(t), the nodal responses x(t) are expressed as the superposition

of the mode shapes fi by the equation xt

P qt

:Having obtained the excitation functions for 80 nodes on the inner surface of the outer ring, the

finite element vibration analysis is performed by using the computer-aided engineering package I-DEAS. The mode superposition method described in the above section is used for the vibration

analysis and the time increment is taken as Dt T10=20 where T10 is the period of the 10th naturalmode (fn10 14 055 Hz) of the bearing structure.

In the case of the constant direction radial load in which the direction of the centreline of the

loading zone remains unchanged during the shaft rotation, the rolling elements periodically enter

and then exit the loading zone. The load amplitude on a rolling element increases and then

decreases as the rolling element passes through the loading zone. In this case, the cage frequency,

the number of balls and the shape of the loading zone determine the loading functions created for

each node on the inner surface of the outer ring. On the other hand, in the case of the rotating

unbalanced force, the shaft frequency should also be considered when creating the nodalexcitation functions since the load zone rotates with the shaft. In this study, a computer code is

developed in order to create the nodal excitation functions in Visual Basic programming language

[31] with a graphical user interface. The developed code is capable of generating the nodal

excitation functions for different cases: constant direction load (gear and pulley forces), rotating

unbalanced force, localised defect on the outer ring, localised defect on the inner ring, localised

defect on the rolling elements and direction/amplitude variable load (joint reactions in a

mechanism). The number of defects, defect locations on the inner surface of the outer ring or on

the inner ring (between 01 and 3601), the label of the defected rolling element, the number of

rolling elements (Z), the load zone parameters (e, f), the shaft speed and the time history of the

rotating force for the case of direction/amplitude variable load are the variables included in the

computer code.The form and the parameters of the bearing loading under the action of an unbalanced force

are shown in Fig. 4. In the finite element model of the outer ring, the raceway is modelled by 80

nodes shown in Fig. 4. These nodes are spaced equally with an angular interval, Dy 4.51. Thenumber of nodes in the load zone is Nz 2N-1, where N is the number of nodes satisfying thecondition jyni yjpfz: In Fig. 4, y represents the shaft rotation, yni (i 1Nz) denotes theangular position of the nodes in the load zone and ybj (j 1Z) denotes the angular position ofthe balls. The nodes which are in the close vicinity of a ball are subjected to a radial load. These

nodes are determined considering the position of the radial load, the position of the balls and the

position of the nodes on the circular path. A node is stored in the group of the excited nodes if the

following condition is satisfied:

jyni ybjjp0:5 Dy; i 12Nz; j 12Zfor a particular instant of rotation. (11)

In the loading model, the ball load is assumed to be carried by the nearest node. If a ball in the

load zone is close to the ith node then the associated force is applied to the ith node, otherwise the

force is applied to the (i+1)th node.

Once the label of the node is determined, then the magnitude of the force acting on this node is

calculated according to its position with respect to the load zone symmetry axis. Two different

stages of the radial loading are shown in Fig. 5. It is observed from the figure that the rolling

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elements rotate with the cage frequency (fc) and different rolling elements transfer the load having

different magnitudes to the outer ring and then to the housing structure.

In the analysed case, the nodes on the inner surface of the outer ring are subjected to different

amplitude loads at any instant of rotation, and the load magnitudes do not repeat themselves

periodically. A sample loading function for the node at 451

is given in Fig. 6 for 9 rotations of theshaft rotating at 1000 rpm.

A detailed view of the load definition on the nodes is shown in Fig. 7. The loading functions are

created for three neighbouring nodes considering the angular position of ball 1 with respect to the

position of the nodes and the load zone. The loading functions are created for 80 circumferential

nodes by the computer code in the same manner given in Fig. 7. The parameters listed below

should be considered when creating the excitation functions

The shaft and the cage frequencies. The number of balls.

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Fig. 4. The form and the parameters of the radial loading.

