Review Article Fault Diagnosis of Rotating Machinery...

ISSN No: 2309-4893 International Journal of Advanced Engineering and Global Technology I Vol-04, Issue-01, January 2016

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Review Article

Fault Diagnosis of Rotating Machinery based on vibration analysis

Dr. F. R, Gomaa Dr. k. M. khader

Faculty of Engineering Faculty of Engineering Shebin EL-Kom -production engineering and Shebin EL-Kom -production engineering Mechanical design department. EGYPT 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,

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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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 condition-

monitoring 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

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 NN-

based 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, model-

based 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,

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 well-

known 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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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 non-

stationary 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]

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.EL-

Mongey]

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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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

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 pre-

acquisition, 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

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.

3.4) Envelope methods

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

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, peak-

value, 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]

4) Technique Analysis

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 well-

known 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 Huang-

Teager (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.

Figure (4.1): Morlet wavelet

Comparison Wavelet Transform with Fourier

Transform

The Fourier transform is less useful in analyzing non-

stationary 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

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 < ∞ (4.1.3)

Where

(w)=∫ 𝝍 (t)exp(-jwt)dt (4.1.4)

𝜙 (w): mother wavelet function

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 - 𝟐𝒋k ) /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.EL-

Mongey]

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]

Some Advantages of Wavelet Theory:

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.

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.

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 Single-

Input/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.

Traditional experimental modal analysis

(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.


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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

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

𝒚(𝒕) =

{

𝒚𝟏(𝒕)

𝒚𝟐(𝒕)...𝒚𝒎 }

(4.3.1)

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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X (t) = {𝒚(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

AC=[𝟎 𝐈

−𝐌−𝟏𝐊 −𝐌−𝟏𝐃] (4.3.5)

B=[𝟎𝑴−𝟏] (4.3.5)

The advantage of this formulation is that the general

solution is directly available,

X (t) =exp (ACt) X (0∫ 𝒆𝒙𝒑(𝑨𝑪(𝒕 −𝒕

𝟎𝛕))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

X (t) =exp (ACK∆t)X0 -AdkX0 (4.3.7)

Ad=exp(AC∆t) (4.3.7)

yk=CAdkX0 (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

RK=𝟏

𝐍−𝐊 Y (1: N-K) Y (K: N) (4.3.9)

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

SSI is simply a gathering of a family of matrices that are

created by shifting the data matrix

Yh=

[ 𝐘(𝟏:𝐍−𝟐𝐬)𝐘(𝟐:𝐍−𝟐𝐬+𝟏)

.

.

.𝐘(𝟐𝐬:𝐍) ]

=[𝒀𝒉𝒑𝒚𝒉𝒇

] (4.3.10)

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 non-

linearity. 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:

B (k,l) =E[X(k ) X(l ) X * (k +l )] (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)|𝟐𝑬[|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)= | ∑ 𝒙𝒊(𝒌)𝒙𝒊(𝒍)𝒙𝒊

∗(𝒌+𝒍)|𝟐𝒌𝒊=𝟏

∑ |𝒌𝒊=𝟏 𝒙𝒊(𝒌)𝒙𝒊(𝒍)|

𝟐∑ |𝒙𝒊(𝒌+𝒍)|𝟐𝒌

𝒊=𝟏 (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]

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 auto-

correlation 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 non-

parametric 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

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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