UNIVERSITY OF CINCINNATI
Date: 05/11/2006
I, _Raghavendran Raghunathan,
hereby submit this work as part of the requirements for the degree of:
Master of Science
in:
Mechanical Engineering
It is entitled:
Development and comparative assessment of CWT
based damage detection techniques on simulated
gearbox signals
This work and its defense approved by:
Chair: Dr.Randall J. Allemang
Dr. Jay Kim
Dr. Allyn Phillips
_______________________________
_______________________________
Development and comparative assessment of CWT based damage detection
techniques on simulated gearbox signals
A thesis submitted to the
Division of Research and Advanced Studies
of the University of Cincinnati
in partial fulfillment of the
requirements for the Degree of
MASTER OF SCIENCE
From the Department of Mechanical, Industrial and Nuclear Engineering
of the College of Engineering
2006
by
Raghavendran Raghunathan
BE (Mechanical Engineering), Birla Institute of Technology and Science (BITS), Pilani
M.Sc. (Mathematics), Birla Institute of Technology and Science (BITS), Pilani
Committee Chair: Dr. Randall J. Allemang
Committee Member : Dr. Allyn Phillips
Committee Member : Dr. Jay Kim
ii
Abstract
The demand for condition monitoring systems has constantly increased with more and
more companies trying to minimize the effect of machine failures, and develop more
efficient ways of using the resources. Over the years the role of damage detection
systems has changed from being a post processing tool to a prognostic tool, which is
expected to deliver information on the current operating condition of the machine and
also predict the available service life. The time and frequency localization properties of
continuous wavelet transform offers a viable and improved option for analyzing the
transient characteristics of defect signals. One of the main limitations in development of
diagnostic applications is the need for large data resources which include all defect
scenarios. Mathematical modeling and computer simulation has been widely used for the
purpose of understanding the dynamics of the system and also to generate the various
defect scenarios encountered during actual operation. This thesis presents an extensive
literature review on existing dynamic models for gearbox systems and attempts to
simulate a simplified model of a single stage gear box system. The research presented in
this thesis attempts to bridge the gap between simulation and diagnostic application
development. A common interface for developing and fine tuning CWT based diagnostic
application for a single stage gearbox is presented. An elaborate mathematical
background of the continous wavelet theory is presented along with a brief overview of
existing CWT diagnostic techniques. A comparative analysis on existing fault detection
algorithms and CWT based diagnostics is presented. Finally, based on the results and
discussion a roadmap for development of CWT based diagnostic application is presented.
1
Acknowledgement
First and foremost I would like to express my gratitude to Dr. Randall J.
Allemang for his constant encouragement and input that has contributed substantially to
my work. He has always been a source of inspiration to me. I appreciate the time he has
taken to discuss and critically examine various aspects of this work.
I would also like to thank Dr. Jay Kim and Dr. Allyn Phillips for being the
members of my thesis advisory and examining committee.
I would like to express my gratitude to Dr. Walter Bartelmus for all his inputs and
the extra effort to mail me valuable research material from Poland. I would also like to
thank my colleagues in the Department of Mechanical, Industrial and Nuclear
engineering for their valuable discussions during this course of work.
Finally, I would like to dedicate this work to my parents, sister, brother-in-law
and grandmother for providing excellent learning opportunity to seek knowledge in
University of Cincinnati, without which this work would not have been possible.
2
TABLE OF CONTENTS
Title
Pg.No.
LIST OF FIGURES
6
LIST OF SYMBOLS
10
CHAPTER 1
11
1.1 MOTIVATION
11
1.2 LITERATURE REVIEW
13
1.2.1 History of Gear Dynamic Simulation:
13
1.2.2 Review of condition monitoring literature
14
1.3 SCOPE OF THE PROJECT
17
1.4 ORGANIZATION OF THE THESIS
17
CHAPTER 2
19
2.1 MATHEMATICAL MODELING OF GEAR DYNAMICS
19
2.2 TORSIONAL DEGREE OF FREEDOM MODEL
20
2.3 TORSIONAL & LATERAL DOF MODEL
22
2.4 MODELING INTER-TOOTH MESHING FORCE AND FRICTIONAL MOMENT
22
2.4.1 Relative Velocities at Contact Point During Meshing
22
2.4.2 Modeling Frictional Force and Moment
25
2.4.3 Moment due to Friction
25
2.4.4 Modeling Profile Error
28
2.5 SOLVING NON-LINEAR DYNAMIC SYSTEM OF EQUATIONS
CHAPTER 3
29
33
3
3.1 BASICS OF TRANSFORMS
33
3.1.1 Vector Space, Basis and Inner Product Space
33
3.2 DISCRETE FOURIER TRANSFORM AND SHORT TIME FOURIER TRANSFORM
35
3.3 THEORY OF WAVELET TRANSFORM
38
3.3.1 Need for Wavelet Transform
38
3.3.2 Discrete Wavelet Transform (DWT)
38
3.4 CONTINUOUS WAVELET TRANSFORM
47
3.4.1 Convergence, completeness & Hilbert spaces:
48
3.5 CLASSIFICATION OF WAVELET FUNCTIONS
53
3.6 GEAR DAMAGE DETECTION TECHNIQUES
55
3.6.1 Time Domain Averaging & Overall Power Level
56
3.6.2 Kurtosis and Pulse Detection Methods
56
3.6.3 Spectrum Analysis
57
3.6.4 Cepstrum Analysis
57
3.6.5 Envelope Detection
57
3.6.6 Time-Frequency Analysis
58
3.6.7 CWT Based Techniques
58
CHAPTER 4
62
4.1 SIMULATION RESULTS
62
4.1.1 Effect of Profile Error
66
4.1.2 Effect of Eccentricity
68
4.1.3 Effect on gear and pinion tooth errors
69
4.1.4 Summary of simulation results and fourier based diagnostic measures
71
4
4.2 CWT BASED DIAGNOSTIC APPLICATION DEMO
71
4.3 DAMAGE DETECTION BASED ON VISUAL INTERPRETATION
76
4.4 KURTOSIS
79
4.5 TIME AVERAGED WAVELET SPECTRUM
80
4.6 WAVELET VARIANCE
81
4.7 SINGLE SCALE IDENTIFICATION
82
4.8 DAMAGE DETECTION RESULT SUMMARY
84
CHAPTER 5
87
5.1 CONCLUSION
87
5.2 FUTURE DIRECTIONS
88
BIBLIOGRAPHY
89
APPENDIX
93
A.1 – MATLAB CODE USED IN DEVELOPING THE CWT BASED DIAGNOSTIC TOOL
93
A.2 – SIMULINK MODEL USED TO SOLVE THE NON-LINEAR SYSTEM OF EQUATIONS 110
A.3 – MATLAB FUNCTION SIMULATING THE VARYING SYSTEM PARAMETERS
5
111
List of Figures
Figure 2.2.1: Simplified model of spur gear meshing contact area. Figure on left shows
the simplified model of a single stage spur gearbox.
20
Figure 2.4.1: Figure showing the resultant velocity on gear and pinion tooth moving
through the contact region.
23
Figure 2.4.2: Free body diagram of spur gear tooth in contact showing the normal and
tangential force on pinion and gear at two different points during contact period
27
Figure 2.4.3 : Plot of change in mesh stiffness with tooth mesh period, simulated signal
attempts to model variation to contact ratio.
28
Figure 2.4.4 : Plot showing variation of tooth profile error with Time, function has
random amplitude over each tooth mesh period
29
Figure 3.2.1 : Figure showing a test signal with time varying harmonic content and
transient impulse signals.
36
Figure 3.2.2 : STFT of the test signal shown in figure, the four figures show the STFT
coefficients using different time windows.
37
Figure 3.3.1 : Time and frequency domain representation of first stage shannon filter
bank sequence.
40
Figure 3.3.2 : Schematic representation of single stage wavelet coefficient extraction and
signal reconstruction
42
Figure 3.3.3 : Schematic representation of multi stage wavelet decomposition
42
Figure 3.3.4 : Schematic representation of multi stage wavelet decomposition using a
parallel structure
44
6
Figure 3.3.5 : Plot showing high pass and low pass Haar filter sequence upto stage 3
45
Figure 3.3.6 : Plot of f and g coefficient of Haar wavelets upto scale 3
46
Figure 3.3.7 : Demonstration of discrete Haar wavelet decomposition of a simple sine
wave.
47
Figure 3.4.1 : Plot showing real and imaginary parts of continous shannon wavelets at
integral scales. The bottom subplot shows the overlapping filter bank structure.
51
Figure 3.4.2 : Comparitive plot of filter bank arrangement of shannon wavelet at integral
and dyadic scales. The bottom subplot shows leakage due to discretization.
52
Figure 3.4.3 : Plot of complex morlet wavelet filter bank at integral and dyadic scales. 53
Figure 3.5.1 : Comparative plot of discrete and continuous wavelet coefficient map of the
test signal.
55
Figure 4.1.1 : Simplified version of SIMULINK model for numerical simulation of the
torsional degree of freedom model.
62
Figure 4.1.2: Plot of simulated displacement, velocity and acceleration signals of pinion
and gear using the parameters listed in table.
64
Figure 4.1.3 : Time and frequency domain representation of simulated pinion angular
velocity signal. The signal represents the baseline condition with nominal tooth
profile variation and tooth meshing stiffness variation.
65
Figure 4.1.4 : Plot of simulated gear angular velocity signal with eccentric mounting
conditon and nominal tooth profile variation.
66
Figure 4.1.5 : comparative plot of simulated gear angular velocity with different profile
error amplitudes.
67
7
Figure 4.1.6 : Comparative plot of gear angular velocity signal with eccentric mounting
condition superimposed over the baseline signal.
69
Figure 4.1.7 : comparative plot of frequency domain representation of simulated gear
angular velocity signal under different error advancement stages.
70
Figure 4.2.1 : Front end of application developed to perform complex continuous wavelet
transform of simulated gear signal. The option available for wavelet and scale
settings, along with the post processing tools are displayed.
72
Figure 4.2.2 : Step 1 - load test data , set analysis parameters.
73
Figure 4.2.3 : Step 2 - Choose data range for analysis
74
Figure 4.2.4 - Step 3 - Perform CWT analysis and explore available post processing tools.
75
Figure 4.3.1 - CWT coefficient of simulated pinion angular velocity signal. Wavelet morlet, Scales - 0 to 30, Error condition - baseline case.
76
Figure 4.3.2 - Plot of coefficients of morlet wavelet analysis for pinion angular velocity
signal with simulated error condition.
77
Figure 4.3.3 - CWT coefficients of morlet wavelet analysis for pinion angular velocity
signal with 1/tooth error on pinion and gear
78
Figure 4.3.4 - Plot of fourth central moment (kurtosis) of wavelet coefficients at
individual scales.
79
Figure 4.4.1 - Comparative plot of kurtosis of pinion angular velocity under different
simulated error conditions. Legend (-. black) Error Advancement 25%, (-. Red)
Error + Eccentricity, (-- Blue) Error Advancement 15%, (- Black) No Error, (Green) No Error, (- Red) Error Advancement 5%.
8
80
Figure 4.5.1 : Comparative plot of time averaged wavelet spectrum of pinion angular
velocity under different simulated error conditions. Legend (-- black) Error
Advancement 25%, (-- Red) Error + Eccentricity, (-- Blue) Error Advancement
15%, (- Black) No Error, (- Green) No Error, (- Red) Error Advancement 5%.
81
Figure 4.6.1 : Comparative plot of wavelet variance of pinion angular velocity signal
under different simulated error conditions. Legend (- Green) Error +Eccentricity, (-.
Red) Error Advancement 25%, (-- Red) Error Advancement 20%, (- Black) Error
Advancement 10%, (- Red) Error Advancement 5%, (- Blue) No Error.
82
Figure 4.7.1 : Plot showing the fault scale identification method on a defect signal.
Legend (Top Left) scale increment = 25, (Top Right) scale increment = 10, (Bottom
Left) scale increment = 0.5, (Bottom Right) scale increment = 0.2.
83
Figure 4.7.2 : Kurtosis of CWT coefficients of simulated pinion angular velocity signal.
84
9
List of Symbols
θp
angular velocity of pinion
θ1
angular velocity of pinion shaft
θg
angular velocity of gear
θ2
angular velocity of gear shaft
rp
radius of pinion
rg
radius of gear
Ig, Mp
inertia and moment on pinion respectively
Ig, Mg
inertia and moment on gear respectively
Kmesh, Qmesh
inter tooth meshing stiffness and damping
K1, Q1
pinion shaft rotational stiffness and damping
K2, Q2
gear shaft rotational stiffness and damping
Mfr
frictional moment acting on pinion and gear
x5, x6
lateral degree of freedom of pinion and gear respectively
DFT(x(m))
discrete Fourier transform of time signal x
STFT(x,τ)
short time Fourier transform of time signal x with window τ
DWT(x(t))
discrete wavelet transform of time signal x
CWT(x(t))
continuous wavelet transform of function x
ST(f)
standard deviation of function f
10
Chapter 1
1.1 Motivation
The application of wavelet transforms for condition monitoring and predictive
maintenance of industrial machinery is being developed at a very rapid rate over the past
10 years. Predictive maintenance itself has been developed extensively over the past 40
years and has paralleled development in industrial machinery and automation. The
demand for condition monitoring systems has constantly increased with more and more
companies trying to minimize the effects of machine failures, and develop more efficient
ways of using the resources. The factors which have lead to the increase in need for
condition monitoring are:
•
The need to cut down unplanned stoppages to improve profitability and
competitiveness.
•
Increased quality expectation and the constant need to set lower prices.
•
To decrease the cost of maintenance, to prevent replacement of under-used
parts.
•
To improve reliability and safety, the need to decrease human
intervention.
The cost effective solutions to automated condition monitoring has been made possible
by reduced cost of instrumentation, faster methods of data processing, low cost
computing and data storage devices. Commonly used condition monitoring techniques
are:
•
Vibration Analysis
•
Oil-debris Analysis
11
•
Thermal Monitoring
•
Performance Monitoring
Vibration analysis is a powerful technique for monitoring rotating machinery and it is
also comparatively easy to implement and cost effective. A number of methods have been
developed for condition monitoring based on vibration signals. Some of the well
established and widely used techniques are cepstrum, time-domain averaging,
demodulation analysis and time-series analysis. All of the above methods suffer from one
common drawback, which is the assumption of stationarity of vibration signals, while
most fault information are essentially non-stationary. Time-frequency analysis and
wavelet analysis are recently developed methods that overcome this difficulty. Most
wavelet based condition monitoring algorithms are based on knowledge systems.
Extensive testing to extract and identify all fault features is necessary to avoid false
alarms and undetected damages. To reduce the cost associated with testing and to develop
a robust system, computer simulation of gear dynamics has been widely employed as an
aid for condition monitoring. Mathematical modeling and computer simulation enable
detail investigation of dynamic properties of gearing systems.
Motivation for this thesis comes from the need to integrate gear dynamic models and
condition monitoring algorithms and to develop a fast method of implementing
continuous wavelet (CWT) based real-time gear diagnostic application. The main focus
of this thesis is on developing a real-time condition monitoring application and a simple
4-DOF torsional gear dynamic model is simulated to validate the fault diagnostic
algorithm.
12
1.2 Literature Review
1.2.1 History of Gear Dynamic Simulation:
There is a vast amount of literature available on mathematical modeling and simulation
of gear systems. Early work on gear system dynamic started as early as 1970, a
classification and review of gear dynamic models present at that time can be found in [1]
by Ozguven et al. Houser [2] has reviewed possible causes of gear noise and has outlined
an analytical technique to predict gear noise excitation. They classified the mathematical
models as simple dynamic factor models, models with tooth compliance, models for gear
dynamics, models for geared rotor dynamics and models for torsional vibration. But only
during the 1990s ([3-6]) a large number of global torsional-lateral models with a large
number of degrees of freedom for simulating a complete gear set was developed.
Kaharam and singh [5] developed a 2-DOF semi-definite model of a spur gear pair
assuming rigidity for shafts and bearings.
Bartelmus [7] to [9] has developed gear dynamic models from a condition monitoring
point of view. He has modeled a single stage torsional DOF system and extended the
system of equations to include lateral degrees-of-freedom and two stage gearbox system.
The error model is divided into design factors, production technology factors, operational
factors and change of condition factors. Using computer simulation, taking Design,
Production Technology, Operation and Change of Condition factors, he has formulated
the DPTOCC inference mechanism. The results from simulation compare well with
experimental data and he has shown that mathematical modeling and computer
