MASTER'S THESIS

Condition Monitoring of Ball BearingsUsing Vibration Analysis

Aziz Kubilay Ovacikli

Master of ScienceEngineering Physics and Electrical Engineering

Luleå University of TechnologyDepartment of Computer Science, Electrical and Space Engineering

Condition Monitoring of Ball Bearings Using Vibration

Analysis

Aziz Kubilay Ovacikli

September 14, 2010

Abstract

In today’s industry, whether it is run-to-failure, preventive, or predictive, mainte-nance is one of the major expenses in the production process. Ball bearings areone of the most vital elements in machinery and maintenance cost for replacementof those elements with interrupting the production is one of the most expensive.Establishing predictive maintenance for those rolling bearings by detecting the pos-sible defects and monitoring the current condition will enable the industry to use themaximum life span of those mechanical devices and reduce the cost of maintenanceconsiderably.

Within the ongoing project of condition monitoring by using vibration analysis atRubico AB, this thesis work aims to understand and implement a new algorithmstep-by-step, first as off-line processing, and then on a fixed-point digital signalprocessor to analyze the measured data from industry. Different approaches formaximizing the performance of the algorithm are tested, compared; and the resultsfrom both off-line floating point precision and fixed-point implementation are eval-uated.

By running the method on different data sets from industry, it has been shownthat the patented algorithm manages to detect the defects on the inner or outerrace of the ball bearings without a priori knowledge about the measurement objectand environment. The concept for implementation on a fixed-point digital signalprocessor is also proven.

Preface

This master thesis in signal processing was carried out within the project consistingof research and design of a new algorithm on vibration analysis for condition moni-toring purpose. The thesis work was held at Rubico AB, with the aim of proving theconcept for the algorithm to be implemented on a fixed-point digital signal processorplatform. The research and the implementation of the thesis with the measurementsfrom industry was done between January 2010 and August 2010.

Acknowledgments

First of all, I am most grateful to my supervisor Patrik Paajarvi for especiallyapproaching me as a friend with understanding and patience; beyond the encour-agement, inspiration and guidance that I was provided with. Thanks a lot to myexaminer James LeBlanc for giving the valuable suggestions and letting me to learnfrom his experiences.

I wish to express my gratitude to all members of Division of Systems and Inter-action at Lulea University of Technology. During the whole period of education, Ihave always been supported by them, and that made it possible for me to come tothis point. It was such a pleasure to have the chance to learn from you.

I also would like to express my special thanks to Rubico AB for giving me theopportunity to work with them, and sharing their great environment and atmo-sphere including the coffee breaks with friendly welcoming. Thanks to OutokumpuTornio Works for letting to use their data in my thesis work.

At the end, thanks to my family, for always backing me at all the steps I take.

Aziz Kubilay Ovacıklı

Lulea, August 2010

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ii

Contents

1 Introduction 1

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Existing Methods and Techniques . . . . . . . . . . . . . . . 2

1.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Ball Bearings and Vibration Analysis 5

2.1 Ball Bearing and Its Geometry . . . . . . . . . . . . . . . . . . . . . 5

2.2 Vibration Patterns, Analysis and Condition Monitoring . . . . . . . 6

2.2.1 Possible Defects on Ball Bearings and Their Vibration Pattern 6

2.2.2 Analysis of Vibrations and Condition Monitoring . . . . . . . 8

3 Measurements and Devices 10

3.1 Measurement Devices . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.2 Data Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4 Vibration Signals and Adaptive Filters 13

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4.2 Statistical Properties of Vibration Signals . . . . . . . . . . . . . . . 14

4.3 The System Identification Problem . . . . . . . . . . . . . . . . . . . 15

4.3.1 Mathematical Representation . . . . . . . . . . . . . . . . . . 17

4.4 Adaptive Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4.4.1 Equalization or System Inversion Problem . . . . . . . . . . . 23

4.4.2 Blind Equalization (Deconvolution) . . . . . . . . . . . . . . . 26

5 Implementation and Results 31

5.1 Hardware and Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

5.2 Fixed-point Implementation . . . . . . . . . . . . . . . . . . . . . . . 32

5.2.1 Parameters and Their Effects on the Algorithm . . . . . . . . 33

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5.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

5.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

6 Future Improvements and Conclusion 40

6.1 Future Improvements . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

6.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

Bibliography 42

iv

List of Figures

2.1 Typical deep-groove ball bearing [17]. . . . . . . . . . . . . . . . . . 5

2.2 Ball bearing components, applied force, load zone and load distribu-tion [6]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.3 Examples of possible defects on a ball bearing and their vibrationpattern. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.4 Fault detection process. . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.1 The accelerometer, DAQ card with measurement software installedon a laptop. (Photographed in Rubico AB.) . . . . . . . . . . . . . . 11

4.1 Probability density functions for stochastic signals with normal dis-tribution, and with high kurtosis. . . . . . . . . . . . . . . . . . . . . 15

4.2 Probability density functions for stochastic signals with normal dis-tribution, and with high skewness. . . . . . . . . . . . . . . . . . . . 16

4.3 System identification problem. . . . . . . . . . . . . . . . . . . . . . 16

4.4 Discrete time model for adaptation. . . . . . . . . . . . . . . . . . . 17

4.5 Frequency responses for unknown plant H and FIR filter F. . . . . . 19

4.6 Block diagram for adaptive system identification. . . . . . . . . . . . 20

4.7 Gradient search for minimum error. . . . . . . . . . . . . . . . . . . . 21

4.8 Frequency responses for unknown plant H and FIR filter F, for time-varying input. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.9 Estimated mean-squared-error for adaptive solution. . . . . . . . . . 22

4.10 The Equalization or system inversion problem. . . . . . . . . . . . . 23

4.11 Training an adaptive equalizer. . . . . . . . . . . . . . . . . . . . . . 24

4.12 Combined delay of systems H and F is determined by the cross-correlation between v(n) and s(n). . . . . . . . . . . . . . . . . . . . 24

4.13 Frequency responses for unknown plant H and FIR filter F, for systeminversion problem with Wiener filter. . . . . . . . . . . . . . . . . . . 25

4.14 Frequency responses for unknown plant H and FIR filter F, for systeminversion problem with time-varying input. . . . . . . . . . . . . . . 25

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4.15 Comparison of MSE cost functions for system inversion problem. . . 26

4.16 Blind equalization model. . . . . . . . . . . . . . . . . . . . . . . . . 26

4.17 Adaptive blind equalization with skewness maximization. . . . . . . 28

4.18 Blind equalization model with noise component. . . . . . . . . . . . 28

4.19 Observed signal with estimate of probability density function. Es-timated kurtosis is 4.86 and estimated skewness is 0.1 for observedsignal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.20 Output signals after MED algorithm. Estimated kurtosis is 34.12 andestimated skewness is 5.02 for output signal. . . . . . . . . . . . . . . 30

4.21 Pdf estimate of output signals after MED algorithm. . . . . . . . . . 30

5.1 An example for convergence of skewness. After certain number ofiterations, skewness will not be increasing significantly for more iter-ations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

5.2 Observed signal and estimate of probability density function for DataSet 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

5.3 Results obtained for Data Set 1. Estimated skewness is 0.05 beforefiltering, and 3.25 after filtering. . . . . . . . . . . . . . . . . . . . . 35

5.4 Results obtained for Data Set 2. Estimated skewness is 0.04 beforefiltering, and 1.83 after filtering. For a better view, samples between1000 and 2000 are given. . . . . . . . . . . . . . . . . . . . . . . . . . 36

5.5 Results obtained for Data Set 3. Estimated skewness is 0.11 beforefiltering, and 3.43 after filtering. For a better view, samples between1000 and 2000 are given. . . . . . . . . . . . . . . . . . . . . . . . . . 37

5.6 Results obtained for Data Set 4. Estimated skewness is -0.21 beforefiltering, and 3.31 after filtering. . . . . . . . . . . . . . . . . . . . . 38

5.7 Results obtained for Data Set 5. Estimated skewness is 0.03 beforefiltering, and 2.24 after filtering. . . . . . . . . . . . . . . . . . . . . 38

vi

Chapter 1

Introduction

1.1 Background

In today’s industry, maintenance is one of the main procedures which has consid-erably high amount of cost regularly. Due to the amount and regularity of thecost, maintenance has been paid a long and dedicated attention by researchers andengineers since the start of automated mass production in industry. In 2000, $1.2trillion was spent on maintaining the plant systems in the US [1]. The scene is notdifferent in other industrialized countries and it can easily be observed that mostof the maintenance budget was spent on ineffective methods. There are apparentreasons of ineffectiveness, which can be given alongside the definitions of methods.Those methods, which have been used in industry so far in order to reduce the cost,can be grouped into categories, which are worth mentioning.

