Non-Invasive Recovery of
Gear Rotation from Machine Vibration
Christopher Allen Lerch
Submitted to the Department of Mechanical Engineeringon January 20, 1995,
in partial fulfillment of the requirements for the degree of
Master of Science
Abstract
A method termed harmonic tracking is developed to recover the rotations of gears as functions of time from machine casing vibration. The harmonic tracking method uses short-time
spectral generation and a subsequent set of algorithms to locate and track gear meshing
frequencies as functions of time. The meshing frequencies are then integrated with respect
to time to obtain the rotation of individual gears. More specifically, spectral generation
is performed using both the discrete fourier transform and autoregressive models, and the
locating and tracking algorithms involve locating tones in each short-time spectrum and
tracking them through successive spectra to recover gear meshing harmonics. The harmonic tracking method is found to be more robust than demodulation-based
methods in
the presenceof measurement noise and signal distortion from the structural transfer function
between gears and the casing.
The harmonic tracking method is tested, both through simulation and experiments
involving motor-operated valves (MOV's) as part of the development of a diagnostic system
for MOV's. In all cases, the harmonic tracking method is found to recover gear rotation at
all velocities down to a level at which the narrowband tones associated with gear meshing
harmonics are indistinguishable from background noise.
The harmonic tracking method should be generally applicable to situations in which a
non-invasivetechnique is required for determining the time-dependent rotations and velocities of gearbox input, intermediary, and output shafts.
Acknowledgments
It may be my name that appears on the cover of this document, but don't be fooled. This
research is very much the collective product of a community of teachers, colleagues, and
friends, without whom none of the work could have taken place. I can't thank everyone
who's touched my life in the last year-and-a-half, but I'll sure try!
We gratefully acknowledge our two funding sources.
on Remedial Action and Nuclear Policy, subcontract
The first is the MIT Program
#9-X51-N8356-1 from Los Alomos
National Laboratory which operates under contract to the United States Department of
Energy. The second is the MIT International Program for Enhanced Nuclear Power Plant
Safety, which is run by the MIT Energy Laboratory under the direction of Kent Hansen.
Special thanks also go to the Electric Power Research Institute (EPRI) for kindly inviting
us to be a part of the testing of Valve 43. In particular, we'd like to acknowledge the help
of Mike Eidson of Southern Nuclear and Neil Estep of Duke Power for their support and
for keeping us in touch with the needs of the industry. Finally, we'd like to thank the
Limitorque Corporation for donating equipment to the project.
Next, I must thank all of the members of the MOV project: Professors Richard Lyon,
Jeffrey Lang, and Jangbom Chai, Dr. Daniel McCarthy, and Wayne Hagman. Professor
Lyon, you've been an absolutely terrific advisor. The fact that you've given me the freedom
to pursue the path(s) I felt were correct, and that you took a minimum direction, maximum
technical support approach to advising has allowed me to grow more as an engineer than I
may be able to fathom at this time. Jeff, acoustics and vibrations may not be your specialty,
but your patience, understanding, and support have certainly enriched my experience at
MIT. Jangbom, our technical discussions on your demodulation-based
method confused,
confounded, and enlightened me. I hope I've done the right thing in changing to the
harmonic tracking method.
Dan, if I could have two advisors for this thesis, your name
would be on the cover. Your technical and personal advice and your passion for acoustics
(not to mention comic relief!) have taught me an immense amount about what it takes to
be a good engineer. Wayne, it's your ability to give me a good kick in the pants when I was
struggling that got this research off the ground. I haven't always agreed with your advice,
but I've never taken it lightly.
Special thanks go to Mary Toscano, who has taken care of all of the necessities of life
that I work so hard to ignore. Thanks for all of your behind-the-scenes help, Mary. I'll
certainly miss those famous cookies!
Two other faculty stand out in my mind as having had a strong impact on my work:
Professors Hamid Nawab and J. Robert Fricke. Rob, it's been great getting to know you
on technical and personal levels. Like Dan, your passion for acoustics has been inspiring.
I look forward to our future technical (but not necessarily engineering-related),
musical,
and two-wheeled conversations. Hamid, through 6.341 and our few discussions, you've had
the greatest impact on the technical content of my research. This thesis is all about signal
processing, and you've played the greatest role in helping me learn enough to tackle this
research problem.
The members of the acoustics and vibration lab have helped me to learn what it really
means to be a part of a tight-knit research group. With John Chi and Sophie Debost
I've shared one of the most difficult, stressful, and rewarding experiences of my life thus
far: getting a master's degree in acoustics and vibration.
John, your dignity and sense
of honor are great to see. I may not agree with your views on politics, but I have great
respect for them.
Sophie, your sheer brilliance has been awe-inspiring.
I hope we have
the chance to work in the same lab again someday. Rama Rao, our talks about matlab,
non-stationary signals, drilling dynamics, and job hunting have left very fond memories.
In-Soo Suh, I wish you the best of luck with your qualifying exams and with the rest of
your PhD. Djamil Boulahbal, thanks for unlimited access to the "library." Without you,
my list of references would be much, much shorter! Hua He, your enthusiasm and kindness
are very much appreciated.
The Conner 2 crew has been a wonderful part of my experience here. Theresa Chiueh, I
can't express strongly enough how much I appreciate your constant support during a very
difficult portion of my life. I'll always treasure your kindness and compassion. To the rest,
including Eugene Chow, Joe Bank, Sylvia Chen, Mike Purcell, Chris Anderson, Jeff Wong,
Yoli Leung, Dave, Janet, Diana Dorinson, and Fe Lam, I must say thanks for the hockey,
unihoc, and sense of community. You guys are great!
Now, on to my closest (unfortunately, non-resident) friends: Mike Katz, Erin Dwyer,
Keith McNeal, and my Mom, Barbara Lerch. Mike, our breaks to play squash, design
automotive greatness, and consider the finer points in life are some of the most enjoyable
I've ever spent. Without you, life here would have been pretty empty. Erin, you more than
anyone, understands who I am and why I am. I can't imagine life without you. Keith,
sadly, we blew our year together in Boston. Hopefully we'll have the chance to make up
for it before too long. It may be unusual, but I include my mom in this list because she
is one of my closest friends. Mom, we've been through more together than I can currently
comprehend (this thesis has thoroughly fried my brain), and your support, encouragement,
understanding, kindness, and love are treasured.
To my dad, Karl Lerch, and my brother, Terry Lerch, I must say thanks for listening
to (or reading) my babbling complaints, triumphs, failures, funny ideas, and for generally
taking an interest in what goes on here in Boston. We may not have that much in common,
but our conversations are still very important to me.
Finally, I'll thank someone who I've only recently begun to know, but who has quickly
become a wonderful part of my life: Lisa Tegeler. Lisa, I can't thank you enough for just
being Lisa. Of course, I must thank you for trundling through the awkward prose that's
become this thesis, but much more importantly, I'd like to express my appreciation for
your insight, intelligence, understanding, and emotion. And, I can't forget to mention my
appreciation for your introduction to the wonderful world of "Cha-Cha Chili." Thanks,
kiddo.
This thesis is dedicated to the memory of my grandfather, Theodore Kowal, and to Jim
Henson, two men who's passion has inspired the research and writing of this document.
Contents
13
1 Introduction
1.1
Motivation, Goals, and Scope ..........................
13
1.2
Strategy ......................................
14
.
15
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
1.3 Executive Summary and a Look at the Chapters Ahead ...............
1.4
Choosing a Methodology
1.4.1
Introduction to Harmonic Tracking Methodology ...............
.
16
1.4.2
Introduction to Demodulation-based Methodology
.
16
1.4.3
Comparison Between Methodologies, Making a Choice .......
.
17
1.4.4
Making a Choice Between the Two Methodologies
.
18
.........
.........
19
2 Source Identification/Characterization
2.1
2.2
Basics of Gear Meshing
. . ...................
.
........
2.1.1
Why Do Gears Create Vibration? ....................
2.1.2
Typical Gear Meshing Spectral Characteristics
Gear Meshing Vibrations in an MOV. . ....................
19
19
...........
.
25
........
..
.....
2.2.1
Introduction to the Mechanical Details of the MOV ........
25
2.2.2
Spectral Characteristics of Gear Meshing Vibrations in an MOV . .
26
2.2.3
Limitations Due to Broadband Noise Sources ............
26
2.2.4
Motor Pinion/Worm Shaft Gear Sideband Characteristics
2.2.5
Time Dependence
of MOV Gear Meshing Speeds
....
2.3 Chapter Summary.
. . .
.
.......................
A priori Information
Required
32
.
33
35
. . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Short-Time Spectral Generation .......................
6
30
.
3 Development of a Harmonic Tracking Signature Recovery Technique
3.1
20
.
35
36
3.2.1 Spectral (Frequency) Considerations ..................
37
3.2.2 Time Consideration: Time Resolution .................
38
3.2.3
39
Making the Tradeoffs
..........................
3.3 Short-time Spectral Analysis ..........................
39
3.3.1
Tone Location .............................
40
3.3.2
Harmonic Tracking ...........................
41
3.3.3
Sideband Relations ...........................
41
3.4
Integrating for Gear Rotation, Valve Travel, and Spring Pack Displacement
46
3.5
Summary and a Look Ahead
47
..........................
4 Predicting Harmonic Tracking Performance Via Simulation
48
4.1 Generating Simulated Casing Vibrations ....................
4.2
48
4.1.1
Modelling Gear Meshing as Phase Modulation
4.1.2
Modelling Transfer Functions as Rational System Functions .....
50
4.1.3
Modelling Measurement Noise as White Noise .............
50
Performing Harmonic Tracking on Simulated Casing Vibrations .......
50
4.2.1
Specifying the Four Simulated Casing Vibration Signals .......
51
4.2.2
Simulation Results ...........................
55
4.2.3
Assessment of Results .........................
61
............
49
5 Application of Harmonic Tracking Method to MOV Vibration Data
5.1
5.2
64
5.2.2
Analysis Goals .....................
.......
.......
.......
.......
.......
.......
.......
.......
5.2.3
Recovery Details, Static Case (No Flow) .......
.......
71
5.2.4
Discussion of SMB-2 Static Recovery Results
5.2.5
Recovery Details, 1800 psi Flow ............
5.2.6
Discussion of SMB-2 1800 psi Recovery Results . . .
.......
.......
.......
76
78
84
Applying Harmonic Tracking to Limitorque SMB-000 Data
5.1.1
Historical Perspective ................
5.1.2
Analysis Goals ....................
5.1.3
Recovery Details ...................
5.1.4
Discussion of Recovery Results
............
Applying Technique to Limitorque SMB-2 Data .......
5.2.1 Historical Perspective ................
7
....
65
65
65
65
69
70
70
71
5.3
Sum m ary
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Spectral
Generation
......................... . 85
5.3.1
5.3.2
Harmonic Tracking and Polynomial Fits ................
85
5.3.3
Results Assessment ............................
85
86
6 Conclusions and Recommendations
6.1
87
Summary of Major Results from Application of Method to MOV's .....
6.1.1
87
Simulation Results ........................................
6.1.2 Experimental Results ....................................
6.2
.
87
Recommendations for Incorporation of Method into MOV Diagnostic System
Diagnostic System Overview
......................
6.2.2
Discussion of Motor Stall During SMB-2 1800 psi Closing Stroke . .
6.2.3
Harmonic Tracking Modifications Necessary for "Black Box" Imple-
96
General Applicability of Harmonic Tracking Method .............
Introduction
98
. . . . . . . . . . . . . . . . . . . . . .
98
.
99
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
101
to Parametric
A.2 Formulation of AR Models
A.3
Zoomed AR Modeling
91
96
A Autoregressive Modeling for Spectral Estimation
A.1
89
89
6.2.1
mentation .................................
6.3
85
Modeling
......................................
8
List of Figures
2-1 A Generic Gear Set
....................
21
2-2
Spectrum of Meshing Forces Produced by Generic Gear Set ........ ...
.
22
2-3
Generic Gear Set with External Load on Shaft 2 .............. .......
.
24
2-4
Spectrum of Gear Meshing Forces with Periodic External Load Applied
. .
25
2-5
Schematic of Major Components in MOV Geartrain
.
27
2-6
The Two Gear Pairs in an MOV ........................
..................
28
2-7 Generic Spectrum of Gear Meshing Forces in an MOV ............
29
2-8 Spectrum of MOV Gear Meshing with Representative Noise Levels Added .
30
2-9 External Loading on Pinion/Worm Shaft Gear Pair in an MOV ......
.
31
2-10 Pinion/Worm Shaft Gear Meshing and Its Sideband Structure in an MOV .
32
.
2-11 Time Dependence of Pinion/Worm Shaft Gear Meshing in an MOV-Typical
of All MOV Harmonics ..............................
33
2-12 Operating Condition Dependence of Pinion/Worm Shaft Gear Meshing as a
Function of Time-Typical of All MOV Harmonics
3-1 Waterfall of Spectra
..............
34
.
...........................................
3-2 Tone Location for a Single Spectrum ......................
3-3
lTracking a Single Harmonic, Waterfall Visualization
40
..................
3-4 Tracking a Single Harmonic, 2D Visualization ................ ........
3-5
Tricking the Simple, Automatic Harmonic Tracker
37
..............
.
42
.
42
43
3-6 Sideband Relations: Carrier Present ......................
44
3-7
Sideband Relations: Carrier Absent ...................
45
4-1
Pseudo Block Diagram of Simulation ......................
49
4-2
Simulated Gear Meshing Frequencies ......................
52
9
4-3 Comparison of Simulated and Experimental Spectra
53
.............
4-4 Simulated Transfer Function ...........................
54
4-5 Tracked Harmonics for Simulation Case 1 (Baseline) and Simulation Case 2
(Realistic TF Distortion) ...........................
.
............
56
4-6 Short-time Spectral Generation and Tone Location for Simulation Case 3
(Minimum Required SNR) ...........................
............
.
56
.
57
4-7 Tracked Harmonics for Simulation Case 4 (Realistic TF and Noise) ....
4-8 Pinion/Worm Shaft Gear Meshing for the Simulation Cases 1 (Baseline) and
58
2 (TF Distortion) .................................
4-9 Worm/Worm Gear Meshing for Simulation Cases 1 (Baseline) and 2 (TF
Distortion) ..................................................
58
4-10 Pinion/Worm Shaft Gear Meshing for Simulation Case 4 (Realistic Case)
.
59
4-11 Worm/Worm Gear Meshing for Simulation Case 4 (Realistic Case) ....
.
59
4-12 Valve Travel for the Simulation Cases 1 (Baseline) and 2 (TF Distortion)
60
4-13 Valve Travel for Simulation Case 4 (Realistic Case) ....................
.
61
4-14 Spring Pack Displacement for Simulation Cases 1 (Baseline) and 2 (TF Dis-
tortion) ......................................
62
4-15 Spring Pack Displacement for Simulation Case 4 (Realistic Case) .....
5-1 Tracked Harmonics for SMB-000 Static Closing Stroke ........... .....
5-2 Recovered Pinion/Worm
.
62
.
66
.
Shaft Gear Meshing for SMB-000 Static Closing
Stroke .......................................
67
5-3 Recovered Worm/Worm Gear Meshing for SMB-000 Static Closing Stroke .
. .
5-4 Valve Travel During Heavily for SMB-000 Static Closing Stroke . .
5-5 Spring Pack Displacement for SMB-000 Static Closing Stroke ..........
68
.
68
.
69
5-6 Short-time Spectral Generation (using AR Models) and Harmonic Tracking
for Lightly Loaded Portion of SMB-2 Static Run .
.
