a wavelet-based rail surface defect prediction and ......a wavelet-based rail surface defect...

A Wavelet-Based Rail Surface Defect Prediction and Detection Algorithm

Brad Michael Hopkins

Dissertation submitted to the faculty of the Virginia Polytechnic Institute and State University in partial fulfillment of the requirements for the degree of

Doctor of Philosophy

In Mechanical Engineering

Saied Taheri, Chair

Mehdi Ahmadian

Robert West

Corina Sandu

Daniel Stilwell

March 26, 2012 Blacksburg, VA

Keywords: rail defect detection, Wavelet Transform, regularity analysis, artificial neural

network, wheel/rail dynamic modeling

A Wavelet-Based Rail Surface Defect Prediction and Detection Algorithm

Brad Michael Hopkins

ABSTRACT

Early detection of rail defects is necessary for preventing derailments and costly damage to

the train and railway infrastructure. A rail surface flaw can quickly propagate from a small

fracture to a broken rail after only a few train cars have passed over it. Rail defect detection is

typically performed by using an instrumented car or a separate railway monitoring vehicle. Rail

surface irregularities can be measured using accelerometers mounted to the bogie side frames

or wheel axles. Typical signal processing algorithms for detecting defects within a vertical

acceleration signal use a simple thresholding routine that considers only the amplitude of the

signal. As a result, rail surface defects that produce low amplitude acceleration signatures may

not be detected, and special track components that produce high amplitude acceleration

signatures may be flagged as defects.

The focus of this research is to develop an intelligent signal processing algorithm capable of

detecting and classifying various rail surface irregularities, including defects and special track

components. Three algorithms are proposed and validated using data collected from an

instrumented freight car. For the first two algorithms, one uses a windowed Fourier Transform

while the other uses the Wavelet Transform for feature extraction. Both of these algorithms use

an artificial neural network for feature classification. The third algorithm uses the Wavelet

Transform to perform a regularity analysis on the signal. The algorithms are validated with the

collected data and shown to out-perform the threshold-based algorithm for the same data set.

Proper training of the defect detection algorithm requires a large data set consisting of

operating conditions and physical parameters. To generate this training data, a dynamic wheel-

rail interaction model was developed that relates defect geometry to the side frame vertical

acceleration signature. The model was generated by using combined systems dynamic

modeling, and the system was solved with a developed combined lumped and distributed

parameter system numerical approximation. The broken rail model was validated with real data

collected from an instrumented freight car. The model was then used to train and validate the

defect detection methodologies for various train and rail physical parameters and operating

conditions.

iii

Dedication____________________________________

To God, for His faithfulness.

“Unless the Lord builds the house, they labor in vain who build it; unless the Lord guards

the city, the watchman keeps awake in vain. It is vain for you to rise up early, to retire late,

to eat the bread of painful labors; for He gives to His beloved even in his sleep”

– Psalm 127:1-2

iv

Acknowledgements __________________________

I thank God for loving me and blessing me, saving me and healing me, redeeming my life

and for being faithful always.

I thank Mom & Dad for their love, support, and encouragement throughout the years.

I thank Dr. Saied Taheri for serving as my advisor and teaching and guiding me over the

past years. I thank Dr. Mehdi Ahmadian for his wisdom and advice in the rail industry. I

thank Dr. Corina Sandu, Dr. Robert West, and Dr. Daniel Stilwell for serving on my

committee and giving me advice along the way.

I thank the Association of American Railroads (AAR) and the Transportation Technology

Center, Inc. (TTCi) for funding this project. I thank Dave Davis for managing this project,

Abe Meddah for supplying data, and Curtis Urban for supplying dynamic models.

I thank Dr. David Russell and Dr. Alireza Farjoud for their advice on numerical

simulation.

v

Contents _____________________________________

A Wavelet-Based Rail Surface Defect Prediction and Detection Algorithm .................................... i

Abstract ..................................................................................................................................................................... ii

Dedication ............................................................................................................................................................... iii

Acknowledgements ............................................................................................................................................. iv

Contents .................................................................................................................................................................... v

List of Figures ........................................................................................................................................................ ix

List of Tables ...................................................................................................................................................... xiv

Chapter 1: Introduction ...................................................................................................................................... 1

1.1 Motivation ......................................................................................................................................... 1

1.2 Research Approach ........................................................................................................................ 4

1.3 Document Outline ......................................................................................................................... 4

1.4 Research Contributions ............................................................................................................... 5

Chapter 2: Review of Literature ...................................................................................................................... 6

2.1 Introduction...................................................................................................................................... 6

2.2 Rail Defects and Defect Detection ............................................................................................ 6

2.2.1 Mechanisms of Derailment ....................................................................................... 7

2.2.2 Train Derailments due to Broken Rails throughout History ....................... 8

2.2.3 Methods of Rail Defect Detection ........................................................................... 9

2.3 Train and Rail Dynamic Modeling ..........................................................................................13

2.3.1 Lumped Parameter System Modeling ................................................................13

2.3.2 Distributed Parameter System Modeling ..........................................................17

2.3.3 Contact Mechanics and Wheel-Rail Interaction ..............................................22

2.3.4 Combined Systems Modeling .................................................................................23

2.4 Wavelet Analysis ...........................................................................................................................30

2.4.1 Continuous Wavelet Transform ............................................................................32

2.4.2 Multiresolution Formulation/ Signal Space Representation ....................33

2.4.3 Discrete Wavelet Transform ..................................................................................35

2.4.4 Filter Bank Representation/ Fast Wavelet Transform ................................36

vi

2.4.5 Local Signal Regularity Calculation using Wavelets .....................................37

2.4.6 Wavelet Families .........................................................................................................39

2.4.7 Wavelets in Event Detection Applications ........................................................42

2.5 Artificial Neural Networks ........................................................................................................49

2.5.1 Artificial Neural Network Basics ..........................................................................49

2.5.2 Neural Network Applications ................................................................................52

2.6 Summary and Conclusions .......................................................................................................54

Chapter 3: Experimental Setup and Data Collection .............................................................................57

3.1 Introduction....................................................................................................................................57

3.2 Experimental Setup ......................................................................................................................58

3.2.1 Sensors, Hardware, and Data Acquisition System .........................................58

3.2.2 Test Facility ...................................................................................................................59

3.3 Data Collection...............................................................................................................................62

3.3.1 Collected Signals and Initial Observations .......................................................62

3.3.2 Defect Signatures/ Signal Content .......................................................................64

3.3.3 Lateral Acceleration Signatures of Irregularities ...........................................68

3.4 Summary and Conclusions .......................................................................................................70

Chapter 4: Rail Defect Detection Algorithms ...........................................................................................72

4.1 Introduction....................................................................................................................................72

4.2 Development of Fourier Transform-Based Algorithm ..................................................73

4.2.1 Feature Extraction: Fourier Transform .............................................................73

4.2.2 Feature Classification: Artificial Neural Network ..........................................74

4.3 Development of Wavelet Transform-Based Algorithm .................................................82

4.3.1 Feature Extraction: Wavelet Transform ............................................................84

4.3.2 Feature Classification: Artificial Neural Network ..........................................88

4.4 Development of Wavelet-Intensity Factor Algorithm .....................................................89

4.5 Simulations and Results ..............................................................................................................91

4.5.1 Results: Threshold-Based Algorithm ..................................................................91

4.5.2 Results: Fourier Transform-Based Algorithm .................................................92

4.5.3 Results: Wavelet Transform-Based Algorithm ...............................................95

4.5.4 Results: Wavelet-Intensity Factor Algorithm ..................................................99

vii

4.6 Summary and Conclusions .................................................................................................... 101

Chapter 5: Dynamic Wheel-Rail Interaction Model ............................................................................ 104

5.1 Introduction................................................................................................................................. 104

5.2 Single-Wheel Model .................................................................................................................. 104

5.2.1 Free Body Diagrams ............................................................................................... 106

5.2.2 Equations of Motion ............................................................................................... 107

5.3 Solution to the System: Single-Wheel Model .................................................................. 108

5.3.1 Solution to the Train Subsystem........................................................................ 109

5.3.2 Solution to the Pad-tie-ballast Subsystems ................................................... 110

5.3.3 Solution to the Rail Subsystem ........................................................................... 111

5.3.4 Solution to the Combined Train-Rail System................................................ 115

5.4 Non-Linear Spring Rates: Single-Wheel Model .............................................................. 118

5.4.1 Application of Hertz Contact Spring ................................................................. 118

5.4.2 Non-linear Pad and Ballast Spring Rates ........................................................ 119

5.5 Simulations and Results: Single-Wheel Model ............................................................... 120

5.6 Broken Rail Model and Simulations ................................................................................... 124

5.6.1 Model of a Broken Rail .......................................................................................... 124

5.6.2 Broken Rail Simulation.......................................................................................... 129

5.6.3 Surface Fracture Simulation ................................................................................ 132

5.7 Development of Two-Wheel Model .................................................................................... 135

5.7.1 Free Body Diagrams and Equations of Motion ............................................ 135

5.7.2 Solution to the Two-Wheel Train-Rail Subsystem ..................................... 138

5.7.3 Calculation of Hertz Contact Stiffness ............................................................. 139

5.7.4 Modeling the Broken Rail for the Two-Wheel System .............................. 141

5.7.5 Effects of Loading from other Bogies ............................................................... 141

5.7.6 Broken Rail Simulation using the Two-Wheel Model ............................... 142

5.7.7 Surface Fracture Simulation using the Two-Wheel Model ...................... 144

5.8 Limitations of the Dynamic Wheel-Rail Interaction Model ....................................... 145

5.9 Effects of Varying Physical Parameters and Operating Conditions on Measured

Vertical Accelerations ........................................................................................................ 146

5.9.1 Effects of Car Payload ............................................................................................ 149

viii

5.9.2 Effects of Forward Speed ...................................................................................... 151

5.9.3 Effects of Rail Cross Section ................................................................................ 153

5.9.4 Effects of Rail Steel Stiffness ............................................................................... 155

5.9.5 Effects of Wheel Contact Point ........................................................................... 157

5.9.6 Observations and Conclusions from Parameter and Operating

Condition Varying Simulations ........................................................................ 159

5.10 Summary and Conclusions .................................................................................................. 163

Chapter 6: Physics-based Tuning of the Defect Detection Algorithm.......................................... 165

6.1 Introduction................................................................................................................................. 165

6.2 Physics-based Tuning of the Defect Detection Algorithm ......................................... 165

6.3 Effects of Filtering the Signals .............................................................................................. 172

6.4 Defect Detection for Simulated Responses ...................................................................... 177

6.5 Limitations of the Dynamic Model and Defect Detection Algorithm ..................... 180

6.6 Summary and Conclusions .................................................................................................... 184

Chapter 7: Conclusions and Future Work .............................................................................................. 186

7.1 Summary of Research .............................................................................................................. 186

7.2 Major Conclusions ..................................................................................................................... 187

7.3 Future Work ................................................................................................................................ 189

References .......................................................................................................................................................... 191

ix

List of Figures_________________________________

Figure 1.1. Train accidents in 2010 by (left) type, (right) cause ....................................................... 2

Figure 1.2. Broken rail accidents per million train miles for Class I mainline tracks ............... 2

Figure 2.1. TTCi Instrumented Freight Car towed by locomotive ..................................................10

Figure 2.2. Hi-rail vehicle equipped with ENSCO RailScan gage and crosslevel inspection

system ......................................................................................................................................................................11

Figure 2.3. Euler-Bernoulli beam model with beam element shown with internal and

external forces and moments acting on it .................................................................................................19

Figure 2.4. (left) example of Fourier transform basis. (right) example of wavelet transform

basis using the Daubechies length 3 wavelet ...........................................................................................32

Figure 2.5. Vector spaces spanned by the scaling function and wavelet function ....................35

Figure 2.6. Filter bank implementation of the wavelet transform .................................................37

Figure 2.7. Haar wavelet and scaling function .........................................................................................40

Figure 2.8. Daubechies wavelet and scaling functions for different support lengths. In each

title, dbd indicates the Daubechies family with d vanishing moments ..........................................41

Figure 2.9. Biorthogonal wavelet and scaling functions for different lengths ............................42

Figure 2.10. Three–layer neural network ................................................................................................49

Figure 3.1. TTCi Instrumented Freight Car: instrumented coal car (left) towed by

locomotive (right) ...............................................................................................................................................58

Figure 3.2. Front right bogie side frame of the IFC with location of accelerometer shown

with red arrow .....................................................................................................................................................59

Figure 3.3. Satellite image of the HTL from Google Earth ..................................................................60

Figure 3.4. Diagram of HTL with sections labeled based on track components tested in

those sections .......................................................................................................................................................60

Figure 3.5. Various track components corresponding to sections of HTL ...................................61

Figure 3.6. Bogie side frame vertical acceleration signals for two consecutive laps around

the HTL. The red dot marks the location of: (left) an impending rail break, (right) a broken

rail .............................................................................................................................................................................63

x

Figure 3.7. Bogie side frame vertical acceleration signals for two consecutive laps. (top)

before broken rail, (bottom) after broken rail ........................................................................................64

Figure 3.8. Vertical acceleration signals at the location of a surface fracture for six laps ......65

Figure 3.9. (top) Acceleration signature of the surface fracture from Lap 1. (bottom) Fast

Fourier Transform of the acceleration signature ...................................................................................66

Figure 3.10. (top) Acceleration signature of the rail break from Lap 5. (bottom) Fast


Figure 3.11. (top) Acceleration signature of the process noise/rail crossing. (bottom) Fast


Figure 3.12. Lateral acceleration signatures at the location of a turnout (end of section 8,

beginning of section 9) on the HTL ..............................................................................................................69

Figure 3.13. Lateral acceleration signatures at the location of a turnout (end of section 27,

beginning of section 28) on the HTL ...........................................................................................................70

Figure 4.1. Frequency content arranged into 6 bins as 6 inputs into the neural network.

From top to bottom: surface fracture, rail break, process noise ......................................................74

Figure 4.2. Visual representation of the design of a neural network classifier. The network

in this example has 6 inputs, 1 hidden layer, 5 neurons in the hidden layer,

and 1 output ..........................................................................................................................................................76

Figure 4.3. (top) Vertical acceleration pattern for a rail break. (bottom) Target output for a

rail break ................................................................................................................................................................81

Figure 4.4. Simulink model of defect detection and classification algorithm ..............................84

Figure 4.5. Block diagram of Wavelet-Intensity Factor algorithm...................................................91

Figure 4.6. Original signal processed by a threshold-based algorithm. Dashed lines show

threshold values for flagging of defects ......................................................................................................92

Figure 4.7. (top) Neural network output and desired (target) values for validation phase.

(bottom) Original vertical acceleration data with rail surface irregularities labeled. ............94

Figure 4.8. Zoomed in neural network output and desired (target) values for validation

phase: (top) process noise, (middle) surface fracture/impending break, (bottom) broken

rail .............................................................................................................................................................................95

Figure 4.9. Original signal fed into the defect detection algorithm .................................................96

Figure 4.10. CWT of original signal using a db3 wavelet .....................................................................97

xi

Figure 4.11. Thresholding of wavelet transformed signal ..................................................................98

Figure 4.12. Third step of the algorithm: classification using artificial neural network .........99

Figure 4.13. Results of the Wavelet-Intensity Factor-algorithm for round 1 of ENSCO V/TI

monitor data. (top) original signal, (second) local Lipschitz exponent/signal regularity,

(third) intensity factor, (bottom) signature type, red bar indicates broken rail .................... 100

Figure 4.14. Results of the Wavelet-Intensity Factor-algorithm for round 2 of ENSCO V/TI

monitor data. (top) original signal, (second) local Lipschitz exponent/signal regularity,

(third) intensity factor, (bottom) signature type, red bar indicates broken rail .................... 101

Figure 5.1. Single wheel dynamic model for rail break simulations: distributed parameter

Euler-Bernoulli beam and discrete lumped parameter inputs across the beam .................... 105

Figure 5.2. Free body diagrams of single wheel dynamic model ................................................. 107

Figure 5.3. Rail deflection response for single wheel model simulation ................................... 122

Figure 5.4. Train subsystem body responses for single wheel model simulation ................. 123

Figure 5.5. Simulation convergence errors for single wheel model simulation ..................... 123

Figure 5.6. Diagram of broken rail model .............................................................................................. 124

Figure 5.7. Wheel driving over broken rail for small broken rail gap size g............................ 125

Figure 5.8. Geometric relationships between wheel and rails at location of broken rail for

small broken rail gap size g .......................................................................................................................... 126

Figure 5.9. Geometric relationships between wheel and rail at location of broken rail for

large broken rail gap size g .......................................................................................................................... 128

Figure 5.10. Left and right rail responses for broken rail simulation ........................................ 130

Figure 5.11. Train body responses for broken rail simulation...................................................... 131

Figure 5.12. Comparison of actual and simulated data for 102 mm broken rail vertical

acceleration signature .................................................................................................................................... 132

Figure 5.13. Left and right rail responses for surface fracture simulation ............................... 133

Figure 5.14. Train body responses for surface fracture simulation ............................................ 134

Figure 5.15. Comparison of actual and simulated data for surface fracture (g = 1 mm)

vertical acceleration signature ................................................................................................................... 135

Figure 5.16. Free body diagrams for the two-wheel model ........................................................... 137

Figure 5.17. Five step process for two-wheel model broken rail simulation .......................... 141

Figure 5.18. Loading of trailing axles of locomotive and IFC ......................................................... 142

xii


vertical acceleration signature from two-wheel model .................................................................... 144


vertical acceleration signature from two-wheel model .................................................................... 145

Figure 5.21. Train body responses for the control simulation ...................................................... 148

Figure 5.22. Side frame vertical acceleration response for the control simulation .............. 149

Figure 5.23. Side frame vertical acceleration response and FFT for the load-varying

simulations ......................................................................................................................................................... 151

Figure 5.24. Side frame vertical acceleration response and FFT for the speed-varying

simulations ......................................................................................................................................................... 153

Figure 5.25. Side frame vertical acceleration response and FFT for the rail section-varying

simulations ......................................................................................................................................................... 155

Figure 5.26. Side frame vertical acceleration response and FFT for the rail steel stiffness-

varying simulations ......................................................................................................................................... 157

Figure 5.27. Diagram of AAR-1B, 1:20 wheel profile showing three different contact

points .................................................................................................................................................................... 158

Figure 5.28. Side frame vertical acceleration response and FFT for the wheel contact point-


Figure 5.29. Side frame vertical acceleration response and FFT for the Hertz contact

stiffness-varying simulations ...................................................................................................................... 163

Figure 6.1. Simplified oscillating single-wheel model ...................................................................... 166

Figure 6.2. Frequency responses of side frame vertical acceleration for load varying

simulations ......................................................................................................................................................... 168

Figure 6.3. Frequency responses of side frame vertical acceleration for speed varying

simulations ......................................................................................................................................................... 169

Figure 6.4. Frequency responses of side frame vertical acceleration for rail section varying

simulations ......................................................................................................................................................... 169

Figure 6.5. Frequency responses of side frame vertical acceleration for rail stiffness


Figure 6.6. Frequency responses of side frame vertical acceleration for wheel contact point


xiii

Figure 6.7. Frequency responses of side frame vertical acceleration for contact stiffness


Figure 6.8. Frequency responses for load varying simulations. The FFT is shown for the

subsampled (256 Hz) response and the subsampled and filtered (30 Hz low pass)

response .......................................................................................................................................................... 173

Figure 6.9. Frequency responses for speed varying simulations. The FFT is shown for the

subsampled (256 Hz) response and the sub-sampled and filtered (30 Hz low pass)

response .............................................................................................................................................................. 174

Figure 6.10. Frequency responses for rail stiffness varying simulations. The FFT is shown

for the subsampled (256 Hz) response and the subsampled and filtered (30 Hz low pass)

response .............................................................................................................................................................. 175

Figure 6.11. Frequency responses for rail section varying simulations. The FFT is shown

for the subsampled (256 Hz) response and the subsampled and filtered (30 Hz low pass)

response .............................................................................................................................................................. 176

Figure 6.12. Frequency responses for wheel contact point varying simulations. The FFT is

shown for the subsampled (256 Hz) response and the subsampled and filtered (30 Hz low

pass) response .................................................................................................................................................. 177

Figure 6.13. Wavelet analysis on simulated signals .......................................................................... 179

Figure 6.14. Thresholded wavelet coefficients of simulated signals .......................................... 180

Figure 6.15. Wavelet analysis on surface fracture (g = 1 mm) simulated signal.................... 181

Figure 6.16. Thresholded wavelet coefficients of surface fracture (g = 1 mm) simulated

signal ..................................................................................................................................................................... 182

Figure 6.17. Wavelet analysis on surface fracture (g = 0.5 mm) simulated signal ................ 183

Figure 6.18. Thresholded wavelet coefficients of surface fracture (g = 0.5 mm) simulated

signal ..................................................................................................................................................................... 184

xiv

List of Tables _________________________________

Table 4-1. Center frequency values of the first 5 scales of the db3 wavelet for a signal

sampled at 256 Hz ..............................................................................................................................................87

Table 4-2. Threshold values at the first four scales of the CWT .......................................................88

Table 5-1. Values for Bi and the first four derivatives of Bi at the nodal points. ..................... 114

Table 5-2. Algorithm for solving for the combined system ............................................................ 116

Table 5-3. Physical parameter values used in wheel/rail interaction model ........................... 120

Table 5-4. Simulation parameter values ................................................................................................. 121

Table 5-5. Simulation parameter values for broken rail simulation ............................................ 130

Table 5-6. Lookup table for determining the value of ξ .................................................................... 140

Table 5-7. Physical parameter values used in the two-wheel model broken rail

simulations ......................................................................................................................................................... 143

Table 5-8. Simulation parameter values for two-wheel model broken rail simulation ....... 143

Table 5-9. Physical parameter values and operating conditions for the control set in the

parameter and operating condition varying simulations ................................................................ 147

Table 5-10. Simulation parameter values for the parameter and operating condition


Table 5-11. Simulation matrix for load varying simulations .......................................................... 150

Table 5-12. Simulation matrix for speed varying simulations ....................................................... 152

Table 5-13. Simulation matrix for rail cross section varying simulations ................................. 154

Table 5-14. Simulation matrix for rail steel stiffness varying simulations ................................ 156

Table 5-15. Simulation matrix for load varying simulations .......................................................... 158

Table 5-16. Results from parameter and operating condition varying simulations with

values of Hertz contact stiffness................................................................................................................. 161

Table 5-17. Parameters for Hertz contact stiffness-varying simulations .................................. 161

1

Chapter 1

Introduction ____________________________________

1.1 Motivation

According to the Federal Railroad Administration (FRA), the freight railroad industry in

the United States (US) generated $49 billion of revenue driving over 1.5 trillion ton-miles in

2009. Seven railroads met the Class I threshold of $378.8 million of freight revenue,

accounting for over 90 percent of the industry’s total revenue in 2009. Railroads comprise

41.2% of the total annual freight ton-miles in the US, followed by trucks (30.7%), water

(14.1%), and oil pipelines (13.6%). Transportation of coal accounts for 42% of rail tonnage

and 21% of rail revenue [1].

Railways spend approximately 40% of their annual revenue on maintaining, renewing,

and expanding their infrastructure. Railroads have decreased their annual train accidents

from 11.5 per million miles in 1980 to 3.2 per million miles in 2007, a 71% decrease over

27 years. Railroads also have the lowest employee injury rates of all the transportation

industries in the US at less than 2 lost workdays per year per 100 employees [2].

In 2010, there were over 703.4 million miles travelled by trains in the US. There were a

total of 1,859 accidents, with 1,307 of the accidents being derailments and 659 of the

accidents being caused by track failures. Figure 1.1 shows the breakdown of the types and

causes of accidents in 2010 [3]. The charts show that derailments are the most prominent

type of accident and track failures are the leading cause of accidents.

2

8%

70%

22%

Train Accidents in 2010: Types

Collisions

Derailments

Other

36%

33%

13%

4%14%

Train Accidents in 2010: Causes

Track Causes

Human Factors

Equipment Causes

Signal Causes

Misc. Causes

Figure 1.1. Train accidents in 2010 by (left) type, (right) cause [3].

As the demand for freight transportation increases, the demand for rail safety increases

as well. Figure 1.2 shows a chart of the broken rail accidents per million train miles for

Class I mainline tracks from October 1998 to September 2010. The chart shows a continual

decrease in broken rail accidents per million train miles since the year 2004, a 65%

decrease from October 2004 to September 2010.

Figure 1.2. Broken rail accidents per million train miles for Class I mainline tracks [4].

2009 was the safest year to date for US railroads, breaking the record set in 2008. There

were 2.04 derailments per million train-miles, a drop of 77% since 1980 and a drop of 30%

since 2000. One of the main reasons for the continual increase in safety is the continual

increase in spending on rail maintenance. It was estimated that in 2009, over $3 billion

0

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was spent on rail by Class I Railroad Companies. Rail quality continues to increase, which

includes cleaner and harder steels, more effective grinding, and better welds. This has led

to more than doubling the life of rails since 1994. However, increased traffic over time

often requires more regular rail replacement. Additionally, early detection of potentially

dangerous rail defects is necessary for reducing accidents, injuries, fatalities, and expenses

[4],[5].

Rail defect detection is currently performed by using an instrumented train car [6], a

railway monitoring vehicle (usually a pickup truck with retractable wheels for driving on

the track), or an instrumented in-service train car [7]. The vehicles contain a wide range of

sensors, including video cameras, accelerometers on the bogies and car body, strain gauges

on the top chord, linear potentiometers in the spring nests, acoustic sensors, LIDAR, and

ultrasonic wave sensors. It has been found that long wavelength track irregularities are

best detected with car body accelerometers, top chord strain gauges, and suspension

displacement measurements, while short wavelength irregularities are best detected with

bogie side frame accelerometers, video cameras, LIDAR, and ultrasonic wave sensors

[6],[8]. In addition, a wide variety of data processing algorithms are in existence for

detecting defects from the sensor signals.

One method of detecting rail surface irregularities, such as rail head fractures and

broken rails, is to use accelerometers mounted to the bogie side frames or wheel axles.

This data is typically processed by applying some pre-determined threshold level to the

signal, and flagging any area of the signal that exceeds this threshold value as a defect. This

method typically performs well in detecting broken rails, but often flags high amplitude

portions of the signal that are not rail defects. Various special track components such as

joints, turnouts, and crossings can generate high amplitude vertical accelerations in the

axles and side frames. Current threshold-based algorithms have no way of distinguishing

between the signatures generated from special track components and rail defects, and

therefore often require unnecessary visual inspection of these areas of track. There is

therefore a need for a more intelligent defect detection and classification algorithm that is

capable of distinguishing between the signatures generated from various rail surface

irregularities. The development and application of such an algorithm is the focus of this

research.

4

1.2 Research Approach

The approach to developing an intelligent rail defect detection algorithm is as follows.

First, data is collected from an instrumented freight car driven around a test track. The

instrumented freight car is equipped with accelerometers on the bogie side frame

measuring vertical acceleration. A thorough visual inspection of the track is performed so

that there is a correlation between rail surface irregularities and defect type or special

track component type. The data is then observed and a frequency analysis is performed on

selected signatures of interest to determine whether or not there is enough information

located in the signal to distinguish between various rail surface irregularities. Rail defect

detection and classification techniques are then developed and applied using frequency

domain feature extraction and pattern classification techniques. The developed algorithms

are then applied to the collected data and the results are compared to those produced from

a typical threshold-based algorithm. Lastly, a dynamic wheel-rail interaction model is

developed to support the defect detection algorithm. Since actual data is time consuming

to collect, there is a limited amount of data available to train the algorithms. The

algorithms therefore have limited capability to classify signals representing a wide range of

physical parameters and operating conditions. A dynamic wheel-rail interaction model is

therefore developed and used to generate artificial training data spanning a wide range of

operating conditions, including various forward speeds, payloads, and rail surface

irregularity types and sizes.

1.3 Document Outline

The remainder of this document is organized as follows. Chapter 2 presents a review of

literature relevant to this research. Topics include mechanisms of train derailment, current

rail defect detection methods, dynamic modeling and solutions, wavelet theory, and

classification techniques using artificial neural networks. Chapter 3 presents the

experimental setup and data collection methods used in this research. Preliminary analysis

is performed on the collected data and initial observations are noted. In Chapter 4, three

frequency-based defect detection algorithms are developed and applied to the collected

data. The first algorithm uses the Fourier Transform for feature extraction and an Artificial

5

Neural Network (ANN) for feature classification. The second algorithm uses the Wavelet

Transform for feature extraction and an ANN for feature classification. The third algorithm

uses the Wavelet Transform to perform a regularity analysis on the signal and then

calculates an Intensity Factor that can be related to defect severity. The results from the

algorithms are compared to the results from a standard threshold-based algorithm.

Chapter 5 presents the development of a dynamic wheel-rail interaction model that can be

used to generate artificial training data for the defect detection algorithms. Chapter 6

presents methodologies for imbedding the physics of the wheel-rail system into the defect

detection algorithm. Chapter 7 presents the final conclusions and discusses the future

work for the completion of this research.

1.4 Research Contributions

This research has resulted in the following contributions to the field of science:

• New algorithms and processes for detecting rail defects from a vertical acceleration

signature that outperform current commercial defect detection algorithms.

• A new dynamic wheel-rail model that details the vertical dynamics of the wheel-rail

interface during enveloping of a broken rail or rail surface fracture.

• A new physical parameter based algorithm tuning process to tune the defect

detection algorithm based on train/wheel/rail physical parameters.

6

Chapter 2

Review of Literature ________________________

2.1 Introduction

This chapter contains a review of some notable literature that can be used to help solve

the problem of developing an intelligent rail defect detection system. In Section 2.2, the

current state of rail defects and rail failure rates is discussed, along with some methods that

are currently used in the industry to detect rail defects. Section 2.3 contains a discussion

on some dynamic modeling methods that can be used to perform train-track interaction

simulations. Section 2.4 contains an overview of wavelet theory, which is a powerful

multiresolution signal processing technique that has been used for event detection in this

research. Section 2.5 gives a presentation of neural networks and their use in classification

applications, which is helpful to addressing the current problem of locating rail defects

within a signal.

2.2 Rail Defects and Defect Detection

In order to develop an intelligent defect detection algorithm and derailment prevention

control strategy, the mechanisms and causes of derailments must be studied. This section

discusses the mechanisms of train derailment and takes a look at some major train

derailments caused by broken rails in the United States throughout history. The current

methods of rail defect detection used in the industry are also discussed.

7

2.2.1 Mechanisms of Derailment

Derailments caused by a lack of lateral guidance between the wheel and rail occur in

four primary mechanisms, which are: wheel flange climb, gauge widening, rail rollover, and

track panel shift. Wheel flange climb occurs when a wheel climbs on top of the rail head

and proceeds to run off the rail. Gauge widening occurs when there is a lateral shift in one

or both of the rails. Rail rollover occurs when a rail rolls over, causing a lateral shift in the

wheel set and subsequent loss of contact between the opposite rail and wheel. Track panel

shift occurs when an entire rail tie shifts with respect to the supporting ballast [9].

Wheel flange climb occurs for high lateral force to vertical force ratios (L/V), which is

usually the result of a large wheelset angle-of-attack. Large wheelset angle-of-attack can be

caused by a misaligned bogie or various types of track irregularities. There are many

different criteria for determining the threshold at which flange climb will occur. One that

utilizes the L/V ratio is the Nadal single-wheel L/V limit criterion. Nadal’s criterion for

wheel flange climb is:

�� = tan � − 1 + tan � (2.1)

where � is the angle of the resultant force acting on the wheel flange from the rail and is

the coefficient of friction between the wheel and the rail. Assuming maximum contact

angle, this equation gives the minimum value of L/V for which flange climb will occur.

A simple geometric criterion for determining whether or not a wheel will drop due to a

shifted rail is given by:

� + + �� − � > � (2.2)

where B is the space between the backs of the wheels on an axle, W is the wheel width, fw is

the flange width, G is the rail gauge, and S is the overlap of the rail and the wheel that could

potentially drop. B and W will typically stay constant, therefore derailment will either

occur for a decrease in fw (worn flange) or an increase in G (gauge widening).

The geometric criterion for a rail rolling over is given by the equation:

�� = �ℎ (2.3)

where d is the lateral distance between the wheel-rail contact point and the pivot point at

the base of the rail, and h is the vertical distance between the wheel-rail contact point and

8

the pivot point. Equation (2.3) assumes that there is no counter moment from the rail

fasteners or the torsion stiffness of the rail.

Track panel shift can be caused by loose or weakly supported ballast, or track with poor

lateral resistance such as newly laid or poorly maintained track. A criterion given by

French National Railroads for the limiting lateral axle load for preventing track panel shift

is:

�� = �� + � (2.4)

where Lc is the critical lateral load, V is the vertical axle load, and a and b are various values

depending on how well the ballast and ties are compacted [9].

Broken rails can play a large role in train derailments. A break in a rail causes a loss of

torsional and lateral stiffness at the break location. This will result in rail shift or rail

rollover at lower L/V values. For a large rail separation due to a break this can also mean

very little or no lateral guidance for the wheel at the location of the break, resulting in a

near instant derailment.

2.2.2 Train Derailments due to Broken Rails throughout History

Below is a list of some train derailments due to broken rails that have occurred in the US

within the last 100 years:

August 25, 1911: two cars from a 14 car train derailed near a bridge in Manchester,

New York. The cars fell about 45 feet into the stream, killing 29 people and injuring 62

others [10].

August 4, 1944: an Atlantic Coast Line passenger train derailed in Stockton, Georgia,

killing 47 and injuring 38. The break occurred in a section of rail that was

manufactured in 1927, but laid in 1943 [11].

August 5, 2010: a suspected broken rail caused 13 cars from the 79 car train to leave

the tracks of a CSX railroad in the Howard Street Tunnel in Baltimore, Maryland. This

was the same site of the July 18, 2001 accident that caused a fire in the tunnel. The

exact cause for this accident was not determined, but was believed by some to be the

result of a broken rail [12].

9

March 7, 2010: 30 cars from a 134 car coal train were derailed on a Union Pacific

Railroad heading from Wyoming to Illinois. There were no injuries [13].

May 9, 2011: an Amtrak train derailed in the East River Tunnel in New York causing

extensive damage to the train and rail [14].

2.2.3 Methods of Rail Defect Detection

The goal of reducing the number of derailments caused by broken rails has been

addressed in many different ways. One of these methods is the increase in the quality of

rail steel. Harder steels lead to longer life and decrease the probability of failure. However,

as the quality of steel has increased over time so has the amount of traffic on railroads.

Failure of rails seems nearly inevitable, and unfortunately, improved quality of materials

has not eradicated the problem of broken rails. Early detection of rail flaws is necessary for

fixing or replacing the track before it becomes a serious problem. Many different

techniques have been researched and implemented for defect detection in rails. Defect

detection requires a vehicle, a sensor suite, a data acquisition system, a signal processing

routine for extracting events from the data, and a means for properly notifying the

necessary parties in the event of a defect.

Transportation Technology Center, Inc. (TTCi) owns an instrumented freight car (IFC)

for track inspection [6]. The IFC is towed by a locomotive, as shown in the photograph in

Figure 2.1. The vehicle is equipped with various sensors for locating both long and short

wavelength rail defects generated under heavy axle loads. Top chord strain gauges, vertical

car body accelerometers, and suspension displacement transducers are used for

determining the bounce and pitch motions of the vehicle, which can be related to long

wavelength track geometry degradation. The top chord strain gauges and suspension

displacement transducers are also beneficial for detecting twist and roll motions. Vertical

side frame accelerometers are used to detect short wavelength irregularities, including

broken rails. The IFC is also equipped with a Global Positioning System (GPS) sensor that

stamps each piece of collected data with longitude and latitude coordinates. The onboard

sensor set and data acquisition system is powered by solar panels and axle generators.

10

The IFC onboard system also consists of software that analyzes the data in real-time and

generates an exception report that is communicated to the office using cellular phone

technology. Exceptions are generated when the magnitude of the sensor signal exceeds

some pre-determined threshold value. The threshold value for each sensor is determined

by the statistical analysis of data over long periods of time correlated with known locations

and types of defects from a visual inspection of the track. For example, if the suspension

displacement transducer has a value greater than 0.5 inches or less than -0.5 inches, the

software will flag this location as a long wavelength track irregularity and calculate the

length of the irregularity. The office will then be notified immediately of the type, size, and

location of the defect.

Figure 2.1. TTCi Instrumented Freight Car towed by locomotive.

