Gearbox Diagnosis

Institutionen för systemteknikDepartment of Electrical Engineering

Examensarbete

Gearbox Diagnosis

Examensarbete utfört i Fordonssystemvid Tekniska högskolan i Linköping

av

Jonas Bengtsson

LiTH-ISY-EX--11/4308--SE

Linköping 2011

Department of Electrical Engineering Linköpings tekniska högskolaLinköpings universitet Linköpings universitetSE-581 83 Linköping, Sweden 581 83 Linköping

Gearbox Diagnosis

Examensarbete utfört i Fordonssystemvid Tekniska högskolan i Linköping

av

Jonas Bengtsson

LiTH-ISY-EX--11/4308--SE

Handledare: Andreas Myklebustisy, Linköpings universitet

Examinator: Mattias Krysanderisy, Linköpings universitet

Linköping, 16 December, 2011

Avdelning, InstitutionDivision, Department

Division of Vehicular SystemsDepartment of Electrical EngineeringLinköpings universitetSE-581 83 Linköping, Sweden

DatumDate

2011-12-16

SpråkLanguage

� Svenska/Swedish� Engelska/English

�

�

RapporttypReport category

� Licentiatavhandling� Examensarbete� C-uppsats� D-uppsats� Övrig rapport�

�

URL för elektronisk versionhttp://www.control.isy.liu.se

http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-ZZZZ

ISBN—

ISRNLiTH-ISY-EX--11/4308--SE

Serietitel och serienummerTitle of series, numbering

ISSN—

TitelTitle

Diagnos av en växellådaGearbox Diagnosis

FörfattareAuthor

Jonas Bengtsson

SammanfattningAbstract

Diagnosis based on vibration analysis is a method that has many benefits to of-fer. It is easy to implement the method on existing transmissions by attachingaccelerometers outside the gearbox housing. If you have knowledge of the gear-box geometry, such as number of tooth on the gears and types of bearings, andany unwanted frequencies can be filtered out a good estimation of the gearboxcondition can be achieved. In this thesis a number of condition indicators havebeen tested to identify and isolate different faults that may appear. All analysinghave been done in the time domain on different synchronously averaged signals.The condition indicators have been used together with diagnosis theory from thedivision of Vehicular systems to create a diagnosis system able to find faults on anumber of modelled signals.

NyckelordKeywords Diagnosis, Gearbox

AbstractDiagnosis based on vibration analysis is a method that has many benefits to of-fer. It is easy to implement the method on existing transmissions by attachingaccelerometers outside the gearbox housing. If you have knowledge of the gear-box geometry, such as number of tooth on the gears and types of bearings, andany unwanted frequencies can be filtered out a good estimation of the gearboxcondition can be achieved. In this thesis a number of condition indicators havebeen tested to identify and isolate different faults that may appear. All analysinghave been done in the time domain on different synchronously averaged signals.The condition indicators have been used together with diagnosis theory from thedivision of Vehicular systems to create a diagnosis system able to find faults on anumber of modelled signals.

SammanfattningDiagnos baserad på vibrationsanalys är en metod som har många fördelar att er-bjuda. Det är enkelt att i efterhand impementera vibrationsanalys på befintligatransmissioner genom att fästa accelerometrar på utsidan av växellådan. Om in-formation finns om växellådans geometri, såsom antal tänder på kuggen och vadför kullager som används, samt att oönskade störningar kan filtreras bort, kan engod uppskattning göras av växellådans skick. I denna uppsats har några olika in-dikatorer för att testa växellådans skick undersökts och använts för att identifieraoch isolera olika typer av fel. All analys har genomförts i tidsdomänen på olikasynkrona medelvärden relaterade till de olika felen. Indikatorerna har använts till-sammans med diagnosteori från avdelningen för Fordsonssystem för att skapa ettdiagnossystem möjligt att identifiera fel på modellerade signaler.

v

Acknowledgments

I would like to thank my supervisor Andreas and my examiner Mattias for theirsupport throughout my work. I would also like to thank my opponent Victor forfeedback and comments on the report and the presentation. A special thanks tomy friends and family for support, understanding and motivation in difficult times.

vii

Contents

1 Introduction 31.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 The PHM Data Challenge . . . . . . . . . . . . . . . . . . . . . . . 31.3 Purpose and Goal . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.5 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 The Apparatus 72.1 Gearbox geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.1 Gears . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1.2 Bearings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Fault Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2.1 Gear Faults . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2.2 Bearing Faults . . . . . . . . . . . . . . . . . . . . . . . . . 102.2.3 Faults on the Shafts . . . . . . . . . . . . . . . . . . . . . . 11

2.3 Data Acquisition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3 Theoretical Background 153.1 Diagnosis Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.1.1 Hypothesis Tests . . . . . . . . . . . . . . . . . . . . . . . . 153.1.2 Decision Structure . . . . . . . . . . . . . . . . . . . . . . . 15

3.2 Theoretically Modelled Signals . . . . . . . . . . . . . . . . . . . . 163.2.1 Modelling the Signals . . . . . . . . . . . . . . . . . . . . . 16

3.3 Monte Carlo Simulation . . . . . . . . . . . . . . . . . . . . . . . . 183.4 Welch’s T-test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.5 Lilliefors Test for Normality . . . . . . . . . . . . . . . . . . . . . . 19

4 Signal Processing 214.1 Calculating the Input Shaft Frequency . . . . . . . . . . . . . . . . 214.2 Gear Mesh Frequencies . . . . . . . . . . . . . . . . . . . . . . . . . 224.3 Estimation of Gearbox Type . . . . . . . . . . . . . . . . . . . . . 224.4 Filtering in the Frequency Domain . . . . . . . . . . . . . . . . . . 224.5 Synchronous Average for the Shafts . . . . . . . . . . . . . . . . . . 234.6 Synchronous Average for Bearing Faults . . . . . . . . . . . . . . . 24

ix

x Contents

4.7 Condition Indicators . . . . . . . . . . . . . . . . . . . . . . . . . . 254.7.1 RMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.7.2 Crest Factor . . . . . . . . . . . . . . . . . . . . . . . . . . 254.7.3 Kurtosis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.7.4 Finding Peak Indexes . . . . . . . . . . . . . . . . . . . . . 26

4.8 FFT Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.8.1 Patterns not Related to the Shaft Frequencies . . . . . . . . 27

5 The Diagnosis System 295.1 Decision Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 295.2 Thresholds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

5.2.1 Data Clustering . . . . . . . . . . . . . . . . . . . . . . . . . 305.2.2 Threshold Calculation . . . . . . . . . . . . . . . . . . . . . 30

5.3 Multiple Faults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315.4 Diagnosis of non Faulty Components . . . . . . . . . . . . . . . . . 325.5 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

6 Parameter and Diagnosis Results 356.1 Evaluation of the Condition Indicators . . . . . . . . . . . . . . . . 356.2 Results from the Monte Carlo simulation . . . . . . . . . . . . . . 35

6.2.1 The Decision Structure . . . . . . . . . . . . . . . . . . . . 366.3 Diagnosis Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

6.3.1 Diagnosis of the Modelled Signals . . . . . . . . . . . . . . . 39

7 Evaluation of Labelled Data 437.1 Helical vs. Spur Gears, Frequency and Load . . . . . . . . . . . . . 437.2 Component Fault Modes . . . . . . . . . . . . . . . . . . . . . . . . 44

7.2.1 Bearing Faults . . . . . . . . . . . . . . . . . . . . . . . . . 447.2.2 Chipped and Broken Gears . . . . . . . . . . . . . . . . . . 457.2.3 The Component Mode Error . . . . . . . . . . . . . . . . . 457.2.4 Bent Axle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477.2.5 Shaft Imbalance . . . . . . . . . . . . . . . . . . . . . . . . 477.2.6 Bad Key . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

7.3 Condition Indicator Results . . . . . . . . . . . . . . . . . . . . . . 497.3.1 RMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497.3.2 Crest Factor . . . . . . . . . . . . . . . . . . . . . . . . . . 507.3.3 Kurtosis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517.3.4 Peak Indexes . . . . . . . . . . . . . . . . . . . . . . . . . . 51

7.4 Diagnosis of the Labelled Data . . . . . . . . . . . . . . . . . . . . 51

8 Conclusions 538.1 Improvements and Future Work . . . . . . . . . . . . . . . . . . . . 53

Bibliography 55

Contents xi

A Synchronous Average 57A.1 Chipped Idle Gear 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 57A.2 Ball Spin Fault Bearing 5 . . . . . . . . . . . . . . . . . . . . . . . 60

B Theoretical signals 64

C Decision Structure 65

D Condition Indicator Results 68D.1 RMS Trms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68D.2 Crest Factor Tcf . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71D.3 Kurtosis Tku . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73D.4 Peak Indexes Tpeaks . . . . . . . . . . . . . . . . . . . . . . . . . . 75D.5 Results From the Labelled Data . . . . . . . . . . . . . . . . . . . . 77

Nomenclature

Fault Modes

NF No Fault

CH Chipped Tooth

BR Broken Tooth

ER Error

BS Ball Spin Fault

IR Inner Race Fault

OR Outer Race Fault

BA Bent Axle

SI Shaft Imbalance

BK Bad Key

Frequencies

fsh,1 Input shaft frequency

fsh,2 Idle shaft frequency

fsh,3 Output shaft frequency

fbs,1 Ball Spin frequency Input shaft

fbs,2 Ball Spin frequency Idle shaft

fbs,3 Ball Spin frequency Output shaft

fir,1 Inner Race frequency Input shaft

fir,2 Inner Race frequency Idle shaft

1

2 Contents

fir,3 Inner Race frequency Output shaft

for,1 Outer Race frequency Input shaft

for,2 Outer Race frequency Idle shaft

for,3 Outer Race frequency Output shaft

fgmH1 Gear Mesh frequency Helical Gears Input Pinion/Idle Gear 1

fgmH2 Gear Mesh frequency Helical Gears Idle Gear 2/Output Gear

fgmS1 Gear Mesh frequency Spur Gears Input Pinion/Idle Gear 1

fgmS2 Gear Mesh frequency Spur Gears Idle Gear 2/Output Gear

fs Sample frequency

fr Resonance frequency

Condition indicators - Test Quantities

Trms RMS (Root Mean Square)

Tcf Crest Factor

Tku Kurtosis

Tpeak Peak Index

Indexes

a Synchronously average

b1...b6 Bearing 1..6

ig1 Idle Gear 1

ig2 Idle Gear 2

ind Indicator

ip Input Pinion

og Output Gear

s Sensor

Other

Jind,a,s,n Threshold for indicator ind synchronous average signal a, sensor s andfrequency case n

Chapter 1

Introduction

1.1 BackgroundGearboxes have been used for many years in different configurations in all sorts ofmachines to scale torque and angular velocity. An emergent failure in one of thosegearboxes is something you want to prevent for economic, safety or other reasons.Diagnosis theory makes it possible to detect and isolate faults at an early stagemaking the monitored systems easier to maintain. In recent years the increase incomputing power have made it possible to develop and implement more and moreadvanced diagnosis systems operating both on- and offline.

1.2 The PHM Data ChallengeThe PHM Data Challenge is a competition held by "The Prognostics and HealthManagement Society" (PHM-Society). The PHM-Society is a non-profit organi-zation dedicated to the advancement of PHM as an engineering discipline. InSeptember 2009, the 2009 PHM Society Conference will bring together the globalcommunity of PHM experts from industry, academia and government in diverseapplication areas such as energy, aerospace, transportation, automotive and in-dustrial automation. Prior to the conference, the competition was held and par-ticipating in the competition has been a part of this thesis.

The challenge is to detect and isolate faults in a generic, industrial gearbox us-ing data from two accelerometers, a tachometer and information about the gearboxgeometry. There are 13 components in the gearbox, four gears, six bearings andthree axels, each with different fault modes. These components except for one axlewill be diagnosed in 560 different cases in the competition. [3]

1.3 Purpose and GoalAt the division of Vehicular systems, at Linköping University, several methodsfor model based diagnosis have been developed. However, no studies have been

3

4 Introduction

focused on gearbox diagnosis using accelerometers which makes participating inthe competition interesting.

The main goal of this thesis has been to study and evaluate different signalprocessing techniques that can be used for gearbox diagnosis. A diagnosis systemusing these techniques and based on modelled signals have been created and im-plemented in Matlab. The system was then used in the data analysis competitionto diagnose the 560 cases.

The ambition was to make the diagnosis system able to detect and isolate amajority of the component faults that may occur in the competition.

1.4 Related WorkThe Shock and Vibration Handbook [9] is a popular handbook, used by studentsand engineers, containing information about different sorts of vibrations. Thebook Machinery Vibration, Measurement and Analysis [16] describes a varietyof techniques for troubleshooting mechanical problems and improve machineryperformance. Vibration Fundamentals [12] is another book describing predictivemaintenance techniques based on vibration analysis and how the techniques canbe used in diagnostic applications. There are numerous articles published on thearea. A good example is the article Condition indicators for gearbox conditionmonitoring systems [15] which describes different condition indicators and howthey can be used in gearbox diagnosis.

1.5 OutlineThe subsequent chapters of this report are arranged as followed

Chapter 2 describes the apparatus that are to be diagnosed. The gearbox withgears and bearings and how data is collected are explained. The different faultmodes are explained as well.

Chapter 3 contains theoretical background to the diagnosis theory. The chapteralso describe the modelled signals that have been used to test and form the diag-nosis system.

Chapter 4 describes the signal processing techniques and condition indicatorsused in the diagnosis system. Synchronous averaging is described as well.

Chapter 5 explains the diagnosis system and how it has been implemented inMatlab.

Chapter 6 presents the results acquired from the diagnosis system when usedon the modelled signals.

Chapter 7 describes some labelled signals, that were revealed at the end of the

1.5 Outline 5

work, and how some of the different faults appear in the data.

Chapter 8 contains conclusions drawn from the work and suggest some improve-ments and future work.

Chapter 2

The Apparatus

This chapter describes the apparatus used in the competition. The gears andbearings in the gearbox and their respective fault modes are described. How, andunder what conditions, data is collected are described as well.

2.1 Gearbox geometryThe gearbox used in the competition is a generic industrial gearbox with a fixed setof gears. Three shafts and two sets of gears transfer power through the gearbox.A schematic of the gearbox, where the bearings are numbered, can be found inFigure 2.1. The input and output shafts, the four gears and the six bearings areall to be diagnosed for possible faults. [1]

2.1.1 GearsThere are two different sets of gears used, one using spur gears and the other spiralcut (helical) gears. The difference between the two types is seen in Figure 2.2. Thenumber of teeth for the two gear sets is found in Table 2.1. The gear reductionratios are the same for the two sets of gears. The ratios become, from input toidle shaft 16

48 = 3296 = 0.33... i.e. 3:1 and from input to output 16

48 ×2440 = 32

96 ×4880 =

0.2 that is 5:1. This means that the input shaft will rotate three times for onerevolution of the idle shaft and five times for one revolution of the output shaft.

Helical SpurInput Pinion 16 32Idle Gear 1 48 96Idle Gear 2 24 48

Output Gear 40 80

Table 2.1. Number of teeth on the two sets of gears

7

8 The Apparatus

Input Shaft

Idle Shaft

Output Shaft

1

2

3 6

5

4Electric Motor

Brake

Bearings

Accelerometer(Sensor 2)

Accelerometer(Sensor 1)

Tachometer

Bearings

Input Pinion

Idle Gear 1

Idle Gear 2

Output Gear

Figure 2.1. Schematic of the gearbox used in the competition.

Figure 2.2. The gears above illustrate the difference between a spur gear (left) and ahelical gear (right). Images courtesy of Precision Gears Inc [4]

.

2.2 Fault Modes 9

2.1.2 BearingsThe bearings used in the gearbox are of type MB Manufacturing ER-10K. Aschematic of a bearing is found in Figure 2.3. In the figure D is the pitch diameter,d the ball diameter and φ the contact angle. The contact angle is the angle betweenthe vertical centre line of the roller element and the line in which the contact forceacts.

D

d

Figure 2.3. Side view and cross-section of a bearing.

2.2 Fault ModesIn the competition there are a number of faults that are to be monitored by thediagnosis system. Each component should be classified into a specific componentmode. The different faults and how they appear in the data are explained in thefollowing sections.

2.2.1 Gear FaultsThe gears in the gearbox are to be diagnosed into one of four different componentmodes. These modes are No Fault (NF), Chipped Tooth (CH), Broken Tooth (BR)and Error (ER). The gear should be in the mode No Fault when there are no faultspresent. When one of the teeth on a gear is either chipped or broken the defectwill hit the meshing gear one time each revolution creating a certain amount ofvibrations. The Broken Tooth mode is when a tooth is heavily damaged or evenmissing. This will create a higher amount of vibrations compared to when a toothis chipped as in the mode Chipped Tooth. The component mode Error contains all

10 The Apparatus

other faults that the gears may have. This mode has not however been includedin the diagnosis system due to the unknown nature of the fault.

