demodulation techniques in gearbox diagnostics539981/fulltext01.pdf · 2012-07-05 · demodulation...

TVE 12 037

Examensarbete 15 hpJuli 2012

Demodulation Techniques in Gearbox Diagnostics

Andreas Meisingseth

Teknisk- naturvetenskaplig fakultet UTH-enheten Besöksadress: Ångströmlaboratoriet Lägerhyddsvägen 1 Hus 4, Plan 0 Postadress: Box 536 751 21 Uppsala Telefon: 018 – 471 30 03 Telefax: 018 – 471 30 00 Hemsida: http://www.teknat.uu.se/student

Abstract

Demodulation Techniques in Gearbox Diagnostics

Andreas Meisingseth

This thesis covers the scope of one out of many ways to diagnose gearboxes, demodulating the excited vibrational signals to enhance fault detection and identification. The topic is not only of academic interest since the achievements that can be made by successful machine condition monitoring in the industry. It has a potential value that is close to be absurd, for example unplanned production stops is commonly known to be one of the worst nightmares for manufacturing companies and if one can detect faults in early stages one can improve the possibilityto plan a production stop and therefore increase the profit. Four demodulation algorithms were developed and implemented in MATLAB on data characterized by close to stationarity and distinctive energy centered around the harmonics of the gearmesh frequency.The resulting algorithms for narrowband phase and amplitude demodulation was shown to outperform Hilbert transform based phase and amplitude demodulation algorithms in gearbox diagnostics. One of the goals with the thesis was therefore reached; demodulation algorithms were developed and implemented on data. A comparison of these algorithms was done and a conclusion of which demodulation technique is superior was done. Experimental work was carried out on a test-rig and both local and distributed faults were introduced to two gearboxes, one kind of fault per gearbox. However, the data acquired from the test-rig showed severe non-stationarity and smeared spectrum properties even when angular resampling was performed and therefore a major drawback of the demodulation techniques was exploited since the methods for demodulation in this thesis are not applicable for signals with smeared spectrums. The other goal was therefore not accomplished; to distinguish a local fault from a distributed fault indata acquired by experimental work by applying the selected demodulation techniques.

ISSN: 1401-5757, TVE 12 037Examinator: Nora MassziÄmnesgranskare: Tadeusz StepinskiHandledare: Adam Jablonski

Popularvetenskaplig sammanfattning

Detta projekt har som syfte att jamfora olika demoduleringsmetoder vid di-agnosticering av roterande kugghjul i vaxellador utifran forandringar i dessvibrationer. Via en litteraturstudie utvecklades och implementerades de-modulerings algoritmer pa data som har stationara och tydliga egenskaper.Detta visade att man med demoduleringstekniker kan utvinna informationom skicket pa kugghjul och avgora vad for typ av fel som ar narvarande.Sedan utfordes ett experimentellt arbete dar vibrationssignaler spelades infran kugghjul i vaxellador med tva olika typer av fel for att vidare provade utvecklade algoritmerna. Resultatet av det experimentella arbetet varatt de utvecklade algoritmerna inte kunde tillampas pa de inspelade signalerda dessa har egenskaper som varierar for mycket med tiden samt att defrekvenserna som ar intressanta for diagnosticering av kugghjul var dranktaav ovriga komponenter i frekvensdoman.

Forord

Detta ar en kandidatuppsats skriven som en del av Civilingenjorsprogrammeti Teknisk Fysik vid Uppsala universitet och handlar om att jamfora sa kalladedemoduleringstekniker for att diagnosticera kugghjul i vaxellador genom dessvibrationer. Arbetet utfordes vid foretaget EC Systems i Krakow, Polen medstod fran det tekniska universitet AGH, Krakow, Polen.

Jag vill speciellt tacka handledare Adam Jablonski vid AGH/EC Systemsoch amnesgranskare Tadeusz Stepinski pa avdelningen for signaler och sys-tem vid Uppsala universtet for stod for och under projektets gang. Jag villaven tacka ovrig personal pa EC Systems for administrativt stod och hjalpmed det experimentella arbetet. Vidare vill jag tacka Emelie Lindgren ochFredrik Wistrom, studenter vid Uppsala universitet, for vardefulla tips ochsynpunkter vid korrekturlasning.

Andreas Meisingseth

Uppsala, SverigeJuni, 2012

Foreword

This is an bachelor thesis written as a part of the Master of Science in En-gineering Physics program at Uppsala University with the main scope ofcomparing selected demodulation techniques with respect to their applica-bility to gearbox diagnostics. The work with the project behind this thesiswas carried out at EC Systems, Cracow, Poland with support from AGHUniversity of Science and Technology, Cracow, Poland.

I want to thank my supervisor Adam Jablonski at AGH/EC Systems and myreviewer Tadeusz Stepinski at Uppsala University for support, including butnot limited to academic support, before and during the project. I would alsolike to thank the stu↵ at EC Systems for administrative support and helpduring the experimental work. I would also like to thank Emelie Lindgrenand Fredrik Wistrom, students at Uppsala University, for valuable input vidkorrekturlasning.

Andreas Meisingseth

Uppsala, SwedenJune, 2012

3

Contents

1 Introduction 1

2 Vibration signals from gearboxes 32.1 Spectrum properties . . . . . . . . . . . . . . . . . . . . . . . 3

2.1.1 Generated frequencies . . . . . . . . . . . . . . . . . . 32.1.2 Generation of harmonics . . . . . . . . . . . . . . . . . 42.1.3 Modulation - Generation of sidebands . . . . . . . . . . 4

2.2 Gear defects and vibration signals . . . . . . . . . . . . . . . . 5

3 Transform theory in gearbox diagnostics 73.1 Fourier transform . . . . . . . . . . . . . . . . . . . . . . . . . 73.2 Hilbert transform . . . . . . . . . . . . . . . . . . . . . . . . . 7

4 Analysis methods 94.1 Time-plot analysis . . . . . . . . . . . . . . . . . . . . . . . . 104.2 Spectrum-plot analysis . . . . . . . . . . . . . . . . . . . . . . 12

5 Demodulation techniques 155.1 Hilbert transform based demodulation . . . . . . . . . . . . . 15

5.1.1 Amplitude Demodulation . . . . . . . . . . . . . . . . 175.1.2 Phase Demodulation . . . . . . . . . . . . . . . . . . . 19

5.2 Narrowband Demodulation . . . . . . . . . . . . . . . . . . . . 245.2.1 Amplitude Demodulation . . . . . . . . . . . . . . . . 255.2.2 Phase Demodulation . . . . . . . . . . . . . . . . . . . 29

5.3 Discussion about The Demodulation Techniques . . . . . . . . 33

6 Data Acquisition 376.1 Test-rig setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 376.2 Introduction of faults to gearboxes and signal recording . . . . 396.3 Indicators of data stability . . . . . . . . . . . . . . . . . . . . 41

6.3.1 Selection of samples for further analysis . . . . . . . . . 46

7 Analysis of Acquired Data 477.1 Time-plot Analysis . . . . . . . . . . . . . . . . . . . . . . . . 477.2 Spectrum-plot Analysis . . . . . . . . . . . . . . . . . . . . . . 507.3 Analysis of Stationarity in Acquired Data . . . . . . . . . . . 53

7.3.1 Angular Resampling . . . . . . . . . . . . . . . . . . . 54

7.3.2 Time-Frequency Analysis . . . . . . . . . . . . . . . . . 55

8 Conclusions 58

A Appendix: Mathematical description of modulation 61

B Appendix: MATLAB-code for illustration of Parseval’s The-orem 63

C Appendix: MATLAB GUI: Initial stability control 65

D Appendix: MATLAB GUI: Save selected datasamples 66

E Appendix: MATLAB GUI: Analysis of seven samples 67

F Appendix: MATLAB GUI: Analysis of two samples 68

G Appendix: MATLAB GUI: Narrowband demodulation of ex-perimental data 69

List of Figures

1 Time-plot for the Academic data. . . . . . . . . . . . . . . . . 102 Zoomed time-plot for the Academic data. . . . . . . . . . . . . 113 Spectrum-plot for the Academic data up to the Nyquist fre-

quency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 Spectrum for the Academic data from up to 3.5 GMF . . . . . 145 Flowchart for amplitude and phase demodulation based on

analytic signals. . . . . . . . . . . . . . . . . . . . . . . . . . . 166 Time-plot of the Academic data after Hilbert transform am-

plitude demodulation . . . . . . . . . . . . . . . . . . . . . . . 187 Spectrum-plot of the Academic data after Hilbert transform

amplitude demodulation up to the Nyquist frequency . . . . . 188 Spectrum-plot of the Academic data after Hilbert transform

amplitude demodulation up to the 3.5*GMF . . . . . . . . . . 199 Graphical illustration of the unwrapping procedure applied to

simulated data. . . . . . . . . . . . . . . . . . . . . . . . . . . 2010 Graphical illustration of the detrending procedure applied to

simulated data. . . . . . . . . . . . . . . . . . . . . . . . . . . 2111 Time-plot of the Academic data after Hilbert transform phase

demodulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 2212 Spectrum-plot of the Academic data after Hilbert transform

phase demodulation up to the Nyquist frequency. . . . . . . . 2213 Spectrum-plot of the Academic data after Hilbert transform

phase demodulation up to the 3.5*GMF. . . . . . . . . . . . . 2314 Band selection of the Academic data for narrowband demod-

ulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2415 Flowchart over algorithm for narrowband amplitude demodu-

lation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2716 Time-plot of the Academic data after narrowband amplitude

demodulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 2817 Spectrum-plot of the Academic data after narrowband ampli-

tude demodulation. . . . . . . . . . . . . . . . . . . . . . . . . 2818 Visualisation of notation of frequency components in selected

bandwidth. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2919 Flowchart over algorithm for narrowband phase demodulation. 3120 Time-plot of the Academic data after narrowband phase de-

modulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3221 Spectrum-plot of the Academic data after narrowband phase

demodulation. Significant result! . . . . . . . . . . . . . . . . 32

22 Zoomed spectrum-plot of the Academic data after narrowbandphase demodulation. . . . . . . . . . . . . . . . . . . . . . . . 33