A sample exception report generated from the IFC onboard software was presented in a

technical paper written by Li [6]. During a night of testing, the software flagged six

locations that it considered to be short wavelength exceptions. Upon visual inspection

following the tests, it was found that three of these locations were broken rails and the

other three were the locations of turnouts and rail joints. One of the primary findings from

this study concerning broken rails was that there were changes in the IFC dynamic

response due to broken weld and broken rail propagation that would require improved

signal processing and further software development for accurate detection of impending

11

rail breakage and the ability to distinguish between the signatures generated from a

broken rail and special track components.

ENSCO, Inc. is a privately owned engineering company that designs, builds, and operates

a number of track inspection vehicles. These include self-propelled vehicles, towed

coaches, and hi-rail vehicles. Self-propelled vehicles and towed coaches have the

advantage of applying wheel loads equivalent to actual operating conditions. Hi-rail

vehicles utilize a truck with retractable wheel sets for driving on both railroads and paved

roads and therefore have the advantage of being cheaper and easier to operate [15]. Figure

2.2 shows a photograph of a hi-rail vehicle equipped with an ENSCO RailScan gage and

crosslevel inspection system. The RailScan system can be easily implemented on a hi-rail

vehicle and plugs directly into a laptop for real-time gage and crosslevel monitoring.

Figure 2.2. Hi-rail vehicle equipped with ENSCO RailScan gage and crosslevel inspection system

[16].

The ENSCO Vehicle/Track Interaction (V/TI) Monitor utilizes tri-axial accelerometers

for detecting rail surface defects. The system can be mounted to any revenue train and

therefore does not require a separate rail inspection vehicle for defect detection.

Additionally, the system operates autonomously and does not require a dedicated crew.

The onboard signal processing routine uses pre-determined threshold values to determine

whether or not a rail surface irregularity is present. If a rail defect is detected, the V/TI

system uses wireless networks to transmit data in real-time so that proper personnel are

notified via email, cell phone, text message, or pager so that damage can be quickly tended

to. The V/TI system is compatible with TrackIT, which is a web-based application that

12

allows summary reports to be viewed in a web browser. The exceptions and their

waveforms can be viewed and analyzed offline, and can also be exported to Google Maps to

view the exact location of the exception. There are currently over 200 V/TI Monitors on

trains throughout North America, travelling on Class I Railroads, Amtrak high speed lines,

and regional and local commuter routes [17].

ENSCO track imaging systems provide non-contact methods for monitoring rail health by

using high resolution video cameras and machine-vision processing. The system is capable

of monitoring a number of track components, including joint bars, ties, fasteners, and rail

heads. The system software automatically identifies track components of interest,

including ties, fasteners, and ballast. These images can then be sent to the office for further

review. This nearly eliminates the need for visual inspection, saving time and increasing

safety [18].

The Federal Railroad Administration (FRA) owns a fleet of track inspection vehicles that

are a part of their Automated Track Inspection Program (ATIP). The fleet consists of five

track geometry cars that are used to assess track safety in the rail industry by comparing

collected data with the Federal Track Safety Standards. The track inspection vehicles are

both towed and self-propelled train cars. The capabilities include: gage measurement and

alignment, Gage Restraint Measurement System (GRMS), rail profile measurements for

wear and cant assessment, VT/I monitors for measuring truck and car body accelerations,

and commercial grade GPS systems for accurate correlation of track geometry

measurements with actual location. In 2010, four of the five track inspection vehicles

averaged 100 days of use each for the year, while one of the vehicles was not used. This

resulted in over 83,000 miles of track surveyed with 24.8 exceptions per 100 miles

reported [19].

The BNSF Derailment Prevention and Resource Protection Solutions Program uses three

track geometry cars and four hi-rail vehicles to monitor the conditions of their tracks. The

three track geometry cars test at a rate of 150,000 track miles per year. Their busiest

routes are inspected daily and most of their other major routes are inspected at least four

times a week. Their track inspection vehicles are equipped with lasers for measuring track

geometry and ultrasound transceivers for detecting internal rail flaws. They also lease

three Holland TrackStar track inspection vehicles to determine track strength on branch

13

lines, sidings, and yard. BNSF uses the data collected from these vehicles with their

developed prediction models to assess the life of rails and predict rail fatigue and failure

[20].

The CN Track Evaluation SysTem (TEST) car is a self-propelled track inspection car

equipped with a multitude of sensors for measuring and analyzing track geometric

imperfections. Among these sensors are high definition video cameras used for identifying

defective track components and recording right-of-ways. The TEST car analyzed over

64,000 track miles in 2009. CN also has several ultrasonic rail flaw cars that are used to

detect internal rail flaws that could potentially lead to broken rails. These cars tested

158,768 miles of track in 2009. CN has also implemented the Track Inspection System, in

which all mainline tracks are inspected visually. The inspectors are able to record the

results of their inspections electronically. For recorded defects, supervisors are

automatically alerted so that proper measures can be taken [21].

2.3 Train and Rail Dynamic Modeling

In this section, several methods for modeling train and track dynamics will be presented.

These methods include the lumped parameter approach for modeling train body dynamics,

distributed parameter methods for modeling rails, contact mechanics theory for modeling

wheel-rail interaction, and combined systems modeling for solving for the coupled train-

track dynamic responses. The following sections will contain a presentation of some

notable literature in the previously mentioned areas.

2.3.1 Lumped Parameter System Modeling

A train can be modeled as a multi-body system with a finite number of degrees-of-

freedom. This method of dynamic modeling is called lumped parameter systems modeling.

Each body in the system is treated as a rigid mass and is assigned a certain number of

degrees-of-freedom. Each rigid mass can have up to six degrees-of-freedom: three

translational and three rotational. Depending on the desired dynamic study of the system,

different degrees-of-freedom are assigned to each body. Of course, the most accurate

model would include as many degrees-of-freedom for each body as are permitted by the

14

physical setup of the system. However, often times the translational and/or rotational

motions in a particular direction are small and can therefore be omitted for simplicity. For

example, if a study is being performed on the response of an automobile suspension while

driving in a straight path at a constant forward speed over a divot in the road, the vertical

and pitch motions would likely be the only two degrees-of-freedom assigned to the vehicle

model. Roll, yaw, longitudinal, and lateral motions of the vehicle will be relatively constant,

i.e., steady state, and can therefore be omitted [22],[23].

Modeling of multiple degree-of-freedom systems is well developed and there are many

different techniques in existence to solve for these systems [22],[23]. The two most

common methods for developing the equations of motion for such systems are Newton’s

second law of motion and Lagrange’s energy equations. Newton’s method simply applies a

force balance on each translational degree-of-freedom and a moment balance on each

rotational degree-of-freedom for every body in the system, which is represented in the

following two equations:

�� = �� (2.5)

�� = !�� (2.6)

where fxi is a force acting in the x direction, m is the mass of the object, M0i is a moment

acting about point 0 with rotational motion given by θ, and J is the moment of inertia of the

object. Lagrange’s method utilizes the concept of conservation of energy to derive the

equations of motion for a system. Lagrange’s energy equation for an undamped system is:

��" #$%$&'�( − $%$&� + $)$&� = 0 (2.7)

where T is the total kinetic energy of the system, U is the total potential energy of the

system, t is the time variable, and q is a degree-of-freedom for the system. Equation (2.7)

must be used for each degree-of-freedom in the system. Additional terms can be added to

this equation to account for energy losses due to viscous damping.

For a multi-body system, both Newton’s and Lagrange’s methods will yield a system of

equations. There should always be as many equations of motion as there are degrees-of-

freedom in the model. These systems can be solved with a number of different techniques.

15

If the forcing inputs into the system are harmonic then they can be represented with simple

equations, and a closed form solution to the problem can be found. However, if the forcing

inputs are non-deterministic then the system can be solved with various numerical

integration approaches. To derive some basic equations for arriving at a numerical

approximation of a system’s response to a forcing input, consider a dynamical system of the

form:

�+� + ,+' + -+ = �./"0+/00 = +�+' /00 = +' � (2.8)

where M is the [3 × 3] mass matrix containing the body masses and moments of inertia, C

is the [3 × 3] damping matrix containing translational and rotational damping coefficients,

K is the [3 × 3] stiffness matrix containing translational and rotational stiffness coefficients,

B is the [3 × �] forcing function coefficient matrix, x is an [3 × 1] vector containing the

translational and rotational degrees-of-freedom, F(t) is an [� × 1] arbitrary forcing input,

x0 is the [3 × 1] initial position vector, and +' � is the [3 × 1] initial velocity vector. The

number of degrees-of-freedom, and hence, number of equations, is denoted by 3, and the

number of forcing inputs into the system is denoted by �.

Equation (2.8) represents a system of second order differential equations. This equation

can be rearranged into a system of first order differential equations so that the equations

are easier to solve. This is accomplished by defining a [23 × 1] state vector:

7/"0 = 879/"07:/"0; = 8+/"0+' /"0; (2.9)

An equation can be written representing the new system of first order differential

equations:

7' /"0 = <7/"0 + =/"0,7/00 = 7� (2.10)

where A is the [23 × 23] state matrix defined by:

< = ? 0 @−�A9- −�A9,B (2.11)

and:

=/"0 = 8 0�A9�./"0;,7�/"0 = 879/007:/00; = 8+/00+' /00; (2.12)

where the superscript -1 denotes a matrix inverse, and I is an [3 × 3] identity matrix.

The numerical solution to Equation (2.10) can be obtained by using the Euler method,

which is:

16

7/"�C90 = 7/"�0 + ∆"<7/"�0 + =/"�0 (2.13)

where ∆" is the time step and ti is the time value " = E∆". When using a numerical solution,

two sources of error will arise. The first is called formula error, which is the difference

between the exact solution (obtained from the closed-form solution, if possible) and the

solution obtained from the numerical approximation. The second type of error is called

round-off error, which is the result of decimal round-off due to computer architecture. For

the Euler formula of Equation (2.13), the formula error accumulates as the time step

increases since the value at each time step is dependent on the value from the previous

time step. The formula error can therefore be reduced by decreasing the step size.

Another approach to arriving at a numerical solution to Equation (2.10) is the Runge-

Kutta method, which produces less formula error than the Euler method without having to

reduce the step size. A Runge-Kutta formulation for the first-order problem �' =�/�, "0, �/00 = ��, where f is any scalar function, is:

�FC9 = �F + ∆"6 /HF9 + 2HF: + 2HFI + HFJ0 (2.14)

where:

HF9 = �/�F, "F0

HF: = � #�F + ∆"2 HF9, "F + ∆"2 (

HFI = � #�F + ∆"2 HF:, "F + ∆"2 (

HFJ = �/�F + ∆"HFI, "F + ∆"0

(2.15)

The accuracy of Equation (2.14) can be increased by adjusting the value of ∆" at each time

step according to the speed at which the value �FC9 is changing. If �FC9 is changing rapidly,

then the value of ∆" can be decreased to improve the accuracy of the solution, and if �FC9 is

not changing rapidly then the value of ∆" can be increased to decrease the computation

time of the simulation. Equation (2.14) can be put into matrix form to accommodate a

system of equations.

Matlab software contains two Runge-Kutta based ordinary differential equation solvers:

ode23 and ode45. The ode23 function uses a second- and third-order pair of equations and

the ode45 function uses a fourth- and fifth-order pair of equations to solve systems of

17

differential equations like those in Equation (2.10). The ode23 function is a more efficient

solver but ode45 provides greater accuracy [24].

2.3.2 Distributed Parameter System Modeling

In order to model the dynamics of a rail accurately, the rail must be treated as an infinite

degree-of-freedom system. The lumped parameter techniques discussed in the previous

section are sufficient for modeling train car dynamics since the different components in a

train car can be treated as rigid bodies. The spring-like and damping components that

connect these bodies dominate the dynamic response as compared to the spring-like and

damping properties of the bodies themselves. However the spring-like and damping

behavior present from the material properties of the rail plays a significant role in its

dynamic response. The rail must therefore be treated as a spatially continuous, i.e.

distributed parameter, system. A popular method for solving for the bending motion or

transverse vibrations of a beam is application of Euler-Bernoulli beam theory, which is

shown in the work of Inman [23] and will be outlined in the following paragraphs.

Figure 2.3 shows a diagram of a cantilevered beam with transverse vibration considered,

with shear deformation and rotary inertia of the beam neglected. For the simplicity of the

derivation of the equations of motion, it is assumed from the beginning that the beam has

spatially constant cross sectional area A, Young’s modulus E, area moment of inertia about

the z axis I, and density ρ. The beam has width hy, thickness hz, and length l. The bending

deflection of the beam in the y direction for time t and location along the length of the beam

x is denoted by w(x,t). Consider also an external time-varying transverse force per unit

length applied along the beam, denoted by f(x,t). Taking a small element of the beam that

has length dx and applying a forced balance in the y-direction gives:

#�/�, "0 +$�/�, "0$� ��( − �/�, "0 + �/�, "0�� = K<�� $:L/�, "0$": (2.16)

where V(x,t) is the shear force at the left end of the element, V(x,t) + Vx(x,t) is the shear

force at the right end of the element, and K<�� is the mass of the element. The small shear

deformation assumption is valid for l/hz ≥10 and l/hy ≥10.

Next, summing the moments about the z axis acting on the small element gives:

18

#�/�, "0 +$�/�, "0$� ��( − �/�, "0 + #�/�, "0 +$�/�, "0$� ��( ��

+/�/�, "0��0 ��2 = 0

(2.17)

where M(x,t) is the moment at the left end of the element and M(x,t) + Mx(x,t) is the

moment at the right end of the element. The right-hand side of the equation is zero

because rotary inertia is assumed to be negligible. Since dx is assumed to be very small, dx2

can be assumed to be zero. The equation is therefore simplified to:

�/�, "0 = −$�/�, "0$� (2.18)

Substituting this into Equation (2.16) gives:

− $:$�: M�/�, "0N�� + �/�, "0�� = K<�� $:L/�, "0$": (2.19)

From mechanics of materials, the bending moment is related to the beam deflection with

the equation:

�/�, "0 = O@ $:L/�, "0$�: (2.20)

where EI acts as the effective stiffness of the beam. Substituting this equation into

Equation (2.19) and dividing by dx gives:

K< $:L/�, "0$": + O@ $JL/�, "0$�J = �/�, "0 (2.21)

For the case of free vibration, the external force is equal to zero and the equation can be

simplified to:

$:L/�, "0$": + P: $JL/�, "0$�J = 0P = QO@K< (2.22)

19

Figure 2.3. Euler-Bernoulli beam model with beam element shown with internal and external forces and moments acting on it.

To arrive at a closed form solution for the vibration w(x,t) of the beam, there must be

two initial conditions specified because of the second order derivative with respect to time

t, and there must be four boundary conditions specified because of the fourth order

derivative with respect to the spatial variable x. The boundary conditions can be put into

equation form by using the following: deflection = w(x,t), slope of the deflection =

∂w(x,t)/∂x, bending moment = EI∂2w(x,t)/∂x2, and shear force = ∂(EI∂2w(x,t)/∂x2)/∂x. As

an easy choice for boundary conditions, it can be determined which of these variables will

be equal to zero for each configuration. For a free end, bending moment and shear force

must equal zero; for a fixed end, deflection and slope must equal zero; for a simply

w(x,t)

x

y

z

f(x,t)

y

x

f(x,t)

V(x,t)

V(x,t) +

x x + dx

∂V(x,t)∂x

dx

M(x,t) +∂M(x,t)

∂xdx

M(x,t)

w(x,t)

20

supported or pinned end, deflection and bending moment must equal zero; and for a

sliding end, slope and shear force must equal zero.

To solve Equation (2.22), the method of separation of variables can be used by assuming

a solution of the form:

w(x,t) = X(x)T(t) (2.23)

where X(x) is the spatial solution and T(t) is the temporal solution. Substituting this into

Equation (2.22) gives:

P: RSSSS/�0R/�0 = −%� /"0%/"0 = T: (2.24)

The temporal equation is therefore:

%� /"0 + T:%/"0 = 0 (2.25)

which is known to have a solution of the form:

%/"0 = < sinT" + � cosT" (2.26)

where A and B are constants of integration determined from the initial conditions. The

spatial equation is:

RSSSS/�0 − YTP Z: R/�0 = 0 (2.27)

For convenience, the following is also defined:

[J = YTP Z: = K<T:O@ (2.28)

By assuming a solution to Equation (2.27) to be of the form Aeσx, the general solution will

be of the form:

R/�0 = �9 sin [� + �: cos [� + �I sinh [� + �J cosh [� (2.29)

where a1, a2, a3, a4 are constants of integration determined from the boundary conditions.

The natural frequencies of the system are denoted by ω and the mode shapes of the

system are denoted by X(x). Since the Euler-Bernoulli beam model assumes an infinite

number of degrees-of-freedom, the true solution to the problem will be an infinite sum of

mode shapes and natural frequencies. Assuming the mode shapes are orthogonal to one

another, it can be shown that the solution to the problem is:

L/�, "0 = �/<F sinTF" +�F cosTF"0RF/�0]F^9 (2.30)

21

where n is the mode index. In practice, an infinite sum of mode shapes is an unrealistic

solution to determine. Solving for the first few mode shapes can often yield an accurate

solution.

The choice of whether or not to use the Euler-Bernoulli model of Equation (2.22) is

dependent on the beam geometry and the desired accuracy. For long, thin beams, rotary

inertia and shear deformation can be neglected and the Euler-Bernoulli beam model can be

used.

Lu developed a computational model for random vertical vibration analysis of a vehicle-

track system [25]. This method models the rail as a periodic Euler-Bernoulli beam with

lumped parameter mass-spring systems at the boundaries. This infinite degree-of-freedom

Euler-Bernoulli beam is repeated several times to simulate the rail. This model showed

large improvements in the accuracy of the bogie response and wheel/rail forces as

compared to the rigid track model.

Manabe also used Euler-Bernoulli beam theory to model the rail for studies on the

vibration response due to multiple wheel sets rolling over a rail surface irregularity [26].

The track was treated as an Euler-Bernoulli beam of infinite length periodically supported

by lumped parameter systems comprised of two springs and a mass. The wheels were

modeled as single moving points. The results of the study showed increased vibration

responses due to wheelbase lengths that caused constructive interference with the

frequency response due to tie spacing and train speed.

Wu approached the problem of solving for the high frequency vibrations of a rail by

using a double Timoshenko beam model [27]. The rail was modeled by considering the

head and web to be one Timoshenko beam attached to the foot which was considered to be

another Timoshenko beam. The two beams were connected with a vertical spring. The use

of Timoshenko beams addressed the problem of cross-sectional deformation which had

been mostly ignored in previous high frequency rail models. The model is also much

simpler and quicker to solve than a finite element (FE) model. The double Timoshenko

beam model showed good agreement with measured data for vibrations up to about 2000

Hz. The model was also used to test the difference between discretely supported and

continuously supported rail models. Results showed that the discretely supported rail

model is much more accurate for the case of vertical vibration.

22

Wu also used Timoshenko beam elements to solve for the rail dynamics in a study that

explored the effects of non-linear pad and ballast models on wheel/rail impact forces [28].

The Timoshenko beam elements were used in a finite element (FE) model to solve for the

rail.

2.3.3 Contact Mechanics and Wheel-Rail Interaction

As discussed previously, the dynamics of a rail can be modeled as a distributed

parameter system and the dynamics of a train can be modeled as a lumped parameter

system. In some applications, it may suffice to assume a rigid connection between the

wheel and rail contact points [26]. For a detailed study of wheel-rail vertical dynamics,

however, the wheel-rail connection cannot be assumed to be rigid because the dynamics of

the wheel-rail interface are complex and tend to dominate the response of some train

components [29],[28]. The study of the interaction between two bodies is called contact

mechanics. There are many different approaches that can be taken when modeling the

dynamics of two bodies in contact with one another. Some of these approaches will be

discussed in the following paragraphs.

The best known contact model is the Hertz contact law, published in 1896 [30]. The

non-linear Hertz law gives the contact force between two spheres constructed of isotropic

materials:

_̀ = -�F (2.31)

where FN is the normal contact force between the two spheres, K is a generalized stiffness

constant dependent on the material of the spheres, � is the relative normal indentation at

the contact point of the two spheres, and the exponent n is a constant dependent on the

material of the two spheres. It is common to use n = 1.5 for metallic surfaces. The model

assumes no adhesion between the two surfaces, therefore FN will be zero for negative

values of �. Additional assumptions of the Hertzian law are: strains are within the elastic

limit, the contact area is much smaller than the radius of each sphere, and frictionless

surfaces.

Other contact-impact models have built on the general concept presented in the Hertz

contact law. Lankarani developed a contact force model that includes the effects of

23

hysteresis damping to the Hertz contact law. This method seeks to account for the energy

dissipation during impact. The equation is:

_̀ = -�F a1 + 3/1 − c:04 �'�'/A0e (2.32)

where e is the restitution coefficient, �' is the relative penetration velocity between the two

spherical objects, and �'/A0 is the initial relative impact velocity.

Dukkipati [29] utilized the non-adhesive non-linear Hertz contact model in Equation

(2.31) to model the wheel-rail interaction in a study on the steady state interactions

between a train and track. In a study on the effects of track non-linearity on wheel/rail

impact, Wu also employed the Hertz contact model [28]. The model allows for loss of

contact between the wheel and rail.

2.3.4 Combined Systems Modeling

When running a simulation, the train and rail systems must be solved for

simultaneously. If the train is modeled as a lumped parameter system and the track as a

distributed parameter system then the entire train-track system is called a combined

system. The methods presented previously will not suffice to solve such a system, so new

techniques must be developed. The Euler-Bernoulli beam with no forcing inputs from

Equation (2.22) has a closed form solution as given by Equation (2.30); however, the Euler-

Bernoulli beam with forcing inputs from Equation (2.21) must be solved in different ways

depending on the form of the forcing function and whether or not the forcing inputs are

deterministic.

Green functions have been used to determine a closed form solution for the free

vibration of a combined dynamical system [31],[22]. An example to illustrate this

technique is a cantilever beam with several spring-mass forcing inputs acting at single

points spaced along the beam. The equation of motion for an Euler-Bernoulli beam with

forcing function fi(t) acting at point hi on a beam of length l is:

O@L�� + K<Lff = ��/"0�/� − ℎ�0g�^9 (2.33)

24

where w is the transverse motion of the beam, the subscripts x and t denote a partial

derivative with respect to that variable of order equal to the number of times that variable

appears in the subscript, i is the lumped parameter system index, R is the total number of

lumped parameter inputs to the distributed parameter beam, and �/� − ℎ�0 is a Dirac

impulse at point x = hi. Continuing with the example, consider several masses attached

directly to the bottom of the beam at points x = hi and a spring attached directly to the

bottom of each of the masses at one end and fixed to a rigid support at the other end. The

equation of motion for each of the appended spring-mass systems is given by:

��h��/"0 + H�h�/"0 = −��/"0 (2.34)

where zi is the vertical motion of the i mass. It can therefore be written:

h�/"0 = L/ℎ�, "0 (2.35)

Combining Equations (2.33) and (2.34) gives:

O@L��/�, "0 + K<Lff/�, "0 = −�[��Lff/�, "0 + H�L/�, "0]�/� − ℎ�0g�^9 (2.36)

This equation can be solved by using separation of variables and assuming a solution of the

form w(x,t) = u(x)a(t), where u(x) is the spatial solution and a(t) is the temporal solution.

Substituting this into Equation (2.36) and rearranging terms gives:

O@iSSSS/�0 + [∑ H��/� − ℎ�0g�^9 ]i/�0[K< + ∑ ��/� − ℎ�0g�^9 ]i/�0 = −�� /"0�/"0 = T: (2.37)

The temporal solution will be of the form:

�/"0 = < sin/T"0 + � cos/T"0 (2.38)

The spatial equation is:

O@iSSSS/�0 + k�/H� − ��T:0�/� − ℎ�0 − K<T:g�^9 l i/�0 = 0 (2.39)

The solution to the spatial equation can be arrived at by using the Green function g(x,m).

The Green function for a beam satisfies:

nSSSS − [Jn = �/� − m0 (2.40)

where [J = K<T:/EI. The solution to this equation for a cantilevered beam is:

25

n/�, m0 = − 14[I/1 + cosh/[o0 cos/[o00 pq/�, m0,0 < � < mq/m, �0,m < � < o (2.41)

where:

q/�, m0 = [s:/[o − [m0s9/[o0 − s9/[o − [m0s:/[o0]sI/[�0+ [s9/[o − [m0s9/[o0 − s:/[o − [m0sJ/[o0]sJ/[�0

(2.42)

for:

s9/�0 = cosh � + cos �

s:/�0 = sinh � + sin �

sI/�0 = cosh � − cos �

sJ/�0 = sinh � − sin �

(2.43)

Using the Green function, the solution to Equation (2.39) is:

i/�0 = 1O@ �[��/T: − Tt�: 0n/�, ℎ�0i/ℎ�0]g�^9 (2.44)

where Tt� are the natural frequencies of the appended oscillators. Writing Equation (2.44)

for the case where x � hj for j = 1, 2, …, R, gives the characteristic equation for the system:

�au��v − ��O@ /T: − Tt�: 0nMℎ�, ℎvNw i/ℎ�0eg�^9 = 0 (2.45)

where ��v is the Kronecker delta. Equation (2.45) can be used to determine the natural

frequencies of the combined system and Equation (2.44) can be used to determine the

mode shapes of the system, which gives the free vibration response.

Abu-Hilal used Green functions to study the forced vibrations of Euler-Bernoulli beams

[32]. Consider first the equation of motion for a beam of length L subjected to a harmonic

forcing input. The equation of motion in complex form is:

O@LSSSS + K<L� + xyL' + x�L' SSSS = �/� − m0c�zf (2.46)

where the prime superscript denotes a partial derivative with respect to the spatial

variable x, the overdot denotes a partial derivative with respect to time, ra is the damping

coefficient for external dampers, ri is the damping coefficient for beam internal damping, �/� − m0 is a Dirac impulse acting at point � = m, i is the imaginary unit, and Ω is the

angular frequency of a periodic forcing function. A solution is then assumed to be of the

form:

26

L/�, "0 = R/�0c�zf (2.47)

where X(x) is the spatial solution and the temporal solution takes on the form of the forcing

function c�zf. Substituting this into Equation (2.46) and reducing gives:

RSSSS − {JR = �/� − m0O@ + Ex�Ω (2.48)

where:

{J = K<Ω: − ExyΩO@ + Ex�Ω (2.49)

The solution to Equation (2.48) will be:

R/�0 = }�/� − m0~�

�/�, m0�m (2.50)

where �/�, m0 is a Green function. To obtain the desired Green function, the Laplace

transform of Equation (2.48) is taken and terms are rearranged to give:

R�/�0 = 1/�J − {J0 a cAt�/O@ + Ex�Ω0 + �IR/00 + �:RS/00 + �RSS/00 + RSSS/00e (2.51)

where the boundary conditions of the beam may be specified by providing values for R/00, RS/00, RSS/00, and RSSS/00. The inverse Laplace transform of Equation (2.51) is then

taken, which gives:

R/�, m0 = �J/� − m0i/� − m0{I/O@ + Ex�Ω0 + R/00�9/�0 + RS/00{ �:/�0+ RSS/00{: �I/�0 + RSSS/00{I �J/�0

(2.52)

where u(x) is the unit step function and:

�9/�0 = 9: /cosh {� + cos {�0

�:/�0 = 9: /sinh {� + sin {�0

�I/�0 = 9: /cosh {� − cos {�0

�J/�0 = 9: /sinh {� − sin {�0

(2.53)

The Green function is then �/�, m0 = R/�, m0. The above relationships can then be used to

take the first three derivatives of R/�, m0 with respect to x for � ≥ m, and used with

Equation (2.52) to write:

27

�� 9/�0 �:/�0{ �I/�0{: �J/�0{I{�J/�0 �9/�0 �:/�0{ �I/�0{:{:�I/�0 {�J/�0 �9/�0 �:/�0{{I�:/�0 {:�I/�0 {�J/�0 �9/�0��

��

�� R/00RSS/00RSSS/00RSSSS/00��

�� = �� R/�0 − �9/m0RSS/�0 − �:/m0RSSS/�0 − �I/m0RSSSS/�0 − �J/m0��

�� (2.54)

where:

�9/m0 = �J/� − m0{I/O@ + Ex�Ω0

�:/m0 = �I/� − m0{:/O@ + Ex�Ω0

�I/m0 = �:/� − m0{/O@ + Ex�Ω0

(2.55)

The Green function for a given beam can be determined with Equation (2.52), Equation

(2.54), and the appropriate boundary conditions.

Green functions have been used for modeling rail track for train-track dynamic

simulations. Specifically, Mazilu used Green function to analyze the dynamic response of

wheel/rail vertical excitation [33]. Equations of motion were developed for a Timoshenko

beam/infinite bar on a discrete pad model. The equation was solved using real Green

functions which were calculated by integrating complex Green functions. These equations

were used to create a Green matrix to study the vertical oscillations of the wheel and rail.

The solution was used to perform studies on steady state driving over vertical

irregularities, including rail corrugation and wheel flats.

Closed form solutions for beams using Green functions are limited in their application.

When considering an external forcing function on the beam, the function must be

deterministic. Also, if it is not periodic then finding a solution can be difficult [32].

Additionally, modeling of lumped parameter inputs is also limited to linear spring rates

[31],[23]. Numerical approximation techniques can solve this problem by providing

accurate solutions for systems composed of a beam and any external forcing functions,

including non-linear spring rate inputs. If the system is modeled as an Euler-Bernoulli

beam with arbitrary forcing function input, then the equation will be a fourth-order

28

parabolic partial differential equation. Caglar determined a solution for fourth-order

parabolic partial differential equations by using a fifth-degree B-spline approximation

[34],[35]. The equation of motion is of the form:

$:i$": + $Ji$�J = �/�, "0,0 ≤ � ≤ 1," > 0 (2.56)

where x is the spatial coordinate and t is the temporal coordinate. The equation has the

following initial conditions and boundary conditions:

i/�, 00 = n�/�0," ≥ 0if/�, 00 = n9/�0," ≥ 0i/0, "0 = ��/"0,i/1, "0 = �9/"0i��/0, "0 = ��/"0,i��/1, "0 = �9/"0 (2.57)

B-splines can be generated by using the Cox-de Boor recursion formula [36]:

�v,�/�0 = p1 if�v ≤ � < �vC90 otherwise ,� = 0, 1, … ,� − 2�v,F/�0 = � − �v�vCF − �v �v,FA9/�0 + �vCFC9 − ��vCFC9 − �vC9 �vC9,FA9/�0,� = 0, 1, … ,� − 2 (2.58)

where the m–n–1 basis B-splines of degree n are defined for n = 0, 1, …, m–2. The fifth-

degree B-splines that Caglar [35] used to solve for Equations (2.56) and (2.57) are:

��/�0 = 99:��,−5�� + 30ℎ�J − 60ℎ:�I + 60ℎI�: − 30ℎJ� + 6ℎ�,10�� − 120ℎ�J + 540ℎ:�I − 1140ℎI�: + 1170ℎJ� − 474ℎ�,−10�� + 180ℎ�J − 1260ℎ:�I + 4260ℎI�: − 6930ℎJ� + 4386ℎ�,5�� − 120ℎ�J + 1140ℎ:�I − 5340ℎI�: + 12270ℎJ� − 10974ℎ�,−�� + 30ℎ�J − 360ℎ:�I + 2160ℎI�: − 6480ℎJ� + 7776ℎ�,

0 ≤ � < ℎ,ℎ ≤ � < 2ℎ,2ℎ ≤ � < 3ℎ,3ℎ ≤ � < 4ℎ,4ℎ ≤ � < 5ℎ,5ℎ ≤ � < 6ℎ, (2.59)

��A9/�0 = ��/� − /E − 10ℎ0,E = 2,3,…

for equally spaced knots of a partition �: � = �� < �9 < ⋯ < �F = � on [a, b]. Using the

spline collocation approach, the approximation of the solution to Equation (2.56) is

considered to be:

�/�0 = � ,v�v/�0FA9v^A� (2.60)

where Cj are coefficients that will be defined later. Then consider n – 1 grid points on the

interval [a, b], which are xi = a + ih, where i is an integer, x0 = a, xn = b, and h = (b – a)/n.

The difference scheme can be used to discretize the second time derivative in Equation

(2.56), giving:

29

i�C: − 2i�C9 + i�/∆"0: + $Ji$�J = �/�, "0 (2.61)

Substituting ∆" = H and rearranging terms gives:

H:i�C:/J0 + i�C: = 2i�C9 − i� + H:�/�, "0 (2.62)

Substituting in the initial conditions from Equation (2.57) gives:

" = 3H,H:iF/J0 + iF = 2iFA9 − iFA: + H:�/�, "0 (2.63)

Substituting Sn as an approximation for un gives:

" = 3H,H:�F/J0 + �F = 2iFA9 − iFA: + H:�/�, "0 (2.64)

The boundary conditions are then written as:

∑ ,v�v/00FA9vÂ� = ��/"0,for� = 0∑ ,v�v/10FA9vÂ� = �9/"0,for� = 1∑ ,v�vSS/00FA9vÂ� = ��/"0, for� = 0∑ ,v�vSS/10FA9vÂ� = �9/"0, for� = 1 (2.65)

The coefficient vector C, which is used in the solution shown in Equation (2.60), is

determined with C = A-1F, where:

, = [,A� ,AJ ,AI … ,FAI ,FA: ,FA9] (2.66)

_ =

�� /3H0��/3H02iFA9/��0 − iFA:/��0 + H:�/��, 3H02iFA9/�90 − iFA:/�90 + H:�/�9, 3H0⋮2iFA9/�FA90 − iFA:/�FA90 + H:�/�FA9, 3H02iFA9/�F0 − iFA:/�F0 + H:�/�F, 3H0�9/3H0�9/3H0 ��

��

(2.67)

30

< =

�� 1120 26120 66120 26120 1120 0 0 ⋯ 020120ℎ: 40120ℎ: −120120ℎ: 40120ℎ: 20120ℎ: 0 0 ⋯ 0�9 �: �I �: �9 0 0 ⋯ 00 �9 �: �I �: �9 0 ⋯ 0⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮0 0 ⋯ �9 �: �I �: �9 00 0 ⋯ 0 �9 �: �I �: �90 0 ⋯ 0 20120ℎ: 40120ℎ: −120120ℎ: 40120ℎ: 20120ℎ:

0 0 ⋯ 0 1120 26120 66120 26120 1120 ��

(2.68)

�9 = H: # 120120ℎJ( + 1120

�: = H: #−480120ℎJ( + 26120

�I = H: # 720120ℎJ( + 66120

(2.69)

Equation (2.60) gives a numerical approximation for a fourth-order parabolic partial

differential equation with an arbitrary forcing input. This fifth-degree B-spline method can

therefore provide a solution to a combined distributed and lumped parameter system. For

example, in railway application, the rail could be modeled as an Euler-Bernoulli beam and

the train system could be modeled as a lumped parameter system, and the interaction

between the two systems would be represented by f(x,t) in Equation (2.56). A combined

approach could then be implemented where the Runge-Kutta method is used to solve for

the train system and fifth-order B-splines are used to solved for the rail system. An

algorithm could be written that employs an iterative procedure where both systems are

solved for simultaneously.

2.4 Wavelet Analysis

Wavelet analysis is a signal processing technique that has emerged over the last 30 years

as an alternative to traditional signal processing methods, like the Fourier transform. The

benefit of wavelet analysis is that it is able to provide a time-frequency or space-frequency

31

representation of a signal, whereas the Fourier transform provides only a frequency

representation of the original signal without any information regarding the location in time

or space of those frequencies. The mathematical operation of generating this time-

frequency or space-frequency representation of a signal is called the wavelet transform.

The difference between the two transforms is that the Fourier transform utilizes all of the

available information to generate the highest possible resolution (limited by the signal

length) frequency representation of the signal, whereas the Wavelet Transform makes a

compromise between the accuracy in the time/space and frequency domains by

decomposing the signal into bands of time/space and frequency content. Since this

research will apply wavelet analysis to rail defect detection, the time and frequency

domains are referred to and the space domain is not mentioned when using the wavelet

transform. The only exception will be in the brief discussion of event detection

applications later in this chapter.

The time-frequency representation produced by the wavelet transform is achieved by

convolving the original signal with a basis of scaled and shifted wavelets. The original

wavelet that is scaled and shifted to form the basis is called the mother wavelet. A

wavelet,¢/"0, is a wave with a finite, concentrated amount of energy that has the

mathematical property:

�|¤/T0|:|T|C]A] �T < +∞ (2.70)

which suggests:

} ¢/"0�" = 0C]A] (2.71)

where ¤/T0 is the Fourier Transform of ¢/"0. Equation (2.71) states that a wavelet must

have zero mean value.

