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BAHIR DAR INSTITUTE OF TECHNOLOGY
Faculty of Mechanical and Industrial Engineering
Department of Mechanical Design
Graduate Seminar Title: - Vibrational Analysis of Ball Bearing
Prepared by Tibebu Meride Id No, BDU07025325PR
Advisors Ass.Prof. Yonas M.
Date 05/2/2008E.C
i
ABSTRACT Vibration signals for two different defect sizes have been extracted and an index for
comparison of different defect sizes has been proposed. Ball bearings are widely used in
industry from home appliances to aerospace industry. Proper functioning of these machine
elements is extremely important in order to prevent catastrophic damages. It is therefore,
important to monitor the condition of the bearings and to know the severity of the defects
before they cause serious catastrophic damages. Hence, the study of vibrations generated by
these defects plays an important role in quality inspection as well as for condition
monitoring of the ball bearing/machine element. This paper describes the vibration analysis
technique to detect the defects in the ball bearing, bearing life estimation, sound in bearing,
application in vibration analysis of ball bearing, bearing damage and finite element analysis.
Keywords- Vibrational Analysis of Ball Bearing, defect, sound in bearing.
ii
ACKNOWLEDGEMENT
Thanks to God for this seminar work properly with time. I would like to express my advisor
Ass.Prof. Yonas Miteku for his guidance and constant support in helping me to conduct
and complete this work.
iii
Contents
ABSTRACT ........................................................................................................................................ i
LIST OF TABLE ............................................................................................................................ iv
LIST OF FIGURE .......................................................................................................................... iv
NOMENCLATURE ....................................................................................................................... vi
1.INTRODUCTION ........................................................................................................................ 0
1.1. Sources of Vibration ............................................................................................................. 1
1.2. Problem Statement ............................................................................................................... 2
1.3. Objectives .............................................................................................................................. 3
1.3.1. Specific objective: .......................................................................................................... 3
2.LITERATURE REVIEW ............................................................................................................ 4
3.TYPES OF VIBRATION AND SOUND IN BEARING ........................................................... 5
3.1. Structural vibration and sound of bearing ..................................................................... 5
3.2 Structural Model of the Outer Ring................................................................................. 7
3.3. Bearing Damage ................................................................................................................ 8
3.4. Ball Bearings and Vibration Analysis ........................................................................... 10
3.5. Vibration Patterns, Analysis and Condition Monitoring ............................................ 12
3.6. Determination of Friction Coefficients in Bearing ...................................................... 13
3.7. Bearing Type and Bearing Material ............................................................................. 14
4.FINITE ELEMENT ANALYSIS .............................................................................................. 14
4.1. Finite Element Model of the Outer Ring .......................................................................... 14
5.RESULTS AND DISCUSSION ................................................................................................. 16
6.CONCLUSION ........................................................................................................................... 21
REFERENCE ................................................................................................................................ 22
iv
LIST OF TABLE
Table 1. RMS values of Polyacetal ball bearing for different defects. ................. Error!
Bookmark not defined.
Table 2: Show Result for 0.5mm outer race defect at 10kg load .................................. 16
LIST OF FIGURE
Figure 1 Influence of rotation speed on race noise [13] ........................................................ 6
Figure 2 Simplified model of the bearing outer ring [11] ..................................................... 7
Figure 3 Typical deep-groove ball bearing ......................................................................... 10
Figure 4 half section deep-groove ball bearing ................................................................... 11
Figure 5: Ball bearing components, applied force, load zone and load distribution [12] ... 11
Figure 6: -The possible lubrication regime distribution within a bearing (Load zone
upwards) [12] ...................................................................................................................... 13
Figure 7 half section Mesh ball bearing .............................................................................. 15
Figure 8 Total deformation of ball bearing ......................................................................... 15
Figure 9:-Graph of speed vs. amplitude for1*0.5outer race defect at 10kg [8] .................. 17
Figure 10 half section of ANSYS result of random vibration of normal elastic strain ..... 17
Figure 11 half section shear elastic strain random vibration result ..................................... 18
Figure 12random vibration result for equivalent stress ....................................................... 18
Figure 13half section random vibration result of normal stress ...................................................... 19
v
vi
NOMENCLATURE
FFT Fast Fourier Transformer
RMS Root Mean Square
FE Finite Element
POM Poly Oxy Methylene
Tm Melting Temperature
1.INTRODUCTION
Rolling elements are widely used as low friction joints between rotating machine
components. Since the rotational motion is often a significant function of the overall system,
such as wheels on a train, rollers in a paper mill, or a rotor on a helicopter, proper functioning
of a bearing over its designed life cycle is of vital importance to ensure product quality,
prevent machine damage or even loss of human life [1].