Different definitions and grouping can be found in different resources, but mainte-nance methods are here divided into three: run-to-failure, preventive and predictivemaintenance.

Run-to-failure method, as the name refers, is to fix the system or machine when itis not able to operate any more due to a failure. This is the most expensive and leastused method in industry today [1]. In the time of microprocessors and computerizedautomation, waiting for a system to fail and terminating the production is notrelevant any more.

Preventive maintenance is the method to run maintenance in every decidedperiod, or after certain amount of operating time of machinery, in order to preventpossible failures on the production line. This approach is especially applied to critical

1

1.1. Background Chapter 1.

machinery, and the process can include lubrication, alignments, adjustments andreplacements. The schedule can be decided upon past performance of equipment,statistical data or similar information. The cost of this maintenance method is stillvery high, since the equipment can be replaced even if there is no failure or defectby interrupting the production.

Predictive maintenance, which depends on current condition of the equipment,is the most efficient and least costly method of maintaining plant systems. By mon-itoring the mechanical condition of the critical equipment using parameters and in-dicators such as heat distribution [1], vibration patterns and acoustic characteristicscontinuously by the help of different sensors and measurement systems, maintenancecan be applied at the exact time that is needed. This exact time can be decidedupon efficiency and operating condition of the equipment that is being monitored.In addition to this, time-to-failure and possible amount of time that the monitoredmachinery will still be able to run can also be approximated.

The algorithm that is implemented in this thesis work aims to achieve an efficient,low-cost, least computational complexity, straightforward and ease of use methodas a new addition to condition monitoring methods for predictive maintenance inindustry. The target equipment is ball bearings, the method on which the algorithmdepends is vibration analysis and the signal processing algorithm is adaptive filtering.

Since the aim is to achieve a predictive maintenance method with condition mon-itoring of ball bearings, it is worthwhile to give brief information about existingmethods.

1.1.1 Existing Methods and Techniques

Rolling element bearings of different sizes are widely used in machinery in variety ofdifferent applications for production in industry. Faults on these equipments such asmisalignment, wear, defects such as cracks and pits, and loss of lubrication can causemalfunction and catastrophic failures, which make up a very high percentage of totalmaintenance cost. For that reason, different methods were developed for diagnosisof ball bearings’ faults in order to reduce this cost. Vibration measurements arethe ones that are most widely used, compared to the other method, acoustic mea-surements [3]. Several examples for acoustic and vibration measurement techniquesin frequency and time-domain can be given as acoustic emission, sound intensity,sound pressure, the shock pulse method (SPM) and high frequency resonance tech-nique (HFRT) [2, 3]. Detailed information with test results for all those methodsare presented in Tandon and Nakra [2], and Tandon and Choudhury [3]. Beyondthese, wavelet transforms, pattern recognition and neural networks are also testedby various researchers. Comparison of these existing methods with the algorithm

2

Chapter 1. 1.2. Motivation

that is implemented in this project, the reason for a new algorithm, advantages anddisadvantages of methods are discussed in the next section.

1.2 Motivation

As mentioned before, different approaches had been developed for diagnosis of faultson ball bearings. SPM, HFRT and acoustic emission are the most known methodsthat had been used in industry so far. Including all other methods, some of thealgorithms can be implemented in time domain, some of them in frequency domainand some in both domains. The presented algorithm in this thesis work is a lineartime-domain approach, which reduces the computational complexity and total costrelatively, while making it easy to implement on a real time fixed-point digital signalprocessor (DSP) platform.

In addition to this, in most known techniques, there can be a need for a prioriknowledge and preprocessing of the measured signal in order to select an appropriate,or interesting frequency band; plus the need for additional filtering to eliminate thenoise, disturbances and distortions caused from both the measurement system itselfand the environment. On the other hand, the presented algorithm does not requireany priori knowledge about the system and any additional operations before theimplementation. Thus, it can enhance the interesting information in the signal whileeliminating the undesired components [5]. This is all done by one of the importantmathematical approaches in signal processing, adaptive filtering.

For a better understanding of the advantages of the presented algorithm, for ex-ample, the popular high frequency resonance technique requires first a band-passfiltering to select the desired frequency range, and then an envelope detection toremove distortions. The other most known method, shock pulse method requires avibration sensor tuned to 32 kHz for similar reasons, that is, to make the systemto enhance shock pulses caused from defects while disregarding the disturbances [5].Also, depending on the application, the techniques that require frequency analysismay be computationally heavy and difficult to implement as a low cost system. Themain reason that those algorithms need a priori knowledge or preprocessing is thatfrequency analysis requires a “clean” signal. When it comes to frequency analysis,non-linearity can be another issue to to deal with. If the designed system is non-linear, the output signal from that system may contain spectral components notpresent in the input, which could make the whole system difficult to test by meansof performance and reliability.

For all those reasons, a new signal processing algorithm, which is a linear time-domain approach, was presented by LeBlanc and Paajarvi [5] to overcome those

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1.2. Motivation Chapter 1.

drawbacks. The aim of this work is to examine the previous researches and develop-ments, understand the needs and requirements of the project for a good conditionmonitoring system and learn the concept of the algorithm by first as off-line process-ing in floating-point platform; then to implement it on a fixed-point digital signalprocessor for possible future real time applications.

The thesis report is organized as follows. Types, properties and basic geometryof ball bearings used in industry, possible defects that can occur and their charac-teristics, measured vibration patterns from those defects and their basic propertiesand analyzing those vibration patterns in time and frequency domain for condi-tion monitoring purposes are given in Chapter 2. In Chapter 3, a brief informationabout measurements and devices those are used during the project are given shortly.Basic theory and the mathematical algorithms alongside the vibration signals arepresented in Chapter 4, Vibration Signals and Adaptive Filters. The platform, hard-ware and tools; implementation issues and results obtained are presented in Chapter5. At the end, in Chapter 6, conclusions derived from the whole work and possiblefuture improvements and additions to the project such as decision making, signalconditioning and real-time implementation issues are given.

4

Chapter 2

Ball Bearings and VibrationAnalysis

2.1 Ball Bearing and Its Geometry

In general, rolling element bearings are designed to carry axial and/or radial loadwhile minimizing the rotational friction by placing rolling elements such as cylindersor balls between inner and outer races. There are different types of rolling elementbearings, among all of them, ball bearings are the cheapest since balls are usedinstead of cylinders in their construction. They are widely used in industry today,in variety of applications in production line, in electric motors, pumps and gearboxes. There are also different types of ball bearings such as thrust, axial, angularcontact and deep groove ball bearings. The measurement data sets used in thisthesis work are mostly from deep groove ball bearings. An example for a typicaldeep groove ball bearing is given in Figure 2.1.

Figure 2.1: Typical deep-groove ball bearing [17].

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2.2. Vibration Patterns, Analysis and Condition Monitoring Chapter 2.

Load Zone and Distrubiton

Cage

Outer Race

Inner Race

Shaft

Force

Balls

Figure 2.2: Ball bearing components, applied force, load zone and load distribution[6].

Ball bearings have smaller sizes and limited load carrying capacity compared to theother rolling element bearings, but they can support both axial and radial loads [15].Axial force is defined as the force applied parallel to the shaft whereas the radialforce is applied perpendicular to the shaft. Correct alignment, placement where itis used, enough lubrication are the important points to take care of to maximize thelife-span of this equipment.

As it can also be observed from Figure 2.2, a ball bearing consists of an inner race,an outer race, balls, a cage holding the balls apart from each other and a shaft. Theload zone and load distribution are also given with the direction of applied force inthe figure. In most cases, the outer race is held stationary where the inner race andthe balls rotate. Most of the defects on the inner side of outer race such as cracksor pits occur on the locations subject to the load zone, since they are directly underthe applied force. The inner race faults on the other hand, can occur anywhere sincethe race is not stationary and rotating.

2.2 Vibration Patterns, Analysis and Condition Moni-

toring

2.2.1 Possible Defects on Ball Bearings and Their Vibration Pat-tern

The ball bearings themselves act as a source of vibration, even if there are no defectspresent and they are perfectly aligned and adjusted [3]. A defect on one of theelements of a ball bearing can cause the vibration level to increase. There are severaltype of defects that can occur on a ball bearing, such as cracks or pits on rotating

6

Chapter 2. 2.2. Vibration Patterns, Analysis and Condition Monitoring

Figure 2.3: Examples of possible defects on a ball bearing and their vibration pat-tern.

surface or rolling elements, distributed defects such as roughness or misaligned races[3]. Those distributed or localized defects form the vibration pattern that can bedetected by a transducer and then analyzed and processed with the algorithm, whichcan enable the condition monitoring system to detect even the occurrence of a failurebefore it damages the machine or interrupt the production.