23 Illustration of how noise ’smear’ the spectrum. . . . . . . . . . 3624 Photo of test-rig setup. . . . . . . . . . . . . . . . . . . . . . . 3725 Photo: Introducing fault to the gearboxes. At the left a local

fault is caused by drilling in one tooth and at the right a localfault is made by pouring sand sand small rocks in the lubrication. 39

26 Photo of gearbox with local faults introduced by drilling inone tooth. At the left the first stage of local fault and at theright the second stage of local fault. . . . . . . . . . . . . . . 40

27 Photo of gearbox with distributed fault introduced by sand inthe lubrication. . . . . . . . . . . . . . . . . . . . . . . . . . . 40

28 Peak-to-peak value for six sets of data. . . . . . . . . . . . . . 4129 Root mean square (RMS) for six sets of data. . . . . . . . . . 4230 PSD for six sets of data. . . . . . . . . . . . . . . . . . . . . . 4331 Crest factor for six sets of data. . . . . . . . . . . . . . . . . 4432 Kurtosis for six sets of data. . . . . . . . . . . . . . . . . . . 4533 Average speed for six sets of data. . . . . . . . . . . . . . . . 4634 Time-plot for all acquired data samples without any demodu-

lation or pre-processing. . . . . . . . . . . . . . . . . . . . . . 4835 Time-plot of the chosen samples. . . . . . . . . . . . . . . . . 4936 Spectrum-plot for acquired data samples without any demod-

ulation or pre-processing. . . . . . . . . . . . . . . . . . . . . . 5137 Spectrum-plot of the chosen samples. . . . . . . . . . . . . . . 5238 Spectrum-plot of the chosen samples. Zoomed to 0-1000 [Hz]. 5239 Spectrum-plot of the chosen samples after resampling. . . . . . 5440 Spectrum-plot of the chosen samples after resampling. Zoomed

to 0-180 [1/rev]. . . . . . . . . . . . . . . . . . . . . . . . . . . 5541 Spectrograms of the Academic data. . . . . . . . . . . . . . . 5542 Spectrograms of raw and resampled time-signals of the chosen

samples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5643 MATLAB GUI for initial stabilization. Requires connection

to EC SYSTEMS database. . . . . . . . . . . . . . . . . . . . 6544 MATLAB GUI for saving data. Requires connection to EC

SYSTEMS database. . . . . . . . . . . . . . . . . . . . . . . 6645 MATLAB GUI for analysis seven samples . . . . . . . . . . . 6746 MATLAB GUI for analysis of two samples . . . . . . . . . . . 6847 MATLAB GUI for narrowband demodulation of experimental

data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

List of Tables

1 Speed and frequency properties for the Academic data. . . . . 92 Speed and frequency properties for experimental data. . . . . 50

1 INTRODUCTION

1 Introduction

It is common knowledge that unplanned production stops has severe impacton the profitability for companies involved in for example manufacturing.Machine condition monitoring is therefore an interesting topic of investiga-tion since there is tremendous potential value in successful condition moni-toring. For example if one can diagnose machines while running to plan theproductions stops, one can decrease the cost of the stops.

Machines in operation do generate vibrations, and thus the correspondingvibration spectra show a characteristic form when the machine is in goodmechanical condition. Changes in these vibration spectra are often an in-dication that the condition of the machine is changing, due to some fault.[1, p.181]. Since many machines contains rotating elements such as shafts,gears and bearings some of the of the vibrations tend to be periodic dueto periodically repeating events such as gear teeth meshing, shaft rotating.Therefore frequency analysis of these vibrations can give an indication ofthe condition of machine elements rotating while running, and thus manypowerful diagnostic techniques are based on frequency analysis. There is ofcourse many other ways to analyze the condition of a machine, among otherslubricant analysis and measurement of acoustic emission, but vibration anal-ysis is by far the most prevalent method for machine condition monitoringas it has a number of advantages compared with other methods. One mainadvantage is that vibration analysis reacts without any intervening time tochanges in machine condition and can therefore be used for both permanentand intermittent monitoring [2, p.3-7]. Hence, vibration analysis has supe-rior characteristics for industrial use and is therefore chosen to be the typeof signals for analysis in this thesis.

The chosen machine elements for analysis are gearboxes, an important partof almost all machines that use some kind of transmission of power fromone shaft to another. When gears mesh the vibration signals may containamplitude and phase modulation, which can be caused by a broad variety offault. For example a gear meshing with an eccentric gear may cause ampli-tude modulation and the speed fluctuations caused by a gear meshing witha gear with a local fault (see Figure 26 for an example) may cause phasemodulation. As stated by Fan and Zou ”Since modulating frequencies arecaused by certain faults of machine components including gear, bearing, andshaft, the detection of the modulating signal is very useful to detect gearboxfault.”[3, p.2] demodulation is an important issue in gearbox fault detectionand therefore in machine condition monitoring, since demodulation is the

1

1 INTRODUCTION

way to detect the modulating signals. Therefore, demodulation is chosen tobe the main scope of this thesis.

The overall purpose of this thesis is to compare selected demodulation tech-niques with respect to their applicability to gearbox diagnostics and thereforetwo goals are set up:

1. Develop and implement several demodulation techniques and comparethem with respect to their applicability to gearbox diagnostics.

2. Use the developed algorithms to identify the di↵erence between localand distributed faults in data acquired by experimental work.

The boundaries for this thesis are

• The project is carried out under ten weeks and is performed by theauthor when still undergraduate. Hence, sophisticated demodulationalgorithms are neglected just because their complexity.

• The narrowband demodulation that is shown in section 5.2 use a band-width chosen by reasoning instead of finding an optimized bandwidthdescribed in [4, p.51-84].

• Signals are presented in time and frequency-domain, and the cepstrumas an analysis tool is neglected.

• The only two out of seven samples of the acquired data were analyzedthoroughly.

• The only pre-processing technique applied to experimental data wasangular resampling.

• The algorithm used for angular resampling is not described due to thefact that the algorithm is intellectual property of EC Systems.

2

2 VIBRATION SIGNALS FROM GEARBOXES

2 Vibration signals from gearboxes

Gears are widely used to in machines to transmit power from one shaft toanother, usually with a change in speed and torque [2, p.40]. To understandwhat causes vibrations in gears, it is necessary to understand the kinematicsof spur gears meshing. During gear meshing teeth, which are compressedelastic bodies, contact each other during their relative motion and forcesacts on both teeth with identical intensity and in opposite directions. Thisexcites vibrations, which may be termed parametrically kinematic, since themagnitude of vibrations is generally determined by the product of time-varying sti↵ness (of gear teeth) and profile error or deformation of a tooth.Despite that research on vibration excitation from gears have a long history,experts still split on what is the principal cause of these excitations. It is awidely believed that one of the principal causes of vibrations in gearings isface-edge impact on tooth surface. However, it is shown that this is not truesince eliminating the tooth edge impact does not reduce the vibrations of anddynamic load on the teeth. It is also shown that the primary exciting factorsin the GMF (see section 2.1.1) are mesh sti↵ness variation, that is due tothat teeth are elastic bodies, pitch error, which is the constant componentof the di↵erence between actual pitches of the driven and driving gears, anddiscrete load on teeth and that tooth edge impact is not among these primaryexciting factors. [5, p.49-54]

2.1 Spectrum properties

This section covers the topic of describing the spectrum for vibration signalsgenerated by gearboxes. This is an important part of gearbox diagnosticssince it is common to describe signals in general, and those from gearboxesin particular, by their frequency properties.

2.1.1 Generated frequencies

One essential frequency generated by the machine is generated by the shaft,to which the gear is attached, and is called shaft rotational speed (SRS),sometimes called gear rotational frequency, and thus is calculated as

SRS =SRS [rpm]

60[Hz] (1)

where SRS is the rotational speed of the gear and the associated shaft. An-other vital frequency in gear diagnostics is the gear mesh frequency (GMF)(sometimes called tooth mesh frequency) and is defined as ”the angular speed

3


of the rotating gear times the number of teeth on the gear”[6, p.246], whichcan be expressed as

GMF = NSRS [rpm]

60[Hz] (2)

where N is the number of teeth on the gear. A fundamental relation be-tween two gears meshing appear from the expression in equation (2) sincethe GMF must be the same regardless which gear (of a particular gear pair)is considered:

N1SRS1 = N2SRS2 (3)

where N1 is the number of teeth of gear one and SRS1 is the shaft rotationalspeed of the associated shaft.

2.1.2 Generation of harmonics

When analyzing response spectra of gear measurements, there may be har-monics present. Harmonics are defined as ”exact integer multiple (wholenumber) of a fundamental frequency” [6, p.246] and the fundamental fre-quency are defined as ”the first harmonic or base frequency, such as gearmesh frequency, shaft speed etc.”[6, p.246]. Some gear problems that maygenerate harmonics of GMF are listed by Taylor in [6, p.86-89] as backlashproblems and oscillating gears, misaligned gears and lead runout, flats andhob marks on the tooth.

2.1.3 Modulation - Generation of sidebands

Sidebands occur when a signal is under the e↵ect of modulation, which isa phenomena that occurs when an otherwise sinusoidal signal, a so-calledcarrier signal, has its amplitude or frequency to vary with time [2, p.96].The first case, when the amplitude varies with time, is known as amplitudemodulation (AM) and the latter case, when the frequency varies with time,is called frequency modulation (FM) or phase modulation (PM), where theFM simply is the time derivative of the PM as shown in Appendix A, wherea mathematical description of modulation is shown. As the name implies,the carrier frequency carries the intelligence. The intelligence is called themodulator. In gear vibration signals, the GMF and its harmonics are thecarriers and the shaft rotating speeds of the meshing gears are the modulators[6, p.44-45]. As stated in section 1 many modulating e↵ects are caused bymachine element faults and it is therefore of highest interest to investigatethe modulation in vibration signals from gearboxes.