The bearing type used in this study is a single row deep groove ball bearing. They are the
most popular of all rolling bearing because it is non-separable, capable of operating at high
even very high speeds, and require little maintenance in service. The bearing model 6204
from SKF is used in study. This bearing has a bore diameter of 20 mm and widely used for
many applications. The work has been extended with Finite Element Analysis of bearing
with artificial defects to study the peaks at its outer ring as well as inner ring defect
frequencies. It is concluded that at constant defect size and constant load with different
speeds of rotation, amplitudes of vibration vary with increase in speed. In this case also
amplitudes of vibration are observed higher for outer ring defected bearings than inner ring
defected bearings for same defect size. U. A. Patel, Shukla Rajkamal [3]. Ball bearing is the
most basic component used in machinery for various engineering applications. Most of the
engineering applications such as electric motors, bicycles and roller skates use these
bearings, which enable rotary motion of shafts apart from complex mechanisms in
engineering such as power transmissions, gyroscopes, rolling mills and aircraft gas turbines.
In general ball bearings are made of four different components, an inner ring, an outer ring,
the ball element and the cage. The cage element helps in separating the rolling elements at
regular intervals and also it holds them in place within the inner and outer raceways to allow
them to rotate freely [5].
Even a newly manufactured bearing may also generate vibration due to components running
at high speeds, heavy dynamic loads and also contact forces which exist between the bearing
components.
1
Bearing defects may be classified as: -
Localized
Distributed.
The localized defects include: - cracks, pits and spalls caused by fatigue on rolling surfaces
[7]. The distributed defects include: - surface roughness, waviness, misaligned races and off
size rolling elements.
The sources of defects may be due to either manufacturing error or abrasive wear. Hence,
study of vibrations generated by these defects plays an important role in quality inspection
as well as for condition monitoring of the ball bearing/machinery [2]
In order to prevent bearing failure there are several techniques in use. such as: -
Oil Analysis,
Wear Debris Analysis,
Vibration Analysis and
Acoustic Emission Analysis.
Among them vibration and acoustic emission analysis [8] is most commonly accepted
techniques due to their ease of application. The time domain and frequency domain analysis
[3] are widely accepted for detecting malfunctions in bearings. The frequency domain
analysis is more useful as it identifies the exact nature of defect in the bearings. These
frequencies of the ball bearing depend on the bearing characteristics and are calculated from
the relations shown below [4].
1.1. Sources of Vibration
Rolling contact bearings represents a complex vibration system whose components – i.e.
rolling elements, inner raceway, outer raceway and cage – interact to generate complex
vibration signatures. Although rolling bearings are manufactured using high precision
machine tools and under strict cleanliness and quality controls, like any other manufactured
part they will have degrees of imperfection and generate vibration as the surfaces interact
through a combination of rolling and sliding. The level of the vibration will depend upon
many factors, including the energy of the impact, the point at which the vibration is
measured and the construction of the bearing.
Variable compliance vibration is heavily dependent on the number of rolling elements
supporting the externally applied load; the greater the number of loaded rolling elements,
the less the vibration. For axially loaded ball bearings operating under moderate speeds the
2
form and surface finish of the critical rolling surfaces are generally the largest source of
noise and vibration. Controlling component waviness and surface finish during the
manufacturing process is therefore critical since it may not only have a significant effect on
vibration but also may affect bearing life. discrete defects refer to damage of the rolling
surfaces due to assembly, contamination, operation, mounting, poor maintenance etc. These
defects can be extremely small and difficult to detect and yet can have a significant impact
on vibration-critical equipment or can result in reduced bearing. A discrete defect on the
inner raceway will generate a series of high energy pulses at a rate equal to the ball pass
frequency relative to the inner raceway. Because the inner ring is rotating, the defect will
enter and leave the load zone causing a variation in the rolling element-raceway contact
force, hence deflections. While in the load zone the amplitudes of the pulses will be highest
but then reduce as the defect leaves the load zone, resulting in a signal.
