Fault Diagnosis of Rotating Machinery based on vibration analysis.

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Fault Diagnosis of Rotating Machinery based on vibration analysis.
ISSN No: 2309-4893
International Journal of Advanced Engineering and Global Technology
I
Vol-04, Issue-01, January 2016
Review Article
Fault Diagnosis of Rotating Machinery based on vibration analysis
Dr. F. R, Gomaa
Faculty of Engineering
Shebin EL-Kom -production engineering and
Mechanical design department. EGYPT
Dr. k. M. khader
Faculty of Engineering
Shebin EL-Kom -production engineering
and Mechanical design department. EGYPT
Eng. M. A. Eissa
Faculty of Engineering
Shebin EL-Kom -production engineering and
Mechanical design department.EGYPT
Abstract -This paper aims to provide abroad review of state of arts n faults diagnosis technique mainly
in rotating machine based on vibrations analysis. Vibrations response measurements give valuable
information on common faults. The general classes of methods are reviewed and particular difficulties
are highlighted in each method to have accurate method for each component.
Keywords: fault diagnosis, Faults Description, Monitoring method, technique analysis
Introduction
The objective of this paper is to provide the reader with an insight into recent developments in the field of fault. Diagnosis in
rotating machines. The different types of faults that are observed in these area of rotating machine and methods of their diagnosis
including (single process, modal Analysis, Stochastic Subspace Identification (SSI), Order Analysis (OHS), Frequency Domain
Decomposition (FDD) Method). The paper is divided into different section, each dealing with various aspects of the subjects: it
begins with a summary of review of faults diagnosis, followed by general overview of faults detection methods. Faults diagnosis
technique in rotating machinery is discussed in detail including technique methods. Special treatment is given of vibration
analysis. One of the major areas of interest in the modern day condition monitoring of rotating machinery is that of vibration. If a
fault developed and goes undetected , then, at best, the problem will not be too serious and can be remedied quickly and cheaply ;
at worst, it may results in expensive damage and down- time , injury , or even loss of life. By measurement and analysis of the
vibration of rotating machinery , it is possible to detect and locate important faults such as mass unbalance , gear faults,
misalignment, crack) .Problem in rotating machinery may also be caused by degradation in the bearing
Pervious literature reviews and surveys
The aim of this section is to provide the reader with an
understanding of the state of the art in fault diagnosis.
Model-based fault detection is, at this time, directly
employed in most areas of fault diagnosis. The model-based
approach involves the establishment of a suitable process
model, either mathematical or signal-based, which can
estimate and predict process parameters and variables.
Described the main principles involved in model-based
procedures and outlined their importance for the realistic
modeling of faults. He concluded that more than one method
of FDI should be utilized, in order to best reach an accurate
diagnosis. Fault trees and forward and backward chaining
are methods of fault diagnosis addressed b A comprehensive
overview of fault diagnosis methodology is first presented,
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based on process measurements, dynamic models and
parameter estimation to generate fault symptoms. Fault trees
and forward and backward chaining.
Then provide the method of fault classification. Fault trees
are a heuristic means of decision making, constructed partly
from knowledge of physical laws and partly from experience
in the field, which may not necessarily be exactly described
By these laws. Forward and backward chaining mimics the
human process of decision making. All possible outcomes
are considered as a first step (forward chaining), the decision
making process is then enhanced by the introduction of
additional information and the most likely outcome due to
this extra input is analyzed, involving the input of yet more
data (backward chaining). The procedure continues until
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ISSN No: 2309-4893
International Journal of Advanced Engineering and Global Technology
I
Vol-04, Issue-01, January 2016
either it is concluded or no more likely outcomes are
possible. Used the definition that a fault will alter the
dynamic behavior of a system, to construct a model to detect
changes in this dynamic behavior and thus identify faults.
Various physical parameters and model sensitivity to fault
size are used to detect and locate faults.[1,2,3,4,5,6,7]
Improving model robustness with respect to these
uncertainties, whilst maintaining sensitivity, helps to provide
the necessary means for inserting FDI models into practical
applications. Begins with a survey of the methods of
improving robustness in model generation and analysis. The
Generalized Observer Scheme (perfect decoupling from
modeling errors achieved by increasing the number of
inputs), robust parity space check, unknown input observer
scheme (state estimation error decoupling), decor relation.
Filter and adaptive threshold selection (uncertainties causing
residual and decision functions to fluctuate) are all described
as means to obtain decoupling from any modeling errors
which may occur. Robustness with respect to nonlinear
systems is also shown to be attainable. A more general
description of robustness and the observer-based FDI
approach is given by Patton and where the passive robust
solution (adaptive threshold method) and active robust
solutions (uncertainty in residual generation) are considered.
Examples are given for the FDI of a jet engine system, a
pumping system and an electric train. [8,9,10,11,12] Since
the analysis and design of rotating machinery is extremely
critical in terms of the cost of both production and
maintenance, it is not surprising that the fault diagnosis of
rotating machinery is a crucial aspect of the subject,
receiving ever more attention. As the design of rotating
machinery becomes increasingly complex, due to rapid
progress being made in technology, so must machinery
condition-monitoring strategies become more advanced in
order to cope with the physical burdens being placed on the
individual components of a machine. Modern conditionmonitoring techniques encompass many different themes,
one of the most important and informative being the
vibration analysis of rotating machinery - a topic which has
prompted much research to be carried out and a
corresponding amount of literature to be produced. Using
vibration analysis, the state of a machine can be constantly
monitored and detailed analyses may be made concerning
the health of the machine and any faults which may be
arising or have already arisen, serious or otherwise.
Common rotor-dynamic faults include self-excited vibration,
due to system instability, and, more often, vibration due to
some externally applied load, such as cracked or bent shafts
and mass unbalance. Vibration condition monitoring as an
aid to fault diagnosis has been examined by in much the
same way as Smith covered the general kinds of faults listed
above and described qualitatively how they may be
recognized from their vibration characteristics, and included
effects caused by nonlinearity. Stewart and Taylor also
included information on the actual data analysis process how measured data should be processed in order for a
diagnosis to be performed. performance efficiency data to
dynamic and static measurements, it is possible to then
control the overall performance level of the machine
presented a method for assessing the severity of vibration in
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terms of the probability of damage by analysis of vibration
signals and its related cost, using the net present value
method. The question of whether or not to shut down a
machine for maintenance was considered and some
guidelines were formulated, comparing maintenance and
down-time costs against the possible costs that would be
incurred
by
damage.[13,14,15,16,17,18,19,20,21,23]
Iwatsubo considered possible errors likely to occur in
vibration analysis and how these errors may influence
calculations of critical speeds, instability and unbalance. A
statistical approach was used to calculate the mean values
and standard deviations of errors, which in turn allows the
calculation of statistical values of unbalance response,
instability and critical speeds. The sensitivity of the model
with respect to errors in the various model parameters was
also determined. It was found that errors in bearing
coefficients have a much larger effect on the variance of
system instability than do errors in mass and stiffness, which
have a predominant effect on the variance of critical speed.
[24, 25, 26, 27, 28, 29, 30]
For the diagnosis of anisotropy and asymmetry in rotating
machinery, [31] Lee and Joh (1994) developed a method
incorporating directional frequency response functions
(dFRFs). Anisotropy and asymmetry may cause whirl,
fatigue and instability, as well as influencing system
characteristics such as unbalance and critical speeds.
Complex modal testing was used to estimate the dFRFs. An
example was presented, showing the proposed method to be
very efficient in identifying anisotropy and asymmetry.
The key factor of the predictive maintenance is diagnostic.
A diagnosis is not an assumption; it is a conclusion reached
after a logical evaluation of the observed symptoms. Then,
the diagnostic is based on a systematic inspection in
vibration signal to find all susceptible defects, which may
affect the machine.