simulation enable detailed investigation of the dynamic properties of the gearing system.
13
Yao and McFadden in [10] have presented a study of modeling for monitoring of gearbox
vibration. A simplified 1-DOF model of gear mesh system is proposed to represent the
gear motion vibration signal and the vibration signal caused by damage due to pitting,
scuffing, wear and tooth crack. Howard et.al [11] have presented a comprehensive
analysis for including friction in dynamic system of equations. A finite element model of
the gears in mesh is developed to model the varying gear mesh stiffness. A 16-DOF
system is simulated using SIMULINK and the results are shown to compare well with
experimental data. Parey and Tandon [12] have presented a comprehensive review of the
current state-of-the-art in gear dynamic modeling and simulation. The paper emphasizes
the importance of defect models for a clear understanding of gear vibrations.
1.2.2 Review of condition monitoring literature
The beginning of wide spread use of the wavelet transform in condition monitoring can
be traced back to 1900. Early work on wavelets was done by Haar in 1909, when his
work on orthogonal system of functions lead to the development of Haar Wavelets. The
concept of continuous wavelet transform was first put forward by Morlet and Grossman
in 1984. In 1989 Mallet [13] unified the theory of wavelets, developed the
multiresolution theory and found the relationship between quadrature mirror filters and
orthogonal basis. Later, Meyers developed a continuously differentiable, non-trivial, noncompact wavelet basis. In 1988, Daubechies [14] formulated compact orthogonal basis
with arbitrary number of vanishing moments. His 10 lectures on wavelets are considered
to be a cornerstone of wavelet applications today. Rioul et.al. [15] have given
comprehensive review of algorithms for discrete and continuous wavelet transforms.
They have developed general guidelines for implementation of discrete and continuous
14
wavelet transforms efficiently and compared various algorithms based on operation
count. Newland in [16] has proposed a harmonic wavelet analysis, which unlike the
wavelets generated by discrete dilation can be expressed in a functional form. He
proposed a wavelet based on “sinc” functions and is similar to the Shannon wavelet basis
which is used widely in transient detection in gear condition monitoring. A discrete
wavelet transform using this wavelet is proposed using Fast Fourier Transform.
One of the first works on the application of wavelet transforms to fault detection in spur
gears was done by Staszewski et.al [17]. He presented a damage detection algorithm
based on similarity analysis of patterns obtained from the modulus of the wavelet
transform. Results from numerical simulation are presented for various seeded trails
including surface wear (spalling, bad tooth contact), cracked tooth and loss of part of
tooth due to breakage. In 1994 Wang and McFadden [18] developed a generalized
formulation for continuous wavelet based gear fault detection technique. They have
emphasized the use of non-orthogonal wavelets like Morlet wavelets for fault detection in
place of orthogonal wavelet transform which might not allow scales fine enough to
capture all the transients in the signal. They have applied the wavelet transform using a
simple algorithm on time averaged vibration signal from a helicopter gear box.
Wang et.al [19] have presented a comparative analysis of different, vibration analysis
based, condition monitoring technique for gearbox. They have compared time domain
and spectral diagnostics procedures with wavelet based fault detection methods,
indication the pros and cons of each method. The good visual inspection property and
overall
robustness
of
the
time-frequency
frequency/cepstrum analysis is emphasized.
15
methods
over
the
time/statistical,
Zheng, Li and Chen have introduced a new approach of continuous wavelet transform
based fault detection in [20]. They have introduced the concepts of Spectrum Comparison
Method (SCM) and Time-Averaged Wavelet Spectrum (TAWS) based on morlet wavelet
transform. They have shown that the TAWS features the fault advancement very well and
is conically proportionally to the gear fault advancement.
Recently in 2004, Tse, Yang and Tam (21) developed the concept of effective exact
wavelet analysis. They have addressed the issue of overlapping and presence of
redundant information in continuous wavelet transforms. To minimize the effect of
smearing and misinterpretation they have formulated an exact wavelet analysis based on
genetic algorithms. They have shown through experimental validation that exact wavelet
analysis not only minimizes overlapping, but also helps operators detect faults and
distinguish the cause of faults.
A review of application of wavelet transform in fault diagnostics is presented by
Badaaoui et.al [22]. The application overview focus on time-frequency analysis, fault
feature extraction, singularity detection, denoising, signal compression and system
identification using wavelet transform. A comparison of performance of different
methods is outlined.
Amy Robertson, Sohn and Farrar [23] have proposed an online damage detection
technique for theme park rides based on wavelet transform. They have developed three
feature extraction method based on Holder exponent, modulus maxima and wavelet
variance. Among the 5 damaged data sets out of 50 total runs, they have shown that the
discrimination of wavelet variance using an auto associative neural network gave the best
results.
16
1.3 Scope of the Project
The main objectives of the project are:
•
To study existing gear dynamic models and to simulate a simple torsional degreeof-freedom system.
•
To study existing wavelet based gear diagnostic procedures.
•
To develop a “fast”, efficient and effective diagnostics procedure based on
complex continuous wavelet transform.
•
To compare performance of different wavelets for gear diagnostic application and
identify “best” suited wavelet for this particular application.
1.4 Organization of the Thesis
This thesis is organized into five chapters. The first chapter serves an introduction to the
thesis work. The literature review section summarizes the research done in the area of
condition monitoring of gearboxes and mathematical modeling and simulation of gear
dynamics. The motivation for this study and the specific objectives of the thesis are
stated.
Chapter 2 presents a study on the existing mathematical model for gear dynamics
simulation. The non-linearities due to varying mesh stiffness, profile error and friction are
modeled based on the system proposed by Bartelmus. The state space expansion of the
system of equation is developed and numerical solution to the system of differential
equation is found using SIMULINK.
In Chapter 3, the theory of the wavelet transform is discussed. The chapter starts with the
discussion of basic of transformation and proceeds to discuss the development of discrete
17
and continuous wavelet transform from a linear algebra point of view. The advantage of
using CWT for analysis of transient signals and the drawbacks in conventional time and
spectral methods is discussed throughout the chapter. The chapter also discusses
commonly used gear damage detection techniques and elaborate on feature extraction
methods employed in CWT based damage detection techniques.
Chapter 4 presents the results and discussions of simulation and complex wavelet
analysis. The simulation results are presented first and a comparative analysis of different
diagnostic techniques is presented.
Chapter 5 is the concluding section which summarizes the research undertaken in this
thesis and provides some future direction for this research.
18
Chapter 2
2.1 Mathematical Modeling of Gear dynamics
A number of gear dynamic system models have been proposed over the years. These
systems can be broadly classified as:
Simple dynamic factors model: The gear tooth load consist of static and dynamic
components. The models under this category include the dynamic effects by
imposing a penalty on the load carrying capacity of the gears.
Models with tooth compliance: These models consider only compliance due to
gear tooth modeled as a cantilever beam. The gear tooth profile and other
components of the gearbox are assumed to be perfect.
Models of gear system dynamics: The models in this category either consider only
the torsional vibration of the rotating gears or they include both torsional, lateral
and horizontal vibration. These models include the effect due to non-linearities
like change in mesh stiffness, profile errors, damaged tooth etc.
The model simulated in this thesis belongs to the third category. The models that simulate
gear system dynamics vary based on the number of degrees of freedom involved in the
simulation, which is based on the actual gearbox that is simulated. Hence the good
correlation obtained between the experimental and simulation results is mainly due to the
fact that the experimental rigs used in such studies satisfied all the basic assumptions
made in the mathematical model.
19
2.2 Torsional Degree of Freedom Model
These models consider only the torsional degrees of freedom in the gearbox system. The
overall system dynamics can be represented using the following set of differential
equations. The dynamic model of the torsional DOF system is shown in Fig [2.2.1].
I sθ&&1 + c1 × (θ&1 − θ&2 ) + k1 × (θ1 − θ 2 ) = M s (θ&1 )
I1 pθ&&2 + c1 × (θ&2 − θ&1 ) + k1 × (θ 2 − θ1 ) + r1 × K mesh × (r1θ 2 − r2θ3 ) + r1 × Qmesh × (r1θ&2 − r2θ&3 ) + M t1 = 0
I 2 pθ&&3 + c2 × (θ&3 − θ&4 ) + k2 × (θ3 − θ 4 ) + r2 × K mesh × (r2θ3 − r1θ 2 ) + r2 × Qmesh × (r2θ&3 − r1θ&2 ) − M t 2 = 0
I l × θ 4 + c2 × (θ&4 − θ&3 ) + k2 × (θ 4 − θ3 ) = −Tout
Figure 2.2.1: Simplified model of spur gear meshing contact area. Figure on left shows the simplified
model of a single stage spur gearbox.
20
Where I s , I1 p , I 2 P , I l are lumped inertia of the motor, pinion, gear and load respectively.
The stiffness of the gear and pinion shafts represented as K1 , K 2 is assumed to constant,
any resonances associated with the shafts are neglected. The damping of the shafts is also
assumed be constant and is indicated by C1 ,C2 in the equation. The meshing stiffness and
damping is represented as K mesh , Cmesh respectively and is modeled as a function varying
over a tooth meshing period. The moment on the pinion and gear due to the tooth
meshing force is represented as M t1 , M t 2 . Finally the motor’s driving moment is M s (θ&) ,
modeled as function of shaft angular velocity and Tout is the load characteristics.
Torsional DOF systems can be used to model physical gearbox systems in which the
lateral and horizontal displacement is negligible.
The above system of equations is based on the following assumptions:
1. Only torsional DOF is considered; the system is assumed to rigid in all
other directions.
2. Static transmission error is assumed to negligible.
3. The gear tooth is assumed to be perfect involute curve. Any static error
due to non-involute shape or improper mounting is introduced using an
error function.
4. Lumped mass model is used for all mass and inertia components.
The non-linearity in the system is introduced by the varying mesh stiffness and profile
error variations. The intertooth meshing force is modeled using the following dynamic
model Fig [2.4.1].
21
2.3 Torsional & Lateral DOF Model
Additional lateral degrees of freedom are allowed for the gears. These models can be
used to study the influence of gear and pinion mounting on the overall system dynamics.
The system of equations is for the 6-DOF model presented below:
I sθ&&1 + c1 × (θ&1 − θ&2 ) + k1 × (θ1 − θ 2 ) = M s (θ&1 )
I θ&& + c × (θ& − θ& ) + k × (θ − θ ) + r × K
1p 2
1
2
1
1
2
1
1
mesh
r1 × Qmesh × (r1θ&2 − r2θ&4 − x&3 + x&5 ) + M t1 = 0
I θ&& + c × (θ& − θ& ) + k × (θ − θ ) + r × K
2p 3
2
4
6
2
4
6
2
× (r1θ 2 − r2θ 4 − x3 + x5 ) +
mesh
× (r2θ 4 − r1θ 2 − x5 + x3 ) +
r2 × Qmesh × (r2θ&4 − r1θ&2 − x&5 + x&3 ) − M t 2 = 0
I × θ&& + c × (θ& − θ& ) + k × (θ − θ ) = −T
l
4
2
6
(
( x&
4
2
6
4
out
)
+ r θ& ) + k ( x
m1 &&
x3 + qmesh x&3 − x&5 − r1θ&2 + r2θ&4 + kmesh ( x3 − x5 − r1θ 2 + r2θ 4 ) = 0
m2 &&
x5 + qmesh
5
− x&3 − r2θ&4
1 2
mesh
5
− x3 − r2θ 4 + r1θ 2 ) = 0
The parameters K mb , qmb represent the stiffness and damping in the lateral direction,
m1 , m2 is the mass of pinion and gear respectively.
2.4 Modeling Inter-tooth Meshing Force and
Frictional Moment
2.4.1 Relative Velocities at Contact Point During Meshing
To understand how frictional force and moment contribute in the dynamic system of
equations, the gear tooth involute action and velocity variation during the tooth meshing
period has to be analyzed. From basic theory of gearing it is well known that the contact
points of the meshing teeth, in a perfect involute gear, lie on a straight line which is
tangential to the base circles of the pinion and gear. The rules of rigid body motion are
22
applied to calculate the velocity components at the point of contact. The following figure
shows velocity components at different locations along the line of contact.
p1
ω1
vg
vp
vp-vg
p
vp
vg
ω2
p2
Figure 2.4.1: Figure showing the resultant velocity on gear and pinion tooth moving through the
contact region.
The linear velocity of the gear and pinion at any point along the involute is expressed as:
v p = rp × ω p , v g = rg × ω g
Where ω g , ω p are the angular velocities of the gear and pinion respectively and rg , rp
are the radius of the gear and pinion at a particular location on the involute. The
components of the velocity components in the tangential and normal directions are:
v p ,t = v p sin(θ ) , v g ,t = v g sin(θ )
The instantaneous radius of curvature of the gear tooth is defined as:
ρ p = rp sin(θ )
23
Hence the constant angular velocity relationship can be used to get:
v p , t = ρ p ω p , v g ,t =
ρ gω p
N
, where N is the gear transmission ratio.
The tangential velocity of the pinion at point P1 will be zero and will attain maximum
value at P2. Similarly, the tangential velocity of the gear at P1 will reach the maximum
value starting from zero at P2. Hence, as an approximation the tangential velocity
variation with meshing period will be assumed to be linear. It can be noted that, at some
point (pitch point) along the line of action the two tangential velocities are equal. Hence
the relative velocity at that point would be zero.
vrel = ρ1ω1 − ρ 2ω2 ,
The relative velocity defined using the above equation will change from being a positive
value to a negative value. Thus contact between the gear teeth will be pure rolling contact
at the pitch point.
The relation between the rotation angle of the gear and the location of the contact point is
approximated using the following relation. For a gear with N teeth, the angle
corresponding to one teeth will be:
θp =
2π
,
N
An approximate relation which relates gear rotation angle to a location on the line of
contact:
Relative contact location = modulus (φ θ p ) .
24
2.4.2 Modeling Frictional Force and Moment
Transmission error in gearing is defined as "the difference between the actual position of
the output gear and the position it would occupy if the gear drives were perfect”. The
force induced in the dynamic system as a result of the transmission error is the inter-tooth
meshing force, given by the following equation:
Fmesh = K mesh ( r1θ1 − r2θ 2 ) + Qmesh ( r1θ&1 − r2θ&2 )
The frictional force acts in a direction perpendicular to the normal contact force and will
apply both translational forces and moments to the gear [Fig.2.4.2]. The frictional force
written in terms of the normal force will be:
(
F f = µ × Fmesh = µ × K mesh ( r1θ1 − r2θ 2 ) + Qmesh ( r1θ&1 − r2θ&2 )
)
Since the tangential velocity changes direction during the meshing period, the frictional
force will also change direction.
2.4.3 Moment due to Friction
The equation of moment equilibrium for the gear and pinion on two points on either side
of the pitch point is derived using Fig 2.4.2. The frictional force acting on the gear and
pinion is represented as F f . The moment balance equations for the point to the right of
pitch point are as follows:
M 1 − r1 × Fmesh + µ × Fmesh × ρ1 = 0
M 2 − r2 × Fmesh + µ × Fmesh × ρ 2 = 0
25
Since the relative velocity becomes zero at the pitch point, tangential velocity changes
direction at the pitch point. The moment balance equation for a point to the left of the
pitch point is:
M 1 − r1 × Fmesh − µ × Fmesh × ρ1 = 0
− M 2 + r2 × Fmesh + µ × Fmesh × ρ 2 = 0
The variation of moment along the line of action is approximated using the following
method developed by Bartelmus [8]:
The maximum value of the moment due to friction will be:
max(M F f ) = (length( P1 P2 ) × µ × ( Fmesh ))
Thus the frictional moment acting on the pinion can be linearly approximated as follows:
 max(M F f ) × mod(φ θ p ) 0 ≤ mod(φ θ p ) ≤ 1 (1 + u ) 
Mp =