When a rolling element strikes to a defect on one of the races, this strike creates animpulse. Since the rolling element bearing rotates, those impulses will be periodicwith a certain frequency [4]. A model to describe the vibration pattern producedby a single point defect on the inner race is described by McFadden and Smith [4].

In case the defect occurs on the inner or outer race, how frequent each rolling elementstrikes to the defect is called “Ball-pass frequency” and determined by the bearinggeometry and rotation speed. Ball pass frequency can also be calculated theoreti-cally, where the formulations are given in [3, 15], and compared to the detected oneafter the signals are processed, which can also be an indicator of the performanceof the algorithm. For further diagnosis, such as determining the size of the defectfor decision making, ball pass frequencies and noise-free vibration pattern can beuseful.

The vibration signals from a fault-free bearing, bearings with inner and outer racefaults are given in Figure 2.3. As illustrated, the characteristics of the impulsesthat occur from a defect on an inner race or outer race are different. Impulsesfrom a defect in outer race have approximately equal amplitudes, since the race isstationary with respect to the load zone; that is, each time a rolling element passesby (or strikes) to the stationary defect on the outer race, equal amplitude impulseswill be created. Impulses created from a defect on the inner race have differentamplitudes and still periodic. The behaviour can be concluded as the impulses areamplitude modulated. Since the impulses occur due to the resonance from bearingelements, the amplitude is directly related to the applied force on the ball bearing.

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2.2. Vibration Patterns, Analysis and Condition Monitoring Chapter 2.

Figure 2.4: Fault detection process.

As the rotating inner race, with a defect, passes through the load zone; that is, asa rolling element strikes to the defect which moves into and out of the load zone,modulated impulses will occur periodically with each shaft rotation. Therefore, theenvelope of the impulses can be described as a function of load distribution [4].

The characteristic signals from a faulty bearing are masked after the measurementsystem, noise, distortions and disturbances. The purpose of the algorithm is torecover and enhance those signals by removing the undesired effects, as illustratedin Figure 2.4. Achieving this with minimum computational power, effort and com-plexity, without additional preprocessing applications such as filtering or envelopedetection, and as a low cost system is surely the desired goal.

2.2.2 Analysis of Vibrations and Condition Monitoring

Initially, one can consider to move to frequency domain by using Fourier Trans-form. It is already mentioned that, depending on the application, frequency domainanalysis may require preprocessing and may be a high cost and complex approach,relative to the time-domain approach. But for research purposes, to analyze thesignal as off-line processing, it is a good starting point. Harmonically related peaksin frequency spectrum such as ball pass frequency and rotational frequency can bedetected in Fourier domain [5]. This detection approach can also be described as amethod for fault detection; whereas it can also be used to analyze the output signal’sfrequency characteristics in order to conclude about performance.

The adaptive filtering approach in time domain enhances the impulses, since theyare the indicators of defects, while suppressing the noise and disturbances at thesame time [5]. The critical point for adaptive filtering is to select the objectivefunction to use, that is; since the aim is to maximize impulses while filtering out theeffect of environment, the algorithm has to check a certain parameter as a feedbackto the whole system, whether the parameter is moving on the desired pattern ornot. Detailed information about adaptive systems will be given in Chapter 4.

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Chapter 2. 2.2. Vibration Patterns, Analysis and Condition Monitoring

The system for condition monitoring of critical components has to be reliable androbust since a false alarm or missing a defect can cause unexpected and catastrophicfailures plus additional costs. For that reason, the algorithm must be proved tooperate at a certain level of confidence, robustness plus being low-complex and low-cost for industrial applications. Thus, making the decision whether the algorithm isgood or bad needs lot of experiments and tests.

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Chapter 3

Measurements and Devices

The measurement data sets with explanations and comments used for analysis andexperiments were taken from two sources. First one is from the project [16], “Au-tomatic Detection of Local Bearing Defects in Rotating Machines”, which was pub-lished in September 11, 2001. The other data sets are collected by Patrik Paajarvifrom Outokumpu Tornio Works in April, 2010. The measurement devices used forcollecting the signals, explanations and properties of the measurement data sets aregiven in this chapter.

3.1 Measurement Devices

The measurement signals in [16] were collected by a PCB triaxial ICP R© accelerom-eter. The sensitivity of the transducer is given to be 100 mV/g. More details aboutthe measurement systems and and equipments with schematics and block diagramscan be found in [16].

For collecting the measurement signals from Outokumpu, another triaxial accelerom-eter, Dytran Instruments 3263A2T R© was used with a magnet that can be mounteddirectly onto the metal surface. With triaxial accelerometer, vibrations from threedifferent directions (vertical, horizontal and axial) can be measured. Those measure-ments from three different directions were collected through three channels of DataTranslation DT9837A DataAcquisition R© (DAQ) Card, which also has a tachometerinput that enables revolution speed measurements.

This DAQ card is being marketed especially for noise and vibration analysis. 24-bit resolution, running on USB power of a PC laptop, software selectable AC or

10

Chapter 3. 3.2. Data Sets

Figure 3.1: The accelerometer, DAQ card with measurement software installed ona laptop. (Photographed in Rubico AB.)

DC coupling, embedded analog-to-digital (A/D) converters with anti-aliasing filtersfor all inputs and one digital-to-analog (D/A) converter for output and 52 kHz ofsampling rate are excellent features for vibration measurements. In addition to this,quickDAQ software was used alongside DT9837A to acquire the data from USBport and then visualize and store the digital data for analysis. It is a simple andstraightforward application for data acquisition and analysis. Figure 3.1 shows theoverall measurement system for collecting vibration data.

3.2 Data Sets

The measurements were taken from machinery including electric motors, pumpsand fans. Details about the measurement data sets with observed faults are givenas a list. For all signals taken from [16], measurements were collected before andafter replacement of the bearing, at three different sampling frequencies: 5.12 kHz,12.8 kHz and 25.6 kHz. Results after implementation of the algorithm for faultybearings, both with floating-point and DSP platform, are discussed for differentsampling frequencies, including down-sampling option, and are given in Chapter 5.

1. Data Set 1 [16]:Machine: 302-510 Silo ScrewMachine Part: Motor Bearing DS (SKF 6313 C3)Comments: Three large defects on the inner race with sizes 4x6 mm, 10x10mm, 6x10 mm respectively. Distance between the defects are equal to therolling element pitch [16].Signals: Measurements collected for three different sampling frequencies. Mo-tor speed was 765 rpm(with accuracy ±2%) [16].

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3.2. Data Sets Chapter 3.

2. Data Set 2 [16]:Machine: 308-511 Condensate PumpMachine Part: Motor bearing DS (NSK 6310 C3)Comments: The inner race has a pitting damage with size 4x2 mm [16].Signals: Measurements collected for three different sampling frequencies. Mo-tor speed was 1470 rpm(with accuracy ±2%) [16].

3. Data Set 3 [16]:Machine: 278-653 Worm screw pumpMachine Part: Motor bearing DS (SKF 6316)Comments: The outer race has a large pitting damage of size 8x5 mm. Also,both races have small pitting damages along the contact zone between theballs and the raceways [16].Signals: Measurements collected for three different sampling frequencies. Mo-tor speed was 1546 rpm(with accuracy ±2%) [16].

4. Data Set 4 (Outokumpu):Machine: FanMachine Part: Fan bearing (FAG 6220)Comments: Suspected unknown type of fault in bearing.Signals: Measurements collected at 22 kHz of sampling frequency. Motorspeed was approximately 1493 rpm.

5. Data Set 5 (Outokumpu):Machine: FanMachine Part: Motor bearing (SKF 6316)Comments: Suspected unknown type of fault in bearing.Signals: Measurements collected at 22 kHz of sampling frequency. Motorspeed was approximately 1493 rpm.

Beside the data sets given here, at the starting point and during the off-line pro-cessing of the algorithm, simulation data sets were used for testing. Those datasets were designed to represent vibration signals as close as possible such as randomsignals plus Gaussian noise and disturbances. The results were used to have a basisfor fixed-point implementation and will be discussed with the theory of the method.

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Chapter 4

Vibration Signals and AdaptiveFilters

4.1 Introduction

With the most general point of view, adaptation can be described as a process for anorganism or a human-made system to adjust itself according to the conditions of theenvironment that it is placed, in order to maximize its performance and/or life-spandue to a prescribed criterion [13]. Adaptive systems – the ones that are human-made- are well-known research area in signal processing, and there are many differentliterature published for the subject. “Adaptive Signal Processing” by Widrow andStearns [13] is the one that is mostly referenced in this chapter. Detailed informationabout the derivations, characteristics and different approaches can also be found inthat reference.