2.2.2 Bearing FaultsThe bearings are to be diagnosed into one of four different modes: No Fault (NF),Ball Spin Fault (BS), Inner Race Fault (IR) and Outer Race Fault (OR). Thedifferent modes are explained in detail below.

All roller bearings gives off specific vibration frequencies and the amplitudesof these frequencies are an indicator of the bearing condition. The interestingfrequencies in this work are known as ball spin frequency fbs, outer race frequencyfor and inner race frequency fir. The inner and outer races are the parts theballs in the bearing rides on. Note that the inner and outer race frequencies arenot the frequencies of the inner and outer races but the frequencies that appearwhen there is damage present on either of these sides. How the frequencies arecalculated can be found in (2.1) - (2.3). In the equations, fsh is the frequency ofthe rotating shaft held by the bearing, N the number of balls and D, d and φ asdescribed earlier in Section 2.1.2. [6], [16].

fbs = Dfsh2d

(1−

(d

D

)2cos2(φ)

)(2.1)

fir = Nfsh2

(1 + d

Dcos(φ)

)(2.2)

for = Nfsh2

(1− d

Dcos(φ)

)(2.3)

When a fault is present the correspondent frequency will become more dom-inant. A fault on one of the roller elements in the bearing will increase the am-plitude of the ball spin frequency. Faults on the inner or outer races will increasethe amplitude of the other frequencies accordingly. In the prerequisites to thecompetition the bearing parameters are specified as N = 8, d = 7.9375 mm,D = 33.5026 mm and the contact angle φ is zero. These numbers gives the vibra-tion frequencies according to Table 2.2

Fault frequencyfbs 1.9919fshfir 4.9477fshfor 3.0523fsh

Table 2.2. Vibration frequencies for the different bearing faults.

As seen in Table 2.2, the vibration frequencies are very close to the first, secondand fourth harmonics of the axle frequency. Further since the gear ratio from the

2.3 Data Acquisition 11

input to idle axle is 3:1 the outer race frequency for for the bearings holding theidle shaft is very close to the input shaft frequency. The same applies to thebearings on the output shaft where the inner race fault frequency fir is very closeto the frequency of the input shaft due to the 5:1 gear reduction ratio. The Table2.4 summarizes the calculated frequencies.

2.2.3 Faults on the ShaftsThe input shaft has, except for the No Fault (NF) mode, two different fault modes,Bent Axle, (BA), and Shaft Imbalance (SI). The shaft should be diagnosed intothese fault modes when the shaft are either bent or not balanced. The outputshaft has the two component modes No Fault (NF) and Bad Key (BK). The badkey fault occurs when the brake controlling the load is not fully in contact withthe output shaft. This will create a partial load condition which may be detectedby differencing frequencies. Under low load conditions this fault is very hard todetect. The idle shaft is not to be diagnosed. Due to time constrains the shaftdiagnosis were excluded from the diagnosis system and no results regarding shaftdiagnosis are presented.

2.3 Data AcquisitionData were sampled synchronously from two accelerometers mounted on each sideof the gearbox housing as seen in Figure 2.1. No information is given about inwhat direction the accelerometers are measuring and the assumption is made thatthey return an absolute value of all directions. There are 14 different systembehavioural modes in the competition that should be diagnosed. Six cases areusing helical gears and the other eight spur gears. Each system behavioral modeis repeated a certain number of times. This information is not however allowedto be a part of the diagnosis system in the competition. The sample frequencyis fs = 200

3 kHz and data is collected at five different shaft speeds 30, 35, 40, 45and 50 Hz under high and low load from the brake. The high and low loads resultin a small difference in shaft speed. Each load case is repeated four times. Thisgives a total of (8 + 6)× 5× 2× 4 = 560 different cases to be diagnosed. Data isrecorded for two seconds in each case thus making the amount of data quite large.Figure 2.4 shows under what conditions data for one system behavioural mode iscollected. The tachometer generates 10 pulses per revolution of the input shaftand data from the tachometer is very accurate.

12 The Apparatus

One system

behavioral mode

35 Hz30 Hz 40 Hz 45 Hz 50 Hz

LH H LL H LH LH

Figure 2.4. Each system behavioural mode is run under five different shaft speeds andunder high and low load (H & L) from the brake. Each load case is repeated four times.

2.4 SummaryThere are 12 components that are to be diagnosed. The components and theircomponent modes are summarized in Table 2.3. The sets of all component modesfor each component are named Ri where i = 1...12 represents the twelve compo-nents that are to be diagnosed. The frequencies related to the different componentsand faults are found in Table 2.4. As seen in the table the frequencies fsh,1, fir,3and for,2 are very close together which could make them more difficult to seperate.

Component Component ModesInput Pinion R1 = {NFip, CHip, BRip, ERip}Idle Gear 1 R2 = {NFig1 , CHig1 , BRig1 , ERig1}Idle Gear 2 R3 = {NFig2 , CHig2 , BRig2 , ERig2}

Output Gear R4 = {NFog, CHog, BRog, ERog}Bearing 1-6 R5...10 = { NFb1...b6 , IRb1...b6 , ORb1...b6 ,BSb1...b6 }Input Shaft R11 = { NFsh,1, BAsh,1, SIsh,1}

Output Shaft R12 = { NFsh,3, BKsh,3}

Table 2.3. The components that are to be diagnosed and their component modes.

2.4 Summary 13

Component Frequency Related to the input shaftInput Shaft fsh,1Idle Shaft fsh,2 = 1

3fsh,1Output Shaft fsh,3 = 1

5fsh,1Ball Spin Input Shaft fbs,1 ≈ 1.9919fsh,1Ball Spin Idle Shaft fbs,2 = 1

3fbs,1 ≈ 0.6640fsh,1Ball Spin Output Shaft fbs,3 = 1

5fbs,1 ≈ 0.3984fsh,1Inner Race Input Shaft fir,1 ≈ 4.9497fsh,1Inner Race Idle Shaft fir,2 = 1

3fir,1 ≈ 1.6499fsh,1Inner Race Output Shaft fir,3 = 1

5fir,1 ≈ 0.9895fsh,1Outer Race Input Shaft for,1 ≈ 3.0523fsh,1Outer Race Idle Shaft for,2 = 1

3for,1 ≈ 1.0174fsh,1Outer Race Output Shaft for,3 = 1

5for,1 ≈ 0.6104fsh,1

Table 2.4. A summary of the frequencies related to different components and faults.

Chapter 3

Theoretical Background

3.1 Diagnosis TheoryThe theory behind the hypothesis tests and decision structure are based on the-ory from the PHD Thesis "Model Based Fault Diagnosis: Methods, Theory, andAutomotive Engine Applications" by Mattias Nyberg [14].

3.1.1 Hypothesis TestsThe diagnosis system consists of a set of hypothesis tests δk. For each hypothesistest there is a test quantity and a rejection region. Let us look at for example a testquantity T1. The input pinion has the four behavioural modes NF, CH, BR and ERshort for No Fault, Chipped Tooth, Broken Tooth and Error. The null hypothesisfor this test quantity on the input pinion can be interpreted as, according to Table3.1, that some of the behavioural modes No Fault or Chipped Tooth can explain theobservations. If the null hypothesis is rejected the alternate hypothesis is assumedto be true. This means that some of the modes Broken Tooth or Error explainthe observations. The null hypothesis is rejected when the test quantity exceeds athreshold i.e. T1 ≥ J1. How the thresholds are created is explained in Section 5.2.The hypothesis test δk for test k will be defined according to equation (3.1) whereS1k is the decision taken when the null hypothesis is rejected and S0

k the decisionwhen it is not. S1

k is the set of component behavioral modes that the test quantityTk is sensitive to. If the null hypothesis is not rejected we cannot assume anythingthus making the set S0

k include all component behavioural modes.

δk = Sk ={S1

k if Tk ≥ JkS0

k if Tk < Jk(3.1)

3.1.2 Decision StructureThe decision structure can be seen as an overview of a diagnosis system based onstructured hypothesis tests. The structure shows what tests that react to different

15

16 Theoretical Background

fault modes. The Table 3.1 shows a simplified decision structure for a gear. AnX in the decision structure means that the test on the current row reacts to somefaults belonging to the behavioural mode in the current column. If for examplethe test T1 reacts, that is T1 ≥ J1, we know that the input pinion are eitherin the Broken Tooth mode or in the mode Error and the diagnosis statementfor this test is S1 = S1

1 = {BR,ER}. If the test does not react however i.e.T1 < J1 we cannot assume anything and thus making the diagnosis statementS1 = S0

1 = {NF,CH,BR,ER}. A zero in the structure means that the test onthat row is insensitive to the fault mode in the current column. The final decision,S, is then calculated as an intersection of the statements from all the tests. Theprocedure is described in the equations (3.2) - (3.3) where the tests T1 and T3have reacted. In the equations Fp denotes the present behavioural mode for thecomponent.

NF CH BR ERδ1 0 0 X Xδ2 0 X 0 0δ3 0 X X 0

Table 3.1. A simplified decision structure for a gear.

T1 ≥ J1 → Fp ∈ S11 = {BR,ER}

T2 < J2 → Fp ∈ S02 = {NF,CH,BR,ER}

T3 ≥ J3 → Fp ∈ S13 = {CH,BR} (3.2)

S = S1 ∩ S2 ∩ S3 = {BR,ER} ∩ {NF,CH,BR,ER} ∩ {CH,BR} = {BR} (3.3)

3.2 Theoretically Modelled SignalsSince the modes from which the data sets are collected are unknown a numberof modelled signals have been created based on some theoretical assumptions.The signals are created to simulate the different faults, how they appear in thesynchronous averaged signals, and how the condition indicators react. Based onthis information the decision structure is created.

3.2.1 Modelling the SignalsHarmonic patterns, not harmonically related to the rotation speed, typically ap-pear when there are faults in rolling-element bearings or on a defective gear. Thesefaults create an impulse that appears at a rate of one of the fault characteristic fre-quencies seen in Table 2.4. The impulse tends to excite any structural resonance

3.2 Theoretically Modelled Signals 17

and the characteristic fault frequency is suppressed by the resonance frequency[9]. Assuming a single dominant excited resonance frequency the impulse can bemodelled according to equation (3.4) [7]. The Figure 3.1 show how the signal in(3.4) may look like.

p(t) = Ae−β(t)cos(2πfrt)u(t) (3.4)

0 0.002 0.004 0.006 0.008 0.01-0.03

-0.02

-0.01

0

0.01

0.02

0.03

Time [s]

Figure 3.1. The impulse p(t) used in the modelled signals with parameters β = 700,A = 0.03 and fr = 2000.

In equation (3.4) A is the amplitude of the impulse, t the time, fr the excitedresonance frequency, β a structural damping coefficient and u(t) a unit step. Themodelled signal y(t) contains a consecutive series of such impulses at a rate ofone of the fault characteristic frequencies listed in Table 2.4. The equations (3.5)describe how the signals are created.

y(t) = s(t) + n(t) n(t) ∈ N(0, σ)s(t) = P (fr, ffault, β, t) P =

∑Nr

k=0 p(t− k1

ffault)

fr = fr0 + vfr(t) vfr

(t) ∈ N(0, σfr)

β = β0 + vβ(t) vβ(t) ∈ N(0, σβ)A = A0 + vA(t) vA(t) ∈ N(0, σA) (3.5)

Different faults are modelled using different parameters but the same equa-tions. For example a fault representing a broken tooth has a wider pulse and

18 Theoretical Background

slightly higher amplitude than a chipped gear. The different bearing faults aremodelled with the same amplitude and pulse width but with different pulse rep-etition frequencies. Signals from the sensor closest to the faulty component aremodelled with a higher amplitude A in equation (3.4) than signals from the sensoron the opposite side. The fault mode Error on the gears is not included in thesimulations due to the unknown nature of the fault. The gear mesh frequency andthe shaft faults are not included neither due to the extra work required modellingthem. The parameters that are used for different faults are found in Table B.1 inAppendix B. The parameters are tuned to make the modelled signals look like theraw data. Figure 3.2 shows a theoretical signal created with an Outer Race faulton one of the bearings on the idle shaft.

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

Time [s]

Figure 3.2. The first 0.1 seconds of a modelled signal with a fault on the Outer Raceon the bearing 4. Data is taken from sensor 2 and the input shaft frequency is 40 Hz.

3.3 Monte Carlo Simulation

A Monte Carlo simulation is a repeated calculation on a set of formulas thatcontains stochastic variables or processes. The simulation is based on the Lawof large numbers that says that the mean values of a number of independentobservations heads towards the expectation value when the number of observationsgrows [5]. The modelled signals described in Section 3.2 are created 1000 times foreach component fault and the result is used to identify how the condition indicatorsreact to the different faults and fill in the decision structure accordingly.

3.4 Welch’s T-test 19

3.4 Welch’s T-testTo determine whether there are a significant difference between the mean values oftwo observations, X1, . . . XnX

and Y1, . . . YnY, from two stochastic variables X and

Y a t-test is used. If the variances σ2X and σ2

Y are unknown and cannot be assumedto be equal and X and Y are normally distributed or the total observation sizenX + nY > 40 the Welch T-test is used. The test variable is calculated accordingto equation (3.6) and is approximately t-distributed with ν degrees of freedomaccording to equation (3.7). In the equations X and Y are the mean valuesfor the two observations, sX and sY the calcualted standard deviation from theobservations and nX and nY the number of samples in each observation. [13]

t = X − Y√s2

X

nX+ s2

Y

nY

(3.6)

ν =

(s2XnX

+ s2YnY

)2

s4X

n2X

(nX −1) + s4Y

n2Y

(nY −1)

(3.7)

A hypothesis test is used to determine if the mean values of the two samplesare equal. The null and alternative hypotheses are formed according to equation(3.8).

H0 : µX ≥ µY

H1 : µX < µY (3.8)

In the implementation the matlab function H=ttest2(X,Y,alpha,’left’,’unequal’)is used. The function performs a t-test using the null hypothesis that mean valueof vector X is greater or equal to the mean value of vector Y against the alternatehypothesis that it is not. The samples in the vectors are assumed to come fromdistributions with unknown and unequal variances. In the implementation alphais the significance level used in the test and it can be used as a tuning parameter.The t-test will be used to determine what tests that have reacted in the MonteCarlo simulations when the decision structure is created.

3.5 Lilliefors Test for NormalityTo test whether a set of observations are from an unspecified normal distributiona Lilliefors test for normality can be used. In the implementation the Matlabfunction lillietest(x) has been used which performs a Lilliefors test of the nullhypothesis that the sample in vector x comes from a normal distribution, againstthe alternative hypothesis that it does not. For a more detailed explanation pleaserefer to [10]. The Lilliefors test is used when the thresholds are calculated inSection 5.2.2

Chapter 4

Signal Processing

A part of this thesis has been to evaluate different signal processing techniques thatcan be used for gearbox diagnosis. A number of condition indicators have beentested and some implemented into the diagnosis system. The condition indicatorsare calculated on synchronously averaged signals for the different components.Some analyzing in the frequency domain was initially made but later excludedfrom the diagnosis system due to time constrains. An overview of the workingprocess for the signal processing can be found in Figure 4.1.

Create synchronous

averages

Remove gear mesh

frequencies

Calculate condition

indicatorsRead raw Data

Figure 4.1. The signal processing workflow

4.1 Calculating the Input Shaft FrequencyThe shaft frequencies for the data sets in the competition are not known in advanceand have to be calculated. By using the sample frequency and the tachometersignal, the input shaft frequency can be calculated by counting the number ofsamples for a revolution of the input shaft. The shaft frequency is known tobe constant for each case and the frequency is calculated as an average of allrevolutions on each case according to equation (4.1). In the equation Nk is thenumber of samples on revolution k, Nr the number of revolutions and Ts thesample time.

fsh,1 =(

1Nr

Nr∑k=1

Nk∑i=1

Ts

)−1

(4.1)

21

22 Signal Processing

4.2 Gear Mesh FrequenciesThe gear mesh frequencies are the vibration frequencies created by the meshinggears. In a fault free case these vibrations are the most apparent coming from thegearbox. The frequencies are calculated by multiplying the number of teeth of agear by the shaft frequency according to (4.2) - (4.5). In the equations fgmH1 andfgmH2 are the gear mesh frequencies for the first and second couple of gears in thegearbox using helical gears. fgmS1 and fgmS2 are the frequencies from the gearboxusing spur gears.

fgmH1 = 16fsh,1 = 48fsh,2 (4.2)

fgmH2 = 24fsh,2 = 40fsh,3 (4.3)

fgmS1 = 32fsh,1 = 96fsh,2 (4.4)

fgmS2 = 48fsh,2 = 80fsh,3 (4.5)

4.3 Estimation of Gearbox TypeNo information is given about what set of gears that is used in the different datasets from the competition. The different gear sets creates different amount of vi-brations and should therefore have its own set of thresholds for when the conditionindicators should react. Initially a calculation was made where the gearbox typewas estimated by calculating the energy in different frequency band. However thefunction was removed when the labelled data were released since it was no longerneeded and the function did not show any good results. In the modelled signalsthe meshing gears are not included and in the labelled data the gearbox type isspecified.