− max(M F f ) × mod(φ θ p ) 1 (1 + u ) < mod(φ θ p ) ≤ 1
26
p1
ω1
P1 : Represents forces at a point
before pitch point
P : Represents Pitch Point
P2 : Represents forces at a point
after pitch point
M1
Fp
Fn
p
P1
Fg
Fg
Fn
P2
Fp
M2
p2
ω2
Figure 2.4.2: Free body diagram of spur gear tooth in contact showing the normal and tangential
force on pinion and gear at two different points during contact period
Similarly, the frictional moment acting on the gear can be approximated as:
 max( M Ff ) × (1 − mod(φ θ p )) 0 ≤ mod(φ θ p ) ≤ 1 (1 + u ) 
Mg = 

− max( M F f ) × (1 − mod(φ θ p )) 1 (1 + u ) < mod(φ θ p ) ≤ 1
Finally, the variation of mesh stiffness along the line of contact depends on the contact
ratio of the gear. Most gears are designed such that when in mesh the rotating gears have
more than one gear tooth in contact, transferring the torque for some of the time. The
ratio of the length of the line-of-action to the base pitch is called the contact ratio. Hence,
change in contact from 1 tooth to 2 teeth leads to change in stiffness as shown in
Fig.2.4.3.
27
Plot of Change in Mesh Stiffness Vs Tooth Mesh Period
9
x 10
2.4
2.38
Mesh Stiffness
2.36
2.34
2.32
2.3
2.28
2.26
1.2789
1.279
1.2791
1.2792
1.2793
1.2794
1.2795
Time
1.2796
6
x 10
Figure 2.4.3 : Plot of change in mesh stiffness with tooth mesh period, simulated signal attempts to
model variation to contact ratio.
2.4.4 Modeling Profile Error
The mathematical models described above assume that the gear tooth surface is a perfect
involute. But practical gears are not involutes, hence there is a need to introduce an error
function which simulates the common errors, due to surface asperities, encountered in
gears. The error model according to Dr. Bartelmus [9] is as follows:
( )

e τ
(0, a]
a


error = 

e (1 − τ )
(a,1] 

(1 − a )


The e is the maximum value of error for the tooth mesh period and is chosen randomly
for every meshing pair [Fig 2.4.4].
28
Plot of Random Error Vs Time
-6
x 10
18
16
14
Error
12
10
8
6
4
2
0
5.894
5.896
5.898
5.9
5.902
5.904
Time
Figure 2.4.4 : Plot showing variation of tooth profile error with Time,
amplitude over each tooth mesh period
5.906
5
x 10
function has random
2.5 Solving non-linear dynamic system of equations
The most popular and robust numerical method to solve differential equations is RungeKutta. The formulation for Runge-Kutta is for first order differential equations, hence the
second order system of differential equations have to converted to a system of first order
differential equation through state-space expansion. The matrix formulation of the
differential equation for 4-DOF and 6-DOF system is given below:
29
 I motor
 0

 0

 0
I pinion
0
0
 q1
 −q
+ 1
 0

 0
− q1
q1 + r12 qmesh
− r1r2 qmesh
0
0
0
0
I gear
0
 θ&&1   k1
  &&   −k
 θ 2  +  1
 θ&&3   0


I load  θ&&4   0
0
0
0
0
r1r2 qmesh
q2 + r2 2 qmesh
− q2
− k1
k1 + r12 K mesh
− r1r2 K mesh
0
0
r1r2 K mesh
K 2 + r2 2 K mesh
−K2
0  θ1 
0  θ 2 
 
− K 2  θ3 

K 2  θ 4 
0  θ&1   Tin 
 
0  θ&2  − M p 
 =

− q2  θ&3   M g 

q2  θ&4   −Tout 
and for the 6-DOF system:
 I motor
 0

 0

 0
 0

 0
 K1
− K
 1
 0

 0
 0

 0
 q1
 −q
 1
 0
+
 0
 0

 0
0
I pinion
0
0
0
0
0
0
m pinion
0
0
0
0
0
0
I gear
0
0
 θ&&1 
 θ&& 
 2
x3 
  &&
  &&  +
 θ 4 
  &&
x 
 5
I load  θ&&6 
 