4


2.2 Gear defects and vibration signals

There is a lot of di↵erent causes of gear defects and the gear defects that theycauses may manifest themselves in di↵erent ways in the vibration signals gen-erated by running gearboxes. Randall [2, p.44-46] depicts a breakdown whichis the base of the description of how gear faults and vibration signals corre-late, described in this section. The first is if the gear defect is a mean e↵ectfor all tooth pairs or a variation from the mean. The reason why that is aproper way to divide the signals is that mean e↵ects for all tooth pairs man-ifest themselves at the GMF and its harmonics and that variations from themean manifest themselves at the harmonics of each gears rotational speedand in the sidebands around the GMF. The second subdivision is in whatcauses these e↵ects, for example a variation from the mean may be causedby a local fault and mean e↵ect for all tooth pairs may be caused by uniformwear.

Mean e↵ects for all tooth pair can be caused by:

• Tooth deflection due to mean torque

• The mean part of initial profile errors resulting from manufacture

• Uniform wear over all teeth.

According to Randall [2, p.44] condition monitoring attempts to separate thee↵ect of uniform wear from the other two, and he points out that the usualinitial indication of wear will be an increase in the second harmonic of theGMF, since the e↵ect at the first harmonic must became greater than thatdue to tooth deflection to become apparent.

Variations from the mean can be caused by

• Random errors

• Local faults

• Slow variations

• Systematic errors.

There is a significant di↵erence how these appear in the spectrum and time-plots of the signals; slow variations a↵ects the low harmonics of the GMFwith high level increase in magnitude and narrowly grouped sidebands mean-while local faults and random errors are identified by low level increase in

5


magnitude of a wide range of harmonics of the GMF with associated side-bands, hence the sidebands are not as narrowly grouped as those caused byslow variations . An example of slow variations is non-uniform wear, thatis a distributed fault and an example of local faults is tooth root cracks.As stated by Randall ”It should be kept in mind that what is illustrated isthe e↵ect close to the source, and that the actual measured response spectrawill be a↵ected by the transfer functions from the source to the measurementpoints.”[2, p.45]. Therefore it is not likely that the actual measured signalswill contain such a clear di↵erence between di↵erent faults. However, it ispointed out that the spectral e↵ects of these faults can be interpreted as thechanges in the spectrum [2, p.45], which in turn may be a useful tool whenanalyzing vibration signals from machinery containing gears.

6

3 TRANSFORM THEORY IN GEARBOX DIAGNOSTICS

3 Transform theory in gearbox diagnostics

3.1 Fourier transform

One assumption in the theory behind the Fourier transform (FT) and Fourierseries (FS) is that the FT or FS is applied to a continuous function. In signalprocessing, it is more common to deal with digital signals that by definitionare discrete. Consider a sequence x(n), periodic in N , in time-domain andthe discrete time Fourier series (DFTS) are defined as

X(k) =1

N

N�1X

n=0

x(n)e�i 2⇡N kn (4)

x(n) =N�1X

k=0

X(k)ei2⇡N kn (5)

where X(k) a periodic sequence in frequency-domain, and is as x(n) periodicin N . In analogy with the transform pairs of the FT, x(n) and X(k) is anone-to-one DTFS pair, denoted as x(n) $ X(k). Due to this analogy, it isnot surprising that one in practice use the term DFT, that is an abbreviationfor discrete Fourier transform, when referring to DTFS.

3.2 Hilbert transform

The Hilbert transform (HT) expresses the relationship between the real andimaginary components of the FT of an one-sided function [8, p.58]. An one-sided function is one which is equal to zero for negative values on the x-axis,i.e x(t) = 0, t < 0. It should be mentioned that all causal time signals areone-sided per definition, and since almost all recordings of physical events intime-domain are causal, the HT is an useful tool in analysis of such signals.The definition of the HT is stated as

x(t) = HT[x(t)] = P.v

1

⇡

Z 1

�1

x(⌧)

t� ⌧

d⌧ (6)

where P.v is the Cauchy principal value. The inverse HT is defined as

x(t) = HT�1[x(t)] = �✓P.v

1

⇡

Z 1

�1

x(⌧)

t� ⌧

d⌧

◆. (7)

It should be stated that the HT does not change the domain as the FT does,the HT of a function in time-domain is also in time-domain. The analytic

7

3 TRANSFORM THEORY IN GEARBOX DIAGNOSTICS

signal xa(t) that corresponds to the real signal x(t) is a complex signal definedby

xa(t) = x(t) + iHT[x(t)] = x(t) + ix(t) (8)

Hence, the analytic signal xa(t) is a complex signal with a real and imaginarypart linked by the HT, A convenient representation of the analytic signal isin the polar form

xa(t) = A(t)ei�(t), A =px

2(t) + x

2(t), �(t) = arctan

x(t)

x(t)

�(9)

where A(t) is called the envelope signal and �(t) the instantaneous phasesignal. Thus, an instantaneous frequency can be defined as

fin =1

2⇡

d�(t)

dt

. (10)

Apply the FT to the HT of a signal x(t)

FT[x(t)] = X(!)(�isgn(!)) (11)

Hence, the HT has the e↵ect of shifting negative frequency components ofx(t) by +90� and positive frequency components by�90�, and can be thoughtof as a 90� quadrature filter [7, p. 291]. It is shown that one can evaluate theHT by the FT using the following algorithm

1. Take the FT of the function x(t)

2. Multiply X(!) by �isgn(!)

3. IFT of the product

4. One now has the HT, x(t)

Which actually is an e�cient algorithm to calculate the HT. The MATLABcommand hilbert() actually use this technique [9, p.748] calculating theFT and IFT with the commands fft() and ifft() respectively and hencethe hilbert() is under the influence of the limitations of the FFT, some ofthem described in section 4.2 .

8

4 ANALYSIS METHODS

4 Analysis methods

During analysis of vibration signals from gearboxes both time and frequency-domain are of interest, hence one needs to use di↵erent analysis methods toperform analysis of signals in di↵erent domains. The time-domain analysis isperformed by simply showing the signal as a function of time (see section 4.1)meanwhile the frequency-domain analysis (see section 4.2) is performed byplotting the spectrum of the signal as a function of frequency.

The author was given a set of data from the supervisor that initially was usedin ’diagnostic project’ in the course MTRN9223-Machine Condition Moni-toring held in 2006 at the University of New South Wales, Sydney, Australia.This set of data is used in this thesis as ’reference’ data and is denoted theAcademic data since it is an example of ’clear’ data where almost no e↵ectsother than the ones caused by the gears are present, hence useful for academicpurposes but not representative for data acquired when other aspects such asbearing faults, transmission path e↵ects etc. a↵ect the recorded data.The theAcademic data consists two .mat-files, one for a good gear pair and one for agear pair where one gear has a local fault. The sampling frequency is 24000[Hz] and the speed and frequency properties are shown in Table 1 where itis shown that the GMF is approximately 192 [Hz] for both recordings. Itshould be mentioned that both gears has 32 teeth.

Sample Average speed [Hz] Speed Std [Hz] GMF [Hz]Fault free 5.9821 0.0025 191.4277Local Fault 6.0109 0.0019 192.3487

Table 1: Speed and frequency properties for the Academic data.

The the Academic data is used to show the di↵erence between time-plotanalysis and spectrum-plot analysis. Hence, the di↵erence between analyzinga signal in time respective frequency-domain. The the Academic data is alsoused as a reference signal in several sections in this thesis.

9

4 ANALYSIS METHODS

4.1 Time-plot analysis

Time-plot analysis is simply to analyze signals in time-domain, and can beinterpreted as the most straightforward way to analyze an signal. In somecases a simple look at time-plot graphs can indicate that a fault is present asshown in Figure 1 where it is clear that there is a periodic fault present. Thisis even clearer in the zoomed plot in Figure 2. But one can not identify whatkind of fault. Hence, the time-plot analysis for this set of of data withoutany pre-processing can detect a fault but not identify the fault.

Figure 1: Time-plot for the Academic data.

10

4 ANALYSIS METHODS

Figure 2: Zoomed time-plot for the Academic data.

11

4 ANALYSIS METHODS

4.2 Spectrum-plot analysis

Spectrum analysis is enables to see how the signal is varying in frequency-domain. Since data usually is recorded in time-domain one needs to trans-form the data from time to frequency-domain before performing the spectrum-plot analysis. The reason why this is interesting in general is to analyze whichfrequencies that are present, and which frequencies that are more dominant,since this may reveal information about signal. In gearbox diagnostics somefrequencies are of particular interest, namely the GMF and its harmonicswith associated sidebands, since this may reveal information about the con-dition of the gearbox and sometimes show the causes of the condition.

Digital signal processing was revolutionized by the fast Fourier transformalgorithm, usually referred to as (FFT), that is an e�cient procedure forcomputing the DFT, defined in equation (4)[7, p. 244]. The FFT is an algo-rithm that reduces the number of operations for DFT, hence the algorithmis consider to be fast. As there is an inverse Fourier transform for continuoustime signals there is also an inverse fast Fourier transform (IFFT). Hence,he IFFT and FFT are related as

IFFT(FFT(x)) = x. (12)

One of the fundamental assumptions of the FFT is that the signal underanalysis is stationary [10, p.1]. However, it is almost impossible to record acompletely stationary signal from a physical event but sometimes the signalscan be interpreted as ’stationary’ enough. This is an important assumptionsince one must make the assumption that the data acquired by the experi-mental work described in section 6 is close to be stationary so that the FFTis applicable. For the purpose of this thesis the FFT and IFFT will be im-plemented in MATLAB using the fft() and ifft()-commands during thesignal processing.

The spectrum of the Academic data is calculated via the two MATLAB-commands abs(), which calculates the absolute value of the argument, andfft(), which calculates the FFT of the argument. Hence, the spectrum S(f)of a time-signal x(t) is calculated as

S(f) = abs(fft(x(t))) (13)

and all spectrums shown in this thesis are calculated in this way. The spec-trum of the Academic data is calculated as depicted above and is plotted inFigure 3 in the frequency band from zero to the Nyquist frequency (i.e half

12

4 ANALYSIS METHODS

the sampling frequency) 12000 [Hz]. Just by looking at Figure 3 one cansee that the most interesting frequency interval is the low frequency band.Therefore, a zoomed spectrum is plotted in Figure 4 where the spectrumfrom zero to 3.5 times the GMF is shown. In Figure 4 it is shown that thegear pair with a local fault has low level increase in sidebands around the firstand second harmonic of the GMF, that is approximately 2*192=384 [Hz], inconformity with the theory in section 2.2, where it is stated that a local faultwill introduce low-level and wide spread increase in sidebands.