Four Stages in Bearing Failure are Detected with Vibration Analysis
The first stage (normal operation) appears at ultrasonic frequencies from about 1,200K to
3,600K CPM (cycles per minute). At this point the frequencies are evaluated by Spike
Energy and Shock pulse instruments which listen to these frequencies. Trending this
information can tell a person if there is a change or not.
The second stage of bearing failure defects begin to ring bearing components natural
frequencies, which are picked up with a spectrum analyzer in the middle of the spectrum,
3OK-12OK CPM.
In the third stage of failure, bearing defect frequencies and harmonics appear on the
spectrum as bearing defect frequencies. At this time if you remove the bearing, you can see
the defects in the rolling elements.
Stage four appears toward the end of bearing life. It shows up as random high frequency
vibration spikes on the spectrum, all running together. With vibration analysis, many other
problems with rotating equipment can be diagnosed without taking equipment out of
service.
1.2. Problem Statement
Vibration analysis of a rolling element passing over a defect on the inner and outer ring of
a ball bearing has been studied.
3
1.3. Objectives
General objective the close vicinity (a particular area; the surrounding or nearby region) of
the fault detection mechanism to the fault source of bearing.
1.3.1. Specific objective: -
Determine the machine vibration response.
The study load impact on the ball bearing
Coupling misalignment of driving system dedicated to vibration and noise.
4
2.LITERATURE REVIEW
The effect of vibration on perfect bearing can be considerably reduced by selecting the
correct preload and number of balls [9]. The vibration monitoring technique is used to
analyses various defects in bearing and it also provides early information in case of
progressive defects [8]. Triaxial vibration measurement was used to capture the signals and
it was found that defect bearing has a strong effect on the vibration spectra [10].
In case of defect on the fixed ring the frequency spectrums generated will appears at its
multiples. If the defect is located on the inner ring or the ball, frequency spectrum is
amplitude modulated. The more is the wear, higher are the amplitudes of the components.
Low speed fault simulation tests were conducted with various defects on the bearing. This
study gives the best frequency bandwidth for early detection of bearing defects running at
lower speeds [11].
5
3.TYPES OF VIBRATION AND SOUND IN BEARING
3.1. Structural vibration and sound of bearing
Even when the most advanced manufacturing technology is used, vibration and sound still
occur naturally in rolling bearings. As such vibration and sound do not degrade bearing
performance, they are accepted as normal bearing characteristics.
a. Race noise
Race noise is the most basic sound in rolling bearings. It is generated in all bearings and is
a smooth and sound. The magnitude of this sound is used to assess bearing quality [12].
The characteristics of race noise are as follows:
The frequency of the sound does not change even when rotational speed changes. Its
frequency is the natural frequency of the raceway rings, as illustrated in Fig. 1.
The faster the running speed, the louder the sound.
If radial clearance is reduced, the sound becomes louder.
As for the lubricant, when its viscosity is higher the sound is reduced. Besides the viscosity
of the grease, the consistency, form and size of the soap fiber in it also affect noise
performance.
The higher the rigidity of the housing, the lower the magnitude of the sound.
6
Figure 1 Influence of rotation speed on race noise [13]
b. Click noise
Click noise tends to occur more often in relatively large bearings under radial loads. It is
generated only at low speeds, disappearing when speed exceeds a certain level. A rough
approximation of this noise used at NSK is “kata kata.
c. Squeal noise
Squeal noise is a metallic noise that can be rather loud in some cases. With squeal noise,
bearing temperature does not generally rise and bearing and grease life are not adversely
affected. Squeal noise tends to occur with relatively large bearings used under a radial load.
The characteristics of squeal noise are:
It tends to occur when radial clearance is large.
It occurs mostly with grease lubrication and only rarely with oil lubrication.
It occurs more often in winter.
It occurs within a certain speed range that tends to become lower as bearing size increases.
Its generation is inconsistent and unpredictable, and depends on the kind and amount of
grease, as well as bearing operating conditions.
d. Cage noise
7
There are two kinds of cage noise: a noise suggestive of the cage colliding with rolling
elements or bearing ring CK noise can be generated in any type of bearing and the magnitude
of it is usually not very high. Characteristics of this noise include:
It occurs with pressed steel cages, machined cages and plastic cages.
It occurs with grease and oil lubrication.