.
.
.
..
73
5-7 Short-time Spectral Generation (using the DFT) and Harmonic Tracking for
73
Heavily Loaded Portion of SMB-2 Static Run .................
5-8 Linear Fits to Pinion/Worm Shaft Gear Meshing for SMB-2 Static Closing
Stroke
......................................
.
74
5-9 Worm/Worm Gear Meshing from Linear Fits for SMB-2 Static Closing Stroke 74
10
5-10 Motor Speed for Entire SMB-2 Static Closing Stroke .............
75
5-11 Motor Speed for Heavily Loaded Portion of SMB-2 Static Closing Stroke..
75
5-12 Valve Travel Over Entire SMB-2 Static Closing Stroke ............
76
5-13 Valve Travel During Heavily Loaded Portion of SMB-2 Static Closing Stroke
77
5-14 Short-time Spectral Generation (using AR Models) and Harmonic Tracking
for SMB-2 1800 psi Run .............................
79
5-15 Short-time Spectral Generation (using DFT) and Harmonic Tracking for
Heavily Loaded Portion of SMB-2 1800 psi Run ................
79
5-16 Pinion/Worm Shaft Gear Meshing for SMB-2 1800 psi Closing Stroke
. . .
81
5-17 Worm/Worm Gear Meshing for SMB-2 1800 psi Closing Stroke .......
81
5-18 Motor Speed for Entire SMB-2 1800 psi Closing Stroke ............
82
5-19 Motor Speed for Heavily Loaded Portion of SMB-2 1800 psi Closing Stroke
82
5-20 Valve Travel Over Entire SMB-2 1800 psi Closing Stroke . . . . .
83
. .. . .
5-21 Valve Travel During Heavily Loaded Portion of SMB-2 1800 psi Closing Stroke 83
6-1 MOV Diagnostic System .....................
....
90
6-2 Estimated Motor Torque for SMB-2 Static Closing Stroke . .
....
91
6-3
Estimated Motor Torque for SMB-2 1800 psi Closing Stroke .
6-4
Recovered Valve Travel for SMB-2 Static Closing Stroke . . .
6-5
Recovered Valve Travel for SMB-2 1800 psi Closing Stroke . .
6-6
Diagnostic Signature for SMB-2 Static Closing Stroke
6-8 Losses Due to Operator for SMB-2 Static Closing Stroke . . .
....
....
....
....
....
....
92
92
93
93
94
94
6-9
. . . .
95
....
6-7 Diagnostic Signature for SMB-2 1800 psi Closing Stroke . . .
Losses Due to Operator for SMB-2 1800 psi Closing Stroke
11
List of Tables
4.1
55
.......................................
12
Chapter 1
Introduction
This thesis develops the harmonic tracking method, an algorithm used for vibration-based
signature extraction.
From the casing vibration signal of a machine containing gears, the
method recovers the angular speed and displacement of each gear as a function of time. This
information may be useful in such applications as vibration-based machinery diagnostics and
controls. This project applies harmonic tracking to the diagnostics of motor-operated valves
(MOV's).
1.1
Motivation, Goals, and Scope
MOV's are applied to fluid piping systems in which remote control of fluid routing is important. Such systems and MOV's are found in nuclear power plants. Because the successful
operation of MOV's (ie. the valve opens or closes on demand under a variety of operating
conditions) is critical to the safe operation of nuclear power plants [4], an initiative is under
way to develop prognostic and diagnostic systems which determine both whether the MOV
will operate successfully and, if not, what components of the valve are faulty. Economically,
it is important that such a system be non-invasive, requiring minimal modifications to ex-
isting hardware, so that a utility company may quickly,easily, and inexpensivelyinstall the
system on any of its MOV's. Our project as a whole is a response to this initiative, and
our overall motivation is to respond to the need for a prognostic and diagnostic system to
ensure the safe operation of MOV's and nuclear power plants.
The scope of my work is not to create the diagnostic system, but rather to develop a
signature extraction method contained in the system. At an earlier stage in the project,
13
it was decided that determining valve travel as a function of time was a critical portion
of our diagnostic system. My task is to develop a robust signal processing method which
determines the angular speeds and displacements of two sets of gears from a casing vibration
signal taken during a valve closing stroke. Diagnostically important valve travel may be
recovered by scaling the angular displacement of one of the gears by an appropriate constant.
1.2
Strategy
Recovery of vibration source characteristics from casing vibration is much like hunting for
lost treasure 1 . For this project, the vibration source is the meshing of gears in an MOV, and
the source information sought (the treasure) is the angular speed and displacement of each
gear. However, this information is buried in a highly complex casing vibration signal. The
fourier transform of this signal, A(f), consists of the the gear meshing force, Xg(f), modified
by the structural transfer function (TF), Hgc(f), between the gears and the accelerometer
on the casing and the summation of other vibration inputs, Ni(f), modified by structural
TF's, Hie(f), between the input and the casing, or
A(f)
-= Hgc(f)Xg(f)
+ E Hic(f)Ni(f).
Because all information sought is contained in Xg(f), the effects of the noise, Ni(f),
(1.1)
and
the TF, Hgc(f), are the "pirates" around which we must maneuver in order to reach the
desired angular speed and displacement information. The first step in finding treasure is to
make a treasure map by defining exactly what information the source contains and where
it is located. The next step is to gather tools and devise a plan to access the treasure by
choosing a methodology and developing a technique for the extraction of source information
from the casing vibration signal. The third step is to develop expertise in using the tools
and to maximize the chances of a successful treasure hunt by predicting the performance of
the method under various conditions using simulated casing vibrations. The final step is to
hunt treasure by applying the method to experimental casing vibrations from two classes
of MOV under two dynamic flow conditions. Only at this stage may the treasure may be
examined to assess its value.
'I can't take credit for developing this analogy. It is the creation of Dr. Dan McCarthy, who used it in
various presentations
to our research group.
14
1.3
Executive Summary and a Look at the Chapters Ahead
Each chapter of this thesis represents a single stage of the treasure hunt outlined above.
Chapter 2, entitled "Source Identification/Characterization", explores the mechanisms
for gear meshing vibration generation, identifies the most appropriate domain for analysis,
characterizes the general form the angular speed and rotation information will take in that
domain, and, finally, narrows that form to the specific case of a valve closing stroke in an
MOV.
The second stage in the treasure hunt, selecting a methodology and developing a technique, contains three parts.
The first concludes Chapter 1, explaining how our choice of
methodology was changed from a demodulation-based method to the harmonic tracking
method based on a comparison of each method's robustness to distortion by the structural
transfer function and masking by measurement noise. Next, Chapter 3, "Development of
a Harmonic Tracking Signature Recovery Technique", details the technique developed to
implement the harmonic tracking methodology. The final step in stage two is Appendix A,
"Autoregressive Modeling for Spectral Estimation", which introduces one of the tools used
in the method.
The third stage of the treasure hunt, developing expertise in using the technique, is contained in Chapter 4, "Predicting Technique Performance Via Simulation", and details how
simulated casing vibrations were used to demonstrate under what conditions the method
may be successfullyimplemented. Three major results are presented. The first is that the
harmonic tracking method is insensitive to distortion by the structural transfer function.
The second is that a rule-of-thumb signal-to-noise ratio of 12 dB is required to perform
harmonic tracking. Finally, using a realistic transfer function and level of operating noise,
the error in total valve travel recovery is predicted to be .1%.
Chapter 5, "Application of Recovery Techniqueto MOV Vibration Data", is the treasure
hunt, cataloging the application of the method to casing vibrations from: (1) a Limitorque
SMB-000 closing stroke under static (no) flow conditions, (2) a Limitorque SMB-2 closing
stroke under static (no) flow conditions, and (3) a Limitorque SMB-2 closing stroke under
1800 psi flow conditions. The harmonic tracking method performed as designed on all three
analyses. Errors in total valve travel of .1% were found for the two static closing strokes.
Because the motor stalled before complete valve closure during the 1800 psi closing stroke,
15
the harmonic tracking method was able to capture total valve travel only to within 1%.
However, this case is held to be exceptional, and should not impact the application of
harmonic tracking to MOV diagnostics.
In the final chapter, "Conclusions and Recommendations", an assessment of technique
performance, recommendations for implementation in an MOV diagnostic system, and a
discussion of the general applicability of the technique are presented.
diagnostic signature is presented for the 1800 psi closing stroke.
In particular, our
It is hoped that this
information will aid in diagnosing the motor stall during this SMB-2 1800 psi closing stroke.
1.4
Choosing a Methodology
This section explains a change in recovery methodology that occurred during this research.
To date, gear rotation has been recovered using a demodulation-based
methodology [3].
Upon consideration of alternative approaches, a different methodology called harmonic
tracking was adopted.
After briefly introducing each approach, the reasons for choosing
harmonic tracking will be presented.
1.4.1
Introduction to Harmonic Tracking Methodology
The harmonic tracking methodology is based on the application of short-time spectral generation methods to gear meshing vibration analysis. Short-time spectral generation is a
time-frequency domain method which enables the user to analyze the frequency content
of slowly time-varying signals by sliding a short window over the longer casing vibration
signal, taking spectra at discrete steps along the way. Short-time spectral analysis thus
produces a series of spectra. Each spectrum represents the average frequency content over
the duration of the window; the spectrum's time-of-occurrence may not be precisely determined. We assign the time-of-occurrence to the center of the window when the spectrum
was taken. By analyzing the content of each spectrum, the angular speed and displacement
of each gear in the machine may be determined as a function of time.
1.4.2
Introduction to Demodulation-based Methodology
The demodulation-based
methodology does not assume that the signal is slowly time-
varying. This recovery methodology employs time-spectrum generation methods such as
16
the Wigner-Ville Distribution which produce spectra at well-definedinstants in time. The
analysis required to determine gear rotations from these "instantaneous" methods is well
documented [3], and differs considerably from the analysis required by harmonic tracking.
The important distinction between the two methodologies is that the demodulation-based
approach does not assume the signal is slowly time-varying while the harmonic tracking
approach does, and, in fact, takes advantage of its time-averaging characteristics.
1.4.3
Comparison Between Methodologies, Making a Choice
The ability of a method to recover the angular speed and displacement of each gear in a
machine is limited primarily by the effects of (1) gear meshing signal distortion due to the
transfer function, Hg(f),
and (2) gear meshing signal masking by other vibration (noise)
sources, ni(t). The methodology of choice is the one which is most robust to these distortion
and masking effects.
Robustness to Operating Noise
The harmonic tracking methodology is more robust to operating noise. Demodulation-based
methods are generally sensitive to the presence of extraneous noise, and require highly favorable signal-to-noise ratios (SNR's) in order to perform acceptably. As a countermeasure,
Chai developed an adaptive-center-frequency
bandpass filter to improve the SNR before
performing demodulation [3]. This tactic, while highly sophisticated, adds considerable
complexity to the method, and does not guarantee that the demodulation method will perform successfully. The harmonic tracking methodology, on the other hand, simply requires
a favorable enough SNR to detect and locate narrowband tones due to gear meshing in
the short-time spectra. Our experience (in the form of a simulation-based comparison)
has indicated that harmonic tracking has a higher tolerance for operating noise than the
demodulation-based method.
Robustness to Transfer Function Distortion
Over the last year-and-a-half, one of our greatest concerns has been whether the recovery
method would be robust to signal distortion resulting from vibration propagation through
the structure. Until very recently, it was expected that experimental measurement of the
structural transfer function and inverse filtering of operating data by the reciprocal of the
17
transfer function [3, 14] would be required to remove signal distortion. A key assumption
was that the transfer function measured on a single MOV would be used to inverse filter
operating data from any MOV of the same class. Unfortunately, past experience [6, 7] indicated that considerable transfer function variability should be expected between nominally
identical MOV's of the same class. It was therefore unclear whether the residual distor-
tion resulting from the mismatch between the inversefilter and the actual transfer function
would negatively impact our ability to recover valve travel.
The harmonic tracking methodology has alleviated these concerns.
While our simu-
lation has indicated that the demodulation-based method is sensitive to transfer function
distortion, failing to accurately recover gear rotations if inverse filtering is not performed,
experience to date indicates that harmonic tracking does not require inverse filtering at all.
This may be due to the fact that short-time spectral analysis averages out distortions due
to structural propagation, whereas the "instantaneous" spectral generation methods of the
demodulation-based method are highly dependent on exact phase information in order to
be accurate. Again, the robust choice is the harmonic tracking methodology.
1.4.4
Making a Choice Between the Two Methodologies
The harmonic tracking methodology is more robust both to distortion from the structural
transfer function and to the masking effects of the noise, and was, therefore, chosen for this
thesis.
18
Chapter 2
Source
Identification/Characterization
In this chapter, we will describe the characteristics of gear meshing vibrations which allow
for determination of the angular speeds of all of the gears in an MOV as functions of
time. The chapter falls logically into two halves. The first half is a basic introduction to
gear meshing vibration.
Some very general questions are asked and answered which set
the stage for the remainder of the chapter.
The second half is MOV-specific, discussing
the information expected to be contained in its gear meshing vibrations, defining where to
look for this information, and introducing many of the concepts that will be used in the
development of a recovery method. Because the chapter builds a logical argument which
depends on each preceding detail, a review of the major findings and a formal problem
statement conclude the chapter.
2.1
2.1.1
Basics of Gear Meshing
Why Do Gears Create Vibration?
Static Transmission Error
Perfectly involute gears with rigid teeth 1 transmit steady angular motion [11]. They do
not create vibration! However, real gears are not perfectly involute, do not have rigid teeth,
and do transmit an unsteady component of relative angular motion. This component is
1
For an introduction
to basic gear terminology and concepts, I recommend Shigley and Mischke [19].
19
known as static transmission error,
, and may be expressed as
09 = Q 9 t + Jo,
(2.1)
where 09 is the angular displacement of one of the gears, Qg is the steady component
of angular speed, and t is time. Because static transmission error modifies the angular
displacement of the gear, it is known as a displacement-type excitation [10].
Sources of Static Transmission Error
At this time, only the three most common sources of static transmission error will be
considered, leaving a fourth source, which is very important to this project, for a discussion
in its own right. The first is due to the finite stiffness of gear teeth. As a result of tooth
flexibility, the contact stiffness of meshing gears is a function of the angle of gear rotation,
since the number of teeth in contact and the stiffness of individual teeth change periodically.
A second source of static transmission error is average tooth-to-tooth variations from perfect
involute. These errors are typically purposefully designed deviations from involute. The
final source is random errors, which include machining errors and tooth damage such as
worn and broken teeth [11]. These three sources combine to create unsteady forces leading
to distinctive vibration characteristics that are easily identified as having been produced by
meshing gears. Such distinctive vibration characteristics are focus of the remainder of the
chapter.
2.1.2
Typical Gear Meshing Spectral Characteristics
As with most rotational sources of vibration, gear meshing is a periodic phenomenon which
displays narrowband frequency characteristics, making it well suited to frequency domain
analysis. The power spectral density (which will generally be referred to as a spectrum)
from the meshing of a generic set of gears (see Figure 2-1) consists of a set of narrowband
tones (see Figure 2-2) [5]. These tones are located at the angular speeds of the gears, fi
and f2, at the gear meshing frequency, fg12, and at sidebands around fg12. The magnitudes
of the tones correspond to the levels of the gear meshing forces. In this thesis, we are
concerned only with angular speeds and displacements, and therefore focus purely on the
location of the tones on the frequency axis.
20
I1
Pinion Gear
Angular speed of Pinion
ear
Gear meshin
N
Number of Teeth
Figure 2-1: A Generic Gear Set
21
fg 2-
2fl
1.