0
0
r1r2 K mesh
r1 K mesh
− r1 K mesh
− K mb
2
K 2 + r2 K mesh
− r2 K mesh
0
0
0
0
mgear
0
− K1
k1 + r12 K mesh
− r1 K mesh
− r1r2 K mesh
r1 K mesh
0
0
− r1 K mesh
K mesh + K mb
rg K mesh
− K mesh
0
− q1
q1 + r12 qmesh
− r1qmesh
−r1r2 qmesh
r1qmesh
0
0
−r1qmesh
qmesh + qmb
rg qmesh
− qmesh
0
0
0
0
0
0
−r2 K mesh
−K2
0
r1r2 qmesh
−r1qmesh
q2 + r2 2 qmesh
− r2 qmesh
− q2
K mesh + K mb
0
0
r1qmesh
−qmb
− r2 qmesh
qmesh + qmb
0
0  θ1 
0  θ 2 
0   x3 
 
− K 2  θ 4 
0   x5 
 
K 2  θ6 
0  θ&1   Tin 
 
0  θ&2  − M p 
0   x&3   0 
 = 

− q2  θ&4   M g 
0   x&5   0 
  

q2  θ&6   −Tout 
The matrix grows with increasing degree of freedom; a mathematical model for up to 16DOF systems is available in the literature [11]. The simulation results are presented later
for the 4-DOF system. If all the values of the parameters are known, increasing the DOF
30
of the system is straight forward. The next step involves converting the system of
differential equations to first order system, which is done using the following procedure:
 [ 0]
A=
[ m ]
Define:
[ m ] ,
[ c ] 
− [ m]
B=
 [ 0]
[ 0] 
[ k ]
then the system of second differential equations:
[ m]{θ&&} + [ c ]{θ&} + [ k ]{θ } = { f }
can be represented using the following first order system of equations:
[ A]{ y&} + [ B ]{ y} = {F }
where { y} , { F } are defined as follows:
{ y&}
 ,
{ y}
{0} 

.
{ f }
For the 4-Dof system the formulation will be:
31






 I
  motor
 0

 0

  0
[ Zeros(4, 4)]
0
I pinion
0
0
0
0
I gear
0





I load 
  I motor
0
0
 
I pinion
0
−  0
  0
0
I gear
 
0
0
  0
+


[ Zeros(4, 4)]



0
0
0
 θ&&1 
  && 
 θ 2 
 θ&&3 
 
 θ&&4 
 
0   θ&1 

0   θ&2 
 
− q2   θ&3 

q2   θ&4 
 θ&1  0

  &  0

θ2  



Zeros
(4,
4)
[
]
 θ&  0


 3 

 θ&4  0
 =

− k1
0
0   θ1  Tin 

k1 + r12 K mesh
r1r2 K mesh
0   θ 2  − M p 

  
− r1r2 K mesh K 2 + r2 2 K mesh − K 2   θ3   M g 

0
K 2   θ 4  −Tout 
− K2
0
0
0 
 I motor
 0
I pinion
0
0 

 0
0
I gear
0 


0
0
I load 
 0
− q1
0
 q1
 −q q + r 2 q
r1r2 qmesh
 1 1 1 mesh
 0
−r1r2 qmesh q2 + r2 2 qmesh

− q2
0
 0





I load 
0
0
0
 k1
 −k
 1
 0

 0
The results for the simulation of the above system are presented in Chapter Four. The
simulation is performed using SIMULINK ode 45 solver.
32
Chapter 3
3.1 Basics of Transforms
A transform is a special kind of function. A function is essentially a map between two
sets (called the domain and co-domain). Hence understanding a particular transform can
be split into a two step process as understanding the domain and co-domain and
understanding the map that relates these two sets. The study and analysis of the set
involved in transformations is a research area in itself. Some of the important definitions
and favorable properties are mentioned here.
3.1.1 Vector Space, Basis and Inner Product Space
A vector space is a set V = {vi} defined over a field of scalar (F) with operations of
vector addition and scalar multiplication satisfying the following properties.
1. For all u,v ∈ V, u + v ∈ V, this property is called closure under addition. Further,
there exist and element in V, denoted 0, such that v + 0 = v for all v ∈ V.
2. For all v ∈ V, α ∈ F, αv ∈ V, this property is called closure under multiplication.
Futher, there exist an element in V, denoted 1, such that 1v = v for all v ∈ V.
3. Additionally, the addition operation should be associative and commutative; the
multiplication operation should be associative and distributive.
All practical signals that are measured and analyzed are vectors in the ℜ n (n-dimensional
real coordinate space) which is also called the Euclidean domain. Any transformation that
is applied on such signals is essentially a change in the way the signal is represented i.e. a
change in basis.
33
A basis of a vector space V over a field F is a subset B of V such that the vectors in B are
linearly independent and every vector in V can be written as a linear combination of
vectors in B. The number of vectors in the basis is called the dimension of the vector
space. A time domain signal with N points belongs to the ℜ n vector space and the basis
used to represent the signal is { e1, e2,………, eN}.
1, i = j 
Where (ei ) j = 

0, i ≠ j 
There are infinitely many basis for a particular vector space. Some basis have
additionally favorable properties which make them more suitable for better understanding
of the signal and easier signal manipulation.
This brings us to the area of orthogonal basis and inner product spaces. A inner product
on a vector space V over C is a map ⋅,⋅ : V×V→C with the following properties:
•
Additivity: u + v, w = u, w + v, w for all u,v,w in V.
•
Scalar homogeneity: αu, v = α u, v for all α ∈ C, u,v in V.
•
Conjugate symmetry: u , v = v, u for all u,v in V.
•
Positive definiteness u, u ≥ 0 for all u in V and u, u = 0 if and only if
u=0.
A set of vectors in V is said to be an orthonormal set if and only if the following
condition is satisfied.
1.
2.
u, v = 0 for all u,v in B and u ≠ kv.
u, u = 1 for all u in B.;
34
The basis used in the Fourier transform {1, e jω , e j 2ω ,..., e jnω } is an orthonormal set.
Orthogonal transforms offer several advantages; one of the most important property from
signal representation point of view is that there is no redundancy in the information in the
transformed domain. Thus theoretically the Fourier transform enables us to view the
individual frequency components of any time domain signal.
3.2 Discrete Fourier Transform and Short Time
Fourier Transform
Discrete Fourier transform represents time domain signals in terms of functions that are
not localized in time domain. They provide the frequency composition of a random
process which is assumed to be periodic or fully observed. The discrete Fourier transform
of a time domain signal of finite length N is defined as:
N −1
DFT ( x( m)) = ∑ x (n)e − 2πimn / N
n=0
This is essentially a transformation from the Euclidean basis to the Fourier basis which
consists of the functions {1, e jω , e j 2ω ,..., e jnω } and is an orthonormal set as mentioned
earlier.
However, many random processes are essentially non-stationary (i.e. frequency
components have time varying characteristics). For example, acoustic signals like speech
and music and occasional transient impulse from faulty machinery are non-stationary.
The spectral density of a non-stationary signal calculated using DFT provides data about
the frequency composition averaged over the duration of the signal. Hence the
reconstructed signal will have a periodic nature with the transient repeating at the
35
fundamental frequency. The following figure shows an example of DFT applied to a nonstationary signal.
fft of a time signal with varying harmonic content
300
250
200
150
100
50
0
0
100
200
300
400
500
600
700
800
900
1000
600
700
800
900
1000
frequency
2
1.5
1
0.5
0
-0.5
-1
0
100
200
300
400
500
time
Figure 3.2.1 : Figure showing a test signal with time varying harmonic content and transient impulse
signals.
One way to adapt Fourier transform to analyze non-stationary signals is by using a
technique called Short Time Fourier Transform (STFT) or moving-window Fourier
transform. The STFT is calculated using the following equation:
N −1
STFT ( x(m),τ ) = ∑ w(t − τ ) × x(n)e − 2πimn / N
n=0
Where w(t) is called the window function, which localizes the signal and Fourier basis
function in time. The window function is normally chosen as the rectangular window. In
the discrete domain this amounts to choosing N samples at a time from the input signal
36
(where N is the length of the rectangular window). Since the time resolution (∆t) is
unaltered the maximum frequency remains the same, but the frequency resolution
decreases as it depends on the length of the time domain signal. The relationship between
time and frequency resolution from Nyquist criterion is:
 1 