Figure 3: Spectrum-plot for the Academic data up to the Nyquist frequency.

13

4 ANALYSIS METHODS

Figure 4: Spectrum for the Academic data from up to 3.5 GMF

14

5 DEMODULATION TECHNIQUES

5 Demodulation techniques

Demodulation is the way to retrieve the information that lies in the modu-lator of the modulated signal. In this thesis two ’classes’ of demodulationtechniques are used; the Hilbert transform based demodulation (section 5.1)and narrowband demodulation (section 5.2). For each ’class’ of demodula-tion, algorithms for amplitude demodulation and phase demodulation aredeveloped.

The demodulation algorithms shown in this thesis has an input signal intime-domain and generates an output signal in time-domain. Thus, to seehow the demodulated signal appear in frequency-domain, the spectrum ofthe demodulated signal is calculated via the algorithm shown in equation(13) in section 4.2.

5.1 Hilbert transform based demodulation

The author denotes the demodulation of the whole frequency band as Hilberttransform based (HTB) demodulation since it is based upon calculating theanalytic time signal of the whole input signal via the Hilbert transform.

Demodulation using an analytic signal is described by Randall in [8, p.178]in a somewhat conceptual way as, a given measured signal x(t) can be de-modulated by the following process:

1. Calculate its HT to get x(t)

2. Form the analytic signal xa(t) = x(t) + ix(t)

3. Decompose this into its amplitude and phase components A(t)ei�(t),where A(t) is the amplitude modulation signal

4. Multiply e

i�(t) by e

�i2⇡fct to remove the carrier frequency componentfc. The resulting phase function �m(t) = �(t) � 2⇡fct is the requiredphase modulation signal. If the phase �(t) is expressed in the range[�⇡, ⇡] it may require unwrapping to give a continuous signal �(t).

This conceptual algorithm is the basis for algorithms developed by the authorfor Hilbert transform based amplitude demodulation (HTAD) and Hilberttransform base phase demodulation (HTPD) shown in a flow chart in Fig-ure 5. The di↵erence between HTAD and HTPD is simply how to processthe data after the analytic signal is obtained as shown both in the conceptualalgorithm above and resulting algorithm in Figure 5. The algorithm shown

15


in Figure 5 is applied to the Academic data and the demodulated signals areshown in Figure 6-13.

Figure 5: Flowchart for amplitude and phase demodulation based on analyticsignals.

16


5.1.1 Amplitude Demodulation

The algorithm developed for HTAD is as follows:

1. A complex analytic signal is formed by applying the Hilbert()-commandon the input signal. One could think that this command should gener-ate the Hilbert transform of the input signal but this is not true sincethe Hilbert()-command generate the complex analytic signal.

2. The generated complex analytic time signal is windowed by a hammingwindow generated by the window(@hamming,)-command.

3. The envelope (i.e the absolute value) of the windowed complex analyticsignal is calculated via the abs()-command.

4. The spectrum of the HTAD-signal is calculated via abs(fft()).

One major modification of the conceptual algorithm described in section 5.1is that the analytic signal is windowed by a hamming window. The reasonwhy this is implemented is described by Jablonski as ”The reason for win-dowing is that a Hilbert transform of a signal composed of many componentsacts like a FIR filter, often called a “Hilbert transformer”, whose impulseresponse extends infinitely in both direction”[4, p.37]. The choice of windowsis not the topic of the thesis, therefore a hamming window is chosen simplybecause it is good enough for the purpose.

The time-plots of the Academic data after implementation of HTAD areshown in Figure 6 and by comparison with the non-demodulated time-plots in Figure 1 the fault appears more distinctive after the demodulation.Nevertheless, there is no possibility to draw conclusions about what kindof fault the gearbox has by simply looking at Figure 6 since there is onlyan indication that a periodic fault is present. The spectrums of the Aca-demic data after implementation of HTAD up to the Nyquist frequency areshown in Figure 7 and by comparison with the corresponding spectrumfor the non-demodulated signals in Figure 3 no significant improvements infrequency-domain is accomplished. However, the zoomed spectrums up to3.5*GMF of the Academic data after implementation of HTAD are shown inFigure 8 and by comparison with the the corresponding spectrums for thenon-demodulated signals in Figure 4 the presence of a local fault is moresignificant.

17


Figure 6: Time-plot of the Academic data after Hilbert transform amplitudedemodulation

Figure 7: Spectrum-plot of the Academic data after Hilbert transform am-plitude demodulation up to the Nyquist frequency

18


Figure 8: Spectrum-plot of the Academic data after Hilbert transform am-plitude demodulation up to the 3.5*GMF

5.1.2 Phase Demodulation

The algorithm developed for HTPD is based upon the algorithm showed byForbes in [11, p.5] combined with the windowing proposed by Jablonski in[4, p.37](described in section 5.1.1 ) and is implemented in MATLAB as:

1. A complex analytic signal is formed by applying the hilbert()-commandon the input signal. One could think that this command should gener-ate the Hilbert transform of the input signal but this is not true sincethe hilbert()-command generate the complex analytic signal.

2. The generated complex analytic time signal is windowed by a hammingwindow generated by the window(@hamming,)-command.

3. The phase of the complex analytic time signal is calculated via theangle()-command, that generates the the argument for a complexsignal expressed in the range of [-⇡, ⇡]. Therefore, the phase of thecomplex time signal is unwrapped by the unwrap()-command.

4. A first degree least square approximation of the linear o↵set in the con-tinuous phase function is performed via the two commands polyval()and polyfit(). The continuous phase function is detrended by sub-tracting the approximated linear o↵set and the HTPD signal in time-domain is now obtained.

19


5. The spectrum of the HTPD signal is calculated via abs(fft()).

Hence, the modifications to the conceptual algorithm described in section 5.1are:

• In analogy with the case of HTAD that the analytic signal is windowed.

• Since the carrier frequency is to be considered unknown, step 4 inthe conceptual algorithm described in section 5.1 is executed by per-forming a least square approximation of the linear o↵set in phase (i.eapproximate the carrier frequency part of the phase) and subtract thiso↵set from the calculated phase, performing so called detrending, anapproach shown by Forbes in [11, p.5].

• The spectrum of the HTPD signal is calculated via abs(fft()).

The phase is calculated via the MATLAB-command angle() which returnsthe phase in radians between -⇡ and ⇡ and hence unwrapping is necessaryas described in the conceptual algorithm described in section 5.1. Unwrap-ping the phase is, in the context of this thesis, the procedure of calculatinga continuous phase function from one expressed in the interval of [-⇡, ⇡] andis implemented in MATLAB via the command unwrap(). As shown in Fig-ure 9 where unwrap() is applied to simulated data, the result is phase as acontinuous function of time. There is a more general description of phaseunwrapping in [2, p.103].

0 0.02 0.04 0.06 0.08 0.1

!4!3.1415�

!2

0

23.1415�

4

6

8

10

12

Time [s]

Phase

[rad]

Wrapped phase

Unwrapped phase

Trendline !

Trendline !!

Figure 9: Graphical illustration of the unwrapping procedure applied to sim-ulated data.

20


The least square approximation (LSA) of the unwrapped phase of a sim-ulated signal is shown in Figure 10. The detrending is performed, as statedabove, by subtracting the LSA from the unwrapped phase. This proceduremay introduce an appearance of quasi-periodicity in the resulting demodu-lated signal (see Figure 11) due to the fact that the LSA is just an approxi-mation.

Figure 10: Graphical illustration of the detrending procedure applied to sim-ulated data.

The time-plot of the Academic data after implementation of HTPD isshown in Figure 11 where it can be observed that the the amplitude isin significant larger range than in the corresponding Figures for the non-demodulated (Figure 1) and HTPD signals (Figure 6). This may be ex-plained as a consequence of the unwrapping and detrending performed. Thespectrum of the Academic data, after implementation of HTPD, up to theNyquist frequency is shown in Figure 12 and in analogy with the case ofhigher amplitude in time-domain for the HTPD signals, there is a significantlarger range than in the corresponding Figures for the non-demodulated (Fig-ure 3) and HTPD signals (Figure 7). The zoomed spectrum up to 3.5*GMF

21


of the Academic data after implementation of HTPD is shown in Figure 13.

Figure 11: Time-plot of the Academic data after Hilbert transform phasedemodulation.

Figure 12: Spectrum-plot of the Academic data after Hilbert transform phasedemodulation up to the Nyquist frequency.

22


Figure 13: Spectrum-plot of the Academic data after Hilbert transform phasedemodulation up to the 3.5*GMF.

23


5.2 Narrowband Demodulation

The author denotes the demodulation techniques in this section as narrow-band (NB) demodulation techniques, as the name implies the idea is to se-lect an interesting frequency band for further analysis instead of analyzingthe whole frequency-domain as done by the HTB techniques showed in sec-tion 5.1. This is performed by plotting the the spectrum and select thefrequency band in frequency-domain. Hence, one is performing filtering ofthe DFT of the signal instead of filtering the signal in time-domain. Anexample of how the selection can look like is shown in Figure 14 where aselected demodulation band is shown on the Academic data.

Figure 14: Band selection of the Academic data for narrowband demodula-tion.

24


5.2.1 Amplitude Demodulation

The algorithm for narrowband amplitude demodulation (NBAD) is shownby Jablonski in [4, p.37-38] and is illustrated in a flowchart in Figure 15. Toimplement this algorithm in MATLAB the following steps are performed:

1. The spectrum of the input signal is plotted by applying the fft()-command.

2. A frequency band of interest is selected and a new, ’zero’, spectrum oftwice the size of the selected bandwidth filled with zeros is generatedby the zeros()-command.

3. The new spectrum is filled with the selected frequency band by shiftingit to the left hand end of the new spectrum, i.e the lower limit of theselected band starts at the zero frequency, and hence one now got theshifted spectrum.

4. The amplitudes of the frequency components from the selected band-width in the shifted spectrum are doubled, and the frequency compo-nents from the upper limit of the selected bandwidth to the end of theshifted spectrum are set to zero.