Figure 2.4 shows a sample comparison between the bases of the Fourier transform and

the wavelet transform. The Fourier transform decomposes a signal into essentially infinite

length sine and cosine waves of various frequencies while the wavelet transform

decomposes the signal into finite-length, scaled and shifted wavelets.

32

Figure 2.4. (left) example of Fourier transform basis. (right) example of wavelet transform basis using the Daubechies length 3 wavelet.

Since the wavelet transform gives localization in both the time and frequency domains, it

is beneficial in applications such as event detection and health monitoring. The following

sections will present more details of wavelet theory and current applications of wavelet

analysis.

2.4.1 Continuous Wavelet Transform

The wavelet transform is the operation that transforms a signal into a set of coefficients

that are the convolution of the original signal with a basis of scaled and shifted wavelets.

The continuous wavelet transform (CWT) of a function f(t) is defined as:

�/�, i0 = 1√� } �/"0¢∗ #" − i� (�"C]A] (2.72)

where ¢∗/"0 is the complex conjugate of the wavelet function, s is the continuous scale

index, u is the continuous position index, and Wf(s,u) are the wavelet coefficients. The

continuous scale index is an integer ranging from zero to the maximum value of desired

scale. The continuous position index is a number ranging from the minimum to the

maximum value of desired scale.

Fourier Transform Basis Wavelet Transform Basis (db3)

33

In some applications, the discrete wavelet transform (DWT) may be used instead of the

CWT. The DWT will be derived in the following sections. Equation (2.72) will reappear

when showing the mathematical proof for the benefit of the wavelet transform in event

detection.

2.4.2 Multiresolution Formulation/ Signal Space Representation

Before defining the discrete wavelet transform (DWT), the signal space representation of

the wavelet transform should first be observed to gain a better understanding of how the

original signal is represented by the (DWT). This multiresolution formulation is important

because it provides equations that are used in the derivation of the fast wavelet transform

(FWT) algorithm. Burrus [37] gives an explanation of this signal space representation.

First, a function space is defined as a linear vector space where the vectors are functions

and the scalars are real numbers. The inner product a of two vectors f(t) and g(t) is

defined as the scalar obtained from the following formula:

� = ⟨�/"0, n/"0⟩ = }�∗/"0n/"0�" (2.73)

The inner product can also be used to define the norm of a vector f(t) by the relationship:

‖�‖ = «|⟨�, �⟩| (2.74)

When two vectors have an inner product equal to zero, they are called orthogonal. When

two vectors are orthogonal and their norm is equal to 1, they are called orthonormal. This

is an important principle in deriving certain wavelet bases, as will be shown in the

following.

A space of particular interest to wavelet analysis is a space called L2(R), which comprises

all square-integrable functions. Mathematically, a function, f(t), is square-integrable /� ∈ �:0 if:

} |�/"0|:�"C]A] < +∞ (2.75)

Next, a basis is defined by starting with a vector space. If any function f(t) that is in

(� ∈ ) can be expressed by the relationship �/"0 = ∑ �®s®/"0® , and the set of functions s®/"0 is unique, then s®/"0 is called a basis. To arrive at a multiresolution representation of

a signal, the space that the scaling function s/"0 spans is first observed. The wavelet

34

function is then defined in terms of the spaces spanned by the scaling function basis. The

basis of scaling functions is defined as:

s®/"0 = s/" − H0 (2.76)

where /s ∈ �:0 and k is a translation in the time domain. The subspace of L2 spanned by the

set {s®/"0} is then defined as:

±� = Span®{s®/"0}´́ ´́ ´́ ´́ ´́ ´́ ´́ ´́ ´́ (2.77)

where the over-bar denotes closure. In order to increase the size of the space spanned by

the scaling function, the time scale of the scaling function is changed through the

relationship:

sv,®/"0 = 2v/:s/2v" − H0 (2.78)

where j is the scaling index for the time scale. The 2j/2 term in Equation (2.78) is for energy

normalization across scales. The subspace of L2 spanned by any scale j of the scaling

function is then defined by:

±v = Span®¶s·,®/"0¸´́ ´́ ´́ ´́ ´́ ´́ ´́ ´́ ´́ ´ (2.79)

A visual representation of the subspace spanned by each scale of the scaling function is

shown in Figure 2.5. In this figure, ±0 ⊂ ±1 ⊂ ±2 ⊂ ±3 … ⊂ ±º ⊂ �2, where J is the

maximum desired scale. From the figure, it can be seen that since s/"0 ⊂ ±1, s/"0 ⊂ ±0.

This means that s/"0can be expressed in terms of a sum of weighted and shifted s/2"0.

This is expressed by the equation:

s/"0 = �ℎ/30√2s/2" − 30F (2.80)

where n is an integer index and h(n) is a vector of coefficients called the scaling filter.

A set of wavelet functions ¢v,®/"0 that spans the difference between the spaces spanned

by the scaling functions is now introduced. This new space is defined as »v , which is the

orthogonal complement of ±v in ±vC1. This requires that the inner product of all wavelet

functions and all scaling functions be zero. The new space is defined such that:

±vC1 = ±v ⊕ »v (2.81)

The visualization of the subspaces spanned by the wavelet functions is also shown in

Figure 2.5. Since »0 ⊂ ±1, the wavelet function can be written in terms of the weighted,

scaled, and shifted scaling function by the relationship:

35

¢/"0 = �ℎ9/30√2s/2" − 30F (2.82)

where h1(n) is a vector of coefficients called the wavelet filter.

Figure 2.5. Vector spaces spanned by the scaling function and wavelet function.

The wavelet function serves the purpose of representing bands of fine scale, high

frequency information present in the signal while the scaling function serves the purpose

of representing all of the course scale, low frequency information in the signal that is not

represented by the wavelet function. Choosing to start with scale 0 as the lowest

resolution, any function�/"0 ∈ �: can be represented by:

�/"0 = � P/��, H0sv½,®/"0C]®^A] + � � �/�, H0¢v,®/"0C]

®^A]C]v^� (2.83)

where c(j0,k) are called the scaling (or approximation) coefficients, d(j,k) are called the

wavelet (or detail) coefficients, and j0 is the coarsest scale represented by the wavelet

transform. Note that in Equation (2.83), c is a function of j0 instead of j and that the

subscript in the scaling function is j0 instead of j. This is because the approximation

coefficients represent only the course scale information not represented by the detail

coefficients.

2.4.3 Discrete Wavelet Transform

The continuous wavelet transform has continuous indices in the wavelet basis, and

therefore the wavelet coefficients contain redundant information. Because of the

redundancy in the wavelet coefficients, the signal will not be able to be perfectly

reconstructed. These additional calculations are also time consuming and can make

±0

»0»1»2

±1±2±3

36

computation of the CWT unrealistic for some real-time applications. As a result, the

discrete wavelet transform (DWT) is introduced.

If Equation (2.83) is considered to be the inverse discrete wavelet transform, then the

DWT is the operation that determines the values for the approximation and detail

coefficients of Equation (2.83). For an orthogonal basis, the approximation and detail

coefficients are determined, respectively, by the equations:

Pv½/H0 = ⟨�/"0, sv½,®/"0⟩ (2.84)

�v/H0 = ⟨�/"0, ¢v,®/"0⟩ (2.85)

The equations show that the approximation and detail coefficients are determined by

calculating the inner product of the original signal with the scaling function basis and the

wavelet function basis, respectively. Since the wavelet basis is able to represent a large

class of signals, the wavelet coefficients decay rapidly as the scale increases. This means

that a majority of the information in the signal can be represented in a minimal number of

DWT scales. This decrease in wavelet coefficient value also allows for mathematical

estimation of local signal regularity, which makes wavelets good for event detection.

2.4.4 Filter Bank Representation/ Fast Wavelet Transform

The equations used to calculate the DWT in real time are arrived at by implementing a

filter bank representation of the DWT. The resulting equations are called the fast wavelet

transform (FWT). The basic idea is that a high pass digital filter and a low pass digital filter

are arrived at from the wavelet function and scaling function, respectively. The wavelet

filter (high pass) is h1(n) from Equation (2.82), and the scaling filter (low pass) is h(n) from

Equation (2.80). The detail and approximation coefficients of the DWT are then

determined as follows. The original signal is passed through both the wavelet and scaling

filters and then down-sampled by 2 by removing all of the odd-numbered samples. This

down-sampling effectively removes redundant information. The resulting wavelet-filtered,

down-sampled signal is the first scale of detail coefficients. The resulting scaling-filtered,

down-sampled signal is then passed through both the wavelet filter and the scaling filter

and the resulting signals are down-sampled by 2. The new wavelet-filtered, down-sampled

signal is the second scale of detail coefficients. The new scaling-filtered, down-sampled

37

signal is once again passed through the filters to repeat this process and obtain the third

scale of detail coefficients. This can be done until a maximum possible number of scales is

reached. The maximum possible scale is = log: À rounded down to the nearest integer,

where J is the max scale and N is the length of the data set. A visual representation of the

filter bank implementation is shown in Figure 2.6, where ↓ 2 represents down-sampling by

2.

Figure 2.6. Filter bank implementation of the wavelet transform.

First in [38] and also in [37] it is shown that the recursive equations for the calculation

of the approximation and detail coefficients using the filter bank representation are:

Pv/H0 = �ℎ/� − 2H0PvC9/�0Â (2.86)

�v/H0 = �ℎ9/� − 2H0PvC9/�0Â (2.87)

Where cj(k) are the approximation coefficients, dj(k) are the detail coefficients, and m = 2k

+ n. Equations (2.86) and (2.87) are the FWT.

2.4.5 Local Signal Regularity Calculation using Wavelets

Mallat [38] showed that the local regularity of a signal can be estimated by using the

wavelet transform to calculate the Lipschitz exponent at each time step in the signal. The

developed formulas can be used to gain insight into the power of the wavelet transform in

locating irregularities in a signal. This method can also be used to calculate the Lipschitz

exponent directly to provide information that can be used in event detection and

classification. The premise is that the decay of wavelet coefficients across wavelet

transform scales can be related to the value of the Lipschitz exponent. Since the wavelet

h(-n)

h1(-n) level 1 detail coefficients

f(t)

2

2

h(-n)


2

2

h(-n)


2

2

38

transform provides localization in the time domain, the time local Lipschitz exponent can

be estimated.

Mallat [38] showed that the regularity of a function at any point t = t0 can be estimated

by observing the decay of the wavelet coefficients of the CWT across scales, s. It was shown

that a function f(t) is uniformly LipschitzÃ if and only if there exists some non-negative

constant A such that

| �/�, i0| ≤ <�Ä (2.88)

where �/�, i0 are the wavelet coefficients calculated from Equation (2.72). Taking the

logarithm of both sides of Equation (2.88) gives:

log| �/�, i0| ≤ log/<0 + Ã log/�0 (2.89)

The modulus maxima of the wavelet coefficients can then be plotted as the ordinate with

scale s as the abscissa with log scale for each location u. The Lipschitz exponent Ã can then

be determined for each location on the time axis by fitting the data to Equation (2.89) with

a linear least squares regression. It was further presented by Huang [39] that the equation

for this least squares method is:

Ã/i0 = ∑ /log:| �/�, i0| log: �0ºt^9 − M∑ log:| �/�, i0|ºt^9 NM∑ log: �ºt^9 N ∑ /log: �0:ºt^9 − M∑ log: �ºt^9 N:

(2.90)

The Lipschitz exponent describes the type of singularity, or in other words, the degree of

regularity. For example, a function that is differentiable once at some point t = t0 has

Lipschitz exponent Ã = 1 at t = t0. A step function has Ã = 0 and a Dirac impulse has Ã = -1.

Therefore, the sharper the irregularity, the lower the value of the Lipschitz exponent. In a

signal, singularities of interest are typically characterized by low-magnitude, positive value

Lipschitz exponent, while noise is typically characterized by a negative value Lipschitz

exponent [38],[40],[41].

As Mallat showed that the degree of a singularity can be calculated by determining Ã in

Equation (2.89), Douka [40],[42] showed that the severity (or intensity) of a defect can be

estimated by determining the value of A in Equation (2.89). Accordingly, A has been called

the intensity factor. Douka found that for a cantilever beam with measured mode shapes,

39

the intensity factor A of a crack has a second order polynomial relationship with crack

depth.

2.4.6 Wavelet Families

Depending on the application, compact representation in the wavelet domain and

perfect signal reconstruction may or may not be desired. The application will influence the

choice of wavelet. Several different wavelet families have been generated that have certain

mathematical properties that are favorable for particular applications. For example, in the

case of de-noising and compression, the DWT with orthogonal wavelet basis will likely be

desired because of its ability to provide perfect signal reconstruction. However, in many

event detection applications, signal reconstruction is not needed and therefore an

orthogonal wavelet transform with orthogonal wavelet basis is not required. Additionally,

for event detection applications, removing the restriction of wavelet choice to an

orthogonal basis gives more freedom to select a wavelet that is representative of the

signatures of interest in the signal. The following presents a short discussion on some

notable wavelet families.

A property that is of key importance in selecting a wavelet that will generate a basis

capable of accurately representing a signal is the number of vanishing moments. The

equation used for determining the number of vanishing moments of a wavelet is [37],[43]:

} "Å¢/"0�"C]A] = 0for0 ≤ � ≤ � (2.91)

If Equation (2.91) is satisfied, a wavelet, ¢/"0, has d vanishing moments, where d is an

integer. Also note that Equation (2.91) shows that a wavelet with d vanishing moments is

orthogonal to a polynomial of degree up to d – 1. A signal with maximum local regularity of Ã can be accurately represented by a wavelet with d≥ Ã .

The first ever wavelet and simplest possible wavelet is called the Haar wavelet. The

Haar wavelet is simply a step function. It was first proposed in 1909 by Alfréd Haar who

used the functions as an example of an orthonormal system capable of spanning the space

of square-integrable functions. This system was not originally generated for the purpose of

forming a wavelet basis but was later recognized as the simplest possible wavelet once

40

wavelet theory was developed. Figure 2.7 shows the scaling function and the wavelet

function for the Haar family. The Haar wavelet has 1 vanishing moment.

Figure 2.7. Haar wavelet and scaling function.

The Daubechies wavelets, named after Ingrid Daubechies, are a family of discrete

wavelets characterized by orthogonality and a maximum number of vanishing moments for

a given support length. The wavelets themselves have no closed form solution, therefore

there is no function written to describe them. Instead, they are defined in terms of their

filter coefficients. For a given filter length N, the Daubechies wavelet will have d=2N

vanishing moments. For a maximum number of vanishing moments at a given support

length, there are 2d-1 possible solutions for wavelet function and scaling function. The

Daubechies family is further characterized by selecting the scaling filter that has extremal

phase. Figure 2.8 shows the wavelet function and scaling function for wavelets with

various numbers of vanishing moments. The title dbd indicates the Daubechies family with

d vanishing moments. The db1 wavelet and scaling function are exactly identical to the

Haar wavelet and scaling function.

0 0.5 1

0

0.5

1

Haar Scaling Function

0 0.5 1

-1

-0.5

0

0.5

1

Haar Wavelet Function

41

Figure 2.8. Daubechies wavelet and scaling functions for different support lengths. In each title, dbd indicates the Daubechies family with d vanishing moments.

Since the Daubechies wavelet and scaling function at a given support length will form

orthogonal bases, they are ideal for applications in which signal reconstruction is needed,

like de-noising and compression. Additionally, a wavelet can be chosen with known

number of vanishing moments so that accurate representation of the signal in the wavelet

domain can be ensured. A wavelet with d vanishing moments can accurately represent a

polynomial of order up to d–1. For example, the db3 wavelet and scaling function are

capable of accurately representing a constant, linear, or quadratic function in the wavelet

domain, and will allow perfect reconstruction of the original signal from the wavelet

coefficients. In selecting a wavelet, it should be understood that there is a tradeoff between

localization and accurate signal representation since the length of the filter increases with

increased number of vanishing moments. An increased number of vanishing moments

improves the accuracy of signal reconstruction, but decreases the time localization since

wavelet length increases with increased number of vanishing moments [44].

0 1 2 3-2

0

2db2 Scaling Function

0 1 2 3-2

0

2db2 Wavelet Function

0 2 4-2

0


0 2 4-2

0


0 2 4 6 8-2

0


0 2 4 6 8-2

0


0 5 10 15-1

0


0 5 10 15-2

0


0 10 20 30-1

0


0 10 20 30-1

0


42

The biorthogonal wavelet family uses different wavelet and scaling functions for

synthesis than the wavelet and scaling functions used for analysis. The wavelet and scaling

functions are characterized by the fact that they have finite length and are compactly

supported. Since different wavelet and scaling functions are used for decomposition than

for reconstruction, there is more design freedom in choosing the functions. The

biorthogonal family is characterized by symmetric or anti-symmetric wavelet and scaling

functions which means that the filters will have linear phase. The wavelet functions are

orthogonal to the dual scaling functions and the scaling functions are orthogonal to the dual

wavelet functions.

Figure 2.9. Biorthogonal wavelet and scaling functions for different lengths.

2.4.7 Wavelets in Event Detection Applications

The wavelet transform provides localization in both the time and frequency domains,

which makes it beneficial for analyzing non-stationary or transient signals [37],[45].

Wavelet analysis has been used in various event detection applications, including locating

cracks and determining crack severity in beams [40],[46] and plates [42],[47], machinery

0 2 4-1

0

1

2

bior1.3 Scaling Function

0 2 40

0.5

1

1.5

bior1.3 Reconstruction Scaling Function

0 2 4-2

0

2

bior1.3 Wavelet Function

0 2 4-2

0

2

bior1.3 Reconstruction Wavelet Function

0 5 10-1

0

1

2


0 5 100

0.5

1

1.5


0 5 10-2

0

2

4


0 5 10-1

0

1

2


0 5 10 15-1

0

1

2


0 5 10 15-1

0

1

2


0 5 10 15-1

0

1

2


0 5 10 15-1

0

1

2


43

health monitoring [39], detection of climate change [48], and event location in pacemakers

[49]. These applications are described in more detail later in this section.

To first understand the ability of the wavelet transform to detect irregular events in a

signal, the role of the scaling function should be observed. This was shown in

[38],[40],[42],[50], where Equation (2.72) is used with the knowledge that the wavelet

function acts as a high-pass filter and the scaling function acts as a low-pass filter. The

behavior of a low-pass filter is to smooth the original signal. Utilizing the fact that the

wavelet function is the first derivative of the scaling function with respect to time, Equation

(2.72) can be re-written as:

�/�, i0 = 1√� } �/"0 ��i s∗ #" − i� (�"C]A] = ��iÆ 1√� } �/"0s∗ #" − i� (�"C]

A] Ç (2.92)

The wavelet transform is therefore a scaled version of the first derivative of f(t) smoothed

by s/"0. By smoothing and differentiating the signal, the resulting signal will contain

information about the regularity of the original signal in its peaks. Therefore, repeated

locations of wavelet coefficient modulus maxima (| �/�, i0|) across several adjacent scales

indicate the locations of sharp irregularities in the original signal.

One of the first applications of wavelet analysis to damage detection was in Wang [51]

where faults were detected in a helicopter gearbox by locating transients in the vibration

signal using the wavelet transform. The time domain average for an individual gear in the

gearbox was extracted from the original vibration signal and a time-scale representation

was given. Despite the fact that the orthogonal wavelet transform (DWT with orthogonal

wavelet basis, like Daubechies family) is beneficial for faster calculations and ensuring no

redundancy in decomposition, the non-orthogonal CWT was used because the time

locations and scale values are limited when the orthogonal wavelet transform is used. If an

orthogonal transform were used in this experiment, certain faults in the gears may have

been missed because exact translations and dilations of the mother wavelet may not be

matched during the transform. The increased redundancy of information across scales

resulting from the non-orthogonal wavelet basis was proven to be beneficial in locating

short-time features in the original signal.

44

Since the non-orthogonal wavelet transform was used, it was not necessary to select an

orthogonal wavelet basis. The authors therefore chose a wavelet based on the signature

that was expected to be generated by a gear fault. The chosen wavelet was a Gaussian-

enveloped oscillation because it seemed to be a good comparison to the components of

interest in the signal. The time-scale map generated from the wavelet transform was

successfully able to show the locations of both small and large size variations in the

vibration signal. The locations of the faults in the gear could be seen in this time-scale

representation. However, no automated routine was developed to extract the exact

location of the gear fault from the time-scale plot.

Wang [52] used the wavelet transform for detecting cracks in beams as an example of

the benefits of using wavelet analysis in structural health monitoring where spatially

distributed structural responses can be measured. The displacement distribution of an

analytical beam model was analyzed with Haar wavelets. Although the location of damage

could not be noted with the eye in the displacement response of the beam, it could be

detected by a sharp change in wavelet coefficient value at fine scales. This change in

wavelet coefficient value at the crack location was observed across several adjacent scales.

The analysis was also performed using Gabor wavelets and the results were slightly better

than those from the analysis that used Haar wavelets.

Since an analytical model was used in this study, no practical regard was given for

spatial resolution in the initial analysis. The number of data points in the original study

was 1024, which would require a costly number of sensors for an experimental analysis

with the same resolution, so studies were performed using 62 data points and 31 data

points. Linear interpolation between data points was used to maintain the same

bandwidth-scale relationship from the Haar basis used in the original study. Results

showed that the crack location could be determined with the wavelet transform with as

few as 31 data points.

Angrisani [41],[53] developed a method for detection and measurement of transients in

a signal. The method is comprised of two steps: 1) use the CWT to determine transient

arrival times and duration, and 2) use the DWT to decompose the transients into frequency

sub-bands and reconstruct the transients using only those frequency sub-bands containing

information of interest. This method seeks to extract a transient from a signal and measure

45

its significant parameters, such as duration, amplitude, and period of oscillation. During

the first step, an efficient algorithm based on the Fast Fourier Transform (FFT) is used to

calculate the CWT. These results are then used to form chains of local modulus maxima

across scales. Chains produced by noise are then removed by a thresholding routine. The

difference in the time values of the remaining chains at the highest time resolution (lowest

frequency resolution) are then used to calculate the duration of the transient in the signal.

The second step is then used to extract the transient from the signal by decomposing the

signal with the DWT and then reconstructing it using only the sub-bands containing

frequencies of interest related to the transient. This method was tested by applying the

algorithm to a signal that contained a single period oscillation transient superimposed on a

carrier sine wave. The method was able to accurately extract the transient from the signal

while retaining vital information, such as amplitude, frequency, and time location.

Åström [49] developed a low-complexity detection algorithm based on the wavelet

transform for locating events in a pacemaker. The algorithm was found to perform well in

moderate to high-level noise environments when noise level estimations could not be

made. However, it was found that noise level estimations could improve the performance

of the algorithm.

Hong [46] applied wavelet analysis to the detection of cracks in a vibrating beam and

utilized the concept of signal local regularity for determining crack severity. For an

analytical model, the Lipschitz exponent was calculated using the CWT with Mexican hat

wavelet and Equation (2.89). The local damage in the beam was found to have a Lipschitz

exponent value between 1 and 2. It was also found that there was a strong correlation

between the Lipschitz exponent values and the size of the damage in the beam. Noise was

added to experimental data for a vibrating beam and it was found that added noise

decreased the value of the Lipschitz exponent, i.e., increased the apparent severity of the

singularity.

Gentile [54] applied the CWT to discrete data to detect cracks in transversely vibrating

beams. Gaussian wavelets with various numbers of vanishing moment were used to

calculate the transform. A focus of interest in this research was the removal of rapid

changing values of wavelet coefficient at the boundaries of the beam due to the

discontinuities present at the boundaries. A Hanning window was used to accomplish this.

46

It was found that if the Hanning window was applied to the signal before the wavelet

transform, then the apparent singularities at the beam boundaries do not show up in the

wavelet coefficients. It was also shown that the application of the Hanning window can

have a negative effect on damage detection by reducing the values of wavelet coefficients at

locations of damage close to the ends of the beam. It was shown that the negative effect of

the Hanning window could be circumvented by using a wavelet with higher number of

vanishing moments.

In this study, the authors also observed the ability of the CWT to detect damages in a

beam in the presence of a noise corrupted signal. It was found that the finest scale, in

general, should not be used in the analysis of damage for a signal with noise because the

damage will be masked. At lower scales (higher frequency), the wavelet coefficient values

are more sensitive to noise than at higher scales. The authors showed that the

mathematical features of the wavelet are able to perform several tasks that are relevant to

damage detection, including: calculation of desired derivatives through proper selection of

number of wavelet vanishing moments, convolution, and the smoothing of noisy data.

Douka used the one-dimensional wavelet transform to identify crack location in beams

[40], and used the two-dimensional wavelet transform to identify crack location in plates

[42]. For the beam problem, the authors applied the continuous wavelet transform to the

fundamental vibration mode of a cantilever beam both analytically and experimentally.

Equation (2.89) was used to calculate both the values of the Lipschitz exponent, α, at

regular spaced intervals across the beam and the depth of the crack by relating it to the

intensity factor A. Since singularities of the same type are characterized by the same α

value, it was expected that all cracks of the same type would have the same α value.

Therefore, for a fixed crack type and fixed α, the intensity factor A is the only variable that

changes in Equation (2.89). Intensity factor was therefore related to crack severity or

depth.

The analytical model of the cantilever beam with known crack location and depth was

analyzed with a “symmetrical 4” wavelet, which has 4 vanishing moments. The beam was

analyzed for several different crack depths and it was shown that a consistent value of α

was calculated for all crack depths, and the exact location of the crack was determined. It

47

was also shown that there was a second order polynomial relationship between the crack

depth and intensity factor.

Noise was added to the vibration response of the analytical model to observe the effect

of noise on the determination of crack location and depth. It was shown that the added

noise shows up as a negative value Lipschitz exponent in the wavelet analysis, allowing for

a distinction between irregularities produced by cracks and irregularities produced by

noise. The results also showed that the noise in the original signal tended to corrupt the

smaller amplitude signal generated from smaller cracks. For small signal to noise ratios, it

is possible for the noise to corrupt the signal to the point that the crack will not be able to

be located. Additionally, the noise nearly doubled the values of the intensity factor for each

respective crack depth. A second order polynomial relationship between intensity factor

and crack depth was still observed, but the large increase in values due to the noise made it

difficult to draw any significant conclusions concerning the exact relationship between

intensity factor and crack depth in the presence of a noise corrupted signal.

Experimental results using this technique showed that it was necessary to perform a

cubic spline interpolation on the original signal in order to provide smoothing between

data points to avoid false detection of singularities by the wavelet transform. Additionally,

to remove singularities caused by noise, a threshold was applied at each scale which was

equal to 50% of the maximum value of the wavelet coefficients at that respective scale.

This procedure allowed for accurate determination of the location of the crack along the

beam. Once again, it was shown that there is a correlation between crack depth and

intensity factor, but an exact relationship was not derived.

These same methods were applied to the vibration modes of an analytical model of a

plate in [42]. The crack in the plate was accurately located and a second order polynomial

relationship between intensity factor and crack depth was observed. No studies were

performed for the addition of noise to the signal or for an experimental case.

Blaszczuk [55] used the discrete wavelet transform to detect damage in a beam and the

Lipschitz exponent to assess the degree of severity of the damage. Up to the point of this

publication, the majority of wavelet based event and damage detection algorithms utilized

the CWT. Although the CWT coefficients contain redundant information making

reconstruction difficult, the time resolution of the original signal is preserved across all

48

scales, making it preferable over the DWT for locating short time events. This study

observed the effectiveness of the DWT in damage detection and the influence of number of

data points on the ability of the DWT to successfully detect damage. It was found that for

more data points, the value of Lipschitz exponent is more accurate and therefore generally

lower since the sharp discontinuities in the signal are better represented for greater time

resolution.

Miao [39] used the CWT for calculating the Lipschitz exponent for machinery health

monitoring applications. A kurtosis health index (KHI) was proposed as a means for

measuring the degree of need for machinery maintenance. The authors used Equation

(2.90) to generate a Lipschitz exponent function Lp(x) in which the value of α at each

discrete location on the temporal axis was calculated from the vector of wavelet coefficient

values across all scales of the wavelet transform. By plotting the values of Lp(x), faults in

the machinery could be noticed by a significant change in the value of Lipschitz exponent.

The KHI was then proposed as a means of establishing quantitative criteria for

condition-based maintenance of the machinery. The KHI was arrived at by calculating the

kurtosis of Lp(x). The authors selected a KHI value of 3 as the threshold for required

maintenance. The KHI method was assessed experimentally by measuring gearbox

vibration and applying the techniques described previously. The KHI method was

compared to other health monitoring methods existing in the literature and the new

method was found to produce excellent results for fault detection in gearboxes.

Gökdağ [44] presented a new algorithm consisting of the DWT and CWT that locates

damages in structures from unhealthy data. This means that no knowledge concerning the

response of a healthy structure is available. The first step in achieving this was to extend

the lengths of the data at the ends to reduce the effects of boundary distortion. The signal

was reflected on both the left and right ends of the data set. An approximation function

(AF) that characterizes a healthy mode shape was then determined by applying the DWT

with suitable decomposition level and number of wavelet vanishing moments. The CWT

was then applied to the AF and the unhealthy data and the difference between the two was

taken to be the damage index. The AF is the approximation coefficients (result of scaling

filter) at the appropriate decomposition level. A symlet wavelet was used to perform the

DWT, and between 10 and 20 vanishing moments was recommended to extract a suitable

49

AF. This method was applied to the first two experimental modes of a damaged beam and

the algorithm was able to accurately locate the damage in the beam.

2.5 Artificial Neural Networks

Artificial neural networks (also known as multi-layer neural networks or multi-layer

perceptrons) are a powerful pattern classification tool. They have the ability to provide

optimal solutions to classification problems by using simple algorithms to map the system

inputs into a non-linear space. Neural networks have been used in many modern day

pattern classification applications, including classification and diagnosis of various types of

cancer [56], structural health monitoring [57], and wayside acoustic train axle bearing fault

detection [58]. Additionally, various feature extraction techniques for generating inputs

into a neural network have been studied, including various frequency domain approaches

[59].

2.5.1 Artificial Neural Network Basics

Figure 2.10 shows a diagram of a three-layer neural network. The inputs to the network

are denoted by x, the outputs from the hidden layer are denoted by y, the outputs of the

neural network are denoted by z, and the target values (desired values) are denoted by t.

The input-to-hidden weights are denoted by wji and the hidden-to-output weights are

denoted by wkj. The subscript i is the index for the input layer, the subscript j is the index

for the hidden layer, and the subscript k is the index for the output layer.

Figure 2.10. Three-layer neural network.

50

The network operates as follows. At each hidden layer neuron, the inputs are weighted

and summed according to the formula:

3c"v = ��Lv�É

�^� (2.93)

where d is the number of inputs. The output from each hidden neuron is then determined

by:

qv = �/3c"v0 (2.94)

where f() is an activation function that limits the values of the output. At each output layer,

all of the outputs from the hidden layers are summed according the formula:

3c"® = �qvL®vFÊv^� (2.95)

where nH is the number of neurons in the hidden layer. The output from each output layer

is determined by:

h® = �/3c"®0 (2.96)

To train the network using backpropagation, a learning rule for adjusting the weights

must be derived. First consider an error function that sums the squared difference

between the desired output tk and the actual output zk. The mean squared error criterion

function is [60]:

/w0 = 12�/"® − h®0:�®^1

(2.97)

where w represents all of the weights in the network (both input-to-hidden and hidden-to-

output). The weights are first initialized to random values and then changed according to a

gradient descent learning rule. The learning rule adjusts the weights in order to reduce the

error calculated from Equation (2.93). The learning rule is:

∆w=-Ë $ $w (2.98)

where η is called the learning rate.

The activation functions in the hidden and output layers are sigmoid functions, which

have the form:

51

Ì/�0 = <1 + cA� (2.99)

The sigmoid function saturates the output P at the value A for large input x. Because of this

saturation, the output of the network is limited, which is beneficial for this application.

Additionally, the nonlinearity of the sigmoid function allows for the generation of complex,

nonlinear decision boundaries for the classes.

A common choice for determining the number of outputs is to assign one output for each

class. The respective target values for each output are then set to the value of +1 if the

input features are a member of that class and a value of -1 if the input features are not a

member of that class. For example, for a four class network, if the input features belonged

to the class represented by output z2, then the target value vector would be t = (-1, +1, -1, -

1).

The training data set should be sufficiently large to provide an accurate representation

of the classes of interest. Often times, artificial training data can be produced using

statistical approaches for more accurate training. One method of generating artificial

training data is to simply add various levels of Gaussian white noise to the input training

patterns. The variance of the noise should be less than 1.0. Another way is to manufacture

data sets that are statistically similar to the training sets. These methods have the ability to

improve the accuracy of the decision boundaries during learning, but do not necessarily

guarantee improved classification.

The number of hidden neurons should be chosen based on the complexity of the decision

boundaries. Less hidden neurons are required for well-defined or linearly separable

decision boundaries. As the boundaries become more complex, more hidden neurons are

needed for accurate representation. However, too many hidden neurons can over-

complicate the network and lead to overfitting of the training data, which can result in poor

classification during feedforward operation. A general rule that is often used in practice is

to set the number of hidden layers such that the number of weights is n/10, where n is the

number of training points. The motivation for this selection is that there should not be

more weights than training points, since the number of weights acts as the number of

degrees-of-freedom for the network.

52

Learning rates should be chosen small enough to ensure convergence of the weights. If

the learning rate is too high, the weights will diverge. If the learning rate is too small, then

the weights will simply take a long time to converge. The theoretically optimal learning

rate is that which causes the local error to reach a minimum in one learning step. The

maximum learning rate that can give convergence is twice that of the optimal learning rate.

For optimal convergence, the learning rate should be set individually for each weight by

calculating the second derivative of the criterion function with respect to the weight

according to the equation:

ËÍÅf = # $ $L:(A9 (2.100)

As an initial choice, the learning rate can be set to Ë = 0.1 and adjusted accordingly during

learning depending on the criterion function speed of convergence or divergence.

2.5.2 Neural Network Applications

In recent years, artificial neural networks (ANN) have emerged as a powerful method of

pattern classification. Theoretically, any nonlinear continuous function can be represented

by a three-layer neural network with a proper number of hidden neurons, weights, and

activation functions [61]. As a result, multi-layer neural networks have been used in a wide

range of applications and fields of research in recent years.

Choe used neural networks to identify train wheel-bearing faults from a wayside

acoustic sensor [59]. This approach was an attempt to improve the defect detection

methods of wayside faulty bearing detection systems. Hot Bearing Detector (HBD) systems

use an infrared (IR) sensor to measure bearing temperature and flag a bearing as defective

if it exceeds some predetermined threshold temperature. Acoustic Defective Bearing

Detector (ADBD) systems use an acoustic signal to determine whether or not a bearing is

overheated and needs to be replaced, which historically have about an 85% success rate for

detecting faulty bearings. This research specifically focused on different feature extraction

methods for use with a neural network classifier. These include the Fast Fourier

Transform (FFT), Continuous Wavelet Transform (CWT), and Discrete Wavelet Transform

(DWT). The results of the study showed between 87% and 100% success rate for FFT

53

feature extraction, between 84% and 98% success rate for CWT feature extraction, and

between 81% and 95% success rate for DWT feature extraction.

Xu continued this research in the area of wayside wheel bearing fault detection with

acoustic sensors [58]. A neural network classifier was designed to be robust to sensor

noise and various operating conditions including different forward speeds, loading

conditions, and bearing types. The DWT was used to extract features as inputs into the

neural network. The first 11 scales of the DWT were determined, and eight different

statistics were taken at each scale, resulting in 88 possible features to select from as inputs

into the network for each signal. The ANN was incorporated into a decision tree, where

different subsets of features were used at different levels of the tree. The ANN used

sigmoid functions as activation functions in the hidden layer and output layer. The

network was trained using standard backpropagation learning with 60% of the data used

as training data, 20% as testing data, and 20% as validation data. The research showed

results of faulty bearing detection success rates as high as 99.9%.

Kirschner explored the use of neural networks in railroad wheel bearing fault detection

using an ultrasonic acoustic signal in the range of 20-120 kHz [62]. One of the benefits of

using ultrasonic acoustic signals is the directionality of the microphones in this frequency

range, which helps when operating in low signal-to-noise ratio environments. Power

spectral density was used as the input feature into the neural network. Artificial training

data was generated because of the limited amount of data available for training the

network. Statistically similar data sets were created using the combination of a normal

distribution and a coordinate transformation. The approaches used in this research

produced good results and suggest promising future use of these techniques in the problem

of pattern classification where an ultrasonic acoustic sensor is used.