After the technological improvements enabled production of fast, economical micro-processors and DSP platforms, adaptive signal processing have found many differentapplication areas in industry such as communications, radar, biomedical engineering,navigation systems [13] and in vibration analysis of mechanical systems for conditionmonitoring purposes, as it is proposed in [5] and in this thesis work.

With brief descriptions and derivations; system identification problem, adaptive so-lution, equalization and deconvolution principles including definitions and propertiesof some statistical theorems such as probability density functions and higher ordermoments will be discussed with a practical approach in forthcoming sections.

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4.2. Statistical Properties of Vibration Signals Chapter 4.

4.2 Statistical Properties of Vibration Signals

Before giving descriptions and basic definitions about the algorithm, it is a mustto mention about statistical properties of the vibration signals, depending on themechanical condition of ball bearings. These statistical properties will enable thedesigner to select an appropriate objective function to use in adaptation process,while adjusting the filter taps to minimize the error.

First of all, some basic definitions for the terms used to define statistical propertieswill be given, where more detailed information can be found in [10].

The cumulative probability distribution function of a random variable X is:

FX(x) = P (X ≤ x), (4.1)

where P (X ≤ x) is defined as the probability of the event (X ≤ x). The probabilitydensity function (pdf) is given as the derivative of the distribution function:

fX(x) =dFX (x)

dx. (4.2)

After that, the expected value (also used as statistical average here) of the randomvariable X can be given:

E[X] =

∫∞

−∞

xfX(x)dx. (4.3)

The mean of the random variable given in Equation 4.3 is also defined as the firstmoment of the random variable X. Hereafter, the random variable X is assumed tohave zero mean. Hence, the nth moment about the origin, denoted by mn can bewritten as:

mn = E[Xn] =

∫∞

−∞

xnfX(x)dx. (4.4)

The variance is defined as the second moment of the zero-mean random variable anddenoted by σ2

X . The third and fourth moments are especially important for thethesis work since they had been used as objective functions for the algorithm. Theirperformance is also compared in this chapter.

The normalized third-order moment is called skewness and is a measure of asym-metry of the probability density function of the random variable:

S(x) =

∫∞

−∞x3fX(x)dx

σ3X

. (4.5)

The normalized fourth-order moment is called kurtosis and can give a measure aboutthe tails of the probability density function of the random variable:

K(x) =

∫∞

−∞x4fX(x)dx

σ4X

. (4.6)

14

Chapter 4. 4.3. The System Identification Problem

x

f(x) f(x)

Normal Distrubiton High Kurtosis

x

Figure 4.1: Probability density functions for stochastic signals with normal distri-bution, and with high kurtosis.

If the random variable has a normal distribution, kurtosis value will be equal to 3.

For the analysis of the measurement signals with presented algorithm, the stochasticsignal must be stationary, or at least its statistics vary slowly over time. Stationarityrequires the statistical properties of the stochastic signal to be invariant to a shiftof the origin [11].

As one of the rotating elements of ball bearing hits a surface with a defect, this strikewill create an impulse in vibration pattern. Since the bearing is rotating, those im-pulses will be periodic. Depending on the location of the defect, inner or outer race,the characteristics of the impulses will change. Regardless of the characteristics ofthe impulses, that “spiky” (impulsive) appearance will effect the probability densityfunction of the desired signal. As there will be dominating impulses that have biggeramplitudes compared to the other points, the pdf of the desired signal, which resultswith a higher kurtosis value compared to a normally distributed signal, can have ashape as given in Figure 4.1.

Another example is given for a data set with statistically asymmetric pdf in Figure4.2, which results with a higher skewness value compared to a normally distributedsignal. Depending on how the measurement device is adjusted, the impulses canoccur as high positive or negative peaks in the original signal, assuming the datais collected from a faulty bearing. Those impulses with relatively high amplitudes,negative or positive, will result an asymmetric distribution on the pdf.

4.3 The System Identification Problem

The goal of the system identification is to find a N-tap Finite Impulse Response(FIR) filter with a transfer function that models an unknown plant, which will becalled as “H” hereafter, as closely as possible. Here, the system “H” is assumed to

15

4.3. The System Identification Problem Chapter 4.

x

f(x) f(x)

High Skewness

x

Normal Distrubiton

Figure 4.2: Probability density functions for stochastic signals with normal distri-bution, and with high skewness.

HInput Signalx(n)

Output Signaly(n)

Unknown Plant

Figure 4.3: System identification problem.

be a linear-time-invariant (LTI) system with a transfer function that is desired to bemodeled. Since it is assumed to be LTI, it can be described by its impulse response.How close is the predicted system to the unknown plant is evaluated by a certainerror criteria that will be given later in this section.

The system in Figure 4.3 is described by its input-output difference equation:

M∑

k=0

aky[n− k] =

N∑

r=0

brx[n− r]. (4.7)

The system is defined as Infinite Impulse Response (IIR) if M is not equal to zero;and it is defined as Finite Impulse Response (FIR) if M is equal to zero.

The advantages of using FIR filters instead of IIR filters are listed and detailedinformation can also be found in [12]:

• First of all, FIR filters are non-recursive, as given in Equation 4.7. There-fore FIR filters always satisfy bounded-input-bounded-output (BIBO) stabil-ity, thus there is no risk for the system to be unstable caused from imple-mentation issues, such as coefficient quantization while working on fixed-pointplatform.

• FIR filters are easier to use in real-time applications with digital signal pro-cessors due to their less sensitive nature to quantization errors.

• FIR filters can have constant group delay and therefore they can be linear-phase systems. With this feature of FIR filters, phase distortion can beavoided.

16

Chapter 4. 4.3. The System Identification Problem

e(n)

H

−

+

d(n)

x(n)F

s(n)

Figure 4.4: Discrete time model for adaptation.

• FIR filters can be efficiently implemented in DSP processors.

Of course, there are disadvantages of FIR systems; but the most important one isthat they require more taps, that is, higher order filter in order to achieve comparableperformance with an IIR filter [12]. This disadvantage causes a need for highermemory and higher cost.

4.3.1 Mathematical Representation

As it is given in Figure 4.4, the FIR system that models unknown plant can berealized by streaming the same input to both F and H, so that the resulting signalsfrom both end are similar with minimum error, again, given that H and F are linear.

For the discrete time model, s(n) is the sampled real signal with discrete-time indexn and F is the FIR parameter vector, which is a N × 1 column vector:

F =[f0 f1 f2 · · · fN−1

]T. (4.8)

In practice, the error signal e(n) can never be equal to zero continuously due tonoise and disturbances. Therefore, the minimum-mean-squared-error (MSE), whichis also called as MSE cost function, is defined as:

ζ = E[e2(n)] = E[(d(n)− x(n))2]. (4.9)

The aim of the procedure is to minimize this cost function.

17

4.3. The System Identification Problem Chapter 4.

Closed-form solution

Let s(n) =[s(n) s(n− 1) . . . s(n−N + 1)

]be the N × 1 input regressor vector to

F. Then, the output signal can be written as:

x(n) =

N−1∑

i=0

fis(n−i) = f0s(n)+f1s(n−1)+. . .+fN−1s(n−N+i) = F T s(n). (4.10)

The result at the end of Equation 4.10 is the inner vector product. This can bereplaced in Equation 4.9 which gives:

ζ = E[(d(n) − x(n))2] = E[(d(n) − F T s(n))2]. (4.11)

Now it can be observed that MSE is a function of the FIR parameter vector F. s(n)and d(n) are termed the input and desired output respectively. After expandingEquation 4.11, the resulting equation will be:

ζ = E[d2(n)]− 2E[d(n)sT (n)]F + F TE[s(n)sT (n)]F . (4.12)

After that, the cross-correlation vector between the vector s(n) and instantaneousvalue of d(n) is given as:

P = E[d(n)s(n)] =[E[d(n)s(n)] E[d(n)s(n − 1)] . . . E[d(n)s(n−N + 1)]

],

(4.13)and the auto-correlation matrix of s(n) will be:

R = E[s(n)sT (n)], (4.14)

which gives:

R =

E[s(n)s(n)] E[s(n)s(n− 1)] . . E[s(n)s(n−N + 1)]E[s(n− 1)s(n)] E[s(n − 1)s(n− 1)] . . E[s(n− 1)s(n −N + 1)]

. . . . .

. . . . .

. . . . .