Fig. 5. Two different stages of the radial loading.



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3. Localised bearing defect cases

Depending on the working conditions, a localised defect may appear on different componentsin a rolling element bearing. Each fault case has its own repetition frequency and defined for

stationary outer ring as follows [1]:

Defect on the outer ring : for Zfs2db 1 dbdm

cos a

: (12)

Defect on the inner ring : fir Zfs2 1 dbdm cos a

: (13)

Defect on the rolling element : fre

fsdmdb

1

d2b

d

2

m

cos2 a : (14)This modelling is given for the ideal case in which there is no sliding and contact angle variation.

3.1. Defect on the outer ring

In the case of a localised defect on the outer ring, successive impulses are produced when a ball

carrying a portion of the radial load contacts with a local defect. The magnitude of the impulses

depends on various parameters such as amplitude of the radial load, size of the defect, material

and velocity. In this study constant defect size is assumed. The defect is modelled by its force

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Fig. 8. x-components of the excitation functions for healthy bearing.



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equivalent. Mainly the defect size affects the impact factor of the ball force. The defect is assumed

to lie between the two neighbouring nodes and the width of the defect is approximately 1.57 mm.

An assumption is made to obtain an impact model as described in Fig. 9. The load magnitude

carried by a ball is multiplied by 3 for the leading edge of the local defect and by 6 for the trailingedge or the impact edge of the local defect in order to model the collision between the ball and the

local defect. The computer code calculates the time of the ball-defect contact and produces the

impulse forces when the impacts occur. In this way, the excitation functions include the effect of

the local defects by an equivalent force model.

The amplification in the nodal force due to the ball-defect impact is shown in Fig. 10. The

magnitude of the impact is different at every contact because the ball which is in the close vicinity

of a local defect carries different load magnitudes due to the rotation of the radial load. The

anomaly in the vibration pattern of the structure is due to these impacts.

The acceleration responses calculated at point P3 for healthy and faulty bearing are given in

Fig. 11. As observed from Fig. 11, the acceleration response has a spiky characteristic for an outer

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Fig. 9. Description of the impact mechanism.

Fig. 10. x- and y-components of the excitation functions with outer ring defect at 2251, 1000 rpm.



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ring defect and this simulated vibration pattern agrees well with the experimental results given in a

previous study [17].

Condition monitoring via vibration analysis is the most popular method in practice. The time,

frequency and joint timefrequency domain analyses are applied to the vibration signals for

condition monitoring of rolling element bearings. The determination of the optimum sensor

position and optimum method are always questioned, and researchers pay great attention to this

subject. In this study, vibration analyses are performed for possible accelerometer positions P1,

P2 and P3. Recently, some results are presented on the determination of the optimum sensor

position and optimum time-domain parameter in Ref. [28].

In this section, the effect of the location of a localised single outer ring defect is investigated.The angular position of a single local defect is changed on the outer ring with the angular interval

of 451. The acceleration response of the structure is calculated for three receiving points for a

broad range of shaft speeds ranging from 1000 to 15 000 rpm. Three different well-known time-

domain parameters rms, crest factor and kurtosis are used to determine the best sensor position

which is more sensitive to a defect. These parameters are defined for a discrete signal x as

rms

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

Ns

XNsi1

x2i ; Fc

vuut maxx minxrms

; Kurtosis XNs

i1 xi meanx 4

Nsrms4, (15)

where Ns stands for the number of samples.The statistical parameters for the acceleration responses at points P1, P2 and P3 are shown in

Figs. 1214. It is observed from Fig. 12 that the rms ratios give information about the existence of

a local defect nearest to the sensor successfully. A defect located at 901 can be detected by a sensor

located at the left- or right-hand side of the housing structure. For points P1 and P3 the rms ratios

are insensitive to a defect in any arbitrary locations at 7000, 13 000 and 140 00 rpm. For point P2,

the rms ratios are insensitive to a defect in any arbitrary locations at 10 000, 13 000 and

14 000 rpm. These insensitive speed regions can be attributed to the dynamic behavior of the

bearing structure. The acceleration rms ratios have a wavy nature indicating the rotational speed

dependency. The point P2 can be used as a sensor location to detect a defect at arbitrary angular

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Fig. 11. Simulated acceleration responses at point P3, 1000 rpm: (a) healthy bearing and (b) faulty bearing (outer ring

defect).