5. Step 4. might seem a bit odd, but it is shown in [2, p. 97] thatthis manipulation of the frequency components will generate a com-plex analytic time signal after applying the ifft()-command to themanipulated shifted spectrum.

6. In analogy with the algorithms for HTB-Demodulation, a hammingwindow generated by the window(@hamming,)-command is applied tothe complex analytic signal.

7. The narrowband amplitude demodulated (NBAD) signal in time-domainis then calculated by taking the absolute value of the windowed complexanalytic signal.

8. The spectrum of the NBAD signal is calculated via abs(fft()).

Applying this algorithm to the Academic data was done by selecting a band-width of 200 [Hz] centered around the second harmonic of the GMF (=384[Hz]) and the selected band is shown in Figure 14. The reason why thisparticular frequency band was selected is that the second harmonic of theGMF is more distinctive than the other harmonics (shown in Figure 3 and4) and that the chosen bandwidth contains su�cient amount of significant

25


sidebands. The resulting time-plot of the amplitude demodulated signal isshown in Figure 16 and by comparing with the non-demodulated time-plot inFigure 1 and HTAD time-plot in Figure 6 improvements are made with theNBAD. Note that the narrowband selection does not change the length ofthe signal in time-domain. The spectrum of the resulting amplitude demod-ulated signal is shown in Figure 17 and show in analogy with the time-plotimprovements when comparing with the corresponding spectrums in Figure 3and Figure 7.

26


Figure 15: Flowchart over algorithm for narrowband amplitude demodula-tion.

27


Figure 16: Time-plot of the Academic data after narrowband amplitudedemodulation.

Figure 17: Spectrum-plot of the Academic data after narrowband amplitudedemodulation.

28


5.2.2 Phase Demodulation

The algorithm for narrowband phase demodulation (NBPD) is developed bythe author by combining two methods:

• The suggested way of setting up a shifted spectrum for narrowbandphase demodulation, described by Randall as ”Where phase demodu-lation is required, the centre of the demodulation band will have to beshifted to zero frequency, and negative frequency components shifted tothe other end of the frequency record ”[2, p.102]. That is, the positivefrequencies (a plot of a selected bandwidth with the notation of thefrequencies component is shown in Figure 18) of the selected band areshifted all the way to the left and the negative frequencies are shiftedall the way to the right.

• Forbes [11, p.5] approach of phase demodulation, including a leastsquare approximation to estimate the linear phase o↵set (i.e the phasecomponent from the carrier signal), described in section 5.1.

Figure 18: Visualisation of notation of frequency components in selectedbandwidth.

29


The developed algorithm for NBPD is illustrated in a flowchart in Figure 19and to implement this algorithm in MATLAB the following steps are per-formed:

1. The spectrum of the input signal is plotted by applying the fft()-command.

2. A frequency band of interest is selected and a new, ”zero”, spectrum oftwice the size of the selected bandwidth filled with zeros is generatedusing the zeros()-command.

3. The new spectrum is filled with the selected frequency band by

• Shift the center frequency and the positive frequencies to the lefthand end, i.e shift the center frequency to 0 [Hz].

• Shift the negative frequencies to the right hand end.

and the shifted spectrum is obtained.

4. The shifted spectrum is inverse transformed to time-domain via theifft()-command giving a complex time signal. Note that this signalis not analytic!

5. The phase of the complex time signal is calculated via the angle()-command, that generates the the argument for a complex signal ex-pressed in the range of [-⇡, ⇡]. Therefore, the phase of the complextime signal is unwrapped by the unwrap()-command.

6. A first degree least square approximation of the linear o↵set in the con-tinuous phase function is performed via the two commands polyval()and polyfit(). The continuous phase function is detrended by sub-tracting the approximated linear o↵set and the NBPD signal in time-domain is now obtained.

7. The spectrum of the NBPD-signal is calculated via abs(fft()).

This algorithm is implemented on the Academic data with the same selectedfrequency band as for the NBAD in section 5.2.1 and the resulting time-plotand spectrum-plot is shown i Figure 20 respectively Figure 21. The time-plotin Figure 20 show improvements in comparison with the non-demodulatedtime-plot in Figure 1 as the previous methods and it is clear that there is aperiodic fault is present.

30


Figure 19: Flowchart over algorithm for narrowband phase demodulation.

31


Figure 20: Time-plot of the Academic data after narrowband phase demod-ulation.

Figure 21: Spectrum-plot of the Academic data after narrowband phase de-modulation. Significant result!

32


5.3 Discussion about The Demodulation Techniques

The time-plot and spectrum-plot after implementation of the selected de-modulation techniques are shown insections 5.1.1-5.2.2. When comparingthe results in this section the author use word improvement in the contextof improving how clear the fault appear in the plots.

The non-demodulated time-plot is shown in Figure 1 and by comparing thetime-plots for the demodulated signals with this non-demodulated time-plotit is shown that applying demodulation enhance the appearance of a faultin time-domain. However, the narrowband demodulation techniques NBADand NBPD both show significant improvements in time-domain comparedto the Hilbert transform based demodulation techniques HTAD and HTPD.Comparison of the NBAD time-plots in Figure 16 and NBPD time-plots inFigure 20 show that there is no significant di↵erence in the enhancementachieved by the two narrowband demodulation techniques, when analyzingin time-domain.

0 5 10 15 20 25 30 35 40 45 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Magnitu

de [ra

d]

Gearbox with a local fault

Frequency [Hz]

SRSx8SRSx7SRSx6SRSx5SRSx4SRSx3SRSx2SRSx1

Figure 22: Zoomed spectrum-plot of the Academic data after narrowbandphase demodulation.

The non-demodulated spectrum-plot is shown in Figure 3 and the zoomednon-demodulated spectrum-plot is shown in Figure 4, and it is the latter thatis the spectrum band of interest. Comparing the resulting zoomed spectrum-

33


plots after implementation of the Hilbert transform based demodulation tech-niques in section 5.1 to the non-demodulated zoomed spectrum show thatthere is a slight improvement, but it is close to be negligible. However thenarrowband demodulation techniques show significant improvements in re-spective spectrum-plot and especially the phase demodulation NBPD showimpressing results, see Figure 21. The zoomed view of the spectrum-plot ofthe gear pair with a local fault after implementation of NBPD is shown inFigure 22 where the harmonics of the SRS are marked and it is clear thata once per revolution fault is present, due to the significant rise of the SRSharmonics.

It should be stated that the improvements made by implementing narrow-band demodulation techniques instead of the corresponding Hilbert trans-form based demodulation techniques have costs, since there ain’t no suchthing as a free lunch1:

• The narrowband demodulation techniques requires that a band is man-ually selected, hence one can not implement narrowband demodulationin the context of gearbox diagnostics without an ad-hoc algorithm.

• The narrowband demodulation do only analyze the selected band andthus it is important to select a proper frequency band around a distinc-tive center frequency. Jablonski points out ”the smaller the bandwidth,the better the quality of the demodulated signal” [4, p.50] which meansthat the engineer or scientist performing the narrowband demodulation(in the context of gearbox diagnostics) needs to select a large enoughdemodulation band so that the significant sidebands are selected, butsmall enough to get as su�cient quality of the demodulated signal.

However, the demodulation techniques in this thesis have a major drawback;they require the signals to have significant energy centered around a harmonicof GMF for the purpose of gearbox diagnostics, as the information about thecondition of the gearbox lies in the sidebands corresponding to modulationof the GMF. This drawback is somewhat intuitive since if the signals energyis not significant around such harmonic the ’information’ about the gearboxthat lies in the sidebands caused by modulation due to gear defects will bedrenched by other e↵ects that appear in the frequency-domain, for exampletransmission path e↵ects, excessive noise, bearing faults etc.

1The phrase There ain’t no such thing as a free lunch also known as the acronym

TANSTAAFL is a popular phrase, popularized by the famous economist Milton Friedman.

34


Even though it is intuitive, a study of Parseval’s formula (sometimes re-ferred to as Parseval’s theorem or Rayleigh’s identity or Rayleigh’s energytheorem) defined, for discrete signals as, in [12, p.325]

1

N

N�1X

n=0

|x(n)|2 =N�1X

k=0

|X(k)|2 (14)

makes it ’clearer’ that the demodulation techniques in this thesis is not ap-plicable if the signals from the gearboxes are filled with for example high-amplitude noise. A short simulation was performed in MATLAB (see thecode in Appendix B for more information) to illustrate how the ratio betweenthe noise and the modulated signal a↵ects the spectrum, see Figure 23, andhence the possibility to use the demodulation techniques depicted in thissection. The ratio between the maximum amplitude of the modulated signaland the amplitude of the noise is denoted noise factor and is calculated as

Noise factor =Amplitude of noise

max|modulated signal| (15)

35


Figure 23: Illustration of how noise ’smear’ the spectrum.

36

6 DATA ACQUISITION

6 Data Acquisition

6.1 Test-rig setup

The measurements was taken on a VIBstand, a test rig for diagnosis of ro-tating machinery, designed and manufactured by EC Systems. The signalsare measured by four sensors connected to an analog input module, whichwas connected to a laptop via USB.

The mechanical part of the test rig includes a steel frame, three-phase syn-chronous motor with a gearbox (which is described more thoroughly below),di↵erential relay, frequency inverter, coupling, steel shaft, two rolling bear-ings with casings, disc with threaded holes and a housing made of organicglass.

The gearbox is a Mechanika Maszyn Kacperek HM14/1, which is a closedparallel helical gearbox that contains a gear with 54 teeth and a pinion with19 teeth and angle of 30 degrees. To be able to collect data from one gearboxwith a local fault and from one gearbox with a distributed fault, one needtwo gearboxes. Hence, two identical types of gearboxes where used in thedata acquisition. The gearbox is attached to three-phase synchronous motorof model Mechanika Maszyn Kacperek Y3-632-2 B14.

Figure 24: Photo of test-rig setup.