Fault diagnosis is essentially pattern recognition. by
analyzing the symptom parameters generated by the
equipment, the faults of the equipment can be known and
the causes of faults can thus be determined. Speed frequency
of the rotor is called the fundamental frequency. When there
is a fault in the rotor-bearing system, the fault will has an
impact on all the frequency bands, and it will change the
distribution of energy. Fault features of the rotor-bearing
system will be mainly in fraction or integer multiple
frequency. Therefore, to diagnose fault, analysis of signal
frequency is adopted to extract features of the fault
parameter and to classify the faults based on these
characteristic parameters. [32, I.H. Witten and E. Frank:
Data Mining2000].
Today's industry uses increasingly complex machines, some
with extremely demanding performance criteria. Attempting
to diagnose faults in these systems is often a difficult and
daunting task for operators and plant maintainers. Failed
machines can lead to economic loss and safety problems due
to unexpected and sudden production stoppages. These
machines need to be monitored during the production
process to improve machine operation reliability and reduce
unavailability. Therefore, conducting effective condition
monitoring brings significant benefits to industry [33, 34],
Altmann, J. (1999), Baillie, D C and Mathew, J (1996),].
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However, condition monitoring requires effective fault
diagnosis, which is labor oriented exercise to this day.
Without efficient diagnosis, one is unable to make reliable
prediction of lead time to failure. A natural progression is
the automation of this labor oriented process of diagnosis by
implementing intelligent diagnosis strategies so that experts
or technicians can be relieved of this relatively expensive
task. Fault diagnosis is conducted typically in the following
phases: data collection, feature extraction, and fault
detection and identification. Fault detection and
identification usually employs artificial intelligent (AI)
approaches for pattern classification. Numerous attempts
have been made to improve the accuracy and efficiency of
fault diagnosis of rotating machinery by employing AI
techniques. Few have attempted to summaries these
techniques comprehensively. [35] Zhong2000 introduced
new developments in the theory and application of
intelligent condition monitoring and diagnostics in China.
He concluded that the trends in intelligent diagnosis are NNbased fault classifiers, NN-based expert systems, NN-based
prognosis, behavior-based intelligent diagnosis, remote
distributed intelligent diagnosis networks and intelligent
multi-agent architecture for fault diagnosis. He provided a
good overview of intelligent fault diagnosis of machinery
but was somewhat general. [36] Pham1999 theoretically
analyses the applicability of artificial intelligence in
engineering problems and predominantly looked at
knowledge-based systems, fuzzy logic, inductive learning,
neural networks and genetic algorithms in different branches
of engineering but not in machinery fault diagnosis. [37]
Tandon1999 mentioned that automatic diagnosis was a trend
in the fault diagnosis of rolling elements. [38] Gao2001
provided an up-to-date review on recent progresses of soft
computing methods-based motor fault diagnosis systems. He
summarized several motor fault diagnosis techniques using
neural networks, fuzzy logic, neural-fuzzy, and genetic
algorithms (GAs) and compared them with conventional
techniques such as direct inspection .also gave a brief
review, which listed 14 papers from experts in the area of
motor fault detection and diagnosis. He grouped those
papers into five main categories: survey papers, modelbased approaches, signal processing approaches, emerging
technology
approaches,
and
experimentation.[39]
Chow2000 The task of fault diagnosis consists of
determining the type, size and location of the fault as well as
its time of detection based on the observed analytical and
heuristic symptoms. If no further knowledge on fault
symptom causalities is available, classification methods can
be applied which allow for a mapping of symptom vectors
into fault vectors. To this end, methods like statistical and
geometrical classification or neural nets and fuzzy clustering
can be used. Note that geometrical analysis, whilst simple to
implement, has a few drawbacks. The most serious is that, in
the presence of noise, input variations and change of
operating point of the monitored process, false alarms are
possible. If a-priori knowledge of fault-symptom causalities
is available, e.g. in the form of causal networks, diagnostic
reasoning strategies can be applied. Forward and backward
chaining, with Boolean algebra for binary facts and with
approximate reasoning for probabilistic or possibility facts,
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are examples. Finally a fault decision indicates the type, size
and location of the most possible fault, as well as its time of
detection. [40], Nicola Orani 2010]
Fault diagnosis is the determination of specific fault that has
occurred in the system. A typical fault detection method
consists of the following stages:
a) Data Acquisition
b) Parameter Extraction
c) Fault Analysis
d) Decision Making
Vibration monitoring is one of the main tools that allow to
determine the mechanical health of various components of a
machine in a non-intrusive manner.
The mechanical components that are the most often
encountered in rotating machinery are bearings - roller
bearings in particular – and gears. Furthermore, these
components are generally the most loaded and consequently
subject to early damages in the machine's life.
[41, Christophe Thirty, Ai-Min Yan, Jean-Claude Golinval
October 2004]
Vibration-based damage detection for rotating machinery
(RM) has been repeatedly applied with success to a
variety of machinery elements such as roller bearings and
gears. In the past, the greatest emphasis has been on the
qualitative interpretation of vibration signatures both in
the frequency and (to a lesser extent) in the time domain.
Numerous summaries and reviews of this approach are
available in textbook form, including detailed charts of
machinery fault analysis, e.g., see [42]-[43]. The
approach taken has generally been to consider the
detection of damage qualitatively on a fault-by-fault basis
by examining acceleration signatures for the presence and
growth of peaks in spectra at certain frequencies, such as
multiples of shaft speed. A primary reason for this
approach has been the inherent nonlinearity associated
with damage in RM and the inability to make
measurements at locations other than the exterior housing
of the machine.
Recently, more general approaches to damage detection
in RM have been developed. These approaches utilize
formal statistical methods to assess both the presence and
level of damage on a statistical basis, e.g., see [44] and
[45]. A particularly detailed and general treatment of
mechanical signature analysis is presented in [46].
The vibration signal analysis is one of the most important
methods used for condition monitoring and fault diagnostics,
because they always carry the dynamic information of the
system. Effective utilization of the vibration signals,
however, depends upon the effectiveness of the applied
signal processing techniques for fault diagnostics. With the
rapid development of the signal processing techniques, the
analysis of stationary signals has largely been based on wellknown spectral techniques such as: Fourier Transform (FT),
Fast Fourier Transform (FFT) and Short Time Fourier
Transform (STFT) [47], [48]. Unfortunately, the methods
based on FT are not suitable for non-stationary signal
analysis [49]. In addition, they are not able to reveal the
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ISSN No: 2309-4893
International Journal of Advanced Engineering and Global Technology
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Vol-04, Issue-01, January 2016
inherent information of non-stationary signals. These
methods provide only a limited performance for machinery
diagnostics [50]. In order to solve these problems, Wavelet
Transform (WT) has been developed. WT is a kind of
variable window technology, which uses a time interval to
analyze the high frequency and the low frequency
components of the signal [51], [52]. The data using WT can
be decomposed into approximation and detail coefficients in
a multi scale, presenting then a more effective tool for nonstationary signal analysis than the FT. Many studies present
the applications of WT to decompose signals for improving
the performance of fault detection and diagnosis in rotating
machinery [53]–[54,55,56,57,58,59,60,61,62,63].
1) Faults Description in Rotating Machine
Machine fault can be defined as any change in a machinery
part or component which makes it unable to perform its
function satisfactorily or it can be defined as the termination
of availability of an item to perform its intended function.
The familiar stages before the final fault are incipient fault,
distress, deterioration, and damage, all of them eventually
make the part or component unreliable or unsafe for
continued use [64], Pratesh Jayaswal,1 A. K.Wadhwani, and
K. B.Mulchandani3,2008]. Classification of failure causes
are as follows:
(i) Inherent weakness in material, design, and
manufacturing;
(ii) Misuse or applying stress in undesired direction;
(iii) Gradual deterioration due to wear, tear, stress fatigue,
corrosion, and so forth.