f max = 
 (2 * ∆t ) 
 1 

∆f = 
 ( N * ∆t ) 
Thus to obtain time resolution, the frequency resolution has to be sacrificed, this relation
is formally stated as the Heisenberg’s uncertainty principle. Another limitation of STFT
is that the frequency resolution is the same throughout. Hence adequate resolution at low
frequencies may result in unnecessarily high resolution at high frequencies.
Window length = 64 Time points
5
10
10
20
Frequency
Frequency
Window length = 32 Time points
15
20
25
30
40
50
30
60
200
400
600
800
1000
200
400
600
800
time
time
Window length = 128 Time points
Window length = 16 Time points
1000
2
20
Frequency
Frequency
4
40
60
80
6
8
10
12
100
14
120
16
200
400
600
800
1000
200
time
400
600
800
1000
time
Figure 3.2.2 : STFT of the test signal shown in figure, the four figures show the STFT coefficients
using different time windows.
37
3.3 Theory of Wavelet Transform
3.3.1 Need for Wavelet Transform
As mentioned earlier Fourier basis elements are not localized in space and hence are not
suited for analyzing non-stationary, transient signals. Any vector v ∈ l 2 ( Ζ N ) is said to be
localized in space near n0 if almost all values away from n0 are zero or very small.
Suppose there is a basis for l 2 ( Ζ N ) which is localized in time. Any discrete time signal
belonging to l 2 ( Ζ N ) can be written as a linear combination of elements from the basis as:
N −1
z = ∑ a n vn
n=0
Now, the difference between these coefficients and the DFT coefficients of the same
signal would be that the coefficients not only provide frequency information but also time
information since the support of the basis functions is localized. One example of a
localized basis is the standard Euclidean basis. Since it has only one nonzero component
it has the maximum time localization possible. However, the drawback here would be
that the frequency domain representation of the basis function will have constant value at
all frequencies and hence there is no frequency resolution. The wavelet transform
provides a balance between the two approaches and gives both frequency and time
resolution.
3.3.2 Discrete Wavelet Transform (DWT)
One of the main reasons for the wide spread use of DFT in signal analysis is because of
the extremely fast computations performed using the FFT algorithm. While constructing
a wavelet basis and computing the DWT coefficient this is a feature that has to be
exploited in order to make DWT a more practical solution.
38
To begin with a first stage wavelet basis for a discrete signal of finite length N is
developed. The fundamental concept which enables the construction of wavelet basis is
[28]:
If v ∈ l 2 (Z N ) such that w(n) = 1 then {Shift( k ) w}
N −1
k =0
is an orthonormal basis for
l 2 (Z N ) . Where Shift(k) is an operator with shifts the vector to the right by k indices.
The above theorem states that if a single vector is to be used to construct a wavelet basis
for l 2 (Z N ) , then the frequency domain resolution cannot be obtained. Hence, instead of
using one vector v whose full set of translates form an orthogonal basis, we will use two
vectors such that their translates by even integers forms an orthonormal basis. Thus we
will effective divide the frequency domain into high pass and low pass regions using the
first stage wavelet basis. The low and high frequency components are also referred to as
the detail and approximation, respectively, of the input signal. A formal definition given
by [28] is:
Let N be an even integer (N = 2M), an orthonormal basis for l2(ZN) of the form
{R2k u}kM=−01 ∪ {R2 k v}Mk=−01
for some u , v ∈ l 2 (Z N ) , is called a first-stage wavelet basis. Where
u and v are called the father and mother wavelets whose translates generate the basis.
To generate the two wavelets such that their translated functions form a basis, the filter
design technique of designing a high pass filter by modifying a low pass filter is used.
The fact that convolution in frequency domain is multiplication in time domain is used to
shift the low pass filter from 0 to π (discrete frequency).
Hence, let N = 2M, u ∈ l 2 (Z N ) , u is defined as
39
u = cos(nπ ) × u (n)
DFT (u (n)) = DFT (u (n + M ))
Therefore starting with a vector u ∈ l 2 (Z N ) , a vector v is formed such that
{u, Shift2 (u ), Shift4 (u),...., Shift N − 2 (u ), v, Shift2 (v),..., Shift N − 2 (v)} forms a basis for l 2 (Z N ) .
First stage Shannon basis developed using the above procedure is shown in Fig.3.2.1.
First Shannon Wavelets low pass filter sequence
First Shannon Wavelets High pass filter sequence
0.8
1
0.6
0.5
0.4
0.2
0
0
-0.2
50
100
150
200
-0.5
250
100
150
200
Discrete Points
Frequency Spectrum of first stage
low pass filter sequence
Frequency Spectrum of first stage
high pass filter sequence
1.5
1.5
1
1
0.5
0.5
0
50
Discrete Points
50
100
150
200
0
250
Frequency
50
100
150
200
250
250
Frequency
Figure 3.3.1 : Time and frequency domain representation of first stage shannon filter bank sequence.
From elementary linear algebra the following can be said about the first stage wavelet
basis:
1. If
{R2k u}kM=−01 ∪ {R2k v}kM=−01
is a first stage wavelet basis, then the
transformation from Wavelet basis to the Euclidean basis can be
performed using the “change-of-basis” matrix (W to E) whose columns
40
are u, R2u,…., RN-2u,v, R2v,…,RN-2V. Further, the W to E “change-ofbasis” matrix is unitary.
2. Instead of performing the matrix multiplication, which requires N2
multiplications, a fast alternative is available by using the following
relationship [28].
•
Coefficients with respect to the basis elements: The first stage
wavelet coefficients are calculated using inner product of the
signal with the wavelet basis functions:
N −1
z = ∑ anu n , which in this case would be
n =0
( N 2 −1)
z=
∑
( N 2 −1)
z , R2 k u R2 k u +
k =0
•
∑
z , R2 k v R2 k v
k =0
Now, the following relation is used to take advantage of fast
computation using FFT.
z , R2 k v = z ∗ v (2k ) where v is the conjugate reflection
of v ( v (n) = (v(− n) ) ).
•
Thus the coefficients are calculated by taking DFT of the signal
and wavelet, multiplying them in the frequency domain and
finally throwing the odd indexed values.
The above mentioned procedure to calculate the first stage wavelet coefficients is
commonly called the filter bank arrangement; the operation that refers to throwing out the
odd sample is called downsampling. The following figure shows the block diagram
structure of a simple filter bank.
41
The block diagram in Figure 3.3.2 shows the reconstruction of the signal from the
coefficients. The downsampling operator in this case will be replaced by an upsampling
operator which adds a zero between adjacent values.
* v~
↓2
↑2
*~
t
x
x
* u~
↓2
↑2
*~
s
Figure 3.3.2 : Schematic representation of single stage wavelet coefficient extraction and signal
reconstruction
The coefficients obtained above can be classified into high frequency terms and low
frequency terms based on the basis function used in their calculation. Thus we have
obtained some degree of frequency localization. To obtain further frequency localization
we iterate the same procedure on the coefficients obtained from the convolution with low
pass filter u.
*v%1
x
*u%1
↓2
*v%2
↓2
*u%2
↓2
↓2
First Stage
Second Stage
Figure 3.3.3 : Schematic representation of multi stage wavelet decomposition
The block diagram form for obtaining the second stage wavelet coefficients iteratively
from the first stage basis is shown in Figure 3.3.3. If the length of the discrete time signal
is divisible by 2p, then we can repeat this process up to p times. The components obtained
through each stage will be
42
N N N
N N
+ + + ... + p + p = N
2 4 8
2
2
The decomposition and reconstruction equation for pth stage wavelet is presented along
with the block diagram. The wavelets used for convolving at different stages are
(
ul , vl ∈ l 2 Z N
2l −1
).
xl = DS ( DS (...( DS ( z * u1 ) * u2 )... * u p −1 ) * u p )
yl = DS ( DS (...( DS ( z * v1 ) * v2 )... * v p −1 ) * v p )
yl = (US ( yl −1 )) * ul −1 + (US ( xl −1 )) * vl −1
Where DS stands for the downsampling operator; US stands for upsampling operator.
As mentioned earlier the implementation of DWT is performed using the FFT algorithm.
The recursive structure though is not the most efficient or fastest way to implement
DWT. Since the pth stage wavelet coefficients can be calculated only if xp-1 & yp-1 are
available. Thus a parallel implementation in which the individual stages can be calculated
directly from the signal would be advantageous. An inductive derivation using the
wavelets upto the pth level is given below [28].
Define
f1 = v1 , g1 = u1
f l = g l −1 *U l −1 (vl )
g l = g l −1 *U l −1 (ul )
Where up & vp are the wavelets filter sequence at pth stage.
Subsequently, the nth stage DWT coefficients can be calculated by:
xl = DS l ( z * f l )
yl = DS l ( z * g l )
43
The following block diagram shows the parallel implementation. A more formalized
derivation consistent with the DWT terminology is presented below [Fig.3.3.4]:
* f%1
↓2
↑2
* f1
* f%2
↓4
↑4
* f2
*g% p
↓ 2p
↑ 2p
*g p
Figure 3.3.4 : Schematic representation of multi stage wavelet decomposition using a parallel
structure
Define,
ψ − j ,k = shift2 k ( f j )
j
ϕ − j ,k = shift2 k ( g j )
j
the pth stage wavelet basis generated by f1 , f 2 ,..., f p , g p has the form
{ψ }(
N / 2 )−1
−1, k k = 0
( N / 4 )−1
∪ {ψ }k =0
(N / 2 )−1
∪ ... ∪ {ψ }k =0
p
(N / 2 )−1
∪ {ϕ }k =0
p
Summarizing, the steps to perform DWT on a discrete signal of finite length N given that
we have a particular wavelet filter sequence upto the required p stages are:
1. The first step involves calculating the wavelet filter sequence upto p
stages. This involves choosing a particular wavelet and calculating the
44
coefficients in time domain through analytical expression or DFT of the
frequency domain representation.
2. Once the filter coefficients are know, f1 , f 2 ,..., f p , g p and ψ − j ,k , ϕ − j ,k as
defined earlier is calculated.
3. The final stage of calculating the DWT coefficients through convolution is
implemented by transforming to the frequency domain and multiplying the
DFT coefficients.
An example illustrating the above procedure is outlined using Haar Wavelets for
analyzing and reconstructing a sine wave (for clarity a only 3 stages are used):
1) The wavelet filter sequence for Haar wavelet is defined as:
Plot of Haar Wavelet Filter Sequence upto 3 stage
Low pass filter sequence phase 1
Low pass filter sequence phase 2
Filter Coefficients
0.8
0.8
0.6
0.6
0.4
0.4
Low pass filter sequence phase 3
2
1.5
1
0.5
0.2
0
0.2
0
2
4
6
8
high pass filter sequence phase 1
0
0
1
2
3
4
high pass filter sequence phase 2
-0.5
Filter Coefficients
1
1
0.5
0.5
0.5
0
0
0
-0.5
-0.5
-0.5
0
2
4
6
8
-1
1
2
3
4
1.2
-1
1
1.2
Figure 1.3.5 : Plot showing high pass and low pass Haar filter sequence upto stage 3
45
1.4
1.6
1.8
2
high pass filter sequence phase 3
1
-1
1
1.4
1.6
1.8
2
2) The f1 , f 2 ,..., f p , g p are calculated using the recursive formula mentioned earlier:
Filter coefficients
Low pass wavelet coefficients scale 1
Low pass wavelet coefficients scale 3
1
1
0.5
0.5
0.5
0
0
0
-0.5
-0.5
-0.5
-1
2
4
6
8
High pass wavelet coefficients scale 1
Filter coefficients
Low pass wavelet coefficients scale 2
1
-1
2
4
6
8
High pass wavelet coefficients scale 2
-1
1
1
0.8
0.8
0.8
0.6
0.6
0.6
0.4
0.4
0.4
0.2
0.2
0.2
2
4
6
8
0
2
4
6
8
4
6
8
High pass wavelet coefficients scale 3
1
0
2
0
2
4
6
8
Recursive wavelet coefficients at different scales
Figure 3.3.6 : Plot of f and g coefficient of Haar wavelets upto scale 3
3) The sine wave is analyzed and re-synthesized using a three stage Haar wavelet is
shown below.
46
First Stage Details D1
0.5
0
-0.5
1
2
3
4
5
6
7
8
3
4
5
6
7
8
3
4
5
6
7
8
3
4
5
6
7
8
3
4
5
6
7
8
Second Stage Details D2
0.2
0
-0.2
1
2
Third Stage Details D3
1
0
-1
1
2
First Stage Details D3
1
0
-1
1
2
Reconstructed signal
1
0
-1
1
2
Figure 3.3.7 : Demonstration of discrete Haar wavelet decomposition of a simple sine wave.
3.4 Continuous Wavelet Transform
One of the main difference between DWT & CWT, apart form the formulation, is that
DFT deals with vector spaces that are finite dimensional (finite & countable). The finite
dimensional vector spaces are in general easier to deal with, since they have a more
predictable measure & bound, functions and transforms can be easily characterized. The
following definitions help to characterize infinite dimensional vector spaces
( l 2 (Z), l 2 (R) ) and are subsequently used to develop CWT theory [28].
47
3.4.1 Convergence, completeness & Hilbert spaces:
∞
Let M ∈ Z . A sequence { xn }n = M in X converges to some x ∈ X if, for all arbitrary small
ε > 0 , there exist k ∈ N such that xn − x < ε ∀n > k .
A sequence
∞
{ xn }n = M
is a Cauchy sequence in X ∀ε > 0 , there is a k ∈ N such that
xn − xm < ε ∀n, m > k .
Hilbert Space: Using the above definitions, a complex inner product space V is said to be
complete if, every Cauchy sequence in V converges, such a vector space is also called
Hilbert Space. l 2 (Z), l 2 (R) are Hilbert spaces on which DFT and CWT are defined
respectively.
Continuous wavelet transform essentially seek to represent a given time domain function
in terms of certain elementary functions which have certain favorable properties. But,
unlike in l 2 (Z N ) a general function belonging to l 2 (Z), l 2 (R ) can never be represented
using a finite sum of elementary functions, due to the infinite dimension of these vector
spaces. Thus the following lemma [28] is useful in the case of infinite dimensional vector
spaces:
{vi }i∈Z
is a complete orthonormal set if and only if f = ∑ f , vi vi .
i∈Z
Therefore, in the case of l 2 (Z) although the set {ei }i∈Z ( as defined earlier) does not
constitute a basis in the vector space sense, it is possible to represent every element of
l 2 (Z) as an infinite linear combination of the set {ei }i∈Z .
Thus using the above definitions the first stage wavelet basis for l 2 (Z) can be defined.
48
Let u, v ∈ l 2 ( Z ) , then the set B
{Shift2 k v}k∈Z ∪ {Shift2 k u}k∈Z
is a first stage
wavelet basis for l 2 ( Z ) if B is complete orthonormal set in l 2 ( Z ) .
Using the filter bank arrangement,

 

z =  ∑ z , Shift2 k v Shift2 k v  +  ∑ z , Shift2 k u Shift2 k u 
 k∈Z
  k∈Z

and similar to l 2 ( Z N )
non-recursive vectors can be defined to enable parallel
implementation.
f l = gl −1 *U l −1 (v)
gl = gl −1 *U l −1 (u )
Since there is no finite number of elements that constitute the basis, an increasing
sequence of subspace is defined as follows [28]:


V−l +1 = ∑ z (k ) Shift2l −1 k gl −1 : z = ( z (k ) )k∈Z ∈ l 2 (Z) 
 k∈Z



V−l = ∑ z (k ) Shift2l k gl : z = ( z (k ) ) k∈Z ∈ l 2 (Z) 
 k∈Z



W− l = ∑ z (k ) Shift2l k f l : z = ( z (k ) )k∈Z ∈ l 2 (Z) 
 k∈Z

such that V−l ⊕ W−l = V− l +1 (Where ⊕ is direct sum of two the subspaces). Thus using an
inductive argument it can be show that:
V0 ⊂ V1 ⊂ V2 ⊂ ... ⊂ L2
and therefore, proceeding in the same way as before, a complete orthonormal system for
l 2 ( Z ) using the standard notation is as follows:
{ψ
− j ,k
: k ∈ Z,1 ≤ j ≤ p} ∪ {ϕ − p ,k : k ∈ Z}
49
The theoretical development for CWT on l 2 (R) is similar to that of l 2 ( Z ) , except that
l 2 (R) is both infinite and uncountable. Hence, the indices in the above expressions are
replaced by continuous variables.
A continuous time signal x(t) is decomposed into a family of functions which are
translations and dilations of a unique-valued function ψ (t ) . The wavelet transform is
defined in the continuous form as:
∞
WT ( x(t )t , s ) =
∫
x(τ )
−∞
1  (τ − t ) 
ψ
 dτ
s  s 
and the signal reconstruction equation is written as:
∞
∞
x(t ) = ∫ ∫ WT ( x(t )τ , s )
∞
0
1  (t −τ ) 
ψ
 dsdτ
s  s 
The main disadvantage of the above form of CWT is that its computationally difficult to
implement and also that it has a lot of redundant information, in a sense that scale and
time shift parameters can be discretized and still the function can be reconstructed
accurately. The formulation for discretization done on a dyadic scale is as follows:
Let ϕi , j ,ψ i , j ∈ L2 (R) , i, j ∈ Z , the wavelets are defined by,
ϕ i , j = 2 2 ϕ ( 2i x − j )
i
ψ i , j = 2 2 ψ ( 2i x − j )
i
The reconstruction equation is
x ( t ) = ∑∑ f ,ψ i , j ψ i , j
j∈Z i∈Z
50
However, the above summation accurately reconstructs the function only if the wavelet
family is orthogonal. More on this continues in the next section.
As an example CWT using Shannon and Morlet wavelets is presented below. The idea of
redundancy
in
information
and
filter
bank
arrangement
is
elaborated
Real Part of complex continous
Scale Shannon wavelet
1
0
-1
-20
-15
-10
-5
0
5
10
15
20
10
15
20
time
Imaginary part of complex continous
Scale Shannon wavelet
1
0
-1
-20
-15
-10
-5
0
5
time
Complex continuous scale Shannon Wavelet in Frequency Domain
150
Amplitude
100
50
0
-2
10
-1
10
0
1
10
10
2
10
3
10
Log(frequency)
log(time)
Figure 3.4.1 : Plot showing real and imaginary parts of continous shannon wavelets at integral scales.
The bottom subplot shows the overlapping filter bank structure.
51
Shannon Wavelet in Frequency Domain at Integral Scales
140
Amplitude
120
100
80
60
40
20
0
-2
10
-1
10
0
1
10
10
2
10
3
10
log(time)
Shannon Wavelet in Frequency Domain at Dyadic Scales
300
Amplitude
250
200
150
100
50
0
-2
10
-1
10
0
1
10
10
2
10
3
10
log(time)
Log(frequency)
Figure 3.4.2 : Comparitive plot of filter bank arrangement of shannon wavelet at integral and dyadic
scales. The bottom subplot shows leakage due to discretization.
52
Morlet Wavelet in Frequency Domain at Integral Scales
2.5
Amplitude
2
1.5
1
0.5
0
-1
10
0
10
1
2
10
10
3
10
4
10
log(time)
Morlet Wavelet in Frequency Domain at Dyadic Scales
1200
Amplitude
1000
800
600
400
200
0
-1
10
0
10
1
2
10
10
3
10
4
10
log(time)
Log(frequency)
Figure 3.4.3 : Plot of complex morlet wavelet filter bank at integral and dyadic scales.
3.5 Classification of Wavelet Functions
Wavelet functions can be classified into two categories as orthogonal and non-orthogonal
wavelets. Based on the theory presented above, it can be seen that basis elements in the
case of wavelet transform in l 2 (Z N ) have to be orthogonal to each other in order to
represent a given discrete signal accurately. Also, even in the case of infinite dimensional
vector spaces l 2 (Z ) , l 2 (R ) a complete orthonormal set is required. Thus orthogonal
wavelets offer perfect reconstruction, non-redundant decomposition, faster algorithms.
Some of the widely used orthogonal wavelets are Haar, Daubechies(Db), Coiflets,
Symlets(Sym) and Meyer. However there are some disadvantages in orthogonal wavelet
functions from a condition monitoring and diagnostic point of view:
53
1. The basis principle in choosing a wavelet function, for condition monitoring, is to
choose a function whose shape is similar to the vibration signal caused by the
machinery. Hence it is difficult to choose a proper wavelet function if the
orthogonality condition is to be satisfied.
2. The scale axis in a orthogonal wavelet transform are discretized on a dyadic grid.
Hence there is a possibility of missing the fault information on such a sparse grid.
3. The time translation variant nature of orthogonal wavelet allows transient at
different time locations to be represented differently.
To overcome the limitations of discretized orthogonal wavelet transform in condition
monitoring applications, continuous wavelet transform using the integral notation
presented earlier can be used. The family of wavelets is represented as:
ψ τ , s (t ) =
1  t −τ 
ψ

s  s 
Where s, t are the chosen scale and time shift parameters respectively.
The following admissibility condition has to be satisfied for a function ψ (t ) ∈ L2 ( R) to
qualify as a wavelet:
2
∞
ψˆ (ω )
Cψ = ∫
dω < ∞ , where ψˆ (ω ) is the Fourier transform of ψ (t ) .
ω
−∞
and the signal reconstruction equation can be written as:
x(t ) =
1
Cψ
∞∞
∫ ∫ WTτ
∞ 0
,s
( x(t ))
1
ψ ( s(t − τ ))dsdτ
s
The scale and time shift parameters can be discretized discretionarily to extract the fault
signal. Morlet, Mexican hat and Shannon wavelets are among the most widely used non-
54
orthogonal wavelets used in condition monitoring applications. The difference between
orthogonal wavelet transform and CWT is shown in the following figure:
Analyzed signal
2
1
0
-1
0
100
200
300
400
500
600
700
800
900
1000
800
900
1000
800
900
1000
level
Discrete Transform, absolute coefficients upto 7th stage.
100
200
300
400
500
600
700
Scale
Continuous Transform, absolute coefficients.
127
120
113
106
99
92
85
78
71
64
57
50
43
36
29
22
15
8
1
100
200
300
400
500
600
time (or space) b
700
Figure 3.5.1 : Comparative plot of discrete and continuous wavelet coefficient map of the test signal.
3.6 Gear Damage Detection Techniques
A number of gear damage detection techniques exist as indicated earlier. Vibration
monitoring is one of the most commonly used damage detection techniques. Vibration
analysis is performed using transducers to measure acceleration, velocity or
displacement. The choice of the transducer depends on the type of application and
frequencies involved in it:
1. Accelerometers are used for the frequency range of 0 – 20 kHz.
2. Velocity transducers work well in the frequency range of 2 Hz to 2 kHz.
55
3. Displacement transducers can be used for a frequency range of 0 to 200
Hz.
The signal measured using the transducer is said to be in the time domain. Thus vibration
measurement based condition monitoring techniques can be classified based on the
domain in which the signal analyzed.
3.6.1 Time Domain Averaging & Overall Power Level
This is a simple technique which involves averaging the time domain signal N number of
times and obtaining the peak, peak-to-peak or RMS values of the signal. The condition of
the gear is assessed from the value based on previous experiments and knowledge
systems. The synchronous time domain averaging not only removes background noise
but also removes non-periodic transient events. The main drawback of this method is that
it tends to be insensitive to large changes in amplitude for certain frequencies which may
occur only for a short period of time.
3.6.2 Kurtosis and Pulse Detection Methods
These methods assess the gear condition by analyzing the peakedness of the vibration
signal. Kurtosis is defined as the normalized form of the fourth central moment of a
distribution. In other words, kurtosis emphasizes the peaks in relation to the overall
average value of the signal. Kurtosis for a normally distributed data is as follows:
N
∑ (Y
i
Kurtosis =
i =1
4
−Y )
(N − 1)s 4
−3
Where Y is the mean of the data set, s is the standard deviation.
56
Shock Pulse is a method developed by SPM Instruments; shock pulse works by
comparing the amplitude of the spike against the back ground level. The gear condition is
determined from further statistical analysis.
3.6.3 Spectrum Analysis
Spectrum analysis is one of most commonly used and reliable diagnostic techniques.
Spectrum analysis breaks the time domain signal into individual frequencies by using
either an analog filter bank or using software. Spectral analysis is a very powerful
technique which relies on linking certain frequencies to particular components on the
machinery. The faults associated with the machinery are identified based on the
frequency content associated with them. The drawback with spectrum analysis from an
implementation point of view is that, it involves a lot of decision making and therefore
requires an expert to interpret results.
3.6.4 Cepstrum Analysis
Cepstrum analysis is a technique which identifies periodicities and repeated patterns in a
signal by calculating the Fourier transform of the spectrum of the signal. It is widely used
in homomorphic systems i.e. signals that comprise a wavelet and multiple echoes. In gear
diagnostic applications, overlapping causes confusion in signals whose spectrum contains
several side bands. Thus cepstrum analysis is used in such situation to detect and separate
families of sideband frequencies.
3.6.5 Envelope Detection
Envelope detection technique is used when the frequency range of interest is small
compared to frequency band width of the signal. Envelope detection works by
57
eliminating the low-frequency vibration, which include the structural properties and
resonances, then taking the frequency plot of the enveloped signal.
3.6.6 Time-Frequency Analysis
Time frequency analysis is becoming more and more important in gear damage detection
and it is also the main topic in this thesis. Time frequency methods describe
simultaneously the frequency content of a signal and how the frequency content varies
with time. The Short Time Fourier Transform (STFT) described before is an important
and widely used time frequency technique. The main drawback with STFT is that it gives
constant resolution in the time and frequency domains since the window width in the time
domain is fixed.
3.6.7 CWT Based Techniques
Recent advances in wavelet transform techniques have provided a powerful technique for
gear condition monitoring. Unlike in STFT, wavelet windows have different support at
different scales; therefore they use narrow time windows at high frequencies and wide
time windows at low frequencies. A number of fault detection algorithms have been
proposed [16] based on the wavelet transform. Visual inference can be made using
wavelet transform if there is not much interference between the features of interest. From
a practical implementation point of view visual inference from CWT plot is not intuitive.
The following feature extraction methods are commonly used for wavelet based
diagnostic decision making [20]:
1. Holder Exponent Analysis: Holder exponent is a measure of the regularity of the
signal. A highly continuous signal has a large value of holder exponent value,
58
while a discontinuous signal has a low value. The Holder regularity is defined as
follows:
If a signal f (t ) has a holder exponent α over [a, b] , then there exist A>0 such
that:
WT (u , s ) ≤ As
α + 12
∀(u , s ) ∈ [a, b]
WT (u, s ) is the modulus of wavelet transform for f (t ) . The exponent α is
calculated at any time by finding the log modulus of that time versus the log of
the scale vector s:
log WT (u, s ) = log( A) + (α + 1 2) log( s )
slope =
log WT (u, s )
log( s )
− 1/ 2 = α
Holder’s exponent is expected to be sensitive to damage that introduces high
frequency components into measured response. Further statistical processing will
be required for canceling measurement noise, otherwise inference made based on
Holder exponent could lead to false alarms.
2. Modulus Maxima: Modulus maxima method attempts to detect discontinuity by
analyzing the trend of the maxima of the modulus of wavelet transform. This
method was originally developed by Mallat [13], the definition given by them is:
•
A local extremum of the wavelet transform of f (t ) is any point
( s 0 , u 0 ) such that:
∂WT ( s 0 , u 0 )
=0
∂x
59
•
A
local
maximum
is
any
point
(s0 , u 0 )
such
that
WT ( s 0 , u ) < WT ( s 0 , u 0 ) when u belongs to either the right or the left
neighborhood of u 0 .
•
A maxima line is any connected curve in the scale-time space along
which all points are modulus maxima.
The defect detection criterion is based on the length of the maxima line, since
high frequency components induced by damage will create maxima at all scales.
3. Wavelet Variance: Wavelet Variance is the commonly used statistical variance
from a mean value applied to wavelet coefficients. This method assumes that any
damage will cause a change in the frequency content of the signal over time. The
standard deviation across the time span at each frequency is:
 n