It tends to occur if a moment load is applied to the outer ring of a bearing.
It tends to occur more often with greater radial clearance and a low-frequency noise (“gaga
gaga”).
3.2 Structural Model of the Outer Ring
To monitor load and vibration within the bearing structure, a piezoelectric sensor is
embedded into a slot cut through the outer ring. The sensor has solid contact with both the
top of the slot and the bearing housing. Each time a rolling element passes over the slot, the
sensor generates an electrical charge output that is proportional to the load applied to the
bearing Fr. Since the outer ring is structurally supported by the bearing housing, it can be
assumed as rigid. The piezoelectric sensor is modeled as a spring with stiffness k that is
related to its material composition. The section of the bearing outer ring where the slot is
cut can be modeled as a beam of varying cross-section, with a spring support at the midpoint.
Since the ends of the beam are solidly connected to the surrounding bearing structure, which
is directly supported by a rigid housing, clamped boundary conditions are considered
appropriate.
Figure 2 Simplified model of the bearing outer ring [11]
8
BEARING LIFE ESTIMATION
Bearing life varies from application to application in accordance with some of the
fundamental influences such as speed, load, temperature etc. A bearings life from new can
be estimated however by using the following equation:
L = 16700(C P⁄ )3
N
Where:
N = rpm
C = bearing life coefficient (obtained from the manufacturer)
P = static load on bearing
This equation as stated gives estimated life. Bearing create vibration that can be used as an
indicator of its health, but the vibration caused by the bearings environment also has an
effect on the bearings health, and we do not mean to confuse this issue with the environments
impact on the complexity of vibration activity. We mean that a bearing subjected to
vibration will last for less time than a bearing that is not. The relationship can in fact be
calculated as below:
L = [C/ (P + .00006773 x MVF)] ^ 3 x (1 6667/RPM)
where:
L = bearing life
P = bearing load
F = frequency (cpm)
C = load capacity
M = mass of vibrating part
V = velocity (in/sec)
3.3. Bearing Damage
It is difficult to name all the causes for bearing damages. Basically the bearing
damages could be attributed to the following main sources:
(a) Wear
9
Wear is a common cause of bearing failure due to dirt and foreign particles entering
the bearing through inadequate sealing. It also occurs often due to the contaminated
lubricant.
(b) Normal fatigue
After certain cycles of rolling, the loaded bearing will accumulate the damage
gradually. Pitting happens in those contact regions when the cracks propagate to the
involved surfaces.
(c) Plastic deformation and brine ling
Due to overloading, sudden vibration or high impact forces could generate apparent
indentation between roller and raceway surfaces. The following operation of bearing
will inevitably face a fluctuating and periodical load mutation.
(d) Corrosion
Corrosion and rust will lead to uneven operation of the bearings. Invasion of water
or acids will also increase the worn off of bearing elements.
(e) Improper mounting
Some bearings need to be preloaded. But preloading may result in nosier running of
the bearings. The operational temperature may increase sharply. Excessive radial
stressing could be formed.
(f) Smearing under transient load
In heavy duty applications, especially when a frequent run-in and braking are
happening, the surfaces of the rollers and raceways will have a smearing due to
slippage.
10
3.4. Ball Bearings and Vibration Analysis
3.4.1 Ball Bearing and Its Geometry
In general, rolling element bearings are designed to carry axial and/or radial load while
minimizing the rotational friction by placing rolling elements such as cylinders or balls
between inner and outer races. There are deferent types of rolling element bearings, among
all of them, ball bearings are the cheapest since balls are used instead of cylinders in their
construction. They are widely used in industry today, in variety of applications in production
line, in electric motors, pumps and gear boxes. There are also deferent types of ball bearings
such as thrust, axial, [16] angular contact and deep groove ball bearings. An example for a
typical deep groove ball bearing is given in Figure 3
Figure 3 Typical deep-groove ball bearing
11
Figure 4 half section deep-groove ball bearing
Figure 5: Ball bearing components, applied force, load zone and load distribution [12]
12
Ball bearings have smaller sizes and limited load carrying capacity compared to the other
rolling element bearings, but they can support both axial and radial loads [15].