E[s(n−N + 1)s(n)] E[s(n −N + 1)s(n− 1)] . . E[s(n−N + 1)s(n −N + 1)]

(4.15)

Placing the cross-correlation vector and auto-correlation matrix into Equation 4.12yields:

ζ(F ) = E[d2(n)]− 2P TF + F TRF. (4.16)

After all, as mentioned before, the F that minimizes MSE cost function must befound. In order to achieve this, an equation has to be given. Equating gradient

18

Chapter 4. 4.4. Adaptive Solution

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−100

−80

−60

−40

−20

0

Normalized Frequency (×π rad/sample)

Pha

se (

degr

ees)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−10

−5

0

5

10

15

20

Normalized Frequency (×π rad/sample)

Mag

nitu

de (

dB)

Frequency Response of H

(a) Frequency response of unknown plant H.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−100

−80

−60

−40

−20

0

Normalized Frequency (×π rad/sample)

Pha

se (

degr

ees)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−10

−5

0

5

10

15

20

Normalized Frequency (×π rad/sample)

Mag

nitu

de (

dB)

Frequency Response of F

(b) Frequency response of FIR filter F.

Figure 4.5: Frequency responses for unknown plant H and FIR filter F.

- which is derived by differentiating ζ with respect to F – to zero will give thatequation:

∂ζ

∂F= 0 ⇐⇒ −2P + 2RF = 0 ⇐⇒ F = R−1P . (4.17)

This filter is also known as Wiener filter. Here, the main concern is that even theinverse of the matrix R exists, that will be computationally costly, and the systemis assumed to be stationary in time.

For the simulation of Wiener filter in floating-point environment, H is designed tobe a second order IIR low-pass system where the input s(n) is uniformly distributedpseudo-random numbers, and without disturbances and noise, the aim is to find theFIR filter F that models H. The result is given for the frequency responses of Fand H in Figure 4.5. As it can be observed, the simulated unknown plant can beidentified by FIR filter F.

4.4 Adaptive Solution

The N-tap FIR filter that is found in previous section can minimize the MSE costfunction, but it is not able to adjust itself to a scenario where H is time varying. Insuch a case, the discrete signal d(n) is still assumed to be stationary, at least, itsstatistics can vary slowly, but the vibration pattern is varying over time. Thus, tominimize the error for a time-varying signal, F has to “adapt” to this situation. Foradaptive solution, instead of calculating the closed-form-solution, the parameters ofF being iteratively adjusted will be imaged while monitoring the error signal e(n).The filter taps will be adjusted in each sampling instant depending on the MSE costfunction, and iterative adjusting is realized in several small steps.

19

4.4. Adaptive Solution Chapter 4.

s(n)

H

F x(n)

+

−

d(n)

e(n)

MSE

Figure 4.6: Block diagram for adaptive system identification.

Steepest-Descent MSE Adaptation

The iteratively adjusted adaptive filter F is a time-varying parameter vector now,where fn

i is the ith tap at time n. At sampling instant n, it can be written as:

Fn =[fn0 fn

1 . . . fnN−1

]T. (4.18)

The next parameter vector at sampling instant n+ 1 is based on:

• the current tap setting Fn,

• the current value of ζ(n).

Therefore, the steepest-descent adaptation can be written as:

Fn+1 = Fn − µ▽ζ(n)

, (4.19)

where ▽ζ(n)

is the gradient of the MSE with respect to F (n). Here, the µ is a small,

positive constant, referred as “stepsize”. In order to obtain a simplified version ofsteepest-descent adaptation, modified MSE cost function can be used, which is aninstantaneous (non-averaged) estimate of the original one:

ζ = e2(n). (4.20)

After the replacement, the gradient becomes:

▽ζ(n)

= −2(d(n)− x(n))s(n) = −2e(n)s(n). (4.21)

After all, replacing this gradient estimate into the steepest-descent algorithm willgive the Least-Mean-Square (LMS) algorithm [13]:

Fn+1 = Fn − µe(n)s(n), (4.22)

20

Chapter 4. 4.4. Adaptive Solution

MSE

Filter Tapsf(n)

0

Figure 4.7: Gradient search for minimum error.

where the constant scalar is absorbed by the adaptive step size µ.

The idea behind this approach can be represented as in Figure 4.7. Imagine anadaptive system with a parameter vector of only one element. The MSE cost functiondepending on this one dimensional system would behave like a curve. In this case, theadaptation process is mainly searching the filter tap that minimizes the MSE, whereit has to be located at the bottom of the curve. The lines representing the tangentsto this curve is the possible error values, which are given as gradient values in thealgorithm. Depending on the direction and the amplitude of the slope, filter tap hasto be adjusted. What the adaptive system does is slightly similar the behaviour ofa ball that is moving along that curve. The ball would be moving backwards andforward since it stops at the bottom of the curve. As it loses its energy, it will slowdown, oscillate faster, and be placed in its final position due to its inertia. The filtertap, on the other hand, moves depending on the direction that the gradient showsand converge monotonically without oscillation. The MSE can be minimum at thebottom, and the purpose is to get there as close as possible. As the error is beingmonitored in each iteration, the filter can be adapted to minimize it. The same ideacan be used for multiple dimensions.

For the simulation of adaptive solution in floating-point environment, as before, s(n)being uniformly distributed pseudo-random numbers, this time time-varying, withH being a second-order IIR low-pass system, the aim is to find the adaptive filterF which models unknown plant H for time-varying input. The input is streamedto get a new sample by shifting the vector in each iteration, and the error criteriais the modified MSE cost function. The result is given in Figure 4.8. As it can beobserved from frequency responses, the unknown plant was modelled by F, which isthe FIR filter parameter vector after the last iteration. The estimated MSE is shownin Figure 4.9, as presented, the error can increase or decrease in different iterationsdue to the time-varying nature of input, but it gets very close to zero after certainnumber of iterations.

21

4.4. Adaptive Solution Chapter 4.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−100

−80

−60

−40

−20

0

Normalized Frequency (×π rad/sample)

Pha

se (

degr

ees)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−10

−5

0

5

10

15

20

Normalized Frequency (×π rad/sample)

Mag

nitu

de (

dB)

Frequency Response of H

(a) Frequency response of unknown plant H.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−120

−100

−80

−60

−40

−20

0

Normalized Frequency (×π rad/sample)

Pha

se (

degr

ees)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−15

−10

−5

0

5

10

15

20

Normalized Frequency (×π rad/sample)

Mag

nitu

de (

dB)

Frequency Response of F

(b) Frequency response of FIR filter F.

Figure 4.8: Frequency responses for unknown plant H and FIR filter F, for time-varying input.

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

0.5

1

1.5

2

2.5

Iterations

MS

E

Estimated mean−squared−error

Figure 4.9: Estimated mean-squared-error for adaptive solution.

22

Chapter 4. 4.4. Adaptive Solution

Hs(n) u(n)

z(n)

+

+

Figure 4.10: The Equalization or system inversion problem.

4.4.1 Equalization or System Inversion Problem

The measurement system H given in Figure 4.10; plus the disturbances and othernoise sources represented as z(n) mask the original signal s(n) that is being measuredas u(n). In order to recover that original signal, the whole system that masks it has tobe identified. Purpose of the adaptive system is to identify and model this unknownplant with unknown characteristics which may be slowly varying with time, as it isin mechanical vibrations.

In the ideal case of a noise-free system, H is successfully identified. s(n) is recoveredby designing and using the inverse filter, H−1, which is referred to as an equalizer.But in practice, where noise is always present, H is rarely known. Even it is knownor approximated, inverse of it may not exist, or can not be realized as a casual andstable filter. In such a realistic situation, trained and adaptive equalizer can bedesigned. In this case, H is unknown and s(n) can only be observed momentarily;therefore an adaptive equalizer F can be trained by using this glimpse of s(n). Forexample, telephone modems include a digital filter that works as an equalizer toremove the effect of distortions caused from the channel, with an accurate modelthat is built by sending a known training signal from transmitter to receiver beforethe regular data transfer [7]. As the distorting channel can be modelled as H,the known training signal will be the “glimpse of s(n)”. Another example can bethe mobile communication systems where the original signal is effected from multi-path propagation. This time, receiving end may not be stationary which means thecharacteristics of the surrounding and channel can vary in time. For that reason, theglimpse can be sent periodically for the equalizer to adapt to the varying conditions.

As given in Figure 4.11, there exists a delay ∆ which corresponds to the combineddelay of both systems H and F. Combined delay caused from the filters can bedetermined by taking the cross-correlation between v(n) and s(n). The point whichthe correlation is maximized will give the amount of delay. As it was the case before,the goal is to minimize the MSE cost function in order v(n) to be as close as possibleto s(n−∆).