 ∑ (WT ( f i , t i ) − W T ( f i ) 

ST ( f i ) =  i =1
n −1
2
where W T ( f i ) is the mean value of wavelet coefficients at a given frequency.
4. Time-Averaged Wavelet Spectrum (TAWS): TAWS is also another method
similar to wavelet variance which provides a good means to reduce the dimension
of the data. Time averaged wavelet spectrum is defined as
W (ai ) =
1
N
N −1
∑W
n
(ai )
2
n =0
where n is the time shift parameter, ai is the ith scale. TAWS was first introduced
and used as a fault detection technique by H.Zheng et.al [19].
60
5. Fault Scale Identification: Fault scale identification is based on the assumption
that it is possible to identify a unique scale associated with every fault feature of
interest. Such that the wavelet transform at that scale clearly indicates the location
of the fault feature. Fault scale identification relies on a knowledge base which
associates different defect conditions with their respective scales for a particular
sampling rate.
61
Chapter 4
4.1 Simulation Results
The torsional degree of freedom model discussed in Chapter Two is numerically
simulated in SIMULINK using the model shown below.
Figure 4.1.1 : Simplified version of SIMULINK model for numerical simulation of the torsional
degree of freedom model.
The blocks labeled MATLAB functions incorporate the non-linear parameters (stiffness
variation and profile error variation) involved in the simulation. The solution parameters
and system parameters are listed below:
62
Solution Parameters:
1. Time Resolution: 4096 points / pinion shaft rotation time period (variable).
2. Solver : Ode 5 fixed time step, 4th order Runge-Kutta solver.
Time varying system Parameter:
1. Meshing stiffness: K mesh , depends on contact ratio and affects the intertooth
meshing force (variable).
2. Profile error: Simulated distributed error in gear tooth due to pitting, wear and
tear.
Table 1: List of solution parameters and time varying system parameters
Other solution parameters are the lumped mass and stiffness of the system as listed in
Table 2 , which remain constant throughout the solution. The parameter list for the base
line measurements simulating the normal operating condition is tabulated below. The
values are chosen from the literature for validation purposes.
Parameter
Value
Stiffness
K 1 - Motor shaft stiffness = 3 e6 N m/rad
K 4 - Load shaft stiffness = 3.5 e7 N m/rad
Mass
I s , I 1 p , I 2 p , I l Inertia of motor, pinion, gear and load
I s = 12.2 kg m2
I1 p = 1 kg m2
I 2 p = 100 kg m2
Gear specifications
r1 , r2 radii of pinion and gear
radius
r1 = 0.122, r2 = 0.4933
63
Nominal profile error E1 = 20 e-6
amplitude
K mesh is function of time with max value 2.4 e9
Gear mesh stiffness
Pinion
Plot of Displacement of Pinion and Gear Vs Time
1500
Gear
1000
500
0
0
2
4
6
8
Time
Plot of Velocity of Pinion and Gear Vs Time
10
12
Pinion
150
Gear
100
Angular Acceleration
(radians/sec2)
Angular Velocity
(radians/sec)
Angular Displacement
(Radians)
Table 2: Parameter list for baseline measurement simulation
50
0
0
2
4
4
4
6
Time
8
10
Pinion
Plot of Acceleration of Pinion and Gear Vs Time
x 10
12
Gear
2
0
-2
0
2
4
6
Time
8
10
12
Figure 4.1.2: Plot of simulated displacement, velocity and acceleration signals of pinion and gear
using the parameters listed in table.
Figure 4.1.2 shows the angular displacement, velocity and acceleration of the gear and
pinion for the base line case. The gear box is driven by a constant torque motor, after the
initial start-up phase, based on the operating frequency and load, the gear box rotates at
constant average angular velocity. The pinion angular velocity at steady state operating
condition is shown in Figure 4.1.3.
The parameter list for the figure 4.1.3 shown below is:
64
Parameter
Value
Nominal stiffness variation
0.06
Error Condition
No Error
Nominal profile error variation
20 e-6
Table 3: Parameter set for Figure 4.1.3.
Plot of angular velocity of defective gear signal
Plot of Angularsuperimposed
Velocity of
onReference
reference signal signal
Angular Velocity
(Radians/sec)
1
Healty Signal
0.5
0
-0.5
Magnitude of Fourier Transform
(Radians/sec)
-1
0
4
2
4
6
8
10
12
Pinion Shaft Rotation Angle (Radians)
Frequency Spectrum of angular velocity of defective gear signal
Frequency Spectrum
of reference
angular velocity signal
superimposed
on reference signal
14
10
Healty Signal
2
10
0
10
-2
10
0
100
200
300
400
Frequency
500
600
700
Figure 4.1.3 : Time and frequency domain representation of simulated pinion angular velocity signal.
The signal represents the baseline condition with nominal tooth profile variation and tooth meshing
stiffness variation.
Harmonics of the gear meshing frequency are clearly seen along with the side bands. The
parameters for the solution are based on already available literature on gear dynamics for
validation of results. Figure 4.1.4 shows the base line signal for healthy gear signal with a
mounting eccentricity. The peak at shaft rotation frequency can be clearly seen.
65
Parameter
Value
Nominal stiffness variation
0.06
Error Condition
No error
Eccentricity amplitude
20 e-6
Profile error amplitude
20 e-6
Table 4: Parameter list for Figure 4.1.4
Plot of angular velocity of defective gear signal
Plot of angular velocity
of reference
gear signal
signal with eccentricity
superimposed
on reference
Angular Velocity
(Radians/sec)
1
Healty Signal
0.5
0
-0.5
Magnitude of Fourier Transform
(Radians/sec)
-1
0
2
Spectrum
4
4
6
8
10
Pinion Shaft Rotation Angle (Radians)
Frequency Spectrum of angular velocity of defective gear signal
of angular velocity
of reference
superimposed
on referencegear
signal signal with
12
14
eccentricity
10
Healty Signal
2
10
0
10
-2
10
0
100
200
300
400
Frequency
500
600
700
Figure 4.1.4 : Plot of simulated gear angular velocity signal with eccentric mounting conditon and
nominal tooth profile variation.
As mentioned earlier, the profile error and meshing stiffness are variables that will used
to simulate different error conditions in the gearbox.
4.1.1 Effect of Profile Error
As explained in chapter two, the profile error accounts for imperfections in the gear tooth
profile deviating from the involute profile. The base line case includes the profile error
66
with in the tolerance level. The profile error can also be used to simulate distributed
errors in the gear due to wear and tear. The simulation incorporates the profile error as a
linear function over the tooth meshing time period with random varying amplitude. The
following figure shows the effect of variation of amplitude of profile error function.
Parameter
Value
Nominal Stiffness Variation
0.06
Error Condition
No error
Eccentricity Condition
No eccentricity
1. 20 e-6 Nominal variation
Profile error amplitude
2. 20 e-5 High precision
Table 5: Parameter list for 4.1.5
Plot of angular velocity of pinion with varying error amplitude
High precision
Nominal
Angular Velocity
(Radians/sec)
1
0.5
0
-0.5
-1
0
2000
4000
6000
8000
10000
12000
14000
16000
18000
Magnitude of Fourier Transform
(Radians/sec)
Time
Frequency spectrum of angular velocity of pinion
4
10
2
10
0
10
-2
10
0
100
200
300
400
500
600
700
Frequency
Figure 4.1.5 : comparative plot of simulated gear angular velocity with different profile error
amplitudes.
67
It can be seen from the above figure that the frequency characteristic does not change
significantly based on amplitude of profile error function. Hereafter all simulated signals,
healthy and defective signals, will be based on the nominal profile error value. As
explained in Chapter Two, a defective gear tooth is modeled as stiffness reduction.
4.1.2 Effect of Eccentricity
The eccentric gear mounting condition is simulated by adding a harmonic term, with
shaft rotation frequency and amplitude based on level of eccentricity, to the profile error
function. Figure 4.1.5 shows the time and frequency representation of eccentric gear
mounting signal superimposed over the baseline case.
Parameter
Value
Nominal stiffness variation
0.06
Error Condition
No error
Eccentricity
1. No eccentricity
2. Eccentricity amplitude = 20 e-6
Table 6: Parameter list for Figure 4.1.6.
68
Plot of angular velocity of defective gear signal
superimposed on reference signal
Angular Velocity
(Radians/sec)
1
0.5
0
-0.5
-1
0
Magnitude of Fourier Transform
(Radians/sec)
Healty Signal
With Pinion and Gear Defect
2
4
10
4
6
8
10
12
14
Pinion Shaft Rotation Angle (Radians)
Frequency Spectrum of angular velocity of defective gear signal
superimposed on reference signal
Healty Signal
With Pinion and Gear Defect
2
10
0
10
-2
10
0
100
200
300
400
Frequency
500
600
700
Figure 4.1.6 : Comparative plot of gear angular velocity signal with eccentric mounting condition
superimposed over the baseline signal.
4.1.3 Effect on gear and pinion tooth errors
The effect of concentrated faults on gear and pinion tooth is modeled as a reduction in
stiffness. As discussed on Chapter Two, the ratio of reduction of stiffness depends on the
intensity of the fault. Figure 4.1.7 shows the effect of error on a single pinion tooth with
varying magnitude.
Error Condition
Value
No Error
Nominal stiffness variation = 0.06
5 % Error Advancement
Nominal stiffness variation = 0.06
Additional stiffness variation = 0.04
25 % Error Advancement
Nominal stiffness variation = 0.06
Additional stiffness variation = 0.1
69
Frequency Spectrum of angular velocity of defective gear signal
superimposed on reference signal
3
10
Baseline signal
5% Error
25% Error
2
Magnitude of Fourier Transform
(Radians/sec)
10
1
10
0
10
-1
10
-2
10
0
100
200
300
400
500
600
700
Frequency
Figure 4.1.7 : comparative plot of frequency domain representation of simulated gear angular
velocity signal under different error advancement stages.
The side bands due to the tooth fault can be clearly seen for the 25 % error case. But the
Fourier transform is limited in the inability to provide time information of the frequencies
and the inability to handle transient signals which limit its usage in damage detection
analysis. There are many methods in the literature which implement condition diagnostic
applications based on frequency data. Though many such attempts have been largely
successful, the results were found to be highly subjective. The following figures show
more comparison of different error scenarios, which indicate that the non-unique
frequency characteristic of different error conditions and hence diagnostic measures
based on frequency domain information often leads to false alarms.
70
4.1.4 Summary of simulation results and Fourier based diagnostic
measures
1. Simulation results agree well with available literature and will be used to develop
a real time wavelet transform based condition diagnostics.
2. FFT based methods are incapable of identifying transients due to the property of
fourier transforms.
3. Early faults will not generate significant side bands and hence error quantification
has not been successfully demonstrated using FFT based methods.
4. STFT cannot be successfully employed when the time window of the transient is
not know.
5. Both FFT and STFT based methods require significant post-processing.
4.2 CWT based diagnostic application demonstration
The following graphical user interface will be used to analyze the simulated data
developed in the previous section. Damage detection methods based on wavelet
transform techniques explained in Chapter Two is implemented in the GUI. The program
enables loading the simulated signal and setting options for continuous wavelet
transform. The program also gives options for further post processing and data reduction.
The following four figures gives an screen shots of the developed applications and a
demonstration of the steps involved in performing CWT based damaged detection.
71
Figure 4.2.1 : Front end of application developed to perform complex continuous wavelet transform of simulated gear signal. The option available for
wavelet and scale settings, along with the post processing tools are displayed.
72
Figure 4.2.2 : Step 1 - load test data , set analysis parameters.
73
Figure 4.2.3 : Step 2 - Choose data range for analysis
74
Figure 4.2.4 - Step 3 - Perform CWT analysis and explore available post processing tools.
75
4.3 Damage Detection based on visual interpretation
Figure 4.3.1 shows the wavelet transform of a healthy gear signal using Morlet wavelets.
This set of wavelet coefficients will be used as the base line data to correlate with various
damage cases.
Figure 4.3.1 - CWT coefficient of simulated pinion angular velocity signal. Wavelet - morlet, Scales 0 to 30, Error condition - baseline case.
The following figure 4.3.2 shows the wavelet spectrum of a defective gear signal with a
1/rev defect on the pinion gear. It can be seen from the figure 4.3.2 that scales 15 to 20
have high coefficient values and hence are more suitable for detecting and locating the
error. Further analysis will proceed by analyzing the scales of interest with finer
resolution.
76
Figure 4.3.2 - Plot of coefficients of morlet wavelet analysis for pinion angular velocity signal with
simulated error condition.
The wavelet spectrum shown in the next figure 4.3.3 represents a defective signal with
pinion and gear defects. The location of the defect gear tooth can be identified using a