A fault is an irregularity in the functioning of the equipment
which results in component damage, energy losses and
reduced efficiency of the machine. The common types of
machine faults are:
_ Unbalance
_ Shaft misalignment or bent shaft
_ Damaged or loose bearings
_ Damaged gears
_ Faulty of misaligned belt drive.
_ Mechanical looseness
_ Increased turbulence
_ Electrical induced vibration
Fault detection using vibration analysis involves analyzing
the vibration signature for signs of fault. Any predominant
fault occurring results in increased vibration level which has
energy concentrated at certain frequency levels.
The relation of the predominant vibration frequencies with
the forcing frequency (input force frequency) gives us an
idea about the source of the fault. The increased amplitude
of the predominant frequencies indicates the severity of the
fault.
Standard relations between common faults and
corresponding fault signatures are available. [65, 65, 66, 67,
68]
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1 .1) Mass unbalance
Mass unbalance is one of the most common causes of
vibration;. Unbalance is a condition where the centre of
mass does not coincide with the centre of rotation, due to the
unequal distribution of the mass about the centre of rotation.
The unbalance creates a vibration frequency exactly equal to
the rotational speed, with amplitude proportional to the
amount of unbalance. [69, Hocine Bendjama, Salah
Bouhouche, and Mohamed Seghir Boucherit, 2012]
Unbalance is a result of uneven distribution of a rotor’s mass
and causes vibration to be transmitted to the bearings and
other parts of the machine during operation. Imperfect mass
distribution can be due to material faults, design errors,
manufacturing and assembly errors, and especially faults
occurring during operation of the machine. By reducing
these vibrations, better performance and more cost-effective
operation can be achieved and deterioration of the machine
and ultimately fatigue failure can be avoided. This requires
the rotor to be balanced by adding and/or removing mass at
certain positions in a controlled manner.
Unbalance may occur due to the following reasons.
_ The shape of the rotor is unsymmetrical.
_ Unsymmetrical mass distribution exists due to machining
or casting error.
_ A deformation exists due to a distortion.
_ an eccentricity exists due to a gap of fitting.
_ An eccentricity exists in the inner ring of a bearing.[70,
Siva Shankar Rudraraju,]
1 .2) Gear fault
The vibrations of a gear are mainly produced by the shock
between the teeth of the two wheels. Gear fault is simulated
with filled between teeth. The vibration monitored on a
faulty gear generally exhibits a significant level of vibration
at the tooth meshing frequency GMF (i.e. the number of
teeth on a gear multiplied by its rotational speed) and its
harmonics of which the distance is equal to the rotational
speed of each wheel.[69, Hocine Bendjama, Salah
Bouhouche, and Mohamed Seghir Boucherit,2012]
Gear faults can be generally classified into two major
categories: distributed faults and local faults. Distributed
faults are those faults that results from poor gear mounting,
or manufacturing inaccuracies such as eccentricities, varying
gear tooth spacing, etc. Meanwhile, local faults are those
resulting from localizing defects that may occur in gear teeth
such as tooth surface wear, cracks in gear teeth, and loss of
part of the tooth due to breakage or loss of the whole teeth.
[71, A.AIbrahim, S. M.Abdel Rahman,M.Z.Zahran,H.H.ELMongey]
1. 3) Misalignment
Misalignment in rotating machinery is one of the most
common faults causing other faults and machine failure. It
causes over 70% of rotating machinery vibration problems
[72, Bognatz, S. R., 1995]. A misaligned rotor generates
bearing forces and excessive vibrations making diagnostic
process more difficult. A perfect alignment can never be
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ISSN No: 2309-4893
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Vol-04, Issue-01, January 2016
achieved practically and misalignment is always present.
[73, Mohsen Nakhaeinejad, Suri Ganeriwala, Sep. 2009]
There are two types of misalignment: parallel and angular
misalignment. With parallel misalignment, the center lines
of both shafts are parallel but they are offset. With angular
misalignment, the shafts are at an angle to each other. The
parallel misalignment can be further divided up in horizontal
and vertical misalignment. Horizontal misalignment is
misalignment of the shafts in the horizontal plane and
vertical misalignment is misalignment of the shafts in the
vertical plane




Parallel horizontal misalignment is where the motor
shaft is moved horizontally away from the pump shaft,
but both shafts are still in the same horizontal plane and
parallel.
Parallel vertical misalignment is where the motor shaft
is moved vertically away from the pump shaft, but both
shafts are still in the same vertical plane and
parallel.Similar, angular misalignment can be divided
up in horizontal and vertical misalignment:
Angular horizontal misalignment is where the motor
shaft is under an angle with the pump shaft but both
shafts are still in the same horizontal plane.
Angular vertical misalignment is where the motor shaft
is under an angle with the pump shaft but both shafts
are still in the same vertical plane.
Errors of alignment can be caused by parallel misalignment,
angular misalignment or a combination of the two.
1 .4) bearing failure
Antifriction bearings failure is a major factor in failure of
rotating machinery. Antifriction bearing defects may be
categorized as localized and distributed. The localized
defects include cracks, pits, and spalls caused by fatigue on
rolling surfaces. The distributed defect includes surface
roughness, waviness, misaligned races, and off-size rolling
elements. These defects may result from manufacturing and
abrasive wear. [74, M. Amarnath, R. Shrinidhi, A.
Ramachandra, and S. B. Kandagal, 2004].
Fatigue: The change in the structure that is caused by the
repeated stresses developed in the contacts between the
rolling elements and the raceways. Fatigue is manifested
visibly as flaking of particles.
Fatigue can be divided into:
*Subsurface Initiated Fatigue
*Surface Initiated Fatigue
Wear: Wear is the progressive removal of material resulting
from the interaction of the asperities of two sliding or
rolling/sliding contacting surfaces during service.
Wear can be divided into:
*Abrasive Wear
* Adhesive Wear
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Corrosion: Corrosion is a chemical reaction on metal
surfaces.
Corrosion can be divided into:
* Moisture Corrosion: When steel used for rolling bearing
components is in contact with moisture (e.g., water or acid),
oxidation of surfaces takes place. Subsequently, the formation of
corrosion pits occurs and finally flaking of the surface .
*Frictional Corrosion: Frictional corrosion is a chemical reaction
activated by relative micro movements between mating surfaces
under certain friction conditions. These micro movements lead to
oxidation of the surfaces and material, becoming visible as
powdery rust and/or loss of material from one or both mating
surfaces. [75, SKF 2008, Chapter 5 – ISO Classification]
1 .5) crack
Cracked rotors are not only important from a practical and
economic viewpoint, they also exhibit interesting dynamics.