4.4 Filtering in the Frequency DomainBy removing the gear mesh frequencies the total vibration level will be reducedmaking smaller faults easier to identify. The frequency bands where the gear meshfrequencies are located are very narrow in relation to the sample frequency henceresulting in very high filter orders. Instead filtering have been done in the frequencydomain by multiplying the signals discrete Fourier transform by a filter accordingto (4.6) where x is the raw signal,H[θ] a band stop filter and y the filtered signal [8].FFT is the fast Fourier transform and IFFT the inverse fast Fourier transform. <is the real part of the complex number achieved after the inverse transform. Sinceno information is given about what gearbox type that is used in the different setsof data and no estimation of the gearbox type is made the same filter is used forthe two types of gears. The gear mesh frequencies are substantially higher thanthe fault characteristic frequencies so this simplification will not affect the result.The first harmonics of the frequencies will be removed as well. The band stopfilter was chosen over a low pass filter to preserve the high resonance frequenciespreviously described in Section 3.2.1. The filtering is performed before creating the

4.5 Synchronous Average for the Shafts 23

synchronously averaged signals. Figure 4.2 shows how the synchronously averagedsignal for the input shaft changes when the original signal is filtered.

X = FFT{x} (4.6)Y = H[θ]X

y = <(IFFT{Y })

0 1 2 3 4 5 60

0.01

0.02

0.03

0.04

0 1 2 3 4 5 60

0.01

0.02

0.03

0.04

Shaft angle [rad]

Figure 4.2. How the synchronously averaged signal for the input shaft changes whenthe sensor signal is filtered. The upper signal is the unfiltered and the lower the filtered.The peak-to-peak value clearly decreases making for example a chipped gear easier toidentify.

4.5 Synchronous Average for the ShaftsBy watching the synchronous averaged signals, periodic faults related to the dif-ferent shafts and gears can be found. The raw data from the accelerometersis averaged over the revolutions for the different shafts using the data from thetachometer. For every revolution on the input, idle and output shafts the tachome-ter generates 10, 30 and 50 pulses respectively. Faults that does not appear as aninteger multiple of the shafts frequencies will for a large number of revolutions beaveraged out. An overview of how different gear faults appear in the synchronouslyaveraged data is found in Table 4.1. In Appendix A figures can be found which


illustrate how some faults in the modelled signals appear in the synchronouslyaveraged data.

Fault Input Pinion Fault Idle Gear Fault Output Gear

Input Shaft Average 1 13

15

Idle Shaft Average 3 1 53

Output Shaft Average 5 35 1

Table 4.1. Number of times, for each revolution, the different faults appears in thesynchronously averaged data.

4.6 Synchronous Average for Bearing FaultsThe bearing frequencies are not integer multiples of the input shaft making thesynchronous averaging a bit more complicated. For example, as seen in Table 2.2,the ball spin frequency fbs,1 for bearings holding the input shaft is 1.9919 times theinput shaft frequency i.e. 1.9919fsh,1. Since the tachometer generates 10 pulsesper revolution of the input shaft 10× fsh,1

fbs,1= 10

1.9919 pulses will pass between ballspin faults for the bearings holding the input shaft. The number of pulses forall revolutions are calculated in advance to prevent rounding errors to propagate.In Figure 4.3 an example using 1.65 pulses are shown. The first four revolutionswill stop after 1.65, 1.65 × 2 = 3.3, 1.65 × 3 = 4.95 and 1.65 × 4 = 6.6 pulses.The fraction of a pulse is calculated as the time passed since the last whole pulseassuming that the current pulse length is the same as the previous one. This isa good enough assumption knowing that the shaft frequency is constant for thewhole set of data.

1:st

2:nd

3:rd

4:th

1 2 3 4 5 6

Pulse nr

Figure 4.3. How the signal is divided into parts containing 1.65 pulses using the datafrom the tachometer.

The number of pulses is calculated for the different bearing faults on the dif-ferent shafts. The condition indicators are then, as with the shafts, calculated on

4.7 Condition Indicators 25

the averaged signals. In Table 4.2 the number of pulses, on the tachometer signal,for each fault is shown.

Ball Spin Inner Race Fault Outer Race Fault

Input Shaft 101.9919

104.9477

103.0523

Idle Shaft 301.9919

304.9477

303.0523

Output Shaft 501.9919

504.9477

503.0523

Table 4.2. How many pulses in the tachometer signal that passes between faults on thebearings.

4.7 Condition IndicatorsThere are a number of condition indicators for gearbox diagnosis described in theliterature [15]. Some of those indicators have been tested and implemented in thediagnosis system. The indicators are calculated on the synchronously averagedsignals for the different components.

4.7.1 RMSThe RMS (Root Mean Square) value is a measure of the total vibration level of asignal. Signals with high values over a large period of time intend to generate alarge RMS values. A narrow peak however will not affect the parameter very muchhence making it less sensitive to small faults as a chipped tooth. The parameter isused in the diagnosis system primary to identify broken teeth. How the parameteris calculated is found in equation (4.7) where si is a sample on a synchronouslyaveraged signal. A high mean value of the measured signal has a big impact onthe Trms indicator.

Trms =

√√√√ 1N

N∑i=1

(s2i ) (4.7)

4.7.2 Crest FactorThe crest factor is sensitive to signals with a high peak to peak value, i.e. thedistance between max and min, and otherwise low signal energy. A chipped toothfor example will not create a large amount of vibrations thus making the Trmsvalue almost unchanged. The peak to peak distance dpeak−to−peak will however belarge and hence result in a large crest factor according to equation (4.8).

Tcf = dpeak−to−peak

Trms(4.8)


4.7.3 KurtosisThe kurtosis is an indicator of how flat or peaked the data is. A signal containingmany sharp peaks will result in a high kurtosis value. The mathematical definitionof the kurtosis value can be found in equation (4.9). The kurtosis value is used toidentify broken or chipped gears as well as the bearing faults. In equation (4.9) siis a sample on a synchronously averaged signal and s̄ the mean value of the samesignal. If the signal is normally distributed the parameter value will be close toTku ≈ 3 [8].

Tku =N∑Ni=1(si − s̄)4(∑N

i=1(si − s̄)2)2 (4.9)

4.7.4 Finding Peak IndexesTo separate faults that appear on different shafts from each other the distancebetween the peak values on the synchronously averaged signals are calculated.If there are three distinct peaks on the synchronous average for the idle shaftseparated by the distance for one revolution of the input shaft it is fair to assumethat the a fault is not located on some of the idle gears but on the input pinion.Likewise if there are five peaks on the synchronous averages for the output shaftthe fault is most probably not located on the output gear. The same applies tothe bearing faults which have the same frequency ratios between the shafts. Firstsome pre-processing of the signal is done by removing the mean value and squarethe signal. Then the k largest peaks, where k = 3 for the idle shaft and k = 5for the output shaft, are calculated by removing the largest peak on at a timewith its closest environment. The peak indexes n are sorted and the distance n∆between them are calculated according to equation (4.10). The mean value of n∆and the estimated variance s2

n∆are calculated according to equations (4.11) and

(4.12) and compared to the length the synchronously averaged signal for the samefault but on the input shaft Navg1 . Figure 4.4 summarizes the procedure. If thedifference n∆ −Navg1 and the variance s2

n∆are small enough the test reacts. The

test is defined according to equation (4.13) where α1 and α2 are tuning parametersused to weight the mean value and variance together. The parameters and thethresholds are set manually after studies of different sets of data.

n∆ = [n[2]− n[1], . . . , n[k]− n[k − 1]] (4.10)

n∆ = 1N

N∑i=1

n∆ (4.11)

s2n∆

= 1N − 1

N∑i=1

(n∆ − n∆)2 (4.12)

Tpeaks =√α1(n∆ −Navgsh,1)2 + α2(s2

n∆)2 (4.13)

4.8 FFT Analysis 27

Sort the stored

indexes

n:=sort(n)

Remove this peak

from the signal y

for peaks=1:k

Calculate distance nD

:=[n2-n1, ... ,nk-nk-1]

between all stored

indexes

Calculate mean vaule

and variance for nD

Start

Find and store index n

for the max value of

the signal y

Remove mean value of

signal and square:

y= (x-x)2

Figure 4.4. How to test if there are k number of peaks on a synchronously averagedsignal.

4.8 FFT AnalysisAnalyzing the signals in the frequency domain is another way to diagnose the gear-box components. This approach has not however been included in the diagnosissystem and may be a subject for future work.

4.8.1 Patterns not Related to the Shaft FrequenciesThe harmonic patterns, not harmonically related to the rotational speed, pre-viously described in Section 3.2.1 tends to dominate the spectrum. The faultcharacteristic frequency for the signal is suppressed by the resonance frequencyand the fault frequency is hard to find in the spectrum. The peaks however is sep-arated by the fault characteristic frequency which makes them easy to identify atleast for the signals created based on theory. The Figure 4.5 shows the estimatedspectrum of the created signal found in Figure 3.2 with a resonance frequency offr = 2000 Hz. The fault characteristic frequency is for,2 = 40.6975 Hz.


0 1000 2000 3000 4000 5000

0

0.005

0.01

0.015

0.02

0.025

Frequency [Hz]

0 50 100 150 2000

0.5

1

1.5

2

2.5x 10

-3

Frequency [Hz]

Figure 4.5. Power spectrum of the signal from Figure 3.2. The resonance frequency fr =2000 Hz dominates the spectrum and the peaks are separated by the fault characteristicfrequency of for,2 = 40.6975 Hz.

Chapter 5

The Diagnosis System

This chapter describes the diagnosis system more in detail. How the decision struc-ture and thresholds are created is explained and an overview of the implementationcan be found in Figure 5.3.

5.1 Decision StructureThe decision structure is generated from the results of the Monte Carlo simulationpreviously described in Section 3.3. A t-test is done to decide whether there is adifference between how the test quantities react on a modelled signal having nofault implemented and one where a fault is present. If there are no significantdifference the correspondent place in the decision structure will be filled with azero. If there is a difference the place will be filled with an X. The three testquantities Trms, Tcf and Tku are calculated on the 12 synchronous averages fromthe two sensors. The Tpeak indicators are calculated on the eight averages relatedto the idle and output shafts. This gives a total of 3 × 12 × 2 + 8 × 2 = 88 rowsin the decision structure. The decision structure is consequently the same for thedifferent shaft speeds and load conditions but the thresholds are different. Thedifferent components with their fault modes are placed as columns next to eachother according to Table 5.1. The Tpeak indicators are used in a different way inthe decision structure. These tests are only used to exclude different faults andtheir respective rows will consist only of X’s except for the component that shouldbe excluded where zeros will be placed. The complete decision structure is foundin Appendix C.

5.2 ThresholdsSince no information is given about the gearbox condition for each set of data,creating thresholds for the diagnosis system is not made with a standard approach.Each sensor and subset described in Section 2.3 has its own thresholds. The 12synchronous averages, five frequencies, two loads, two sensors and three condition

29

30 The Diagnosis System

Input Pinion . . . Bearing 6 Input Shaft Output ShaftNF CH BR ER . . . NF IR OR BS NF BA SI NF BK

Trms(a,s) 0 0/X 0/X 0 . . . 0 0/X 0/X 0/X 0 0 0 0 0Tcf(a,s)

0 0/X 0/X 0 . . . 0 0/X 0/X 0/X 0 0 0 0 0

Tku(a,s)0 0/X 0/X 0 . . . 0 0/X 0/X 0/X 0 0 0 0 0

Tpeak 0/X 0/X 0/X 0/X . . . 0/X 0/X 0/X 0/X X X X X X

Table 5.1. Outline of the decision structure. The index a = 1 . . . 12 are the 12 syn-chronous averages and s = 1, 2 are the two sensors. The shaft faults are included in thedescision structure but not diagnosed.

indicators Trms, Tcf and Tku gives a total of 12×5×2×2×3 = 720 thresholds. Inaddition to this 8× 5× 2× 2 = 160 for the peak indicators Tpeak. The thresholdsare created from the data that are to be diagnosed and labelled Jind,a,s,n wherethe indexes stands for indicator, synchronous average, sensor and frequency casewhich also includes the two brake loads. Table 5.2 summarizes all the thresholds.

Indicator Synchronous averages Sensors Frequencies Brake Loads SumTrms 12 × 2 × 5 × 2 = 240Tcf 12 × 2 × 5 × 2 = 240Tku 12 × 2 × 5 × 2 = 240Tpeak 8 × 2 × 5 × 2 = 160Total: 880

Table 5.2. The number of thresholds used in the diagnosis system.

5.2.1 Data ClusteringA Matlab cluster algorithm is used to divide the calculated indicator values for thedifferent cases described in Section 2.3 into two subgroups. The function computesthe distances between all values in the data set and, based on the result, createsa hierarchical cluster tree. Thereafter two clusters are created from the clustertree as specified as an argument in the function. More information about thefunction can be found in the Matlab online manual [2]. The cluster algorithm isalso used initially when dividing the data into the five frequency and two loadcases by changing the input argument of the function. The function divides alldata into five different groups decided by the input shaft frequency and thereafterall of these groups are split into two subgroups that symbolises the high and lowload from the brake. In the following sections the two clusters created for each ofthe ten cases are named cll and clh, where cll is the group containing the lowestvalues and clh the highest.

5.2.2 Threshold CalculationTo decide if there is a significant difference between the two groups separatedby the cluster algorithm a Lilliefors test for normality is used. Data from both

5.3 Multiple Faults 31

clusters are used with the null hypothesis that the combined clusters comes froman unspecified normal distribution according to equation (5.1) with the alternatehypothesis that the data does not (5.2). If the null hypothesis is rejected thethreshold is placed between the two clusters according to equation (5.4) and if notthe threshold is placed above the highest cluster clh (5.3)

H0 : {cll, clh} ∈ N(µ, σ) (5.1)

H1 : {cll, clh} /∈ N(µ, σ) (5.2)

H0 → Jind,a,s,n = max(clh)−min(cll)2 + max(clh) (5.3)

H1 → Jind,a,s,n = min(clh)−max(cll)2 + max(cll) (5.4)

Figure 5.1 shows an example using a histogram when the null hypothesis hasbeen rejected and Figure 5.2 when it has not.

0 0.5 1 1.5 2 2.5 3 3.5 40

5

10

15

Indicator Value

Figure 5.1. The null hypothesis (5.1) has been rejected and the threshold has beenplaced between the clusters according to (5.4).

5.3 Multiple FaultsThe gears in the gearbox are assumed to only have one fault mode present ata time. If a tooth on a gear is chipped it cannot be broken and the other wayaround. A gear is also assumed only to have one tooth chipped or broken at atime. This assumption is not made on the bearings that consequently may haveall fault modes present at a time. There are no assumptions regarding the numberof faults present in the system simultaneously.


0.5 1 1.5 2 2.5 3 3.5 4 4.50

1

2

3

4

5

6

7

8

9

10

Indicator Value

Figure 5.2. The null hypothesis has not been rejected and the threshold has been placedabove the highest cluster according to equation (5.3)

5.4 Diagnosis of non Faulty ComponentsIn the competition, and hopefully in real cases as well, the No Fault mode isthe most likely behavioural mode for every component. The only test quantitysensitive to the No Fault mode for any component is the indicator Tpeak. Toget a diagnosis also for non faulty components an intersection operation is madebetween all component modes for each component and the computed diagnosis S.If the result is an empty set, i.e. no component mode for the current component isincluded in the diagnosis S, the result is updated with the first component modeRi(1) for this component which is the No Fault mode. The procedure is performedfor all components and is described in the equations (5.5). For example if no testreact, the diagnosis will be S = ∪Ri, i = 1 . . . 12 where Ri previously describedin Section 2.4 is all component behavioural modes for one component. Or if onlythe test Tku4,1 reacts the diagnosis, according to the theory in (3.1.2), will becomeS = {BSb1, BSb2, BSb4}. In these cases we do not know anything about thecomponents having all or none component modes in the diagnosis statement S.Since the No Fault mode is the most likely for every component, it is fair to assumethat these components have no faults present.

for i = 1 : 12 (5.5)if Ri ∩ S = ∅

S := S ∪Ri(1)elseif S ∩Ri = Ri

S := S�{Ri�Ri(1)}end

end

5.5 Implementation 33

5.5 ImplementationThe diagnosis system is implemented in Matlab. The decision structure is gener-ated based on the results from the Monte Carlo simulations in Matlab and savedinto an Excel worksheet. The structure is then read into Matlab when the diag-nosis is made. This extra step is made to get an easy overview of the decisionstructure and simplify for possible manual changes. A schematic of the systemcan be found in Figure 5.3. The data object seen in Figure 5.3 contains all in-formation about one case of data from the competition or from some modelledsignal. The object reads data from memory, filter the gear mesh frequencies, cre-ates the synchronous averages and finally calculate all condition indicators. A loopruns through all cases and creates the thresholds according to the results from thecondition indicators. Finally the decision is taken based on the reactions of thetests.