The derivations for this approach is the same as it is in Section 4.3, where u(n)replaces s(n). Since the equalization problem can be formulated as a task of mini-

23

4.4. Adaptive Solution Chapter 4.

s(n)H

+

+

z(n)

u(n)

z− d(n)=s(n− )

e(n)

v(n)

MSE

F

Trainingon/off

Figure 4.11: Training an adaptive equalizer.

0 10 20 30 40 50−2

0

2

4

6

8

10

12

14Cross−correlation between s(n) and v(n)

Sample(n)

Am

plitu

de

Figure 4.12: Combined delay of systems H and F is determined by the cross-correlation between v(n) and s(n).

mizing the MSE with respect to the taps of F, the Wiener filter can also be calculatedhere, as well as LMS algorithm.

With the same unknown plant and input signal as before, the simulation of Wienerfilter and LMS algorithm for system inversion problem is presented. This time, thereis also noise component in the system which is designed to be normally distributedrandom numbers. As mentioned before, the combined delay for both systems Hand F is determined by the cross-correlation, which is plotted in Figure 4.12. Theresult for Wiener filter is presented in Figure 4.13, where magnitude response of FIRsystem F is approximately the inverse of the magnitude response of H.

The result for another simulation for time-varying input s(n) is presented in Figure4.14 with the frequency responses of F and H. As it is the case for simulation ofWiener filter, there is noise component in the system, and the inverse of unknownplant is given by F. The MSE cost function is plotted in Figure 4.15, with itsestimated value. Comparing the true value of the cost function with the estimatedone, it can be observed that the estimated MSE does not converge to a minimumvalue in the presence of noise within the iterations.

24

Chapter 4. 4.4. Adaptive Solution

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−100

−80

−60

−40

−20

0

Normalized Frequency (×π rad/sample)

Pha

se (

degr

ees)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−10

−5

0

5

10

15

20

Normalized Frequency (×π rad/sample)

Mag

nitu

de (

dB)

Frequency Response of H

(a) Frequency response of unknown plant H.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−3500

−3000

−2500

−2000

−1500

−1000

−500

0

Normalized Frequency (×π rad/sample)

Pha

se (

degr

ees)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−20

−15

−10

−5

0

5

10

Normalized Frequency (×π rad/sample)

Mag

nitu

de (

dB)

Frequency Response of F

(b) Frequency response of FIR filter F.

Figure 4.13: Frequency responses for unknown plant H and FIR filter F, for systeminversion problem with Wiener filter.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−100

−80

−60

−40

−20

0

Normalized Frequency (×π rad/sample)

Pha

se (

degr

ees)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−10

−5

0

5

10

15

20

Normalized Frequency (×π rad/sample)

Mag

nitu

de (

dB)

Frequency Response of H

(a) Frequency response of unknown plant H.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1000

−800

−600

−400

−200

0

Normalized Frequency (×π rad/sample)

Pha

se (

degr

ees)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−20

−15

−10

−5

0

5

10

Normalized Frequency (×π rad/sample)

Mag

nitu

de (

dB)

Frequency Response of F

(b) Frequency response of FIR filter F.

Figure 4.14: Frequency responses for unknown plant H and FIR filter F, for systeminversion problem with time-varying input.

25

4.4. Adaptive Solution Chapter 4.

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 1000010

−2

10−1

100

101

102

Iterations

MS

E

Mean−squared−error

(a) MSE cost function.

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

0.5

1

1.5

2

2.5

Iterations

MS

E

Estimated mean−squared−error

(b) Estimated MSE cost function.

Figure 4.15: Comparison of MSE cost functions for system inversion problem.

v(n)u(n) FHs(n)

Figure 4.16: Blind equalization model.

4.4.2 Blind Equalization (Deconvolution)

In equalization, or system inversion problem, it is assumed that there is an accessto training; but in practice, this is generally not possible, that is, the desired signalcan not be observed at any time. This can be defined as a blind equalization (alsoreferred as deconvolution) problem, where “blind” is used to define the method sincethere is no information about the signal that is desired to be recovered. The blindequalization model is given in the Figure 4.16 with disregarding the presence ofnoise. Here, the objective is to obtain the equation:

v(n) = ±αs(n−∆), (4.23)

where α and ∆ is an arbitrary scale and delay respectively. The question here ishow to find F that satisfies Equation 4.23 without a model of H and any observationof s(n). One possible solution is minimum entropy deconvolution (MED) algorithmproposed by Wiggins [8].

Minimum Entropy Deconvolution (MED) Algorithm

Here, it is assumed that the source signal has an impulsive appearance, due to apossible defect on one of the elements of ball bearing. This impulsive occurrence willcause the pdf of the original signal to be heavy-tailed. In a realistic case where H is

26

Chapter 4. 4.4. Adaptive Solution

non-trivial, as argued by Donoho [14], the output after H should have a pdf which ismore Gaussian, due to the central limit theorem [10]: ”The probability distributionfunction of the sum of a large number of random variables approaches a Gaussiandistribution.” Since filtering is just sum of products, the central limit theorem alsoholds here. In addition to the disturbances and noise, this is one of the major reasonfor the original signal to be masked.

After those assumptions and information, the strategy can be to measure the spik-iness of v(n) using kurtosis and adapt F to maximize the impulsiveness of v(n),in order to restore the high kurtosis value. Thus, maximum spikiness can be con-cluded as maximum kurtosis, whereas minimum Gaussianity can be concluded as“minimum entropy”.

The Kurtosis of the zero-mean signal v(n) is defined as the normalized fourth mo-ment:

K(v) =E[v4]

(E[v2])2. (4.24)

For the adaptation algorithm for kurtosis maximization, as before, let:

Fn =[fn0 fn

1 . . . fnN−1

]T, (4.25)

and the observed signal:

u(n) =[u(n) u(n− 1) . . . u(n−N + 1)

]. (4.26)

So that, the output signal can be written as:

v(n) = F Tu(n). (4.27)

Therefore, kurtosis of the output signal is a function of kurtosis of the filter F. Then,using steepest ascent:

Fn+1 = Fn + µ∇K(F ). (4.28)

And, the gradient is defined as the derivative:

∇K(F ) =∂K(F )

∂F. (4.29)

After that the steepest ascent algorithm becomes:

Fn+1 = Fn + µ(E[v3u]

E[v2]−K(v)E[vu]). (4.30)

In a recent patent application from Rubico AB, an alternative approach for rollingbearings based on statistical asymmetry was proposed [9]. According to that ap-proach, Skewness can also be used for gradient estimate with the same method.Normalized third-order moment for zero-mean v(n) is given as:

27

4.4. Adaptive Solution Chapter 4.

s(n)

S(.)Learning Rule

v(n)FH

u(n)

Figure 4.17: Adaptive blind equalization with skewness maximization.

+H v(n)s(n) +x(n) u(n)

z(n)

F

Figure 4.18: Blind equalization model with noise component.

S(v) =E[v3]

(E[v2])3/2, (4.31)

taking the derivative will give the steepest ascent algorithm with skewness as objec-tive function:

Fn+1 = Fn + µ(E[v2u]− S(v)√

E[v2]E[vu]). (4.32)

During the derivations, any scalar can be absorbed in µ. Adaptive blind equalizationwith skewness maximization is shown in Figure 4.17.

With the same system for blind equalization problem with noise as given in Fig-ure 4.18, the performance of skewness and kurtosis as objective functions can becompared. Noise component is assumed to be symmetric, white and uncorrelatedwith x(n). The first advantage of third order moment is that the convergence formaximizing the objective function can be faster [7], which is also tested during theimplementation. The second advantage is, which is the most important one andalso discussed with details in [5] and [7], in the output v(n) after adaptive filtering,noise components including sinusoidal disturbances can be suppressed, whereas afterkurtosis maximization the output may be more noisy. Another drawback of kurtosisis that the algorithm may lock onto a strong sinusoid present in the measured sig-nal, since the kurtosis values of impulses and sinusoids are not easy to distinguish[5]. Therefore, using the skewness as objective function can enable the system toeliminate the interference and disturbances without any additional operation.

28

Chapter 4. 4.4. Adaptive Solution

2600 2700 2800 2900 3000 3100 3200 3300 3400 3500 3600

−3

−2

−1

0

1

2

3

Observed Signal u(n)

Sample(n)

Am

plitu

de

(a) Observed signal.

−4 −3 −2 −1 0 1 2 3 40

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

x

fx(x

)

PDF estimate for observed signal u(n)

(b) Pdf estimate for observed signal.

Figure 4.19: Observed signal with estimate of probability density function. Esti-mated kurtosis is 4.86 and estimated skewness is 0.1 for observed signal.