37

6 DATA ACQUISITION

The system part part of the test rig is made up by the frequency inverterdrive SV-iE5 made by LS Industrial Systems that is a device controlling thespeed of the motor by controlling the frequency of the electrical power sup-plied to the the motor. The vibration signals are measured with accelerome-ters IMI 623C01 with a sensitivity of 100mV/g, an accuracy of ±5%, a rangeof measuring of±50 g and a valid frequency band of 2.4-8000[Hz](±3dB). Thebinary shaft rotational signals are measured with hall sensors SM12-31010PAwith a working voltage of DV 4.5-24V, detection distance of 10mm, outputform of PNP and output state of N0. One of these hall sensors is mountedinside the engine cover to detect the engine running speed and one that ismounted on the shaft on the output side of the gearbox. The accelerometersthat measures the excited vibrations are mounted on the gearbox cover anda photo of how the sensors were mounted is in Figure 24. The signals fromthe sensors are amplified by a factor of ten in a EC SYSTEMS PA3000, thatis a triple channel, portable ICP signal conditioner with a voltage range of±10PP and a frequency range of 3[Hz]to 100 KHz. The amplified signals arethen connected to to an analog input module National Instruments NI9233,that is a four channel IEPE accelerometer input module characterized by aninput range of ±5V with 50kS/s per-channel maximum sampling rate and24-bit resolution. The data from the input module is then transferred to anotebook, HP Compaq 6510b, via USB 2.0 ports.

The signals are processed in VIBex, a machine condition monitoring softwaremade by EC Systems, and uploaded to EC Systems database. A web-enableddatabase system is used to enable the author to access the data in MATLABand an ad-hoc MATLAB GUI (see Appendix C) is built to perform an initialvalidation of the collected data. For each set of data, except for the thirdstage of distributed fault, 100 samples are chosen for a more thorough datastability analysis, depicted in section 6.3, and is stored (via the MATLABGUI shown in Appendix D) first as .mat-files and then as one structure builtup by nested structures in a .mat-file at the computer where the analysisis taking place. Hence, no connection with the database is required for thefurther data analysis.

38

6 DATA ACQUISITION

6.2 Introduction of faults to gearboxes and signal record-ing

The experimental work is about to record vibration and hall sensor signalsfor two particular cases of faults:

• Local fault - introduced by drilling away pieces of one tooth at onegear.

• Distributed fault - introduced by pouring sand and small stones intothe lubrication that covers the gear pair.

The initial measurement plan for the experimental work was as follows: thedata acquisition can be subdivided into three phases, where all samples areof ten seconds each and all measurements are taken with constant load andthe frequency inverter drive setting an input speed of 20 [Hz] to the engine.The goal was to collect 100 samples of each setup and thus approximately130 samples was recorded to get some margin. First, a set of samples wasrecorded for each gearbox when in a good condition, to acquire a referencesignal. For both cases (that is, local and distributed fault) respective gear-box (distributed fault for gearbox 2 and local fault for gearbox 1) was to bedamaged into two stages and the photo of causing two of these faults areshown in Figure 25.

Figure 25: Photo: Introducing fault to the gearboxes. At the left a localfault is caused by drilling in one tooth and at the right a local fault is madeby pouring sand sand small rocks in the lubrication.

Then signals were recorded for the di↵erent stages with the intention torecord data were it is possible to see how di↵erent magnitude of fault manifestthemselves. Unfortunately the second stage of the distributed fault was notas severe as desirable, and this is the reason why a third stage of distributedfault was introduced. This fault was so severe that the gearbox could just

39

6 DATA ACQUISITION

run for less than a minute at each time before it stopped due to that stonesgot stuck between the gears, hence it was not possible to record 100 samplesin a row for this stage. The two stages of local fault is showed in Figure 26and the third stage of distributed fault is showed in Figure 27, and it is visi-ble that the gearboxes are under severe damage just by looking at the photos.

Figure 26: Photo of gearbox with local faults introduced by drilling in onetooth. At the left the first stage of local fault and at the right the secondstage of local fault.

Figure 27: Photo of gearbox with distributed fault introduced by sand in thelubrication.

40

6 DATA ACQUISITION

6.3 Indicators of data stability

Six sets of data, each set with 100 samples of ten seconds, was selected in theinitial validation GUI and these sets are analyzed by computing five di↵erentproperties that are showed in Figure 28-33 to ensure that six proper samplesare chooses for more thorough analysis.

0 50 100

2468

10

Gearbox ! 1

Fault free

PT

P [

g]

0 50 100

2468

10

Gearbox ! 2

Fault free

0 50 100

2468

10

Local fault 1

PT

P [

g]

0 50 100

2468

10

Distributed fault 1

0 50 100

2468

10

Local fault 2

Sample [number]

PT

P [

g]

0 50 100

2468

10

Distributed fault 2

Sample [number]

Figure 28: Peak-to-peak value for six sets of data.

The peak-to-peak value (PTP), is shown in Figure 28 and is a measure-ment of the range of the amplitude of each sample. To ensure that theselected sample is not under the influence of vast amplitude deviation it isnecessary to select a sample that has a PTP that is su�cient representativefor the particular set of data. It is notable that Figure 28 seems to implythat a distributed fault causes both more fluctuation in PTP and a generalhigher value of PTP than the corresponding stage of local fault.

41

6 DATA ACQUISITION

0 50 100

0.2

0.4

Gearbox ! 1

Fault free

RM

S [g]

0 50 100

0.2

0.4

Gearbox ! 2

Fault free

0 50 100

0.2

0.4

Local fault 1

RM

S [g]

0 50 100

0.2

0.4

Distributed fault 1

0 50 100

0.2

0.4

Local fault 2

Sample [number]

RM

S [g]

0 50 100

0.2

0.4

Distributed fault 2

Sample [number]

Figure 29: Root mean square (RMS) for six sets of data.

The root mean square, is shown in Figure 29 where it is showed that thedistributed fault causes both more fluctuation in RMS and a general highervalue of RMS, in analogy with the result in Figure 28. This is of coursenotable since it implies that the gear when having a distributed fault notonly having a larger range of amplitude but also a mean value of amplitudethat is higher than the corresponding case of local fault.

42

6 DATA ACQUISITION

Figure 30: PSD for six sets of data.

Power Spectral Density (PSD) estimate calculated in MATLAB by evok-ing the command pwelch(tv,[ ],[ ],250,25000) that uses 250 FFT pointsfor calculating the PSD-estimate and the specified sampling frequency of25000 Hz. By setting the second and third input arguments to empty vec-tors, MATLAB divides the input signal tv into eight sections with 50 % over-lap and each of these eight sections are windowed with a Hamming window.These eight sections are then used to calculate eight modified periodogramsthat are averaged and one now got the PSD estimate which is showed inFigure 30. The reason why it is important to calculate PSD for each sampleand plot the 100 PSD’s in one Figure is to see if there is any deviation forany particular sample, which would a↵ect the validity when later calculatingthe spectrum for one sample. As one can see in Figure 30 all six sets ofdata are without any particular deviation, hence when to decide which sam-ples are to be analyzed further the PSD calculations will not a↵ect the choice.

43

6 DATA ACQUISITION

0 50 1005

10152025

Gearbox ! 1

Fault free

Cre

st [

]

0 50 1005

10152025

Gearbox ! 2

Fault free

0 50 1005

10152025

Local fault 1

Cre

st [

]

0 50 1005

10152025

Distributed fault 1

0 50 1005

10152025

Local fault 2

Sample [number]

Cre

st [

]

0 50 1005

10152025

Distributed fault 2

Sample [number]

Figure 31: Crest factor for six sets of data.

The crest factor is calculated for the six sets of samples and is shown inFigure 31 where it is showed that there is significant fluctuation in all setsof data without any apparent correlation with either the stage of fault ortype of fault. Note that the crest factor is the maximum absolute value ofamplitude divided by the RMS, hence a low value of the crest factor impliesthat the deviation from the mean amplitude in the sample is low which is adesirable property of a sample.

44

6 DATA ACQUISITION

0 50 100

5

10

Gearbox ! 1

Fault free

Kurt

osi

s [ ]

0 50 100

5

10

Gearbox ! 2

Fault free

0 50 100

5

10

Local fault 1

Kurt

osi

s [ ]

0 50 100

5

10

Distributed fault 1

0 50 100

5

10

Local fault 2

Sample [number]

Kurt

osi

s [ ]

0 50 100

5

10

Distributed fault 2

Sample [number]

Figure 32: Kurtosis for six sets of data.

The sample kurtosis measures how outlier-prone a set of data is. It iscalculated in MATLAB by evoking the command kurtosis(tv) where tv isthe vibration signal in time-domain and is shown in Figure 32. By analogywith the result in Figure 31 there seems to be no correlation between faultof the gearboxes in the experiments and the sample kurtosis. Neverthelessit is desirable to select samples that are characterized by a low value of thekurtosis since it implies that the presence of outliers are minimized.

The average speed for the six sets of data is calculated via the hall sen-sor signal, and is shown in Figure 33. The result is that there is a speedfluctuation present in all cases but that the fluctuation seems to rise whenthe damage of the gear rise. By looking at the graphs ( and especially lookingat the second stage of local and distributed fault) a trend of rising speed withhigher sample number seems to appear present, and a possible explanationof this is that the lubrication is heated while the gearbox is running.

45

6 DATA ACQUISITION

0 50 100

380

385

390

Gearbox ! 1

Fault free

Sp

ee

d [

rpm

]

0 50 100

380

385

390

Gearbox ! 2

Fault free

0 50 100

380

385

390

Local fault 1

Sp

ee

d [

rpm

]

0 50 100

380

385

390

Distributed fault 1

0 50 100

380

385

390

Local fault 2

Sample [number]

Sp

ee

d [

rpm

]

0 50 100

380

385

390

Distributed fault 2

Sample [number]

Figure 33: Average speed for six sets of data.

6.3.1 Selection of samples for further analysis

The samples that are selected are to be typical for the set of data. By lookingat Figures 28-33, one can select six samples (one from each combination ofgear and damage state) that are typical for each set according to statisticalproperties. As a coincidence, sample number 50 is a valid choice for all sets.Thus, sample number 50 from each set were to be analyzed more thoroughin the section below. In addition, the fourth sample for Gearbox 2 is addedto this new set of data, in total seven samples. The reason why the fourthsample from Gearbox 2 is absent in Figures 28-33 is that with that severedamage, the gearbox could just run for less than one minute each time.Hence, there was no possibility to collect hundred samples for that case.However, that fourth sample is interesting as a reference signal of how acompletely damaged gearbox behave.