Frequency
Figure 2-2: Spectrum of Meshing Forces Produced by Generic Gear Set
Correspondence of Spectrum Impulses to Sources of Static Transmission Error
Having identified three of the major sources of gear meshing vibration and shown a typical
spectrum of the vibration, we ought to take a brief moment to explain how each source
contributes to the spectrum. Tooth deflection and average tooth-to-tooth variations from
perfect involute are periodic with period T, the amount of time required for a single tooth
to mesh. Thus the period is
1
1
= Nlf 1
-
N 2f 2
1
-
fgl2'
(2.2)
and these errors contribute to the narrowband tone at the gear meshing frequency, fg12.
Random tooth-to-tooth variations from perfect involute, on the other hand, are periodic at
integer multiples of each complete rotation of the gears. Therefore, there are narrowband
tones at
= fi
T1
22
(2.3)
with sidebands on fg12 spaced from fg12 by
n
-=
nf
(2.4)
1,
and
1
-=f2
T2
(2.5)
with sidebands on fg12 spaced from fg12 by
n
-=
nf2,
T2
(2.6)
where n is an integer. Note that any shaft imbalances will also contribute to the tones at
fi and f2.
Effect of Time-varying External Torques on Spectral Characteristics
Time-varying external torques are not typically included in model-based gear meshing analyses, primarily because of the mathematical complexities involved. They are, however, of
critical importance to this project, though they will only be described at an empirical level.
For the sake of clarity, we will initially limit ourselves to the case in which one of the shafts
is subject to an angular position-dependent torque, r(0), which is periodic with period equal
to TT (see Figure 2-3). This external torque amplitude and phase modulates gear meshing
frequency, fgl2, resulting in additional sideband structure around fg12 at integer multiples
of
1
- = fload
(2.7)
TT
(see Figure 24).2
In order to see why external loads are important
to this project, the
mechanical details of the MOV must first be described.
2
For many years, the dominant thought in the theoretical study of gear meshing was that the sidebands
around gear meshing frequency are purely due to amplitude and phase modulation effects [11, 12, 17].
Recently, because of such experimental observations as a decided lack of symmetry around the "carrier,"
researchers have been reconsidering whether the sidebands due to random errors are created by amplitude
and phase modulation [1]. Our position is that whether the sidebands due random errors are modulation
phenomena remains an open question. However, our evidence [3] suggests that the sidebands due to timedependent external loads are, in fact, modulation phenomena.
23
no-n.,
| s ....
D|..,
Angular speed of Pinion
3e.ar
Gear mesh
of TeINumber
Number of Te,
External load on shaft 2
Figure 2-3: Generic Gear Set with External Load on Shaft 2
24
4)
`C$
4.4
. -4
0
.4
1'
,0
_T
91
Ik
!
f2
fl/
I
w
-
fg12 3 fload fg12 2fioadf 12 fload fgl2
2
fgl2+fload fgl2+ fload fg12+3fload
Frequency
Figure 2-4: Spectrum of Gear Meshing Forces with Periodic External Load Applied
2.2
Gear Meshing Vibrations in an MOV
2.2.1 Introduction to the Mechanical Details of the MOV
The MOV consists of three major subsystems: the motor, the operator, and the valve (see
Figure 2-5). The motor drives the valve through an operator. The operator consists of two
gear reductions and a "nut-to-bolt" type conversionof rotation to translation. The first gear
reduction is a pair of helically-cut gears, known as the motor pinion gear and the worm shaft
gear. The second gear reduction is of the worm and worm gear type, converting rotation
about one axis to rotation about a perpendicular axis. The "nut-to-bolt" conversion of
rotation to translation is achievedby internally splining the worm gear to the outer diameter
of the "nut", appropriately known as the stem nut. The inner diameter of the stem nut
is threaded to the "bolt" or the valve stem. Because the stem nut is constrained not to
translate, the valve stem translates up or down depending on the direction of rotation of
the stem nut, allowing the valve to open and close. One additional detail has been left out
which plays a very important role in system dynamics. The worm is splined to the worm
shaft, and is therefore able to slide axially along the shaft, though in order to do so, it must
25
compress a spring pack consisting of a stack of belleville washers. This scope of this research
is limited to valve closing strokes, and for these strokes the worm's ability to slide axially
is intended to "soften the blow" for the motor when the valve impacts its seat. During a
closing stroke, the valve stem translates downward with a nearly constant speed until the
valve impacts its seat. At that point, the load on the motor increases abruptly, and it will
stall if it is not turned off. The back loading of the motor is made more gradual by the
worm sliding along its shaft when the load applied to the worm by the worm gear exceeds
the force required to compress the spring pack. Thus the motor can slow down and have
its back load increased more gradually, allowing it to be turned off at an appropriate time
before motor stall. An important note is that during spring pack compression, the angular
speed of the worm gear decreases more quickly than that of the worm. The two are no
longer constrained by the relation
fg = Nlfl = N2f2.
(2.8)
The importance of this fact will become clear as we continue.
2.2.2
Spectral Characteristics of Gear Meshing Vibrations in an MOV
As discussed in the previous section, the two gear pairs in an MOV are: the motor pinion/worm shaft gear and the worm/worm gear (see Figure 2-6). The meshing of both gear
pairs create the combined spectrum shown in Figure 2-7, in which there are narrowband
tones at the shaft rotation rates, f, f2, and f3, and at the gear meshing frequencies, fgl2
and
fg23.
An as yet unidentified sideband structure exists around fgl2. When the worm is
not translating axially, it engages one tooth of the worm gear during each complete rota-
tion, making fg23 equal to f2. When the worm is undergoing axial translation,
fg23 < f2,
because, as noted above, the angular speed of the worm gear slows more quickly during
spring pack compression.
2.2.3
Limitations Due to Broadband Noise Sources
Two additional broadband vibration sources have the potential to mask vibrations from gear
meshing: (1) environmental noise and (2) dynamic flow noise of the piped fluid past the
valve, both of which are broadband sources. The exact characteristics of dynamic flow noise
26
Operator
Stem Nut
lorm Gear
.
Stem
Worm Shaft Gear
Spring Pack
Valve
Talve Seat
Figure 2-5: Schematic of Major Components in MOV Geartrain
27
Number of Teeth
Gear
Gear
Shaft Spe
N1
Frequency
fgt2-
.
.
A.
As
Figure
2 .......ee12
Shaft Speed
Figure 2-6: The Two Gear Pairs in an MOV
28
Spring Pack
It
'4
l
.
f3 f2
I
fl
1
I
[
l
fg12
II (nominally)
fg23
Frequency
Figure 2-7: Generic Spectrum of Gear Meshing Forces in an MOV
spectra are not known, but, in practice, have not been found to mask the narrowband tones
due to gear meshing. Environmental noise is typically high in magnitude at low frequencies
and decays at high frequencies (see Figure 2-8), effectively masking the narrowband tones
at f 3 , f2, fg23, and fl, but leaving the tones at fg12 and its sidebands unmasked. Therefore,
fj and f2 cannot be directly measured, but may be calculated from
fl = f2/N1
(2.9)
f2 = fgl2/N2
(2.10)
and
In addition, when the worm is not translating axially, fg23 and f may be calculated from
fg23 = f2
(2.11)
and
f3 = f
23 /N3
29
(2.12)
Broadband Noise Source Levels
a)
.-
t
i
II (nominally)
fg23
Frequency
Figure 2-8: Spectrum of MOV Gear Meshing with Representative Noise Levels Added
However, when the worm is translating axially, no knowledge of f
or
f23
is possessed,
leaving unsatisfied our goal of determining the angular speeds of all gears. However, we still
have not considered the information contained in the sideband structure around f2.
2.2.4
Motor Pinion/Worm Shaft Gear Sideband Characteristics
The motor pinion/worm shaft gear pair is subject to angular-position dependent external
loading on both of its shafts (see Figure 2-9). The load on the input shaft, rl, is created by
the motor and is periodic with period
T = 1/f1.
(2.13)
More importantly, the load on the output shaft, r2 , is created by the meshing of the worm
and worm gears [3] and is periodic with period
T2 = 1/fg23.
30
(2.14)
Pinion Glear
Angular spe
ear
N
External load
Figure 2-9: External Loading on Pinion/Worm Shaft Gear Pair in an MOV
Recalling our earlier discussion of time-varying external loads, the expected sideband struc-
ture contains narrowband tones at integer multiples of fi and fg23 from fg12 (see Figure 210). However, the sideband structure also contains tones at integer multiples of f2 from
fg12 due to random errors. Because the sidebands at f2 and fg23 are coincident when the
worm is not translating and separate only when the worm is translating, the stronger of the
two will dominate. As long as the gears are undamaged, we expect the sidebands at fg23 to
dominate.
fg23
may, therefore, be calculated by subtracting the sideband(s) at fgl2 ± nfg23
from the tone at fg12, and f3 may be calculated by dividing fg23 by N 3 . The angular speed
of each gear in the MOV can thus be determined.
31
.U
'e~
~11
12-
fG
fg12- 2f2
^r
I
3f 2
fg 122f2
*4
II
-
\I fj
+
fg 12 +fl
fg 2 +f
4J
fg 12 3f2
fgl22fi
..
12 -
tgl2- 3fg23
2fl
f2
i
I
fgl2
-
2
g23
fg12- fg23
J
fgl2
-
fgl2+ fg23
>
-
-
+
fgl12+2fg23 fg 12 3 fg23
Frequency
Figure 2-10: Pinion/Worm Shaft Gear Meshing and Its Sideband Structure in an MOV
2.2.5
Time Dependence of MOV Gear Meshing Speeds
The gear meshing signal is not stationary. Our earlier discussion of the axial motion of the
worm stated that the motor speed and the rotational speed of each gear decreases when
the valve impacts its seat. Logically, then, the valve closing stroke may be broken into
two major portions: lightly loaded running and heavily loaded running (see Figure 2-11).
During lightly loaded running, all of the rotational speeds are varying slowly because valve
motion is opposed only by mechanical losses in the operator and by fairly steady dynamic
flow effects. During heavily loaded running, on the other hand, all of the rotation speeds
are decreasing rapidly because the valve is about to or has impacted its seat. A critical goal
of this project to track these changes in speed throughout the closing stroke.
Operating Condition Dependence
MOV's are designed to open and close under a variety of fluid flow conditions through the
attached piping. These flow conditions have a considerable effect on the angular speed of
each gear during heavily loaded running. In particular, as the change in pressure across the
32
Time Dependenceof Pinion/WormShaft Gear Meshingin an MOV
I
I
I
I
I
I
I
I
I
I
I
I
I
0
(3
Cr
t-
O'
eL
-
- -
LightlyLoaded Running
HeavilyLoaded Running
I
Time
Figure 2-11: Time Dependence of Pinion/Worm Shaft Gear Meshing in an MOV-Typical
of All MOV Harmonics
valve face increases, the rate of angular deceleration decreases, resulting in a more gradual
decline in the angular speed of each gear (see Figure 2-12).
2.3 Chapter Summary
This chapter has presented a fairly extensive body of information. The following is a brief
summary of the major findings:
* The vibrations produced by gear meshing are narrowband phenomena.
These nar-
rowband tones are located at frequencies corresponding to the angular speeds and
meshing rates of the gears. Gear meshing forces are most insightfully analyzed in the
frequency domain.
* Due to low frequency masking from environmental noise, all gear meshing information
in an MOV must be contained in the first motor pinion/worm shaft gear harmonic,
fgl2, and its sidebands.
* Fortunately, fg12 and its sidebands contain information from which the angular speed
of each gear in an MOV may be determined.
33
OperatingConditionDependenceof Pinion/WormShaftGear Meshing
II
C,
C
a)
D_
E?
U-
I I
I
I
I
-
II
-
I~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Time
Figure 2-12: Operating Condition Dependence of Pinion/Worm
Function of Time-Typical of All MOV Harmonics
Shaft Gear Meshing as a
* These angular speeds are not constant, but may change throughout the valve closing
stroke (on a "long-time" basis-these are global changes not due to phase modulation).
To conclude this chapter, we'll present a formal statement of the problem, and briefly
introduce some of the critical issues we must address in arriving at a solution.
The data we can expect to be given includes:
* Physical parameters of the MOV being analyzed. Examples include: number of teeth
on particular gears and the pitch of the stem.
* The casing vibration signal taken during a closing stroke of the valve.
The information to be recovered is the angular speed of each gear in the MOV as a function
of time. Of primary importance is our ability to develop a method which produces spectral
information as a function of time. Once this has been accomplished and the speed information recovered, gear rotation may be calculated by integrating gear speeds with respect
to time.
34
Chapter 3
Development of a Harmonic
Tracking Signature Recovery
Technique
The bases of the harmonic tracking method are short-time spectral generation and analysis,
which allow for the recovery of spectral information, in the form of the angular speed of
each gear, as a function of time from the casing vibration signal. Two supporting steps are
also performed, one preceding short-time spectral generation and one immediately following
short-time spectral analysis. The first supporting step is to gather a priori information by
defining all of the constants in the problem statement discussed at the end of Chapter 2.
The other supporting step is to perform a simple set of calculations to determine rotations
from the recovered angular speeds. As a black box, the harmonic tracking method requires
a casing vibration signal and a set of a priori information as inputs, and produces outputs
including the angular speeds and displacements of each gear in the machine.
3.1 A Priori Information Required
Two major sets of information are required before proceeding with the recovery. The first
is the following MOV structural constants:
* Number of teeth on the motor pinion gear
* Number of teeth on the worm shaft gear
35
* Number of teeth on the worm gear
* Pitch of the worm
* Pitch of the stem
These constants are used at various points during the method to relate meshing frequencies
to gear angular speeds.
The second required piece of information is motor speed during light loading, which enables
us to locate the narrowband tone in the spectrum at motor pinion/worm shaft gear meshing
frequency. To date, we have not determined how this information will be acquired for typical
applications of the harmonic tracking method.
A likely possibility is that motor current
and voltage data, which is already used to estimate motor torque in another portion of the
MOV diagnostic system, will be used to estimate motor slip, and, therefore, motor speed
during lightly loaded running.
3.2 Short-Time Spectral Generation
Short-time spectral generation is generally employed to perform spectral analysis of nonstationary signals. To generate short-time spectra, the long casing vibration signal is multiplied by a much shorter window (eg, a hanning window), resulting in a signal which is
non-zero only over the length of the window. By sliding this window through time along the
casing vibration signal, a series of small segments of the signal is produced. The spectrum
of each segment may then be estimated and assigned to a time roughly corresponding to
the position of the center of the window at that segment. The result is a "waterfall" of
spectra (see Figure 3-1), displaying the spectral content of the signal as a function of time.
However, there are issues in the frequency and time domains which must be considered
in order to effectively implement a short-time spectral generation method. We begin by
considering an issue of practical importance: what spectral estimation method to use. This
discussion will lead to an issue of more fundamental importance in the use of short-time
spectral generation methods: time resolution versus spectral resolution. Both issues lead
to tradeoffs which must be balanced in practical implementation.
36
250
200
"' 150
: 100
50
0
600
...
a~
~Frequency
,.,.v~~
(Hz)
Figure 3-1: Waterfall of Spectra
3.2.1
Spectral (Frequency) Considerations
Two spectral estimation methods have been used in this research:
the discrete fourier
transform (DFT) and autoregressive (AR) models. In comparing the two methods relative
to ease of implementation and spectral resolution, some familiarity with the DFT is assumed
[15],while an introduction to AR modeling is presented in the Appendix and in more detail
in the literature
[13, 16, 2].