NF1 F11 F21 ... NFn F1n F2n

d1 0 X X ... 0 X O

d2 0 0 X ... 0 O O

... ... ... .. ..

dk 0 0 X ... 0 X X

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02

0

2

4

6

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01-0.1

0

0.1

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01-0.1

0

0.1

Create synchronous

averages

Remove gear mesh

frequencies

Calculate condition

indicators

Read raw data

Create theoretical

signals

Fill in decision

structure based on

results

Create synchronous

averages

Calculate condition

indicators

Diagnosis

Statement

Dataobject – one case

Loop through all cases

and save the condition

indicators Tk

Categorize the cases

by frequency and load

Calculate thresholds:

Jk

What tests have

reacted?

Read decision

structure

Monte Carlo simulation

Force each set into

two subsets

Calculate diagnosis

using intersection

operations

Update diagnosis by

adding non faulty

components

Figure 5.3. The diagnosis system

Chapter 6

Parameter and DiagnosisResults

This chapter describes the achieved results when running the diagnosis system onthe modelled signals. The condition indicators, thresholds and diagnosis state-ments are calculated.

6.1 Evaluation of the Condition IndicatorsThe condition indicators have been evaluated on modelled signals with single faultspresent and the results can be found in Appendix D. The indicators react asexpected on the modelled signals and tables showing the indicator values can befound in Appendix D as well.

6.2 Results from the Monte Carlo simulationThe signals previously described in Section 3.3 were created 1000 times in orderto fill in the decision structure correctly. To decide wheter a significant differencebetween the two cases exists a t-test is made with the null hypothesis H0 : µF ≤µNF against the alternate hypothesis H1 : µF > µNF where µF is the meanvalue of the data coming from the tested row in the decision structure and µNFis the mean value of the no fault case. The t-test is for large populations verysensitive. When using all 1000 observations in the test, the slightest difference inthe mean value even with very high significance levels affect the results. This givesan unrealistic number of X’s in the decision structure. Instead only 50 samplesfrom the fault case have been chosen in the t-test. The Figures 6.1-6.3 illustratesthe differences. Figure 6.4 show histograms of how the condition indicator Tcf hasreacted in the simulation when a fault on the inner race on bearing 1 is present.In the figures H = 1 means that the null hypothesis have been rejected and H = 0that it has not. The light bars are from the fault free case and the darker fromwhen a fault is present.

35

36 Parameter and Diagnosis Results

0.1 0.15 0.2 0.250

20

40

60

80

100H=1

0.1 0.15 0.2 0.250

20

40

60

80

100H=1

0.1 0.15 0.2 0.250

20

40

60

80

100H=1

Figure 6.1. All 1000 observations are used with a significance level of α = 1%. Thet-test reacts and the null hypothesis H0 : µF ≤ µNF is rejected in all cases (H=1).

0.1 0.15 0.2 0.250

20

40

60

80

100H=0

0.1 0.15 0.2 0.250

20

40

60

80

100H=1

0.1 0.15 0.2 0.250

20

40

60

80

100H=1

Figure 6.2. When raising the significance level to α = 0.001% the t-test does not rejectthe null hypothesis in the left graph (H=0) but still in the middle graph and to the right.

0.1 0.15 0.2 0.250

20

40

60

80

100H=0

0.1 0.15 0.2 0.250

20

40

60

80

100H=0

0.1 0.15 0.2 0.250

20

40

60

80

100H=1

Figure 6.3. In this figure only 50 samples are used from the fault case and the signifi-cance level is α = 1%. The null hypothesis is only rejected in the right graph.

6.2.1 The Decision Structure

Most of the decision structure is generated from the results of the t-tests on theresults from the Monte Carlo simulations. The rows for the test quantities Tpeakhowever are filled in manually due to the different implementation of this testquantity. Some tests get the same row in the structure. This does not renderone of the tests useless but they may react under different circumstances. The

6.2 Results from the Monte Carlo simulation 37

3 3.5 4 4.5 5 5.50

50

100

150Input Shaft Sensor 2, H=0

5 5.5 6 6.5 7 7.50

50

100

150Idle Shaft Sensor 2, H=0

5 6 7 8 9 100

100

200Output Shaft Sensor2, H=0

2 2.5 3 3.5 4 4.50

100

200Input Shaft BS S2, H=0

3 4 5 6 70

50

100

150Idle Shaft BS S2, H=0

4 5 6 7 80

50

100

150Output Shaft BS S2, H=0

2 4 6 8 10 120

100

200Input Shaft IR S2, H=1

2 4 6 8 100

100

200Idle Shaft IR S2, H=1

2 4 6 8 100

100

200Output Shaft IR S2, H=1

4 5 6 7 80

50

100

150Input Shaft OR S2, H=0

5 6 7 8 9 100

50

100

150Idle Shaft OR S2, H=1

4 5 6 70

100

200Output Shaft OR S2, H=0

Figure 6.4. Histogram over how the Tcf test reacts on the different synchronous averagesfrom sensor 2 when a fault is present on the inner race of bearing 1. The indicator reactson the synchronous averages related to the IR fault on the three axles (only 50 samplesare used in the t-test). The hypothesis rejections can be found in the decision structurein column 18 (Bearing 1 IR) and rows Tcfa,2 a = 1 . . . 12

complete decision structure is found in Appendix C. The number of X’s in thedecision structure can be changed slightly by adjusting the significance level αor the number of samples used in the t-test. A smaller α-value or fewer samplesmeans that the number of X’s will decrease. Too many X’s results in a bad faultisolation and too few may result in incorrect or empty diagnosis for single faults.

The Figure 6.5 illustrates what columns in the decision structure that cannotbe separated from each other. The figure is created by checking what columnsin the decision structure that are a subset of another. If column ci is a subset ofcolumn cj i.e ci ⊆ cj a mark will be placed on row i and column j in Figure 6.5.This implies that the component mode related to column ci cannot be separatedfrom the mode related to cj . The ideal case would be the identity matrix whereall columns in the decision structure only would be a subset of themselves.

For example row 18, i.e. i = 18 in Figure 6.5 have marks in column 18 and30. This means that the column 18 in the decision structure i.e. an inner racefault on bearing 1, IRb1 cannot be separated from column 30 an inner race faulton bearing 4. The Table 6.1 shows the interpretation of Figure 6.5.


0 5 10 15 20 25 30 35 40

0

5

10

15

20

25

30

35

40

Column cj

Colu

mn c

i

Figure 6.5. What columns in the decision structure that are a subset of another column.The columns related to the No Fault modes the mode Error on the gears and all shaftmodes have been removed from the matrix.

6.3 Diagnosis Results

Diagnosis was performed both on the theoretically modelled signals as well as onthe data used in the competition. No information or result is given about thedata from the competition which makes it hard to draw any conclusions fromthe results. Three different diagnoses were sent into the competition before itclosed. The best result was achieved when all components were diagnosed to benon faulty which was far from correct. All parts of the diagnosis system were notcompleted at this time and the system was only searching for single faults. Dueto the lack of information given about the data used in the competition, and thebad result achieved, the diagnosis results from the competition are not presentedin this report.

6.3 Diagnosis Results 39

Component Mode Cannot be separated fromCHig1 CHip, BRip, BRig1

CHig2 BRip, BRig2

CHog BRogIRb1 IRb4ORb1 ORb4BSb1 BSb4ORb2 ORb1 , ORb4BSb2 BSb1 , BSb4ORb3 ORb1 , ORb4ORb5 ORb4ORb6 ORb4

Table 6.1. Interpretation of Figure 6.5 Some single faults cannot be separated fromeach other. For some reason there are many faults that cannot be separated from theouter race fault on bearing 4 ORb4 which can be seen as the large number of marks incolumn 31 (cj = 31) representing this component mode.

6.3.1 Diagnosis of the Modelled Signals

Theoretical signals were created according to the prerequisites in the competition.Different faults were simulated under the different shaft frequencies described inSection 2.3. The parameters used on the signals can be found in Table B.1. Table6.2 describes the fault signals that were created and the diagnosis results from thewhole system as well as from the individual components are presented. In everysystem mode there are 12 component modes. So for 40 system diagnoses there are480 component diagnoses. The diagnosis result is divided into five cases on boththe system and component level. These cases are called correct, sub diagnosis,false alarm, missed diagnosis and incorrect and are found as columns in Table 6.2.The different cases are described below.

Correct

If the diagnosis statement is completely correct on either system or componentlevel the diagnosis is counted as correct. One correct system diagnosis gives 12correct component diagnoses.

Sub Diagnosis

If the diagnosis statement is wrong but the correct diagnosis is included in thediagnosis statement the diagnosis is counted as a Sub Diagnosis. For example, ifthe correct diagnosis for the input pinion is {CH} but the diagnosis statement forthis component is {CH,BR} the result will be counted as a sub diagnosis.


False Alarm

If a component or the whole system is in the No Fault mode, i.e. the system orthe component is correct but the diagnosis statement says otherwise the result iscounted as a false alarm.

Missed Diagnosis

If the diagnosis statement for a component or the whole system says that there isno fault present when the correct statement would be that there are, the result iscounted as a missed diagnosis.

Incorrect

If the diagnosis statement on system or component level is incorrect and the cor-rect diagnosis cannot be found within the diagnosis statement, as in the case withsub diagnosis, the result is counted as incorrect.

System Diagnosis Component Diagnosis

Case Nr Simulated Fault Cor

rect

Su

bD

iagn

osis

Fal

seA

larm

Mis

sed

Inco

rrec

t

Cor

rect

Su

bD

iagn

osis

Fal

seA

larm

Mis

sed

Inco

rrec

t

1-80 No Fault 77 0 3 0 0 930 0 30 0 081-120 Input Pinion Chipped 0 40 0 0 0 424 40 16 0 0

121-160 Input Pinion Broken 0 40 0 0 0 440 0 40 0 0161-200 Idle Gear 1 Chipped 0 0 0 0 40 344 0 96 0 40201-240 Idle Gear 1 Broken 0 40 0 0 0 357 0 123 0 0241-280 Idle Gear 2 Chipped 0 37 0 0 3 360 37 80 0 3281-320 Idle Gear 2 Broken 0 40 0 0 0 292 0 188 0 0321-360 Output Gear Chipped 0 40 0 0 0 323 40 117 0 0361-400 Output Gear Broken 0 36 0 0 4 405 0 71 4 0401-440 Inner Race Fault Bearing 1 0 39 0 0 1 439 0 40 1 0441-480 Outer Race Fault Bearing 1 0 39 0 0 1 409 15 55 1 0481-520 Ball Spin Fault Bearing 1 0 39 0 0 1 439 0 40 1 0521-560 Inner Race Fault Bearing 2 0 24 0 0 16 420 0 44 16 0561-600 Outer Race Fault Bearing 2 0 37 0 0 3 265 37 175 3 0601-640 Ball Spin Fault Bearing 2 0 26 0 0 14 347 0 119 14 0641-680 Inner Race Fault Bearing 3 0 23 0 0 17 423 0 40 17 0681-720 Outer Race Fault Bearing 3 0 39 0 0 1 439 0 40 1 0721-760 Ball Spin Fault Bearing 3 0 38 0 0 2 351 0 127 1 1761-800 Inner Race Fault Bearing 4 40 0 0 0 0 480 0 0 0 0801-840 Outer Race Fault Bearing 4 0 40 0 0 0 440 40 0 0 0841-880 Ball Spin Fault Bearing 4 40 0 0 0 0 480 0 0 0 0881-920 Inner Race Fault Bearing 5 0 29 0 0 11 429 0 40 11 0921-960 Outer Race Fault Bearing 5 0 35 0 0 5 300 35 140 5 0

961-1000 Ball Spin Fault Bearing 5 0 35 0 0 5 435 0 40 5 01001-1040 Inner Race Fault Bearing 6 0 38 0 0 2 438 0 40 2 01041-1080 Outer Race Fault Bearing 6 0 40 0 0 0 421 0 59 0 01081-1120 Ball Spin Fault Bearing 6 0 40 0 0 0 389 20 71 0 0

Total: 157 834 3 0 126 11219 264 1831 82 44︸︷︷︸︸︷︷︸Total: 1120 13340

Table 6.2. The results from the diagnosis system when used on the modelled signals.

6.3 Diagnosis Results 41

As seen in Table 6.2 sub diagnosis is the most common result on the systemlevel. The simulated fault is found but the result also includes some false alarmor sub diagnosis on the component level. Some of these sub diagnosis can berelated to the decision structure that does not have a full Rank according toFigure 6.5. Every extra mark outside the diagonal in the figure will give an extra40 false alarms or sub diagnosis on component level. Other false alarms on thecomponent level are related to components with the same rotational speed or thesame component fault but on different axles. The two shafts are included in thenumbers but they will always be diagnosed into the No Fault Mode.

Chapter 7

Evaluation of Labelled Data

Late into the work labelled data were released to illustrate how some of the differentfaults appear in the data. At this time most of the work on the diagnosis systemwere already completed so no new modelling of the signals used to form the decisionstructure were made. The diagnosis system however was tested on the labelleddata but the results were not very accurate. The labelled data is specified in Table7.1. Each case is repeated under high and low load from the brake and at fivedifferent shaft speeds as described in Section 2.3. Each load and speed case isrepeated two times making a total of (6 + 8)× 2× 5× 2 = 280 runs.

7.1 Helical vs. Spur Gears, Frequency and Load

The vibration levels of the fault free signals differ between the helical and the spurgears. The only cases where the two gear boxes have the same component modes,and therefore useful for comparison are the no fault cases. Figure 7.1 shows how theRoot Mean Square value for the complete signal increases with higher frequenciesfor the two sets of gears. Under low load from the brake the RMS-value is higherthan when the load is high. This is the opposite of the assumed in the previoussections that a higher load would generate more vibrations. The same pattern isseen in the fault cases where the low load cases have higher RMS-values than whenthe load is high. This should not affect the diagnosis however since the high andlow load have different thresholds. The spur gears have, in all low load cases andin most of the high load cases, a slightly higher value than the helical gears. TheRMS-value of the signals from the spur gears increase more with higher frequenciesthan the value from the signals using helical gears. The same applies to the lowload cases where the RMS-value increases more with higher frequencies than whenthe load from the brake is high.

43

44 Evaluation of Labelled Data

Gears Bearings Shafts

Inpu

t

Idle

1

Idle

2

Outpu

t

1 2 3 4 5 6 Inpu

t

Outpu

t

Helical 1 NF NF NF NF NF NF NF NF NF NF NF NFHelical 2 NF NF CH NF NF NF NF NF NF NF NF NFHelical 3 NF NF BR NF NF NF NF Comb IR NF BA NFHelical 4 NF NF NF NF NF NF NF Comb BS NF SI NFHelical 5 NF NF BR NF NF NF NF NF IR NF NF NFHelical 6 NF NF NF NF NF NF NF NF NF NF BA NF

Spur 1 NF NF NF NF NF NF NF NF NF NF NF NFSpur 2 CH NF ER NF NF NF NF NF NF NF NF NFSpur 3 NF NF ER NF NF NF NF NF NF NF NF NFSpur 4 NF NF ER BR BS NF NF NF NF NF NF NFSpur 5 CH NF ER BR IR BS OR NF NF NF NF NFSpur 6 NF NF NF BR IR BS OR NF NF NF SI NFSpur 7 NF NF NF NF IR NF NF NF NF NF NF BKSpur 8 NF NF NF NF NF BS OR NF NF NF BA NF

Table 7.1. The labelled data that was released. Comb is a combination of faults notspecified.

.

7.2 Component Fault ModesMost of the fault modes are present either separately or in some combination inthe labelled data. The following sections describe how some of these faults appearwhen studying the data.

7.2.1 Bearing FaultsThe synchronous averaging for bearing faults did not show any good result andare probably only useful in theory. The averaging itself was accurate with verylittle variation in relation to the frequencies on the modelled signals. For higherfrequencies the number of pulses on the tachometer signal that pass between dif-ferent faults are low. For an inner race defect on either bearing one or four, asseen in Table 4.2, only 10

4.9477 ≈ 2.02 pulses will pass between the recurring defecton the bearing. This will make the synchronous averaging very sensitive for smallfrequency variations. If a ball in a bearing slides the frequency in the synchronousaveraging will be wrong. Overall the bearing faults are very hard to find in thesynchronous averaged signals for the different fault modes. A higher resolutionon the tachometer signal would maybe result in a somewhat better result. Nocase does only have bearing faults present. This makes it hard to identify single

7.2 Component Fault Modes 45

30 35 40 45 500

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Frequency [Hz]

Spur Gears

Low load sensor 1

High load sensor 1

Low load sensor 2

High load sensor 2

30 35 40 45 500

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Frequency [Hz]

Helical Gears

Low load sensor 1

High load sensor 1

Low load sensor 2

High load sensor 2

Figure 7.1. The RMS-value of the complete sensor signals from the two sensors, gearsets and under high and low load from the brake. The value increases more with higherfrequencies on the signals from the gear box using spur gears than from the gear boxusing helical gears.

bearing faults since they appear to generate lower amplitudes on their fault char-acteristic frequencies than fault on the other components. Figure 7.2 shows thebearing synchronous averages from sensor 2 from the case Helical 5. In this casethe second idle gear is broken as well as bearing 5 which has an inner race faultpresent. The input shaft frequency is 45 Hz and the load from the brake is high.The inner race fault does not appear in the synchronous average for inner racefaults on the idle shaft as expected.