Lopes used an ANN to perform impedance based structural health monitoring, which

uses the electromechanical coupling property of piezoelectric materials to measure high

frequency structural vibrations greater than 30 kHz [57]. The general idea is that the

electrical impedance of the piezoelectric sensor can be related to the mechanical

impedance of the structure on which the sensor is mounted. The impedance method is

used to detect faults, which are then used as input features into the neural network for

classification. The purpose of the research was to generate a structural health monitoring

54

technique capable of detecting, locating, and classifying structural damage with no prior

knowledge of the structure’s dynamic model. The network used in this research utilized

one hidden layer, sigmoid functions for activation functions in the hidden layer, and linear

functions for activation functions in the output layer. Backpropagation learning was used

to train the network. The damage detection scheme was tested on a quarter scale bridge

section and a space truss structure for proof of concept. Damage was induced on the

quarter scale bridge section by loosening some of the bolts at joint locations. Damage was

induced on the space truss structure by loosening some of the connector threads. In both

instances, the damage was successfully located.

ANN’s have also been used in the medical field for cancer diagnosis [56]. Cancer gene

expression signatures from cDNA microarrays were used as inputs into the neural network

for classification into one of four diagnostic categories. The network was trained and

validated using samples of signatures from small, round blue-cell tumors. For all samples

introduced to the network, the ANN was able to accurately classify the signatures.

2.6 Summary and Conclusions

This chapter contained a presentation of some literature that was helpful in developing a

solution to an advanced rail surface irregularity detection and classification system.

Existing techniques for locating rail defects were first explored. Then, some methods for

modeling a train-rail system were presented. Train-rail dynamic simulations play an

important role in the development of the defect detection algorithm because they can

provide useful data representing a wide range of operating conditions that may be difficult

to collect. A review of wavelet analysis was presented which suggests promising results for

being used as a basis for an intelligent defect detection algorithm. A review of artificial

neural networks was presented as this technique has potential to solve the problem of rail

surface irregularity classification. The major conclusions from each of these sections are

shown below.

55

The major conclusions drawn from Section 2.2 are:

• Track failures are the leading cause of train accidents, and train derailments are the

most prominent type of accident.

• There was a 65% decrease in broken rail accidents per million train miles from

October 2004 to September 2010.

• Current rail defect detection methods used by major railroads include:

instrumented in-service vehicles, instrumented rail inspection vehicles, and hi-rail

vehicles.

• Accelerometers mounted to the bogie side frame or wheel axle box are sensitive to

rail surface irregularities.

• Current commercially available signal processing techniques for extracting rail

defect information from a vertical acceleration signal are basic and typically only

consider the magnitude of the signal.

The major conclusions drawn from Section 2.3 are:

• Lumped parameter modeling techniques are useful for modeling the dynamics of a

train.

• Distributed parameter modeling techniques are useful for modeling the dynamics of

a rail.

• Methods from contact mechanics can be used to model wheel-rail interactions.

• There are several combined systems approaches in existence that allow for the

modeling of a combined lumped and distributed parameter system. The use of B-

spline collocation provides useful results in this area.

The major conclusions from Section 2.4 are:

• Wavelet analysis is a multiresolution technique that can provide a simultaneous

time-frequency representation of a time domain signal.

• Wavelet analysis has proven itself to be useful in event detection applications.

• One of the powers of the wavelet transform in event detection is its ability to

mathematically calculate the local signal regularity.

56

• Repeated locations of wavelet coefficient modulus maxima across several adjacent

scales indicate the locations of sharp irregularities in the original signal.

The major conclusions from Section 2.5 are:

• Neural networks have been used extensively in a wide range of classification

applications.

• Theoretically, any continuous non-linear function can be represented by a three-

layer network with appropriate selection of weights, number of hidden units, and

activation functions.

• There are several heuristic approaches in the literature that can be used to design a

neural network.

• There are several statistical approaches that can be used to generate artificial

training data for training a neural network.

57

Chapter 3

Experimental Setup and Data Collection _____________

3.1 Introduction

The purpose of this research is to develop a rail surface defect detection system that can

be implemented on an in-service train. Such a system should be sensitive to rail surface

irregularities and should be able to provide information that allows for the distinguishing

of various types of surface irregularities. Accelerometers are a good choice for a sensor

since rail surface irregularities will generate high and/or abnormal impact loads on the car.

There are already several accelerometer based systems in existence that are implemented

on in-service cars, commissioned track inspection vehicles, and research vehicles used on

closed tracks. One of these systems is the ENSCO Vehicle/Track Interaction (V/TI) Monitor

system, which is implemented on over 200 in-service trains and many track inspection

vehicles [17]. The V/TI system uses accelerometers to measure axle and car body

accelerations. This is beneficial for calculating impact loads which can be correlated to

surface irregularities. Transportation Technology Center, Inc. (TTCi) uses an Instrumented

Freight Car (IFC) for research purposes. This vehicle is equipped with accelerometers

mounted to the bogie side frame, which are used to calculate impact loads and detect

broken rails [6].

The following sections will discuss the experimental setup for this research, including

sensors, hardware, and data acquisition system, as well as test vehicle and test track

description. Samples of the collected signals will be presented, with different signatures

corresponding to different rail surface irregularities based on a visual inspection of the test

track. A frequency analysis of some of the collected signals will be presented along with

observations of the features present in the signals.

58

3.2 Experimental Setup

The following sections describe the experimental setup for the initial testing of this

research. Section 3.2.1 contains information on the sensors, hardware, and data

acquisition system used, and Section 3.2.2 contains an explanation of the test sites and

facilities used to collect data.

3.2.1 Sensors, Hardware, and Data Acquisition System

The initial data used in this research was provided by TTCi. The data was collected using

the Instrumented Freight Car (IFC) on the High Tonnage Loop (HTL) at the Facility for

Accelerated Service Testing (FAST) in Pueblo, Colorado. The IFC is an instrumented Union

Pacific (UP) coal car loaded to a gross weight of 285,375 lbs. using sand and gravel, and is

towed by Norfolk Southern (NS) locomotive 2595. Figure 3.1 shows photographs of the UP

coal car and NS locomotive. The IFC is equipped with top chord strain gauges, bolster

strain gauges, suspension displacement transducers, car body accelerometers, bogie side

frame accelerometers, and GPS sensors for position measurements and speed calculations.

The IFC utilizes these sensors to assess vehicle performance which can be related to

various track geometries [6].

Figure 3.1. TTCi Instrumented Freight Car: instrumented coal car (left) towed by locomotive (right).

It was found that out of the sensor suite, the side-frame mounted accelerometers were

the most sensitive to short wavelength rail head surface irregularities and were therefore

most effective in detecting broken rails. Figure 3.2 shows a photograph of the

59

accelerometer mounted to the front right side frame, which measures acceleration in the

vertical direction. The onboard data acquisition system collects data at a sampling rate of

256 Hz and processes it in real time by applying a pre-determined threshold value to each

of the sensor signals. If the magnitude of the sensor signal exceeds the threshold, then an

exception will be generated and communicated to the office via cellular phone technology.

The threshold values are determined from extensive statistical analysis on sensor signal

values over a long range of time.

Figure 3.2. Front right bogie side frame of the IFC with location of accelerometer shown with red arrow.

3.2.2 Test Facility

The IFC is run on the HTL at FAST for the purpose of studying track geometry

degradation under heavy axle loads. The HTL is a 2.7 mile loop consisting of tangents,

spirals, five and six degree curves, and turnouts. Figure 3.3 shows a satellite image of the

HTL from Google Earth. The track is used to test the reliability, wear, and fatigue of various

track components [63]. While the IFC is driving around the HTL, it sends an exception

report to the office on a per lap basis. The personnel viewing the report can then assess the

repeatability of the IFC measurements and real time analysis, and observe the propagation

of various rail defects over time.

60

Figure 3.3. Satellite image of the HTL from Google Earth.

Different sections of the HTL are used to perform research on different track

components. Figure 3.4 shows a diagram of the HTL with different sections labeled

according to the track components tested there. Some of these track components include

ties constructed of different materials, joints, turnouts, different kinds of rail, concrete and

steel bridges, and ballast. Knowledge of the locations of various track components along

with a visual inspection of them is important for relating information in the IFC signal to

the rail surface irregularity that it represents.

Figure 3.4. Diagram of HTL with sections labeled based on track components tested in those sections [64].

61

a)

b)

c)

d)

e)

f)

g)

h)

i)

j)

Figure 3.5. Various track components corresponding to sections of HTL.

62

Figure 3.5 shows several photographs of track components corresponding to the various

sections of track shown in Figure 3.4, which are: a) flange bearing turnout at end of section

8, b) wooden ties from section 25, c) steel ties from section 25, d) concrete ties from section

25, e) turnout at end of section 27, f) joint from section 29, g) moveable point frog at end of

section 35, h) electric flash head repair weld from section 3, i) concrete bridge at end of

section 3, j) steel bridge from section 5. Differences in rail, ties, and presence of special

track components can heavily influence the wheel impact loads and the accelerometer

signal. Therefore it is important to identify these track components and relate them to the

features present in the sensor signal, as will be discussed in the following sections.

3.3 Data Collection

The following sections contain a discussion about some of the collected data. Section

3.3.1 contains a description of the data collection process and presents some initial

observations based on a visual inspection of the data. Section 3.3.2 shows some of the

vertical acceleration signatures generated from various rail surface irregularities and

includes a frequency analysis of those signatures. Section 3.3.3 discusses the significance of

side frame lateral accelerations.

3.3.1 Collected Signals and Initial Observations

The data collected from the IFC for this research was a bogie side frame vertical

acceleration signal at a sampling rate of 256 Hz. The IFC was travelling at a relatively

constant forward speed of 35 mph. Longitude and latitude GPS coordinates were recorded

with each data point. Detection of impending rail breaks in the form of small rail head

surface fractures and detection of broken rails is the primary purpose of this research.

However, surface fractures and broken rails occur in an unpredictable manner during

testing. Therefore in order to develop a rail defect detection algorithm, it is necessary to

collect data until a broken rail occurs. After the break occurs, data sets from previous laps

can be observed with known location of the broken rail to analyze the signature generated

from an impending rail break. Figure 3.6 shows plots of two consecutive laps with the first

lap showing an impending rail break and the second lap showing a broken rail. The bogie

63

38.445

38.45

38.455

104.33

104.34

104.35

104.36

-5

0

5

LatitudeLongitude

Acceleration (g)

38.445

38.45

38.455

104.33

104.34

104.35

104.36

-5

0

5

LatitudeLongitude

Acceleration (g)

side frame vertical acceleration is plotted against longitude and latitude coordinates. The

location of the rail defect is shown with a red dot. Comparison of the two laps shows

repeatability in the vertical acceleration measurements from lap to lap, with the only major

difference between the two signals being the increase in amplitude at the location of the

rail defect in lap 2.

Figure 3.6. Bogie side frame vertical acceleration signals for two consecutive laps around the HTL. The red dot marks the location of: (left) an impending rail break, (right) a broken rail.

Figure 3.7 shows one dimensional signal plots of the data from Figure 3.6. The top plot

is the signal before the broken rail and the bottom plot is the signal after the broken rail.

These plots further illustrate the sensitivity of the accelerometer to rail surface

irregularities and repeatability of the vertical acceleration measurements. Figure 3.7

illustrates how quickly a surface fracture can turn into a broken rail. In the top plot, the

signature of the impending break around data point 284,000 cannot be distinguished from

the signature of the turnout around data point 568,000 by simply looking at the amplitude.

After only a few more passes of train cars, the surface fracture quickly becomes a broken

rail, which is shown in the bottom plot around data point 284,000. A standard threshold

based algorithm for processing these signals [17] would flag the broken rail in the bottom

plot due to its 5.2 g magnitude peak, but would miss the impending rail break from the top

plot. If the threshold criteria were lowered enough to catch the impending break, then the

algorithm would also flag the turnout around data point 568,000, issuing a false positive.

This data set illustrates the necessity for a more intelligent defect detection algorithm

64

capable of locating both impending rail breaks and broken rails while minimizing or

eliminating false positives.

Figure 3.7. Bogie side frame vertical acceleration signals for two consecutive laps. (top) before

broken rail, (bottom) after broken rail.

3.3.2 Defect Signature/ Signal Content

Before proceeding with the development of an intelligent defect detection algorithm, it is

beneficial to look at the signatures generated from various rail surface irregularities and

make observations and draw conclusions. A preliminary analysis has been performed on a

few sets of collected data to determine whether or not there is enough information located

in the vertical acceleration signatures to distinguish between process noise, surface

fractures, and breaks. Data was collected for six laps of the IFC driving around the HTL

with known location of a surface fracture. The train was traveling at 35 mph and the

vertical acceleration data was time stamped and GPS coordinate stamped. Figure 3.8

shows the vertical acceleration signatures for each of the six laps at the location of the

surface fracture. The rail defect progresses from a small surface fracture in Lap 1 to a

break with a gap of about 4 inches in Lap 6. The fast propagation of the surface crack to a

rail break further reinforces the need for an early detection algorithm.

0 1 2 3 4 5 6

x 104

-5

0

5

Acceleration (g)

0 1 2 3 4 5 6

x 104

-5

0

5

Data Point Number

Acceleration (g)

65

Figure 3.8. Vertical acceleration signals at the location of a surface fracture for six laps.

The vertical acceleration signatures for a surface fracture, a rail break, and a rail crossing

were analyzed by taking a windowed Fast Fourier Transform (FFT) of the signal. It was

observed that a window of size 100 data points was sufficient for capturing the full

signature generated by defects and process noise for a sampling rate of 256 Hz. Figure 3.9,

3.10, and 3.11 show the vertical acceleration signature (top) and FFT (bottom) for a surface

fracture, rail break, and process noise, respectively. Figure 3.9 shows that there are

dominant frequencies in the surface fracture signature around 30 Hz and 40 Hz. Figure

3.10 shows that there are dominant frequencies in the rail break signature around 30 Hz

and 40 Hz as well, but with larger amplitude. Figure 3.11 shows that there are dominant

frequencies in the process noise signature around 40 Hz and 50 Hz.

Comparison of Figure 3.10 and Figure 3.11 shows that the acceleration signatures for the

rail break and the process noise are similar in amplitude but have different frequency

content. The frequency spectrum of the process noise appears to be similar to the

5.59 5.592 5.594 5.596 5.598

x 104

-5

0

5Lap 1, speed at break = 35.4 mph

5.334 5.336 5.338 5.34 5.342

x 104

-5

0


3.902 3.904 3.906 3.908 3.91

x 104

-5

0


Vertical Acceleration (g's)

3.89 3.892 3.894 3.896 3.898

x 104

-5

0


3.762 3.764 3.766 3.768 3.77

x 104

-5

0


Time (s)

3.65 3.652 3.654 3.656 3.658 3.66

x 104

-5

0


Time (s)

66

frequency spectrum of the rail break, but with a 10 Hz frequency shift. Figures 3.9, 3.10,

and 3.11 illustrate the ability of a frequency domain technique to extract features that can

be sent into a classifier.

Data point number

Figure 3.9. (top) Acceleration signature of the surface fracture from Lap 1. (bottom) Fast Fourier

Transform of the acceleration signature.

0 20 40 60 80 100 120-2

-1

0

1

2

Time (s)

Acceleration (g)

Lap 3

0 20 40 60 80 100 1200

0.2

0.4

0.6

Frequency (Hz)

Modulus

67

Data point number

Figure 3.10. (top) Acceleration signature of the rail break from Lap 5. (bottom) Fast Fourier


Data point number

Figure 3.11. (top) Acceleration signature of the process noise/rail crossing. (bottom) Fast Fourier


0 20 40 60 80 100 120-4

-2

0

2

4

6

Acceleration (g)

0 20 40 60 80 100 1200

0.2

0.4

0.6

0.8

1

Frequency (Hz)

Modulus

0 20 40 60 80 100 120-4

-2

0

2

4

Acceleration (g)

0 20 40 60 80 100 1200

0.2

0.4

0.6

0.8

1

Frequency (Hz)

Modulus

68

3.3.3 Lateral Acceleration Signatures of Irregularities

For some of the data collected at the HTL, lateral acceleration values were recorded. The

first sets of tests performed utilized uni-axial accelerometers on the left and right side

frames of the IFC, both measuring vertical acceleration. Later tests utilized a bi-axial

accelerometer on the right side frame, measuring vertical and lateral accelerations, and a

uni-axial accelerometer on the left side frame, measuring vertical acceleration.

Longitudinal acceleration values were not recorded for any of the tests. Because rail

breaks at the HTL are not induced, but rather allowed to occur naturally, no lateral

acceleration values were recorded at locations of broken rails or impending rail breaks. As

a result, only vertical acceleration values are taken into account in the defect detection

algorithms presented in the following chapter.

However, since lateral acceleration values were recorded for the IFC driving over some

special track components, it is worthwhile to observe these signatures in order to gain

insight into the physical system and also to consider the inclusion of lateral acceleration in

the development of future algorithms. Figure 3.12 shows the lateral acceleration

signatures for five passes over the location of a turnout. The turnout is located at the end

of section 8 and beginning of section 9 from Figure 3.4. In the plots, the IFC drives over the

turnout around 0.48 s, and a lateral acceleration oscillation with amplitude of about 4 g is

observed. The plots of laps 1 through 5 show the repeatability of this measurement.

Figure 3.13 shows the lateral acceleration signatures for five passes over a turnout located

at the end of section 27 and beginning of section 28. The train passes over the turnout at

about 0.28 s and a lateral acceleration oscillation with amplitude of about 1.7 g is observed.

Once again, the repeatability of the measurement can be seen.

Since a limited amount of data was collected for lateral acceleration signatures

generated by special track components, and since no data was collected for lateral

acceleration signatures generated from rail defects, there is not a significant amount of

information available to include lateral acceleration values in the classification of rail

surface irregularities. However, the observations noticed from Figure 3.12 and Figure 3.13

suggest that lateral acceleration may contain relevant information for irregularity

69

classification, and it is therefore suggested that tri-axial accelerometers be used on both

side frames for all future testing.

Figure 3.12. Lateral acceleration signatures at the location of a turnout (end of section 8, beginning of section 9) on the HTL.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-4

-2

0

2Lap 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-4

-2

0

2Lap 2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-4

-2

0

2Lap 3

Lateral Acceleration (g)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-4

-2

0

2Lap 4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-4

-2

0

2Lap 5

Time (s)

70

Figure 3.13. Lateral acceleration signatures at the location of a turnout (end of section 27, beginning of section 28) on the HTL.


This chapter presented an overview of the experimental setup and data collection

procedures for this research. An instrumented freight car (IFC) towed by a locomotive was

equipped with an accelerometer on the front right and front left bogie side frames. The IFC

collects vertical acceleration data at a sampling rate of 256 Hz while driving around the

0 0.1 0.2 0.3 0.4 0.5 0.6-4

-2

0

2Lap 1

0 0.1 0.2 0.3 0.4 0.5 0.6-4

-2

0

2Lap 2

0 0.1 0.2 0.3 0.4 0.5 0.6-4

-2

0

2Lap 3

Lateral Acceleration (g)

0 0.1 0.2 0.3 0.4 0.5 0.6-4

-2

0

2Lap 4

0 0.1 0.2 0.3 0.4 0.5 0.6-4

-2

0

2Lap 5

Time (s)

71

High Tonnage Loop (HTL) at the Facility for Accelerated Service Testing (FAST) at

Transportation Technology Center, Inc. (TTCi) in Pueblo, CO. Data was collected for a train

forward speed of 35 mph. The vibration signatures generated at the side frame were

observed and correlated with different types of rail surface irregularities based on a visual

inspection of the track. It was found that broken rails can occur quickly and unexpectedly.

A small surface fracture can propagate and turn into a broken rail with only a few train car

passes. A frequency analysis was performed on signatures generated from a rail surface

fracture (impending rail break), a broken rail, and a rail crossing. Results showed that

there is different frequency content in all of these signatures, suggesting potential for a

frequency-based defect detection and classification algorithm. Observation of the side

frame lateral acceleration signatures suggests that lateral acceleration may contain

relevant information for irregularity classification, and it is therefore suggested that tri-

axial accelerometers be used on both side frames for all future testing.

72

Chapter 4

Rail Defect Detection Algorithms___________________

4.1 Introduction

Currently, railroad defect detection is performed by using an instrumented car [6], a

railway monitoring vehicle, or an instrumented in service train [7]. Using these methods,

sensors include video cameras, accelerometers, strain gauges, acoustic sensors, LIDAR, and

ultrasonic wave sensors [8], and various methods are used to process the collected data.

This chapter presents two data processing routines for use with a vertical acceleration

signal from an accelerometer mounted to the side frame of a freight car bogie. The defect

detection algorithms are designed to detect and classify rail defects using a limited amount

of data. Since the proposed system has a limited number of sensors, it can be installed on a

car in a relatively short amount of time, enabling it to be placed on a train that is going

about its normal day-to-day operation. This can reduce or eliminate the need for testing

with separate instrumented cars or railway monitoring vehicles which can be a time

consuming and costly process.

The data processing routines in this chapter operate in two primary steps: feature

extraction and feature classification. During feature extraction, information of interest is

obtained from the original signal. The form of this information is dependent on the feature

extraction technique that is used, which may or may not be easily interpreted. During

feature classification, the features are sent as inputs into a classifier to determine the class

that each pattern of inputs lies in. This chapter presents three separate routines for

processing the bogie side frame vertical acceleration data. The first routine uses the

windowed Fast Fourier Transform in the feature extraction step and an Artificial Neural

Network in the feature classification step. The second routine uses the Wavelet Transform

73

in the feature extraction step and an artificial neural network in the feature classification

step. The third algorithm uses wavelets to perform a local regularity analysis on the signal

and calculates a local intensity factor that can be related to defect severity. The details of all

three algorithms are presented, followed by simulations using the proposed algorithms and

a basic commercial threshold-based algorithm. The results of the algorithms are compared.

4.2 Development of Fourier Transform-Based Algorithm

A proposed solution to the problem of classifying rail surface fractures, rail breaks, and

process noise from a bogie side frame vertical acceleration signal is to design an Artificial

Neural Network (ANN) as a classifier. Proper design of the ANN includes determination of

the inputs into the neural network, the number of hidden layers, the number of neurons in

each hidden layer, appropriate target values for training, values for weights, and activation

functions for the hidden and output layers. The first algorithm uses the windowed Fast

Fourier Transform (FFT) to extract features as inputs into the neural network.

4.2.1 Feature Extraction: Fourier Transform

As shown in Chapter 3, the frequency content of the vertical acceleration patterns for

various rail surface irregularities are noticeably different. Additionally, from a physical

understanding of the system, it can be understood that different rail surface irregularities

will excite different natural frequencies of the train system. Data processed by the Fourier

Transform thus serves as a good candidate for input features to the neural network. The

vertical acceleration patterns are first pre-processed with a moving FFT window. From

observation of the signatures generated from various rail surface irregularities, it is seen

that a window size of 100 data points is sufficient to capture the signature for a signal

sampled at 256 Hz. For a windowed FFT of length 100, there will be 50 pieces of useful

information available to input into the neural network. Since 50 different frequency values

may not best highlight the dominant features in the signal and since 50 inputs may also

over-complicate the network, the FFT information is arranged into six bins representing six

different frequency ranges. The bins and respective frequency ranges are: bin 1, 5-15 Hz;

bin 2, 15-25 Hz; bin 3, 25-35 Hz; bin 4, 35-45 Hz; bin 5, 45-55 Hz; bin 6, 55-65 Hz. Figure

74

4.1 shows the frequency content of the signatures of a surface fracture, a broken rail, and

process noise arranged into the 6 bins.

Figure 4.1. Frequency content arranged into 6 bins as 6 inputs into the neural network. From top to bottom: surface fracture, rail break, process noise.

4.2.2 Feature Classification: Artificial Neural Network

Artificial Neural Networks implement simple equations with adjustable weights that can

be learned from training data. For a sufficiently large set of training data, the network can

be trained to generate complex, non-linear class decision boundaries. The network can be

trained by using a learning technique called backpropagation. Backpropagation is a

method of supervised learning, which uses a set of data to train the network during the

learning mode of operation before the network is used in classification during the

feedforward mode of operation.

10 20 30 40 50 600

0.5

1

1.5

2

2.5Surface Fracture

10 20 30 40 50 600

0.5

1

1.5

2

2.5Broken Rail

10 20 30 40 50 600

0.5

1

1.5

2

2.5Process Noise

Bin Center Frequency (Hz)

75

A visual representation of the general design of a three-layer neural network classifier is

shown in Figure 4.2. The inputs to the network are denoted by x, the outputs from the

hidden layer are denoted by y, the outputs of the neural network are denoted by z, and the

target values (desired values) are denoted by t. The input-to-hidden weights are denoted

by wji and the hidden-to-output weights are denoted by wkj. The subscript i is the index for

the input layer, the subscript j is the index for the hidden layer, and the subscript k is the

index for the output layer.

The inputs into the network are the features of interest from the original signal. Feature

extraction is an important part of classification and often times complex algorithms must

be developed to perform this step accurately. The number of outputs of the network is

typically equal to the number of classes of interest. For example, the network may be

trained so that each output corresponds to one class of signal, with the desired output

value set equal to 1 if the input features represent that class and desired output value set

equal to -1 if the input features do not represent that class. Another option is to have just

one output for the network with different desired output value ranges corresponding to

different classes. The activation functions at the hidden and output layers can be chosen

appropriately to give non-linear decision boundaries. The weights (both input-to-hidden

and hidden-to-output) can be tuned by using training data and backpropagation learning to

give application specific decision boundaries for accurate classification. Each circle in

Figure 4.2 represents a unit, or “neuron,” which derives its name from the function of a

biological neuron. Since these multi-layer networks are comprised of large numbers of

neurons, the classifier gets the name “Artificial Neural Network.”

76

Figure 4.2. Visual representation of the design of a neural network classifier. The network in this example has 6 inputs, 1 hidden layer, 5 neurons in the hidden layer, and 1 output.

During feedforward operation, which is when input features are being classified, the

network in Figure 4.2 operates as follows. At each hidden layer neuron, the inputs, x, are

weighted and summed according to the formula:

3c"v = ��Lv�É

�^9 + Lv� = ��Lv�É

�^� (4.1)

where d is the number of inputs. The bias weight, wj0, applies a constant offset value to the

summation. The output from each hidden neuron is then determined by:

qv = �/3c"v0 (4.2)

where f() is an activation function that limits the values of the output and, if desired,

applies some nonlinearity to the input. The outputs from the hidden layer become the

inputs to the output layer of the network. At each output layer, all of the outputs from the

hidden layer are summed according the formula:

3c"® = �qvL®v + L®�FÊv^9 = �qvL®v

FÊv^� (4.3)

y5y1y2 y3 y4

x1 x2 x3 x4 x5

x1 x2 x3 x4 x5

z1

t1

output

input

hidden

target

wkj

wji input-to-hidden weights

hidden-to-output weights

x6

x6

77

where nH is the number of neurons in the hidden layer. The output from each output unit is

determined by:

h® = �/3c"®0 (4.4)

Assuming the same activation function in both the hidden and output layers, Equations

(4.1)–(4.4) can be combined to get what is called the discriminate function, gk(x), which is

equal to the output of the network:

n®/+0 ≡ h® = � Æ�L®vFÊv^9 � Ð�Lv�� + Lv�

É�^9 Ñ + L®�Ç (4.5)

This equation effectively maps the input feature vector x to the classes, which are the

output values. It has been shown that any continuous function can be represented in terms

of a multi-layer neural network for proper number of hidden neurons, proper

nonlinearities in the activation functions, and proper selection of weights [61].

Before operating the network in feedforward mode, the weights must be properly

adjusted by training the network. To train the network there must be a learning rule for

adjusting the weights. The problem of adjusting the hidden-to-output layer weights is

fairly straightforward, requiring minimization of an error function that calculates the

difference between the desired output values and the actual output values. Since the

desired values of the outputs from the hidden layer are not known, adjustment of the input-

to-hidden layer weights requires a different approach. Backpropagation solves this

problem by calculating an effective error at each hidden layer and using it with an error

criterion function to minimize the error and adjust the weights accordingly.

First consider an error function that sums the squared difference between the desired

output tk and the actual output zk. The mean squared error criterion function is [60]:

/w0 = 12�/"® − h®02

�®^1

(4.6)

where w represents all of the weights in the network (both input-to-hidden and hidden-to-

output) and c is the number of units in the output layer. The weights are first initialized to

random values and then changed according to a gradient descent learning rule. The

learning rule adjusts the weights in order to reduce the error calculated from Equation

(4.6). The learning rule is:

78

∆w=-Ë $ $w (4.7)

where η is called the learning rate, which can be adjusted to effect how much the weights

are changed for a given amount of error. At each step in the learning process, the weights

are adjusted by adding ∆w to the values of the weights (w) from the previous step. This is

done until the error is minimized and the actual output values converge to the desired

output values for the given set of training data.

Equation (4.7) must be evaluated for both the input-to-hidden and hidden-to-output

weights. Starting first with the hidden-to-output weights and using the chain rule:

$ $L®v = $ $3c"® $3c"®$L®v = −�® $3c"®$L®v (4.8)

where �® is the sensitivity of output unit k. Assuming the activation function is

differentiable, the chain rule can be used to rewrite the sensitivity as:

�® = − $ $3c"® = − $ $h®$h®$3c"® = /"® − h®0�S/3c"®0 (4.9)

The last derivative in Equation (4.8) can be evaluated using Equation (4.3), which gives:

$3c"®$L®v = qv (4.10)

The results shown in Equations (4.7) – (4.10) can then be combined to give the learning

rule for the hidden-to-output weights, which is:

ΔL®v = Ë/"® − h®0�S/3c"®0qv (4.11)

Equation (4.7) is once again used to derive the learning rule for the input-to-hidden

weights. Using the chain rule gives:

$ $Lv� = $ $qv

$qv$3c"v $3c"v$Lv� (4.12)

The first term on the right-hand side of this equation can be rewritten as:

$ $qv = $$qv 812� /"® − h®0:�®^9 ;

= − �/"® − h®0 $h®$qv�

®^9

(4.13)

79

= −�/"® − h®0 $h®$3c"® $3c"®$qv�

®^9

= −�/"® − h®0�S/3c"®0L®v�

®^9

This equation shows how the outputs from the hidden layer, yj, are related to the error at

the output layer. The sensitivity for a hidden unit is defined as:

�v ≡ �S/3c"v0� L®v�®�

®^9 (4.14)

Combining Equation (4.7) and (4.12) – (4.14) yields the learning rule for the input-to-

hidden layers:

ΔLv� = Ë k� L®v�®�

®^9 l �S/3c"v0�� (4.15)

Equation (4.11) and (4.15) can then be used with various training protocols to tune the

network for the given classification problem. The training set, with known desired output

values called target values, can be used in different configurations to adjust the weights

during the backpropagation training. Three common training protocols are stochastic,

batch, and on-line. Stochastic training is when the training patterns are randomly selected

from the training set like a random variable. Batch training is when all of the training data

is fed into the network before any training takes place. The training data is usually

presented to the network several times. In on-line training each pattern from the training

set is presented only once and learning takes place after every pattern is presented.

Often times, the details of the design of a network are determined using several heuristic

approaches. There are also several general guidelines published in the literature that have

proven to produce good results in many applications and therefore should be noted. Some

design considerations include selection of the activation functions, target values, training

data, number of hidden neurons, initial values for weights, and learning rates.

A common selection for the activation functions in the hidden and output layers is the

sigmoid function, which has the form:

�/�0 = � tanh/�3c"0 = � acCÓFÔf − cAÓFÔfcCÓFÔf + cAÓFÔfe (4.16)

80

The sigmoid function saturates the output f at the value a for large magnitude input b net.

Because of this saturation, the output of the network is limited, which is often beneficial.

Additionally, the nonlinearity of the sigmoid function allows for the generation of complex,

nonlinear decision boundaries for the classes. The sigmoid is also smooth and

differentiable, which means that it can be used with Equations (4.11) and (4.15).

The data collected from the HTL was used to train and validate the ANN. Supervised

learning was used since training data was obtained for known locations of rail surface

irregularities. A backpropagation learning algorithm was used since the network has one

hidden layer. A hidden layer with 5 neurons was chosen since multilayer neural networks

have proven to be beneficial in recognizing and classifying patterns from nonlinear

functions [61]. It was found that 5 neurons in the hidden layer provided good accuracy

during the validation phase. Using less neurons did not allow the network to properly map

the nonlinearities present in the network inputs, and more neurons overcomplicated the

network and caused the final results to lose accuracy.

The input layers to the neural network are the 6 bins from the windowed FFT of the

vertical acceleration data, with frequency ranges centered around 10 Hz for x1, 20 Hz for x2,

30 Hz for x3, 40 Hz for x4, 50 Hz for x5 , and 60 Hz for x6. There is one output in the network

with its value being dependent on the surface irregularity. During training, the target

vector value was assigned as follows: t1 = 25 for a broken rail, t1 = 15 for a surface

fracture/impending break, t1 = 5 for process noise, and t1 = 0 for nothing. Since a rail

surface pattern is characterized by a series of data points in the time domain rather than a

single point, and since a moving FFT window is used, there will be some overlap in the

input patterns. This means that many of the windows of input data will contain portions of

patterns from two classes. This gives rise to transition ranges, where the data window is

transitioning from one class of pattern to another. In order to account for this in the

training of the network, values in the target vector were assigned based on a Gaussian

distribution rather than a square distribution. Figure 4.3 illustrates this technique. The top

plot shows the vertical acceleration pattern from a broken rail and the bottom plot shows

the target value for the output of the neural network. Since the FFT window spans 100

data points (about 0.39 s for a 256 Hz sampling rate), the window will begin to pick up

some of the contents of the rail break while analyzing some of the data before the rail

81

break. During training of the network, the gradual ascent to the maximum value rather

than a quick jump in the target vector helps the network to handle the problem of class

overlap. The Gaussian envelope is thus set in the target vector so that the maximum value

of 25 corresponding to a rail break coincides with the starting point of the rail break. Note

that in Figure 4.3, the maximum value of the target output coincides with the start of the

broken rail pattern around 0.19 s. The same procedure was used to set the target values

for surface fractures and process noise.

Figure 4.3. (top) Vertical acceleration pattern for a rail break. (bottom) Target output for a rail break.

During the experimental phase of this study, one hour and seven minutes of bogie side

frame vertical acceleration data was collected. With the 256 Hz sampling rate, this yielded

1,029,221 pieces of data. Since the pre-processing/feature extraction technique uses a

moving FFT window of size 100 points, this resulted in 1,029,121 pieces of available

information for training and validating the network. 800,000 data points were used to

train the network using an on-line training protocol. The other 229,221 data points were

used to validate the network. The known events within the entire data set are 22 locations

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-10

-5

0

5

10

Vertical Acceleration (g)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40

5

10

15

20

25

Target Output

Time (s)

82

of process noise, 3 locations of surface fractures/impending breaks, and 5 locations of

broken rails. The training set consisted of 17 locations of process noise, 2 locations of

surface fractures/impending breaks, and 4 locations of broken rails. The validation set

consisted of 5 locations of process noise, 1 location of a surface fracture/impending break,

and 1 location of a broken rail. The network was developed in the Matlab Simulink

environment by using the Adaptive Neural Network library [65]. The network was trained

using backpropagation learning. The simulation results from this algorithm are presented

in Section 4.4.2.

4.3 Development of Wavelet Transform-Based Algorithm

As mentioned previously, the two steps to the defect detection algorithm are feature

extraction and feature classification. For the Wavelet-based algorithm, these two steps can

be equated to: 1) a defect detection phase, and 2) a defect classification phase. In [66], a

generic defect detection and classification algorithm using wavelets was presented. The

detection phase utilizes the Wavelet Transform to locate irregularities in the signal. In

recent years, wavelet analysis has shown to be useful in event detection. Wavelets are

compactly supported, which allows them to be used for decomposing a signal into a time-

scale (time-frequency) representation. Traditional digital signal processing makes use of

the Fourier Transform, which gives the frequency content of a signal but yields no

information regarding the location in time or space at which each frequency occurs. The

Wavelet Transform provides localization in both the time and frequency domains, which

makes it extremely beneficial in the analysis of non-stationary or transient signals

[37],[45], like those expected in rail defect detection. Wavelet analysis has been used in

various event detection applications, including crack identification in beams [40] and plates

[42], machinery health monitoring [39], detection of climate change [48], and event

location in pacemakers [49].