Ease of Implementation
Ease of implementation is important both to expediently generate "one-off" laboratory results and to develop an automated system. The DFT is relatively straightforward to implement. The potential pitfalls of using the DFT (especially aliasing) are common knowledge,
and are fairly easy to avoid. In addition, a highly computationally efficient algorithm, the
fast fourier transform' (fft) exists to decrease computational
time. Implementation of AR
modeling is more complex. In particular, the performance of the algorithm (both in terms
of accuracy and computational
time) is governed by the choice of model order. Adjusting
this parameter, especially in a fully automated system, requires fairly novel, sophisticated
methods which make implementation formidable. There are clearly costs incurred by using
37
AR modeling. However, these costs are balanced by advantages, one of which is spectral
resolution.
Spectral Resolution
Spectral resolution may be loosely defined as the amount of frequency information contained
in the spectrum of a sequence. The amount of frequency information contained depends on
the length of the sequence. As the length increases, the frequency resolution also increases.
For example, spectral resolution for the DFT is related to the length of the sequence (N, a
measure of time) and sampling rate ()
in the following manner:
i\
A~~f
~(3.1)
f~s,~
where /Af is the spacing between frequency samples. The smaller Af is, the more highly
resolved the spectrum is. Theoretically, AR models offer superior spectral resolution [13]. In
practice, we've found AR models to offer measurably better spectral resolution.1 In addition
to this practical comparison between spectral generation methods, spectral resolution also
has an important impact on the general implementation and of any short-time spectral
generation method, as will now be discussed.
3.2.2
Time Consideration: Time Resolution
Time resolution for short-time spectral generation methods is dependent on the following
concept: short-time stationarity. Strictly speaking, both the DFT and AR modeling assume
that the sequenceto be transformed is stationary. Our assumption is that the windowlength
may be decreased until the windowed signal is essentially stationary. If this assumption is
valid, the signal is said to be slowly time-varying, or short-time stationary. Time resolution
may be defined as the amount of time information contained in the waterfall of spectra.
This amount of information is inversely related to the length of the time sequence to be
transformed. Thus, by decreasing the windowlength to satisfy the requirement of short-time
stationarity, time resolution is increased. However, this requirement is in direct opposition
'It should also be noted that DFT's are all-zero models, while AR models axe all-pole models. Recall that
gear meshing spectra contain tones which are appropriately modeled as poles. AR models should therefore
be more appropriate in estimating gear meshing spectra. In practice, however, the DFT may prove to be
the better choice, as will be discussed in Chapter 5.
38
to increasing the window length to obtain better spectral resolution. Therefore, a conflict
is faced between time resolution and spectral resolution.
3.2.3
Making the Tradeoffs
Spectral Resolution Versus Time Resolution
In applying short-time spectral generation, a balance must be reached between time resolution and spectral resolution. The window must be long enough to capture the required
frequency information, but short enough to satisfy short-time stationarity. In practice, we
relax the requirement of short-time stationarity. Recall that gear-meshing spectra consist of
tones. If the frequency of these tones changes too rapidly over the length of a single window,
the tones will smear in frequency, eventually becoming indistinguishable from background
noise. Therefore, a balance is maintained between keeping the window length short enough
to make the tones detectable and long enough not to miss the tones between frequency
samples.
AR Versus DFT
Which is more important, ease of implementation or spectral resolution? Practically speak-
ing, they are both important. In this research, the DFT and AR modeling are used interchangeably for spectral estimation. AR modeling is always the first choice for implementation because of its spectral resolution advantage. However, if a proper balance is not easily
found between the AR model order and the window length, the DFT may be substituted
without jeopardizing the recovery.
3.3
Short-time Spectral Analysis
Having gathered a priori information and generated short-time spectra, the next step is to
perform short-time spectral analysis. Three analysis steps are required. First, narrowband
tones are located within each short-time spectrum. Then, using lightly loaded motor speed
and the structural constants, the located tones corresponding to gear meshing and each of
its sidebands are tracked as they evolve through successivespectra (and time) to construct
harmonics as functions of time. Finally, known sideband relations may be used on these
harmonics to calculate gear meshing frequencies as functions of time.
39
0)
510
515
520
525
530
Frequency(Hz)
535
540
545
Figure 3-2: Tone Location for a Single Spectrum
3.3.1
Tone Location
Tone location is performed by an algorithm which finds the maxima of a function. Our
algorithm simply looks for points which satisfy the requirement that the previous and next
points have lower values:
Si > Si+l AND Si > Si- 1 .
(3.2)
A single spectrum and its located maxima are shown in Figure 3-2. The most important
assumption made by this approach is that the generated spectra are smooth enough to
capture only the tones due to gear meshing. Otherwise, the tone locator may find extraneous
maxima that corrupt harmonic tracking. This assumption is easily satisfied when using AR
models, which generally produce smooth spectra for which the model order may be tuned to
capture only tones due to gear meshing. The smoothness of the DFT spectra are influenced
by spectral resolution. In practice, we've found that the required spectral resolution is low
enough that only the dominant tones due to gear meshing are captured.
40
3.3.2
Harmonic Tracking
The most challenging stage in short-time spectral analysis is to track the located tones
through time. A fairly simple, automatic routine for tone tracking was developed by Chai
for his demodulation-based method [3]. The inputs to this routine are an estimation of
the lightly loaded frequency of a particular harmonic and a set of tones from all of the
spectra in the waterfall. The routine then employs a set of user-defined constraints such as
the maximum frequency change between successivespectra to produce a tracked harmonic
(see Figures 3-3 and 3-4).2 Unfortunately, this simple, automatic tone tracking routine is
not highly reliable; the routine is easily "tricked" into tracking the wrong harmonic (see
Figure 3-5). An alternative approach is to track the harmonics manually, which requires
the user to choose each tone by hand, tracking the harmonic time step by time step. As
may be imagined, this technique is tedious, time-consuming, and unacceptable for inclusion
in a diagnostic system. All of the results presented in this thesis are a mix of the simple
automatic tracking routine and the manual approach. Two alternative approaches are under
currently under consideration. One is to develop a more sophisticated automatic tracking
routine, perhaps one which tracks only the tones which fall within a window around the
expected harmonic. The other replaces or combines short-time spectral generation and
analysis with short-time cepstral generation and analysis [15, 10, 17]. Because the sidebands
to be tracked are periodic in frequency, the power cepstrum may include impulses at gear
meshing periods which are less sensitive to operating noise and, therefore, easier to track.
Both alternatives will be considered in future work.
3.3.3
Sideband Relations
The following two situations were encountered when determining the gear meshing fre-
quencies of an MOV: pinion/worm shaft gear meshing present and pinion/worm shaft gear
meshing absent.
2
Figure 3-3 displays a tracked harmonic in waterfall format. A simpler scheme which allows the user to
more clearly see transients is to remove the magnitude axis, resulting in the two-dimensional time-frequency
plot format of Figure 3-4. This format is used for the remainder of the thesis.
41
250
200
I
150
0)
CE100
50
0
600
0
Time (s)
480
Frequency(Hz)
Figure 3-3: Tracking a Single Harmonic, Waterfall Visualization
0
0.2
0.4
0.6
Time (sec)
0.8
1
Figure 3-4: Tracking a Single Harmonic, 2D Visu,alization
42
1.2
, ,~o 'o
54v
o
o
o
0
5440
o
o
35
o ov 0
o
o
o
0
c~~~0
UIPC
0
.,U
,~~~~X
a)5,30
C
LL
5:,25
~°:~=~c ~
~=
m
0
o
cdo
·'
~~~~0
o
%0 0.O
o
5;20
0
0.2
'
0.4,
0.6
- -,
0o
,,i0we
I
0.8
Time (sec)
o
I,
1I
0
1.2
Figure 3-5: Tricking the Simple, Automatic Harmonic Tracker
Pinion/Worm Shaft Gear Meshing Present
When pinion/worm shaft gear meshing, f1l2, is one of the tracked harmonics, a minimum
of one additional harmonic is required to determine worm/worm gear meshing frequency,
fg23
(see Figure 3-6). If the additional harmonic is an adjacent sideband,
fand(t),
due to
modulation by worm/worm gear meshing, worm/worm gear meshing, fg23(t), is simply
fg 2 3(t) = IfbLand(t)- fg12 (t)I-
(3.3)
For the Ith sideband, fband(t), due to modulation by worm/worm gear meshing, fg 23 (t)
may be calculated from
fg23 (t)= fbd(t) I-fg2(t)
(3.4)
If more than one sideband due to modulation from worm/worm gear meshing has been
tracked, fg23(t) is taken to be the average of the fg23(t)'s calculated.
43
SidebandRelations: Carrier Present
565
560
555
550
N
-r-
' 545
o 540
V)
LI
535
530
Pinion/Worm Gear Meshing
- - First Upper Sideband
525
rgon
-
0
.
.
0.2
i
i
0.4
0.6
.
I
0.8
I
1
.
I
.
1.2
1.4
Time (s)
Figure 3-6: Sideband Relations: Carrier Present
Pinion/Worm Shaft Gear Meshing Absent
When pinion/worm shaft gear meshing is not one of the tracked harmonics, a minimum
of two harmonics is required to determine both worm/worm gear meshing, fg23(t),
and
(see Figure 3-7). If the harmonics recovered are
pinion/worm shaft gear meshing, fgl2(t)
the first and second upper sidebands, ful(t) and f2(t), due to worm/worm gear meshing,
fg23(t), may be calculated as
fg23(t) = f2 (t) - f (t).
(3.5)
For the Ith and Kth upper sidebands, worm/worm gear meshing is
fg(t)
-
IfS(t) - f (t)
~g23( I-I-KI
(3.6)
Pinion/worm gear meshing frequency may then also be calculated as
fg 12 (t) = fI(t)-
44
Ifg9 23 (t) or
(3.7)
SidebandRelations: CarrierAbsent
590
580
_
570
I_
.560
I
N
r 550
0~
{3'
E
U---=I.,
530 K
- - First UpperSideband
- - SecondUpper Sideband
520
6,
Carrier(for Reference)
all-
0
i
i
i
0.2
0.4
0.6
i
0.8
Time (s)
i
1
i
1.2
1.4
Figure 3-7: Sideband Relations: Carrier Absent
fg12(t) = fI(t) - Kfg23(t).
(3.8)
If one upper and one lower sideband have been tracked, these formulas may be easily
modified to calculate both fgl2(t)
and fg23(t).
Also, if more than two sidebands have been
recovered, averaging may be employed to calculate fg 23 (t).
Calculating the Angular Speed of Each Gear
After recovering fgl2(t)
and
fg23(t),
the speeds of all three gears may be calculated from
f 1 (t) = f 9 12 (t)/N 1, (Motor Pinion Gear Angular Speed [Hz]),
(3.9)
f 2 (t) = fg12 (t)/N 2, (Worm Shaft Gear Angular Speed [Hz]), and
(3.10)
f3 (t) = fg23 (t)/N3 , (Worm Gear Angular Speed [Hz]),
(3.11)
where N 1 , N 2 , and N 3 are the numbers of teeth on the motor pinion gear, worm shaft gear,
45
and worm gear, respectively.
3.4
Integrating for Gear Rotation, Valve Travel, and Spring
Pack Displacement
The final step in the harmonic tracking method is to calculate the angular displacements
of the pinion gear, 0 1 (t), worm shaft gear,
2 (t),
and worm gear, 03 (t), by integrating the
angular speeds with respect to time as follows:
l(t) = j
02(t) =
//
t
fl(T)d,
(3.12)
f 2 (T)d, and
(3.13)
f 3 (T')dr.
(3.14)
03 (t) =
For MOV diagnostics, an additional step is taken to calculate valve travel and spring
pack displacement. Valve travel, z(t), is calculated by the following scaling operation:
z(t) = PsA3 (t),
where Ps is the pitch of the stem in units of length/thread.
(3.15)
Spring pack displacement, y(t),
is calculated by scaling and subtracting in this way:
y(t) = (02- 03N 3 )P,
where Pw, is the pitch of the worm in units of length/thread.
(3.16)
y(t) corresponds to the total
number of times the worm rotates without engaging worm gear teeth (the difference between
the number of rotations of the worm shaft (02) and the number of tooth engagements of
the worm gear (03 N 3 )) multiplied by the pitch of the worm. The diagnostic significance of
these quantities will become evident as we continue.
46
3.5
Summary and a Look Ahead
The details of the harmonic tracking method may appear formidable, but the fundamental
logic and steps required to implement it are quite simple. The followingfour steps must be
performed:
* A Priori Information Gathering
- Structural constants
- Lightly loaded motor speed
* Short-Time Spectral Generation
* Short-Time Spectral Analysis
- Tone location
- Harmonic tracking through time to recover gear meshing harmonics
- Calculation of pinion/worm shaft gear meshing and worm/worm gear meshing
using known sideband relations
* Simple Calculations for Gear Rotations, Valve Travel, and Spring Pack Displacement
- Gear rotations: Scale by the number of gear teeth and integrate with respect to
time.
- Valve travel: Scale worm gear rotations by stem pitch.
- Spring pack displacement: Subtract the number of worm gear tooth engagements
from the number of worm rotations and scale by the pitch of the worm.
Therefore, by consulting the treasure map from Chapter 2, it was found that the casing
vibration signal contains the angular speed of each shaft. By implementing short-time
spectral generation and analysis methods in the time-frequency domain, we have gathered
the appropriate tools and developed a method to recoverthe angular speed and displacement
of each gear as a function of time. The remainder of the thesis applies this method to the
MOV: first by performance prediction using simulated casing vibrations (Chapter 4), then
by application to experimentally obtained casing vibrations (Chapter 5), and finally by
application to an MOV diagnostics problem (Chapter 6).
47
Chapter 4
Predicting Harmonic Tracking
Performance Via Simulation
In this chapter, performance of the harmonic tracking method is predicted using simulated
casing vibrations. The two important performance measures introduced in Chapter
are:
(1) the robustness of the method to masking by measurement noise and (2) the robustness
to signal distortion from the structural transfer function (TF) between the gears and the
casing. This chapter builds on the qualitative descriptions of Chapter 1 by quantitatively
documentingthe performance of the harmonic tracking method. The first part of the chapter
describes the production of a general simulated casing vibration signal, which consists of
gear mesh induced vibrations, measurement noise, and a TF. The remainder of the chapter
presents and assesses the sensitivity of the harmonic tracking method to distortion by the
structural TF and to masking by measurement noise, and predicts technique performance
when subjected to a realistic TF and level of measurement noise.
4.1
Generating Simulated Casing Vibrations
The simulated casing signal consists of three components: gear mesh induced vibrations,
xg(t), measurement noise, n(t), and signal distortion due to the TF, Hgc(f), between the
gears and the casing. The casing vibration, a(t), is constructed from these components the
following way (see Figure 4-1):
a(t) = xg(t) * h9gc(t)+ n(t),
48
(4.1)
fg 2 (t)
f 23(t)
G
_-.-4"L=
Figure 4-1: Pseudo Block Diagram of Simulation
where * means the convolution of xg(t) with the impulse response, hgc(t), between the gears
and the accelerometer location on the casing.1
4.1.1
Modelling Gear Meshing as Phase Modulation
Real gear meshing forces are phase and amplitude modulated, resulting in sideband structure around pinion/worm shaft gear meshing, fgl2.
important to capture the tones at
fg12
For simulation purposes, it is only
and its sidebands, a task which is accomplished by
modeling the signal as pure phase modulation of the form
xg(t) = J(eqO(t)), where
q(t) = foj 27rfgl2(t)dt + 31sin
[ft 27rfg2 3 (t)dt
(4.2)
+
2
sin
[fo2rfl(t)dt],
(4.3)
where fg23 (t) is worm/worm gear meshing, f (t) is motor speed, and 31and 32 are modulation indices [17, 18].