7.2.2 Chipped and Broken GearsIn the case Helical 2 the only fault present is a chipped tooth on idle gear 2. Figure7.3 shows the synchronous averages for the three shafts from the two sensors. Inthe avergage for the idle shaft, especially on the output side, there is a clear peakwhich looks like the modelled signal in Figure 3.1. The input shaft frequency is45 Hz and the load from the brake is low. When there is more than one faulton the gears the faults are harder to identify since they will be mixed up in thesynchronous averaging due to the gearbox ratios.

7.2.3 The Component Mode ErrorThe component mode Error was not included in the diagnosis system due to theunknown nature of the fault. In all cases having this fault mode in the labelleddata the faulty gear was eccentric. Figure 7.4 shows the synchronous averages forthe shafts from the case Spur 3 where the only fault present is an eccentric second


0 200 400 600

-0.02

0

0.02

Ball Spin Fault

Input

Shaft

0 100 200 300

-0.02

0

0.02

Inner Race Fault

0 200 400

-0.02

0

0.02

Outer Race Fault

0 1000 2000

-0.02

0

0.02

Idle

Shaft

0 200 400 600 800

-0.02

0

0.02

0 500 1000

-0.02

0

0.02

0 1000 2000 3000

-0.02

0

0.02

Outp

ut

Shaft

0 500 1000 1500

-0.02

0

0.02

0 1000 2000

-0.02

0

0.02

Figure 7.2. Synchronous averages for the bearings from sensor 2. It is not possible tosee the inner race fault on bearing 5 on the average from the idle shaft as expected.

0 2 4 6

-0.02

0

0.02

0.04

0.06

Sensor 1

Input

Shaft

0 2 4 6

-0.05

0

0.05

Sensor 2

0 2 4 6

-0.02

0

0.02

0.04

0.06

Idle

Shaft

0 2 4 6

-0.05

0

0.05

0 2 4 6

-0.02

0

0.02

0.04

0.06

Shaft Angle [rad]

Outp

ut

Shaft

0 2 4 6

-0.05

0

0.05

Shaft Angle [rad]

Figure 7.3. Synchronous averages for the shaft from the two sensors on case Helical 2.The second idle gear is chipped which can be seen on the synchronously averaged signalfor the idle shaft.

idle gear. The fault leaves a clear mark in the synchronous average for the idleshaft from the second sensor. In this plot the input shaft rotates at 45 revolutions

7.2 Component Fault Modes 47

per second and the load from the brake is high.

0 2 4 6

-0.05

0

0.05

0.1

Sensor 1In

put

Shaft

0 2 4 6

-0.05

0

0.05

Sensor 2

0 2 4 6

-0.05

0

0.05

0.1

Idle

Shaft

0 2 4 6

-0.05

0

0.05

0 2 4 6

-0.05

0

0.05

0.1

Shaft Angle [rad]

Outp

ut

Shaft

0 2 4 6

-0.05

0

0.05

Shaft Angle [rad]

Figure 7.4. Synchronous averages for the shafts from the two sensors in the case Spur3. The second idle gear is eccentric and the fault have a distinct appearance in thesynchronously averaged data.

7.2.4 Bent Axle

The fault mode Helical 6 does only have the fault mode Bent Axle, which meansthat the input shaft is bent. Figure 7.5 shows how the synchronous averages lookslike for the shafts under high load from the brake and the input shaft rotating at45 Hz. For some reason every fifth revolution of the input shaft leaves a biggermark in the signal which can be seen as a big peak in the average for the outputshaft. The reason behind this has not been investigated further since the faultmode was not included in the diagnosis system.

7.2.5 Shaft Imbalance

The fault mode Shaft Imbalance is in fault case Helical 4 present at the same timeas some bearing faults on bearing 4 and bearing 5. The synchronous averages forthe shafts are found in Figure 7.6. From the figure however it seems like there isa fault on one of the idle gears. This fault mode has not been investigated furthereither.


0 2 4 6

-0.02

0

0.02

0.04

0.06

Sensor 1

Input

Shaft

0 2 4 6

-0.04

-0.02

0

0.02

0.04

Sensor 2

0 2 4 6

-0.02

0

0.02

0.04

0.06

Idle

Shaft

0 2 4 6

-0.04

-0.02

0

0.02

0.04

0 2 4 6

-0.02

0

0.02

0.04

0.06

Shaft Angle [rad]

Outp

ut

Shaft

0 2 4 6

-0.04

-0.02

0

0.02

0.04

Shaft Angle [rad]

Figure 7.5. Synchronous averages for the shafts from the two sensors in case Helical6 where the input shaft is bent. Some pattern can be seen in the average for the inputshaft and a very high peak can be seen in the output shaft average.

0 2 4 6-0.02

0

0.02

0.04

0.06

Sensor 1

Input

Shaft

0 2 4 6

-0.04

-0.02

0

0.02

0.04Sensor 2

0 2 4 6-0.02

0

0.02

0.04

0.06

Idle

Shaft

0 2 4 6

-0.04

-0.02

0

0.02

0.04

0 2 4 6-0.02

0

0.02

0.04

0.06

Shaft Angle [rad]

Outp

ut

Shaft

0 2 4 6

-0.04

-0.02

0

0.02

0.04

Shaft Angle [rad]

Figure 7.6. Synchronous averages for the shaft from the two sensors in case Helical 4.The speed of the input shaft is 45 Hz and the load from the brake is low. The fault hasan appearance as an expected fault on some of the idle gears.

7.3 Condition Indicator Results 49

7.2.6 Bad KeyWhen the Bad Key fault appeared the frequency of the shaft was the same asin the no load case. The Figure 7.7 shows the input shaft frequency for all 160spur cases sorted by frequency. The cases which have the fault mode Bad Keypresent are highlighted. As seen in the figure the frequency is the same when thefault is present as when there is no load from the brake. Hence the fault cannot beidentified using differencing frequencies as previously assumed. The fault case Spur7 does have a Bad Key fault on the output shaft and the synchronous averagedsignals for the shafts can be found in Figure 7.8. In the figure the average for theinput shaft does look like an arch which is repeated three and five times in theidle and output shaft averages on the input side.

0 20 40 60 80 100 120 140 16025

30

35

40

45

50

Fre

quency [

Hz]

Figure 7.7. Frequencies for the 160 spur cases. The cases where the Bad Key fault ispresent are highlighted.

7.3 Condition Indicator ResultsThe condition indicators have been calculated on the labelled data and the resultare discussed in the following sections. The calculated values can be found inAppendix D. Since the synchronous averaged signals did not look like expected itis hard to read too much from the results.

7.3.1 RMSThe indicator Trms is in most cases not very useful on its own. Only when bigfaults occur a significant difference can be noticed compared to the no fault case.


0 2 4 6

-0.02

0

0.02

0.04

0.06

Sensor 1

Input

Shaft

0 2 4 6

-0.02

0

0.02

Sensor 2

0 2 4 6

-0.02

0

0.02

0.04

0.06

Idle

Shaft

0 2 4 6

-0.02

0

0.02

0 2 4 6

-0.02

0

0.02

0.04

0.06

Shaft Angle [rad]

Outp

ut

Shaft

0 2 4 6

-0.02

0

0.02

Shaft Angle [rad]

Figure 7.8. Synchronous averages for the shaft from the two sensors on case Spur 7.The speed of the input shaft is 45 Hz and the load from the brake is high.

Table D.9 and D.10 show the calculated values. Some larger values can be foundfor cases Spur 2-5 in the parameters calculated on the synchronous averages fromthe shafts. In those cases the second idle gear is eccentric which generates a largeamount of vibrations. The bearing faults are too small to affect the parametersignificantly which can be seen by comparing the no fault case Spur 1 with caseSpur 8 where only bearing faults are present. Further the high mean value on thesignal from sensor 1 makes those values more close together than the values fromsensor 2. To make the indicator more useful the mean value of the signal shouldprobably be removed before creating the averaged signals.

7.3.2 Crest FactorThe Tables D.11 and D.12 show the calculated values from all synchronous av-eraged signals when the input shaft rotates at 40 revolutions per second and thebrake is applied. Some observations can be made. For example on the helical caseson the second idle gear the largest value is found in the case Helical 2 which havea chipped tooth on the second idle gear which is in line with theory from previouschapters. For the spur cases large values can be found especially in the cases 4 and5 in the synchronous averages related to the shafts from sensor 1. These cases havesome different faults on the shafts that may explain the higher values. The valuesfrom the second sensor do not have the same high values in these cases which canbe explained by the higher Trms values for the same averages. The values fromthe first sensor are more close to a peak-to-peak value due to the more equal valueof the Trms indicator used when calculating the indicator. In the case Spur 3 the

7.4 Diagnosis of the Labelled Data 51

third largest values from the averages related to the bearing faults can be found inthe column Output Shaft BS. This case does not however have any bearing faultpresent. The conflicting results are a consequence of the unsatisfying outcome ofthe synchronous averages.

7.3.3 KurtosisThe kurtosis value does react as expected from the signals. But since the syn-chronous averaging does not show the faults as expected it is hard to read toomuch from the results. The Tables D.13 and D.14 in Appendix D show indicatorvalues from when the indicator is calculated on the synchronous averaged signals inthe different cases. The indicator has values close to 3 on most of the synchronousaverages for the bearing faults. On the averages related to the inner race faultlarger values can be seen on the input and ouput shafts for case Spur 6. This casedoes have different bearing faults on the first three bearings including an innerrace fault on the first bearing on the input shaft. The indicator does however nothave big values on the same averages in the cases Spur 5 and Spur 7 which havean inner race fault on bearing 1 as well.

7.3.4 Peak IndexesAs for the other condition indicators the results from the Tpeak indicators do notgive very much useful information. The calculated indicator values can be foundin the Tables D.15 and D.16 in Appendix D. Some lower values in Table D.16can be found in the first two columns for the case Spur 1. This indicates thatthere are three peaks in the synchronous averages for the idle shaft and five in theaverages for the output shafts separated by the length of the average for the inputshaft. Since there are no faults present in this case this is conflicting with previousassumptions. The signal however does contain this pattern in the synchronousaverages for the shafts. Figure 7.9 shows synchronous averages for the shafts fromthe case Spur 1. Since the indicator is only used for excluding possible faultsthis would mean that there are no faults at either the idle or output shafts whichactually is true.

7.4 Diagnosis of the Labelled DataThe diagnosis system was tested on the labelled data but the result was not veryaccurate. Since the faults did not appear as expected in the synchronous averagedsignals the diagnosis statements were in most cases incorrect and not very consis-tent. Due to the bad outcome no results from the diagnosis system are presented.If the decision structure would be created based on real data having only singlefaults present a better diagnosis result would probably be achieved.


0 2 4 6

-0.02

0

0.02

0.04

0.06

Sensor 1

Input

Shaft

0 2 4 6

-0.02

0

0.02

Sensor 2

0 2 4 6

-0.02

0

0.02

0.04

0.06

Idle

Shaft

0 2 4 6

-0.02

0

0.02

0 2 4 6

-0.02

0

0.02

0.04

0.06

Shaft Angle [rad]

Outp

ut

Shaft

0 2 4 6

-0.02

0

0.02

Shaft Angle [rad]

Figure 7.9. Synchronous averages for the shafts for the case Spur 1. In this non faultycase it looks like there could be a fault with the same frequency as the input shaft.

Chapter 8

Conclusions

The idea was to create and implement a diagnosis system only by using unlabelleddata and having knowledge about the gearbox dimensions. Both fault free andfault cases should be included in the unlabelled data and the system should identifythe fault cases and isolate the faults. However the diagnosis system was only ableto correctly diagnose modelled signals and not labelled real data.

There are several reasons why the diagnosis results of the data given after thecompetition were not very accurate. One of the main reasons is the modelling ofthe signals. The bearing faults were much smaller than predicted which made themhard to detect. The bearing faults are probably easier to find by analyzing thesignal in the frequency domain and when there are no other faults present whichgenerate a lot more vibrations. The fault mode Error on the gears was one of themost apparent fault modes. All gears having this fault mode were eccentric andappeared very clearly in the synchronously averaged signals. This fault mode wasnot however included in the signal modelling when forming the decision structure.The gear ratios, which makes the input shaft rotate three times for every rotationof the idle shaft and five times for every rotation of the output shaft, creates sometroubles as well especially if there are more than one fault present. The t-test usedwhen creating the decision structure is maybe a bit too sensitive which results ina very large number of X’s in the decision structure. The significance level andthe number of samples can be used as tuning parameters.

8.1 Improvements and Future WorkBy having labelled data for all single fault modes a more correct decision struc-ture can be created. The decision structure, based on the modelled signals, isprobably only useful for just the modelled signals. When the appearance of thedifferent faults is known, better signal models of the different faults, can be madeto achieve a better diagnosis result. If the diagnosis system is used on a gear boxthat uses gear ratios that does not make the shafts rotate at speeds that are aninteger multiple of the other shafts a better result could probably be achieved.Faults not related to the same shaft would be averaged out when the common

53

54 Conclusions

divisor of the shaft speeds would be much higher. The bearing diagnosis could beseparated from the gear and shaft diagnosis and only be computed if there are nogear or shaft faults present which generate a lot of vibrations. There are severaltechniques for analyzing the signal in the frequency domain which also may bea subject for future work. More data from cases having no fault would give thesystem more information to calculate the thresholds correctly and a failure easierto isolate from the fault free cases. The sample frequency is much higher than theinteresting fault characteristic frequencies. By low pass filtering and down sam-pling the data initially a faster calculation of the following synchronous averagesand test quantities can be achieved since the amount of data is reduced.

Bibliography

[1] Apparatus | phm society. URL: http://www.phmsociety.org/competition/09/apparatus, August 2009.

[2] Matlab online documentation. URL: http://www.mathworks.com/access//helpdesk/help/toolbox/stats/clusterdata.html., October 2009.

[3] Phm society. URL: http://www.phmsociety.org., August 2009.

[4] Precision gears, inc. URL: http://www.precisiongears.com., October 2009.

[5] Gunnar Blom, Jan Enger, Gunnar Englund, Jan Grandell, and Lars Holst.Sannolikhetsteori och statistikteori med tillämpningar. Studentlitteratur, 5edition, 2005. ISBN 91-44-02442-8.

[6] Li Bo, Chow Mo-Yuen, Tipsuwan Yodyium, and Hung James C. Neural-network-based motor rolling bearing fault diagnosis. Industrial Electronics,IEEE Transactions, Vol. 47, 2000.

[7] I. Soltani Bozchalooi and Ming Liang. Identification of the high snr frequencyband for bearing fault signature enhancement. 2007.

[8] Fredrik Gustafsson, Lennart Ljung, and Mille Millnert. Signalbehandling.Studentlitteratur, 2 edition, 2001. ISBN 978-91-44-01709-9.

[9] Cyril M. Harris. Shock and Vibration Handbook. McGraw-Hill, Inc, 4 edition,1996. ISBN 0-07-026920-3.

[10] Hubert W. Lilliefors. On the kolmogorov-smirnov test for normality withmean and variance unknown. Journal of the American Statistical AssociationVol. 62, No. 318 , pp. 399- 402, 1967.

[11] Jan Lundgren, Mikael Rönnqvist, and Peter Värbrand. Optimeringslära. Stu-dentlitteratur, 2 edition, 2003. ISBN 91-44-03104-1.

[12] R.Keith Mobley. Vibration Fundamentals. Butterworth-Heinemann, 1999.ISBN 0-7506-7150-5.

[13] Douglas C. Montgomery and George C. Runger. Applied Statistics and Prob-ability for Engineers. John Wiley & Sons, Inc., 4 edition, 2006. ISBN 978-0-471-74589-1.

55

56 Bibliography

[14] Mattias Nyberg. Model Based Fault Diagnosis: Methods, Theory, and Auto-motive Engine applications. McGraw-Hill, Inc, 1999. ISBN 91-7219-521-5.

[15] Večeř P., Kreidl M., and Šmíd R. Condition indicators for gearbox conditionmonitoring systems. Acta Polytechnica, 45, 2005.

[16] Victor Wowk. Machinery Vibration, Measurement and analysis. McGraw-Hill, Inc, 1991. ISBN 0-07-071936-5.

Appendix A

Synchronous Average

The following sections illustrate how different modelled faults appear in the syn-chronous averaged signals.

A.1 Chipped Idle Gear 1The following figures show how the synchronous averages look like for a modelledsignal with a fault on the first idle gear. The input shaft frequency is the same inall cases i.e. 40 Hz

0 0.5 1 1.5 2-0.05

0

0.05

0.1

0.15Sensor 1

0 0.5 1 1.5 2-0.05

0

0.05Sensor 2

Time [s]

Figure A.1. The sensor signals.