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positions by using the acceleration rms ratios especially at low speeds. Points P1 and P3 can alsobe used. A defect located at 2251 or 3151 is difficult to detect for every receiving points. The

acceleration crest factor ratios are given in Fig. 13 and they can be used to detect the defects

located at any arbitrary positions on the outer ring for relatively low speeds up to 3000 rpm. The

effectiveness of the crest factor reduces as the shaft speed increases. The acceleration kurtosis

ratios are shown in Fig. 14 and it is observed that the acceleration kurtosis ratios can be a good

defect indicator at low speeds. The kurtosis ratios lose their ability to point out the existence of a

defect at the high-speed region. This is due to the decreasing duration between two successive

impulses and the interference of the impulse responses.

It can be concluded from Figs. 1214 that the crest factor and the kurtosis ratios can be used for

defect detection at relatively low shaft speeds. The rms ratios can be used in a broad shaft speedregion except some speeds depending on the bearing structure. In general, the defects located at

the upper part of the bearing structure are easy to monitor.

3.2. Effect of the number of the outer ring defect

At the further stage of the defect formation in a rolling element bearing, the number of defect

may increase during the operation. In this section, the effect of the defect number on the time-

domain parameters is investigated by locating up to four defects on the outer ring. The number of

defects on the outer ring is increased step by step to show the effect of the defect number on the

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0

2

4

6

8

10

12

1000 3000 5000 7000 9000 11000 13000 1 500 0

Shaft Speed (RPM)

rmsfaulty

/rmshealthy

rmsfaulty

/rmshealt

hy

rmsfaulty

/rmshealthy

0

45

90

135

180

225

270

315

0

2

4

6

8

10

12

1000 3000 5000 7000 9000 11000 13000 1 500 0

Shaft Speed (RPM)

0

45

90

135

180

225

270

315

0

2

4

6

8

10

12

1 00 0 3 00 0 5 00 0 7 00 0 9 00 0 1 10 00 13 000 1 50 00

Shaft Speed (RPM)

Shaft Speed (RPM)

0

45

90

135

180

225

270

315

Defect angle

(degree)

Defect angle

(degree)

Defect angle

(degree)

(a) (b)

(c)

Fig. 12. Rms ratios for the acceleration responses: (a) P1, (b) P2 and (c) P3.



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0

2

4

6

8

10

12

10 00 300 0 5 000 700 0 9 000 110 00 1 3000 1 5000

Shaft Speed (RPM)

1 000 30 00 5000 70 00 9000 11 000 13000 1 5000

Shaft Speed (RPM)

1000 3 000 500 0 7 000 900 0 1 1000 130 00 1 500 0

Shaft Speed (RPM)

Kurtosisfaulty

/Kurtosishealthy

Kurtosisfaulty

/Kurtosishealthy

Kurtosisfaulty

/Kurtosishe

althy

0

45

90

135

180

225

270

315

0

2

4

6

8

10

12

0

45

90

135

180

225

270

315

0

2

4

6

8

10

12

0

45

90

135

180

225

270

315

Defect angle

(degree)

Defect angle

(degree)

Defect angle

(degree)

(a) (b)

(c)

Fig. 14. Kurtosis ratios for the acceleration responses: (a) P1, (b) P2 and (c) P3.