The defect classification phase of the algorithm uses the output from the defect detection

phase as an input into an ANN. A similar method has been used in [59] and [58] to detect

faulty wheel bearings with a wayside acoustic system. Features are extracted using the

discrete wavelet transform and subsequently classified using a genetic algorithm. However,

83

with the current problem of locating rail defects in real time with a limited accelerometer

signal, the features present in the signal are much different than the features present in an

acoustic signal for faulty wheel bearings. The neural network in this classification

algorithm uses the original signal and the first four scales of the wavelet transform as

inputs, and classifies defects as either a rail break or an impending rail break.

The algorithm is tested using a real signal collected from an accelerometer mounted to

the bogie side frame of a freight car. The signal contains an instance of an impending rail

break/surface crack and a rail break. The proposed wavelet-based defect detection

algorithm takes into account the amplitude of the signal as well as the local regularity and

frequency content of the signal, and is therefore able to distinguish between an impending

break and process noise.

A Simulink model of the defect detection and classification algorithm is shown in Figure

4.4. There are three main steps to this algorithm: the Wavelet Transform, thresholding of

the wavelet transformed signal, and classification using the ANN. The Wavelet Transform

and thresholding steps are responsible for locating irregularities in the original

accelerometer signal and distinguishing them from all other process noise and sensor noise

in the signal. The neural network step is responsible for classifying the defects from the

first two steps, which in this case is distinguishing between an impending rail break and a

rail break.

84

Figure 4.4. Simulink model of defect detection and classification algorithm.

4.3.1 Feature Extraction: Wavelet Transform

The feature extraction phase of this algorithm is performed using the Wavelet Transform.

A wave with a finite, concentrated amount of energy is mathematically considered to be a

wavelet, ¢/"0, if and only if its Fourier Transform, Ψ/T0, satisfies:

} |Ψ/T0|:|T| �TC]A]

< +∞ (4.17)

Equation (4.17) suggests that:

} ¢/"0�"C]A]

= 0 (4.18)

which states that a wavelet must have zero mean value. Therefore, when wavelets are used

as a basis to represent a signal, they are able to provide bands of both time and frequency

information because of their compact support. The classical signal processing method of

the Fourier Transform uses theoretically infinite length sine and cosine waves as a basis,

and therefore is only able to provide the frequency content of the signal.

85

The Wavelet Transform is the name of the operation that generates a set of coefficients

representing the convolution of a signal with a set of scaled and shifted versions of a

mother wavelet. The scaled and shifted versions of the mother wavelet form the set of

basis functions over which the signal is decomposed. The Continuous Wavelet Transform

(CWT) of a function f(t) is defined as:

�/�, i0 = 1√� } �/"0¢∗ #" − i� ( �"C]A]

(4.19)

where ¢∗/"0 is the complex conjugate of the wavelet function. Equation (4.19) shows that

the CWT is the inner product of the signal, f(t), with scaled (s) and shifted (u) versions of

the mother wavelet function ¢/"0. The continuous scale index, s, is an integer ranging from

zero to the maximum value of desired scaled. The continuous position index, u, ranges

from zero to N – 1 where N is the length of the signal f(t).

The Wavelet Transform is able to accurately detect changes and irregularities in the

signal. This can be seen by investigating the use of the scaling function,s/"0. For every

wavelet function, ¢/"0, which acts as a detail, high-pass filter, there exists also a scaling

function, s/"0, that acts as smoothing, low-pass filter. If the wavelet function is chosen to be

the first derivative of the scaling function with respect to time, Equation (4.19) can be

written as:

�/�, i0 = 1√� } �/"0 ��i s∗ #" − i� ( �"C]A]

= ��i Æ 1√� } �/"0s∗ #" − i� ( �"C]A]

Ç (4.20)

The CWT is therefore proportional to the first derivative of f(t) smoothed by s/"0.

Smoothing and differentiating the signal results in a new signal that contains information

about the regularity of f(t) in its peaks. This means that the modulus maxima of Wf(s,u) are

the locations of the sharp irregularities in f(t).

The CWT is a non-orthogonal Wavelet Transform. Since the position (u) is continuous,

the wavelet coefficients Wf(s,u) contain redundant information. For applications such as

de-noising and compression, this redundancy is not desirable because perfect signal

reconstruction is needed. In such applications, the discrete wavelet transform (DWT) with

an orthogonal wavelet basis would be used. The DWT can be implemented using a fast,

recursive filter bank algorithm, and produces a set of wavelet coefficients of the same

86

length as the original signal, which is a powerful tool. However, in event detection

applications where signal reconstruction is not necessary, the redundancy of information in

the CWT wavelet coefficients does not produce any major drawbacks, other than increased

computation times. Therefore, the CWT is used in the wavelet-based algorithm. Choosing

to use the CWT instead of the DWT in event detection applications has two main benefits:

1) the continuous position index, although producing redundant information, allows the

same time resolution across all scales, and 2) since there is no requirement to select an

orthogonal wavelet basis, there is freedom to choose a wavelet that matches well with the

signatures of interest in the signal.

The ability of the CWT to locate irregular events in a signal can further be seen in its

ability to estimate the local Lipschitz exponent of a signal, which is a mathematical measure

of the degree of regularity in the signal. Mallat [38] showed that the regularity of a point in

a signal can be estimated by observing the decay of the wavelet coefficient modulus across

Wavelet Transform scales. Mallat showed that a function f(t) is Lipschitz Ãat point u if and

only if there exists some non-negative constant A such that:

| �/�, i0| ≤ <�Ä (4.21)

Taking the logarithm of both sides of this equation gives:

log| �/�, i0| ≤ log/<0 + Ã log/�0 (4.22)

The wavelet coefficient values with scale values can then be fitted to Equation (4.22) using

a linear least-squares regression to determine the value of Ã at each location u.

In performing the Wavelet Transform, a wavelet should be chosen with the proper

number of vanishing moments, which is related to its ability to represent a signal with

certain regularity. A wavelet has d vanishing moments if it satisfies the criteria:

} "Å¢/"0�"C]A]

= 0for0 ≤ � ≤ � (4.23)

where d is an integer. A wavelet with d vanishing moments is orthogonal to a polynomial of

degree up to d – 1, and can accurately represent a signal with regularity Ã for � ≥ Ã.

The Daubechies family of wavelets has proven itself to be effective in event detection

because it provides bases of orthonormal wavelets with a maximum number of vanishing

moments for a given support length [45]. To determine a suitable number of vanishing

87

moments for the analyzing wavelet, the local regularity of the test signal was calculated by

using Equation (4.22). It was found that a Daubechies wavelet with three vanishing

moments (db3) was sufficient to span the regularity of the test signal (sampling rate of 256

Hz) in this study. The center frequency values at the first 5 scales of the db3 wavelet family

for a signal sampled at 256 Hz are shown in Table 4-1. From observation of the test signal,

it was found that the vertical acceleration signals from an impending rail break (surface

crack) and a rail break contain dominant frequencies around 40 Hz (see Section 3.3.2).

Additionally, after performing the CWT and observing the results, it was found that no

significant information concerning rail defects was located in scale 5 or higher. Therefore,

the first 4 scales of the CWT were used in identifying the defects present in the signal.

Table 4-1. Center frequency values of the first 5 scales of the db3 wavelet for a signal sampled at 256 Hz

Scale Center Frequency (Hz)

1 300 2 150 3 75 4 37.5 5 18.75

Thresholding of the wavelet transformed signal serves the purpose of removing sensor

noise and process noise from the signal, leaving only information concerning the location of

defects. Process noise can be considered anything that causes a significant vertical

acceleration signal but is not a defect, such as frogs, switches, joints, and other special track

components. It was observed from the test signal that hard threshold values could be

applied at each scale to successfully remove all sensor and process noise and leave only the

signatures generated from the impending break and the break. The value of the required

threshold at each CWT scale was observed to a have a sigmoidal shape. The equation for

determining the hard threshold value at each scale is:

%/�0 = Ì� + cAÂv (4.24)

where T is the threshold value, j is the discrete scale, and the constants in the equation are

P = 0.22, a= 0.025, and m = 1.25. Table 4-2 shows the threshold values calculated for the

first four scales of the CWT using Equation (4.24).

88

Table 4-2. Threshold values at the first four scales of the CWT. Scale (j) Threshold (T(j)) 1 0.71 2 2.05 3 4.53 4 6.93

The threshold values from Table 4-2 are applied to the wavelet transformed signal by using

the following criteria:

qv,® = Ö×�v,®×, ×�v,®× ≥ %/�00, ×�v,®× < %/�0 (4.25)

where yj,k is the output of the thresholding step that is sent to the next step as an input to

the neural network.

The threshold values in Table 4-2 are not necessarily universal for all signals that will be

encountered during typical operation. It can only be conclusively said that these threshold

values are accurate for the class of signals represented by the test signal used in this study.

For a more comprehensive set of signals spanning a wider range of operating conditions,

including various forward speeds and payloads, new threshold values would have to be

developed.

4.3.2 Feature Classification: Artificial Neural Network

The neural network used for classification in the wavelet-based algorithm has the same

general architecture as described in Section 4.2.2. The neural network receives five inputs:

the raw signal and the first four thresholded scales of the CWT. The network has one

hidden layer with 100 neurons and uses a radial basis function as an activation function

[67]. The network has one output. The network was trained with learning data to produce

an output of 0 when there is no defect, an output of 0.7 when there is an impending rail

break/surface crack, and an output of 1.0 when there is a rail break.

Artificial training data was generated by using the test signal and adding various levels

of Gaussian white noise to it. Four sets of training data were produced with signal to noise

ratios of 97.86, 24.62, 10.86, and 6.13. The network was trained until the error converged

to a value close to zero, and the resulting weights were used in the final neural network.

89

4.4 Development of Wavelet-Intensity Factor Algorithm

Mallat [38] showed that the local regularity of a signal can be estimated by using the

wavelet transform to calculate the Lipschitz exponent at each time step in the signal. The

developed formulas can be used to gain insight into the power of the wavelet transform in

locating irregularities in a signal. This method can also be used to calculate the Lipschitz

exponent directly to provide information that can be used in event detection and

classification. The premise is that the decay of wavelet coefficients across Wavelet

Transform scales can be related to the value of the Lipschitz exponent. Since the Wavelet

Transform provides localization in the time domain, the time local Lipschitz exponent can

be estimated.

Mallat [38] showed that the regularity of a function at any point t = t0 can be estimated

by observing the decay of the wavelet coefficients of the CWT across scales, s. It was shown

that a function f(t) is uniformly LipschitzÃ if and only if there exists some non-negative

constant A such that

| �/�, i0| ≤ <�Ä (4.26)

where �/�, i0 are the wavelet coefficients calculated from Equation (4.19). Taking the

logarithm of both sides of Equation (4.26) gives:

log| �/�, i0| ≤ log/<0 + Ã log/�0 (4.27)

The modulus maxima of the wavelet coefficients can then be plotted as the ordinate with

scale s as the abscissa with log scale for each location u. The Lipschitz exponent Ã can then

be determined for each location on the time axis by fitting the data to Equation (4.27) with

a linear least squares regression. It was further presented by Huang [39] that the equation

for this least squares method is:

Ã/i0 = ∑ /log:| �/�, i0| log: �0ºt^9 − M∑ log:| �/�, i0|ºt^9 NM∑ log: �ºt^9 N ∑ /log: �0:ºt^9 − M∑ log: �ºt^9 N:

(4.28)

The Lipschitz exponent describes the type of singularity, or in other words, the degree of

regularity. For example, a function that is differentiable once at some point t = t0 has

Lipschitz exponent Ã = 1 at t = t0. A step function has Ã = 0 and a Dirac impulse has Ã = -1.

Therefore, the sharper the irregularity, the lower the value of the Lipschitz exponent. In a

90

signal, singularities of interest are typically characterized by low-magnitude, positive value

Lipschitz exponent, while noise is typically characterized by a negative value Lipschitz

exponent [38],[40],[41].

As Mallat showed that the degree of a singularity can be calculated by determining Ã in

Equation (4.27), Douka [40],[42] showed that the severity (or intensity) of a defect can be

estimated by determining the value of A in Equation (4.27). Accordingly, A has been called

the intensity factor. Douka found that for a cantilever beam with measured mode shapes,

the intensity factor A of a crack has a second order polynomial relationship with crack

depth.

As the train wheel envelopes a rail surface irregularity, a sharp jump in vertical

acceleration will appear in the accelerometer signal. This sharp irregularity in the signal

will show up as a low value Lipschitz exponent calculated from Equation (4.28).

Regardless of the type of rail surface irregularity, whether it be a broken rail, a surface

fracture, or a special track component, a low value of local Lipschitz exponent will be

present but may not be noticeably different for different types of surface irregularities. The

intensity factor, however, provides a means for measuring the magnitude of an irregularity

for a given Lipschitz exponent. The relative intensity factor for a given set of data is

therefore observed as a means for determining the locations of rail defects within a signal.

Figure 4.5 shows a diagram of the Wavelet-Intensity Factor algorithm. The raw

accelerometer signal is sent to the algorithm located on a computing device. The algorithm

first calculates the first n scales of the CWT. These values are then used to calculate the

local Lipschitz exponent for each increment of time. The Lipschitz exponent values are

then used to calculate the local intensity factor values. Finally, the local intensity factor

values are thresholded and an appropriate classification is made.

91

Figure 4.5. Block diagram of Wavelet-Intensity Factor algorithm.

4.5 Simulations and Results

The following sections present the results of the previously described defect detection

algorithms. Section 4.5.1 first shows the results of a standard commercial threshold-based

defect detection algorithm, Section 4.5.2 shows the results of the Fourier Transform-based

defect detection algorithm, Section 4.5.3 shows the results of the Wavelet Transform-based

defect detection algorithm, and Section 4.5.4 shows the results of the Wavelet-Intensity

Factor algorithm.

4.5.1 Results: Threshold-Based Algorithm

Currently, threshold-based defect detection algorithms are used commercially in

products such as the ENSCO V/TI monitor [17]. A typical threshold-based algorithm for

processing the signal in this study would simply monitor the amplitude of the acceleration

and flag any incidents where the acceleration jumps above some predetermined threshold

value. In the case of the signal in Figure 4.6, a typical threshold value may be around 5 g,

which is shown in the figure by the dotted lines. In this case, only the broken rail would be

flagged and the impending break would be missed since the impending break signature

does not have as high amplitude as the peaks at 16.5 s and 253 s, which are only process

noise. Reducing the threshold value to allow detection of the impending break would also

cause the process noise to be flagged as well. Flagging of these peaks would be considered

92

a false positive, which is not desirable as this would require time and money to perform an

unnecessary inspection of these locations.

Figure 4.6. Original signal processed by a threshold-based algorithm. Dashed lines show threshold

values for flagging of defects.

4.5.2 Results: Fourier Transform-Based Algorithm

Figure 4.7 shows a comparison of the output of the neural network to the desired

(target) values for the validation step. The results show that the neural network output

closely matches the desired values. The bottom plot of Figure 4.7 shows the original bogie

side frame vertical acceleration signal with rail surface irregularities labeled. The locations

of the rail surface irregularities were determined by a thorough visual inspection of the

track before, during, and after experimental data collection. Comparison of the neural

network output to the original signal reveals the power of the feature extraction technique

and neural network to accurately classify the patterns present in the original signal. It is

seen that all rail surface irregularities have been located and clearly distinguished from one

another.

For example, the impending break present at data point 826,500 shows up as a small

amplitude oscillation in the original signal and initially is not noticeably different than, say,

the process noise oscillation at data point 888,500. However, the neural network output

0 100 200 300 400 500-6

-4

-2

0

2

4

6

8

Time (s)

Acceleration (g's)

impending break

break

93

clearly distinguishes between these two classes of patterns by accurately assigning a value

of approximately 16 to the impending break and approximately 0 to the process noise.

Similarly, based on amplitude inspection, the broken rail at data point 937,000 is not

noticeably different than the high amplitude oscillation at data point 837,100, which

corresponds to nothing of interest. In fact, this random oscillation generates a maximum

vertical acceleration of 5.13 g while the broken rail generates a maximum amplitude of

4.74 g. The proposed neural network classifier gives an accurate classification of the

vertical acceleration patterns. The final results show no Type I or Type II errors in the

neural network output.

Figure 4.8 shows zoomed in portions of the neural network output with target values for

a selected portion of process noise, the surface fracture, and the broken rail from the

validation data. This shows that the neural network is able to accurately locate the events

in the signal: process noise at data point 888,469, surface fracture at data point 826,472,

and broken rail at data point 937,648.

94

Figure 4.7. (top) Neural network output and desired (target) values for validation phase. (bottom)

Original vertical acceleration data with rail surface irregularities labeled.

8.5 9 9.5 10

x 105

-5

0

5

10

15

20

25

30Nothing = 0 ; Process Noise = 5 ; Impending Break = 15 ; Break = 25

ANN Output

Actual Output

Desired Output

8.5 9 9.5 10

x 105

-8

-6

-4

-2

0

2

4

6

8Original Signal

Acceleration (g)

Data Number

impending break break

process noise

95

Figure 4.8. Zoomed in neural network output and desired (target) values for validation phase:

(top) process noise, (middle) surface fracture/impending break, (bottom) broken rail.

4.5.3 Results: Wavelet Transform-Based Algorithm

The test signal fed into the Wavelet Transform-based defect detection algorithm was

from the IFC driven around the HTL twice. During the first run, there was an impending

rail break that showed up as a surface crack and in the second run the rail had broken in

8.881 8.882 8.883 8.884 8.885 8.886 8.887 8.888

x 105

-1

0

1

2

3

4

5Process Noise

Actual Output

Desired Output

8.26 8.261 8.262 8.263 8.264 8.265 8.266 8.267 8.268 8.269

x 105

0

5

10

15

Surface Fracture/Impending Break

ANN Output

9.373 9.374 9.375 9.376 9.377 9.378 9.379 9.38

x 105

0

5

10

15

20

25

Broken Rail

Data Number

96

the same location of the surface crack. The two signals were combined into one signal to

feed into the defect detection algorithm for validation purposes. Figure 4.9 shows the

signal that was fed into the algorithm. It is known that the impending rail break (surface

fracture) is located around 142 s and the rail break is located around 373 s. The original

signal is shown in Figure 4.9.

Figure 4.9. Original signal fed into the defect detection algorithm.

As described in the previous section, the first step of the algorithm is to calculate the first

four scales of the CWT of the original signal using a db3 wavelet. The result of this step is

shown in Figure 4.10. The top plot is the original signal, the next is the scale 1 detail

coefficients, then the scale 2 detail coefficients, and so on.

0 100 200 300 400 500-6

-4

-2

0

2

4

6

8

Time (s)

Acceleration (g's)

97

Figure 4.10. CWT of original signal using a db3 wavelet.

The next step is to pass the wavelet transformed signal through the thresholding phase,

as described previously. Figure 4.11 shows the results of the thresholding step. The

results show that the thresholding successfully removes all unnecessary information

(sensor and process noise) from the signal and leaves only information concerning the

impending break at 142 s and the break at 373 s.

All of the information in Figure 4.11 (raw signal and thresholded CWT) is passed on to

the next step for classification, the ANN. The output from the neural network step is shown

in Figure 4.12. The blue line is the actual output from the ANN and the red line is the

desired output based on the characterization that no defect gives output of 0, an impending

break gives an output of 0.7, and a broken rail gives an output of 1.0. The sensor and

process noise from the signal show up as small amplitude output, but are nowhere near the

value of 0.7 for an impending break. The impending break and break are also distinguished

0 100 200 300 400 500-10

0

10

D0

0 100 200 300 400 500-1

0

1

D1

0 100 200 300 400 500-5

0

5

D2

0 100 200 300 400 500-10

0

10

D3

0 100 200 300 400 500-10

0

10

D4

Time (s)

98

from one another by a difference in value. These values are very close to the desired

output values. Therefore, both the impending break and the break in this data set are

successfully detected and distinguished from one another without issuing any false

positives for the given set of data.

Figure 4.11. Thresholding of wavelet transformed signal.

0 100 200 300 400 500-10

0

10

D0

0 100 200 300 400 5000

0.5

1

D1

0 100 200 300 400 5000

2

4

D2

0 100 200 300 400 5000

5

10

D3

0 100 200 300 400 5000

5

10

D4

Time (s)

99

Figure 4.12. Third step of the algorithm: classification using artificial neural network.

4.5.4 Results: Wavelet-Intensity Factor Algorithm

Figure 4.13 shows the results of the Wavelet-Intensity Factor algorithm for a set of data

consisting of 20 high amplitude signatures collected from an ENSCO V/TI monitor at

various forward speeds for a sampling frequency of 200 Hz. The V/TI monitor was

mounted to a wheel axle box and the signal shows wheel vertical acceleration for the

enveloping of various rail surface irregularities. The signatures were combined to make

one signal, which is shown in the top plot of Figure 4.13. The second plot shows the local

signal regularity, α, calculated using Equations (4.27) and (4.28) with a Daubechies wavelet

with four vanishing moments. The third plot shows the local intensity factor, A, also

calculated using Equations (4.27) and (4.28). The bottom plot shows the type of each of the

signatures in the original signal. A blue bar is used to denote an axle impact corresponding

to nothing of interest, a yellow bar is used to denote a broken frog, and a red bar is used to

denote a broken rail. The results of the analysis show a high intensity factor for each of the

three locations of a broken rail (sample #7, #12, #19) and relatively low intensity factor for

all other signatures.

Figure 4.14 shows the results of the Wavelet-Intensity Factor algorithm for a second

round of data collected from an ENSCO V/TI monitor. The data set consists of 13 wheel

0 100 200 300 400 500-0.4

-0.2

0

0.2

0.4

0.6

0.8

1Nothing = 0 ; Impending = 0.7 ; Defect = 1

Time (s)

ANN Output

Actual Output

Desired Output

100

vertical acceleration signatures. The data set contains four locations of broken rails, which

are indicated by the red bars in the bottom plot (sample #2, #3, #7, #11). The intensity

factor plot shows that the four locations of broken rails are distinguished from the other

signatures by the highest amplitude intensity factor. The results of the Wavelet-Intensity

Factor algorithm display the capability of the proposed methodologies to locate rail defects

within a vertical acceleration signal.

Figure 4.13. Results of the Wavelet-Intensity Factor-algorithm for round 1 of ENSCO V/TI monitor

data. (top) original signal, (second) local Lipschitz exponent/signal regularity, (third) intensity factor, (bottom) signature type, red bar indicates broken rail.

101

Figure 4.14. Results of the Wavelet-Intensity Factor-algorithm for round 2 of ENSCO V/TI monitor

data. (top) original signal, (second) local Lipschitz exponent/signal regularity, (third) intensity factor, (bottom) signature type, red bar indicates broken rail.


This chapter presented two defect-detection algorithms for locating rail surface

irregularities from a bogie side frame vertical acceleration signal and one algorithm for

locating rail surface irregularities from an ENSCO V/TI Monitor signal. The first two

algorithms use an artificial neural network for feature classification, but utilize different

102

frequency-domain approaches for feature extraction from the original signal. The Fourier

Transform-based algorithm uses a moving FFT window to perform frequency analysis on a

section of data. The Wavelet Transform-based algorithm uses the CWT to perform a

simultaneous time-frequency analysis on the signal. Both algorithms show successful

results and out-perform the threshold-based algorithm. Both algorithms successfully

detect the broken rail and the impending rail brake without issuing any false positives. The

Fourier Transform-based algorithm performs a frequency analysis on all of the data and is

able to classify signatures into four different classes: broken rail, impending break, process

noise, and nothing of interest. The Wavelet Transform-based algorithm classifies

signatures into three different classes: broken rail, impending break, and everything else.

The Wavelet Transform-based algorithm specifically targets the signatures of rail surface

defects, whereas the Fourier Transform-based algorithm considers all signatures and then

seeks to classify them accordingly.

The third algorithm uses the CWT as a basis to calculate a local intensity factor. The

CWT coefficients are used with Mallat’s method to calculate local signal regularity and local

intensity factor. It has been shown that for proper wavelet selection, the local intensity

factor is strongly related to defect type. All of the broken rails in the ENSCO V/TI Monitor

data sets were successfully detected with no false positives issued.

The major conclusions from this chapter are:



• Current signal processing techniques for extracting rail defect information from a

vertical acceleration signal are basic and typically only consider the magnitude of

the signal, resulting in frequent false positives and false negatives.

• Frequency analysis shows that there is sufficient information in the bogie side

frame vertical acceleration signal to distinguish between signatures generated from

various rail surface irregularities.

• A small rail head surface fracture can quickly propagate and turn into a broken rail

after only a few train cars have passed over it. Therefore, early detection of rail

surface defect is necessary.

103

• The developed Fourier Transform-based feature extraction and neural network-

based feature classification algorithm analyzes the signal with a moving FFT

window and classifies signatures into one of four categories: broken rail, impending

break, process noise (special track), and nothing of interest. The algorithm

successfully detects the impending break and broken rail from the HTL data

without issuing any false positives or false negatives.

• The developed Wavelet Transform-based feature extraction and neural network-

based feature classification algorithm uses wavelets to target the defect signatures

present in the signal. The algorithm successfully detects the impending break and

broken rail from the HTL data without issuing any false positives or false negatives.

• Both the Fourier Transform-based and Wavelet Transform-based algorithms

outperform a standard commercial threshold-based algorithm for the HTL data.

However, the algorithms require more extensive training in order to be used for a

wider range of operating conditions.

• The proposed Wavelet-Intensity Factor algorithm successfully located all instances

of broken rails within the ENSCO V/TI Monitor data sets without issuing any false

positives. The intensity factor approach targets defect signatures and locates them

by using the wavelet coefficients to estimate signal local regularity and intensity

factor and relating these to defect signature class.

The algorithms presented in this chapter are by no means a definite and/or final solution

to this problem. The algorithms were trained with a limited amount of data representing a

limited range of train operating conditions. The current results simply show the capability

and potential for such algorithms to detect and locate impending rail breaks and broken

rails where current algorithms are not able to. In order for the algorithms to be accurate

for a wider class of signals, they would need to be trained using signals generated from a

broader range of operating conditions, including various forward speeds and payloads.

104

Chapter 5

Dynamic Wheel-Rail Interaction Model ______________

5.1 Introduction

Because of the limited amount of data available for training the defect detection

algorithms presented in Chapter 4, artificial training data must be generated in order to

cover the range of operating conditions necessary to implement the algorithms in an

onboard defect detection system in real-time. The defect detection and classification

methods should be able to accommodate a wide range of signal classes. This chapter

presents a dynamic wheel-rail interaction model that can be used to generate training data

for various train operating conditions, including different forward speeds and payloads, as

well as different track characteristics and surface irregularities. The following sections

discuss the development and results of this model.

5.2 Single-Wheel Model

In developing the dynamic wheel-rail model, a single-wheel model is first considered

and equations of motion for the system are developed. This begins with a train driving in a

straight line at constant speed over a continuous rail. Complexity is then added to the

model to account for additional dynamics and various rail surface irregularities. Figure 5.1

shows a diagram of a single-wheel, single-rail model. This model considers only the

vertical dynamics of the system and neglects all other translational and rotational motions.

The train is modeled as a mass (car body) sitting on top of a spring and damper in parallel,

connected to a bolster sitting on top of a spring and damper in parallel, connected to a

wheel. The mass represents 1/8 of the car body total mass, the bolster represents ¼ of the

mass of a single bolster, and the wheel represents the sum of ½ of the mass of a side frame

105

plus ½ of the mass of an axle plus the mass of a single wheel. The spring and damper

connecting the wheel to the bolster is called the primary suspension, and the spring and

damper connecting the bolster to the car body is called the secondary suspension. Each

body in the train model has a single translational degree-of-freedom in the vertical

direction. The rail is modeled as a distributed parameter cantilevered Euler-Bernoulli

beam. The pads, ties, and ballast that support the rail are modeled by lumped parameter

systems. The rail sits on pads equally spaced along the length of the beam. The pads are

modeled as a spring and damper in parallel. Each pad is connected to a tie, which is

modeled as a rigid mass. Each tie sits on top of ballast, which is modeled as a spring and

damper in parallel. Each tie has a single translational degree-of-freedom in the vertical

direction. The contact forces between the wheel and rail are modeled with a contact

spring.

Figure 5.1. Single wheel dynamic model for rail break simulations: distributed parameter Euler-Bernoulli beam and discrete lumped parameter inputs across the beam.

106

5.2.1 Free Body Diagrams

To derive the equations of motion for the model shown in Figure 5.1, the free body

diagrams for each of the bodies are considered, which are shown in Figure 5.2. The

symbols in the diagram are as follows: uc(t) is the car body vertical displacement, uB(t) is

the bolster vertical displacement, uw(t) is the wheel vertical displacement, ur(x,t) is the rail

vertical deflection, ut,i(t) is the rail tie vertical displacement, i is the tie index, Mc is the mass

of the car body, MB is the mass of the bolster, Mw is the mass of the wheel, Mt is the mass of a

rail tie, k1 is the primary suspension stiffness coefficient, c1 is the primary suspension

damping coefficient, k2 is the secondary suspension stiffness coefficient, c2 is the secondary

suspension damping coefficient, Ch is the contact stiffness between the wheel and rail, kp is

the pad stiffness coefficient, cp is the pad damping coefficient, kb is the ballast stiffness

coefficient, cb is the ballast damping coefficient, Vx is the forward speed of the train, and hi is

the distance from the left free end of the rail to the location of the ith pad-tie-ballast system.

It is assumed that all pads have the same stiffness and damping properties, all ties have the

same mass, and all ballast supports have the same stiffness and damping properties.

The rail has Young’s Modulus E, cross-sectional area Ac, area moment of inertia I, and

density ρ. The model assumes that the wheel is free-rolling with no slip and has single

point contact with the rail. It is also assumed that the forward speed is constant and the

initial position of the wheel is x = 0, which means that the location of the wheel with

respect to the left-hand side of the rail is Vxt for all time. The model will have N pad-tie-

ballast lumped parameter systems supporting the rail. N should be large enough such that

a fixed boundary condition can be assumed at the right end of the rail. The rail is also

assumed to be free at the left end. With these boundary conditions, it can be determined

from Section 2.3.2 that the left end will have zero bending moment and zero shear force,

and the right end will have zero deflection and zero slope.

107

Figure 5.2. Free body diagrams of single wheel dynamic model.

5.2.2 Equations of Motion

Using Figure 5.2 and the methods outlined in Section 2.3, the equations of motion for the

train-rail system can be developed. Newton’s force balance (Equation (2.5)) on a rigid

mass is used to derive the equations of motion for the lumped parameter systems (car

Mc

MB

Mw

k2 (uB(t) – uc(t))

k1 (uw(t) – uB(t))

uc(t)

uB(t)

uw(t)

c2 (uB(t) – uc(t))

c1 (uw(t) – uB(t))

. .

. .

ur(x,t)

Ch (ur(Vxt,t)–uw(t))

ut,i(t)

kp,i(ut,i(t)–ur(hi,t)) cp,i(ut,i(t)–ur(hi,t)). .

kb,iut,i(t) cb,iut,i(t).

xhi

z

x… …

Vx

Mt

Vxt

ut,1(t) ut,N(t)

108

body, bolster, wheel, and pad-tie-ballast), and Euler-Bernoulli beam theory is used to

derive the equation of motion for the distributed parameter rail. The equation for the

vertical motion of the car body is:

��i� �/"0 = P:/i'Ø/"0 − i' �/"00 + H:/iØ/"0 − i�/"00 (5.1)

The equation for the vertical motion of the bolster is:

�Øi�Ø/"0 = P:Mi' �/"0 − i'Ø/"0N + H:Mi�/"0 − iØ/"0N

+P9Mi'�/"0 − i'Ø/"0N + H9Mi�/"0 − iØ/"0N (5.2)

The equation for the vertical motion of the wheel is:

��i��/"0 = P9Mi'Ø/"0 − i'�/"0N + H9MiØ/"0 − i�/"0N + ,�MiÙ/��", "0 − i�/"0N (5.3)

The equation of motion for each lumped parameter pad-tie-ballast system is:

�fi� f,�/"0 = PÅ,� Yi' Ù/ℎ�, "0 − i' f,�/"0Z + HÅ,� YiÙ/ℎ�, "0 − if,�/"0Z− PÓ,�i' f,�/"0 − HÓ,�if,�/"0

(5.4)

The equation of motion for the rail is:

O@ $JiÙ/�, "0$�J + K<� $:iÙ/�, "0$": = ,�/i�/"0 − iÙ/�, "00�/� − ��"0

+�?YHÅ,�Mif,�/"0 − iÙ,�/�, "0N + PÅ,�Mi' f,�/"0 − i' Ù,�/�, "0NZ �/� − ℎ�0B`�^9

(5.5)

where �/� − m0 is the Dirac delta function acting at point � = m and representing a force

acting at a discrete location. Equations (5.1) – (5.5) are a system of coupled differential

equations representing the dynamics of the system shown in Figure 5.2. The equations

must be solved simultaneously in order to determine the system response.

5.3 Solution to the System: Single-Wheel Model

In order to solve for the equations of motion given by Equations (5.1) – (5.5), they must

first be rearranged into a form that is easier to solve mathematically. The procedure for

arriving at a solution for each of the individual subsystems is first determined. These

solutions are then combined using an iterative approach at each time step to arrive at a

numerical solution for the system. Section 5.3.1 contains the procedure for solving for

everything above the rail, i.e., the train subsystem, Section 5.3.2 contains the procedure for

109

solving for the pad-tie-ballast subsystems, Section 5.3.3 contains the procedure for solving

for the rail subsystem, and Section 5.3.4 contains a procedure for solving for the combined

system, including interactions between all subsystems.

5.3.1 Solution to the Train Subsystem

As discussed in Section 2.3.1, the state space form is a convenient representation of a

multi-body system because it can easily be solved numerically with Runge-Kutta

integration methods. First consider the train subsystem that sits above the rail, including

the car body, bolster, and wheel. The state vector and the first derivative with respect to

time of the state vector are then defined as:

ÚÛ/"0 =��&Û,9/"0&Û,:/"0&Û,I/"0&Û,J/"0&Û,�/"0&Û,Ü/"0��

��=

��i�/"0i' �/"0iØ/"0i'Ø/"0i�/"0i'�/"0��

��,Ú' Û/"0 =

��&'Û,9/"0&'Û,:/"0&'Û,I/"0&'Û,J/"0&'Û,�/"0&'Û,Ü/"0��

�� (5.6)

where the subscript A denotes the lumped parameter system above the rail. Substitution of

the relationship shown in Equation (5.6) into Equations (5.1) – (5.5) and rearrangement

of terms yields the three state equations:

&'Û,:/"0 = − H:�� &Û,9/"0 − P:�� &Û,:/"0 + H:�� &Û,I/"0 + P:�� &Û,J/"0 (5.7)

&'Û,J/"0 = H:�Ø &Û,9/"0 + P:�Ø &Û,:/"0 − /H9 + H:0�Ø &Û,I/"0− /P9 + P:0�Ø &Û,J/"0 + H9�Ø &Û,�/"0 + P9�Ø &Û,Ü/"0

(5.8)

&'Û,Ü/"0 = H9�� &Û,I/"0 + P9�� &Û,J/"0 − /H9 + ,�0�� &Û,�/"0− P9�� &Û,Ü/"0 + ,�� iÙ/��", "0

(5.9)

Equation (5.6) also yields the following relationships:

&'Û,9/"0 = &Û,:/"0 (5.10)

&'Û,I/"0 = &Û,J/"0 (5.11)

110

&'Û,�/"0 = &Û,Ü/"0 (5.12)

Taking Equations (5.7) – (5.12) and putting them in matrix form, gives the state equation:

Ú' Û/"0 = <ÛÚÛ/"0 + =Û/"0,ÚÛ/00 = ÚÛ,� (5.13)

where:

<Û =

�� 0 1 0 0 0 0− H:�� − P:��

H:��P:�� 0 00 0 0 1 0 0H:�Ø

P:�Ø −/H9 + H:0�Ø − /P9 + P:0�ØH9�Ø

P9�Ø0 0 0 0 0 10 0 H9��P9�� −/H9 + ,�0�� − P9��

�� (5.14)

and:

=Û/"0 =�� 00000,�� iÙ/��", "0��

��,ÚÛ,� =

��&Û,9/00&Û,:/00&Û,I/00&Û,J/00&Û,�/00&Û,Ü/00��

��=

��i�/00i' �/00iØ/00i'Ø/00i�/00i'�/00��

�� (5.15)

Equation (5.15) shows that the only forcing input into the train subsystem is the vertical

motion of the rail at the location of the wheel, ur(Vx,t). For the purpose of solving equation

(5.13), it is assumed that ur(Vx,t) is known. This system of equations can be solved using

the Runge-Kutta approach as explained in Section 2.3.1. This approach can be easily

implemented by utilizing the ode45 function in Matlab. The function uses the Dormand-

Prince pair based on an explicit Runge-Kutta formula [68].