Cracks in rotor machine is greatest danger and research in
crack detection has been ongoing for the past 30 years. A
crack in rotor will change the dynamic behaviour of the
system but in practice it has been found that small or
medium size cracks make such a small change to the
dynamics of the machine system that they are undetectable
by this means. Only if the crack grows a potentially
dangerous size it can be readily detected. So we use high
resolution frequency and filter. Crack detection methods fall
into two groups, model updating and pattern recognition
(see, for example, [76, and 77]). In the former method, the
dynamic behavior of the rotor is used to update a model of
the rotor and in the process determine both the severity and
location of any crack. Clearly the crack model used must be
adequate for the task. If the pattern recognition approach is
used, then whilst a crack model is not directly required, it is
desirable to have some idea of the dynamic behavior that
will result from a cracked rotor in order that it can be
recognized in the pattern of behavior. There are a number of
approaches to the modeling of cracks in beam structures
reported in the literature, that fall into three main categories;
local stiffness reduction, discrete spring models, and
complex models in two or three dimensions. [78]
Dimarogonas and [79] Ostachowicz and Krawczuk gave
comprehensive surveys of crack modeling approaches. [80]
Friswell and Penny considered the performance of various
crack models in structural health monitoring. If the vibration
due to any out-of-balance forces acting on a rotor is greater
than the static deflection of the rotor due to gravity, then the
crack will remain either opened or closed depending on the
size and location of the unbalance masses. In the case of the
permanently opened crack, the rotor is then asymmetric and
this condition can lead to stability problems. If the vibration
due to any out-of-balance forces acting on a rotor is less
than the static deflection of the rotor due to gravity the crack
will open and close (or breathe) as the rotor turns. [81,Jerzy
T. Sawiciki , Michael I. Friswell, ,Zbingniew Kulesza,
Adam Wroblewski , John D. Lekki, 2011]
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2) Technique of Monitoring For Fault Detection
Based On Vibration
Machine monitoring, or early detection of incipient faults,
aims to survey the machine health, or condition, at critical
locations, e g, gears and bearings, and possibly predict a
future failure. At a certain stage of defect progress or
severity, a scheduled stop for maintenance can be made, the
damaged element replaced, and production can then
continue without unnecessary delays. [82, a Barkov, N
Barkova and a Azovtsev]
Actually developments still deal with minimization of
measurement equipment and analysis techniques
implementing worldwide standards for data processing and
acquisition, with the possibility of central data acquisition.
The supply of more cost-effective monitoring tools has been
made possible by technical advances such as:
* reduced costs of instrumentation,
* increased capability of instrumentation such as data preacquisition, data storage, radio transmission direct by the
sensors with integrated electronically circuits,
* improved data storage media in combination with low cost
computation,
* faster and more effective data analysis using specialist
software tools.[83, Wilfried Reimche, Ulrich Südmersen,
Oliver Pietsch, Christian Scheer, Fiedrich-Wilhelm Bach,
2003]
3) Monitoring Methods
The defect signature energy is usually distributed over a
wide frequency interval and is therefore easily masked by
energy from other sources. Several time domain as well as
frequency domain methods have been developed over the
years to cope with this problem and provide an accurate
defect detection method. Time domain methods dealing with
localized defects involve indicators sensitive to impulsive
oscillations. Well-known examples are the peak value, root
mean square (r.m.s) value, crest factor and Kurtosis.
Intelligent signal filtering is crucial for the success of all
these methods. Early methods, developed before FFT
analyzers appeared, used peak and r.m.s detecting
instruments in combination with various filters. After birth
of the FFT analyzer, methods focused on frequency domain
properties like harmonic sequences of the characteristic
defect frequencies. Finally, during the last two decades of
the 20th century, methods such as high frequency envelope
detection, relying on advanced signal processing and
stochastic signal analysis have emerged.[84,85,86,87]
3.1) Early Time Domain Methods
The classical machine monitoring method, used by machine
operators since the 1920’s, is to press a screw driver tip to
the machine casing and the handle to the skull bone and
listen to the machine vibrations, taking advantage of bone
conduction. This is basically an instrument which uses the
screw driver as vibration sensor and the operator's auditory
system as the signal analyzer. In the 1950´s, efforts were
made to summaries these experiences in automatic
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measurement systems. At that time the available instruments
were simple accelerometers, analog filters, root mean square
indicating voltmeters and oscilloscopes indicating peak
values. So the early machine monitoring methods used
various combinations of vibration r.m.s and peak values.
Several methods used either the peak or the r.m.s -value;
others used their ratio, the crest factor. Unfortunately
indicators based on the r.m.s -value are insensitive to
incipient defects. Crest factor indicators, on the other hand,
are sensitive for early defects when the peak value is high
and the r.m.s-value is low. At later stages of defect life the
overall r.m.s value increases significantly and the crest
factor is reduced. This fact has led to the common
misconception that defect severity decreases after the
early stage when it in reality is progressing towards the
final failure. The early methods suffer from some severe
disadvantages: - early defect detection is difficult in a noisy
environment, - defect localization and - defect identification
are in practice impossible. A solution to these problems
requires intelligent filtering and signal processing.
3.2) Spectral methods
The character of the vibration signature is most prominent in
the frequency domain. For this reason many monitoring
techniques are based on frequency domain characteristics,
the spectrum, of the vibration signature. Spectral methods
are most successfully applied to the low and intermediate
part of the frequency range. The strategy is to extract the
signature from a single gear or bearing from the total
vibration measured on the machine casing and to monitor
the characteristic defect frequency amplitudes. The basic
assumption is, of course, that a growing defect implies
increasing amplitudes at the defect frequencies. Some defect
indicators, like the defect severity index, focus on the
difference in amplitude at the defect frequency and the
amplitude of the background. Other alternative methods
focus on the side bands and seek to construct indicators
measuring the relative amplitudes of the side bands. The
main problem of spectral methods is, as always, to suppress
the contributions from other vibration sources to such an
extent that they do not interfere with the investigated
signature. In some cases the signature dominates the
measured vibrations whereas in other cases the signature
may be hidden in other contributions. In such cases some
signal processing is needed to extract the signature and
suppress contributions from other sources. One useful signal
processing tool is time domain averaging or synchronised
time domain averaging, Synchonised averaging means that
the signal is ensemble averaged in the time domain with
each time record synchronised with a tachometer signal
tracking the shaft rotation period. This averaging will, if
properly performed, cancel out all frequency components
except the ones synchronous with the shaft, i e the ones
periodic in the time window. It is common practice to
normalize the frequency axis with a characteristic frequency,
e g the shaft rotation frequency, to get an order spectrum.
The use of order spectra has several advantages as compared
to standard frequency spectra.
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3.3) Cepstral methods
The signal cepstrum is useful for detecting repetitive
patterns in a signal. In principle, the signal cepstrum is
defined as the Fourier transform of the magnitude of the
signal spectrum, Hence, vibration signatures including
harmonic side bands are relatively easy to detect in a signal
cepstrum. On the other hand several experimental
investigations have shown that monitoring the cepstrum may
give ambiguous results, especially regarding the defect
progress.
detect, identify and diagnose the defect. Several indicators
exist. In the time-domain, abrupt changes in the envelope
can be detected and quantified using r.m.s-value, peakvalue, crest factor or statistical measures like Kurtosis, in the
frequency domain peaks in the envelope spectrum
coinciding with characteristic defect frequencies can identify
and localize a defect. A damaged gear tooth can be localized
using phase information. Experience shows that phase is
more sensitive than amplitude in detecting and localizing
gear tooth cracks. A frequently used indicator is for
instance. [89, 90, 91, 92]
3.4) Envelope methods
4) Technique Analysis
The main difficulty with spectral and cepstral methods is
that they focus on low and mid frequencies where the
influence from other sources is usually large. Thus, it may
be very difficult to extract the correct signature from the
large number of frequency lines found at low frequencies.
Envelope methods aims to avoid this problem by focusing
on high frequencies, e.g. for bearings from 20 kHz to 30
kHz. The fundamental idea of high frequency envelope
methods is to use the high frequency part of the defect
modulated vibration signal and demodulate it to obtain the
vibration signature. To ensure that the vibration signal is
dominated by contributions from the correct element, the
accelerometer is placed as closes possible to the monitored
element. Contributions from other distant sources will then
be attenuated before they reach the accelerometer.
Obviously, high frequency envelope methods require the
excitation to have a substantial amount of energy at high
frequencies. This is true for localized defects, such as cracks
and spalls, but also for surface wear in fluid film bearings.