0.5

1

1.5

2

2.5

3

1 00 0 3 00 0 50 00 7 000 9 00 0 1 10 00 1 30 00 1 50 00

Cffaulty

/Cfhealthy

Cffaulty

/Cfhealthy

Cffaulty

/Cfhealthy

0

45

90

135

180

225

270

315

0.5

1

1.5

2

2.5

3

0

45

90

135

180

225

270

315

0.5

1

1.5

2

2.5

3

Shaft Speed (RPM)

1 000 3 000 5 00 0 7 00 0 9 00 0 1 100 0 13 00 0 15 000

Shaft Speed (RPM)

10 00 3 00 0 5 00 0 7 000 9 00 0 11 00 0 130 00 1 50 00

Shaft Speed (RPM)

0

45

90

135

180

225

270

315

Defect angle

(degree)Defect angle

(degree)

Defect angle

(degree)

(a) (b)

(c)

Fig. 13. Crest factor ratios for the acceleration responses: (a) P1, (b) P2 and (c) P3.



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time-domain parameters. The results of a single defect located at 2251 are presented in the

previous section. In this section, multiple defects are introduced on the outer ring. Firstly, two-

defect case located at 451 and 2251 is analysed. Then the number of defects is increased to three

and the locations of the local defects are 451, 1351 and 2251. Finally, the case with four defectslocated on the outer ring at the angular positions 451, 1351, 2251 and 3151 as shown in Fig. 15 is

considered. The results of the time domain analyses are given in Figs. 1618. The computer code

creates the excitation functions for all the analysed cases.

The statistical parameter ratios calculated from the acceleration responses for two-defect case

are given in Fig. 16. The acceleration rms ratio can be used as a defect indicator for point P2 at

low speeds, and it can be used for points P1 and P3 at high speeds. At some speeds, rms ratios are

far away from being a defect indicator. The acceleration crest factor and kurtosis ratios can be

used at low speeds as seen in Fig. 16(b) and Fig. 16(c). The acceleration crest factor ratios are only

suitable at some speeds for the points located on both sides of the bearing structure ( P1 and P3)

for defect detection and the acceleration kurtosis ratios can be used for low speeds for threereceiving points.

The statistical parameters for three-defect case located at 451, 1351 and 2251 are given in

Fig. 17. The same trend is recorded for rms, crest factor and kurtosis ratios. But, the values of

crest factor and kurtosis ratios decrease as the rms ratios increase. This is expected because the

vibration energy increases as the defect number increases. The crest factor and the kurtosis ratios

decrease due to the increasing uniformity of the vibration pattern.

The statistical parameter ratios for four-defect case are given in Fig. 18. The trend in the ratios

is the same as two-defect and three-defect cases. The acceleration rms ratios given in Fig. 18(a) can

be used to detect the defects for four-defect case like the other defect cases. The acceleration crest

factor ratios are only suitable at some speeds for the points located on both side of the bearing

structure (P1 and P3) for defect detection and the acceleration kurtosis ratios can be used at lowspeeds for three receiving points. It can be concluded for varying defect number cases that, only

the rms ratios are influenced by the number of defect located in the bearing structure. The crest

factor and kurtosis values are insensitive to the changes in the defect number.

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Fig. 15. Multiple defects on the outer ring.



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3.3. Defect on the inner ring

Depending on the working conditions, a local defect may appear on the inner ring of a rolling

element bearing. In this section, a single defect located at 01 on the inner ring is considered and the

excitation functions are created accordingly. The rotational speed of the inner ring is greater than

the cage speed and an impulse occurs when the defect on the inner ring strikes a ball. The inner ring

defect produces periodic impulses if the defect is in the load zone. But, the magnitude of the impulse

is different at every contact due to the changing ball load. A strike occurs when ydi ybj and at thistime the node satisfying the condition jyni ybjjpDy is excited by the impulse force. The impactmodel for the inner ring defect case is given for two different instants of the shaft rotation in Fig. 19.