5.3.2 Solution to the Pad-tie-ballast Subsystems

To solve for the pad-tie-ballast subsystems, the same approach as used in Section 5.3.1 is

used. To arrange Equation (5.4) into a set of first-order differential equations, first

consider the state vector for pad-tie-ballast subsystem number i:

ÚÝ,�/"0 = a&Ý,�,9/"0&Ý,�,:/"0e = aif,�/"0i' f,�/"0e,Ú' Ý,�/"0 = 8&'Ý,�,9/"0&'Ý,�,:/"0; (5.16)

111

where the subscript B denotes everything beneath the rail. Substitution of the

relationships shown in Equation (5.16) into Equation (5.4) and rearrangement of terms

gives the equation:

&'Ý,�,:/"0 = −MHÓ,� + HÅ,�N�f &Ý,�,9/"0 − MPÓ,� + PÅ,�N�f &Ý,�,:/"0

+HÅ,��f ix/ℎ�, "0 + PÅ,��f i' x/ℎ�, "0 (5.17)

From Equation (5.16) the following relationship can also be determined:

&'Ý,�,9/"0 = &Ý,�,:/"0 (5.18)

Putting Equations (5.17) and (5.18) into matrix form gives the state equation:

Ú' Ý,�/"0 = <ÝÚÝ,�/"0 + =Ý,�/"0,ÚÝ,�/00 = ÚÝ,�,� (5.19)

where:

<Ý = Þ 0 1−MHÓ,� + HÅ,�N�f −MPÓ,� + PÅ,�N�f

ß (5.20)

and:

=Ý,�/"0 = Þ 0HÅ,�iÙ/ℎ�, "0 + PÅ,�i' Ù/ℎ�, "0�fß,ÚÝ,�,� = a&Ý,�,9/00&Ý,�,:/00e = aif,�/00i' f,�/00e (5.21)

Equation (5.21) shows that the forcing inputs into the pad-tie-ballast subsystems are

dictated by the vertical deflection and velocity of the rail at the location of each respective

subsystem. In solving for Equation (5.19) it is assumed that iÙ/ℎ�, "0 and i' Ù/ℎ� , "0 are

known. Just like the train subsystem, the pad-tie-ballast subsystems can be solved by using

the Runge-Kutta based ode45 solver in Matlab.

5.3.3 Solution to the Rail Subsystem

To solve for the distributed parameter rail represented by Equation (5.5), consider the

B-spline collocation approach presented in Section 2.3.4 [34],[35]. The equation of motion

of the vibrating rail is:

O@ $JiÙ/�, "0$�J + K<� $:iÙ/�, "0$": = �/�, "0,0 ≤ � ≤ �," > 0 (5.22)

112

where L is the length of the rail and f(x,t) is a forcing function containing the right-hand

side of Equation (5.5), which is the sum of all pad-tie-ballast subsystem and train

subsystem connections to the rail. The assumption is that the beam is initially un-deflected

and not moving, which gives the initial conditions:

iÙ/�, 00 = 0," ≥ 0i' Ù/�, 00 = 0," ≥ 0 (5.23)

For the free-fixed beam considered in this problem, the boundary conditions are:

iÙSS/0, "0 = 0,iÙSSS/0, "0 = 0iÙ/�, "0 = 0,iÙS /�, "0 = 0 (5.24)

where the apostrophe denotes a partial derivative with respect to the spatial variable x.

The fourth-order spatial partial derivative in Equation (5.22) is isolated so that the fifth-

degree B-splines can be used to approximate the spatial solution. This is accomplished by

dividing the equation by EI, which gives:

$JiÙ/�, "0$�J + K<�O@ $:iÙ/�, "0$": = �/�, "0O@ (5.25)

An approximation to the solution of iÙ/�, "0 is assumed to be:

�/�0 = � ,v�v/�0FA9v^A� (5.26)

where the B-splines, Bj(x), are defined for equally spaced knots of a partition �: � = �� <�9 < ⋯ < �F = � on [a, b] as:

��/�0

= 1120ℎ��,−5�� + 30ℎ�J − 60ℎ:�I + 60ℎI�: − 30ℎJ� + 6ℎ�,10�� − 120ℎ�J + 540ℎ:�I − 1140ℎI�: + 1170ℎJ� − 474ℎ�,−10�� + 180ℎ�J − 1260ℎ:�I + 4260ℎI�: − 6930ℎJ� + 4386ℎ�,5�� − 120ℎ�J + 1140ℎ:�I − 5340ℎI�: + 12270ℎJ� − 10974ℎ�,−�� + 30ℎ�J − 360ℎ:�I + 2160ℎI�: − 6480ℎJ� + 7776ℎ�,

0 ≤ � < ℎ,ℎ ≤ � < 2ℎ,2ℎ ≤ � < 3ℎ,3ℎ ≤ � < 4ℎ,4ℎ ≤ � < 5ℎ,5ℎ ≤ � < 6ℎ, (5.27)

�à̂A9/�0 = ��/� − /â̂ − 10ℎ0,â̂ = 2,3,…

Considering 3ã − 1 grid points on the interval [a, b], which are �à̂ = � + â̂ℎ where â ̂ is an

integer, x0 = a, �Fã= b, and h = (b – a)/3ã.

The difference scheme is then used to discretize the second order partial derivative with

respect to time in Equation (5.25):

113

$JiÙ/�, "0$�J + #K<�O@ ( #iÙ,à̅C: − 2iÙ,à̅C9 + iÙ,à̅/∆"0: ( = �/�, "0O@ (5.28)

where ∆" is the size of the time step and â ̅ is the time step index. Using the substitution ∆" = H and rearrangement of terms gives:

H:iÙ,à̅C:/J0 + #K<�O@ ( iÙ,à̅C: = #K<�O@ ( M2iÙ,à̅C9 − iÙ,à̅N + H:�/�, "0O@ (5.29)

Equation (5.23) shows that iÙ,� = iÙ,9 = 0. For the following time steps, it can then be

written:

" = 0 + 2H, H:iÙ,:/J0 + #K<�O@ ( iÙ,: = #K<�O@ ( M2iÙ,9 − iÙ,�N + H:�/�, "0O@" = 0 + 3H, H:iÙ,I/J0 + #K<�O@ ( iÙ,I = #K<�O@ ( M2iÙ,: − iÙ,9N + H:�/�, "0O@⋮ ⋮" = 0 + 3́H, H:iÙ,F́/J0 + #K<�O@ ( iÙ,F́ = #K<�O@ ( M2iÙ,F́A9 − iÙ,F́A:N + H:�/�, "0O@ (5.30)

The solution to Equation (5.30) is determined by using a B-spline approximation according

to Equation (5.26), so the substitution iÙ,F́/�0 = �F́/�0 can be made:

" = 0 + 2H, H:�:/J0 + #K<�O@ ( �: = #K<�O@ ( M2iÙ,9 − iÙ,�N + H:�/�, "0O@" = 0 + 3H, H:�I/J0 + #K<�O@ ( �I = #K<�O@ ( M2iÙ,: − iÙ,9N + H:�/�, "0O@⋮ ⋮" = 0 + 3́H, H:�F́/J0 + #K<�O@ ( �F́ = #K<�O@ ( M2iÙ,F́A9 − iÙ,F́A:N + H:�/�, "0O@ (5.31)

Substitution of Equation (5.24) into Equation (5.26), allows the boundary conditions to be

written as:

114

� ,v�vSS/00FãA9vÂ� = 0� ,v�vSSS/00FãA9vÂ� = 0� ,v�v/�0FãA9vÂ� = 0� ,v�vS/�0FãA9vÂ� = 0

(5.32)

Bi and the first four derivatives of Bi must be evaluated at the nodal points using Equation

(5.27) in order to determine the spatial solution. These values are shown in Table 5-1.

Table 5-1. Values for Bi and the first four derivatives of Bi at the nodal points. �� C9 ��C: ��CI ��CJ ��C� ��CÜ �� 0 1 26 66 26 1 0 ��′ 0 5 ℎ⁄ 502 ℎ⁄ 0 −502 ℎ⁄ −5 ℎ⁄ 0 ��′′ 0 20 ℎ:⁄ 40 ℎ:⁄ −120 ℎ:⁄ 40 ℎ:⁄ 20 ℎ:⁄ 0 ��′′′ 0 60 ℎI⁄ −120 ℎI⁄ 0 120 ℎI⁄ −60 ℎI⁄ 0 ��′′′′ 0 120 ℎJ⁄ −480 ℎJ⁄ 720 ℎJ⁄ −480 ℎJ⁄ 120 ℎJ⁄ 0

Using Table 5-1, Equations (5.31) and (5.32) are put into the matrix form:

<, = _ (5.33)

where:

, = [,A� ,AJ ,AI … ,FAI ,FA: ,FA9] (5.34)

_ =

�� 00#K<PO@ ( M2iF́A9/��0 − iF́A:/��0N + H:�/��, 3́H0O@#K<PO@ ( M2iF́A9/�90 − iF́A:/�90N + H:�/�9, 3́H0O@⋮#K<PO@ ( M2iF́A9/�FãA90 − iF́A:/�FãA90N + H:�/�FãA9, 3́H0O@#K<PO@ ( M2iF́A9/�Fã0 − iF́A:/�Fã0N + H:�/�Fã , 3́H0O@00 ��

��

(5.35)

115

< =

�� 20120ℎ: 40120ℎ: −120120ℎ: 40120ℎ: 20120ℎ: 0 0 ⋯ 060120ℎI −120120ℎI 0 120120ℎI −60120ℎI 0 0 ⋯ 0�9 �: �I �: �9 0 0 ⋯ 00 �9 �: �I �: �9 0 ⋯ 0⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮0 0 ⋯ �9 �: �I �: �9 00 0 ⋯ 0 �9 �: �I �: �90 0 ⋯ 0 1120 26120 66120 26120 11200 0 ⋯ 0 5120ℎ 502120ℎ 0 −502120ℎ −5120ℎ��

��

(5.36)

�9 = H: # 120120ℎJ( + #K<PO@ ( # 1120(�: = H: #−480120ℎJ( + #K<PO@ ( # 26120(�I = H: # 720120ℎJ( + #K<PO@ ( # 66120( (5.37)

By inverting the matrix A, the coefficients Cj can be solved for to arrive at a numerical

approximation according to Equation (5.26). This is the solution for the dynamic response

of the rail, assuming the value of f(x,t) is known, which is:

�/�, "0 = ,�/i�/"0 − iÙ/�, "00�/� − ��"0

+�?YHÅ,�Mif,�/"0 − iÙ/�, "0N + PÅ,�Mi' f,�/"0 − i' Ù/�, "0NZ �/� − ℎ�0B`�^9

(5.38)

5.3.4 Solution to the Combined Train-Rail System

The solutions for the three subsystems explained in the previous sections require

knowledge of the forcing inputs into those respective subsystems: for the train subsystem,

knowledge of /,� ��⁄ 0iÙ/��", "0; for the pad-tie-ballast subsystem, knowledge of

YHÅ,�iÙ/ℎ�, "0 + PÅ,�i' Ù/ℎ� , "0Z �fæ ; and for the rail subsystem, knowledge of f(x,t). From

Equations (5.15), (5.21), and (5.38), it is seen that the three subsystems are dynamically

coupled, and therefore must be solved simultaneously. An iterative approach is therefore

used to solve for the combined system. This is accomplished by utilizing a Matlab script

116

that incorporates the solutions described in Sections 5.3.1 – 5.3.3. Table 5-2 presents the

general approach to solving for the combined system.

Table 5-2. Algorithm for solving for the combined system. 1 T = length of simulation 2 k = time step 3 m = maximum allowable error 4 ç = maximum number of iterations at each time step 5 ��/�0 = 0 6 �9/�0 = 0 7 for 3́ = 3:1:T/k 8 railerror = 1; 9 count = 0; 10 �F́/�0 = �F́A9/�0 11 �F,è éytf/�0 = 0 12 qA = 0 13 qB,i = 0 14 while (railerror > m) 15 count = count + 1; 16 qA,0 = last(qA) 17 qA = ode45(‘Trainsubsystem.m’,�F́/�0, qA,0) 18 qB,i,0 = last(qB,i) 19 qB,i = ode45(‘PTBsubsystem.m’,�F́/�0, qB,i,0) 20 �F́/�0 = Bsplinesolver(last(qA), last(qB,i), �F́A9/�0, �F́A:/�0) 21 railerror = abs((�F́/�0 − �F́,éytf/�0)/�F́/�0) 22 �F,è éytf/�0 = �F́/�0 23 if count > ç , break, end 24 end 25 end

The basic approach outlined in Table 5-2 for solving for the combined system is to

iteratively solve all three subsystems at each time step until the solutions from all

subsystems converge. In Lines 1-4, values are assigned to the length of simulation, the size

of the time step, the maximum allowable error, and the maximum number of iterations at

each time step. The maximum allowable error, m, dictates how well the solutions for all

subsystems converge at each time step. The maximum number of iterations at each time

step, ç, limits the number of times the subsystems can be solved for in a single time step

before moving on to the next time step. Lines 5-6 initialize the values of the rail deflection

S(x) for the first two time steps. The deflection of the rail is set equal to zero for the first

two time steps according to Equation (5.23). Lines 7-25 contain a ‘for’ loop that runs all the

subsystem solvers from the third time step until the final time step. Lines 12-24 contain a

117

‘while’ loop that continuously solves for the three subsystems at each time step until their

solutions converge to an error value lower than m. Line 8 initializes the value for the

calculated rail error to 1. Line 9 initializes the value for count to 0. The variable count is

responsible for counting the number of iterations at each time step. At Line 10, the value

for �F́/�0 is initialized to the rail solution from the previous time step. At Line 11, the value

for �F,è éytf/�0 is initialized to the value of 0. �F,è éytf/�0 is the value of �F́/�0 calculated from

the last iteration and is used for calculating the error to ensure convergence of solutions.

Lines 12 and 13 give initial conditions for the train subsystem and pad-tie-ballast

subsystems, respectively. Here, both systems are assumed to be at zero displacement and

at rest. Line 16 sets the initial conditions for the train subsystem to be equal to the values

of the states of the train subsystem from the last iteration. Line 17 solves for the train

subsystem using an ordinary differential equation solver as explained in Section 5.3.1.

‘Trainsubsystem.m’ is a separate file containing the equations of motion for the train

subsystem. The current value of �F́/�0 is also an input into the differential equation solver,

as it acts as the forcing input into the system. Lines 18 and 19 solve for all of the pad-tie-

ballast subsystems in the same way that Lines 16 and 17 solved for the train subsystem.

When the train and pad-tie-ballast subsystems are solved using the ode45() function in

Matlab, they are solved for a time period ranging from 0 to k. The only values of interest in

this solution are the last time values, as they correspond to the states of the subsystems at

the current time step. As a result, the last() function is used to obtain these values, which

are input as forcing functions into the rail subsystem in Line 20. The function

Bsplinesolver() is an algorithm that implements the B-spline approximation of the rail as

outlined in Section 5.3.3. The solution to the rail deflection also uses the values of the rail

deflection from the previous two steps. In Line 21, the convergence error is calculated as

the difference between the values of the rail approximation for the current iteration and

the values of the rail approximation from the last iteration. Line 22 sets the new value of �F́,éytf/�0 to the value of �F́/�0 just calculated. Line 23 immediately exits the ‘while’ loop if

the number of iterations exceeds the maximum allowable value.

118

5.4 Non-Linear Spring Rates: Single-Wheel Model

The algorithm that was developed to solve for the train-rail system described in Section

5.3 uses linear spring rates to model the contact forces between the wheel and rail as well

as the spring rates for the pad and ballast. In reality, it is inaccurate to assume linear

springs, and more details must be added to the model to account for the complex forces

that arise at the wheel-rail interface and the inherent non-linearity in the pad and ballast.

In this section, complexity is added to the single-wheel model by utilizing some methods

from contact mechanics theory. Non-linear spring rates for the pad and ballast are then

considered as well.

5.4.1 Application of Hertz Contact Spring

Consider the non-linear Hertz contact spring as a candidate to model the contact

mechanics between the wheel and rail. It is assumed that there is no adhesion between the

wheel and rail, therefore the contact spring is only capable of applying a force when in

compression. The equation for the force generated by the contact spring is:

��/"0 = Ö 0, foriÙ/��", "0 − i�/"0 < 0,êMiÙ/��", "0 − i�/"0NI :⁄ , foriÙ/��", "0 − i�/"0 ≥ 0 (5.39)

where fc(t) is the contact force and CH is a nonlinear contact stiffness. The equations of

motion for the system then change as follows. For the train subsystem, the state matrix

and input force vector in Equation (5.13) will change, replacing Equations (5.14) and

(5.15). The new equations are:

<Û =

�� 0 1 0 0 0 0− H:�� − P:��

H:��P:�� 0 00 0 0 1 0 0H:�Ø

P:�Ø −/H9 + H:0�Ø −/P9 + P:0�ØH9�Ø

P9�Ø0 0 0 0 0 10 0 H9��P9�� − H9�� − P9��

�� (5.40)

and:

119

=Û/"0 =�� 00000��/"0��

��,ÚÛ,� =

��&Û,9/00&Û,:/00&Û,I/00&Û,J/00&Û,�/00&Û,Ü/00��

��=

��i�/00i' �/00iØ/00i'Ø/00i�/00i'�/00��

�� (5.41)

The input force to the rail shown on the right hand side of Equation (5.22), replacing

Equation (5.38), is:

�/�, "0 = −��/"0�/� − ��"0

+�?YHÅ,�Mif,�/"0 − iÙ/�, "0N + PÅ,�Mi' f,�/"0 − i' Ù/�, "0NZ �/� − ℎ�0B`�^9

(5.42)

5.4.2 Non-linear Pad and Ballast Spring Rates

Wu [28] studied the effects of pad and ballast non-linearity on wheel-rail impact forces,

which is considered in this study. The equation for the non-linear pad stiffness is:

HÅ,� = Y52 + /6.24 × 10ë0Mif,�/"0 − iÙ/ℎ� , "0N:Z × 10Ü (5.43)

The equation for the non-linear ballast stiffness is:

HÓ,� = Y22.75 + /2.60 × 10ë0Mif,�/"0N:Z × 10Ü (5.44)

Replacing Equation (5.42), the new input force into the rail subsystem is:

�/�, "0 = −��/"0�/� − ��"0

+�?#?Y52 + /6.24 × 10ë0Mif,�/"0 − iÙ/ℎ�, "0N:Z × 10ÜB Mif,�/"0 − iÙ/�, "0N`�^9

+ PÅ,�Mi' f,�/"0 − i' Ù/�, "0N( �/� − ℎ�0B (5.45)

Making the substitutions of these non-linear spring rates into the state equations for the

pad-tie-ballast system, Equation (5.17) is also adjusted to account for the non-linear spring

rates.

120

5.5 Simulations and Results: Single-Wheel Model

The results of a simulation using the model presented in Figure 5.1 can now be observed

and compared to an actual vehicle response. In this simulation, a single-wheel train model

with non-linear Hertz contact spring moves along a free-fixed rail to the right at a constant

forward speed of 15.65 m/s (35 mph). All subsystems in the model are initially at zero

vertical deflection and zero vertical velocity. The physical parameters from the train car

are obtained from a NUCARS model of the IFC used at the FAST track at TTCi [6]. The

physical parameters for the rail are the values for UIC 60 rail taken from a publication by

Wu [28]. All of these values are shown in Table 5-3. A 2 meter length of rail is used in this

simulation with three pad-tie-ballast subsystem supports placed at 0.2 m, 0.85 m, and 1.5

m from the left end of the rail.

Table 5-3. Physical parameter values used in wheel/rail interaction model.

Parameter Symbol Value Units 1/8 carbody mass Mc 16,596 kg ¼ bolster mass MB 184 kg ½ axle + ½ sideframe + wheel mass Mw 1,072 kg ¼ primary suspension stiffness k1 4,515,179 N/m ¼ primary suspension damping c1 5.5 x 104 N/m/s ¼ secondary suspension stiffness k2 1,751,264,214 N/m ¼ secondary suspension damping c2 8,851 N/m/s Hertzian contact spring coefficient CH 3.5 x 108 N/m3/2 Young’s Modulus of rail E 2.1 x 1011 N/m2 Area moment of inertia of rail I 30.55 x 10-6 m4 Cross sectional area of rail Ac 7.69 x 10-3 m2 Density of rail ρ 7,850 kg/m3 Pad stiffness kp 52 + 6.24 x 108 xp

2 MN/m Pad damping cp 3 x 106 N/m/s ½ tie mass Mt 40 kg Ballast stiffness kb 22.75 + 2.60 x 108 xb

2 MN/m Ballast damping cb 6 x 105 N/m/s Wheel radius r 0.4826 m Forward speed Vx 15.65 m/s Length of rail L 2 m Location of rail tie #1 h1 0.2 m Location of rail tie #2 h2 0.85 m Location of rail tie #3 h3 1.5 m

121

The simulation parameters are shown in Table 5-4. The length of the simulation is T =

0.1 s and the time step is k = 0.001 s. The rail is divided into 200 equally spaced partitions

for the fifth-degree B-spline solution. The maximum allowable error is ξ = 10-6 and the

maximum number of iterations is σ = 35. During the simulation, the rail vertical deflection

response is observed, the vertical displacement, vertical velocity, and vertical acceleration

of all bodies in the train subsystem (car body, bolster, wheel) are observed, and the error

convergence between subsystems is observed.

Table 5-4. Simulation parameter values. Parameter Symbol Value Units Simulation time T 0.1 s Time step k 0.001 s Knots (number of beam elements) n 200 Maximum allowable error ξ 10-6 Maximum iterations σ 35

Figure 5.3 shows the deflection response of the rail. The x-axis shows the spatial step

(divided into 200 beam elements), the y-axis shows the time step (101 steps for a 0.1 s

simulation time with 0.001 size time step), and the z-axis shows the rail deflection. The rail

is initially at rest, and the left end is free and oscillates while the right end is fixed and

remains un-deflected. The train moves to the right at a constant forward speed and

reaches x = 1.788 m at the end of the simulation. Figure 5.4 shows the vertical responses of

the bodies in the train subsystem. In this simulation, it is assumed that the primary and

secondary suspensions are initially at zero displacement. Therefore, when the simulation

begins, the train subsystem will move to a position of static equilibrium, as can be seen in

the response plots. From these plots the natural frequencies of the train subsystem can

also be observed. The wheel, bolster, and car body all oscillate out of phase with one

another at about 30 Hz. The plot of wheel vertical acceleration is of primary interest in this

study since this is what is measured by the sensor in the defect detection system. Figure

5.5 shows plots of the error convergence at each time step during the simulation. The

maximum error of ξ = 10-6 is for the error in the rail deflection response at the x location of

the wheel at each time step. This is shown in the upper left plot. Rather than including

multiple error criteria for all the variables in the system, the rail response at the wheel

122

location is the only convergence error criteria for the simulation. Therefore, this error

value is set strictly to ensure convergence in the other variables as well. The plots show

that the error converges to 10-6 within ten iterations at each time step. The convergence

error in the rail deflection at each location in the rail remains below 5 × 10AI for all time

steps. The convergence error for rail tie displacement remains below 4 × 10AJ for all rail

ties at all time steps.

Figure 5.3. Rail deflection response for single wheel model simulation.

123

Figure 5.4. Train subsystem body responses for single wheel model simulation.

Figure 5.5. Simulation convergence errors for single wheel model simulation.

0 0.02 0.04 0.06 0.08 0.1-0.01

-0.005

0

CarbodyD

isp

lace

me

nt

(m)

0 0.02 0.04 0.06 0.08 0.1-0.1

-0.05

0

Carbody

Ve

loci

ty (

m/s

)

0 0.02 0.04 0.06 0.08 0.1-5

0

5

Carbody

Acc

ele

rati

on

(m

/s2)

0 0.02 0.04 0.06 0.08 0.1-0.01

-0.005

0

Bolster

Dis

pla

cem

en

t (m

)

0 0.02 0.04 0.06 0.08 0.1-0.1

-0.05

0

Bolster

Ve

loci

ty (

m/s

)

0 0.02 0.04 0.06 0.08 0.1-10

0

10

Bolster

Acc

ele

rati

on

(m

/s2)

0 0.02 0.04 0.06 0.08 0.1-0.01

-0.005

0

Wheel

Dis

pla

cem

en

t (m

)

Time (s)0 0.02 0.04 0.06 0.08 0.1

-1

0

1

WheelV

elo

city

(m

/s)

Time (s)0 0.02 0.04 0.06 0.08 0.1

-200

0

200

Wheel

Acc

ele

rati

on

(m

/s2)

Time (s)

124

5.6 Broken Rail Model and Simulations

The models and approaches developed in the previous sections are now expanded to

develop a broken rail model. Section 5.6.1 details the model and explains how the wheel-

rail interactions are modeled for a rail break, and Section 5.6.2 shows the results of a

simulation using the broken rail model.

5.6.1 Model of a Broken Rail

Figure 5.6 shows the approach to modeling a broken rail. The broken rail consists of a

fixed-free beam of length L1 on the left side and a free-fixed beam of length L2 on the right

side with a gap of size g between the two. Both the left and right beams are supported by a

finite number of lumped parameter pad-tie-ballast systems. The vibration responses of the

left and right rails are completely independent of one another. As before, the broken rail

model assumes single point contact between the wheel and rail at all times. Therefore, the

wheel can only be in contact with the left or right rail, and not both. It is also assumed that

there is no externally applied drive or brake torque to the wheel and zero slip between the

wheel and rail. During simulation, the train starts at some initial position on the left rail

(xw0) and travels right at a constant forward speed. It is assumed that both the left and

right rails are un-deflected at time t = 0. Therefore, as the train moves to the right, the end

of the left rail will gradually become more deflected in the negative direction. Figure 5.7

shows an enlarged view of the wheel driving along the rail at the location of the rail break.

Figure 5.6. Diagram of broken rail model.

125

Consider first the case of small break gap size g. Figure 5.7 shows the wheel as it is

driving along the left rail just as it comes into contact with the right rail. The wheel

remains in contact with the left rail until it reaches a point at distance xe from the end of the

rail. At this point the x location of the wheel on the left rail is xw1 = L1–xe and the deflection

of the left rail is ur1(L1–xe,t). At this point, the wheel comes into contact with the right rail

at point B and immediately loses contact with the left rail at point A.

Figure 5.7. Wheel driving over broken rail for small broken rail gap size g.

Figure 5.8 shows the geometric relationships at the point the wheel contacts the right

rail as shown in Figure 5.7. This diagram can be used to determine equations for the

broken rail model. Although the Hertz contact model utilizes the deformation at the wheel-

rail contact to determine the contact force, this deformation is small and can be considered

negligible in deriving the geometric relationships. Therefore, it is assumed that OA´́ ´́ = OB´́ ´́ = x. The angle ! = ∠AOB and the distance ℎÓ = iÙ9/�9 − �Ô , "0 are also assigned.

By viewing AOB as a triangle in a Cartesian coordinate plane, the following coordinates can

be assigned to each point: A, (0,0); B, (xe+g,hb); O, (0,r). Then, using Pythagoras’ theorem

the relationship for the distance between point O and point B can be written:

x: = M0 − /�Ô + n0N: + /x − ℎÓ0: (5.46)

Multiplying out and rearranging terms gives the polynomial:

ur1(xw1,t)

ur2(xw2,t)

xe g

ur1(L1–xe,t)

rθ

O

A

B

126

�Ô: + /2n0�Ô + /n: + ℎÓ: − 2xℎÓ0 = 0 (5.47)

The quadratic equation is then used to solve for xe:

�Ô = −n + ð2xℎÓ − ℎÓ: (5.48)

Figure 5.8. Geometric relationships between wheel and rails at location of broken rail for small broken rail gap size g.

Once the wheel comes in contact with the rail at point B, there is a loss of contact at point

A and perfect adhesion at point B. The wheel then remains in contact with the right rail at

point B until it rotates an angle ! about point B. At this point, the wheel then begins

moving to the right along the right rail at a constant forward speed. The time that the

wheel remains in contact with the right rail at point B is given by the amount of time it

takes the wheel to rotate through the angle ! in degrees, which is:

"Ø = 2�!T ,T = 2� ��x (5.49)

where:

! = cosA9 a1 − /�Ô + n0: + /ℎÓ0:2x: e (5.50)

Since the rotational dynamics of the wheel are neglected and single point contact is

assumed, as soon as the wheel contacts point B, the wheel is stationary in the longitudinal

θ

r

r

O (0,r)

A (0,0)

B (xe+g,hb)

xe g

hb = -ur1(L1–xe,t)

127

direction at point B for a time tB. After this period of time, the wheel moves to the right at a

speed of Vx. Since the contact force is modeled with a spring, and the wheel contacts point

B at an angle !, there will be a component of contact force in the longitudinal direction. In

this study, only the vertical dynamics of the train subsystem are of interest. Additionally,

since x ≫ n, ! ≪ 1, and cos ! ≈ 1, the total force in the Hertz contact spring is

approximately equal to the vertical component of the force in the spring. Equations (5.48)

— (5.50) are valid for a small broken rail gap size, i.e., n ≤ «2xℎÓ − ℎÓ:.

For the case of a large broken rail gap size, i.e., n > «2xℎÓ − ℎÓ:, the wheel will not

contact the right rail until it reaches the end of the left rail and rotates through some angle

γ. Figure 5.9 shows a diagram for the wheel-rail relationship at the location of the rail

break for large values of g. The wheel drives along the left rail until xw1 = L1, where it

begins to rotate about point A. The wheel contacts the free end of the right rail at point B

after the wheel has rotated an angle of γ about point A. Once the wheel contacts point B,

there is loss of contact at point A and the wheel begins to rotate about point B. There is

perfect adhesion at point B until the wheel rotates through an angle ϕ. At this point, the

wheel begins to move to the right at constant forward speed. Using the law of cosines, the

equation for determining the angle θ is:

! = cosA9 a1 − n: + ℎÓ:2x: e (5.51)

Using basic trigonometry, the following relationships can also be determined:

Ã = tanA9 #ℎÓn ( (5.52)

and

[ = � − !2 (5.53)

The angle through which the wheel rotates about point A is then:

ô = �2 − /[ + Ã0 = !2 − Ã = 12 acosA9 u1 − n: + ℎÓ:2x: we − tanA9 #ℎÓn ( (5.54)

Assuming constant angular velocity with no slip, the time that it takes the wheel to rotate

about point A through the angle γ is:

128

"õ = 2�ôT ,T = 2� ��x (5.55)

The angle through which the wheel rotates about point B is:

� = � − /ô + !0 = � −32 ! + Ã = � − 32 cosA9 u1 − n: + ℎÓ:2x: w + tanA9 #ℎÓn ( (5.56)

The time that it takes the wheel to rotate about point B through the angle ϕ is:

"Ø = 2��T ,T = 2� ��x (5.57)

Figure 5.9. Geometric relationships between wheel and rail at location of broken rail for large broken rail gap size g.

After the wheel rotates about point A and contacts point B, there is a vertical velocity

built up because of the rotational motion. This rotational motion causes an additional

impact force that lasts the duration of the wheel rotation about point B. Using simple

conservation of energy and work-energy principles, this additional impact force can be

derived. With acceleration due to gravity acting at point O, the impact velocity will be:

ö = «2nℎ (5.58)

θ

rr

O

A (0,0)

B (g,hb)

g

hb = -ur1(L1,t)

γ

ϕ

ββ

α

129

where g is the acceleration due to gravity and h is the height from which the wheel has

“fallen,” which is:

ℎ = x/1 − cos ô0 (5.59)

When the wheel contacts point B, the kinetic energy will be:

KE = 12�ö: (5.60)

The impact force can then be determined by assuming that all of the kinetic energy is

absorbed by the rail, which applies a force to the wheel. The distance over which the

impact force acts is the indentation between the wheel and rail surfaces. It can then be

written:

KE = 12�ö: = _�ÂÅ� (5.61)

where d is the depth of penetration of the wheel into the rail. Combining Equations (5.58)

– (5.61) gives an expression for the impact force:

_�ÂÅ = �nx� /1 − cos ô0 (5.62)

5.6.2 Broken Rail Simulation

Using the broken rail model from the previous section, a simulation is run to compare

the signature generated from the model to an actual broken rail signature. The collected

data was for the IFC driving across a broken rail of size g = 102 mm (4 in) at a constant

forward speed of 15.65 m/s (35 mph). The physical parameters used in the simulation are

obtained from the IFC NUCARS model and UIC 60 rail model as listed in Table 5-3. The

length used for the rails in these simulations is L1=L2=3 m. Starting from the left end of

each rail, the locations of the rail ties are: xw1 = [0.35,1.0,1.65, 2.3, 2.95], xw2 = [0.6, 1.25,

1.9,2.55]. Both the left and right rails are initially at rest with zero deflection. The

simulation parameters are shown in Table 5-5.

130

Table 5-5. Simulation parameter values for broken rail simulation. Parameter Symbol Value Units Simulation time T 0.25 s Time step k 0.001 s Knots (number of beam elements) per rail n 200 Maximum allowable error ξ 10-7 Maximum iterations σ 35

Figure 5.10 shows the left and right rail responses for the broken rail simulation. The

initial position of the wheel on the left rail is xw1(0) = 0 m. At t = 0.191 s, the wheel comes

in contact with the right rail and leaves the left rail. Therefore the left rail response is

shown from 0 s to 0.191 s and the right rail response is shown from 0.191 s to 0.38 s.

Figure 5.11 shows the vertical displacement, vertical velocity, and vertical acceleration

responses of the wheel, bolster, and car body. The wheel encounters the broken rail at

0.191 s, which is seen by the discontinuities in the train body responses at this time.

Figure 5.10. Left and right rail responses for broken rail simulation.

131

Figure 5.11. Train body responses for broken rail simulation.

Figure 5.12 shows the wheel vertical acceleration response as the wheel passes over the

broken rail. The plot is overlaid with the actual response of the IFC passing over a broken

rail of approximately 102 mm gap size at approximately 15.65 m/s. The original signal was

collected at a sampling rate of 256 Hz, therefore the simulation response is down-sampled

from 1000 Hz to 256 Hz for better comparison. Since the IFC collects data at the bogie side

frame, one oscillation occurs for each pass of the front and rear axle. Since the simulation is

for a single wheel, the oscillation generated from the passing over the rail was repeated

0 0.2-0.0445

-0.044

-0.0435

-0.043

-0.0425

-0.042Carbody

Displacement (m

)

Time (s)

0 0.2-0.02

-0.01

0

0.01

0.02Carbody

Velocity (m/s)

Time (s)

0 0.2-1

-0.5

0

0.5

1Carbody

Acceleration (m/s2)

Time (s)

0 0.2-0.044

-0.0435

-0.043

-0.0425

-0.042Bolster

Displacement (m

)

Time (s)

0 0.2-0.02

-0.01

0

0.01

0.02Bolster

Velocity (m/s)

Time (s)

0 0.2-3

-2

-1

0

1

2

3Bolster

Acceleration (m/s2)

Time (s)

0 0.2-8.5

-8

-7.5

-7

-6.5

-6

-5.5x 10

-3Wheel

Displacement (m

)

Time (s)

0 0.2-0.4

-0.2

0

0.2

0.4Wheel

Velocity (m/s)

Time (s)

0 0.2-100

-50

0

50

100Wheel

Acceleration (m/s2)

Time (s)

132

twice in the figure in order to compare to the actual response. The bottom plot of Figure

5.12 shows a Fast Fourier Transform of the break signatures of the actual and simulated

responses. The results show similar amplitude and frequency content for the two

signatures.

Figure 5.12. Comparison of actual and simulated data for 102 mm broken rail vertical acceleration signature.

5.6.3 Surface Fracture Simulation

To model a surface fracture, a broken rail with small rail break gap size is considered.