When a ball in a ball bearing rolls over a spall on the
bearing inner race an abruptly changing contact force is
experienced. This contact force has impact character and
excites vibrations from low frequencies up to ultrasound
frequencies. One of the crucial elements of envelope
methods is to select a suitable high frequency region and
band-pass filter the accelerometer signal. Some methods
suggest that the filter should be centered on a structural
resonance frequency. Others recommend regions free from
strong tones due to resonances or harmonic components. A
second important element is to demodulate or calculate the
vibration signature from the band-pass filtered vibration
signal. High frequency resonance techniques use the large
amplitude vibration in the neighborhood of a structural high
frequency resonance. After band-pass filtering the vibration
signal its time history consists of a series of exponentially
decaying impulse responses. Each impulse response consists
of a narrowband resonant vibration signal acting as carrier
modulated with a series of impacts produced by the
defect. This implies that information on the defect, its
periodicity etc., is available in the signal modulation. The
defect signature is obtained by removing the carrier wave,
that is, by extracting the envelope, from the band-passed
signal. Below describes the defect signature extraction
procedure schematically. When the residual error signature
has been extracted by demodulating the band-pass filtered
vibration signal it can be processed in various ways to
1577
Basically, machine vibration monitoring uses so called
signature analysis, i.e. the characteristic vibration signature
of the monitored machine element is investigated. In
practice, machine monitoring requires some physically
measureable signal to monitor. It might be vibration, sound
pressure or a temperature signal. Vibrations are the most
commonly used monitoring signals. Vibrations are not
subject to background disturbances to the same extent as
acoustic noise. Vibration sensors, accelerometers, can be
placed closer to the source than microphones. [84, 85, 86,
87]
4.1) single process
The vibration signal analysis is one of the most important
methods used for condition monitoring and fault diagnostics
because they always carry the dynamic information of the
system. Effective utilization of the vibration signals,
however, depends upon the effectiveness of the applied
signal processing techniques for fault diagnostics. With the
rapid development of the signal processing techniques, the
analysis of stationary signals has largely been based on wellknown spectral techniques such as: Fourier Transform (FT),
Fast Fourier Transform (FFT) and Short Time Fourier
Transform STFT) [87, K. Shibata, A. Takahashi, T.
Shirai,2000], 88, S. Seker, E. Ayaz,2000]
Unfortunately, the methods based on FT are not suitable for
non-stationary signal analysis [89, C. Cexus, “Analyse des
signaux non-stationnaires par Transformation de Huang,
Opérateur de Teager-Kaiser, et Transformation de HuangTeager (THT, 2005]. In addition, they are not able to reveal
the inherent information of non-stationary signals. These
methods provide only a limited performance for machinery
diagnostics [90, J. D. Wu, C.-H. Liu, 2008]. In order to
solve these problems, Wavelet Transform (WT) has been
developed.
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Wavelet Transform
A wavelet means a small wave (the sinusoids used in
Fourier analysis are big waves) and in brief, a wavelet is an
oscillation that decays quickly.
The wavelet analysis is done similar to the STFT analysis.
The signal to be analyzed is multiplied with a wavelet
function just as it is multiplied with a window function in
STFT, and then the transform is computed for each segment
generated. However, unlike STFT, in Wavelet Transform,
the width of the wavelet function changes with each spectral
component. The Wavelet Transform, at high frequencies,
gives good time resolution and poor frequency resolution,
while at low frequencies, the Wavelet Transform gives good
frequency resolution and poor time resolution. In
comparison to the Fourier transform, the analyzing function
of the wavelet transform can be chosen with more freedom,
without the need of using sine-forms. A wavelet function (t)
is a small wave, which must be oscillatory in some way to
discriminate between different frequencies. The wavelet
contains both the analyzing shape and the window. Shows
an Figure (4.1) example of a possible wavelet, known as the
Morlet wavelet has been extensively used for impulse
isolation and mechanical fault diagnosis [71, A.A Ibrahim,
S.M.Abdel-Rahman,M.Z.Zahran,H.H.EL-Mongey] use in
Adaptive wavelet analysis. For the CWT several kind of
wavelet functions are developed which all have specific
properties.
Wavelets is much larger than that of the Fourier transform.
In fact, the mathematics of wavelets encompasses the
Fourier transform. The size of wavelet theory is matched by
the size of the application area. Initial wavelet applications
involved signal processing and filtering. However, wavelets
have been applied in many other areas including nonlinear
regression and compression. An offshoot of wavelet
compression allows the amount of determinism in a time
series to be estimated. The main difference is that wavelets
are well localized in both time and frequency domain
whereas the standard Fourier transform is only localized in
frequency domain. The Short-time Fourier transform
(STFT) is also time and frequency localized but there are
issues with the frequency time resolution and wavelets often
give a better signal representation using Multi resolution
analysis[91, M. Sifuzzaman1, M.R. Islam1 and M.Z.
Ali,2009]
Wavelet decomposes a signal in both time and frequency in
terms of a wavelet, called mother wavelet. The WT includes
Continuous Wavelet Transform (CWT) and Discrete
Wavelet Transform (DWT). Let s (t) is the signal; the CWT
of s(t) is defined as.[92,Hocine Bendjama, Salah
Bouhouche, and Mohamed Seghir Boucherit,2012]
∞
CWT (a,b) = 1/ ( √|𝒂| )∫−∞ 𝒔( 𝒕 ) 𝝍∗ ((t - b)/ a )d t
(4.1.1)
Where ψ*(t) is the conjugate function of the mother wavelet
ψ (t) (1.2), a and b are the dilation (scaling) and translation
(shift) parameters, respectively. The factor 1/√|𝑎| is used
to ensure energy preservation.
Ψ (t) = 1/ (√𝒂 ) ψ ((t −b) / a)
(4.1.2)
The mother wavelet must be compactly supported and
satisfied with the admissibility condition
+∞
∫−∞ |𝝓(𝒘)|𝟐 /|𝒘| 𝒅𝒘 < ∞
Where
(w)=∫ 𝝍 (t)exp(-jwt)dt
Figure (4.1): Morlet wavelet
(4.1.3)
(4.1.4)
𝜙 (w): mother wavelet function
Comparison
Transform
Wavelet
Transform
with
Fourier
The Fourier transform is less useful in analyzing nonstationary signal (a non-stationary signal is a signal where
there is change in the properties of signal). Wavelet
transforms allow the components of a non-stationary signal
to be analyzed. Wavelets also allow filters to be constructed
for stationary and non-stationary signal. The Fourier
transform shows up in a remarkable number of areas outside
of classic signal processing. Even taking this into account,
we think that it is safe to say that the mathematics of
1578
The DWT is derived from the discretization of CWT. The
most common discretization is dyadic. The DWT is given by
∞
DWT (j,k ) =1/( √𝟐𝒋 )∫−∞ 𝒔( 𝒕 ) ψ ((t - 𝟐𝒋 k ) /𝟐𝒋 )d t (4.1.5)
Where a and b are replaced by 2j and 2jk, j is an integer.
A very useful implementation of DWT, called multi
resolution analysis [93, S.G.Mallat, 1989], is demonstrated
DWT analyzes the signal at different scales. It employs two
sets of functions, called scaling functions and wavelet
functions [93, S. G. Mallat, 1989], [94, N.Lu,F. Wang, F.
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Gao,2003], which are associated with low pass and high
pass filters, respectively.
In engineering application , the square of the modulus of the
CWT is often called scalogram which is a time - scale
distribution , and is defined :
SGX (a,b) = |WX(a,b)|2
(4.1.6)
Compared to SHFT, whose time - frequency resolution is
fixed, the time -frequency resolution of the wavelet
transform depends on the frequency of the signal. At high
frequencies, the wavelet uses narrow time windows yielding
high time resolution but a low frequency resolution.
Whereas, at low frequencies, wide time windows are used so
the frequency resolution is high while the time resolution is
low. Thus, Wavelet transform is very suitable for the
analysis of the analysis of the transient and non-stationary
signals.