The frequency of the inner ring defect is greater than the outer ring defect frequency because adefect on the inner ring strikes a ball more times in one revolution of the inner ring. The rolling

elements are assumed to be perfectly rigid and the impulse force is transferred to the outer ring

directly. The excitation function for the associated outer ring node is created accordingly. The

node label and the time at the strike are stored for function generation. The parameters which are

used while creating the nodal excitation functions for the inner ring defect case are as follows:

The position of the defect/s on the inner ring. The position of the rolling elements (balls) with respect to load zone. The label and the angular position of the node on the outer ring at the defect-ball strike.

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0

0.5

1

1.5

2

2.5

1000 3000 5000 7000 9000 11000 13000 15000

Cff

aulty

/Cfh

ea

lthy

P1

P2

P3

0

0.5

1

1.5

2

2.5

3

3.5

4

1000 3000 5000 7000 9000 11000 13000 15000

Kurtosisfaulty

/Kurtosishealthy

P1

P2

P3

0

1

2

3

4

5

6

7

1000 3000 5000 7000 9000 11000 13000 15000

Shaft speed (RPM) Shaft speed (RPM)

Shaft speed (RPM)

rmsfaulty

/rmshealthy

P1

P2

P3

(a) (b)

(c)

Fig. 16. Time-domain parameters for the acceleration responses, two-defect: (a) rms, (b) crest factor and (c) kurtosis.



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The functions including the impulses are created separately and then added into the excitation set

in which the excitation functions for healthy case are defined. The excitation functions including a

single inner ring defect are given in Fig. 20. The vibration pattern for the inner ring defect case is

shown in Fig. 21. This vibration pattern agrees well with the experimental result given in Ref. [14].

The ratios of the statistical parameters are given in Fig. 22. The acceleration rms ratios given in

Fig. 22(a) for three receiving points are good indicator while defect detection. The inner ring

defect can be detected at all rotational speeds for point P2. Points P1 and P3 can also be used

except 7000 rpm by using rms ratios. The acceleration crest factor ratios may be used in the middle

speed region for points P2 and P3 but fail at low and high speeds (Fig. 22(b)). The acceleration

kurtosis ratios may be used only at low rotational speeds ( Fig. 22(c)). The rms ratios seem to be agood defect indicator for the inner ring defect case.

3.4. Defect on the rolling element

The rolling element defect is another type of rolling element bearing failure. A local defect is

introduced on a rolling element for the analysis. The rolling elements rotate with the cage about

the bearing axis and spin about their own axis with a rotational speed fr dmfs=2db1 d2b=d2mcos2 a simultaneously. Therefore, a defect strikes both the inner ring and the outer ring.The impulses are created for the outer ring nodes which are determined according to the position

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0

2

4

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10

1000 3000 5000 7000 9000 11000 13000 15000

Shaft speed (RPM)

1000 3000 5000 7000 9000 11000 13000 15000

Shaft speed (RPM)

1000 3000 5000 7000 9000 11000 13000 15000

Shaft speed (RPM)

P1

P2

P3

0.5

0.8

1.1

1.4

1.7

2

P1

P2

P3

0

0.5

1

1.5

2

2.5

3

P1

P2

P3

(a) (b)

(c)

Cff

aulty

/Cfh

ealthy

Kurtosisfaulty

/Kurtosishealthy

rmsfaulty

/rmshealthy

Fig. 17. Time-domain parameters for the acceleration responses, three-defect: (a) rms, (b) crest factor and (c) kurtosis.



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of the defect and the inner or the outer ring strike. The parameters which are used to define the

impulses are as follows:

The position of the defect with respect to the ball axis. The position of the load zone. The position of the defected ball with respect to the position of the load zone. The label and the angular positions of the nodes on the outer ring at the defect-inner ring or

defect-outer ring strike.