1hgc(t)
is the inverse fourier transform of the transfer function, Hgc(f).
49
4.1.2 Modelling Transfer Functions as Rational System Functions
The structural TF distorts the gear meshing signal through the physical processes of dispersion and reverberation [10]. TF's of continuous structures are typically analyzed as rational
system functions possessing the poles and zeros of their lumped parameter equivalents. For
simulation, this convention is followed by modeling the Z-transform of hgc(t) as a rational
function of the form
(1- Rzejwzlz-l)(1 - Rz 2 eIwz2z-1)...(1
Hgc(Z) = (1 - RpleJwPlz-l)(1
- Rp 2 eJW2Z-l)...(1
-
Rzmewzmzl)
- Rpnejwpnz - 1 )
(4.4)
where the Wpk and wzi are the discrete frequencies of the poles and zeros, and the Rpk and
Rzi represent a measure of the damping of the poles and zeros.2
4.1.3 Modelling Measurement Noise as White Noise
There are two primary sources of broadband noise in an MOV: environmental (plant) noise
and noise from fluid flow past the valve. Theses sources are modeled as a white noise source
using a random number generator.
4.2
acking on Simulated Casing
Performing Harmonic
Vibrations
Four simulation cases were performed for each of which a unique casing vibration signal
was produced and harmonic tracking recovery was performed. The first simulation case is
a baseline test in which no TF distortion is performed and no measurement noise is added.
The second and third simulation cases separately test for distortion and masking effects.
For the second simulation case, a realistic structural TF is used, but no noise is added.
For the third simulation case, no TF distortion is applied, but the noise level is increased
until the harmonics may no longer be tracked. The final simulation case is designed for
performance prediction.
The realistic TF and a realistic level of measurement noise are
applied to create a simulated casing vibration signal which matches an experimental signal
2In the continuous-time domain, represented by the Laplace transform, the damping coefficient, , ranges
from 0 to 1, with zero representing the undamped case in which the pole or zero lies on the frequency axis.
However, in the discrete time domain, represented by the Z-transform, damping, R, ranges from 1 to 0, with
1 representing the undamped case in which the pole or zero lies on the unit circle.
50
as closely as possible.
4.2.1
Specifying the Four Simulated Casing Vibration Signals
The simulated casing vibrations are modeled after experimental data obtained from a Limitorque SMB-000 MOV closing stroke performed under static (no) flow conditions [3]. Pa-
rameters required to specify the form of the three components of the casing vibration signal
(gear meshing forces, structural TF, and measurement noise) were adjusted to provide a
reasonable match between the simulated and experimental vibration signals.
Gear Meshing Forces (All Simulation Cases)
The same gear meshing force was used for all simulation cases. The first step in creating
the gear meshing force was to specify the a priori information required for any application
of harmonic tracking. The specification was completed by modeling the time dependence
and modulation indices of the SMB-000 experimental data.
Standard A Priori Information
The MOV structural constants for this SMB-000 are:
* Number of teeth on the motor pinion gear, N1 = 18
* Number of teeth on the worm shaft gear, N 2 = 27
* Number of teeth on the worm gear, N 3 = 50
* Pitch of worm, P, = .18 in/thread
* Pitch of stem, PS = .167 in/thread.
It was found that, for this particular operator, lightly loaded motor speed, fi, is the rated
operating speed, 1800 rpm or 30 Hz. From the a priori information, lightly loaded pin-
ion/worm shaft gear meshing frequency, f2,
and lightly worm/worm gear meshing, f23
were calculated from
f1 2 = f lN 1= 540 Hz and
f3 2
fgl2
N2
f923
51
-
20
Hz(4.6)
2 Hz.
20
(4.5)
~~~~~~(4.6)
Generated Pinion/WormShaft Gear MeshingFrequency(Hz)
I
I
I
=An
I
i0'530
C
a)
L 520
I
1 n
0
0.2
I
4U.0
0.4
I
0.6
0.8
Time (s)
1
1.2
GeneratedWorm/WormGear MeshingFrequency(Hz)
I
I
I
I
1.4
1.6
I
20
N
v
819.5
a_
(1)
19
c
(D
iL
18.5
I
1,9
0
0.2
0.4
0.6
0.8
Time (s)
1
1.2
1.4
1.6
Figure 4-2: Simulated Gear Meshing Frequencies
Completing Specification of Gear Meshing Force To completethe specification of
the gear meshing force, the gear meshing frequencies, fg 12 (t) and fg23(t), motor speed,
fl(t), and the modulation indices, 31 and /32 were determined. f12(t) and fg23(t) were
generated by making linear fits to the meshing frequencies recovered using a demodulationbased method [3] (see Figure 4-2). Because no additional information would be gained by
modeling the entire lightly loaded portion of the signal, a truncated lightly loaded portion
and the complete heavily loaded portion of the experimental signal were modeled.
For
convenience,the first data point of the truncated lightly loaded portion is assigned to time
t
=
0. Motor speed, fi (t), is determined by the relation
fl (t) = fg12 (t)/N1.
(4.7)
The modulation indices, l31and /32, were set so that the relative magnitudes of the gear
meshing tones in the simulated and experimental signals matched as closely as possible (see
Figure 4-3).
52
30
S
230-
0,
cm
10
Spectrum of Lightly Loaded, Experimental Casing Acceleration
l
l
l
l
l
Za
co
5
0
350
400
450
500
550
Frequency (Hz)
600
650
700
Spectrum of Lightly Loaded, Simulated Gear Force with Noise Added
)O
Frequency(Hz)
Figure 4-3: Comparison of Simulated and Experimental Spectra
Structural Transfer Function
Two different structural TF's were used in the simulation. The first was used in simulation
cases 1 and 3, for which no TF distortion was required. To perform no distortion, the TF
may be set to any convenient constant, and we chose
HND=
1(4.8)
The second TF was used in simulation cases 2 and 4, for which a realistic level of TF
distortion was required. Because the narrowband tones produced by the gear meshing force
are located within a limited band of frequencies, only the poles and zeros of the structural
TF that are within or near this band of frequencies will distort the signal. Based on a
statistical energy analysis (SEA) calculation of the modal density of the Limitorque SMB000, it was found that a maximum of one pole and one zero of the structural TF are likely to
be located in the gear meshing frequency band. Therefore, the general TF may be simplified
to the form
HReal(z) = (1 - RzeWzz
gC
9
(1
-
53
RpeJwpz-l)
1).
(4.9)
Magnitudeof TransferFunction
m
a
Frequency(Hz)
Phase of TransferFunction
0)
CU
-a
CU
460
480
500
520
Frequency(Hz)
540
560
580
Figure 4-4: Simulated Transfer Function
The frequency of the pole and the zero, wp and wz, were arbitrarily assigned to values
corresponding to continuous-time frequencies of 510 Hz and 530 Hz, respectively, falling
within the gear meshing frequency band of 480 to 600 Hz. The measures of damping, Rz
and Rp, were set by mimicking an experimentally measured transfer function which is not
shown in the figure [3] (see Figure 4-4).
Measurement Noise Levels
Three different noise levels were used in the simulation. The first was used in simulation
cases 1 and 2, for which no measurement noise was required. To perform no measurement
noise masking, the noise was set to zero (nNM(t) = 0). The second noise level was used in
simulation case 3, and was determined by increasing the noise level until harmonic tracking
could no longer be performed. The third different noise level was used in simulation case
4, for which a realistic noise level was required. To simulate a realistic level of operating
noise, the signal-to-noise ratio (SNR) of the experimental signal was estimated.
From the
spectrum of the lightly loaded, experimental casing vibration signal (see Figure 4-3), the
broadband noise floor was found to be approximately 20 dB below the magnitude of the
largest narrowband tone, resulting in an SNR of 20 dB. The simulated noise level was then
54
Simulation
Case
Transfer
Function
Measurement
Noise
1
0
1
2
Realistic
3
4
0
1
Realistic
Maximum
Realistic
Table 4.1: Comparison of Four Simulation Cases
set to match a 20 dB SNR (again see Figure 4-3).
Summary of Simulation Cases
To briefly review, the simulation cases are: (1) the baseline case, in which neither TF
distortion nor measurement noise are included, (2) the case in which a realistic TF is
used and the measurement noise is set to zero, (3) the case in which no TF distortion is
performed and measurement noise is increased until harmonic tracking may no longer be
performed, and (4) the realistic case in which a representative TF and representative level
of measurement noise are included. The simulation cases are summarized in Table 4.1.
4.2.2
Simulation Results
Short-time Spectral Generation, Tone Location, and Harmonic Tracking
For all four simulation cases, short-time spectral generation was performed using the DFT. 3
For simulation case 3, after an appropriate window length was chosen for short-time spectral
generation, measurement noise was increased to an SNR of 12 dB, at which point harmonic
tracking could not be performed (see Figure 4-6). For the other three simulation cases,
performance of the tone location and harmonic tracking stages of spectral analysis (using
the automatic algorithms described in Chapter 3) produced recoveries of pinion/worm shaft
gear meshing,
fg23
fg12,
and four sidebands due to modulation by worm/worm gear meshing,
(see Figures 4-5 and 4-7).
3
At the time the simulation was performed, AR modeling was not considered because we possessed
insufficient confidence in our ability to reliably use it.
55
SpectralGeneration,Tone Location,and HarmonicTracking
580
560
540
I
N
rC.
0*
a) 520
.-a)
LL
500
fgl 2
480
.Generated
Case 1 (Baseline)
-
460
II I I II
-
Case 2 (TF Distortion)
_
0
0.2
0.4
0.8
0.6
1
1.2
Time (s)
Figure 4-5: Tracked Harmonics for Simulation Case 1 (Baseline) and Simulation Case 2
(Realistic TF Distortion)
SpectralGenerationand Tone Location
Time (s)
Figure 4-6: Short-time Spectral Generation and Tone Location for Simulation Case 3 (Minimum Required SNR)
56
Spectral Generation,Tone Location,and HarmonicTracking
I
I
I
I
0.8
1
- I
580
560
540
I
C520
IJ.
500
480
Generatedfg12
Case 4 (RealisticTF and Noise)
-
460
:
0
0.2
0.4
0.6
1.2
Time (s)
Figure 4-7: Tracked Harmonics for Simulation Case 4 (Realistic TF and Noise)
Short-time Spectral Analysis: Sideband Relations
For simulation cases 1 (Baseline), 2 (TF Distortion), and 4 (Realistic Case), fgl2 and fg23
were determined using the carrier present, multiple sideband relations discussed in Chapter
3 (see Figures 4-8, 4-9, 4-10, and 4-11). For simulation cases 1 and 2, f12 and fg23 are
nearly identical, indicating that the harmonic tracking method is insensitive to distortion
from the realistic TF. For simulation case 4, only the first upper and two lower sidebands
were used to calculate fg23. fgl2 and fg23 are not as accurately recovered for case 4 as they
were for cases 1 and 2.
Calculating Valve Travel and Spring Pack Displacement
For MOV diagnostics, gear rotations are not of direct importance, though a measure of the
accuracy of gear rotation recovery is obtained by calculating valve travel, z(t), from the
multiplication of the rotation of the worm gear,
3, by stem pitch, P (see Figures 4-12
and 4-13). For simulation cases 1 (Baseline) and 2 (TF distortion), the generated and
recovered valve travels are nearly indistinguishable.
For simulation case 4 (Realistic Case),
the curves are barely distinguishable, and values of generated and recovered valve travel
at the end of the recovery are .082663 in and .082354 in, respectively. The relative error
57
545
_~
~~_~~~~~~~~~
~1I
~~~- -
545
_
I'540
-53
0c
a)
-y 535
O3
._
0)
-s
a)
530
(0 525
-.
I
Generated
- - Recovered--No Distortion
en
E
E 520
.
O
.o
( 515
AwnJ
ho}
~,,
III
0
Distortion
- Recovered--TF
I
I
I
I
0.2
0.4
0.6
0.8
1
1.2
1.4
Time (s)
Figure 4-8: Pinion/Worm Shaft Gear Meshing for the Simulation Cases 1 (Baseline) and 2
(TF Distortion)
I
a)
I'0l-
c-)
0)
SC
o3
(3
CD
LL
e
03
a
O
Ei
0i
C!,
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Time (s)
Figure 4-9: Worm/Worm Gear Meshing for Simulation Cases 1 (Baseline) and 2 (TF Dis-
tortion)
58
-1-
I
C
cia)
Q>
LL
c-
0
C'
0
._o
co
Q_
4
Time (s)
Figure 4-10: Pinion/Worm
0
0.2
Shaft Gear Meshing for Simulation Case 4 (Realistic Case)
0.4
0.6
0.8
1
1.2
1.4
Time (s)
Figure 4-11: Worm/Worm Gear Meshing for Simulation Case 4 (Realistic Case)
59
0.1
0.09
0.08
0.07
=0.06
a)
' 0.05
ci)
> 0.04
0.03
- - Recovered--No Distortion
- - Recovered--TFDistortion
0.01
01, ,
0
Generated
-
0.02
0.2
0.4
I
I
0.6
0.8
I,
1
1.2
1.4
Time (s)
Figure 4-12: Valve Travel for the Simulation Cases 1 (Baseline) and 2 (TF Distortion)
between generated and recovered values is .10168%.
Spring Pack Displacement
Spring pack displacement, y(t), is an acid-test of the harmonic tracking method because
it is the difference between two quantities, worm shaft rotation and the number of tooth
engagements of the worm gear, which are nearly identical, and thus amplifies recovery
errors. This quantity is of peripheral importance to both MOV diagnostics and the harmonic
tracking method, and is included here primarily because it is one of the quantities which was
measured on the SMB-000 experiments, and it was central to previous work performed by
Chai [3]. Spring pack displacement was calculated for simulation cases 1 (Baseline), 2 (TF
Distortion), and 4 (Realistic Case) (see Figures 4-14 and 4-15). Comparing simulation cases
1 and 2, it is apparent that the recovered spring pack displacements are not as identical
as expected.
In fact, a single data point in the recovered worm/worm gear meshings is
responsible for this difference (see Figure 4-9, paying particular attention to two data points
which occur at approximately .9 seconds, and which are separated by about 1.3 Hz). On
average, the spring pack displacement recovery for simulation case 4 is reasonable, but not
highly accurate. If the recovery of spring pack displacement is so sensitive to single data
60
c
IF>E
4
Time (s)
Figure 4-13: Valve Travel for Simulation Case 4 (Realistic Case)
points, is there any way to improve the recovery by smoothing the data? One way is to
perform least squares polynomial fits to the recovered sidebands. By doing so, the effect of
single data points is minimized, and the spring pack displacement recovery is smoother, and,
hopefully, more accurate. This approach will be tried in the next chapter during analysis
of experimental data from an SMB-000 static (no) flow closing stroke.
4.2.3
Assessment of Results
This chapter has produced three major results:
* The harmonic tracking method is insensitive to signal distortion due to the transfer
function between the gears and the accelerometer on the casing.
* In order to use the harmonic tracking method, the operating data should have an
SNR of at least 12 dB.
* The simulation predicts that for SMB-000static (no) flowdata, the harmonic tracking
method should recover a maximum valve travel within .102% of the actual value.