For the simulation of MED algorithm in floating-point environment, real data fromindustry, Data Set 3 given in Section 3.2 is used. MED algorithm is simulated withboth objective functions for fixed number of iterations. The observed signal with itsestimate of probability density function is shown in Figure 4.19. The output signalsafter filtering is given in Figure 4.20. There is not much difference in the outputin this case, noise and disturbances are eliminated and impulses are enhanced. Theestimates of probability density functions of both outputs are also given in Figure4.21. Note that the Gaussianity in the pdf of input signal is reduced in the outputpdf with MED algorithm with both objective functions.

29

4.4. Adaptive Solution Chapter 4.

2600 2800 3000 3200 3400 3600

−0.5

0

0.5

1

1.5

2

2.5

Output Signal v(n)

Sample(n)

Am

plitu

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(a) Output signal after kurtosis maximization.

2600 2800 3000 3200 3400 3600

−0.5

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Output Signal v(n)

Sample(n)

Am

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(b) Output signal after skewness maximization.

Figure 4.20: Output signals after MED algorithm. Estimated kurtosis is 34.12 andestimated skewness is 5.02 for output signal.

−1 −0.5 0 0.5 1 1.5 2 2.5 30

0.02

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x

fx(x

)

PDF estimate for filtered signal v(n)

(a) Pdf estimate of output signal after kurtosismaximization.

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fx(x

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PDF estimate for filtered signal v(n)

(b) Pdf estimate of output signal after skewnessmaximization.

Figure 4.21: Pdf estimate of output signals after MED algorithm.

30

Chapter 5

Implementation and Results

In this chapter, first the hardware platform for fixed-point application and the toolsfor software programming are given. Design rules and issues for fixed-point im-plementation follow before the results from all data sets are presented, both fromfloating point and DSP platform. Effect of different parameters such as numberof filter taps, adaptive step size, norm of the adaptive FIR filter and condition forbreaking the loop is discussed in Section 5.2. At the end, discussion about theconcept and results obtained are given.

5.1 Hardware and Tools

Digital signal processing can be described as processing of sampled, quantized sig-nals on a specialized hardware. This specialized hardware selected for this projectis a digital signal processor (DSP), which is a platform for implementing real- anddiscrete-time systems. As the Equation 4.7 states, digital filtering consists of addi-tions, multiplications and importing new data samples at each sampling instant. Adigital signal processor is a microprocessor specially designed for fast implementa-tion of these operations, such as, performing one multiplication, one addition, readtwo and write one value to memory at one core clock cycle.

The DSP platform used for the project is a Blackfin ADSP-BF533 DSP from AnalogDevices. It is a 16-bit fixed-point DSP with 148 Kbytes of on-chip memory, and ithas 600 MHz of core clock frequency. This is a general purpose high performanceprocessor, which is a good choice for research applications. Detailed informationabout the hardware can be found in Analog Devices’ home page [18] and hardwarereferences. Since the implementation of the algorithm mainly concerns about fixed-

31

5.2. Fixed-point Implementation Chapter 5.

point software development, analysis of hardware is beyond the scope of this project.

The fixed-point development of the algorithm was done by using EZ-KIT Lite eval-uation kit for ADSP-BF533. The evaluation board has the DSP, external memoriesfor DSP, analog-to-digital and digital-to-analog converters. Internal SRAM of DSPwas enough for the program instructions and the data set, and the recorded data isstreamed through that internal memory. Since the data is not streamed in real-time,there is no need to consider real-time implementation requirements and issues.

EZ-KIT Lite can be connected to the PC via USB, which allows downloading theprogram and controlling the DSP, by the specific software called VisualDSP++.VisaulDSP++ is the development environment for Analog Devices’ DSPs. Thesoftware program, written with C/C++ and Assembly language, can be edited andcompiled within this environment. It has many features that are used during theimplementation such as monitoring the memory, registers, flags and variables andplotting the data from memory.

5.2 Fixed-point Implementation

Implementation with floating point processors gives better accuracy and enablesthe algorithm to be implemented in an easier manner, compared to the fixed-pointprocessors. But the speed of such computations in floating point format requiresmuch more time for implementation, which is a very critical drawback in real-timesystems.

A fixed-point 16-bit DSP processor can operate only on integers, with differentformats. Most common way of representing signed binary numbers in fixed-pointformat is to use 1.15 fractional format, with one sign bit and 15 fractional bits.This approach gives the opportunity for very fast implementation of digital filteringalgorithms. But, it brings its own consequences, such as:

• The numbers represented in a fixed-point algorithm must lie within the range−1 ≤ d ≤ 1 − 2−15. In order to avoid overflow, the input data and/or filtercoefficients have to be scaled.

• This quantization of coefficients causes loss of accuracy, which can result un-desired errors in the output.

• The overflow is not the only problem, since there are only 16-bits for represent-ing the numbers, and since those numbers must lie between the range [−1, 1),multiplication of them results even in smaller numbers in magnitude. On the

32

Chapter 5. 5.2. Fixed-point Implementation

other hand, accumulation of certain amount of those small numbers can causeoverflow.

Solutions for the specific problems occurred due to those limitations in the imple-mentation process were proposed and tested. In the next section, the effects ofparameters such as number of taps, adaptive step size, norm of the adaptive filterand condition for breaking the loop are examined.

5.2.1 Parameters and Their Effects on the Algorithm

For the implementation and testing of the algorithm with the consideration of the-oretical results and better comparison of floating-point and fixed-point platform,fixed number of samples from measurement data were used. Step size and numberof iterations were also held fixed.

To achieve the best performance from the algorithm for industrial applications, thepossible modifications can be listed as follows:

• By streaming the input data to the Wiener filter for different number of filtertaps, the optimum filter length can be determined, if and only if there is anaccess to the original signal and an approximate model of distorting systemcan be designed. For the MED algorithm where there is no information aboutthe distorting system, optimum filter length can be determined by consideringthe criteria such as delay, sampling frequency and number of samples used.

• The step size can be recalculated in each iteration, to adapt to the gradientsearch. This adaptive step size, having an upper and a lower bound, canincrease the rate of convergence rapidly.

• The norm of the filter, which has an effect on the variance of the output, mustbe kept stationary around unity. Otherwise, the filter length can increase insampling instants and cause undesired errors in the output.

• Instead of breaking the loop after certain number of iterations, a more intuitiveapproach can be suggested. The skewness of the output can be calculatedin each iteration and be recorded for certain length after certain number ofiterations. Differentiating this vector to get the difference between consecutiveskewness values of the output would give an idea about the convergence. Ifthe skewness is not increasing significantly for more iterations, the loop canbe broken. Figure 5.1 is an example of convergence for skewness.

33

5.3. Results Chapter 5.

0 200 400 600 800 1000 1200 1400 1600 18000

0.5

1

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2

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3

3.5

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Iterations

Ske

wne

ss

Skewness vs Iterations

Figure 5.1: An example for convergence of skewness. After certain number of itera-tions, skewness will not be increasing significantly for more iterations.

5.3 Results

In this section, all the results from both floating-point and fixed-point implementa-tion will be given with figures of output, skewness values of data before and afteradaptive filtering and the error between the filter vectors with MSE criteria. Detailedinformation about the data sets are given in Section 3.2.

1. Data Set 1:

The observed signal and the estimate of probability density function is givenin Figure 5.2. This can also be assumed as a typical observed signal with aGaussian pdf, where the original impulsive signal is masked. The observedsignals and their distributions are similar. After enhancement of statisticalasymmetry of the input signal, the output from both floating point and fixed-point platforms are given in Figure 5.3. Measurements from one axis of theaccelerometer is used for all experiments.From the modulated characteristic of the impulses, it can be concluded thatthere is an inner race fault on the bearing. However, it was not possible todetect all three inner race defects. Since the algorithm aims to maximize skew-ness, it possibly combines the peaks close to each other. That can be given asa reason for this behavior. The MSE between the filter vectors from floating-point and fixed-point platform is calculated to be 1.96 × 10−4.

34

Chapter 5. 5.3. Results

0 500 1000 1500 2000 2500 3000 3500−1

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fx(x

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(b) Estimate for pdf of observed signal.

Figure 5.2: Observed signal and estimate of probability density function for DataSet 1.

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(b) Output signal from fixed-point platform.

Figure 5.3: Results obtained for Data Set 1. Estimated skewness is 0.05 beforefiltering, and 3.25 after filtering.

35

5.3. Results Chapter 5.

1100 1200 1300 1400 1500 1600 1700 1800 1900

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Figure 5.4: Results obtained for Data Set 2. Estimated skewness is 0.04 beforefiltering, and 1.83 after filtering. For a better view, samples between 1000 and 2000are given.