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2

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8

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12

1000 3000 5000 7000 9000 11000 13000 15000

P1

P2

P3

0

0.4

0.8

1.2

1.6

2

1000 3000 5000 7000 9000 11000 13000 15000

Shaft speed (RPM)Shaft speed (RPM)

Shaft speed (RPM)

P1

P2

P3

0

0.5

1

1.5

2

2.5

1000 3000 5000 7000 9000 11000 13000 15000

P1

P2

P3

(a) (b)

(c)

Cff

aulty

/Cfh

ealthy

Kurtosisfaulty

/Kurtosishealthy

rmsfaulty

/rmshe

althy

Fig. 18. Time-domain parameters for the acceleration responses, four-defect: (a) rms, (b) crest factor and (c) kurtosis.

Fig. 19. Inner ring defect and impact mechanism.



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It should be noted that, the cage frequency must be taken into account while calculating the

angular position and the time of the defect-inner ring or the defect-outer ring strike. A strike

occurs when the rolling element spin angle yrd i180-ybj (i 13) and at this time, the nodewhich is in the close vicinity of a faulty ball and satisfying the condition jyni ybjjpDy is excitedby an impulse force. The impact mechanism for the rolling element defect case is given in Fig. 23.

The excitation functions for the rolling element defect case are seen in Fig. 24. The number of

impact is less than the other two defect cases since the defected ball enters the loading zone in a

longer period or in other words the load zone passes over the defected ball quickly. In the case of

the ball slippage, the time of the occurrence of each impulse changes and the excitation functionsslightly differ from the normal case. In this study, the effect of the ball slippage is omitted. The

vibration pattern for the rolling element defect case is shown in Fig. 25. This vibration pattern

agrees well with the experimental result given in Ref. [17]. The ratios of the statistical parameters

for the acceleration response are given in Fig. 26. The acceleration rms ratios are good defect

indicators for the high-speed region and at low speeds the rms ratios are not suitable for defect

detection. The crest factor ratios work well especially for point P2, but, points P1 and P3 are not

suitable for the defect detection. The kurtosis ratios are suitable for the rolling element

defect detection in a broad range of shaft speeds as opposite to the other defect cases especially for

point P2.

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Fig. 21. Simulated acceleration response at point P2, inner ring defect, 1000 rpm.

Fig. 20. x- and y-components of the excitation functions with inner ring defect.



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4. Frequency domain analyses

Another conventional approach is processing the vibration signals in the frequency domain.

The basic indicator is the characteristic defect frequencies in the frequency domain analysis. The

characteristic defect frequencies depend on the rotational speed and the location of the defect in a

bearing. The existence of one of the defect frequencies in the direct or processed frequency

spectrum is the main indicator of the fault.

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1000 3000 5000 7000 9000 11000 13000 15000

Shaft speed (RPM)

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P2

P3

0.6

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1.4

1.6

1000 3000 5000 7000 9000 11000 13000 15000

Shaft speed (RPM)

P1

P2

P3

0.6

0.8

1

1.2

1.4

1.6

1.8

1000 3000 5000 7000 9000 11000 13000 15000

Shaft speed (RPM)

P1

P2

P3

(a) (b)

(c)

Cffaulty

/Cfhea

lthy

Kurtosisfaulty

/Kurtos

ishealthy

rmsfaulty

/rmshealthy

Fig. 22. Time-domain parameters for the acceleration responses, inner ring defect: (a) rms, (b) crest factor and (c)

kurtosis.

Fig. 23. Rolling element defect and impact mechanism.



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Fig. 27 shows the frequency spectra of the vibration (displacement) signal for both healthy and

faulty case (single defect on the outer ring). The outer ring defect frequency is shown in Fig. 27(b).