61
F
o
a)
E
a)
CZ
.
aCO
0)
CL
=)
4
.Time(s)
Figure 4-14: Spring Pack Displacement for Simulation Cases 1 (Baseline) and 2 (TF Dis-
tortion)
S
c
E
a)
0)
0.
to
C)
c
C
aCL
CO
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Time (s)
Figure 4-15: Spring Pack Displacement for Simulation Case 4 (Realistic Case)
62
Sensititivity to Transfer Function Distortion
The recoveries for the simulation cases 1 (Baseline) and 2 (TF Distortion) are essentially
identical (with the exception of a single, outlying data point in the no distortion case),
indicating that the harmonic tracking method is insensitive to this type of signal distortion.
As a result, we conclude that no inverse filtering will have to be performed in order to remove
the distortion effects of the transfer function. It is not currently known why the harmonic
tracking method is insensitive to transfer function distortion. It is believed that short-time
spectral generation's averaging of the meshing signal over long enough time intervals to
produce sidebands may, in turn, result in averaging out of the phase distortion from the
transfer function, producing an effectivelyundistorted recovery.
Sensititivity to Measurement Noise
By adding various levels of broadband noise to the gear meshing signal (simulation case 3),
we found that a minimum SNR of 12 dB should be present in operating data. However, this
value will depend on both the spectral generator and the windowlength used for short-time
spectral generation, and it should be considered a rule-of-thumb value only. For example, if
the signal being analyzed contains a rapidly changing transient, the window length must be
very short, and the SNR must be relatively high. For general application of the harmonic
tracking method, the allowable SNR will have to be judged on a case by case basis, with the
rule-of-thumb value providing an initial indication about whether the harmonic tracking
method should be considered for use.
Realistic Case Harmonic Tracking Performance
Simulation case 4, representing SMB-000 casing vibration under static flow conditions,
produced an error in recovered valve travel of .102%. If recovery errors from analysis of
SMB-000 experimental data are similar, the simulation may be considered realistic and
generally applicable.
An additional question is whether .102% error is acceptable.
For
MOV diagnostics, we expect this degree of error to be acceptable. However, final judgment
will be postponed until Chapters 5 and 6, after application to experimental data has been
performed.
63
Chapter 5
Application of Harmonic Tracking
Method to MOV Vibration Data
This chapter has two purposes. The first is to apply the harmonic tracking method to
a generic machine to determine whether the method performs as designed, recovering the
angular speed and displacement of each gear in the machine. The second is to address
MOV-specificconcerns, including whether the harmonic tracking method recovers accurate
valve travel (1) on MOV's of different classes and (2) on MOV's under various operating
conditions. To perform these two tasks, three sets of experimental data have been analyzed.
The first data set is from a Limitorque SMB-000 class MOV under static (no) flow conditions. The second and third sets are from a Limitorque SMB-2 class MOV under static
and 1800 psi (pressure gradient) flow conditions, respectively. These three data sets form
the minimum, but complete set required to address the two MOV-specific concerns, and
are also a representative application of the harmonic tracking method to a generic machine.
The chapter contains a brief presentation of the historical perspective, analysis goals, and
recovery details for each of the three data sets. A brief discussion of the results concludes
the chapter.
64
5.1
Applying Harmonic Tracking to Limitorque SMB-OOO
Data
5.1.1
Historical Perspective
The SMB-000 is the smallest MOV produced by Limitorque [9], weighing approximately 200
lbs when equipped with a 2.5 inch Edwards gate valve. The SMB-000 tested was obtained
by this research project, forming the basis of Chai's stand-alone valve experiment [3].
5.1.2
Analysis Goals
The primary goal is of this analysis is to take the first step beyond the SMB-000-based
simulation, determining whether harmonic tracking accurately recovers valve travel and the
angular speed and displacement of each gear from an experimental data set. In addition, detailed improvements to the method will be tested. In the previous chapter, it was found that
recovery of spring pack displacement is highly sensitive to single, "outlying" data points.
To alleviate this sensitivity, polynomials have been fit to all of the harmonics, resulting in
a much smoother and, hopefully, more accurate recovery of spring pack displacement.
5.1.3
Recovery Details
The operating data analyzed in this section was obtained by Chai for inclusion in his PhD
thesis [3]. At the time, he was testing his method's performance when obstructions such as
steel and aluminum rods were placed in the valve's path of travel. We've chosen to analyze
the operating data in which an aluminum rod obstructs the valve's path.
It should also be noted that the analysis includes the same truncated
lightly loaded
portion and complete heavily loaded portion of the signal examined in Chapter 4 for the
simulations. Chai's analysis focused purely on this portion of the signal, and due to the
constraints of reanalyzing his data, this analysis will do the same.
A Priori Info
The structural constants are identical to those stated in chapter 4.
As in chapter 4, the lightly loaded motor speed, fl, is the rated operating speed of the
motor, 1800 rpm or 30 Hz, which, when scaled by the number of teeth on the motor pinion
65
560
550
--
_% - - -
*W
9=--
q%
-
.540
Static Run
Tracked Harmonics, SMB-O000O
1W _V
Q01
0
C
'l-a)
v
530
'e A"'-
9D
],_
DeW
0I
510
fgl2
- - 1st Lower Sideband
500
490
0
- -
0
1st Upper Sideband
.....
0.5
1
1.5
Time (s)
2
2.5
3
Figure 5-1: Tracked Harmonics for SMB-000 Static Closing Stroke
gear, yields a lightly loaded pinion worm shaft gear meshing frequency of 540 Hz.
Spectral Analysis
Short-time Spectral Generation, Tone Location, and Harmonic Tracking Shorttime spectral generation was performed using a particular form of AR modeling we call
zoomed AR modeling (see Appendix A). Performance of the tone location and harmonic
tracking stages of short-time spectral analysis produced recoveries of pinion/worm shaft
gear meshing, fgl2, and two sidebands due to modulation by worm/worm gear meshing,
fg23
(see Figure 5-1).
Short-time Spectral Analysis: Polynomial fits
Subsequent analysis used polynomi-
als fit to the harmonics, which are also shown in Figure 5-1. Two piecewise continuous
polynomials were fit to each harmonic. The first is a least-squares constant fit to the lightly
loaded portion of the harmonic, and the second is a least-squares quadratic fit to the heavily
loaded portion of the harmonic. To determine the transition point between the lightly and
heavily loaded portions, the most clearly defined harmonic-in this case, upper sideband-is
analyzed, first by choosing an initial guess by eye. The lightly loaded constant should be
66
Linear/QuadraticFitsto P/WSG Meshing,SMB-000 StaticRun
I.,
---
I
538
I->,536
0
C
ci)
534
ID
U.
a
r_
532
C,)
M 530
co
ci)
a
%528
U,
526
f 524
.2
522
5 20
0
0.5
I
I
I
I
1
1.5
Time (s)
2
2.5
3
Figure 5-2: Recovered Pinion/Worm Shaft Gear Meshing for SMB-OOOStatic Closing Stroke
the same as the first data point in the heavily loaded quadratic. If the constant and the first
point in the quadratic are not equal, iteration is performed until the minimum difference
between the two values is obtained.
The same transition point is then used on all other
harmonics.
Short-time Spectral Analysis: Sideband Relations
fgl2 and fg23 were determined
using the carrier present, multiple sideband relations discussed in Chapter 3 (see Figures 5-2
and 5-3). The recovered fg23 is the average of the fg23's calculated from both sidebands.
Calculating Gear Rotation, Valve Travel, and Spring Pack Displacement
Gear rotations were calculated by integrating gear angular speeds with respect to time.
Because these quantities were not measured directly, they will not be plotted. Valve travel,
too, was not measured, and was calculated by scaling the rotations of the worm gear by
the pitch of the valve stem (see Figure 5-4). Finally, spring pack displacement was measured and is compared to the recovered quantity (see Figure 5-5). Note that the curve fits
have smoothed recovered spring pack displacement considerably compared to the simulation
recoveries of chapter 4, thus fulfillingtheir intended purpose.
67
StaticRun
W/WG Meshingfrom Linear/QuadraticFits, SMB-O000O
I
C)
C
IL
0)
c.
0)
:)
Ca
Ua
c-
00o3
O3
0
0.5
1
1.5
Time (s)
2
2.5
3
Figure 5-3: Recovered Worm/Worm Gear Meshing for SMB-OOOStatic Closing Stroke
Static
Recovered(Not Measured)HeavilyLoadedValve Travel,SMB-O000O
>
a)
3
Time (s)
Figure 5-4: Valve Travel During Heavily for SMB-OOOStatic Closing Stroke
68
n h-
Spring Pack DisplacementUsingSmoothingFits, SMB-000 StaticRun
.rC
E
M
C.
u
C,
--
0C
C
I
Time (s)
Figure 5-5: Spring Pack Displacement for SMB-000 Static Closing Stroke
5.1.4 Discussion of Recovery Results
Spectral Generation by Zoomed AR Modeling
AR models were only used for analysis after direct comparison with DFT-based recoveries.
It was found that, after finding a suitable model order and window length, AR models
produced marginally more distinct harmonics which were easily tracked with the simple,
automatic algorithm. When the model order and window length may be properly tuned,
AR modeling is a marginally better choice for spectral estimation.
Harmonic Tracking and Polynomial Fits
The polynomial fits were originally intended to smooth the harmonics for more accurate
calculation of spring pack displacement. They successfully performed this role, primarily
because the harmonics for this static run were well approximated by low orderpolynomials,
an issue to which we will return during analysis of the flow data. A second potential use of
polynomial fits was discovered during this analysis: to fillgaps in the harmonics at timesteps
for which a tone was not located (notice from Figure 5-1 that, especially during the lightly
loaded portion of the 560 Hz sideband, there are many gaps during which the band's tone
69
was not able to be located). Polynomial fits are one way of solving this problem. Another
approach will be discussed and used on the 1800 psi flow data.
Results Assessment
Results assessment is made more difficult by the fact that only one measured quantity is
available for comparison: spring pack displacement.
Nonetheless, from observations and
comparisons between this recovery and that obtained from the realistic simulation case
from Chapter 4, a reasonable assessment will be made.
On average, harmonic tracking
recovers spring pack displacement well, though a perfect match is not expected because the
method produces a cubic resulting from the integration of the heavily loaded quadratics
that will never exactly match a generic curve. The polynomial fits have resulted in a better
spring pack displacement recovery than that obtained for the realistic simulation case. It is
expected that, because the fits match the general shape of the harmonics well, the accuracy
of valve travel recovery will also improve. The relative error in maximum valve travel is,
therefore, expected to be no greater than the .102% reported for the realistic simulation
case.
5.2
Applying Technique to Limitorque SMB-2 Data
5.2.1 Historical Perspective
The Limitorque SMB-2 is a 900-lb class MOV, and is a mid-to-large sized MOV [9]. The
SMB-2 motor operator used for this research was equipped with a 10 inch Edwards gate
valve. It is owned by the Electric Power Research Institute (EPRI), and is known as EPRI
Valve 43. In August 1993, under contract by EPRI, Valve 43 was put through a series of
tests at the Wyle Laboratories in Hunstville, Alabama. These tests included the opening
and closing of the valve under motor operation against a variety of flow conditions, and
were fully instrumented in terms of conventional valve measurements.
We were given the
opportunity by EPRI to take vibration measurements during these tests. This opportunity
was accepted, and Wyle Laboratories was contracted to take the vibration measurements
on our behalf. During the tests, however, the MOV did not perform as designed; the motor
stalled on the 1800 psi flow run before the valve had fully closed. This occurrence presents
a unique diagnostic opportunity for our project: to aid in the determination of what went
70
wrong with EPRI Valve 43. Determining valve travel by harmonic tracking is the first step
in performing these diagnostics.
5.2.2
Analysis Goals
Analyses of static and 1800 psi closing strokes are the bases for determining whether the
harmonic tracking method recovers valve travel (1) on MOV's of different classes and (2) on
an MOV under different flow conditions. An assumption we make is that if the harmonic
method successfully recovers valve travel on both static data (0 percent design-rated flow)
and 1800 psi data (100 percent design-rated flow), it should work for any flow condition in
between. An additional goal of these two analyses is to apply the technique to data acquired
by an outside party, Wyle Laboratories. We have had far less control over the acquisition
of this data than is typical of a university research project. These analyses represent our
ability to perform harmonic tracking recovery on "generic" data sets.
5.2.3
Recovery Details, Static Case (No Flow)
A Priori Information
The structural constants for the SMB-2 MOV under static conditions are:
* Number of teeth on the motor pinion gear, N1 = 28
* Number of teeth on the worm shaft gear, N 2 = 42
* Number of teeth on the worm shaft gear, N 3 = 33
* Pitch of stem, Ps = .33 in/thread.
Spring pack displacement was not calculated, so worm pitch, Pw, is not included among the
structural constants. Motor speed was measured as part of the test, and our estimate of
lightly loaded motor speed, fl, was obtained directly from those measurements. Its value
is 59.18 Hz, which, when scaled by the number of teeth on the pinion gear yields a lightly
loaded pinion/worm shaft gear meshing frequency, f
2,
of 1657 Hz. Recall that knowledge
of fgl2 enables easy location of its corresponding tone and sidebands in the spectra.
71
Spectral Analysis
Short-time Spectral Generation, Tone Location, and Harmonic Tracking The
closing stroke was analyzed in two, distinct parts: light loading and heavy loading. During
light loading, spectral estimation was performed using zoomed AR modeling. fgl2 and the
first lower sideband due to modulation by
f23
were recovered by the tone location and
harmonic tracking stages in short-time spectral analysis (see Figure 5-6). During heavy
loading, spectral estimation was performed using the DFT. fg12 could not be recovered
during heavy loading. Instead, the first and seventh lower sidebands due to modulation
by
f23
were recovered using the tone location and harmonic tracking stages of short-time
spectral analysis (see Figure 5-7).
static analysis,
As for the SMB-OOO
Short-time Spectral Analysis: Polynomial Fits
polynomials were fit to the harmonics. For this data, the lightly and heavily loaded portions
of the harmonics were approximated by least-squares linear fits (see Figures 5-6 and 57).
Note that the heavily loaded harmonics contain an initial period of light loading,
necessitating the location of a transition point between two piecewise continuous linear
fits. This point was found on the first lower sideband using the approach discussed for the
SMB-OOOstatic analysis.
Short-time Spectral Analysis: Sideband Relations
recovered directly, and
modulation by
fg23
fg23
During light loading, fg12 was
was calculated by subtracting the first lower sideband due to
from fg12. During heavy loading,
fg23 - f
f
6
fg23
was calculated from the relation
",7 and fg12 was calculated from
fgl2 = fl + fg23.
See Figures 5-8 and 5-9. In addition, motor speed, f,
(5.1)
(5.2)
was calculated by scaling fg12 by
the number of teeth on the pinion gear, and compared with the motor speed measurement
made during the test (see Figures 5-10 and 5-11).
72
Lightly Loaded Harmonics, SMB-2 Static Closing Stroke
10/u
I
1660
1650
IN
0
0)
1640
Pinion/Worm Shaft Gear Meshing
- - 1st Lower Sideband
-
1L
1630
I I1T
1620
I
/.../*~,l
i~',
..
i6f.