The same exact operating conditions, physical parameters, and simulation parameters used

from the broken rail simulation are used, the only difference is the broken rail gap size is g

= 1 mm (0.039 in). Figure 5.13 shows the rail deflections for the simulations. Figure 5.14

shows the train body responses for the simulation. Figure 5.15 shows the vertical

acceleration signature and frequency content for the simulation. The results show that the

broken rail simulation produces results that closely match the collected data. It should be

noted that for the broken rail simulation the case is that of Figure 5.9 where n >«2xℎÓ − ℎÓ:, which means that the additional impact force due to wheel vertical velocity is

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

-5

0

5

Time (s)

Vertical Acceleration (g) Vertical Acceleration Signature

Data

Simulation

0 20 40 60 80 100 1200

0.5

1

1.5

Frequency (Hz)

Modulus

Fast Fourier Transform

133

present according to Equation (5.62). For the case of the surface fracture simulation, the

case shown in Figure 5.8 is present, where n ≤ «2xℎÓ − ℎÓ:, which means that the

additional impact force due to wheel vertical velocity is not present. This accounts for the

major difference in amplitude between the two cases.

Figure 5.13. Left and right rail responses for surface fracture simulation.

134

Figure 5.14. Train body responses for surface fracture simulation.

0 0.2-0.0445

-0.044

-0.0435

-0.043

-0.0425

-0.042Carbody

Displacement (m

)

Time (s)

0 0.2-0.02

-0.01

0

0.01

0.02Carbody

Velocity (m/s)

Time (s)

0 0.2-1

-0.5

0

0.5Carbody

Acceleration (m/s2)

Time (s)

0 0.2-0.0445

-0.044

-0.0435

-0.043

-0.0425

-0.042Bolster

Displacement (m

)

Time (s)

0 0.2-0.02

-0.01

0

0.01

0.02Bolster

Velocity (m/s)

Time (s)

0 0.2-3

-2

-1

0

1

2

3Bolster

Acceleration (m/s2)

Time (s)

0 0.2-8.5

-8

-7.5

-7

-6.5

-6

-5.5x 10

-3Wheel

Displacement (m

)

Time (s)

0 0.2-0.3

-0.2

-0.1

0

0.1

0.2

0.3Wheel

Velocity (m/s)

Time (s)

0 0.2-40

-20

0

20

40

60Wheel

Acceleration (m/s2)

Time (s)

135

Figure 5.15. Comparison of actual and simulated data for surface fracture (g = 1 mm) vertical

acceleration signature.

5.7 Development of Two-Wheel Model

To add further complexity and accuracy to the dynamic wheel-rail interaction model, a

two-wheel model is now considered. The two-wheel model assumes a rigid connection

between the two wheels via a side frame with given mass and pitch moment of inertia. The

vertical motions of the wheels are considered as well as the vertical and pitch motions of

the side frame. The following sections discuss the development of the two-wheel model,

beginning with free body diagrams and equations of motion. The procedure for modeling a

broken rail is also discussed.

5.7.1 Free Body Diagrams and Equations of Motion

The free body diagrams for the two-wheel model are shown in Figure 5.16. Performing a

force and moment balance on the system yields the equations of motion. The equations of

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-5

0

5

Time (s)


Vertical Acceleration Signature

Data

Simulation

0 20 40 60 80 100 1200

0.2

0.4

0.6

0.8

1

Frequency (Hz)

Modulus


136

motion for the train subsystem are as follows. The equation governing the vertical motion

of the carbody is:

��i� �/"0 = P:/i'Ø/"0 − i' �/"00 + H:/iØ/"0 − i�/"00 (5.63)

The equation for the vertical motion of the bolster is:

�Øi�Ø/"0 = P:Mi' �/"0 − i'Ø/"0N + H:Mi�/"0 − iØ/"0N

+P9Mi'�/"0 − i'Ø/"0N + H9Mi�/"0 − iØ/"0N (5.64)

The equation for the vertical motion of the side frame is:

M�tù + ��9 + ��:Ni� tù/"0 = P9 Yi'Ø/"0 − i' tù/"0Z + H9 YiØ/"0 − itù/"0Z

+,�9MiÙ/��", "0 − i�9/"0N + ,�:MiÙ/��" − �, "0 − i�:/"0N (5.65)

The equation for the pitch motion of the side frame is:

tù!�tù/"0 = #�2( Y,�9MiÙ/��", "0 − i�9/"0N − ,�:MiÙ/��" − �, "0 − i�:/"0NZ (5.66)

The spring coefficients Ch1 and Ch2 are defined by:

,�9 = Ö 0, foriÙ/��", "0 − i�9/"0 < 0,ê9MiÙ/��", "0 − i�9/"0N9 :⁄ , foriÙ/��", "0 − i�9/"0 ≥ 0 (5.67)

,�: = Ö 0, foriÙ/��" − �, "0 − i�:/"0 < 0,ê:MiÙ/��" − �, "0 − i�:/"0N9 :⁄ , foriÙ/��" − �, "0 − i�:/"0 ≥ 0 (5.68)

Since the side frame is rigidly attached to the wheels, the vertical motions of the wheels

can be derived from the vertical and pitch motions of the side frame using simple

geometric relationships. The vertical motions of the wheels are:

i�9/"0 = #�2( sin !tù + itù (5.69)

i�:/"0 = − #�2( sin !tù + itù (5.70)

137

Figure 5.16. Free body diagrams for the two-wheel model.

Mc

MB

Mw1

k2 (uB(t) – uc(t))

k1 (uw(t) – uB(t))

uc(t)

uB(t)

uw1(t)

c2 (uB(t) – uc(t))

c1 (uw(t) – uB(t))

. .

. .

ur(x,t)

Ch1 (ur(Vxt,t)–uw1(t))

ut,i(t)

kp,i(ut,i(t)–ur(hi,t)) cp,i(ut,i(t)–ur(hi,t)). .

kb,iut,i(t) cb,iut,i(t).

xhi

z

x… …

Vx

Mt

Vxt

ut,1(t) ut,N(t)

Mw2

uw2(t)

Ch2 (ur(Vxt– L,t)–uw2(t))

L

Msf ,Jsf θsf

usf(t)

138

5.7.2 Solution to the Two-Wheel Train-Rail Subsystem

To solve for the train subsystem, the equations of motion are rearranged into state-space

form. The state vector and the first derivative with respect to time of the state vector are

then defined as:

ÚÛ/"0 =��&Û,9/"0&Û,:/"0&Û,I/"0&Û,J/"0&Û,�/"0&Û,Ü/"0&Û,ú/"0&Û,ë/"0��

��=

�� i�/"0i' �/"0iØ/"0i'Ø/"0itù/"0i' tù/"0!tù/"0!'tù/"0��

��,Ú' Û/"0 =

��&'Û,9/"0&'Û,:/"0&'Û,I/"0&'Û,J/"0&'Û,�/"0&'Û,Ü/"0&'Û,ú/"0&'Û,ë/"0��

�� (5.71)

where the subscript A denotes the lumped parameter system above the rail. Substitution of

the relationship shown in Equation (5.71) into Equations (5.63)–(5.66) and rearrangement

of terms yields the four state equations:

&'Û,:/"0 = − H:�� &Û,9/"0 − P:�� &Û,:/"0 + H:�� &Û,I/"0 + P:�� &Û,J/"0 (5.72)

&'Û,J/"0 = H:�Ø &Û,9/"0 + P:�Ø &Û,:/"0 − /H9 + H:0�Ø &Û,I/"0− /P9 + P:0�Ø &Û,J/"0 + H9�Ø &Û,�/"0 + P9�Ø &Û,Ü/"0

(5.73)

&'Û,Ü/"0 = u 1��9 + ��: + �tùw ûH9&Û,I/"0 + P9&Û,J/"0

− H9&Û,�/"0 − P9&Û,Ü/"0 + _9 + _:ü (5.74)

&'Û,ë/"0 = u �2 tùw [_9 − _:] (5.75)

where:

_9 = ,�9 YiÙ/��", "0 − /� 2⁄ 0&Û,ú/"0 − &Û,�/"0Z (5.76)

_: = ,�: YiÙ/��" − �, "0 + /� 2⁄ 0&Û,ú/"0 − &Û,�/"0Z (5.77)

Equation (5.71) also yields the following relationships:

&'Û,9/"0 = &Û,:/"0 (5.78)

&'Û,I/"0 = &Û,J/"0 (5.79)

139

&'Û,�/"0 = &Û,Ü/"0 (5.80)

&'Û,ú/"0 = &Û,ë/"0 (5.81)

Taking Equations (5.72) – (5.81) and putting them in matrix form, gives the state equation:

Ú' Û/"0 = <ÛÚÛ/"0 + =Û/"0,ÚÛ/00 = ÚÛ,� (5.82)

where:

<Û =

�� 0 1 0 0 0 0 0 0− H:�� − P:��

H:��P:�� 0 0 0 00 0 0 1 0 0 0 0H:�Ø

P:�Ø −/H9 + H:0�Ø −/P9 + P:0�ØH9�Ø

P9�Ø 0 00 0 0 0 0 1 0 00 0 H9�tù,�P9�tù,� −/H9 + ,�9 + ,�:0�tù,� − P9�tù,�

�/,�: − ,�902�tù,� 00 0 0 0 0 0 0 10 0 0 0 �/,�: − ,�902 tù 0 �:/,�9 + ,�:04 tù 0��

��

(5.83)

and:

=Û/"0 =

�� 00000,�9�tù,� iÙ/��", "0 + ,�:�tù,� iÙ/��" − �, "0

0�,�92 tù iÙ/��", "0 − �,�: tù iÙ/��" − �, "0 ��,ÚÛ,� =

��&Û,9/00&Û,:/00&Û,I/00&Û,J/00&Û,�/00&Û,Ü/00&Û,ú/00&Û,ë/00��

��=

�� i�/00i' �/00iØ/00i'Ø/00itù/00i' tù/00!tù/00!'tù/00��

�� (5.84)

Equation (5.84) shows that the only forcing input into the train subsystem is the vertical

motion of the rail at the locations of the wheels, ur(Vxt,t) and ur(Vxt—L,t). This system of

equations can be solved using the Runge-Kutta approach as explained in Section 2.3.1.

5.7.3 Calculation of Hertz Contact Stiffness

To add accuracy to the model, calculation of the Hertz contact stiffness, CH, is considered.

The Hertz contact stiffness can be calculated with the equation:

,ý = 2m # 23OS:_�þ�(A9 I⁄

(5.85)

where F0 is the normal load, and E’ is the plain strain elastic modulus determined by:

140

OS = O1 − �: (5.86)

R0 is the effective radius of curvature of the wheel-rail surfaces, and is determined by:

þ� = 812 # 1þ� + 1þ�f + 1þÙ + 1þÙf(;A9

(5.87)

where Rw is the effective wheel radius, Rwt is the wheel transverse radius of curvature, Rr is

the rail radius of curvature in the rolling direction (usually infinity), and Rrt is the rail

transverse radius of curvature.

The variable ξ from Equation (5.85) is determined from the lookup table shown in Table

5-6 and the equation:

cos ! = −þ�2 # 1þ� − 1þ�f + 1þÙ − 1þÙf( (5.88)

Table 5-6. Lookup table for determining the value of ξ. cosθ ξ/2

0 1 0.1711 0.9934 0.3329 0.9741 0.4781 0.9436 0.6022 0.9036 0.7036 0.8566 0.7836 0.8048 0.8446 0.7503 0.8900 0.6949 0.9231 0.6398 0.9467 0.5861 0.9634 0.5346 0.9750 0.4859 0.9831 0.4401 0.9886 0.3974 0.9923 0.3580 0.9949 0.3217 0.9966 0.2885 0.9977 0.2582 0.9985 0.2307 0.9990 0.2058

Using Equations (5.85) – (5.88), with available wheel and rail parameters, the Hertz

coefficient can then be calculated by using the values of the normal load, F0, on the spring.

141

5.7.4 Modeling the Broken Rail for the Two-Wheel System

Modeling a broken rail for the two-wheel system involves more details than the single

wheel model since both wheels will encounter a transition phase in which they will be

traversing the rail break. This involves a five-step process in the solver, which is illustrated

in Figure 5.17. The five steps are as follows. Step 1: from the beginning of the simulation

until the leading wheel contacts the right rail; Step 2: from the leading wheel contacting the

right rail to the leading wheel fully rotating onto the right rail; Step 3: from the leading

wheel leaving the left end of the right rail to the trailing wheel contacting the right rail; Step

4: from the trailing wheel contacting the right rail to the trailing wheel fully rotating onto

the right rail; Step 5: from the trailing wheel leaving the left end of the right rail to the end

of the simulation. In each step of the simulation the vertical motions of each wheel is

solved for as well as the vertical motions of the left and right rails. The five steps are

combined to form the final solution to the broken rail simulation.

Figure 5.17. Five step process for two-wheel model broken rail simulation.

5.7.5 Effects of Loading from other Bogies

To further improve the accuracy of the two-wheel model, the effects of loading from the

trailing bogie of the locomotive and the trailing bogie of the IFC are also considered. Figure

5.18 shows a diagram to illustrate this. The dynamic responses of the two wheels on the

lead bogie of the IFC are determined as described in the previous sections. To incorporate

the effects of other wheels contacting the rail in the locomotive-IFC system, constant

weights from the trailing bogie of the locomotive and the trailing bogie of the IFC are

applied to the rail as well. The constant weight from the trailing bogie of the IFC is applied

at a fixed distance, Ltrail, behind the trailing wheel of the leading bogie of the IFC. The

constant weight from the trailing bogie of the locomotive is applied at a fixed distance, Llead,

in front of the leading wheel of the leading bogie of the IFC.

Step 1 Step 2 Step 3 Step 4 Step 5

142

Figure 5.18. Loading of trailing axles of locomotive and IFC.

5.7.6 Broken Rail Simulation using the Two-Wheel Model

This section presents the setup and results of a single simulation of the two-wheel

railcar model traversing a broken rail. Table 5-7 shows the physical parameters for the

simulation. The physical parameters for the train and rail are obtained from the IFC

NUCARS model and 136 lbs./yard rail model. The length used for the rails is L1=L2=3 m

with a broken rail gap size of g = 0.1 m and tie spacing of Ltie = 0.65 m. Both the left and

right rails are initially at rest with zero deflection. The simulation parameters are shown in

Table 5-8.

L LleadLtrail

IFC Locomotive

F1F2 WlocoWIFC

143

Table 5-7. Physical parameter values used in the two-wheel model broken rail simulations.

Table 5-8. Simulation parameter values for two-wheel model broken rail simulation. Parameter Symbol Value Units Simulation time T 0.28 s Time step k 0.001 s Knots (number of beam elements) per rail n 300 Maximum allowable error ξ 10-7 Maximum iterations σ 100

Figure 5.19 shows a comparison of the simulated side frame vertical acceleration

response to actual data collected for a broken rail at the HTL. The FFT of both signals are

Parameter Symbol Value Units ¼ carbody mass Mc 29,845 kg ½ bolster mass MB 368 kg 1 axle + 1 sideframe + 2 wheel mass Msf,w 2,148 kg ½ primary suspension stiffness k1 4,515,179 N/m ½ primary suspension damping c1 5 x 104 N/m/s ½ secondary suspension stiffness k2 3,502,528,428 N/m ½ secondary suspension damping c2 17,701 N/m/s Wheel radius r , Rw 0.4572 m Wheel transverse radius of curvature Rwt 0.0143 m Distance between two wheels L 1.59 m Distance from rear axle to trailing force Ltrail 11.4 m Distance from front axle to leading force Llead 5.3 m Forward speed of train Vx 15.65 m/s Length of left rail L1 3 m Length of right rail L2 3 m Size of rail break g 0.1 m Young’s Modulus of rail E 1.93 x 1011 N/m2 Area moment of inertia of rail I 39.20 x 10-6 m4 Cross sectional area of rail Ac 8.7 x 10-3 m2 Density of rail ρ 7,850 kg/m3 Poisson’s ratio of rail ν 0.3 Rail radius of curvature (rolling direction) Rr ∞ m Rail transverse radius of curvature Rrt 0.0318 m Initial Hertzian contact spring coefficient CH0 1.071 x 109 N/m3/2 Pad stiffness kp 52 + 6.24 x 108 xp

2 MN/m Pad damping cp 3 x 106 N/m/s ½ tie mass Mt 40 kg Tie spacing Ltie 0.65 m Ballast stiffness kb 22.75 + 2.60 x 108 xb

2 MN/m Ballast damping cb 6 x 105 N/m/s

144

also compared. The response from the simulation was post-processed in order to mimic

the data collected at the HTL. It was first down-sampled from 1000 Hz to 256 Hz to match

the sampling frequency. Next, the signal was filtered with a 5th order low pass Butterworth

filter. The results show that the simulated response comes close to matching the collected

data.

Figure 5.19. Comparison of actual and simulated data for broken rail (g = 100 mm) vertical acceleration signature from two-wheel model.

5.7.7 Surface Fracture Simulation using the Two-Wheel Model

As discussed previously, a surface fracture is simulated by using the broken rail model

with small break gap size. In this case, g = 1 mm was used. All of the same physical

parameters, operating conditions, and simulation parameters were used as the broken rail

simulation, which are shown in Table 5-7 and 5-8. The results of the simulation are shown

in Figure 5.20. The plot shows that the actual and simulated side frame vertical

acceleration responses match one another well in terms of amplitude and frequency

content.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

-5

0

5

Time (s)



Data

Simulation

0 20 40 60 80 100 1200

0.5

1

1.5

Frequency (Hz)

Modulus


145

Figure 5.20. Comparison of actual and simulated data for surface fracture (g = 1 mm) vertical

acceleration signature from two-wheel model.

5.8 Limitations of the Dynamic Wheel-Rail Interaction Model

The dynamic wheel-rail interaction model has limitations in terms of the physical system

that it can represent. Figure 5.19 shows a validation for a broken rail with gap size of 100

mm and Figure 5.20 shows a validation for a surface fracture (broken rail) with gap size of

1 mm. As a result, it can only safely be said that the dynamic wheel-rail interaction model

is valid for broken rails ranging from 1 mm to 100 mm. For a given forward speed and rail

break gap size, it is recommended that the time resolution of the simulation be chosen such

that there will be at least three calculations during the enveloping of the broken rail. This

condition will hold true if H ≤ 3n �⁄ . It was also found that there are no limitations for the

wheel radius, as long as the value is physically realistic.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-5

0

5

Time (s)



Data

Simulation

0 20 40 60 80 100 1200

0.2

0.4

0.6

0.8

1

Frequency (Hz)

Modulus


146

5.9 Effects of Varying Physical Parameters and Operating Conditions

on Measured Vertical Accelerations

The two-wheel dynamic wheel-rail interaction model is now used to perform

simulations to study the effects of varying track and railcar physical parameters and

operating conditions on measured side frame vertical accelerations. This section considers

varying the parameter values in five different categories and observing the responses. The

categories of interest are: payload, forward speed, rail cross section, rail steel stiffness, and

wheel contact point. In each category, the upper and lower bounds of the parameters are

selected based on physically realistic operating conditions for freight cars in the U.S. The

tests consider three different payloads (mass of the carbody, Mc, is varied), four different

forward speeds (Vx is varied), three different rail cross sections (rail cross sectional area,

Ac, moment of inertia, I, and transverse radius of curvature, Rrt, are varied), four different

rail steel stiffness values (Young’s modulus, E, is varied), and three different rail contact

points (wheel transverse radius of curvature, Rwt, is varied).

The simulations begin with a control run. The physical parameters and operating

conditions of the control run are shown in Table 5-9. For the parameters and operating

conditions varied for each of the five studied categories, one of the runs is the control run.

The control run is the IFC running over a broken rail loaded to a gross weight of 268,000

lbs. (Mc = 27,875 kg), driving at the maximum safe speed of 40 mph on Class 3 rail (Vx =

17.88 m/s), driving on 132 lb./yard rail (Ac = 8.4 x 10-3 m2, I = 3.67 x 10-5 m4, Rrt = 0.03175

m), standard rail steel stiffness (E = 190 GPa), and AAR-1B 1:20 wheel profile with contact

point near the flange (Rwt = 0.01429 m). These parameters and operating conditions are

highlighted in Table 5-9 with a bold border. For all simulations, only the parameters and

operating conditions previously mentioned and shown with a bold border in Table 5-9 are

varied. All other parameters and operating conditions shown in Table 5-9 remain constant

for all simulations. The simulation parameters are the same for all simulations, and are

shown in Table 5-10.

147

Table 5-9. Physical parameter values and operating conditions for the control set in the parameter and operating condition varying simulations.

Table 5-10. Simulation parameter values for the parameter and operating condition varying simulations.

Parameter Symbol Value Units Simulation time T 0.25 s Time step k 0.001 s Knots (number of beam elements) per rail n 300 Maximum allowable error ξ 10-7 Maximum iterations σ 100

Parameter Symbol Value Units ¼ carbody mass Mc 27,875 kg ½ bolster mass MB 368 kg 1 axle + 1 sideframe + 2 wheel mass Msf,w 2,148 kg ½ primary suspension stiffness k1 4,515,179 N/m ½ primary suspension damping c1 5 x 104 N/m/s ½ secondary suspension stiffness k2 3,502,528,428 N/m ½ secondary suspension damping c2 17,701 N/m/s Wheel radius r , Rw 0.4572 m Wheel transverse radius of curvature Rwt 0.01429 m Distance between two wheels L 1.59 m Distance from rear axle to trailing force Ltrail 11.4 m Distance from front axle to leading force Llead 5.3 m Forward speed of train Vx 17.88 m/s Length of left rail L1 3 m Length of right rail L2 3 m Size of rail break g 0.1 m Young’s Modulus of rail E 1.90 x 1011 N/m2 Area moment of inertia of rail I 3.67 x 10-5 m4 Cross sectional area of rail Ac 8.4 x 10-3 m2 Density of rail ρ 7,850 kg/m3 Poisson’s ratio of rail ν 0.3 Rail radius of curvature (rolling direction) Rr ∞ m Rail transverse radius of curvature Rrt 0.03175 m Initial Hertzian contact spring coefficient CH0 1.038 x 109 N/m3/2 Pad stiffness kp 52 + 6.24 x 108 xp

2 MN/m Pad damping cp 3 x 106 N/m/s ½ tie mass Mt 40 kg Tie spacing Ltie 0.65 m Ballast stiffness kb 22.75 + 2.60 x 108 xb

2 MN/m Ballast damping cb 6 x 105 N/m/s

148

Figure 5.21 shows the train body responses for the control simulation. Vertical

displacement, velocity, and acceleration are shown for the carbody, bolster, side frame, and

wheels. Rotational displacement, velocity, and acceleration are shown for the side frame.

Of all of these measured values, the side frame vertical acceleration is of primary interest

because that is where the sensor is placed on the IFC. Figure 5.22 shows a larger plot of the

side frame vertical acceleration (in g’s) for the broken rail simulation. The plot shows a

maximum vertical acceleration amplitude of 4.71 g. The side frame oscillates at a

frequency around 45 Hz.

Figure 5.21. Train body responses for the control simulation.

0 0.1 0.2-0.066

-0.065

-0.064Carbody

Displacement (m

)

0 0.1 0.2-10

-5

0

5x 10

-3Carbody

Velocity (m/s)

0 0.1 0.2-0.5

0

0.5Carbody

Acceleration (m/s2)

0 0.1 0.2-0.066

-0.065

-0.064Bolster

Displacement (m

)

0 0.1 0.2-10

-5

0

5x 10

-3Bolster

Velocity (m/s)

0 0.1 0.2-2

-1

0

1Bolster

Acceleration (m/s2)

0 0.1 0.2-4

-3.5

-3

-2.5x 10

-3Side Frame

Displacement (m

)

0 0.1 0.2-0.2

0

0.2Side Frame

Velocity (m/s)

0 0.1 0.2-50

0

50Side Frame

Acceleration (m/s2)

0 0.1 0.2-2

0

2x 10

-3Side Frame

Roation (rad)

0 0.1 0.2-1

0

1Side Frame

Rot. Velocity (rad/s)

0 0.1 0.2-500

0

500Side Frame

Rot. Acceleration (rad/s2)

0 0.1 0.2-6

-4

-2

0x 10

-3Wheels

Displacement (m

)

Time (s)

Front

Rear

0 0.1 0.2-1

0

1Wheels

Velocity (m/s)

Time (s)

0 0.1 0.2-500

0

500Wheels

Acceleration (m/s2)

Time (s)

149

Figure 5.22. Side frame vertical acceleration response for the control simulation.

The following sections describe the details and show the results of the other simulations

in the parameter and operating condition varying tests. These sections show the side

frame vertical acceleration response for changes in car payload, forward speed, rail cross

section, rail steel stiffness, and wheel contact point. After the results are presented,

observations are made and conclusions are drawn.

5.9.1 Effects of Car Payload

The simulations in this section seek to explore the effects of car payload on side frame

vertical acceleration response. The lower and upper bounds were chosen based on typical

loads that may be seen by a freight car, which range from 220,000 lbs. to 315,000 lbs. To

achieve this effect in the dynamic wheel-rail interaction model, the carbody mass was

changed accordingly. The simulation matrix for these tests is shown in Table 5-11.

0 0.05 0.1 0.15 0.2 0.25-5

-4

-3

-2

-1

0

1

2

3

4

5

Time (s)


150

Table 5-11. Simulation matrix for load varying simulations. Simulation # Description Varied parameter: Mc (kg)

1 Light load (220,000 lbs/ 99,790 kg) 22,432 2* Medium load (268,000 lbs/ 121,563 kg) 27,875 3 Heavy load (315,000 lbs/ 142,882 kg) 33,205

*control

Figure 5.23 shows the results from these simulations. Both the side frame vertical

acceleration signatures and the frequency content (FFT) of the vertical acceleration

signatures are shown. The results show that the payload has a fairly significant effect on

the side frame response. The maximum amplitude of the vertical acceleration signature is

3.74 g for the light load simulation, 4.71 g for the medium load simulation, and 5.69 g for

the heavy load simulation. The FFT plots also show that the dominant frequencies in the

acceleration signature increases with increased load. The low load signature shows

frequency content concentrated around 45 Hz, the medium load signature shows frequency

content concentrated in the 45-50 Hz range, and the heavy load signature shows frequency

content concentrated in the 45-50 Hz range with significant content between 50-55 Hz as

well.

151

Figure 5.23. Side frame vertical acceleration response and FFT for the load-varying simulations.

5.9.2 Effects of Forward Speed

To assess the effects of railcar forward speed on side frame vertical acceleration while

traversing a rail surface irregularity, standard forward speeds for freight trains are used in

the dynamic wheel-rail interaction broken rail model. The Federal Railroad Administration

(FRA) sets maximum forward speeds for freight trains travelling on different classes of

track. The classes are chosen based on the quality of the rail. For freight rail, the class 1

maximum forward speed is 10 mph, the class 2 maximum forward speed is 25 mph, the

0 0.05 0.1 0.15 0.2 0.25-6

-4

-2

0

2

4

6

Mc = 22,432 kg

Time (s)


0 20 40 60 80 100 1200

0.5

1

1.5

Mc = 22,432 kg

Frequency (Hz)

Modulus

0 0.05 0.1 0.15 0.2 0.25-6

-4

-2

0

2

4

6

Mc = 27,875 kg

Time (s)


0 20 40 60 80 100 1200

0.5

1

1.5

Mc = 27,875 kg

Frequency (Hz)

Modulus

0 0.05 0.1 0.15 0.2 0.25-6

-4

-2

0

2

4

6

Mc = 33,205 kg

Time (s)


0 20 40 60 80 100 1200

0.5

1

1.5

Mc = 33,205 kg

Frequency (Hz)

Modulus

152

class 3 maximum forward speed is 40 mph, and the class 4 maximum forward speed is 60

mph. These four values of forward speed are used in the simulations. These simulations

are summarized in Table 5-12.

Table 5-12. Simulation matrix for speed varying simulations. Simulation # Description Varied operating condition: Vx (m/s)

1 Class 1 rail: 10 mph 4.47 2 Class 2 rail: 25 mph 11.18 3* Class 3 rail: 40 mph 17.88 4 Class 4 rail: 60 mph 26.82

*control

Figure 5.24 shows the side frame vertical acceleration responses for the IFC traveling

over a broken rail at the four different forward speeds. Since the IFC is traveling at a

different speed in each simulation, the time span between the leading and trailing wheels

traversing the broken rail is different. As a result, the lengths of the vertical acceleration

signatures are different for each simulation and comparison of the results is less

straightforward. Since all simulations were run at the same sampling frequency, and since

slower speed for equal length rail means longer simulation time, the lower speed

simulations have higher resolution FFT’s than the higher speed simulations. Nevertheless,

careful observation of the results yields valuable information. The maximum amplitude of

the vertical acceleration signature is 4.85 g for the class 1 speed, 4.80 g for the class 2

speed, 4.71 g for the class 3 speed, and 4.78 g for the class 4 speed. These results suggest

that there is no specific correlation between forward speed and vertical acceleration

amplitude. For all speeds, the dominant frequencies are between 45 and 50 Hz, and there

is no noticeable trend between forward speed and frequency content.

153

Figure 5.24. Side frame vertical acceleration response and FFT for the speed-varying simulations.

5.9.3 Effects of Rail Cross Section

To assess the effects of rail cross section on the side frame vertical acceleration

response, three common rail cross sections used in freight transportation are considered.

The rail is described by its weight per unit length. The three rail sections used in this study

are 115 lb/yard, 132 lb/ yard, and 136 lb/yard. For each rail section, the dimensions of the

head, web, and base are different, which lead to differences in rail cross sectional area, area

moment of inertia, and rail transverse radius of curvature. The parameter values for each

of the three rail sections are shown in Table 5-13.

0 0.2 0.4 0.6 0.8 1

-5

0

5

Time (s)

Vertical Acceleration (g) V

x = 4.47 m/s

0 20 40 60 80 100 1200

0.5

1

1.5

Vx = 4.47 m/s

Frequency (Hz)

Modulus

0 0.1 0.2 0.3 0.4

-5

0

5

Time (s)


x = 11.18 m/s

0 20 40 60 80 100 1200

0.5

1

1.5

Vx = 11.18 m/s

Frequency (Hz)

Modulus

0 0.05 0.1 0.15 0.2 0.25

-5

0

5

Time (s)


x = 17.88 m/s

0 20 40 60 80 100 1200

0.5

1

1.5

Vx = 17.88 m/s

Frequency (Hz)

Modulus

0 0.05 0.1 0.15

-5

0

5

Time (s)


x = 26.82 m/s

0 20 40 60 80 100 1200

0.5

1

1.5

Vx = 26.82 m/s

Frequency (Hz)

Modulus

154

Table 5-13. Simulation matrix for rail cross section varying simulations. Simulation

#

Description Varied parameter:

Ac (m2)

Varied parameter:

I (m4)

Varied parameter:

Rrt (m)

1 115 lb/yard rail 7.3 x 10-3 2.73 x 10-5 0.0381 2* 132 lb/ yard rail 8.4 x 10-3 3.67 x 10-5 0.0318 3 136 lb/ yard rail 8.7 x 10-3 3.92 x 10-5 0.0318

*control

The results of the rail section-varying simulations are shown in Figure 5.25. The

maximum amplitude of the vertical acceleration signature is 5.23 g for the 115 lb/ yard rail,

4.71 g for the 132 lb/ yard rail, and 4.58 g for the 136 lb/ yard rail. The results show higher

amplitude vertical acceleration signatures for lower weight rail. In all simulations, the

dominant frequencies are between 43 and 47 Hz. There is not any noticeable difference in

the frequency responses of the three rail sections.

155

Figure 5.25. Side frame vertical acceleration response and FFT for the rail section-varying simulations.

5.9.4 Effects of Rail Steel Stiffness

To assess the effect of rail steel stiffness on the vibration response of the side frame, four

different values of Young’s Modulus for the rail steel are considered. Young’s modulus

values for steel are typically in the range of 180-210 GPa. As a result, this range is used

with the following labels for each value: 180 GPa for low stiffness, 190 GPa for medium-low

0 0.05 0.1 0.15 0.2 0.25-6

-4

-2

0

2

4

6

Time (s)


Rail: 115 lb/yard

0 20 40 60 80 100 1200

0.5

1

1.5Rail: 115 lb/yard

Frequency (Hz)

Modulus

0 0.05 0.1 0.15 0.2 0.25-6

-4

-2

0

2

4

6

Time (s)


Rail: 132 lb/yard

0 20 40 60 80 100 1200

0.5

1


Frequency (Hz)

Modulus

0 0.05 0.1 0.15 0.2 0.25-6

-4

-2

0

2

4

6

Time (s)


Rail: 136 lb/yard

0 20 40 60 80 100 1200

0.5

1


Frequency (Hz)

Modulus

156

stiffness, 200 GPa for medium-high stiffness, and 210 GPa for high stiffness. This is

outlined in Table 5-14.

Table 5-14. Simulation matrix for rail steel stiffness varying simulations. Simulation # Description Varied parameter: E (GPa)

1 Low stiffness 180 2* Medium-low stiffness 190 3 Medium-high stiffness 200 4 High stiffness 210

*control

Figure 5.26 shows the results of the rail steel stiffness varying simulations. The results

show that the maximum amplitude of the side frame vertical acceleration signature is 4.70

g for E = 180 GPa, 4.71 g for E = 190 GPa, 4.71 g for E = 200 GPa, and 4.71 g for E = 210 GPa.

The results of the FFT on the vertical acceleration signatures shows that the frequency

content is concentrated between 43 and 47 Hz for all values of Young’s Modulus. For E =

180 GPa and 190 GPa, the dominant frequency is at 43 Hz, and the modulus at 47 Hz is

slightly lower. For E = 200 GPa, the modulus at 43 Hz and 47 Hz is close to equal. For E =

210 GPa, the dominant frequency is at 47 Hz, and the modulus at 43 Hz is slightly lower.

Therefore, it is seen that there is similar frequency content in the vertical acceleration

responses generated from all values of Young’s Modulus, but with a slight shift to higher

frequency content as the value of Young’s Modulus increases.

157

Figure 5.26. Side frame vertical acceleration response and FFT for the rail steel stiffness-varying simulations.

5.9.5 Effects of Wheel Contact Point

To assess the effects of wheel contact point on the side frame vertical acceleration

response, three different contact points were selected on a standard AAR-1B wheel profile

with 1:20 taper. Figure 5.27 shows the three contact points on the wheel profile, labeled A,

B, and C. Each of these points has a different value for the wheel transverse radius of

curvature. These values are listed in Table 5-15.

0 0.05 0.1 0.15 0.2 0.25

-5

0

5

Time (s)


E = 180 GPa

0 20 40 60 80 100 1200

0.5

1

1.5E = 180 GPa

Frequency (Hz)

Modulus

0 0.05 0.1 0.15 0.2 0.25

-5

0

5

Time (s)


E = 190 GPa

0 20 40 60 80 100 1200

0.5

1

1.5E = 190 GPa

Frequency (Hz)

Modulus

0 0.05 0.1 0.15 0.2 0.25

-5

0

5

Time (s)


E = 200 GPa

0 20 40 60 80 100 1200

0.5

1

1.5E = 200 GPa

Frequency (Hz)

Modulus

0 0.05 0.1 0.15 0.2 0.25

-5

0

5

Time (s)


E = 210 GPa

0 20 40 60 80 100 1200

0.5

1

1.5E = 210 GPa

Frequency (Hz)

Modulus

158

Table 5-15. Simulation matrix for load varying simulations. Simulation # Description Varied parameter: Rwt (m)

1* AAR-1B, 1:20, point A 0.0143 2 AAR-1B, 1:20, point B 0.0381 3 AAR-1B, 1:20, point C 0.2540

*control

Figure 5.27. Diagram of AAR-1B, 1:20 wheel profile showing three different contact points.

Figure 5.28 shows the results of the wheel contact point-varying simulations. The

maximum amplitude of the side frame vertical acceleration response is 4.71 g for contact

point A, 4.86 g for contact point B, and 5.02 g for contact point C. For all three contact

points, the dominant frequencies in the vertical acceleration signatures lie between 43 Hz

and 47 Hz. At contact point A, the modulus at 43 Hz is higher than at 47 Hz, at contact point

B the modulus at 47 Hz is slightly higher than at 43 Hz, and at contact point C the modulus

at 47 Hz is higher than at 43 Hz. The results of the FFT analysis show that the frequency

content of the vertical acceleration signatures is similar for all three contact points, but

with a slight shift to higher frequency content as the contact point moves closer to the field

side of the wheel profile, i.e., as the wheel transverse radius of curvature increases.

AB C

159

Figure 5.28. Side frame vertical acceleration response and FFT for the wheel contact point-varying simulations.