57

58 Synchronous Average

0 500 1000 1500

0

0.02

0.04

0.06

Sensor 1

Input

Shaft

0 500 1000 1500

-0.02

0

0.02

0.04

Sensor 2

0 2000 4000

0

0.02

0.04

0.06

Idle

Shaft

0 2000 4000

-0.02

0

0.02

0.04

0 2000 4000 6000 8000

0

0.02

0.04

0.06

Outp

ut

Shaft

0 2000 4000 6000 8000

-0.02

0

0.02

0.04

Figure A.2. Synchronous averages for the three shafts.

0 200 400 600 800

0

0.02

0.04

0.06

Sensor 1

Input

Shaft

0 200 400 600 800

-0.02

0

0.02

0.04

Sensor 2

0 1000 2000

0

0.02

0.04

0.06

Idle

Shaft

0 1000 2000

-0.02

0

0.02

0.04

0 2000 4000

0

0.02

0.04

0.06

Outp

ut

Shaft

0 2000 4000

-0.02

0

0.02

0.04

Figure A.3. Synchronous averages for the ball spin faults on the three shafts.

A.1 Chipped Idle Gear 1 59

0 100 200 300

0

0.02

0.04

0.06

Sensor 1

Input

Shaft

0 100 200 300

-0.02

0

0.02

0.04

Sensor 2

0 500 1000

0

0.02

0.04

0.06

Idle

Shaft

0 500 1000

-0.02

0

0.02

0.04

0 500 1000 1500

0

0.02

0.04

0.06

Outp

ut

Shaft

0 500 1000 1500

-0.02

0

0.02

0.04

Figure A.4. Synchronous averages for the inner race faults.

0 200 400

0

0.02

0.04

0.06

Sensor 1

Input

Shaft

0 200 400

-0.02

0

0.02

0.04

Sensor 2

0 500 1000 1500

0

0.02

0.04

0.06

Idle

Shaft

0 500 1000 1500

-0.02

0

0.02

0.04

0 1000 2000

0

0.02

0.04

0.06

Outp

ut

Shaft

0 1000 2000

-0.02

0

0.02

0.04

Figure A.5. Synchronous averages for the outer race faults.


A.2 Ball Spin Fault Bearing 5The following figures show the synchronous averages for a modelled signal havinga ball spin fault on bearing 5 on the idle shaft. The input shaft frequency is 40Hz.

0 0.5 1 1.5 2-0.02

0

0.02

0.04

0.06Sensor 1

0 0.5 1 1.5 2-0.05

0

0.05Sensor 2

Time [s]

Figure A.6. The raw data from the sensor signals.

A.2 Ball Spin Fault Bearing 5 61

0 500 1000 1500

0

0.02

0.04

Sensor 1

Input

Shaft

0 500 1000 1500

-0.02

0

0.02

0.04

Sensor 2

0 2000 4000

0

0.02

0.04

Idle

Shaft

0 2000 4000

-0.02

0

0.02

0.04

0 2000 4000 6000 8000

0

0.02

0.04

Outp

ut

Shaft

0 2000 4000 6000 8000

-0.02

0

0.02

0.04

Figure A.7. Synchronous averages for the gear faults on the shafts.

0 200 400 600 800

0

0.02

0.04

Sensor 1

Input

Shaft

0 200 400 600 800

-0.02

0

0.02

0.04

Sensor 2

0 1000 2000

0

0.02

0.04

Idle

Shaft

0 1000 2000

-0.02

0

0.02

0.04

0 2000 4000

0

0.02

0.04

Outp

ut

Shaft

0 2000 4000

-0.02

0

0.02

0.04

Figure A.8. Synchronous averages for ball spin faults.


0 100 200 300

0

0.02

0.04

Sensor 1

Input

Shaft

0 100 200 300

-0.02

0

0.02

0.04

Sensor 2

0 500 1000

0

0.02

0.04

Idle

Shaft

0 500 1000

-0.02

0

0.02

0.04

0 500 1000 1500

0

0.02

0.04

Outp

ut

Shaft

0 500 1000 1500

-0.02

0

0.02

0.04

Figure A.9. Synchronous averages for inner race faults.

0 200 400

0

0.02

0.04

Sensor 1

Input

Shaft

0 200 400

-0.02

0

0.02

0.04

Sensor 2

0 500 1000 1500

0

0.02

0.04

Idle

Shaft

0 500 1000 1500

-0.02

0

0.02

0.04

0 1000 2000

0

0.02

0.04

Outp

ut

Shaft

0 1000 2000

-0.02

0

0.02

0.04

Figure A.10. Synchronous averages for outer race faults.

A.2 Ball Spin Fault Bearing 5 63

Find First Up & Down Flank

Set starting position as last flankflankNumber:=-1

i=i+1

Up or Down Flank?

flankNumber := +1prevFlankDist:=

(flankIndex-prevFlankIndex)prevFlankIndex:=flankIndex

flankIndex:=iflankDist:=flankIndex-

prevFlankIndexperiodLength:=

prevFlankDist+flankDist

Even(flankNumber) & flankNumber>0

samplesLeft:=(nextPulse-pulseNr)*periodLength

Locked:=1nextPosition:=samplesLeft+i

INIT NextPulse:=rev*PulsesPerRevolution

i-nextPosition =0?

Extract data(lastPosition:i)lastPosition:=i

revolution:=revolution+1Locked:=0

nextPulse=pulsesPerRevolution*revolution

Locked=1?

One Pulse

i

pulseNr:=pulseNr+1

PulseNr=nextPulse?

Extract data(lastPosition:i)lastPosition:=i

Revolution:=revolution+1nextPulse:=revolution*PulseNr

nextPulse-pulseNr <1

Figure A.11. How the synchronous average is implemented. The algorithm works forfraction of pulses.

Appendix B

Theoretical signals

Sensor 1 Sensor 2Component Component Mode A0 β0 A0 β0 ffault

Input Pinion Chipped 0,06 1500 0,01 1500 fsh,1Broken 0,07 200 0,02 200 fsh,1

Idle Gear 1 Chipped 0,03 1500 0,01 1500 fsh,2Broken 0,04 200 0,02 200 fsh,2

Idle Gear 2 Chipped 0,01 1500 0,03 1500 fsh,2Broken 0,02 200 0,04 200 fsh,2

Output Gear Chipped 0,01 1500 0,06 1500 fsh,3Broken 0,02 200 0,07 200 fsh,3

Bearing 1Ball Spin 0,04 1500 0,005 1500 fbs,1

Inner Race 0,04 1500 0,005 1500 fir,1Outer Race 0,04 1500 0,005 1500 for,1











Table B.1. Parameters used when forming the modelled signals.

64

Appendix C

Decision Structure

65

66 Decision Structure

Gea

rsB

eari

ngs

Sh

afts

Inp

ut

Pin

ion

Idle

Gea

r1

Idle

Gea

r2

Ou

tpu

tG

ear

12

34

56

Inp

ut

Ou

tpu

tN

FC

HB

RE

RN

FC

HB

RE

RN

FC

HB

RE

RN

FC

HB

RE

RN

FIR

OR

BS

NF

IRO

RB

SN

FIR

OR

BS

NF

IRO

RB

SN

FIR

OR

BS

NF

IRO

RB

SN

FB

AS

IN

FB

KT

rm

s1,

1X

XX

XX

XX

XX

XT

rm

s2,

1X

XX

XX

XX

XX

XT

rm

s3,

1X

XX

XX

XX

XX

XX

Tr

ms4,

1X

XX

XX

XX

XX

XT

rm

s5,

1X

XX

XX

XX

XX

XX

Tr

ms6,

1X

XX

XX

XX

XX

XT

rm

s7,

1X

XX

XX

XX

XX

XT

rm

s8,

1X

XX

XX

XX

XX

XT

rm

s9,

1X

XX

XX

XX

XX

XT

rm

s10

,1X

XX

XX

XX

XX

XX

Tr

ms11

,1X

XX

XX

XX

XX

XT

rm

s12

,1X

XX

XX

XX

XX

XT

rm

s1,

2X

XX

XX

XX

XX

XT

rm

s2,

2X

XX

XX

XX

XX

XX

XT

rm

s3,

2X

XX

XX

XX

XX

XX

XX

Tr

ms4,

2X

XX

XX

XX

Tr

ms5,

2X

XX

XX

XX

XX

XT

rm

s6,

2X

XX

XX

XX

XX

XX

XT

rm

s7,

2X

XX

XX

XX

XX

XX

XX

Tr

ms8,

2X

XX

XX

XX

XX

Tr

ms9,

2X

XX

XX

XX

XX

Tr

ms10

,2X

XX

XX

XX

XX

XX

Tr

ms11

,2X

XX

XX

XX

XX

XX

XX

XT

rm

s12

,2X

XX

XX

XX

XX

XX

Tc

f1,

1X

XX

XX

XX

XX

XX

XX

Tc

f2,

1X

XX

XX

XX

XX

XX

XX

XX

XT

cf3,

1X

XX

XX

XX

XX

XX

XX

XX

XX

Tc

f4,

1X

XX

XX

XX

XX

XX

XX

XX

XX

Tc

f5,

1X

XX

XX

XX

XX

XX

XX

XX

XX

XT

cf6,

1X

XX

XX

XX

XX

XX

XX

XX

XX

XT

cf7,

1X

XX

XX

XX

XX

XX

XX

XX

XX

Tc

f8,

1X

XX

XX

XX

XX

XX

XX

XX

XT

cf9,

1X

XX

XX

XX

XX

XX

XX

XX

XT

cf10

,1X

XX

XX

XX

XX

XX

XX

XX

XX

XX

Tc

f11

,1X

XX

XX

XX

XX

XX

XX

XX

XX

XT

cf12

,1X

XX

XX

XX

XX

XX

XX

XX

XX

XT

cf1,

2X

XX

XX

XX

XX

XX

XX

XX

XT

cf2,

2X

XX

XX

XX

XX

XX

XX

XX

XX

Tc

f3,

2X

XX

XX

XX

XX

XX

XX

XX

XX

XT

cf4,

2X

XX

XX

XX

XX

XX

XX

XX

XT

cf5,

2X

XX

XX

XX

XX

XX

XX

XX

XX

XX

Tc

f6,

2X

XX

XX

XX

XX

XX

XX

XX

XX

XT

cf7,

2X

XX

XX

XX

XX

XX

XX

XX

XT

cf8,

2X

XX

XX

XX

XX

XX

XX

XX

Tc

f9,

2X

XX

XX

XX

XX

XX

XX

XX

Tc

f10

,2X

XX

XX

XX

XX

XX

XX

XX

Tc

f11

,2X

XX

XX

XX

XX

XX

XX

XX

XX

XT

cf12

,2X

XX

XX

XX

XX

XX

XX

XX

XX

67

Gea

rsB

eari

ngs

Sh

afts

Inp

ut

Pin

ion

Idle

Gea

r1

Idle

Gea

r2

Ou

tpu

tG

ear

12

34

56

Inp

ut

Ou

tpu

tN

FC

HB

RE

RN

FC

HB

RE

RN

FC

HB

RE

RN

FC

HB

RE

RN

FIR

OR

BS

NF

IRO

RB

SN

FIR

OR

BS

NF

IRO

RB

SN

FIR

OR

BS

NF

IRO

RB

SN

FB

AS

IN

FB

KT

ku

1,1

XX

XX

XX

XX

XT

ku

2,1

XX

XX

XX

XT

ku

3,1

XX

XX

XX

XX

XT

ku

4,1

XX

XT

ku

5,1

XX

XX

XX

XT

ku

6,1

XX

XX

XX

XT

ku

7,1

XX

XX

XX

Tk

u8,

1X

XX

XT

ku

9,1

XX

XX

XT

ku

10,1

XX

XX

XX

XX

XT

ku

11,1

XX

XX

XX

XX

XX

XX

XX

XX

XX

Tk

u12

,1X

XX

XT

ku

1,2

XX

XX

XX

XX

XX

Tk

u2,

2X

XX

XX

XX

XX

Tk

u3,

2X

XX

XX

XX

XX

XT

ku

4,2

XX

XT

ku

5,2

XX

XX

XX

XT

ku

6,2

XX

XX

XX

XX

Tk

u7,

2X

XX

XX

XT

ku

8,2

XX

XT

ku

9,2

XX

XT

ku

10,2

XX

XX

XX

XX

XX

Tk

u11

,2X

XX

XX

XX

XX

XX

XX

XX

XX

XT

ku

12,2

XX

XX

XT

pe

ak2,

1X

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

Tp

ea

k3,

1X

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XT

pe

ak5,

1X

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

Tp

ea

k6,

1X

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

Tp

ea

k8,

1X

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

Tp

ea

k9,

1X

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

Tp

ea

k11

,1X

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

Tp

ea

k12

,1X

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

Tp

ea

k2,

2X

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

Tp

ea

k3,

2X

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XT

pe

ak5,

2X

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

Tp

ea

k6,

2X

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

Tp

ea

k8,

2X

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

Tp

ea

k9,

2X

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

Tp

ea

k11

,2X

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

Tp

ea

k12

,2X

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

XX

Tab

leC

.1.The

decision

structure

Appendix D

Condition Indicator Results

D.1 RMS Trms

If the tested signal have a high mean value the indicator Trms is not very use-ful. Table D.1 shows how the test reacts on the synchronous averaged signals fordifferent single faults. The data is collected from sensor 1 under low load at theinput shaft frequency 40 Hz. As seen in the table the indicator values do notdiffer from the case No Fault very much. The signal offset on sensor 1 makes thevalue rather constant for all single faults. When the input pinion is broken a slightincrease of the value can be noticed for the synchronous averages related to theshafts. The second sensor has a lower offset thus making the values from the Trmsindicator having a wider spread. Table D.2 shows the indicator values when datacomes from the second sensor. The frequency and load are the same as in TableD.1. In these tables and in the following D.3-D.6 values where a bigger differencetheoretically could be observed have been highlighted. The high values related tothe synchronous average Idle Shaft OR is a consequence of the high noise in theaveraged signal for which no explanation have been found.

68

D.1 RMS Trms 69

Synchronous averages

Fa

ult

Mo

de

Inp

ut

Sh

aft

Idle

Sh

aft

Ou

tpu

tS

haf

t

Inp

ut

Sh

aft

BS

Idle

Sh

aft

BS

Ou

tpu

tS

haf

tB

S

Inp

ut

Sh

aft

IR

Idle

Sh

aft

IR

Ou

tpu

tS

haf

tIR

Inp

ut

Sh

aft

OR

Idle

Sh

aft

OR

Ou

tpu

tS

haf

tO

R

NF 2.159 2.161 2.162 2.159 2.160 2.161 2.160 2.159 2.159 2.165 2.186 2.160CHip 2.233 2.235 2.237 2.181 2.182 2.186 2.186 2.181 2.181 2.198 2.244 2.183BRip 2.680 2.685 2.685 2.182 2.183 2.298 2.189 2.182 2.183 2.241 2.540 2.194

CHig1 2.167 2.171 2.169 2.165 2.165 2.166 2.169 2.165 2.165 2.170 2.190 2.166BRig1 2.184 2.226 2.187 2.163 2.164 2.169 2.164 2.163 2.164 2.167 2.203 2.169

CHig2 2.161 2.163 2.164 2.161 2.161 2.162 2.159 2.161 2.161 2.166 2.188 2.161BRig2 2.167 2.179 2.170 2.161 2.162 2.164 2.159 2.161 2.162 2.165 2.197 2.163CHog 2.161 2.163 2.165 2.161 2.162 2.162 2.162 2.161 2.161 2.168 2.191 2.162BRog 2.164 2.166 2.175 2.163 2.163 2.166 2.159 2.162 2.162 2.162 2.190 2.163BSb1 2.222 2.224 2.227 2.222 2.223 2.223 2.338 2.334 2.334 2.241 2.291 2.223

IRb1 2.202 2.203 2.206 2.200 2.201 2.202 2.204 2.200 2.200 2.281 2.303 2.271

ORb1 2.186 2.187 2.197 2.231 2.231 2.232 2.189 2.185 2.186 2.201 2.253 2.187

BSb2 2.176 2.178 2.180 2.176 2.179 2.178 2.181 2.198 2.184 2.180 2.211 2.177

IRb2 2.169 2.171 2.173 2.169 2.170 2.171 2.176 2.169 2.169 2.181 2.217 2.174

ORb2 2.167 2.168 2.171 2.170 2.176 2.171 2.168 2.167 2.167 2.175 2.202 2.167

BSb3 2.166 2.167 2.169 2.165 2.166 2.167 2.168 2.166 2.172 2.171 2.199 2.166

IRb3 2.164 2.166 2.168 2.164 2.165 2.166 2.159 2.164 2.164 2.159 2.181 2.168

ORb3 2.164 2.166 2.168 2.165 2.165 2.168 2.168 2.164 2.164 2.171 2.187 2.165

BSb4 2.190 2.192 2.194 2.190 2.191 2.192 2.225 2.219 2.219 2.204 2.233 2.191

IRb4 2.180 2.182 2.184 2.179 2.180 2.181 2.183 2.179 2.180 2.209 2.231 2.198

ORb4 2.173 2.175 2.178 2.184 2.185 2.186 2.177 2.173 2.173 2.180 2.209 2.174

BSb5 2.166 2.168 2.169 2.166 2.167 2.167 2.170 2.168 2.167 2.177 2.201 2.167

IRb5 2.164 2.165 2.167 2.164 2.164 2.165 2.164 2.164 2.164 2.169 2.196 2.165

ORb5 2.164 2.165 2.167 2.164 2.165 2.165 2.162 2.164 2.164 2.167 2.187 2.164

BSb6 2.163 2.165 2.166 2.162 2.163 2.164 2.167 2.162 2.163 2.170 2.194 2.163

IRb6 2.161 2.163 2.164 2.161 2.162 2.162 2.164 2.161 2.161 2.169 2.188 2.162

ORb6 2.163 2.165 2.166 2.163 2.163 2.164 2.167 2.163 2.163 2.172 2.194 2.163

Table D.1. Values of the condition indicator Trms at input shaft frequency 40 Hz underlow load at sensor 1. The values should be multiplied by 10−2