46

7 ANALYSIS OF ACQUIRED DATA

7 Analysis of Acquired Data

A MATLAB GUI shown in Appendix E was built to analyze the seven se-lected samples in time and frequency-domain. Another MATLAB GUI shownin Appendix F was built to analyze two out of these seven selected samplesin enlarged format.

7.1 Time-plot Analysis

The first and most intuitive topic of analysis is to analyze the time-plot ofthe vibration signals. This takes place by comparing the vibration signals intime-domain of gears in good and bad condition and is showed in Figure 34where it is clear that a local or distributed fault cause higher vibrations thana gear in good condition.

47


Figure 34: Time-plot for all acquired data samples without any demodulationor pre-processing.

It seems that the vibration excited by a gearbox increases by the stage offault but by looking at the graphs in for gearboxes in fault free stage to stagenumber two in Figure 34 it is not clear what type of fault that correspondsto each graph. Analyzing the graph for the distributed fault of stage threethat corresponds to a stage where the gearbox is under severe damage fromthe vast amount of rocks and sand in the lubrication, one can see that thevibrations excited are of a non-periodic nature and are of a high magnitude.It should be mentioned that the range of the accelerometers used in the dataacquisition is from -5 to 5 [g] and hence the vibrations excited by the thirdstage of distributed fault seems to exceeds this range.

48


Due to limitations in time of the project two samples are selected for morethorough analysis: Gearbox 1 - Fault Free andGearbox 1 - Local FaultStage 2, in the rest of this thesis denoted as the chosen samples, and thetime-plot of these are shown in an enlarged version in Figure 35 where theabsence of a clear periodic pattern is shown in contradictory to the corre-sponding view for the Academic data in Figure 1.

Figure 35: Time-plot of the chosen samples.

49


7.2 Spectrum-plot Analysis

The second analysis method applied to the seven samples is spectrum analysisand is shown in Figure 36. The speed and frequency properties was calculatedfor the data acquired by experimental work and is shown in Table 2. Theaverage speed (average SRS) is calculated via the key marker signal on thegear side of the gearbox (i.e the side of the gearbox where the faulty gear aremounted).

Sample Average speed [Hz] Speed Std [Hz] GMF [Hz]Gearbox 2 - Fault free 6.4506 0.0148 348.3341Gearbox 1 - Fault free 6.4422 0.0060 347.8801Distributed Fault - Stage 1 6.3795 0.0076 344.4942Local Fault - Stage 1 6.4321 0.0063 347.3360Distributed Fault - Stage 2 6.4424 0.0104 347.8872Local Fault - Stage 2 6.4373 0.0059 347.6128Distributed Fault - Stage 3 6.1855 0.0749 334.0192

Table 2: Speed and frequency properties for experimental data.

There is one frequency component in Figure 36 that is especially present;the peak at 6000 [Hz]. This is the second harmonic of carrier frequency ofthe frequency inverter drive. This is an example that frequencies generatedby the system around the gearbox can be present in the spectrum of thevibration signal. In Figure 36 it is also shown that the distributed and localfault seems to appear in di↵erent ways in the spectrum, but not in way thatis in total conformity with section 2.2. The distributed fault seems to lackthe growth of narrowly grouped sidebands but it causes higher levels of ex-citation if frequency-domain than the corresponding local faults. The localfault however, causes a widespread and low-level rise in the spectrum, thatis in conformity with section 2.2.

50


Figure 36: Spectrum-plot for acquired data samples without any demodula-tion or pre-processing.

The spectrum of the two chosen samples are once again shown, but in anenlarged format in Figure 37 and the corresponding zoomed spectrums areshown in Figure 38. The conclusion one can draw after viewing Figure 37and 38 is that the distinctive presence of a carrier frequency with excessiveenergy in the frequency band of relevance (i.e the low frequencies) is absentand that the spectrum-plots are somewhat smeared.

51


Figure 37: Spectrum-plot of the chosen samples.

Figure 38: Spectrum-plot of the chosen samples. Zoomed to 0-1000 [Hz].

52


7.3 Analysis of Stationarity in Acquired Data

When looking at the plots in Figure 36 one cannot see a such a clear trend inthe spectrum corresponding to more excessive damage. The most significantconclusion when looking at the plots in Figure 36-38 is that all plots hasa smeared spectrum. This will be the next topic on analysis since none ofthe demodulation techniques described in section 5 are applicable if thereis no distinctive carrier frequency in frequency band of interest , i.e for thedemodulation techniques to be applicable there must be a carrier frequencywith excessive energy present.

The speed and frequency properties was calculated for the data acquiredby experimental work and is shown in Table 2 and the result is that thestandard deviation of speed is remarkably higher for the data acquired bythe experimental work than for the Academic data shown in Table 1, hencethe acquired data is more ’non-stationary’ than the Academic data. In thecontext of this thesis, a stationary signal is one that has constant frequen-cies and since the frequencies of interest are the harmonics of the SRS andthe GMF together with the associated sidebands it means that a station-ary signal has constant speed which is equivalent to zero standard deviation.However, this will never be the case since there is always some deviation dueto that all gearbox signals are generated by non-ideal gears (i.e real gears)meshing and thus ’good’ data, such as the Academic data in Table 1, willshow a non-zero standard deviation of speed.

53


7.3.1 Angular Resampling

Angular resampling (AR), also called order tracking since the e↵ect is thatthe signal is resampled to orders of shaft rotational speed, is one of the mostimportant vibration analysis techniques for diagnosing faults in rotating ma-chinery. The main advantage of (AR) over other vibration analysis tech-niques lies in the analysis of non-stationary noise and vibration, which varyin frequency with the rotation of a reference shaft or shafts [13, p.803].Thisis the frequency analysis method that uses multiples of the running speed,Order [1/rev], instead of absolute frequencies [Hz], as the frequency base.AR is useful for machine condition monitoring because it can easily identifyspeed-related vibrations such as shaft defects and bearing wear [14, p.187].To use multiples of shaft rotational speed instead of frequency on the x-axisin frequency-domain can be to avoid smearing of discrete frequency compo-nents due to speed fluctuations, or can be to see how the strength of thevarious harmonics changes over a greater speed range.

The AR algorithm implemented in this thesis is an algorithm developed bythe authors supervisor at EC Systems, Cracow, Poland and is proprietaryinformation of same company and is therefore is not described explicitly.The spectrum of the resampled chosen data are shown in Figure 39 and thezoomed spectrum are shown in Figure 40 and the smearing a↵ects still ap-pear after resampling. This is a remarkable result, since it shows that a vast’amount of non-stationarity’ is caused by events that is independent of thespeed fluctuations.

Figure 39: Spectrum-plot of the chosen samples after resampling.

54


Figure 40: Spectrum-plot of the chosen samples after resampling. Zoomedto 0-180 [1/rev].

7.3.2 Time-Frequency Analysis

The time-frequency analysis is performed by calculating the spectrograms ofthe chosen samples before and after angular resampling. The spectrogramis calculated in MATLAB via the spectrogram()-command which uses theshort time Fourier transform (STFT) defined by Randall in [2, p.130] as

S(f, ⌧) =

Z 1

�1x(t)w(t� ⌧)exp(�j2⇡ft)dt (16)

where w(t) is window which is moved along the record, to calculate how thespectral density of a signal varies with time.

Figure 41: Spectrograms of the Academic data.

55


For a perfect stationary signal, the spectrogram would have perfect straightvertical lines since a stationary signal per definition has non-varying frequen-cies. The spectrograms for the Academic data are shown in Figure 41 andshow close-to-stationary properties even though no AR is applied (i.e one sim-ply does not need to perform AR on such ’good’ data). As shown in Figure 42where the spectrograms of the chosen data before and after resampling areshown there is a time-varying spectral density due to the non-stationarity inthe signals, which implies that there is some kind of excessive non-stationaritythat occurs due to events not periodic with shaft revolutions.

Figure 42: Spectrograms of raw and resampled time-signals of the chosensamples.

When comparing the spectrograms for the Academic data (Figure 41)with the spectrograms for the chosen data (Figure 42) one can draw twoconclusions, the experimental data is once again shown to be ’more non-stationary’ and the spectrum is smeared in such a way as distinctive har-monics of the GMF with associated sidebands are absent, in other words;the energy around interesting carrier frequencies are to low in comparison

56


with the rest of the spectrum. Hence, none of the demodulation techniquespresented in section 5 are applicable to the data acquired by experimen-tal work, according to whats stated in section 5.3. Therefore the analysisof the acquired data ends here, with the only conclusion about diagnosingthe conditions of gearboxes in the experimental data drawn in time-domain(Figure 34). However, a MATLAB GUI shown in Appendix G was built toperform Narrowband demodulation on experimental data.

57

8 CONCLUSIONS

8 Conclusions

This thesis has two parts; the first part is about developing demodulationalgorithms and apply these to the benchmarked data denoted the academicdata, the second part is about the experimental work carried out to generatesignals from the test-rig with the intention to use the demodulation algo-rithms to show the di↵erence between local and distributed faults.

Narrowband demodulation and Hilbert transform demodulation techniqueswas compared and the narrowband demodulation techniques was shown tobe superior when analyzing vibration signals from gearboxes. Especially thenarrowband phase demodulation technique showed significant results. How-ever, this was done on ’good’ data characterized by close to stationarity fre-quency properties and distinctive energy centered around harmonics of theGMF. Experimental work was carried out with the intention to use demod-ulation techniques to show the di↵erence of how distributed and local faultsmanifest themselves in frequency-domain but due to severe non-stationarityand lack of distinctive energy centered around the harmonics of the GMF thedemodulation techniques developed in this thesis was not applicable to doso. This exploits one of the most important drawbacks with the developeddemodulation techniques in this thesis. However, the one out of two goalswith this thesis: develop and compare demodulation techniques with respectto their applicability to gearbox diagnostics was accomplished.

The author suggest future work related to scope of this thesis to be donein two ways:

1. Develop algorithms for more sophisticated demodulation techniquesand compare them to the demodulation techniques showed in this the-sis. The author propose among others an implementation of the adap-tive amplitude and phase demodulation techniques showed by Brie et.al in [15].