[71,A.AIbrahim,S.M.AbdelRahman,M.Z.Zahran,H.H.ELMongey]
Some Application of Wavelets
Wavelets are a powerful statistical tool which can be used
for a wide range of applications, namely
• Signal processing
• Data compression
• Smoothing and image denoising
• Fingerprint verification
• Biology for cell membrane recognition, to distinguish the
normal from the pathological membranes
• DNA analysis, protein analysis
• Blood-pressure, heart-rate and ECG analyses
• Finance (which is more surprising), for detecting the
properties of quick variation of value
• In Internet traffic description, for designing the services
size
• Industrial supervision of gear-wheel
• Speech recognition
• Computer graphics and multiracial analysis
• Many areas of physics have seen this paradigm shift,
including molecular dynamics, astrophysics, optics,
turbulence and quantum mechanics. Wavelets have been
used successfully in other areas of geophysical study [91, M.
Sifuzzaman1, M.R. Islam1 and M.Z. Ali, 2009]
d) A wavelet transform can be used to decompose a signal
into component wavelets
e) In wavelet theory, it is often possible to obtain a good
approximation of the given function f by using only a few
coefficients which is the great achievement in compare to
Fourier transform.
f) Most of the wavelet coefficients { j k }j k N d , , ≥ vanish
for large N.
g) Wavelet theory is capable of revealing aspects of data that
other signal analysis techniques miss the aspects like trends,
breakdown points, and discontinuities in higher derivatives
and self-similarity.
h) It can often compress or de-noise a signal without
appreciable degradation. [91, M. Sifuzzaman1, M.R. Islam1
and M.Z. Ali, 2009]
Wavelets are an incredibly powerful tool, but if you can’t
understand them, you can’t use them. Up till now, wavelets
have been generally presented as a form of Applied
Mathematics. Most of the literature still uses equations to
introduce the subject. [95, preview of wavelets, wavelet
filters, and wavelets transform space and signal's
Technology, 2009]
4.2) Modal Analysis
Modal analysis has been widely applied in vibration trouble
shooting, structural dynamics modification, analytical model
updating, optimal dynamic design, vibration control, as well
as vibration-based structural health monitoring in aerospace,
mechanical and civil engineering. Traditional experimental
modal analysis (EMA) makes use of input (excitation) and
output (response) measurements to estimate modal
parameters, consisting of modal frequencies, damping ratios,
mode shapes and modal participation factors. EMA has
obtained substantial progress in the last three decades.
Numerous modal identification algorithms, from SingleInput/Single-Output (SISO), Single-Input/Multi-Output
(SIMO) to Multi-Input/Multi-Output (MIMO) techniques in
Time Domain (TD), Frequency Domain (FD) and Spatial
Domain (SD), have been developed
.
Some Advantages of Wavelet Theory:
Traditional experimental modal analysis
a) One of the main advantages of wavelets is that they offer
a simultaneous localization in time and frequency domain.
b) The second main advantage of wavelets is that, using fast
wavelet transform, it is computationally very fast.
(EMA), however, suffers from several limitations, as
described below.
1– It requires artificial excitation to evaluate frequency
response functions (FRF) or impulse response functions
(IRF). In some cases, such as civil structures, providing
adequate excitation is difficult if not impossible.
2– Operational conditions are often different from those
adopted in tests because traditional EMA is conducted in a
laboratory environment.
c) Wavelets have the great advantage of being able to
separate the fine details in a signal. Very small wavelets can
be used to isolate very fine details in a signal, while very
large wavelets can identify coarse details.
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3– The boundary conditions are simulated because tests are
usually conducted in a laboratory environment on
components instead of with complete systems.
4– Structural modification and sensitivity analysis to
evaluate the effect of changes on the dynamics of a structure
without actual modifications;
5– Structural health monitoring and damage detection by
comparing modal parameters from the current state of a
structure with those at a reference state to obtain information
about the presence, location and severity of damage;
6– Performance evaluation, if modal parameters and mode
shapes are used to evaluate the dynamic performance of a
system.
7– Force identification starting with only structural response
measurements.
Operational modal analysis
Although most operational modal analysis techniques are
derived from traditional EMA procedures, the main
difference is related to the basic assumptions about the
inputs. In fact, EMA procedures are developed in a
deterministic framework, while OMA methods are based on
random responses and, therefore, a stochastic approach.
Thus, many OMA techniques can be seen as the stochastic
counterparts of the deterministic methods used in classical
EMA, despite the availability of new hybrid
deterministic-stochastic techniques. [96, Carlo Rainieri and
Giovanni Fabbrocino, 2011]
This technique has been successfully used in civil
engineering structures (buildings, bridges, platforms,
towers) where the natural excitation of the wind is used to
extract modal parameters. It is now being applied to
mechanical and aerospace engineering applications (rotating
machinery, on-road testing, and in-flight testing). The
advantage of this technique is that a modal model can be
generated while the structure is under operating conditions.
That is, a model within true boundary conditions and actual
force and vibration levels. Another advantage of the
technique is the ability to perform modal testing in-situ, i.e.,
without removing parts under test.[97, Mehdi Batel, Brüel &
Kjær, Norcross, Georgia,2002]
Operational modal analysis techniques are based on the
following assumptions:
1– Linearity: the response of a system to a certain
combination of inputs is equal to the same combination of
corresponding outputs;
2– stationary: the dynamic characteristics of a structure do
not change over time, and the coefficients of the differential
equations are constant with respect to time; and
3– observability: the test setup must be defined to enable
measurements of the dynamic characteristics of interest; for
instance, nodal points must be avoided to detect a certain
mode.[96, Carlo Rainieri and Giovanni Fabbrocino.2011]
The technique of modal analysis can't be used in
rotating element because modal analysis techniques are
based on assumptions (linearity, stationary) so
1580
vibration analysis is the best in case of rotating
machine element.
4.3) Stochastic Subspace Identification (SSI)
Stochastic Subspace Identification (SSI) modal estimation
algorithms have been around for more than a decade by
now. The real break-through of the SSI algorithms happened
in 1996 with the publishing of the book by van Overschee
and De Moor [98, Peter van Overschee and Bart De Moor,
1996]. A set of MATLAB files were distributed along with
this book and the readers could easily convince themselves
that the SSI algorithms really were a strong and efficient
tool for natural input modal analysis. Because of the
immediate acceptance of the effectiveness of the algorithms
the mathematical framework described in the book where
accepted as a de facto standard for SSI algorithms.
However, the mathematical framework is not going well
together with normal engineering understanding. The reason
is that the framework is covering both deterministic as well
as stochastic estimation algorithms. To establish this kind of
general framework more general mathematical concepts has
to be introduced. Many mechanical engineers have not been
trained to address problems with unknown loads enabling
them to get used to concepts of stochastic theory, while
many civil engineers have been trained to do so to be able to
deal with natural loads like wind, waves and traffic, but on
the other hand, civil engineers are not used to deterministic
thinking. The book of van Overschee and De Moor [98,
Peter van Overschee and Bart De Moor, 1996] embraces
both engineering worlds and as a result the general
formulation presents a mathematics that is difficult to digest
for both engineering traditions. take a another ride with the
SSI train to discover that most of what you will see you can
recognize as generalized procedures well established in
classical modal analysis. [99, Rune Brincker, Palle
Andersen]
The discrete time formulation
We consider the stochastic response from a system as a
function of time
𝒚𝟏 (𝒕)
𝒚𝟐 (𝒕)
.
𝒚(𝒕) =
(4.3.1)
.
.
{ 𝒚𝒎 }
The system can be considered in classical formulation as a
multi degree of freedom structural system
M y..(t)+D y. (t) +k y (t) =f (t)
(4.3.2)
Where Κ, D, Μ, is the mass, damping and stiffness matrix,
and where f (t) is the loading vector. In order to take this
classical continuous time formulation to the discrete time
domain the easiest way is to introduce the State Space
formulation.