'
ql~l~t
,ty
,
II,
I~~~~~~~
I"
I1
m,
i-1
i
0
5
10
15
20
25
Time (s)
Figure 5-6: Short-time Spectral Generation (using AR Models) and Harmonic Thacking for
Lightly Loaded Portion of SMB-2 Static Run
Heavily Loaded Harmonics, SMB-2 Static Run
"r'
I0~
a)
U-
3
Time (s)
Figure 5-7: Short-time Spectral Generation (using the DFT) and Harmonic Tracking for
Heavily Loaded Portion of SMB-2 Static Run
73
P/WSG Meshing from Linear/Linear Fits, SMB-2 Static Run
1700
' 1650
>.,
0
C
(- 1600
0
LL
0)
-
s 1550
eq)
oa) 1500
C1
c
E
8 1450
C
.14
E 1400
1 '_rn
00o
5
15
10
I
20
25
Time (s)
Figure 5-8: Linear Fits to Pinion/Worm Shaft Gear Meshing for SMB-2 Static Closing
Stroke
W/WG Meshing with Linear/Linear Fits, SMB-2 Static Closing Stroke
1.U
.'
38
N
- 36
0
c
-34
0
32
eCD)
a)
u 30
30
E 28
28
o 26
24
fJ
....
J
0
5
I
10
I
15
I
20
25
Time (s)
Figure 5-9: Worm/Worm Gear Meshing from Linear Fits for SMB-2 Static Closing Stroke
74
Motor Speed, SMB-2 Static Closing Stroke
-I
:
U)
U-
0
10
5
15
20
25
Time (s)
Figure 5-10: Motor Speed for Entire SMB-2 Static Closing Stroke
Heavily Loaded Motor Speed, SMB-2 Static Closing Stroke
60
58
56
I
0,54
0"
C
a:52
50
48
46
20
Figure 5-11: Motor
20.5
21
21.5
Time (s)
22
22.5
23
peed for Heavily Loaded Portion of SMB-2 Static Closing Stroke
75
Total Valve Travel, SMB-2 Static Run
._
I7>
an
CZ
-5
0
5
10
15
20
25
30
Time (s)
Figure 5-12: Valve Travel Over Entire SMB-2 Static Closing Stroke
Calculating Gear Rotation and Valve Travel
Gear rotation is calculated by integrating shaft speeds with respect to time, but because
these values were not measured, gear rotations were not plotted. However, a measure of the
accuracy of gear rotation recovery is demonstrated by valve travel, which was calculated by
scaling worm gear rotation, 03(t), by stem pitch, Ps, and was compared to the experimentally
measured quantity (see Figures 5-12 and 5-13).
5.2.4
Discussion of SMB-2 Static Recovery Results
Spectral Generation
During light loading, the model order and window length for the zoomed AR models were
easily chosen, allowing us to use AR modeling. However, during heavy loading, where time
resolution is of critical importance, we were unable to find suitable AR model order and
window length.
Therefore, the DFT was used instead.
The resulting, marginal loss of
spectral resolution during the heavily loaded portion of the stroke did not adversely affect
our ability to locate and track the required sidebands.
76
HeavilyLoadedValve Travel, SMB-2 StaticRun
Z-
15
20
20.5
21
21.5
Time (s)
22
22.5
23
Figure 5-13: Valve Travel During Heavily Loaded Portion of SMB-2 Static Closing Stroke
Harmonic Tracking and Polynomial Fits
As was the case for the SMB-000 data, low order polynomial fits were good approximations
to the harmonic shapes, smoothing discontinuities that were likely due to random errors in
spectral generation. The polynomials also filled gaps in the harmonics during which tones
were not recovered. In general, it appears that recoveries performed on static closing strokes
may be effectively smoothed by using polynomial fits.
Results Assessment
For this data set, comparisons between recovered and measured motor speed and valve
travel were performed.
During light loading, recovered and measured motor speeds are
nearly indistinguishable.
During heavy loading, the recovery was a good approximation to
the best linear fit to measured motor speed. The followingcalculations were performed to
determine the accuracy of the recovery of diagnostically important valve travel:
* Maximum measured valve travel: 8.94602 in
* Maximum recovered valve travel: 8.93364 in
* Difference between maximum values: .01238 in
77
* Percent error (Difference/max measured value): .1384%
For MOV diagnostics, these levels of error are likely to be acceptable. In addition, note that,
particularly during heavy loading, the shape of the recovered valve travel curve matches the
measured curve nearly exactly.
5.2.5
Recovery Details, 1800 psi Flow
A Priori Information
The structural constants for the SMB-2 MOV under 1800 psi operating conditions are:
* Number of teeth on the motor pinion gear, N1 = 23
* Number of teeth on the worm shaft gear, N 2 = 47
* Number of teeth on the worm shaft gear, N 3 = 33
* Pitch of stem, P = .33 in/thread.
Note that, due to a motor stall on the 600 psi closing stroke, the pinion/worm shaft gear
pair was changed between the static and 1800 psi closing strokes. Spring pack displacement
was not calculated, so worm pitch, Pw, is not included among the structural constants. A
lightly loaded motor speed, f , of 58.15 Hz was estimated from the measurement taken
during the test, yielding, after scaling by the number of teeth on the pinion gear, a lightly
loaded pinion/worm shaft meshing frequency,
fg/12,
of 1337.5 Hz.
Spectral Analysis
Short-time Spectral Generation, Tone Location, and Harmonic Tracking During light loading, short-time spectral generation was performed using zoomed AR modeling.
The tone location and harmonic tracking stages of short-time spectral analysis produced re-
covered harmonics at fg12 and at the first lower sideband due to modulation by worm/worm
gear meshing,
fg23
(see Figure 5-14). During the initial five seconds of heavy loading, four
sidebands due to modulation by
fg23
were located and tracked. During the final one-half
second of heavy loading, only two sidebands due to
Figures 5-14 and 5-15).
78
fg23
modulation were recovered (see
Tracked Harmonics, SMB-2 1800 psi Closing Stroke
1
! I I,
1400
1300
I-- .1200
v
1
:3
11 100
IL
1000
900
-1st
*.__._2nd
Upper/LowerS-bands
Upper/LowerS-bands
i
~llll
.
.
I
.
.
5
0
!
,
I
I
,
I
,
I
I
10
15
20
25
30
35
Time (s)
Figure 5-14: Short-time Spectral Generation (using AR Models) and Harmonic Tracking
for SMB-2 1800 psi Run
T....¢v
Heavily Loaded Harmonics, SMB-2 1800 psi Closing Stroke
5UUll
I*_vv
1400
r
1300
-
-.
-,
-
*
-
f * *
i
.
\11
I. 1200
r-
*
1100 f
IL
..
1000
_
900
_
-Cal
~~~~~~~~x
's~
\ ..
culated P/WSG Meshing
1st Upper/Lower S-bands
--
- - 2nd Upper/Lower S-bands
UgU~
ntJU
_
-- 25
urvl
-
-
_ I
-
26
I
I
27
28
Time (s)
.
29
l
~~~~~~~L
.
30
31
Figure 5-15: Short-time Spectral Generation (using DFT) and Harmonic Tracking for Heavily Loaded Portion of SMB-2 1800 psi Run
79
Polynomials were initially fit to
Short-time Spectral Analysis: Polynomial Fits?
all harmonics. However, especially during heavy loading, it was found that the harmonic
shapes were too complex to be modeled using low order polynomials, and that reasonably
high order polynomials contained extraneous oscillations that did not correspond well to
the shapes of the curves. Therefore, it was decided that no polynomials should be fit.
In order to compensate for the loss of smoothing provided by polynomial fits, the time
interval between spectra was decreased. It was hoped that the higher density of spectral
data would reduce the effect of random errors due to operating noise. The decision not to
fit polynomials also necessitated the development of an alternative approach to filling the
gaps in the harmonics during which no tones were located. A simple algorithm was written
which fills these gaps by fitting a line between the last recovered data point before the gap
and the first recovered data point after the gap. "Filled" harmonics are shown in all figures.
Short-time Spectral Analysis: Sideband Relations
recovered directly, and
to modulation by
fg23
was calculated by subtracting
f23
from fgl2.
During light loading, fgl2 was
the first lower sideband due
During the initial 5 seconds of heavy loading,
f23
was
calculated by averaging the two fg23's calculated by subtracting the two upper sidebands and
the two lower sidebands.
due to modulation by
fg12
fg23.
was then calculated by adding
During the final .5 seconds,
fg23
fg23
to the first lower sideband
and fgl2 were calculated from
the two remaining sidebands (see Figures 5-16 and 5-17). Motor speed was calculated by
dividing fgl2 by the number of teeth on the pinion gear, N 1 , and was compared to the
experimentally measured quantity (see Figures 5-18 and 5-19).
Calculating Gear Rotation and Valve Travel
Gear rotation is calculated by integrating shaft speeds with respect to time, but, again, is
not plotted. As before, a measure of the accuracy of gear rotation recovery is provided by
comparing valve travel recovery to the experimentally measured values (see Figures 5-20
and 5-21).
80
P/WSG Meshing with NO Fits, SMB-2 1800 psi Closing Stroke
1500
'1400
0
9 1300
IL
= 1200
0 1100
(5
9 1000
.c
a
900
.- v0
5
10
15
20
25
30
35
Time (s)
Figure 5-16: Pinion/Worm Shaft Gear Meshing for SMB-2 1800 psi Closing Stroke
W/WSG Meshing from NO Fits, SMB-2 1800 psi Closing Stroke
5
Time (s)
Figure 5-17: Worm/Worm Gear Meshing for SMB-2 1800 psi Closing Stroke
81
Motor Speed, SMB-2 1800 psi Closing Stroke
6
5
IN, 4
3
C)
0)
c o3
U
U-
2
1
5
Time (s)
Figure 5-18: Motor Speed for Entire SMB-2 1800 psi Closing Stroke
Motor Speed, SMB-2 1800 psi Closing Stroke
N
0
CF
U-
2
Time (s)
Figure 5-19: Motor Speed for Heavily Loaded Portion of SMB-2 1800 psi Closing Stroke
82
Total Valve Travel, SMB-2 1800 psi Closing Stroke
IZ
0
5
10
15
20
25
30
35
Time (s)
Figure 5-20: Valve Travel Over Entire SMB-2 1800 psi Closing Stroke
Heavily Loaded Valve Travel, SMB-2 1800 psi Closing Stroke
.I_
ua
28
28.5
29
29.5
30
Time (s)
30.5
31
31.5
32
Figure 5-21: Valve Travel During Heavily Loaded Portion of SMB-2 1800 psi Closing Stroke
83
5.2.6
Discussion of SMB-2 1800 psi Recovery Results
Spectral Generation
The discussion presented for SMB-2 static analysis applies to the 1800 psi closing stroke
as well. During light loading, zoomed AR modeling was used because a suitable model
order and window length were found. During heavy loading, these suitable quantities were
not found for AR modeling, and the DFT was substituted. The DFT produced adequate
spectral resolution to track the necessary number of harmonics.
Harmonic Tracking
No polynomial fits were used for this run, primarily because dynamic flow effects result
in complexly shaped heavily loaded sidebands which are not well approximated by low
order polynomial fits. The lack of fits allows more of the detailed characteristics of the
recovered quantities to be captured, which is particularly evident during the heavily loaded
motor speed recovery (see Figure 5-19). In general, polynomial fits are not advisable for
closing strokes under dynamic flow conditions because even at low changes in pressure, the
harmonics display complex shapes due to flow effects.
Results Assessment
Results assessment for the SMB-2 1800 psi closing stroke is made more difficult by the fact
that the motor stalled during the run. As a result of motor stall, the motor speed dropped
to zero before being turned off by the controller. Using casing vibrations, it was not possible
to track harmonics below a motor speed of approximately 38 Hz which corresponds to a
test time of approximately 31 seconds (see Figure 5-19). Up to this point, motor speed is
tracked well, but information is lost by not being able to track harmonics to zero motor
speed. In particular, all valve travel that occurs between 31 and 32 seconds is not recovered.
The maximum measured and recovered valve travels are therefore 8.7928 in and 8.6965 in,
respectively. Again, up to loss of ability to track harmonics, valve travel is recovered well,
with a percent error between recovered and measured valve travels at the time of harmonic
loss of .0274%. The effect on EPRI Valve 43 diagnostics of losing valve travel between 31
and 32 seconds will be assessed in Chapter 6.
84
5.3
Summary
5.3.1 Spectral Generation
We've found that, as a rule of thumb, when the AR model order and window length can be
determined for effective implementation, AR models should be used.' Use of the DFT does
not adversely affect the recovery results, and, because of its ease of implementation, is still
an attractive candidate for future implementations.
5.3.2
Harmonic Tracking and Polynomial Fits
For static runs, polynomial fits to harmonics are acceptable, and in certain cases, such
as the SMB-000 static closing stroke, fitting may improve recovery results.
For general
implementation of the harmonic tracking method, however, no fits are likely to be used in
order to capture the finer details of the harmonics. This approach was used satisfactorily
on the SMB-2 1800 psi run. Gaps in harmonics resulting from not generating and locating
tones during certain time periods were filled using an automatic routine.
5.3.3 Results Assessment
During each of the three analyses, the harmonic tracking method performed as designed.
The two static runs were particularly successful, producing maximum valve travel errors
on the order of .1%. Extenuating circumstances, in the form of a stalled motor, prevented
recovery of complete valve travel for the 1800 psi case. It is important to note, however,
that harmonic tracking underestimates valve travel, very clearly predicting that the valve
did not close completely. Overall, the harmonic tracking method demonstrated that it
is capable of recovering valve travel on MOV's of different classes under different flow
conditions. Additional discussion, summary, and an assessment of the effect on EPRI Valve
43 diagnostics of underestimating
valve travel on the 1800 psi run will be presented in the
next chapter.
1
There are techniques for automatically determining AR model order. They were beyond the scope of
this research, but should be considered for future implementation.
85
Chapter 6
Conclusions and
Recommendations
This thesis has purposely had dual themes. The first is the development of a method to
contribute to MO V diagnostics by creating a set of algorithms that recover valve travel from
a casing vibration signal. The second is the development of a generally applicable method
in machine operation information gathering and diagnostics by non-invasive recovery of
the angular speed and displacement of each gear in the machine.
The first section in
this chapter addresses both themes, discussing primarily general issues which arose in the
application of harmonic tracking to MOV diagnostics. The second section focuses purely
on MOV diagnostics, including a diagnostic system overview, presentation of our diagnostic
signature for an MOV closingstroke during which the operator did not perform as designed,
and a discussion of recommendations for improving the harmonic tracking method. The
final section addresses general applicability, focusing on extending the applicability of the
harmonic tracking method into areas not considered in MOV diagnostics.
86
6.1 Summary of Major Results from Application of Method
to MOV's
6.1.1 Simulation Results
Method Unaffected by Transfer Function
Three accelerometers placed at three different locations on EPRI Valve 43 produced equiva-
lent casing vibration signals, indicating that no detectable structural transfer function (TF)
distortion occurred. However, harmonic tracking performance is unaffected even when TF
distortion occurs, as was illustrated by the nearly identical valve travel recoveries obtained
from two simulation cases, one of which contained no distortion and the other of which
contained distortion purely from a structural transfer function with one pole and one zero
in the frequency range of interest. Therefore, harmonic tracking will provide an accurate
recovery of valve travel, without inverse filtering, regardless of the accelerometer location
on the MOV.
Minimum Signal-to-Noise Ratio Required
By gradually increasing the amount of broadband noise added to the simulated gear meshing
signal, it was found that a minimum signal-to-noise ratio (SNR) of 12 dB is required to track
harmonics.
12 dB should be considered a rule-of-thumb value because it depends on the
spectral generation method (for example, DFT or AR modeling) and the window length
used for short-time spectral generation. It is expected that for MOV-related applications,
12 dB is a representative value. For non MOV-related applications, it should simply be
considered a guideline to be tested using representative data.