Axial force is defined as the force applied parallel to the shaft whereas the radial force is
applied perpendicular to the shaft. Correct alignment, placement where it is used, enough
lubrication are the important points to take care of to maximize the life-span of this
equipment. As it can also be observed from Figure 2, a ball bearing consists of an inner race,
an outer race, balls, a cage holding the balls apart from each other and a shaft. The load zone
and load distribution are also given with the direction of applied force in the figure. In most
cases, the outer race is held stationary where the inner race and the balls rotate. Most of the
defects on the inner side of outer race such as cracks or pits occur on the locations subject
to the load zone, since they are directly under the applied force. The inner race faults on the
other hand, can occur anywhere since the race is not stationary and rotating.
3.5. Vibration Patterns, Analysis and Condition Monitoring
3.5.1. Possible Defects on Ball Bearings and Their Vibration Pattern
The ball bearings themselves act as a source of vibration, even if there are no defects present
and they are perfectly aligned and adjusted [3]. A defect on one of the elements of a ball
bearing can cause the vibration level to increase. There are several type of defects that can
occur on a ball bearing, such as cracks or pits on rotating surface or rolling elements,
distributed defects such as roughness or misaligned races [3]. Those distributed or localized
defects form the vibration pattern that can be detected by a transducer and then analyzed
and processed with the algorithm, which can enable the condition monitoring system to
detect even the occurrence of a failure before it damages the machine or interrupt the
production. When a rolling element strikes to a defect on one of the races, this strike creates
an impulse. Since the rolling element bearing rotates, those impulses will be periodic with
a certain frequency [4]. In case the defect occurs on the inner or outer race, how frequent
each rolling element strikes to the defect is called “Ball-pass frequency” and determined by
the bearing geometry and rotation speed. Ball pass frequency can also be calculated
theoretically, where the formulations are given in [3, 15], and compared to the detected one
after the signals are processed, which can also be an indicator of the performance of the
algorithm. For further diagnosis, such as determining the size of the defect for decision
making, ball pass frequencies and noise-free vibration pattern can be useful.
13
3.6. Determination of Friction Coefficients in Bearing
In multi-body-simulation, based on the same tendency as the Stribeck curve, Hersey
number is replaced by the film parameter for identifying the different lubrication
regimes since the Stribeck curve may not properly address the switch from
hydrodynamic lubrication to elasto- hydrodynamic lubrication. The film parameter
A is
defined as:
A=ℎ
√𝑅𝑟𝑜2 +𝑅𝑟𝑎
2
Where:
Rro: Root mean square surface finish of roller surface
Rar: Root mean square surface finish of raceway surface
h: Minimum/central film thickness for elasto-hydrodynamic/non-elasto-
hydrodynamic lubrication
Figure 6: -The possible lubrication regime distribution within a bearing (Load zone
upwards) [12]
Figure 5 shows the possible lubrication regimes in a downwards loaded bearing.
Meanwhile some aspects should be noticed:
(1) In load-zone an elasto-hydrodynamic lubrication is normally formed.
14
(2) In non-load-zone, roller-raceway contact force comes from centrifugal force. In
this case a hydrodynamic lubrication is dominant.
3.7. Bearing Type and Bearing Material
The bearing type used in this study is a single row deep groove ball bearing with bearing
model 6204 series. The ball bearing is made with thermoplastic material called Poly-acetal
shown in Figure 5. The Polyacetal has melting temperature (Tm) 182-2180o C, density 1.41
g/cc and carbon Polyacetal known as poly oxy methylene (POM), is a high strength,
crystalline engineering thermoplastic material having a desirable balance of excellent
properties and easy processing [6].
Polyacetal is one of the thermoplastic materials that can replace metals and thermosets
because of its long-term performance over a wide range of temperature conditions and harsh
environments. It retains properties such as creep resistance, fatigue endurance, wear
resistance and solvent resistance under demanding service conditions. Also, it is a
lubricious, strong, and has good dimensional stability.
4.FINITE ELEMENT ANALYSIS
4.1. Finite Element Model of the Outer Ring
Technically the complete bearing structure can be modeled using a complex three-
dimensional finite element model. However, by observing the nature of the boundary
conditions and loads on the outer ring, it was determined that the FE model could be
simplified by using symmetry of the system. Loads on the bearing structure are applied to
the outer ring through small ellipsoidal contact areas between the rolling elements and the
outer ring groove. Assuming a pure radial force, the resulting Hertzian stress distribution is
located at the base of the groove. Defect at inner race and outer race are created in ANSYS
itself. In order to define bearing model assembly of three components is made those are,
inner race, balls, outer race. These components are made of nodes. It means that each
component is defined as a group of nodes. A finite element analysis of the outer ring was
done in ANSYS using Plane elements. The necessary boundary conditions and forces are
applied to the model. A modal analysis using Subspace.