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𝒚(𝒕)
X (t) = { . }
𝒚 (𝒕)
(4.3.3)
Here we are using the rather confusing terminology from
systems engineering where the states are denoted x (t) (so
please don’t confuse this with the system input, the system
input is still f (t)). Introducing the State Space formulation,
the original 2nd order system equation given by eq. (3.2)
simplifies to a first order equation
X∙ (t)=ACX(t)+Bf(t)
(4.3.4)
y (t)=CX(t)
(4.3.4)
Where the system matrix AC in continuous time and the load
matrix B is given by
𝟎
𝐈
AC=[
]
(4.3.5)
−𝐌 −𝟏 𝐊 −𝐌 −𝟏 𝐃
B=[
𝟎
]
𝑴−𝟏
(4.3.5)
The advantage of this formulation is that the general
solution is directly available,
𝒕
X (t) =exp (ACt) X (0∫𝟎 𝒆𝒙𝒑(𝑨𝑪 (𝒕 − 𝛕))𝐁𝐟(𝛕)𝒅𝒕
(4.3.6)
Where the first term is the solution to the homogenous
equation and the last term is the particular solution. To take
this solution to discrete time, we sample all variables like
yk=y (k∆t) and thus the solution to the homogenous equation
becomes
(ACK∆t)X0 -AdkX0
X (t) =exp
Ad=exp(AC∆t)
yk=CAdkX0
(4.3.7)
(4.3.7)
(4.3.7)
The Block Henkel Matrix
In discrete time, the system response is normally represented
by the data matrix
𝒀 = [ 𝒚𝟏
𝒚𝟐 … … . .
𝒚𝑵 ]
(4.3.8)
Where N is the number of data points. To understand the
meaning of the Block Henkel matrix, it is useful to consider
a more simple case where we perform the product between
two matrices that are modifications of the N data matrix
given by eq. (2.5.3.7). Let Y (1: N-K) be the data matrix where
we have removed the last K data points, and similarly, let Y
(K: N) be the data matrix where we have removed the first K
data points, then
𝟏
RK=
Y (1: N-K) Y (K: N)
(4.3.9)
SSI is simply a gathering of a family of matrices that are
created by shifting the data matrix
𝐘(𝟏:𝐍−𝟐𝐬)
𝐘(𝟐:𝐍−𝟐𝐬+𝟏)
𝒀𝒉𝒑
.
Yh=
=[ 𝒚 ]
(4.3.10)
.
𝒉𝒇
.
[ 𝐘(𝟐𝐬:𝐍) ]
The upper half part of this matrix is called “the past” 𝒀𝒉𝒑
and denoted and the lower half part of the matrix is called
“the future” and is denoted 𝒚𝒉𝒇 . The total data shift 2sis
and is denoted “the number of block rows” (of the upper or
lower part of the Block Hankel matrix). The number of rows
in the Block Hankel matrix is 2sM, the number of columns
is N-2S. [99, Rune Brincker, Palle Andersen]
4.4) Order Analysis (OHS)
One of the most popular analysis methods for
engineers/scientists has been harmonic analysis. Here, the
term harmonic refers to frequencies that are integer (or
fractional) multiples of a fundamental frequency. In the
automobile industry, such harmonics are traditionally
referred to as orders. Accordingly, the harmonic analysis is
called as order analysis. [100, Shie Qian,] .Order tracking
(OT) is one of the most important vibration analysis
techniques for diagnosing faults in rotating machinery. The
main advantage of OT over other vibration analysis
techniques lies in the analysis of non-stationary noise and
vibration, [101, K. S. Wang and P. S. Heyns]
Bispectral Analysis
The statistical properties of a stationary random process are
completely described by its mean value m (first order
moment) and variance s2 (second order central moment) if
and only if its probability density function has a Gaussian
distribution. HOS measures, such as higher order moments,
are extensions of second-order measures to higher orders
and cannot provide additional information about a signal if it
is Gaussian. However, many signals found in the industrial
field are non-Gaussian, e.g. vibration signals of rotating
machines. Therefore, HOS may be used to extract
information about signals and systems which cannot be
obtained from conventional statistics. The most used
traditional signal processing measure is the power spectrum
that is the decomposition over frequency of the signal power
and is therefore related to the signal variance s2. The
bispectrum (third order spectrum) can be viewed as a
decomposition of the third moment (skewness) of a signal
over frequency and as such can detect non-symmetric nonlinearity. For a stationary random process, the discrete
bispectrum B (k,l) can be defined in terms of the signal's
Discrete Fourier Transform X(k) as:
𝐍−𝐊
B (k,l) =E[X(k ) X(l ) X * (k +l )]
Is an unbiased estimate of the correlation matrix at time lag
K. This follows directly from the definition of the
correlation estimate, The Block Henkel Yh matrix defined in
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(4.4.1)
Where E [] denotes the expectation operator. It should be
noted that the bispectrum is complex-valued (it contains
phase information) and that it is a function of two
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independent frequencies, k and l. Furthermore, it is not
necessary to compute B (k, l) for all (k, l) pairs, due to
several symmetries existing in the (k, l) plane. In particular
there exists a non-redundant region called the Principal
Domain which is defined as:
{K, l}: 0 ≤ k ≤
𝒇𝒔
𝟐
, l ≤ k, 2k + l ≤ fs
(4.4.2)
Where fs is the sampling frequency the bispectrum is able to
detect the non-linear interactions between spectral
components and to measure the extent of their dependencies.
Second-order measures contain no phase information and,
consequently, cannot be used to identify phase coupling,
which are often associated with system non-linearity. This is
in contrast to third order methods which are sensitive to
certain types of phase
Moreover, since the signals
encountered in practice are often contaminated by Gaussian
measurement noise, to which HOS measure are theoretically
insensitive, they are potentially powerful tools for analyzing
real-life signals.
Bicoherence
Bispectral analysis often does not deal with the bispectrum,
as given by equation (4.4.1), but with a normalized form of
bispectrum, e.g. the bicoherence, b2 (k, l), which is defined
by:
b2(k,l) =
|𝐄[𝐗(𝐊)𝐗(𝐋)𝐗 ∗ (𝐊+𝐋)]|𝟐
𝑬[| 𝑿(𝑲)𝑿(𝑳)|𝟐 𝑬[|𝑿(𝑲+𝑳)|𝟐 ]
(4.4.3)
The reason for normalizing the bispectrum is due to the fact
that this estimator has a variance which is proportional to the
triple product of the power spectra, which can result in the
second order properties of the signal dominating the
estimate. The advantage of normalization is to make the
variance approximately flat across all frequencies. For
stochastic (random) signals the bicoherence is a measure of
the signal skewness (the third order moment). An important
feature of the bicoherence is that it is always restricted to
vary between 0 and 1 .The bicoherence, as well as the
bispectrum, can be computed from a signal by dividing it
into K segments, applying an appropriate window to each
segment to reduce leakage, computing the quantities in
Equations (4.4.1) and (4.4.3) for each segment by using the
Discrete Fourier Transform and then obtaining the statistical
estimate by averaging over all segments. Segment averaging
is used in order to achieve a consistent estimate.