70 Condition Indicator Results


Fa

ult

Mo

de

Inp

ut

Sh

aft

Idle

Sh

aft

Ou

tpu

tS

haf

t

Inp

ut

Sh

aft

BS

Idle

Sh

aft

BS

Ou

tpu

tS

haf

tB

S

Inp

ut

Sh

aft

IR

Idle

Sh

aft

IR

Ou

tpu

tS

haf

tIR

Inp

ut

Sh

aft

OR

Idle

Sh

aft

OR

Ou

tpu

tS

haf

tO

R

NF 0.887 1.193 1.457 0.798 0.969 1.125 1.258 0.826 0.885 1.855 3.560 0.992CHip 1.228 1.474 1.707 0.856 1.018 1.193 1.345 0.882 0.941 1.981 3.686 1.051BRip 4.542 4.637 4.721 0.884 1.070 2.390 1.452 0.912 0.974 2.511 5.319 1.262

CHig1 0.953 1.302 1.510 0.828 1.002 1.155 1.342 0.859 0.919 1.911 3.645 1.032BRig1 1.758 2.852 2.124 0.824 0.999 1.352 1.345 0.855 0.920 1.986 3.990 1.269


IRb1 0.993 1.307 1.592 0.906 1.078 1.247 1.684 1.289 1.341 2.045 3.882 1.105

ORb1 0.957 1.272 1.556 0.859 1.033 1.195 1.376 0.888 0.944 2.114 3.841 1.255

BSb2 0.914 1.220 1.492 0.910 1.182 1.207 1.239 0.861 0.915 1.902 3.624 1.024

IRb2 0.953 1.261 1.529 0.872 1.074 1.198 1.409 1.353 1.116 1.912 3.739 1.057

ORb2 0.935 1.247 1.517 0.848 1.020 1.173 1.286 0.877 0.942 1.953 3.702 1.149

BSb3 0.931 1.240 1.538 0.937 1.082 1.525 1.321 0.854 0.916 1.999 3.629 1.032

IRb3 0.947 1.269 1.564 0.868 1.091 1.212 1.480 1.146 1.863 1.922 3.720 1.070

ORb3 0.947 1.276 1.534 0.858 1.021 1.192 1.372 0.875 0.929 2.003 3.780 1.642

BSb4 1.016 1.316 1.821 2.442 2.501 2.574 1.393 0.970 1.011 2.217 4.203 1.177

IRb4 1.175 1.467 1.751 1.105 1.269 1.390 3.887 3.734 3.744 2.422 4.583 1.299

ORb4 1.145 1.421 1.713 0.989 1.152 1.312 1.486 1.010 1.068 3.444 4.694 3.035

BSb5 0.959 1.270 1.594 1.420 2.201 1.619 1.386 0.968 1.015 2.012 3.892 1.087

IRb5 1.037 1.401 1.631 0.960 1.433 1.285 2.141 3.235 2.054 2.128 4.213 1.148

ORb5 1.006 1.331 1.658 0.902 1.082 1.249 1.353 0.974 0.989 2.210 4.403 1.766

BSb6 1.013 1.321 1.720 1.248 1.353 2.363 1.360 0.895 0.961 2.094 3.599 1.092

IRb6 1.004 1.367 1.714 0.923 1.269 1.295 1.827 1.704 3.343 2.133 4.091 1.133

ORb6 1.032 1.437 1.643 0.920 1.102 1.292 1.419 0.920 0.992 2.085 3.992 2.714

Table D.2. Values of the condition indicator Trms at input shaft frequency 40 Hz underlow load at sensor 2. All values should be multiplied by 10−3.

D.2 Crest Factor Tcf 71

D.2 Crest Factor Tcf

The Table D.3 and D.4 show values of the crest factor for different single faults.The crest factor reacts more clearly to the different faults than the indicator Trms.The values related to the second sensor in Table D.4 have a higher value due tothe lower value of the Trms from the second sensor. As for the Trms indicatorthe crest factor gets a higher value on the averages with the same faults but ondifferent shafts.


Fa

ult

Mo

de

Inp

ut

Sh

aft

Idle

Sh

aft

Ou

tpu

tS

haf

t

Inp

ut

Sh

aft

BS

Idle

Sh

aft

BS

Ou

tpu

tS

haf

tB

S

Inp

ut

Sh

aft

IR

Idle

Sh

aft

IR

Ou

tpu

tS

haf

tIR

Inp

ut

Sh

aft

OR

Idle

Sh

aft

OR

Ou

tpu

tS

haf

tO

R

NF 0.176 0.335 0.472 0.117 0.227 0.300 0.288 0.135 0.181 0.483 1.044 0.238CHip 3.692 3.885 3.994 0.132 0.229 0.459 0.343 0.145 0.182 0.746 2.729 0.328BRip 4.869 5.029 5.156 0.148 0.265 1.521 0.471 0.166 0.223 1.315 4.523 0.619

CHig1 0.685 2.029 0.934 0.118 0.228 0.346 0.299 0.138 0.182 0.517 1.078 0.250BRig1 1.193 3.560 1.467 0.118 0.232 0.486 0.331 0.145 0.195 0.578 1.870 0.459


IRb1 0.182 0.368 0.548 0.132 0.270 0.346 2.587 2.444 2.508 0.797 2.132 0.283

ORb1 0.255 0.403 0.595 0.138 0.263 0.344 0.364 0.152 0.191 2.645 3.336 2.606

BSb2 0.172 0.335 0.504 0.690 1.999 0.809 0.299 0.167 0.197 0.493 1.370 0.263

IRb2 0.198 0.380 0.502 0.140 0.314 0.329 0.764 1.933 0.732 0.615 1.919 0.241

ORb2 0.184 0.381 0.544 0.130 0.232 0.328 0.312 0.157 0.190 0.709 2.473 0.788

BSb3 0.185 0.322 0.513 0.287 0.365 1.424 0.292 0.141 0.174 0.546 1.089 0.256

IRb3 0.180 0.340 0.485 0.122 0.268 0.328 0.387 0.322 1.353 0.489 1.473 0.237

ORb3 0.194 0.343 0.480 0.138 0.223 0.321 0.307 0.129 0.174 0.517 1.276 1.410

BSb4 0.189 0.336 0.545 1.295 1.406 1.498 0.318 0.150 0.183 0.639 1.661 0.257

IRb4 0.171 0.332 0.489 0.129 0.238 0.329 1.381 1.358 1.405 0.620 1.667 0.251

ORb4 0.198 0.376 0.490 0.119 0.244 0.335 0.327 0.143 0.183 1.575 2.230 1.453

BSb5 0.178 0.331 0.465 0.267 0.732 0.423 0.302 0.143 0.176 0.524 1.147 0.235

IRb5 0.184 0.348 0.499 0.122 0.247 0.318 0.411 0.685 0.334 0.490 1.324 0.238

ORb5 0.175 0.333 0.473 0.128 0.216 0.293 0.299 0.143 0.181 0.536 1.337 0.378

BSb6 0.177 0.337 0.474 0.141 0.235 0.522 0.292 0.136 0.179 0.512 1.152 0.245

IRb6 0.179 0.322 0.447 0.120 0.221 0.306 0.306 0.151 0.399 0.529 1.249 0.239

ORb6 0.184 0.341 0.434 0.119 0.220 0.290 0.272 0.140 0.173 0.515 1.152 0.443

Table D.3. Values of the condition indicator Tcf at input shaft frequency 40 Hz underlow load from sensor 1.



Fa

ult

Mo

de

Inp

ut

Sh

aft

Idle

Sh

aft

Ou

tpu

tS

haf

t

Inp

ut

Sh

aft

BS

Idle

Sh

aft

BS

Ou

tpu

tS

haf

tB

S

Inp

ut

Sh

aft

IR

Idle

Sh

aft

IR

Ou

tpu

tS

haf

tIR

Inp

ut

Sh

aft

OR

Idle

Sh

aft

OR

Ou

tpu

tS

haf

tO

R

NF 4.278 6.068 7.007 3.157 5.064 5.765 4.933 3.530 4.411 5.631 6.411 5.174CHip 12.640 13.093 12.820 3.079 4.620 5.851 4.726 3.251 3.899 5.662 7.395 5.147BRip 8.624 9.506 9.938 3.396 5.010 5.975 4.836 3.733 4.521 6.239 8.819 5.578

CHig1 6.423 13.569 8.094 3.026 4.951 6.184 4.821 3.482 4.192 5.865 6.468 4.672BRig1 7.925 14.618 9.321 3.165 5.065 5.708 4.717 3.373 4.243 5.511 7.601 5.366


IRb1 3.885 5.700 6.830 2.932 4.712 5.782 6.597 7.048 7.771 5.994 7.419 4.998

ORb1 4.278 6.325 6.839 2.969 4.843 5.579 4.604 3.785 4.168 7.218 8.166 9.023

BSb2 3.949 5.670 6.331 6.668 13.047 7.240 5.135 3.553 4.033 5.494 6.714 5.172

IRb2 4.230 6.081 7.170 3.328 4.949 5.844 5.708 11.060 6.469 5.796 7.684 4.991

ORb2 3.928 6.185 7.177 3.174 4.706 5.803 4.971 3.428 4.262 5.579 8.200 7.714

BSb3 4.167 5.675 7.444 6.654 7.234 20.446 4.797 3.517 4.216 5.927 6.572 5.298

IRb3 4.092 5.930 6.468 3.115 5.459 5.851 5.445 6.166 15.536 5.421 7.533 4.803

ORb3 4.384 5.867 6.671 3.350 4.850 5.877 5.053 3.228 4.021 5.541 8.167 18.741

BSb4 3.995 5.470 6.439 11.527 12.250 12.981 5.054 3.427 3.923 6.119 9.572 4.750

IRb4 3.385 5.181 6.357 2.659 4.121 5.120 8.080 7.914 8.242 5.464 7.975 4.273

ORb4 3.978 5.742 6.397 2.712 4.569 5.492 4.678 2.946 3.758 10.088 9.937 10.702

BSb5 4.005 5.992 6.944 10.215 19.443 10.991 5.009 3.904 4.310 6.070 9.150 4.776

IRb5 3.957 5.867 6.804 2.879 4.845 5.521 7.557 13.359 7.831 5.803 9.535 4.790

ORb5 3.974 5.919 6.851 2.981 4.597 5.412 5.275 3.599 4.000 7.330 12.388 9.802

BSb6 4.529 6.263 8.430 9.285 9.942 25.339 5.017 3.302 4.192 6.422 7.063 5.230

IRb6 3.833 5.617 6.803 2.972 4.778 5.729 6.361 7.498 17.103 6.305 9.925 4.867

ORb6 4.377 6.709 6.882 3.141 4.934 5.830 4.877 3.186 3.960 5.894 9.816 20.651

Table D.4. Values of the condition indicator Tcf at input shaft frequency 40 Hz underlow load from sensor 2.

D.3 Kurtosis Tku 73

D.3 Kurtosis Tku

When calculating the kurtosis indicator the mean value is removed from the signal.Hence a high signal offset does not affect the value. The indicator reacts in a similarway as the crest factor. Signals which are normally distributed get a value closeto three.


Fa

ult

Mo

de

Inp

ut

Sh

aft

Idle

Sh

aft

Ou

tpu

tS

haf

t

Inp

ut

Sh

aft

BS

Idle

Sh

aft

BS

Ou

tpu

tS

haf

tB

S

Inp

ut

Sh

aft

IR

Idle

Sh

aft

IR

Ou

tpu

tS

haf

tIR

Inp

ut

Sh

aft

OR

Idle

Sh

aft

OR

Ou

tpu

tS

haf

tO

R

NF 2.985 2.970 2.995 2.914 2.955 2.954 2.953 3.017 3.152 2.864 3.051 2.926CHip 28.877 28.871 28.874 2.929 4.574 2.425 3.755 2.425 3.800 6.988 22.527 1.692BRip 3.478 3.478 3.478 1.645 2.295 2.251 1.625 2.158 2.094 1.912 3.505 2.140

CHig1 27.275 39.025 26.193 3.051 3.130 2.817 4.705 3.000 2.895 3.061 2.976 2.847BRig1 3.471 5.384 3.580 2.118 4.736 2.251 1.767 2.352 3.567 2.373 7.265 1.680


IRb1 4.614 2.445 3.831 1.962 1.579 1.744 5.867 3.596 3.526 2.069 4.159 2.710

ORb1 1.552 2.032 1.932 4.003 2.303 5.068 1.757 1.933 2.331 9.565 9.590 4.764

BSb2 5.612 5.535 2.296 7.009 21.345 6.914 3.561 1.523 1.982 8.937 25.989 2.479

IRb2 8.825 2.507 10.013 1.571 1.807 2.507 5.638 10.875 3.417 3.616 9.666 2.879

ORb2 1.683 2.398 1.994 1.680 1.720 3.186 2.519 1.539 2.789 9.248 28.697 4.697

BSb3 2.465 3.019 6.647 6.577 5.963 30.572 3.114 2.966 3.124 5.093 2.971 2.954

IRb3 3.945 3.070 3.522 2.953 3.232 3.617 5.298 3.335 14.162 5.656 18.203 2.496

ORb3 2.281 3.709 3.608 2.455 2.215 3.302 3.755 2.847 2.702 6.238 18.424 23.444

BSb4 6.265 10.666 1.766 7.351 7.217 7.221 2.367 1.990 2.469 3.548 11.086 1.831

IRb4 4.655 2.491 3.808 2.006 1.589 1.778 5.940 3.627 3.554 2.060 4.156 2.801

ORb4 1.560 2.057 1.965 3.664 2.299 4.968 1.775 2.189 2.456 9.538 9.548 4.764

BSb5 3.003 2.897 2.882 6.263 17.049 4.648 3.032 2.468 2.788 3.056 3.924 2.909

IRb5 3.009 2.971 3.476 2.194 1.830 2.915 5.799 10.799 3.533 3.193 7.564 2.942

ORb5 2.911 2.937 2.632 2.527 2.868 2.962 3.001 2.264 3.006 5.152 15.379 4.207

BSb6 2.966 2.910 3.017 3.052 3.012 3.164 2.970 3.081 3.052 2.974 3.028 2.979

IRb6 2.963 3.038 2.996 3.028 2.921 2.956 3.219 3.112 7.572 2.999 3.059 2.997

ORb6 3.032 2.993 3.094 2.998 2.934 2.937 2.892 3.072 3.010 3.103 3.021 3.966

Table D.5. Values of the condition indicator Tku at input shaft frequency 40 Hz underlow load from sensor 1.