15
Figure 7 half section Mesh ball bearing
Figure 8 Total deformation of ball bearing
16
5.RESULTS AND DISCUSSION
When a bearing is running properly, the vibrations generated are very small and generally
constant. But, due to some of the dynamic processes that act in the machine, defects develop
causing the changes in the vibration spectrum.
Vibration signals collected in the form of time domain are converted into frequency domain
by processing Fast Fourier Transform (FFT) on each of the four bearings. From the vibration
data, the amplitude of vibration spectra is relatively small for new bearing and defect ball
bearing cases, whereas vibration spectra are comparatively larger in both the cases of load
i.e., 30N and 60N for defects on inner ring and outer ring.
Vibration and sound generated in bearings are related to the natural frequency of the
raceway rings. The two raceway rings, the natural frequency of the outer ring becomes a
problem more often than the inner ring due to a loose fit between it and the housing.
Two cases are considering i.e. 0.5mm defect on outer race and inner race with constant load
of 10 kg.
Table 1: Show Result for 0.5mm outer race defect at 10kg load
Speed FEA
Outer race 1*0.5mm
Frequency Hz Amplitude m/s2
600 279 3.839
900 258 1.04
1200 164 1.77
1500 165 2.59
When the speed varied from 600rpm to1500rpm a constant load of 10kg.
17
Figure 9:-Graph of speed vs. amplitude for1*0.5outer race defect at 10kg [8]
It is found that the amplitude values for the case of outer race defect are more than that for
the inner race defect. It is because of defect present on the outer race is remained in the load
zone at maximum position as in second case, inner race moves in and out of the load zone
during each revolution of the shaft. The strong fault vibration spectrum produced while the
defect is in the load zone and weaker fault vibration spectrum produced while the defect is
outside the load zone.
The difference between the predicted sensor output and experimental output is due to two
reasons.
Unlike the sensor output predicted by the FE model, the experimental data did not
reach exactly the zero output line due to vibrational noise, which was observed from
the machine environment during the experiments.
In the FE model, the housing was assumed to be infinitely rigid. However, the
experimental housing has a certain degree of flexibility since it was made of
aluminum which is an elastic material.
The effect frequency and velocity of ball bearing increase the output of the vibration
increase. Due to the effect of misalignment of the shaft high vibration of the bearing occurs.
Based on ANSYS workbench software to analysis of symmetrical ball bearing the effect of
inter race exerted load equivalent distribution of the bearing.
Figure 10 half section of ANSYS result of random vibration of normal elastic strain
18
Random vibration of ball bearing shear elastic strain occurs at the contact region of the ball
in the inner and outer race bearing.so within harmonic repetition of rotation the shear elastic
value will be maximum.
Figure 11 half section shear elastic strain random vibration result
Due to the effect of random vibration the contact of ball and inner race of the ball bearing
equivalent stress induced relatively larger than outer race.
Figure 12random vibration result for equivalent stress
Due to the effect of random vibration normal stress large value induced in the ball contact
region and minimum value produced in inner and outer race of the bearing.
19
Figure 13half section random vibration result of normal stress
Random vibration the same effect of directional velocity, directional acceleration and
directional deformation. but different results. The maximum value will be produced in the
inner race of the bearing part.
Figure half section of ball bearing random vibration for directional velocity
Figure half section of ball bearing random vibration for directional acceleration
20
Figure half section of ball bearing random vibration of directional deformation
21
6.CONCLUSION
Vibration analysis of ball bearing it consider different conditions with deferent defects
factors, damages, types of vibration sounds and bearing life with finite element methods.
The structural integration of a load sensor into the outer ring of a rolling element bearing
provides an effective means for assessing the time varying load conditions within the
bearing structure identifying the vibration of the bearing. The variation of forces exerted by
the rolling element on the outer ring in the vicinity of the defect. Though the values for the
forces employed in this analysis are not exact, the aim has been to understand the trend of
vibration signatures. By observing the nature of the boundary conditions and loads on the
outer ring, it was determined that the FE model could be simplified by using symmetry of
the system.
22
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