Accordingly, the bicoherence can be calculated using the
following estimator:
B^2 (k,l)= ∑𝒌
∗
𝟐
| ∑𝒌
𝒊=𝟏 𝒙𝒊 (𝒌)𝒙𝒊 (𝒍)𝒙𝒊 (𝒌+𝒍)|
𝟐 𝒌
𝟐
𝒊=𝟏 |𝒙𝒊 (𝒌)𝒙𝒊 (𝒍)| ∑𝒊=𝟏 |𝒙𝒊 (𝒌+𝒍)|
(4.4.4)
It is important to observe that the bicoherence is only a
normalization of the bispectrum and it should not be
confused with the second order coherence function;
moreover, its computation requires a single signal
measurement, whilst the ordinary coherence function needs
two measures. [102, A. RIVOLA]
1582
4.5) Frequency Domain Decomposition (FDD) Method
The FDD method [103 Brincker R, Zhang LM,
Andersen200] can be viewed as an extension of the
traditional basic frequency domain method. It is
performed using the output power spectral density (PSD),
and based on the assumption that the excitation is pure
Gaussian white noise and that all natural modes are
lightly damped [53].
A singular value decomposition (SVD) is carried out for
each PSD matrix and all modes contributing to the
vibratory signature of a structure at a given frequency are
separated into principal values and orthogonal vectors.
When a single mode identified by peak picking at a
Given frequency prevails in the spectrum, the first vector
obtained by the SVD will constitute an estimate of the
mode shape. The first singular value corresponding to
this mode should be approximately equal to the sum of
the terms on the diagonal of the PSD matrix, which
means that most of the power of the measured signals at
this frequency can be attributed to the vibratory signature
of this particular mode. Other singular values that are not
associated with any mode will consist of decomposed
noise initially contained in the signals before the SVD
was performed. Once natural frequencies have been
roughly identified by peak picking and mode shapes have
been estimated using the singular vector matrices,
equivalent single degree of freedom ‘spectral bells’ are
identified for each mode. This step is achieved by
comparing the estimated mode shape of interest with all
vectors previously estimated throughout the spectrum by
SVD of all the PSD matrices. A comparison of the mode
shapes is then carried out by computing the modal
assurance criterion (MAC). All singular values
corresponding to a MAC value superior to a user
specified parameter (which is called the MAC rejection
level) are kept, thus forming an equivalent single degree
of freedom spectral bell. Then, by inverse fast Fourier
transform (IFFT) of that spectral bell, the resulting autocorrelation function can be used to reevaluate the
frequency by counting the number of zero crossings in a
finite time interval. Damping ratios are also estimated
using the logarithmic decrement of the auto-correlation
function. More details on the theory and implementation
of the FDD method can be found in Reference [103].
The Frequency Domain Decomposition (FDD) is an
extension of the Basic Frequency Domain (BFD) technique,
or more often called the Peak-Picking technique. This
approach uses the fact that modes can be estimated from the
spectral densities calculated, in the condition of a white
noise input, and a lightly damped structure. It is a nonparametric technique that estimates the modal parameters
directly from signal processing calculations, Refs.
[103,104]. The FDD technique estimates the modes using a
Singular Value Decomposition (SVD) of each of the data
sets. This decomposition corresponds to a Single Degree of
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Freedom (SDOF) identification of the system for each
singular value. The relationship between the input x(t), and
the output y(t) can be written in the following form, Refs.
[105,106]
The Enhanced Frequency Domain Decomposition (EFDD)
technique is an extension to the Frequency Domain
Decomposition (FDD) technique. FDD is a basic technique
that is extremely easy to use. You simply pick the modes by
locating the picks in SVD plots calculated from the spectral
density spectra of the responses. Animation is performed
immediately. As the FDD technique is based on using a
single frequency line from the FFT analysis, the accuracy of
the estimated natural frequency depends on the FFT
resolution and no modal damping is calculated. Compared to
FDD, the EFDD gives an improved estimate of both the
natural frequencies and the mode shapes and also includes
damping.
In EFDD, the SDOF Power Spectral Density function,
identified around a peak of resonance, is taken back to the
time domain using the Inverse Discrete Fourier Transform
(IDFT). The natural frequency is obtained by determining
the number of zero-crossing as a function of time, and the
damping by the logarithmic decrement of the corresponding
SDOF normalized auto correlation function. The SDOF
function is estimated using the shape determined by the
previous FDD peak picking - the latter being used as a
reference vector in a correlation analysis based on the Modal
Assurance Criterion (MAC). A MAC value is computed
between the reference FDD vector and a singular vector for
each particular frequency line. If the MAC value of this
vector is above a user-specified MAC Rejection Level, the
corresponding singular value is included in the description
of the SDOF function. The lower this MAC Rejection Level
is, the larger the number of singular values included in the
identification of the SDOF function will be. [106] N-J.
Jacobsen, P. Andersen, R. Brincker]
The advantages of the FDD are that the technique is easy
and fast to use. There is a “snap to peak” feature that can be
applied on the Averaged Normalized Singular Value
function. The corresponding singular vector, which is an
approximation to the mode shape, is extracted from all
dataset at the selected frequency. In fact, the singular vectors
can be extracted from any singular value at any frequency,
which may lead to better understanding of the structural
behavior.
The disadvantages are that no damping is estimated and the
frequency resolution is no better than the FFT line spacing.
The main advantages of the EFDD technique are that both
Frequency and Damping are estimated. The “snap to peak”
is applied to the maximum Singular Values for each dataset.
Then the averaged frequency and damping values are
estimated as well as their standard deviations from all data
sets.
Major disadvantage is that the algorithm does not work
properly if no distinguished peak is found in some of the
data sets. It is also required to fine tune the three modal
estimation parameters, MAC Rejection Level, maximum
and minimum correlation (i.e. correlation interval) for each
resonance frequency in each dataset, which may be a very
1583
time consuming procedure, when there are many data sets
and/or many modal frequencies present.
Conclusion
This paper presents a fault diagnosis method. Many methods
have been developed to monitor machine condition. and
describe the main types defect which occur in rotating
element and in this paper has present technique of
monitoring for fault diagnosis and compare with this method
and choose the best way for monitoring .we also talk about
technique of analysis (signal processing , modal analysis)
the technique of modal analysis can't be use in rotating
element because modal analysis techniques are based on
assumptions(linearity, stationary) so vibration analysis is
the best in case of rotating machine element, (SSI) Many
mechanical engineers have not been trained to address
problems with unknown loads enabling them to get used to
concepts of stochastic theory, while many civil engineers
have been trained to do so to be able to deal with natural
loads like wind, waves and traffic and ( HOS) It is important
to observe that the bicoherence is only a normalization of
the bispectrum and it should not be confused with the
second order coherence function; moreover, its computation
requires a single signal measurement, whilst the ordinary
coherence function needs two measures) ,paper shows that
The disadvantages of (FDD) Method are that no damping is
estimated and the frequency resolution is no better than the
FFT line spacing and Major disadvantage of (EFDD) is that
the algorithm does not work properly if no distinguished
peak is found in some of the data sets. It is also required to
fine tune the three modal estimation parameters, MAC
Rejection Level, maximum and minimum correlation (i.e.
correlation interval) for each resonance frequency in each
dataset, which may be a very time consuming procedure,
when there are many data sets and/or many modal
frequencies present.
We will notice from comparison that the best way for
analysis is signal process use in the most application. The
vibration signal analysis is one of the most important
methods used for condition monitoring and fault diagnostics.
Unfortunately, the methods of vibration analysis based on
FT are not suitable for non-stationary signal analysis so we
use the wavelet transform for non-stationary signal as shown
in the paper the technique of wavelet .Short Time Fourier
Transform (SHFT) has fixed resolution that depends upon
the selection of the window half-width . Continuous
Wavelet transform (CWT) is very efficient in localizing
impulses and detecting periodic events as well due to its
multi-resolution property. Complex CWT is very sensitive
to any abrupt changes in time signal so it can detect
impulses. Adaptive Wavelet (DWT) is efficient in locating
impulses but it is difficult to specify frequencies of periodic
events.
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ISSN No: 2309-4893
International Journal of Advanced Engineering and Global Technology
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Vol-04, Issue-01, January 2016
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