Fa

ult

Mo

de

Inp

ut

Sh

aft

Idle

Sh

aft

Ou

tpu

tS

haf

t

Inp

ut

Sh

aft

BS

Idle

Sh

aft

BS

Ou

tpu

tS

haf

tB

S

Inp

ut

Sh

aft

IR

Idle

Sh

aft

IR

Ou

tpu

tS

haf

tIR

Inp

ut

Sh

aft

OR

Idle

Sh

aft

OR

Ou

tpu

tS

haf

tO

R

NF 2.985 2.970 2.995 2.914 2.955 2.954 2.953 3.017 3.152 2.864 3.051 2.926CHip 26.531 25.653 24.642 3.037 3.014 2.564 3.123 2.960 2.977 4.059 9.733 2.471BRip 3.467 3.467 3.467 1.664 2.324 2.248 1.626 2.209 2.128 1.913 3.491 2.140

CHig1 7.585 6.749 3.718 3.009 2.977 3.026 2.951 2.996 2.933 3.058 2.954 2.957BRig1 3.526 5.419 3.633 2.401 3.737 2.242 1.794 2.622 3.346 2.363 7.272 1.705


IRb1 3.133 2.942 3.097 2.979 2.460 2.873 5.656 3.508 3.440 2.457 3.751 2.963

ORb1 2.776 2.964 2.985 3.087 2.967 3.003 2.769 3.317 3.087 7.782 6.327 4.412

BSb2 3.059 3.041 2.912 5.371 14.953 4.279 2.882 2.279 2.520 3.161 4.401 3.029

IRb2 3.064 2.976 3.544 2.000 1.940 2.824 5.085 10.365 3.143 3.091 7.587 2.985

ORb2 2.860 2.937 2.738 2.522 2.813 2.963 2.888 2.035 2.964 5.644 16.509 4.369

BSb3 2.474 3.000 6.521 6.492 5.848 29.960 3.080 2.954 3.113 5.021 3.025 2.971

IRb3 3.942 3.056 3.553 3.003 3.236 3.612 5.331 3.350 14.300 5.696 18.372 2.527

ORb3 2.314 3.768 3.609 2.454 2.207 3.308 3.738 2.857 2.679 6.285 18.572 23.495

BSb4 6.423 10.612 1.767 7.325 7.191 7.194 2.388 1.981 2.481 3.552 11.091 1.839

IRb4 4.679 2.494 3.786 2.008 1.590 1.776 5.945 3.625 3.552 2.058 4.159 2.820

ORb4 1.562 2.057 1.977 3.738 2.304 4.994 1.769 2.192 2.439 9.569 9.579 4.775

BSb5 6.645 6.164 2.384 7.469 22.379 7.355 3.752 1.530 2.050 9.340 26.687 2.391

IRb5 8.340 2.442 9.983 1.578 1.775 2.593 5.905 10.997 3.549 3.684 9.834 2.898

ORb5 1.699 2.418 1.930 1.668 1.729 3.294 2.648 1.549 2.837 9.351 29.177 4.745

BSb6 1.933 2.740 8.917 6.851 6.820 35.545 5.519 2.886 3.280 8.829 3.090 2.383

IRb6 17.080 3.343 3.550 2.934 3.372 4.431 5.750 3.602 14.997 6.918 22.172 2.513

ORb6 2.014 4.348 4.380 1.910 1.904 3.894 4.202 2.524 2.209 9.032 27.894 23.879

Table D.6. Values of the condition indicator Tku at input shaft frequency 40 Hz underlow load from sensor 2.

D.4 Peak Indexes Tpeaks 75

D.4 Peak Indexes Tpeaks

The Tpeaks indicator is only tested on the synchronous averages related to the idleand output shafts. Low values indicate that the test has reacted. Table D.7 andD.8 show the calculated values.


Fa

ult

Mo

de

Idle

Sh

aft

Ou

tpu

tS

haf

t

Idle

Sh

aft

BS

Ou

tpu

tS

haf

tB

S

Idle

Sh

aft

IR

Ou

tpu

tS

haf

tIR

Idle

Sh

aft

OR

Ou

tpu

tS

haf

tO

R

NF 1022.175 841.424 652.347 342.428 237.324 186.165 201.646 429.993CHip 0.707 0.645 427.241 420.052 136.235 147.445 472.052 189.539BRip 0.707 0.645 428.364 305.845 263.190 255.306 463.500 313.637

CHig1 763.675 1.443 685.516 351.004 296.534 196.071 156.630 191.639BRig1 1001.971 0.645 908.025 503.949 277.576 433.844 456.070 559.537

CHig2 975.146 18.642 532.093 425.810 56.996 136.921 186.651 143.255BRig2 989.950 0.000 1028.792 319.444 194.612 432.541 463.502 339.164CHog 220.193 647.081 455.151 449.554 21.378 109.028 307.673 268.644BRog 16.508 2530.864 479.114 699.278 159.726 430.646 448.001 627.268BSb1 830.000 830.719 381.838 233.875 116.932 95.308 372.374 206.530

IRb1 855.029 888.494 695.489 457.687 0.000 27.331 190.606 241.871

ORb1 890.861 1003.247 602.854 291.000 148.553 159.205 1.000 43.734

BSb2 598.109 596.749 1026.860 443.007 219.094 202.587 481.001 372.930

IRb2 544.869 730.595 327.296 246.360 279.576 160.529 130.920 107.080

ORb2 122.066 601.718 333.041 467.857 155.322 148.711 463.500 287.929

BSb3 873.284 1490.250 209.305 617.490 257.317 267.396 255.726 282.061

IRb3 1304.528 120.503 36.366 315.919 1.000 463.983 467.532 268.118

ORb3 694.192 833.353 856.574 434.914 143.967 208.750 478.001 420.176

BSb4 831.000 695.165 381.838 0.433 164.685 81.640 383.710 229.168

IRb4 941.245 1011.938 505.025 336.406 0.000 27.331 231.410 363.549

ORb4 821.879 573.285 439.245 247.752 54.562 179.856 0.707 43.734

BSb5 720.302 461.432 270.500 273.118 158.090 245.918 31.851 298.006

IRb5 1395.092 422.200 282.931 604.267 279.601 160.678 133.135 402.876

ORb5 848.409 1294.297 342.905 442.744 198.081 162.150 340.847 210.375

BSb6 807.592 585.442 262.851 588.380 117.184 181.219 376.906 257.147

IRb6 1346.338 1222.892 71.694 610.035 25.933 307.419 275.754 228.745

ORb6 828.341 865.569 339.299 330.831 115.365 118.930 304.973 280.431

Table D.7. Values of the condition indicator Tpeaks at input shaft frequency 40 Hzunder low load from sensor 1.



Fa

ult

Mo

de

Idle

Sh

aft

Ou

tpu

tS

haf

t

Idle

Sh

aft

BS

Ou

tpu

tS

haf

tB

S

Idle

Sh

aft

IR

Ou

tpu

tS

haf

tIR

Idle

Sh

aft

OR

Ou

tpu

tS

haf

tO

R

NF 1022.175 841.424 652.347 342.428 237.324 186.165 201.646 429.993CHip 1.000 0.577 353.910 510.968 205.822 287.711 426.130 326.254BRip 0.707 0.645 439.010 315.107 260.156 251.849 728.593 304.808

CHig1 1.414 625.047 357.100 261.052 48.166 179.073 391.710 225.812BRig1 990.657 0.645 696.046 496.439 208.127 435.495 456.070 568.717

CHig2 520.431 1.000 511.610 746.484 208.400 140.450 297.292 230.034BRig2 989.950 0.433 958.361 320.388 353.272 434.423 463.000 565.297CHog 0.707 626.358 106.885 728.031 83.537 189.821 245.277 310.680BRog 0.707 2529.081 34.504 703.250 246.376 411.701 455.070 469.544BSb1 1110.608 707.542 383.960 389.493 207.183 221.053 297.595 338.172

IRb1 598.222 686.803 327.226 335.652 0.707 180.806 216.120 240.186

ORb1 891.119 856.734 407.384 565.852 93.408 176.916 1.581 42.493

BSb2 138.232 1058.185 632.096 323.778 211.084 158.225 103.259 273.867

IRb2 659.001 383.286 350.738 638.068 279.088 223.722 132.501 256.576

ORb2 1026.646 921.488 262.900 510.043 180.546 100.921 465.507 287.746

BSb3 873.284 1467.410 209.305 617.349 7.649 274.474 166.433 269.909

IRb3 1531.488 66.986 65.349 331.417 1.581 464.453 467.532 173.453

ORb3 1075.000 833.353 715.868 445.488 165.941 140.347 428.169 420.176

BSb4 587.633 704.491 381.838 0.645 90.028 155.021 383.710 245.167

IRb4 744.775 946.059 499.009 240.711 0.000 27.331 149.482 246.115

ORb4 546.770 573.674 348.477 247.017 139.004 72.942 1.000 0.000

BSb5 595.383 558.820 1023.614 327.895 197.717 250.961 479.500 332.253

IRb5 1100.286 364.024 234.307 269.348 279.576 160.603 129.027 96.787

ORb5 82.671 814.668 293.082 493.410 155.726 142.267 471.086 293.889

BSb6 76.922 1481.618 435.845 1191.298 212.712 112.521 334.141 289.762

IRb6 51.624 67.417 17.393 177.572 0.000 463.983 476.017 294.144

ORb6 1090.304 921.188 390.516 444.458 110.494 187.598 479.000 407.099

Table D.8. Values of the condition indicator Tpeaks at input shaft frequency 40 Hzunder low load from sensor 2.

D.5 Results From the Labelled Data 77

D.5 Results From the Labelled DataIn the following tables, the condition indicators are calculated on the labelled dataspecified in Table 7.1. The same case, 40 Hz low load, is used in all tables.


Fa

ult

Ca

se

Inp

ut

Sh

aft

Idle

Sh

aft

Ou

tpu

tS

haf

t

Inp

ut

Sh

aft

BS

Idle

Sh

aft

BS

Ou

tpu

tS

haf

tB

S

Inp

ut

Sh

aft

IR

Idle

Sh

aft

IR

Ou

tpu

tS

haf

tIR

Inp

ut

Sh

aft

OR

Idle

Sh

aft

OR

Ou

tpu

tS

haf

tO

R

Helical 1 2.169 2.177 2.176 2.164 2.164 2.164 2.165 2.164 2.164 2.165 2.165 2.164Helical 2 2.163 2.171 2.176 2.157 2.153 2.154 2.153 2.155 2.155 2.156 2.153 2.153Helical 3 2.159 2.165 2.171 2.152 2.149 2.150 2.162 2.151 2.149 2.152 2.149 2.152Helical 4 2.171 2.180 2.179 2.163 2.163 2.165 2.163 2.165 2.163 2.166 2.163 2.169Helical 5 2.164 2.169 2.175 2.154 2.155 2.154 2.153 2.156 2.155 2.154 2.153 2.153Helical 6 2.155 2.158 2.166 2.146 2.146 2.146 2.146 2.148 2.146 2.146 2.146 2.147

Spur 1 2.164 2.171 2.169 2.159 2.158 2.158 2.157 2.158 2.157 2.158 2.158 2.158Spur 2 2.212 2.259 2.223 2.199 2.199 2.205 2.203 2.214 2.204 2.203 2.199 2.230Spur 3 2.172 2.197 2.185 2.166 2.168 2.176 2.165 2.166 2.166 2.164 2.164 2.164Spur 4 2.201 2.361 2.247 2.161 2.171 2.236 2.158 2.163 2.164 2.162 2.162 2.171Spur 5 2.259 2.517 2.328 2.172 2.175 2.212 2.194 2.186 2.177 2.176 2.174 2.176Spur 6 2.178 2.187 2.183 2.173 2.174 2.175 2.178 2.173 2.200 2.174 2.175 2.174Spur 7 2.173 2.186 2.182 2.160 2.160 2.161 2.156 2.159 2.160 2.159 2.171 2.160Spur 8 2.181 2.190 2.191 2.173 2.172 2.173 2.177 2.173 2.172 2.175 2.176 2.172

Table D.9. Calculated values of the condition indicator Trms from sensor 1. All valuesshould be multiplied by 10−2.


Fa

ult

Ca

se

Inp

ut

Sh

aft

Idle

Sh

aft

Ou

tpu

tS

haf

t

Inp

ut

Sh

aft

BS

Idle

Sh

aft

BS

Ou

tpu

tS

haf

tB

S

Inp

ut

Sh

aft

IR

Idle

Sh

aft

IR

Ou

tpu

tS

haf

tIR

Inp

ut

Sh

aft

OR

Idle

Sh

aft

OR

Ou

tpu

tS

haf

tO

R



Table D.10. Calculated values of the condition indicator Trms from sensor 2. All valuesshould be multiplied by 10−3.


Synchronous averagesF

au

ltC

ase

Inp

ut

Sh

aft

Idle

Sh

aft

Ou

tpu

tS

haf

t

Inp

ut

Sh

aft

BS

Idle

Sh

aft

BS

Ou

tpu

tS

haf

tB

S

Inp

ut

Sh

aft

IR

Idle

Sh

aft

IR

Ou

tpu

tS

haf

tIR

Inp

ut

Sh

aft

OR

Idle

Sh

aft

OR

Ou

tpu

tS

haf

tO

R



Table D.11. Calculated values of the condition indicator Tcf from sensor 1. All valuesshould be multiplied by 10−1.


Fa

ult

Ca

se

Inp

ut

Sh

aft

Idle

Sh

aft

Ou

tpu

tS

haf

t

Inp

ut

Sh

aft

BS

Idle

Sh

aft

BS

Ou

tpu

tS

haf

tB

S

Inp

ut

Sh

aft

IR

Idle

Sh

aft

IR

Ou

tpu

tS

haf

tIR

Inp

ut

Sh

aft

OR

Idle

Sh

aft

OR

Ou

tpu

tS

haf

tO

R



Table D.12. Calculated values of the condition indicator Tcf from sensor 2.

D.5 Results From the Labelled Data 79

Synchronous averagesF

au

ltC

ase

Inp

ut

Sh

aft

Idle

Sh

aft

Ou

tpu

tS

haf

t

Inp

ut

Sh

aft

BS

Idle

Sh

aft

BS

Ou

tpu

tS

haf

tB

S

Inp

ut

Sh

aft

IR

Idle

Sh

aft

IR

Ou

tpu

tS

haf

tIR

Inp

ut

Sh

aft

OR

Idle

Sh

aft

OR

Ou

tpu

tS

haf

tO

R



Table D.13. Calculated values of the condition indicator Tku from sensor 1.


Fa

ult

Ca

se

Inp

ut

Sh

aft

Idle

Sh

aft

Ou

tpu

tS

haf

t

Inp

ut

Sh

aft

BS

Idle

Sh

aft

BS

Ou

tpu

tS

haf

tB

S

Inp

ut

Sh

aft

IR

Idle

Sh

aft

IR

Ou

tpu

tS

haf

tIR

Inp

ut

Sh

aft

OR

Idle

Sh

aft

OR

Ou

tpu

tS

haf

tO

R



Table D.14. Calculated values of the condition indicator Tku from sensor 2.



Fa

ult

Mo

de

Idle

Sh

aft

Ou

tpu

tS

haf

t

Idle

Sh

aft

BS

Ou

tpu

tS

haf

tB

S

Idle

Sh

aft

IR

Ou

tpu

tS

haf

tIR

Idle

Sh

aft

OR

Ou

tpu

tS

haf

tO

R

Helical 1 506.550 876.651 388.809 298.248 156.208 160.596 227.607 217.752Helical 2 591.500 810.727 412.229 577.829 77.175 74.485 368.679 218.577Helical 3 494.559 774.455 359.111 455.610 120.451 143.068 493.407 245.175Helical 4 447.138 978.814 667.782 453.058 151.463 123.475 263.746 174.158Helical 5 469.159 1062.586 439.318 401.871 221.798 252.889 196.577 257.634Helical 6 418.116 647.973 290.506 929.069 111.521 130.215 324.581 328.700

Spur 1 39.579 653.098 371.295 448.062 199.812 204.626 213.992 314.185Spur 2 1065.580 1520.661 538.238 414.656 194.182 195.896 251.465 446.749Spur 3 1067.105 782.590 308.183 670.845 196.897 160.233 69.112 401.045Spur 4 1058.662 643.755 406.503 476.678 21.024 160.185 244.861 101.823Spur 5 891.267 120.700 418.151 384.604 190.042 141.172 395.238 52.170Spur 6 1123.000 402.306 462.507 221.150 56.939 228.282 183.914 241.718Spur 7 732.007 753.205 695.471 357.042 184.923 154.261 60.519 112.260Spur 8 1199.088 793.060 181.069 347.832 51.420 197.184 192.774 267.918

Table D.15. Calculated values of the condition indicator Tpeak from sensor 1.


Fa

ult

Mo

de

Idle

Sh

aft

Ou

tpu

tS

haf

t

Idle

Sh

aft

BS

Ou

tpu

tS

haf

tB

S

Idle

Sh

aft

IR

Ou

tpu

tS

haf

tIR

Idle

Sh

aft

OR

Ou

tpu

tS

haf

tO

R

Helical 1 816.094 792.025 367.400 347.769 167.489 156.414 186.254 226.899Helical 2 60.270 635.392 384.047 457.848 168.671 100.832 175.103 245.383Helical 3 246.082 914.476 344.001 554.769 186.254 106.068 423.281 202.152Helical 4 1244.882 962.474 486.310 415.350 76.763 228.890 329.667 61.868Helical 5 834.022 1019.188 311.147 471.832 116.062 177.181 279.236 276.133Helical 6 422.200 611.162 293.466 521.890 121.511 110.393 240.700 262.240

Spur 1 29.732 28.601 24.668 435.260 161.419 143.164 135.781 128.013Spur 2 1170.000 934.729 439.073 456.020 108.759 180.394 223.939 366.108Spur 3 980.972 821.104 528.766 578.640 43.618 212.965 210.705 113.074Spur 4 1131.638 577.299 331.308 321.372 164.774 120.237 140.801 232.678Spur 5 1091.572 36.072 506.502 269.645 182.000 155.034 346.561 227.642Spur 6 91.220 423.191 352.040 323.757 219.528 122.057 224.589 104.038Spur 7 783.558 811.105 373.739 226.049 120.484 80.368 200.116 91.897Spur 8 256.695 48.482 676.681 297.175 141.437 126.280 167.100 111.320

Table D.16. Calculated values of the condition indicator Tpeak from sensor 2.

Gearbox Diagnosis

Documents

Transcript of Gearbox Diagnosis