2. Analyze the experimental data with more sophisticated frequency-domainmethods, such as the variable amplitude Fourier series shown in byYuan and Cai in [16] and applied to experimental data by the sameauthors in [17].

The combination of these two suggested ways of future research and imple-mentation will probably overcome much of the problems that occurred dueto the smeared spectrums, and thus accomplish the second goal of this thesis:to distinguish the local and distributed faults.

58

REFERENCES

References

[1] A. de Kraker and M.J.L Stakenborg. Cepstrum analysis as a usefulsupplement to spectrum analysis for gear-box monitoring. Experimentalstress analysis: proceedings of the 8th international conference, Amster-dam, Netherlands, May 12-16: 181-190, 1986.

[2] Robert Bond Randall. Vibration-based condition monitoring: industrial,aerospace, and automotive applications. Wiley, Chichester, West Sussex,U.K., 2011.

[3] Xianfeng Fan and Ming J. Zuo. Gearbox fault detection using hilbert andwavelet packet transform. Mechanical Systems and Signal Processing,20(4):966 – 982, 2006.

[4] Adam Jablonski. Development of algorithms of generating an enve-lope spectrum of a vibration signal in the frequency domain for rollingelement bearing fault detection. Master’s thesis, Akademia GorniczoHutnicza (AGH), 2008.

[5] K.V Frolov and O.I Kosarev. Control of gear vibrations at their source.International Applied Mechanics, 39(1):49–54, 2003.

[6] James I Taylor. The gear analysis handbook: a practical guide for solvingvibration problems in gears. VCI, 1st ed edition, 2000.

[7] Simon Braun. Discover signal processing: an interactive guide for engi-neers. Wiley, Chichester, England, 2008.

[8] R. B. Randall. Frequency analysis. Bruel & Kjær, Nærum, 3. ed. edition,1987.

[9] Michael Feldman. Hilbert transform in vibration analysis. MechanicalSystems and Signal Processing, 25(3):735 – 802, 2011.

[10] P. Masri and A. Bateman. Identification of nonstationary audio signalsusing the ↵t, with application to analysis-based synthesis of sound. IEEColloquium on Audio Engineering, 1995/089, 1995.

[11] Gareth Forbes. Phase demodulation using the hilbert transform in thefrequency domain. Technical report, University of New South Wales, 012010.

[12] Lennart Rade and Bertil Westergren. Mathematics handbook for scienceand engineering. Springer, Berlin, 5th ed edition, 2004.

59

REFERENCES

[13] K.S. Wang and P.S. Heyns. The combined use of order tracking tech-niques for enhanced fourier analysis of order components. MechanicalSystems and Signal Processing, 25(3):803 – 811, 2011.

[14] K.R. Fyfe and E.D.S. Munck. Analysis of computed order tracking.Mechanical Systems and Signal Processing, 11(2):187 – 205, 1997.

[15] D. Brie, M. Tomczak, H. Oehlmann, and A. Richard. Gear crack de-tection by adaptive amplitude and phase demodulation. MechanicalSystems and Signal Processing, 11(1):149 – 167, 1997.

[16] Xiaohong Yuan and Lilong Cai. Variable amplitude fourier series withits application in gearbox diagnosisa¤”part i: Principle and simulation.Mechanical Systems and Signal Processing, 19(5):1055 – 1066, 2005.

[17] Xiaohong Yuan and Lilong Cai. Variable amplitude fourier series with itsapplication in gearbox diagnosisa¤”part ii: Experiment and application.Mechanical Systems and Signal Processing, 19(5):1067 – 1081, 2005.

60

A APPENDIX: MATHEMATICAL DESCRIPTION OF MODULATION

A Appendix: Mathematical description of mod-

ulation

This appendix covers the mathematical description of modulation of signalsdescribed by Braun in [7, p.238-42]. Amplitude modulation, as mentionedin section 2.1.3, is when the amplitude of the carrier is variant with time.To obtain amplitude modulation, there must be a continuous wave carrier.Consider a carrier signal of frequency fc that is being modulated by a signalof frequency fm, then the amplitude modulation can expressed as

XAM(t) = A [1 + kAM cos (2⇡fmt)] cos (2⇡fct) (17)

and using trigonometric identities this can be decomposed into a sum ofharmonic signals

xAM(t) = A cos (2⇡fct) +AkAM

2{cos [2⇡(fc + fm)] + cos [2⇡(fc � fm)]}

(18)

That has three frequenciesfc and fc ± fm present. By evoking the FT ofa sinusoidal it is clear that amplitude modulation causes sidebands aroundthe carrier frequency that are equally spaced with the modulator frequency.One could also directly looked at the FT of a product of a function and asinusoidal, such as equation (17).

Phase modulation is the deviation in phase, which is angular displacement,from the linearly increasing phase of the carrier. Frequency modulation is thedi↵erence in instantaneous frequency, which is the angular rotation, from theconstant carrier frequency fc. Thus, frequency modulation is the derivativeof phase modulation. For FM or PM, the signal is modeled as

xPM(t) = A cos(2⇡fct+ kPM cos(2⇡fmt)) (19)

that can be decomposed into a sum of harmonic components using the Besselfunctions of the first kind, J(kPM)

xPM = A

1X

k=�1

J(kPM) cos[2⇡(fc + kfm)t] (20)

According to the same arguments as for AM do PM generate sidebands, butnot just two as for AM but a multitude number of them.

61

A APPENDIX: MATHEMATICAL DESCRIPTION OF MODULATION

Consider that one gets a combination of AM and PM, and the analogy toequations (17) and (20) is

xAM+PM(t) = A [1 + kAM cos (2⇡fmt)]1X

k=�1

J(kPM) cos[2⇡(fc + kfm)t].

(21)

However, the interaction between the AM and PM will destroy the symmetryof the sideband amplitudes around the carrier.

62

B APPENDIX: MATLAB-CODE FOR ILLUSTRATION OFPARSEVAL’S THEOREM

B Appendix: MATLAB-code for illustration

of Parseval’s Theorem

1 % Illustration of modulation appearence2 % Andreas Meisingseth3 % Uppsala, June 20124

5 %Generate time vector6 Fs = 1000; % Sampling frequency [Hz]7 dt = 1/Fs; % Time resolution [s]8 N = 10000; % Length of signals9 df = Fs/N; % Frequency resolution [Hz]

10 t = dt*(0:N�1); % Time vector11 f = df*(0:N�1); % Frequency vector12

13 %Modulator signal14 modulator = 1+sin(2*pi*20*t)+0.5*sin(2*pi*40*t);15

16 %Carrier signal17 carrier = sin(2*pi*100*t);18

19 %Perform the modulation by multiplication20 %of the modulator and the carrier, i.e21 %amplitude modulation22 modulated signal = modulator.*carrier;23 max amplitude mod sig = max(max(modulated signal))'24 k = 0; %Noise factor25 noise coeff = k*max amplitude mod sig;26 % Add Gaussian (normal distributed) noise to the27 % modulated signal.28 modulated w noise = modulated signal + noise coeff*randn(1,N);29 subplot(4,1,1)30 spectrum = abs(fft(modulated w noise))/(N/2);31 plot(f(1:N/2+1),spectrum(1:N/2+1),'k')32 xlim([5 Fs/2])33 title(['Noise factor = ',num2str(k),' '])34 ylabel('Amplitude []')35

36 subplot(4,1,2)37 k = 2 %Noise factor38 noise coeff = k*max amplitude mod sig;39 modulated w noise = modulated signal + noise coeff*randn(1,N);40 spectrum = abs(fft(modulated w noise))/(N/2);41 plot(f(1:N/2+1),spectrum(1:N/2+1),'k')42 xlim([5 Fs/2])43 title(['Noise factor = ',num2str(k),' '])

63

B APPENDIX: MATLAB-CODE FOR ILLUSTRATION OFPARSEVAL’S THEOREM

44 ylabel('Amplitude []')45

46 subplot(4,1,3)47 k = 10 %Noise factor48 noise coeff = k*max amplitude mod sig;49 modulated w noise = modulated signal + noise coeff*randn(1,N);50 spectrum = abs(fft(modulated w noise))/(N/2);51 plot(f(1:N/2+1),spectrum(1:N/2+1),'k')52 xlim([5 Fs/2])53 title(['Noise factor = ',num2str(k),' '])54 ylabel('Amplitude []')55

56 subplot(4,1,4)57 k = 20 %Noise factor58 noise coeff = k*max amplitude mod sig;59 modulated w noise = modulated signal + noise coeff*randn(1,N);60 spectrum = abs(fft(modulated w noise))/(N/2);61 plot(f(1:N/2+1),spectrum(1:N/2+1),'k')62 xlim([5 Fs/2])63 title(['Noise factor = ',num2str(k),' '])64 ylabel('Amplitude []')65 xlabel('Frequency [Hz]')

64

C APPENDIX: MATLAB GUI: INITIAL STABILITY CONTROL

C Appendix: MATLAB GUI: Initial stabil-

ity control

Figure 43: MATLAB GUI for initial stabilization. Requires connection toEC SYSTEMS database.

65

D APPENDIX: MATLAB GUI: SAVE SELECTED DATASAMPLES

D Appendix: MATLAB GUI: Save selected

datasamples

Figure 44: MATLAB GUI for saving data. Requires connection to EC SYS-TEMS database.

66

E APPENDIX: MATLAB GUI: ANALYSIS OF SEVEN SAMPLES

E Appendix: MATLAB GUI: Analysis of seven

samples

Figure 45: MATLAB GUI for analysis seven samples

67

F APPENDIX: MATLAB GUI: ANALYSIS OF TWO SAMPLES

F Appendix: MATLAB GUI: Analysis of two

samples

Figure 46: MATLAB GUI for analysis of two samples

68

G APPENDIX: MATLAB GUI: NARROWBAND DEMODULATIONOF EXPERIMENTAL DATA

G Appendix: MATLAB GUI: Narrowband

demodulation of experimental data

Figure 47: MATLAB GUI for narrowband demodulation of experimentaldata

69

demodulation techniques in gearbox diagnostics539981/fulltext01.pdf · 2012-07-05 · demodulation...

Documents