But in most cases, these defect frequencies are not seen in the frequency spectrum of a raw

vibration signal (velocity or acceleration) due to the dominant high frequencies related to the

resonant ringings. In practice, the envelope method is being used to overcome this problem. In

this section, the envelope method is employed for all the studied cases. The basic understandings

of the envelope method are given in Ref. [12]. In this method, the vibration signal dominated by

the high-frequency ringings carrying the information about the impulse response of the structure

is filtered around a resonant frequency. The envelope of the filtered signal is obtained taking the

Hilbert transform. The Hilbert transform of a real-valued signal x(t) is defined as [7]

~xt 1p

Z11

xtt t dt: (16)

The signal envelope of x(t) is 9z(t)9.

zt xt i~xt, (17)where i

ffiffiffiffiffiffiffi1

p:

The velocity signals are used in the envelope analyses. The acceleration and velocity signals

behave in the same manner in the frequency domain analysis [28]. The velocity signals calculated

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Fig. 24. x- and y-components of the excitation functions with rolling element defect.

Fig. 25. Simulated acceleration response at point P2, rolling element defect, 1000 rpm.



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for point P2 are processed by using the envelope procedure given in Fig. 28. The vibration signals

are band-pass filtered around the one of the resonant frequency of the structure (7340 Hz) for

which the modal displacements are purely in the y-direction using a fourth-order Butterworth

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0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

P1

P2

P3

0.8

1

1.2

1.4

1.6

1.8

2

2.2

1000 3000 5000 7000 9000 11000 13000 150001000 3000 5000 7000 9000 11000 13000 15000

1000 3000 5000 7000 9000 11000 13000 15000

P1

P2

P3

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

P1

P2

P3

Cff

aulty

/Cfh

ealthy

Kurtosisfaulty

/Kurtosishealthy

Shaft speed (RPM) Shaft speed (RPM)

Shaft speed (RPM)

rmsfaulty

/rmshe

althy

(a) (b)

(c)

Fig. 26. Time-domain parameters for the acceleration responses, rolling element defect: (a) rms, (b) crest factor and (c)

kurtosis.

Fig. 27. Frequency spectra of the vibration signals with outer ring defect: (a) healthy bearing and (b) faulty bearing.



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filter. The bandwith of the band-pass filter is selected as 4fdefect where fdefect is the related defect

frequency, for, fir or fre. The band energy or envelope energy is used as the area under the

frequency spectrum of the enveloped signal for the frequency interval (fdefect-0.1fdefect)p

fp(fdefect+0.1fdefect) in this study, and it is calculated from the FFT of the enveloped signal for

the given frequency interval.

The band energy ratios are given in Fig. 29 to show the usefulness of the envelope technique for

different number of defects and different defect locations. It can be observed from Fig. 29 that theband energy ratios are very successful for defect detection especially at low speeds for all outer

ring defect cases. The envelope method can also be used successfully to detect the inner ring

defects. The envelope method gives a better response in the relatively low-peed region for the

outer ring defects and in the middle speed region for the inner ring defects. The rolling element

defects are relatively difficult to detect by using the envelope method except some rotational

speeds. Attention must be paid while using the envelope method to detect the rolling element

defects. It should be noted that different resonant frequencies can be excited by different type of

faults at different rotational speeds. The efficiency of the envelope method may be improved by

starting the envelope procedure around the proper resonant frequency.

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Fig. 28. Envelope analysis procedure.



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5. Conclusions

In the studies related to the condition monitoring of the rolling element bearings, generally thedirection of the radial force is assumed to be constant, and the bearing structure is assumed to be

excited from the same region defined by the load distribution expression. In addition, the artificial

vibration signals do not include the real structural information. The use of the finite element

vibration analysis with the proper loading model, which is proposed in this study, produces

simulated vibration signals including the structural information in order to find the most efficient

analysis method. The results obtained from the statistical parameters show that the defect position

on the outer ring, number of defects and defect locations, on the inner ring or on the rolling

element, and the shaft speed affect the statistical indices. The envelope method can be used

efficiently in order to detect the outer and inner ring defects, but rolling element defects are not

easy to detect via envelope and band energy ratio procedures.

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Fig. 29. Results of the envelope analysis.



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vibration analysis of bearings chapter 2

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