5.9.6 Observations and Conclusions from Parameter and Operating Condition

Varying Simulations

The results of the physical parameter and operating condition varying simulations show

the following results: increased payload results in increased side frame vertical

acceleration amplitude and higher frequency content; changing the forward speed does not

0 0.05 0.1 0.15 0.2 0.25-6

-4

-2

0

2

4

6

Time (s)


Rwt = 0.0143 m

0 20 40 60 80 100 1200

0.5

1

1.5

Rwt = 0.0143 m

Frequency (Hz)

Modulus

0 0.05 0.1 0.15 0.2 0.25-6

-4

-2

0

2

4

6

Time (s)


Rwt = 0.0381 m

0 20 40 60 80 100 1200

0.5

1

1.5

Rwt = 0.0381 m

Frequency (Hz)

Modulus

0 0.05 0.1 0.15 0.2 0.25-6

-4

-2

0

2

4

6

Time (s)


Rwt = 0.254 m

0 20 40 60 80 100 1200

0.5

1

1.5

Rwt = 0.254 m

Frequency (Hz)

Modulus

160

significantly affect the side frame vertical acceleration amplitude or frequency content;

increased rail cross sectional area and increased rail moment of inertia both result in

decreased amplitude of the side frame vertical acceleration response; increased values of

rail Young’s modulus results in an increase in the frequency content of the side frame

response, and a slight increase in amplitude in the side frame vertical acceleration

response; increase in wheel transverse radius of curvature results in increased side frame

vertical acceleration amplitude and higher frequency content.

The results from the simulations suggest that the value of the Hertz contact stiffness

tends to dominate the side frame vertical acceleration response. In all of the simulations,

the parameter values that changed the side frame response the most are all present in the

calculation of the Hertz contact stiffness. Equations (5.85)–(5.87) show that there will be

an increase in Hertz contact stiffness for increased values of normal load, Young’s modulus,

and effective radius of curvature at the contact point between the wheel rail. The

conclusions drawn from the tests and the values of the Hertz contact stiffness for each

simulation are summarized in Table 5-16. The results show a direct correlation between

increase in Hertz contact stiffness and increase in side frame vibration amplitude as well as

increase in frequency content. The only exception is the simulation for the 136 lb/ yard

rail where there is no change in Hertz contact stiffness but a decrease in the vibration

amplitude of the side frame. The reason for this is that the increase in cross sectional area

and the increase in area moment of inertia result in an increase in the rail stiffness. As a

result, there is less rail deflection at the rail break, resulting in a smaller amplitude

excitation into the train system.

161

Table 5-16. Results from parameter and operating condition varying simulations with values of Hertz contact stiffness.

Simulation

type

Description

Effect on

amplitude

(compared

to control)

Effect on

frequency

(compared

to control)

CH

(Nx109/m3/2)

Payload Light load (220,000 lbs) decrease decrease 0.972 Payload Medium load (268,000 lbs)* - - 1.038 Payload Heavy load (315,000 lbs) increase increase 1.096 Forward speed Class 1 rail: 10 mph no change no change 1.038 Forward speed Class 2 rail: 25 mph no change no change 1.038 Forward speed Class 3 rail: 40 mph* - - 1.038 Forward speed Class 4 rail: 60 mph no change no change 1.038 Rail section 115 lb/yard rail increase increase 1.043 Rail section 132 lb/ yard rail* - - 1.038 Rail section 136 lb/ yard rail decrease no change 1.038 Steel stiffness Low stiffness decrease decrease 1.001 Steel stiffness Medium-low stiffness* - - 1.038 Steel stiffness Medium-high stiffness increase increase 1.074 Steel stiffness High stiffness increase increase 1.110 Contact point AAR-1B, 1:20, point A* - - 1.038 Contact point AAR-1B, 1:20, point B increase increase 1.090 Contact point AAR-1B, 1:20, point C Increase increase 1.143

*control

To verify the conclusion that the contact stiffness at the wheel-rail interface dominates

the side frame response, two more simulations are run. The first simulation uses a

combination of parameters from the previous simulations that results in a low Hertz

contact stiffness, and the second simulation uses a combination of parameters from the

previous simulations that results in a high Hertz contact stiffness. The parameters for

these two simulations are outlined in Table 5-17.

Table 5-17. Parameters for Hertz contact stiffness-varying simulations. Simulation CH

(Nx109/m3/2)

Mc

(kg)

Vx

(m/s)

Ac

(m2)

I

(m4)

Rrt

(m)

E

(GPa)

Rwt

(m)

Low CH 0.938 22,432 17.88 8.7 E-3 3.92 E-5 0.032 180 0.0143 Control 1.038 27,875 17.88 8.4 E-3 3.67 E-5 0.032 190 0.0381 High CH 1.290 33,205 17.88 8.4 E-3 3.67 E-5 0.032 210 0.2540

162

The results from the Hertz contact stiffness-varying simulations are shown in Figure

5.29. The control set shows a maximum value of 4.71 g for the side frame vertical

acceleration and frequency content concentrated around 43 Hz to 47 Hz. The low contact

stiffness simulation shows a vertical acceleration response with maximum value of 3.65 g

and frequency content concentrated around 43 Hz. The high contact stiffness simulation

shows a vertical acceleration response with maximum value of 6.14 g and frequency

content concentrated around 47 Hz to 55 Hz. The results clearly show that the value of the

Hertz contact stiffness strongly affects the side frame response. The dominance of the

wheel-rail contact in the dynamic response of the system suggests that a defect detection

algorithm using side frame vertical acceleration signatures should be tuned using

parameters that affect the calculation of CH. From Equations (5.85) – (5.87) it is seen that

these parameters are: vehicle normal load, Young’s modulus of the steel, Poisson’s ratio of

the steel, wheel radius, wheel conicity/contact point (wheel transverse radius), and rail

contact point (rail transverse radius of curvature).

163

Figure 5.29. Side frame vertical acceleration response and FFT for the Hertz contact stiffness-varying simulations.


This chapter presented a dynamic wheel-rail interaction model for generating data to

support the defect detection algorithm. A combined system approach was used where the

train and pad-tie-ballast rail supports were all modeled as discrete parameter systems, and

the rail was modeled as a distributed parameter system. Because of the complexity of the

system, a numerical approximation was used. The Runge-Kutta method was used to solve

0 0.05 0.1 0.15 0.2 0.25

-5

0

5

Time (s)


Low Contact Stiffness, CH = 0.93775 E9 N/m

3/2

0 20 40 60 80 100 1200

0.2

0.4

0.6

0.8

1

Low Contact Stiffness, CH = 0.93775 E9 N/m

3/2

Frequency (Hz)

Modulus

0 0.05 0.1 0.15 0.2 0.25

-5

0

5

Time (s)


Control Contact Stiffness, CH = 1.0382 E9 N/m

3/2

0 20 40 60 80 100 1200

0.2

0.4

0.6

0.8

1

Control Contact Stiffness, CH = 1.0382 E9 N/m

3/2

Frequency (Hz)

Modulus

0 0.05 0.1 0.15 0.2 0.25

-5

0

5

Time (s)


High Contact Stiffness, CH = 1.29 E9 N/m

3/2

0 20 40 60 80 100 1200

0.2

0.4

0.6

0.8

1

High Contact Stiffness, CH = 1.29 E9 N/m

3/2

Frequency (Hz)

Modulus

164

for the discrete parameter subsystems and a fifth-degree B-spline approximation was used

to solve for the distributed parameter subsystem. An iterative approach was then used to

simultaneously determine the solutions for all subsystems.

The model was expanded to include non-linear dynamics in the wheel-rail contact

mechanics and in the pad and ballast spring rates. A broken rail model and surface fracture

model were developed for the single-wheel model. A vertical acceleration response of the

wheel was obtained using the broken rail model and was then compared to an actual

vertical acceleration response from the IFC driving over a broken rail at similar operating

conditions. The results showed similar amplitude and frequency content for the two

responses for a broken rail simulation with break gap size of 100 mm and a surface

fracture simulation with break gap size of 1 mm.

The model was then expanded to include two wheels and pitch motion of the side frame.

The model was then validated by showing similar amplitude and frequency content

between the collected data and simulated response for a broken rail simulation with break

gap size of 100 mm and a surface fracture simulation with break gap size of 1 mm. This

model was then used to perform an analysis on the effect of changing physical parameters

and operating conditions on the side frame vertical acceleration response. It was found

that the value of the Hertz contact stiffness, CH, tends to dominate the side frame response.

Therefore, parameters affecting the value of the contact stiffness most affect the side frame

response. It was found that the car payload and rail steel stiffness (value of Young’s

modulus) affect the value of CH the most. As a result, these are the parameters of interest

for tuning the defect detection algorithm.

165

Chapter 6

Physics-based Tuning of the Defect Detection Algorithm

6.1 Introduction

Chapter 4 presented three frequency-based algorithms for detecting rail defects from a

side frame and wheel vertical acceleration signal. Chapter 5 presented a validated dynamic

wheel-rail interaction model that is capable of simulating vertical acceleration responses

for various values of physical parameters and operating conditions. The varied physical

parameters and operating conditions lie within typical ranges of these values for normal

freight train operation. This chapter considers the results of the previous two chapters in

order to present a physics-based method for tuning the defect detection algorithms.

Section 6.2 presents a simplified physical model that can be used to select the

frequencies to be targeted by the defect detection algorithm. Section 6.3 shows the effects

of applying a low pass filter to the signal and discusses the effects that it may have on

defect detection. Section 6.4 shows the results of a wavelet analysis on all of the vertical

acceleration signals from the parameter and operating condition varying simulations from

Chapter 5. Section 6.5 presents the limitations of the defect detection algorithm. Section

6.6 draws conclusions on the results.

6.2 Physics-based Tuning of the Defect Detection Algorithm

In order to imbed the physics of the train-rail system into the defect detection algorithm,

the results from the previous chapter are first considered. It was found (and summarized

in Table 5-16) that the vertical acceleration response at the side frame is dominated by the

contact mechanics, i.e., the value of the Hertz contact stiffness, CH, between the wheel and

the rail. As a result, the proposed method for including the effects of the train-rail system

166

into the defect detection methodologies is an algorithm that targets the dominant

frequencies that arise from the presence of the wheel-rail contact stiffness. These

dominant frequencies are calculated from a simplified physical model, which is shown in

Figure 6.1. In the diagram, k1 is the primary suspension stiffness, CH,lin is the Hertz contact

stiffness linearized about some nominal operating point, and Mw is the mass of the wheel +

½ axle + ½ side frame.

Figure 6.1. Simplified oscillating single-wheel model.

For the single degree-of-freedom oscillating system shown in Figure 6.1, it is known that

the natural frequency of the mass is determined by:

TF = QHÔùù�Ôùù = 2��F (6.1)

where ωn is the natural frequency in rad/s, fn is the natural frequency in Hz, keff is the

effective spring stiffness, and meff is the effective mass. To determine the value of keff, the

linearization of the Hertz contact spring is considered. The force in the contact spring is

given by the equation:

_t = ,ê�I :⁄ (6.2)

To linearize the contact stiffness, the first derivative with respect to displacement is taken:

#�_t�� (�^�½ = 32,ê��9 :⁄ = ,ê,é�F (6.3)

167

where x0 is the nominal operating point at which the force-displacement relationship is

linearized. Plugging this into Equation (6.1) gives an equation for the calculation of the

natural frequency of the single degree-of-freedom system:

�F = 12� QH9 + ,ê,é�F�� = �/H9, ��, _�, O, �, þÙ , þÙf, þ�, þ�f0 (6.4)

The right hand side of Equation (6.4) shows that the natural frequency calculation is a

function of train primary suspension stiffness, wheel mass, payload, wheel and rail material

properties (Young’s modulus and Poisson’s ratio), rail radius of curvature, rail transverse

radius of curvature, wheel radius of curvature, and wheel transverse radius of curvature.

The majority of these physical parameters come from the calculation of the linearized

Hertz contact stiffness as given by Equation (6.3) and Equations (5.85) – (5.88).

The assumption in the single-wheel oscillator is that the natural frequency calculated

from Equation (6.4) will match the dominant frequency in the FFT of the simulated

response, thus suggesting that the contact mechanics between the wheel and the rail

dominate the data recorded by the accelerometer. In order to assess the validity of this

claim, the frequency content of the side frame vertical acceleration responses from the

parameter and operating condition varying simulations is compared to the natural

frequency calculated with equation (6.4). Figure 6.2 shows the frequency responses for the

load varying simulations, Figure 6.3 shows the frequency responses for the speed varying

simulations, Figure 6.4 shows the frequency responses for the rail section varying

simulations, Figure 6.5 shows the frequency responses for the rail stiffness varying

simulations, Figure 6.6 shows the frequency responses for the wheel contact point varying

simulations, and Figure 6.7 shows the frequency responses for the CH varying simulations.

The results show that for each simulation, the natural frequency calculated with the

proposed model closely matches the dominant frequency present in the frequency

response. It should be noted that when calculating the FFT, the vertical acceleration

signatures were padded with zeros to increase the frequency resolution, thus resulting in

the smooth curves shown in the plots. Padding with zeros affects the amplitude, as seen by

the low values of modulus. However, this is unimportant since the relative change in

168

amplitude is of interest. The results from the simulations and natural frequency

calculations are summarized in Table 6-1.

Figure 6.2. Frequency responses of side frame vertical acceleration for load varying simulations.

0 20 40 60 80 100 1200

0.005

0.01

0.015

0.02

0.025

0.03

0.035

Frequency (Hz)

Modulus

Load Varying Simulations

Sim 1 freq

Sim 1 fn,calc

Sim 2 freq

Sim 2 fn,calc

Sim 3 freq

Sim 3 fn,calc

169

Figure 6.3. Frequency responses of side frame vertical acceleration for speed varying simulations.

Figure 6.4. Frequency responses of side frame vertical acceleration for rail section varying

simulations.

0 20 40 60 80 100 1200

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Frequency (Hz)

Modulus

Speed Varying Simulations

Sim 4 freq

Sim 5 freq

Sim 6 freq

Sim 7 freq

0 20 40 60 80 100 1200

0.005

0.01

0.015

0.02

0.025

0.03

Frequency (Hz)

Modulus

Rail Section Varying Simulations

Sim 8 freq

Sim 8 fn,calc

Sim 9 freq

Sim 9 fn,calc

Sim 10 freq

Sim 10 fn,calc

170

Figure 6.5. Frequency responses of side frame vertical acceleration for rail stiffness varying

simulations.

Figure 6.6. Frequency responses of side frame vertical acceleration for wheel contact point

varying simulations.

0 20 40 60 80 100 1200

0.005

0.01

0.015

0.02

0.025

0.03

Frequency (Hz)

Modulus

Rail Stiffness Varying Simulations

Sim 11 freq

Sim 11 fn,calc

Sim 12 freq

Sim 12 fn,calc

Sim 13 freq

Sim 13 fn,calc

Sim 14 freq

Sim 14 fn,calc

0 20 40 60 80 100 1200

0.005

0.01

0.015

0.02

0.025

0.03

Frequency (Hz)

Modulus

Wheel Contact Point Varying Simulations

Sim 15 freq

Sim 15 fn,calc

Sim 16 freq

Sim 16 fn,calc

Sim 17 freq

Sim 17 fn,calc

171

Figure 6.7. Frequency responses of side frame vertical acceleration for contact stiffness varying

simulations.

Table 6-1. Summary of results comparing the dominant frequency of each simulation to the natural frequency calculated with the single degree-of-freedom model.

Sim

#

Description

F0

(kN)

CH

(Nx109/m3/2)

x0

(mm)

CH,lin

(MN/m)

fn,calc

(Hz)

fn,sim

(Hz)

1 Light load (220,000 lbs) 122.4 0.972 2.512 73.08 42.15 43.97 2 Medium load (268,000 lbs)* 149.1 1.038 2.742 81.55 44.46 45.14 3 Heavy load (315,000 lbs) 175.2 1.096 2.946 89.21 46.45 45.85 4 Class 1 rail: 10 mph 149.1 1.038 2.742 81.55 44.46 45.12 5 Class 2 rail: 25 mph 149.1 1.038 2.742 81.55 44.46 43.68 6 Class 3 rail: 40 mph* 149.1 1.038 2.742 81.55 44.46 45.14 7 Class 4 rail: 60 mph 149.1 1.038 2.742 81.55 44.46 47.49 8 115 lb/yard rail 149.1 1.043 2.733 81.81 44.52 44.91 9 132 lb/ yard rail* 149.1 1.038 2.742 81.55 44.46 45.14

10 136 lb/ yard rail 149.1 1.038 2.742 81.55 44.46 45.14 11 Low stiffness 149.1 1.002 2.809 79.61 43.94 44.91 12 Medium-low stiffness* 149.1 1.038 2.742 81.55 44.46 45.14 13 Medium-high stiffness 149.1 1.074 2.680 83.43 44.96 45.38 14 High stiffness 149.1 1.110 2.623 85.26 45.43 45.61 15 AAR-1B, 1:20, point A* 149.1 1.038 2.742 81.55 44.46 45.14 16 AAR-1B, 1:20, point B 149.1 1.090 2.654 84.25 45.17 45.38 17 AAR-1B, 1:20, point C 149.1 1.143 2.571 86.97 45.87 45.61 18 Low CH simulation 122.4 0.938 2.573 71.35 41.67 43.73 19 Medium CH simulation* 149.1 1.038 2.742 81.55 44.46 45.14 20 High CH simulation 175.2 1.290 2.642 99.47 48.98 47.26

*control

0 20 40 60 80 100 1200

0.005

0.01

0.015

0.02

0.025

0.03

Frequency (Hz)

Modulus

CH Varying Simulations

Low CH freq

Low CH fn,calc

Control freq

Control fn,calc

High CH freq

High CH fn,calc

172

6.3 Effects of Filtering the Signal

The results shown in Table 6-1 show dominant frequencies between 40 and 50 Hz for all

simulations. The FFT for each run is calculated from the raw, unfiltered response.

However, the signal collected at the HTL and used in the training and validation of the

defect detection algorithms was filtered with a 30 Hz low pass filter. In order to use the

data from the simulations with the defect detection algorithms from Chapter 4, the data

must be processed in the same way as the data collected at the HTL. As a result, a fifth

order Butterworth filter with cutoff frequency of 30 Hz is applied to the vertical

acceleration signatures from the simulations. To visualize this, the FFT of the unfiltered

signal and the filtered signal are compared. In both cases, the signals are subsampled from

the 1000 Hz sampling frequency of the simulation to the 256 Hz sampling frequency of the

IFC data acquisition system. Figure 6.8 shows these results for the load varying

simulations, Figure 6.9 shows these results for the speed varying simulations, Figure 6.10

shows these results for the rail stiffness varying simulations, Figure 6.11 shows these

results for the rail section varying simulations, and Figure 6.12 shows these results for the

wheel contact point varying simulations. It should be noted that a gain is applied to the

filter in order to preserve the amplitude of the filtered signal for means of comparison.

The results show that the frequency content of the filtered signal is very similar for all

simulations. The effect of the high-frequency content (40-50 Hz), which is the

distinguishing difference for all the parameter and operating conditions varying

simulations, is drastically reduced by the low pass filter. After filtering, the remaining low

frequency content (below 40 Hz) then contains the dominant features of the signal.

However, for all simulations, the low frequency content remains consistent. This suggests

that there are inherent low-frequency values excited by a rail surface fracture or broken

rail present regardless of the physical parameters or operating conditions. These low

frequencies are targeted by the defect detection algorithm and therefore have the potential

to be detected.

173

Figure 6.8. Frequency responses for load varying simulations. The FFT is shown for the subsampled (256 Hz) response and the subsampled and filtered (30 Hz low pass) response.

0 20 40 60 80 100 1200

0.2

0.4

0.6

0.8

1

Mc = 22,432 kg (subsampled)

Frequency (Hz)

Modulus

0 20 40 60 80 100 1200

0.2

0.4

0.6

0.8

1

Mc = 22,432 kg (subsampled, filtered)

Frequency (Hz)

Modulus

0 20 40 60 80 100 1200

0.2

0.4

0.6

0.8

1


Frequency (Hz)

Modulus

0 20 40 60 80 100 1200

0.2

0.4

0.6

0.8

1


Frequency (Hz)

Modulus

0 20 40 60 80 100 1200

0.2

0.4

0.6

0.8

1


Frequency (Hz)

Modulus

0 20 40 60 80 100 1200

0.2

0.4

0.6

0.8

1


Frequency (Hz)

Modulus

174

Figure 6.9. Frequency responses for speed varying simulations. The FFT is shown for the subsampled (256 Hz) response and the sub-sampled and filtered (30 Hz low pass) response.

0 20 40 60 80 100 1200

0.5

1

Vx = 4.47 m/s (subsampled)

Frequency (Hz)

Modulus

0 20 40 60 80 100 1200

0.5

1

Vx = 4.47 m/s (subsampled, filtered)

Frequency (Hz)

Modulus

0 20 40 60 80 100 1200

0.5

1


Frequency (Hz)

Modulus

0 20 40 60 80 100 1200

0.5

1


Frequency (Hz)

Modulus

0 20 40 60 80 100 1200

0.5

1


Frequency (Hz)

Modulus

0 20 40 60 80 100 1200

0.5

1


Frequency (Hz)

Modulus

0 20 40 60 80 100 1200

0.5

1


Frequency (Hz)

Modulus

0 20 40 60 80 100 1200

0.5

1


Frequency (Hz)

Modulus

175

Figure 6.10. Frequency responses for rail stiffness varying simulations. The FFT is shown for the subsampled (256 Hz) response and the subsampled and filtered (30 Hz low pass) response.

0 20 40 60 80 100 1200

0.5

1E = 180 GPa (subsampled)

Frequency (Hz)

Modulus

0 20 40 60 80 100 1200

0.5

1E = 180 GPa (subsampled, filtered)

Frequency (Hz)

Modulus

0 20 40 60 80 100 1200

0.5


Frequency (Hz)

Modulus

0 20 40 60 80 100 1200

0.5


Frequency (Hz)

Modulus

0 20 40 60 80 100 1200

0.5


Frequency (Hz)

Modulus

0 20 40 60 80 100 1200

0.5


Frequency (Hz)

Modulus

0 20 40 60 80 100 1200

0.5


Frequency (Hz)

Modulus

0 20 40 60 80 100 1200

0.5


Frequency (Hz)

Modulus

176

Figure 6.11. Frequency responses for rail section varying simulations. The FFT is shown for the subsampled (256 Hz) response and the subsampled and filtered (30 Hz low pass) response.

0 20 40 60 80 100 1200

0.2

0.4

0.6

0.8

1Rail: 115 lb/yard (subsampled)

Frequency (Hz)

Modulus

0 20 40 60 80 100 1200

0.2

0.4

0.6

0.8

1Rail: 115 lb/yard (subsampled, filtered)

Frequency (Hz)

Modulus

0 20 40 60 80 100 1200

0.2

0.4

0.6

0.8


Frequency (Hz)

Modulus

0 20 40 60 80 100 1200

0.2

0.4

0.6

0.8


Frequency (Hz)

Modulus

0 20 40 60 80 100 1200

0.2

0.4

0.6

0.8


Frequency (Hz)

Modulus

0 20 40 60 80 100 1200

0.2

0.4

0.6

0.8


Frequency (Hz)

Modulus

177

Figure 6.12. Frequency responses for wheel contact point varying simulations. The FFT is shown for the subsampled (256 Hz) response and the subsampled and filtered (30 Hz low pass) response.

6.4 Defect Detection for Simulated Responses

The results from Section 6.3 suggest that there is inherent low-frequency content

present in all side frame vertical acceleration signatures generated from rail defects

regardless of the physical parameters or operating conditions. These low-frequency values

are highlighted by the low pass filter. The next step is to run the simulated signals through

the defect detection algorithm. To ensure consistency between the analyses of the real data

0 20 40 60 80 100 1200

0.2

0.4

0.6

0.8

1

Rwt = 0.0143 m (subsampled)

Frequency (Hz)

Modulus

0 20 40 60 80 100 1200

0.2

0.4

0.6

0.8

1

Rwt = 0.0143 m (subsampled, filtered)

Frequency (Hz)

Modulus

0 20 40 60 80 100 1200

0.2

0.4

0.6

0.8

1


Frequency (Hz)

Modulus

0 20 40 60 80 100 1200

0.2

0.4

0.6

0.8

1


Frequency (Hz)

Modulus

0 20 40 60 80 100 1200

0.2

0.4

0.6

0.8

1


Frequency (Hz)

Modulus

0 20 40 60 80 100 1200

0.2

0.4

0.6

0.8

1


Frequency (Hz)

Modulus

178

in Chapter 4 and simulated data from Chapter 5, the subsampled (256 Hz) and filtered (30

Hz low pass) vertical acceleration signatures from the simulations are analyzed. The top

plot of Figure 6.13 shows the 20 vertical acceleration signatures from the simulations

combined into one signal. Low-level Gaussian white noise was added to the pre-filtered

signal in order to simulate sensor noise and rail roughness. The same exact wavelet

analysis that was performed in Section 4.5.3 on the collected data is now performed on the

simulated responses. A Daubechies wavelet with 3 vanishing moments is used and the first

four scales of the Continuous Wavelet Transform are observed. Figure 6.13 shows these

results.

To proceed with the analysis, the wavelet coefficients from Figure 6.13 are then

thresholded at the same levels as the signals presented in Section 4.5.3. These threshold

values are shown in Table 4-2. The thresholded wavelet coefficients are shown in Figure

6.14. The results show that the low frequency content (scales 3 and 4) is highlighted by the

wavelet analysis. All 20 vertical acceleration signatures from the simulated responses are

identified by the wavelet analysis and their exact locations in the time domain are

identified.

179

Figure 6.13. Wavelet analysis on simulated signals.

0 5 10 15 20 25 30 35 40 45-10

0

10

D0

0 5 10 15 20 25 30 35 40 45-1

0

1

D1

0 5 10 15 20 25 30 35 40 45-2

0

2

4

D2

0 5 10 15 20 25 30 35 40 45-10

0

10

D3

0 5 10 15 20 25 30 35 40 45-10

0

10

20

D4

Time (s)

180

Figure 6.14. Thresholded wavelet coefficients of simulated signals.

6.5 Limitations of the Dynamic Model and Defect Detection Algorithm

As a final look at the defect detection algorithm, the case is considered for generating a

surface fracture signature with the dynamic wheel-rail interaction model and running it

through the wavelet analysis. In Section 5.8 it was concluded that the minimum rail break

gap size that the dynamic model can mimic is 1 mm. A broken rail of this size is considered

to fall in the category of a surface fracture for this study. Figure 6.15 shows the results of

0 5 10 15 20 25 30 35 40 45-10

0

10

D0

0 5 10 15 20 25 30 35 40 45-1

0

1

D1

0 5 10 15 20 25 30 35 40 450

1

2

3

D2

0 5 10 15 20 25 30 35 40 450

5

10

D3

0 5 10 15 20 25 30 35 40 450

5

10

15

D4

Time (s)

181

the wavelet analysis on the surface fracture signature generated from the dynamic model.

White noise was added to the original signal to simulate sensor noise and rail roughness.

The vertical acceleration signature was post-processed by down-sampling the signal to 256

Hz and filtering it with a fifth order Butterworth filter with cutoff frequency of 30 Hz.

Figure 6.16 shows the results of the thresholding of wavelet coefficients. The results show

that the wavelet analysis successfully identifies the exact location of the surface fracture.

Figure 6.15. Wavelet analysis on surface fracture (g = 1 mm) simulated signal.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5-50

0

50

D0

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5-0.5

0

0.5

1

D1

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5-1

0

1

2

D2

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5-5

0

5

D3

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5-10

0

10

20

D4

Time (s)

182

Figure 6.16. Thresholded wavelet coefficients of surface fracture (g = 1 mm) simulated signal.

To further test the limitations of the dynamic wheel-rail interaction model and the defect

detection algorithm, a simulation is run for g = 0.5 mm. The results of the wavelet analysis

are shown in Figure 6.17 and Figure 6.18. The small fracture is recognized only in the

fourth scale of the wavelet transform. Nevertheless, the exact location of the defect is still

identified.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5-50

0

50

D0

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5-1

0

1

D1

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5-1

0

1

D2

0 0.5 1 1.5 2 2.5 3 3.5 4 4.50

2

4

6

D3

0 0.5 1 1.5 2 2.5 3 3.5 4 4.50

5

10

15

D4

Time (s)

183

Figure 6.17. Wavelet analysis on surface fracture (g = 0.5 mm) simulated signal.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5-50

0

50

D0

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5-0.5

0

0.5

D1

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5-1

0

1

2

D2

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5-5

0

5

D3

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5-10

0

10

D4

Time (s)

184

Figure 6.18. Thresholded wavelet coefficients of surface fracture (g = 0.5 mm) simulated signal.


This chapter presented a methodology for locating the dominant frequency in a side

frame vertical acceleration signature for a train traversing a broken rail. The methodology

assumes that the contact mechanics between the wheel and rail interface dominate the side

frame response. As a result, a single degree-of-freedom wheel oscillator model was used to

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5-50

0

50

D0

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5-1

0

1

D1

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5-1

0

1

D2

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5-1

0

1

D3

0 0.5 1 1.5 2 2.5 3 3.5 4 4.50

5

10

D4

Time (s)

185

calculate the natural frequency. This method proved to work well, and simplifies the

physical model by placing emphasis on the parameters that affect the dynamic response

the most.

The simulated responses were then post-processed with a 30 Hz low pass filter in order

to mimic the processing of the data collected at the HTL. Filtering of the simulated

responses showed a near removal of all of the high frequency content that proved to be the

distinguishing characteristic for the parameter and operating condition variation.

Nevertheless, the filtered signal retained vital information in the low frequency values that

could be used by the wavelet analysis to indicate a broken rail. All 20 vertical acceleration

signatures were processed by the same wavelet analysis presented in Chapter 4, and all 20

vertical acceleration signatures were exactly located with no false positives. Additionally, it

was shown that the wavelet analysis is able to successfully identify the exact location of a

broken rail as small as 0.5 mm in size.

186

Chapter 7

Conclusions and Future Work ______________________

7.1 Summary of Research

This research focused on the development of a rail defect detection and classification

algorithm suitable for use with a data acquisition system installed onboard a train car

equipped with accelerometers mounted to the bogie side frames. The research began by

collecting data using the instrumented freight car (IFC) at the High Tonnage Loop (HTL) at

Transportation Technology Center, Inc. (TTCi). Vertical acceleration data from

accelerometers mounted to the bogie side frame was collected for several laps around the

HTL. The data was analyzed using the Fourier Transform. The time-domain signatures and

frequency content for various rail surface irregularities were correlated with photographs

from a visual inspection of the track. It was found that there is enough information

contained in the vertical acceleration signatures to distinguish between different rail

surface irregularities, including various special track components and various defect types.

Three defect detection and classification algorithms were then developed and presented.

The first algorithm uses the Fourier Transform for feature extraction and an Artificial

Neural Network for classification. The second algorithm uses the Wavelet Transform for

feature extraction and an Artificial Neural Network for classification. The third algorithm

uses the Wavelet Transform to perform a regularity analysis on the signal to detect defects.

All of the algorithms were trained and validated using the data collected from the HTL. The

results of the algorithms were compared to the results produced from a standard

commercial threshold-based algorithm.

A dynamic wheel-rail interaction model was then developed for the purpose of

generating artificial training data for the defect detection algorithms. The model uses

187

combined systems modeling to treat the train and rail supports as lumped parameter

models and the rail as a distributed parameter Euler-Bernoulli beam. A solution was found

by developing a solver that simultaneously solves for the combined lumped-distributed

parameter system. The wheel-rail interaction model was validated by comparing the side

frame vertical acceleration response due to a broken rail from the model to broken rail

data collected at the HTL.

Finally, the dynamic wheel-rail interaction model was used to perform a sensitivity

analysis on the physical system to determine the physical parameters and operating

conditions that most affect the dynamic response of the train as measured by an

accelerometer. It was found that the contact mechanics between the wheel and rail tend to

dominate the wheel and side frame vertical acceleration responses. Therefore, physical

parameters and operating conditions that affect the value of the wheel-rail Hertz contact

stiffness play the largest role in the dynamic response. A method was proposed that uses a

single wheel oscillator to calculate the dominant natural frequency of the system. The

defect detection algorithm can then be tuned to target this natural frequency.

7.2 Major Conclusions

The major conclusions from this research are:



• Current signal processing techniques for extracting rail defect information from a

vertical acceleration signal are basic and typically only consider the magnitude of

the signal, resulting in frequent false positives and false negatives.

• Frequency analysis shows that there is sufficient information in the bogie side

frame vertical acceleration signal to distinguish between signatures generated from

various rail surface irregularities.

• A small rail head surface fracture can quickly propagate and turn into a broken rail

after only a few train cars have passed over it. Therefore, early detection of rail

surface defects is necessary.

188

• The developed Fourier Transform-based feature extraction and neural network-

based feature classification algorithm analyzes the signal with a moving FFT

window and classifies signatures into one of four categories: broken rail, impending

break, process noise (special track), and nothing of interest. The algorithm

successfully detects the impending break and broken rail from the HTL data

without issuing any false positives or false negatives.

• The developed Wavelet Transform-based feature extraction and neural network-

based feature classification algorithm uses wavelets to target the defect signatures

present in the signal. The algorithm successfully detects the impending break and

broken rail from the HTL data without issuing any false positives or false negatives.

• The developed Wavelet Transform-Intensity Factor approach uses the Wavelet

Transform to perform a regularity analysis on the signal. For a given degree of

regularity, the degree of severity was also calculated, which is called intensity

factor. The algorithm uses the value of the intensity factor to determine whether or

not a signature was generated from a defect. The algorithm successfully detects the

impending break and broken rail from the HTL data without issuing any false

positives or false negatives.

• All three algorithms outperform a standard commercial threshold-based algorithm

for the HTL data. However, the algorithms require more extensive training in order

to be used for a wider range of operating conditions.

• A dynamic wheel-rail interaction model was developed to generate artificial

training data for the defect detection algorithms.

• The developed dynamic wheel-rail interaction model was validated by comparing

the results from a broken rail simulation to actual broken rail data collected from

the HTL. The results showed similar amplitude and frequency content for the two

responses. The model was validated with real data for a broken rail/ surface

fracture with defect sizes of 100 mm and 1 mm.

• The dynamic wheel-rail interaction model was used to perform a sensitivity

analysis on the physical system to determine which physical parameter and

operating conditions most affect the measurements of the accelerometer. It was

189

determined that those variables affecting the value of the wheel-rail Hertz contact

stiffness most affect the train dynamic response. These parameters are: train

primary suspension stiffness, wheel mass, payload, wheel and rail material

properties (Young’s modulus and Poisson’s ratio), rail radius of curvature, rail

transverse radius of curvature, wheel radius of curvature, and wheel transverse

radius of curvature.

• A simple single degree-of-freedom wheel oscillator was used as a basis to calculate

the dominant natural frequency of the train-rail system. The physics of the wheel-

rail system can then be imbedded in the defect detection methodologies by tuning

the algorithm to target this natural frequency.

• The data collected at the HTL was processed with a 30 Hz low-pass filter, therefore,

the data generated from the physical parameter and operating condition varying

simulations was also post-processed with a 30 Hz low-pass filter. It was found that

after filtering the data, regardless of the physical parameters or operating

conditions, the resulting signals contained low frequency content that could be used

to detect broken rails.

• The defect detection algorithm is capable of detecting a broken rail from data

generated by the dynamic wheel-rail interaction model for defects as small as 0.5

mm.

7.3 Future Work

The future work in this research is as follows:

• Collect data with no filter or a low-pass filter with higher cutoff frequency from the

IFC at the HTL for different physical parameters and operating conditions and use

this data to further validate the dynamic wheel-rail interaction model.

• Mount tri-axial accelerometers to the left and right side frames of the IFC and record

all accelerations. Use the collected data to determine whether or not lateral and

longitudinal accelerations contain relevant information for detecting defects.

• Based on the results from the unfiltered data, improve the dynamic wheel-rail

interaction model as necessary.

190

• Determine whether or not the data should be filtered and if so, at what cutoff

frequency.

• Use the dynamic wheel-rail interaction model to generate data for different physical

parameters and operating conditions and use it to imbed the physics of the train-rail

system into the defect detection algorithm.

• Pending permission from TTCi, test the defect detection system in real time by

installing it on the IFC and performing testing at the HTL.

• Implement the defect detection algorithm on a commercial defect detection system,

such as the ENSCO V/TI Monitor. The proposed algorithm should be able to out-

perform the signal processing routine currently used in the commercial system in

terms of detecting and locating impending rail breaks and broken rails.

191

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a wavelet-based rail surface defect prediction and ......a wavelet-based rail surface defect...

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