6.1.2 Experimental Results
Small vs Large MOV's
No major differences were found between harmonic tracking performance on a small MOV
(Limitorque SMB-000 class) and on a larger MOV (Limitorque SMB-2 class). One minor
difference-of-note is that the lightly loaded motor speed on the SMB-000 MOV could be
estimated accurately enough by the rated motor speed, while the SMB-2 lightly loaded
motor speed needed to be measured more accurately.
87
Otherwise, analyses of the static
closing strokes on the two valves were nearly identical.
Dynamic vs Static Flow Conditions
The primary difference between static and flow conditions is the rate of speed decrease
during heavy loading. The natural expectation is that the more slowly decreasing speeds
from flow conditions would be easier to track if, as appears to be the case, flow noise does
not significantly decrease the SNR during heavy loading. By comparing static flow and
1800 psi flow closing strokes on an SMB-2 class MOV, these expectations were confirmed.
It appears that the 1800 psi harmonics were easier to track both because of more gradually
decreasing speeds and because of a more favorable SNR resulting from higher loads on the
motor and attendant increases in gear meshing forces.
Overall Assessment of Results
The most important assessment is whether the harmonic tracking method recovers valve
travel accurately enough to perform MOV diagnostics. Both static recoveries resulted in
percent errors1 in valve travel measurement of .1%. As a result of an unexpected motor stall
before complete valve closure during the analyzed SMB-2 1800 psi stroke, total valve travel
was underestimated by .1 inches or 1%. The 1800 psi closing stroke analysis was exceptional,
forcing the harmonic tracking method to perform a task for which it was not designed: to
track harmonics from operating speed to stall. Up to the time at which harmonic tracking
was unable to track harmonics (which was well before the frequencies had dropped to 0), the
method performed as designed (see Figure 5-21). Therefore, we conclude that the harmonic
tracking method should be applicable to MOV diagnostics.
'Percent error is defined as
recovered valve travels, respectively.
z;:
l Xwhere
88
ZX,,
and ZC,
are the maximum measured and
6.2
Recommendations for Incorporation of Method into MOV
Diagnostic System
6.2.1
Diagnostic System Overview
Diagnostic Signature
At an earlier stage in this project, a diagnostic signature was developed. It includes valve
travel measured from casing vibration and motor torque estimated from motor voltage and
current, both measured as functions of time. Results from simulations using an electro-
mechanical model of the MOV (a separate study from the simulation discussed in this
thesis) indicate that faults such as a tapered or bent stem, tight stem packing, poor stem
nut lubrication, and poor seating,2 show unique fault signatures when displayed on a motor
torque versus valve travel curve [3]. The motor torque versus valve travel curve was therefore
chosen as our diagnostic signature.
Total Diagnostic System
The MOV diagnostic system consists of three major parts: the MOV, the data acquisition
system, and the data analysis system (see Figure 6-1). For data acquisition, current probes,
a voltage meter, and an accelerometer are mounted. The charge output from the accelerometer is first passed through a charge amplifier. All three signals are then passed through
anti-aliasing filters and an analog-to-digital (A/D) converter. During data analysis, signa-
ture extraction and a signature database input information to a failure analysis system.
Signature extraction currently consists of a motor torque estimator and harmonic tracking
valve position recovery system. A future addition is the extraction of gear torques from
casing vibration data using a meshing force analysis. By comparing the valve travel versus
motor torque diagnostic signature to a model-based, diagnostic signature database, diagnosis of mechanical faults and prognosis of the effects of such faults on system performance
may be performed.
2
Diagnostics on the gears themselves was not performed because this mode of failure is uncommon in
MOV's
89
MOV
]
Current
Data
Acquisition
System
V
I
Harmonic
Tracking
Torque
Estimator
I
T
Data Analysis
I ..................................................................................................................................
Figure 6-1: MOV Diagnostic System
90
--_uu
i
~
i
Estimated Motor Torque, SMB-2 Static Closing Stroke
i
i
I
I
180
160
140
E
z 120
E'100
0
I-
60
40
.I...r.- '
20
0
0
I
5
I
10
15
I
20
Time (s)
i
25
I
30
I
35
40
Figure 6-2: Estimated Motor Torque for SMB-2 Static Closing Stroke
6.2.2
Discussion of Motor Stall During SMB-2 1800 psi Closing Stroke
No attempt will be made to explicitly diagnose the motor stall during the SMB-2 1800 psi
closing stroke. The diagnostic signature and various other MOV performance indicators
will be generated and discussed. It is hoped that this information will ultimately aid in
diagnosis of the motor stall. However, a complete diagnosis is beyond our level of expertise
at this time.
Diagnostic Signatures-Static
and 1800 psi Closing Strokes
To create the diagnostic signatures for both static and 1800 psi closing strokes, motor
torque was estimated (see Figures 6-2 and 6-3) and plotted against normalized valve travel
recoveries from Chapter 5 (see Figures 6-4 and 6-5 and Figures 6-6 and 6-7). In addition,
the losses due to the operator were calculated by subtracting measured stem thrust (scaled
to match the units of torque) from estimated motor torque (see Figures 6-8 and 6-9).
Discussion
The following observations were made from the estimated torques, valve travels, diagnostic
signatures, and losses curves:
91
Estimated Motor Torque, SMB-2 1800 psi Closing Stroke
zE
ci
0
i_e
w
0
5
10
15
25
20
Time (s)
30
35
40
Figure 6-3: E stimated Motor Torque for SMB-2 1800 psi Closing Stroke
Recovered Valve Travel, SMB-2 Static Closing Stroke
v
[-
0
0
5
10
15
20
25
35
30
Time (s)
Figure 6-4: Recovered Valve Travel for SMB-2 Static Closing Strc)ke
v
92
Recovered Valve Travel, SMB-2 1800 psi Closing Stroke
la
I>0
;>
5
0
10
15
20
25
30
35
Time (s)
Figure 6-5: Recovered Valve Travel for SMB-2 1800 psi Closing St]roke
Diagnostic Signature, SMB-2 Static Closing Stroke
V.
ll
180
160
140
z120
0
I-O27100
i2
·
0
LEI
80
60
40
1
20
0
0
10
20
I
30
I
40
50
60
Closure of the Gate (%)
I
70
80
90
100
Figure 6-6: Diagnostic Signature for SMB-2 Static Closing Stroke
93
Diagnostic Signature, SMB-2 1800 psi Closing Stroke
200
180
160
140
zE 120
120
v
cii
E100
0
I. 80
a)
w
- I
60 I-
40
20
61w-
..
.
- - - ... . -- --
.
---- - -1 - I
- --
I
. -- -- -1
-
-
- , - '.
... -11--
0
I
0
F 'igure
10
20
30
I
40
50
60
Closure of the Gate (%)
70
80
90
100
6-7: Diagnostic Signature for SMB-2 1800 psi Closing Stroke
Losses Due to Operator, SMB-2 Static Closing Stroke
z
U)
0
-j
0
10
20
30
40
50
60
Closure of the Gate (%)
70
80
90
100
Figure 6-8: Losses Due to Operator for SMB-2 Static Closing Stroke
94
LossesDue to Operator,SMB-2 1800 psi ClosingStroke
AA
zU
0
-20
-40
z -60
0>
CO
30 -80
-100
-120
-140
I
-1n
0
10
~~~~~~vi
20
30
40
50
60
70
80
90
100
K~~~~~~~~~~
Closureof the Gate (%)
I
I
I
I
I
I
Figure 6-9: Losses Due to Operator for SMB-2 1800 psi Closing Stroke
* From the 1800 psi estimated torque, it is evident that the motor stalled at its rated
torque of approximately 110 Nm (see Figure 6-3). Therefore, the motor performed as
designed.
* The 1800 psi diagnostic signature indicates (conservatively) that the valve did not
close completely (see Figure 6-7).
* The losses due to the operator in the 1800 psi stroke increased by approximately 35
Nm between 96% and 98% valve closure (see Figure 6-9). No such increase in losses
was observed for the static stroke (see Figure 6-8).
In addition, by referring to a plot of spring pack displacement contained in the EPRI report
on the testing of Valve43 [8],it has been determined that the spring pack began to compress
at the same percent valve closure that lossesbegan to increase, indicating that motor torque
was lost during axial sliding of the worm to compress the spring pack. It is hoped that this
information will aid in diagnosing the cause of motor stall in the EPRI Valve 43.
95
6.2.3
Harmonic Tracking Modifications Necessary for "Black Box" Im-
plementation
The ultimate goal for implementation is to simply plug harmonic tracking algorithms into a
diagnostics system and have it automatically produce valve travel from casing acceleration.
This goal has not been reached yet, but with fairly minor modifications, it should be possible.
Need to Determine Lightly Loaded Motor Speed
Experience on the SMB-2 MOV has indicated that a fairly accurate estimate of motor speed
during light loading is necessary to locate the pinion/worm shaft gear meshing harmonic in
casing vibration spectra. By estimating motor slip from motor currents and voltages, we
expect to arrive at a sufficiently accurate estimation of lightly loaded motor speed.
Need for Automatic Tracking Algorithm
Perhaps the greatest stumbling block to automatic implementation is the development of
a robust, automatic harmonic tracking algorithm.
A number of possible approaches are
being considered, including combining harmonic tracking in the time-frequency domain
with short-time cepstral tracking in a cepstrum time-time domain.
Need for More Experience with Flow Data and Diagnostics
While this work has demonstrated proof-of-concept, there are still questions and potential
downfalls, some of which we may not yet be able to imagine. Further development, including
building our own flow loop to facilitate quickly gathering larger amounts of data under
various operating conditions, is necessary to transform a promising method into a feasible
diagnostic system.
6.3
General Applicability of Harmonic Tracking Method
Many methods exist for determining shaft speeds and rotations as functions of time. Why
would one consider using the casing vibration-based harmonic tracking method? The first
reason is non-invasiveness and ease of experimental setup. When using harmonic tracking,
all that is required is to find a convenient location on the machine casing for mounting an
accelerometer. Data may then immediately be taken and processed. A second reason is that
96
harmonic tracking offers the potential for combining diagnostics and information gathering
for controls. While the implementation presented here is not real-time, there are no conceptual blocks to modifying it to produce shaft speed data on a nearly real-time basis. On
MOV's alone, we may imagine a system that produces shaft speed information for control,
valve travel for MOV diagnostics, and spectra and cepstra to perform diagnostics on the
gears themselves. The potential for creating a mechanically straightforward, electronically
sophisticated diagnostic/control system is there. Implementation is not far away.
97
Appendix A
Autoregressive Modeling for
Spectral Estimation
A.1 Introduction to Parametric Modeling
Autoregressive models are one in the more general family of parametric models. The power
spectral density (PSD) is defined as the discrete-time fourier transform of an infinite autocorrelation sequence (ACS). This transform relationship between the PSD and ACS is
considered to be a non-parametric description of the second-order statistics of a random
process. Alternatively, a parametric description of the second-order statistics may created
by forming a PSD that is a function of model parameters rather than the ACS. Autoregressive (AR), moving average (MA), and autoregressive-moving average (ARMA) models
comprise a special class of these parametric models that are driven by white noise and
possess rational system functions.
The ARMA models a process with a rational system
function containing both poles and zeros (the most general case), while AR models use
rational system functions with all poles, and MA models use system functions with all zeros
[13].
Parametric modeling is used in closely related forms in a number of fields. In speech
processing, parametric models are typically referred to as linear predictive models, providing
a robust, reliable, and accurate method for short-time spectral estimation of the slowlytimevarying production of speech [16]. In controls and information theory, parametric modeling
has been used in system estimation and identification.
98
In our research, parametric modeling (and, in particular AR modelingl ) was considered
as a spectral estimator primarily because it should achieve superior spectral resolution to
classical estimators such as the DFT. When the DFT is employed, the ACS is typically
truncated to be zero outside of a certain range. An AR PSD, however, extrapolates the
ACS outside of the original range, resulting in the signal's energy being more accurately
concentrated near the poles. The location of the maximum value of the PSD should be
the same for the DFT and AR model, so gear meshing tones should be locatable using
either method.
However, AR models are expected to be more robust to the presence of
measurement noise.
This appendix is not intended to be an exhaustive treatment of the subject of parametric
modeling. Only the most general theory will be presented; for the most part, we'll leave
algorithmic computational details to be found in the literature [16, 13, 2]. We will, however,
present some algorithmic details of zoomed AR modeling, which is a modification to the
typical implementation that was developed specifically for this project by Jangbom Chai.
A.2
Formulation of AR Models
The basic time-series model representing an AR process of order p is a linear difference
equation of the form
p
x[n] =-
a[k]x[n- k] + u[n],
(A.1)
k=1
where x[n] is the output sequence from a filter with input u[n] and a[k] are the model
coefficients. If we assume that the input is white noise with zero mean and variance, p,,
the PSD, P(f), of the AR process may be expressed as
Tpw
P(f) = IA(f) l2'
(A.2)
where
p
- j2 fkT
A(f) = 1 + E a[k]e
k=1
Gear meshing spectra contain tones which are most appropriately modeled as poles.
99
(A.3)
and T is the sampling interval. The AR PSD is therefore a function of the model coefficients,
a[k], and the input variance, pw,.
Two unique methods have been developed to calculate the model coefficients assuming
Pw is known. The first is the autocorrelation-based
method, which relates the model coef-
ficients to the autocorrelation sequence, RXX. This method assumes that it is given an n
point sequence which is zero outside this interval. Estimation errors are expected at the
beginning and end of the sequence because the model is estimating the signal from data
points that have been set to zero. Such starting and ending transient errors are typically
minimized by (hanning) windowing the original sequence. The second method is the covariance method, which solves for the model coefficients by solving a matrix equation containing
a covariance matrix. This method assumes that it is given a sequence of length n + p (p is
the model order). This method does not require windowing of the original sequence. How-
ever, the solution of the matrix equation is not as straightforward as for the autocorrelation
method. In addition, covariance-based estimations are considered to be highly sensitive to
measurement noise [20]. Because of these two potential downfalls of the covariance based
algorithm.
calculation, we've chosen to use an autocorrelation-based
To calculate the model coefficients using the autocorrelation-based
method, the following
relations between the autocorrelation sequences of the input and output and the model
coefficients have been derived: [13]
rux[i] =
0
for i > 0
Pw
for i = 0
pwh* [-i]
for i < 0
- Ek=l a[k]rxx[m- k]
rxx[m]=
MPla[k]rxx[-k]
+p
for m > 0
for m = 0
(A.4)
(A.5)
for m < 0
r*x[-m ]
which may be evaluated for the p + 1 lag indices 0 < m < p, and formed into the matrix
expression
100
rxx[O]
rxx[-1]
rxx[1]
rx.[O]
.
...
rxx[-p]
rxx[-p
Pw
a[1]
+ 1]
rxx[p
-
1]
...
r[O]
-
a[p]
(A.6)
~~~~~~~~~~(A.6)
,
,
rxip]
0
0
where the matrix of Toeplitz form. A number of fast, efficient algorithms exist for the
solution of these equations. We use the Levinson-Durbin recursion.
A.3
Zoomed AR Modeling
Zoomed AR modeling was developed because the tones we'd like to find in the gear meshing
spectra are located in a small, well-defined frequency range. Therefore, we'd only like to
model the signal in that particular range of frequencies. The algorithm is fairly straightforward, consisting of four major steps. First, the windowed sequence is extracted using
a hanning window. Next, the power spectrum of the sequence is created using the DFT,
and samples of the spectrum in the frequency range of interest are extracted. Then, the
autocorrelation sequence of the extracted portion of the power spectrum is created. Finally,
the AR model coefficients corresponding to the autocorrelation sequence are found using
the Levinson-Durbin algorithm, and a power spectrum of the AR model is made.
101
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