2. Data Set 2:

After enhancement of statistical asymmetry of the input signal, the outputfrom both floating point and fixed-point platforms are given in Figure 5.4. Theimpulses acquired after adaptive filtering are amplitude modulated. It can beconcluded that there is an inner race defect on the bearing. The observedsignal for this data set was already impulsive, therefore skewness was not in-creased significantly. Nevertheless, the statistical asymmetry is enhanced. TheMSE between the filter vectors from floating-point and fixed-point platform iscalculated to be 6.41 × 10−4.

3. Data Set 3:

After enhancement of statistical asymmetry of the input signal, the outputfrom both floating point and fixed-point platforms are given in Figure 5.5. Theperiodic impulses after running the algorithm is a sign of outer race defect.The disturbances are well suppressed, and impulsiveness is recovered. TheMSE between the filter vectors from floating-point and fixed-point platform iscalculated to be 4.67 × 10−5.

4. Data Set 4:

After enhancement of statistical asymmetry of the input signal, the outputfrom both floating point and fixed-point platforms are given in Figure 5.6.The data set is down-sampled by 2 for better analysis. From the outputsignal after filtering, it may be concluded that there are two inner race defects

36

Chapter 5. 5.4. Discussion

1100 1200 1300 1400 1500 1600 1700 1800 1900 2000

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(b) Output signal from fixed-point platform.

Figure 5.5: Results obtained for Data Set 3. Estimated skewness is 0.11 beforefiltering, and 3.43 after filtering. For a better view, samples between 1000 and 2000are given.

following each other. For a better result, algorithm can be implemented formore iterations to clear the impulses from disturbances. The MSE betweenthe filter vectors from floating-point and fixed-point platform is calculated tobe 1.68 × 10−4.

5. Data Set 5:

After enhancement of statistical asymmetry of the input signal, the outputfrom both floating point and fixed-point platforms are given in Figure 5.7.The data set is down-sampled by 4 for better analysis. The impulses in theoutput data are amplitude modulated, therefore it appears as if there is aninner race fault on the bearing. The MSE between the filter vectors fromfloating-point and fixed-point platform is calculated to be 1.29 × 10−4.

5.4 Discussion

The concept of implementing the algorithm on a fixed-point platform is proven.The impulses caused from defects on bearings had been successfully enhanced, byapplying the proposed algorithm on the data sets from industry. The procedureis straightforward, low-complex and low computational cost. The performance issatisfying on the fixed-point platform with short time of implementation, and theerror is considerably small. The dynamic range of 90 dB of a fixed-point DSP isproven to be enough for the desired output, however the trade-off between preventing

37

5.4. Discussion Chapter 5.

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Figure 5.6: Results obtained for Data Set 4. Estimated skewness is -0.21 beforefiltering, and 3.31 after filtering.

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Figure 5.7: Results obtained for Data Set 5. Estimated skewness is 0.03 beforefiltering, and 2.24 after filtering.

38

Chapter 5. 5.4. Discussion

the overflow and reducing the signal-to-quantization noise ratio is a challenge. Thealgorithm has a huge potential for condition monitoring purposes, even though thereis still a need for research and testing. For more realistic applications in industry, adecision making procedure has to be designed in order to let the user have an ideaabout the current condition of the ball bearing. In case the revolution speed can bean input to the system, the interesting frequencies can be estimated and the datacan be down-sampled accordingly, which can simplify the analysis.

The loss of accuracy caused from quantization of input data, filter vector and otherparameters did not effect the performance on the output. The key futures of theproposed algorithm are there is no need for priori knowledge about the disturbingsystem, and disturbances are suppressed without any additional filtering. Defectsfrom outer race and inner race are successfully distinguished in the output. But, thedefects on rotating elements are not investigated in this thesis work since they occurrarely. Possible solutions to the problems and improvements are suggested, and asa starting point in this long run of research, the algorithm is proved to be useful.

39

Chapter 6

Future Improvements andConclusion

6.1 Future Improvements

Possible future improvements on the algorithm for a final system to be used in indus-try can be suggested. For example, after the desired output is given by the adaptivefiltering, a decision making procedure is required. This procedure must be reliableand robust, and there is a potential for a complex algorithm which includes stochas-tic processes. The location of the defect can successfully be detected, but there isa need for detection of the size and characteristic of the defect which can be donewith future extraction algorithms, in order to tell how long the bearing can surviveunder certain varying conditions such as load, revolution speed and lubrication. Forthe real time implementation, the accelerometer, and other hardware devices mustbe chosen and designed for variety of different industrial environments. If possible,the vibration sensor must be placed in or near load zone for better performance onthe adaptive filtering side.

6.2 Conclusion

Beyond the proof of the concept that the algorithm can be implemented with fixed-point platform, the thesis work was a great experience how to learn, test and applyan algorithm in a real project going on within industry. Specification of the re-quirements were well described, the theory was investigated, improvements wereexamined, problems and possible solutions were suggested. From theory to prac-

40

Chapter 6. 6.2. Conclusion

tice, the thesis project made possible to achieve the knowledge of engineering workrequired for also different research and application areas. Searching for other ap-proaches to the same problem can let the designer to improve his point of view, andmakes the analysis more straightforward.

The presented algorithm, which is a novel time-domain approach, performed wellwith different data sets collected from industry. The ease of use, automatic suppres-sion of disturbances make it a promising algorithm for future. Interest from differentbranches of industry has already been noticed. The engineering side for selection ofthe hardware, such as an appropriate accelerometer, low-cost DSP and creating arobust end product for different environments has started. On the algorithm side,there is still a need for research on future extraction, and decision making.

As a result, whole education in signal processing had come to an end with a promis-ing project in applied statistical signal processing to be implemented directly inindustry for condition monitoring purposes. Ongoing research will hopefully createopportunities for more efficient production with less wasting of resources, which canbe depicted as a good step for technology.

41

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[2] N. Tandon and B. C. Nakra, “Comparison of vibration and acoustic measure-ment techniques for the condition monitoring of rolling element bearings”, Tri-bology International, vol. 25, no. 3, 1992.

[3] N. Tandon and A. Choudhury, “A review of vibration and acoustic measurementmethods for the detection of defects in rolling element bearings”, TribologyInternational 32, pp. 469-480, 1999.

[4] P. D. McFadden and J. D. Smith, “Model for the vibration produced by a singlepoint defect in a rolling element bearing”, Journal of Sound and Vibration, pp.69-82, 1984.

[5] Patrik Paajarvi, James P. LeBlanc, “Fault-impact enhancement using adaptivefiltering for condition monitoring of ball bearings”, The Seventh InternationalConference on Condition Monitoring and Machinery Failure Prevention Tech-nologies, 22-24 June 2010

[6] N. Sawalhi, R. B. Randall, H. Endo, “The enhancement of fault detection anddiagnosis in rolling element bearings using minimum entropy deconvolutioncombined with spectral kurtosis”, Mechanical Systems and Signal Processing,(21) pp. 2616-2633, 2007.

[7] Patrik Paajarvi, “Blind Equalization Using Third-Order Moments” LuleaUniversity of Technology, Department of Computer Science and Electri-cal Engineering, Division of Systems and Interaction, 2008:29—ISSN:I402-I544—ISRN:LTU-DT–08/29–SE

[8] Ralph A. Wiggins, “Minimum Entropy Deconvolution”, Computer-Aided Seis-mic Analysis and Discrimination. Geoexploretaion, vol. 16, pp. 21-35, 1978.

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[9] Patrik Paajarvi, James P. LeBlanc, “Method for Rolling Bearing Fault Detec-tion Based on Enhancing Statistical Asymmetry”, Swedish patent application,1000313-5, March 30, 2010.

[10] Peyton Z. Peebles, JR., Probability, Random Variables and Random SignalPrinciples, 4th Edition,Mcgraw-Hill series in electrical and computer engineering, ISBN 0-07-366-8

[11] Athanasios Papoulis, Probability, Random Variables and Stochastic Processes,2nd Edition,Mcgraw-Hill series in electrical engineering, ISBN 0-07-048468-6

[12] Sen M. Kuo, Woon-Seng Gan, Digital Signal Processors, Architectures, Imple-mentations and Applications,Pearson Prentice Hall 2005, ISBN 0-13-127766-9

[13] Bernard Widrow, Samuel D. Stearns, Adaptive Signal Processing,Pearson Prentice Hall 1985, ISBN 0-13-004029-1

[14] David F. Findley, Applied Time Series Analysis,Academic Press 1981, ISBN 0-12-256420-0

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