tach signal. Using the simulation program explained above, the vibration signal for
different error conditions simulating varying levels of damage can be obtained. The
wavelet analysis program helps in identifying concentrated errors in pinion and gear
tooth.
77
Figure 4.3.3 - CWT coefficients of morlet wavelet analysis for pinion angular velocity signal with
1/tooth error on pinion and gear
Though the wavelet transform map provides detailed information on different errors
spectrums and their location in the time domain, the large amount data makes it
unsuitable for real-time implementation. Hence the following figures employ various data
reduction techniques explained in Chapter Three to quantify and locate gear tooth
defects. The following figure uses kurtosis, fourth central moment of a distribution to
quantify error conditions. As explained in Chapter Three, a comparative analysis of
results from all post processing methods have to be performed and the “best method” for
the defect of interest has to identified.
78
Kurtosis
Figure 4.3.4 - Plot of fourth central moment (kurtosis) of wavelet coefficients at individual scales.
4.4 Kurtosis
Kurtosis of a defective gear tooth is shown in figure.4.3.4. It can be seen in comparison
with the baseline signal that the deviation lies in the 20 to 25 scale range. Based on
further trails, it is seen that kurtosis can be used for detection of concentrated single tooth
errors. Based on the following figure 4.4.1, a relation can be formulated between the error
percentage and the amplitude of kurtosis at certain scales.
79
Comparitive plot of kurtosis of CWT coefficients
1000
Error Advancement = 5%
No Error
No Error
Error Advancement = 15%
Error Advancement = 25%
Error + Eccentricity
900
800
700
Kurtosis
600
500
400
300
200
100
0
0
5
10
15
20
25
30
35
40
Scales
Figure 4.4.1 - Comparative plot of kurtosis of pinion angular velocity under different simulated error
conditions. Legend (-. black) Error Advancement 25%, (-. Red) Error + Eccentricity, (-- Blue) Error
Advancement 15%, (- Black) No Error, (- Green) No Error, (- Red) Error Advancement 5%.
4.5 Time Averaged wavelet spectrum
The following figure 4.5.1 uses time averaged wavelet spectrum as a means of data
reduction. The results indicate that error can be quantified by comparing with baseline
data. Though TAWS offers fast implementation, based on literature, it is not well
equipped for novelty detection. Another apparent disadvantage is that, time resolution
will be lost and the insignificant response to different error conditions make it unsuitable
for real-time damage detection.
80
Plot of Time averaged CWT coefficients under different error conditions
0.7
0.6
Mean Amplitude
0.5
0.4
0.3
0.2
0.1
0
0
5
10
15
20
25
30
35
40
Scales
Figure 4.5.1 : Comparative plot of time averaged wavelet spectrum of pinion angular velocity under
different simulated error conditions. Legend (-- black) Error Advancement 25%, (-- Red) Error +
Eccentricity, (-- Blue) Error Advancement 15%, (- Black) No Error, (- Green) No Error, (- Red)
Error Advancement 5%.
4.6 Wavelet Variance
The following plot 4.6.1 shows wavelet variance of CWT coefficients. Results indicate
that wavelet variance can be used to successfully detect errors like pitting and broken
tooth. As is observed from Figure.4.6.1, a threshold detection routine, for different error
conditions, can be implemented based on maximum or average value of wavelet variance.
However, as is the case with most data detection, the time information will be lost at the
cost of significantly less processing time and less data storage requirements.
81
Variance of CWT coefficients under different error conditions
0.03
No Error
Error Advancement 5%
Error Advancement 10%
Error Advancement 20%
Error Advancement 25%
Error + Eccentricity
0.025
Variance
0.02
0.015
0.01
0.005
0
0
5
10
15
20
25
30
35
40
Scales
Figure 4.6.1 : Comparative plot of wavelet variance of pinion angular velocity signal under different
simulated error conditions. Legend (- Green) Error +Eccentricity, (-. Red) Error Advancement 25%,
(-- Red) Error Advancement 20%, (- Black) Error Advancement 10%, (- Red) Error Advancement
5%, (- Blue) No Error.
4.7 Single Scale identification
Figure 4.7.1 shows the idea behind fault scale identification, the problem scale for every
particular defect is identified in an iterative fashion. A comparative analysis of the
number of operation required using each the above discussed methods will presented
later. The figure presents analysis of the simulated signal at increasing scale refinement.
The successive range of scale and number of steps required can be estimated using visual
analysis.
82
CWT plot of defect signal at different scale refinements
10
6
5
Scale 10
Scale = 25
8
6
4
2
3
2
1
2000
4000
6000
8000 10000
TIme
12000
14000
0
0
16000
20
20
15
15
scale 0.2
scale 0.5
0
0
4
10
5
0
0
2000
4000
6000
8000 10000
Time
12000
14000
16000
2000
4000
6000
8000 10000
Time
12000
14000
16000
10
5
2000
4000
6000
8000 10000
Time
12000
14000
0
0
16000
Figure 4.7.1 : Plot showing the fault scale identification method on a defect signal. Legend (Top Left)
scale increment = 25, (Top Right) scale increment = 10, (Bottom Left) scale increment = 0.5, (Bottom
Right) scale increment = 0.2.
The following figure 4.7.2 shows the kurtosis of CWT coefficients of the defective
signal, the range of scales and scale refinement are estimated using fault scale
identification method. It can be observed from the figure that distinction between faulty
signal and healthy gear signals is easily identified, since the defective signals marked in
dashed lines is clearly separated from the healthy signals at multiple scales.
83
1
Normalized Kurtosis
0.95
0.9
0.85
No Error case 1
No Error case 2
0.8
No Error case 3
0.75
Erro Advancement 25 %
Error Advancement 15%
0.7
0.65
0
Error Advancement 25%
5
10
15
20
25
Scale
Figure 4.7.2 : Kurtosis of CWT coefficients of simulated pinion angular velocity signal.
4.8 Damage detection result summary
The above plots Figures 4.4.1-4.7.2 demonstrate some of the robust and low cost damage
detection methods that can be implemented using continuous wavelet transform. As
discussed in Chapter Three damage detection techniques require significant fine tuning
for specific applications. Hence thorough testing and troubleshooting has to be carried
out to improve operator confidence and damage detection rate.
Simulation and detection of concentrated errors: Based on the simulated signals the
following figure presents a road-map for developing CWT based condition diagnostic
utility.
84
System Knowledge
- Thorough understanding of system base signal and
expected defect conditions is required
- Knowledge of unique signal features is required.
Diagnostic requirements identification
- Identify critical operational requirements.
- Perform cost-benefits analysis.
- Investigate pros and cons of existing condition
diagnostic applications
Transient Signals
Defect localization
No
Conventional
techniques
Yes
CWT based diagnostic application
- Choose wavelets based on signal knowledge
- Perform trade-off analysis on favorable wavelets
and available fast implementation techniques.
Real time implementation
- Data reduction has to be employed to achieve realtime implementation
- Extensive training to be carried to achieve
improved detection rate.
It can be observed, based on the results of investigation undertaken in this thesis, that
each diagnostic method implemented here has its own merits and demerits. The fault
scale identification method gives better threshold variations compared to the rest of the
techniques. Although the CWT plot of the simulated signal requires handling large
85
amount of data, it is the most reliable and robust technique to detect and locate most
anomalies in the signal. The following table (Table 7) summarizes the merits and
demerits of CWT based diagnostic techniques.
Techniques
Merits
Demerits
Fault Scale
Highly Reliable
Amount of Data involved is
Identification
No post processing
comparatively high
required
Averaged Wavelet
Less memory requirements Need for extensive testing and good
Spectrum
knowledge base
Kurtosis
Fast Implementation
Results are subjective
Wavelet Variance
Less Memory requirements Increased processing time
Detailed post-processing required
Table 7: Table of merits and demerits of CWT based of analysis techniques.
86
Chapter 5
5.1 Conclusion
This research presents an integrated platform for performing simulation of various
operating conditions of a single gearbox and developing CWT based condition
monitoring algorithms. The detailed literature survey presents the state-of-the-art in both
gearbox simulation and CWT based condition monitoring algorithms. The SIMULINK
model enables simulation of different error conditions and as suggested in Chapter Two
the program can be easily extended to include additional degrees of freedom to closely
approximate any given test stand. The increased availability of accurate mathematical
models, which closely approximate actual test stand, makes simulation a viable option to
better understand the system dynamics and minimize experimental cost.
The mathematical theory of wavelets discussed in Chapter Three highlights the flexibility
and robustness inherent in the wavelet theory. The GUI developed for CWT analysis and
post processing enables performing several iterations back and forth between simulation
and diagnostic inference, thus creating a platform for implementation of generalized
condition detection algorithms. The discussion presented in Chapter Four indicates the
subjectivity of condition monitoring algorithms apart from discussing the merits and
demerits of CWT based condition monitoring. The results outline the relative
performance of the implemented damage detection methods for different error conditions,
including the newly developed error scale identification algorithm. The increasing
availability of large number of fast implementation algorithms makes CWT based
condition monitoring a viable and attractive alternative to existing frequency domain
based techniques.
87
5.2 Future Directions
As discussed in Chapter Two there are four classes of gear dynamic models, each with
their merits and demerits. Future gear dynamic models are expected to integrate the
existing dynamic models, thereby creating a unified overarching theory which accounts
for overall system dynamics as well as localized system changes. Future research in CWT
based damage detection will focus on the following three areas:
1. New wavelet development: As discussed in Chapter Three, any function which
satisfies a set of conditions can qualify as a wavelet. Hence a quantitative study to
optimize the choice of wavelet functions for specific applications could be
undertaken.
2. Neural network based knowledge systems: The data compression property of
wavelet transforms can be exploited to develop neural network based systems.
The system could be trained to make decisions based on analysis results of
various CWT based damage detection procedures.
3. Operational life prediction: Existing condition monitoring algorithms are largely
unsuccessful in predicting the available service life of the system; most systems
are taken out of operation well before they reach their maximum service life. The
time localization property of wavelets can be used to develop new wavelets that
can quantify damage propagation and thereby predict the service life.
4. Real-time implementation: As more and more industrial processes move towards
sophisticated automation, to achieve better quality and lower cost, the need for an
accurate real-time monitoring system becomes all the more important. The
88
robustness integral to CWT theory can be exploited to develop algorithms which
out perform existing frequency domain based techniques.
5. Integration with self healing materials: Self healing materials are being developed
at a rapid pace for aerospace structure. Real-time CWT based diagnostics can be
integrated with such structures to provide knowledge for an intelligent healing
mechanism.
6. Extension to generalized diagnostics: Medical diagnostics is one of the latest
areas in which applications based on CWT is being developed. Future CWT
applications can be expected to be robust enough to handle generalized systems
and signals.
7. Data mining and data compression: Data compression properties of wavelets
highlighted in Chapter Three have been used widely in image compression
applications. Future diagnostic systems can be expected to build knowledge bases
of various system characteristic based on wavelet data compression properties.
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an effective exact wavelet analysis”, Journal of Sound and Vibration, 277, 10051024.
25. Wang, W., Wong, A.K., (2002), “Autoregressive Model-Based Gear Fault
Diagnosis”, Transactions of the ASME, Vol.124, april 2002.
26. Le, Thien-Phu, Argoul, P, “Continuous wavelet transform for modal identification
using free decay response”, Journal of Sound and Vibration, 277 (2004), 73-100.
27. Frazier, M.W. (1999), “An Introduction to wavelets through linear algebra”, Book,
Springer, edition 1, ISBN: 0387986391.
92
Appendix
A.1 – Matlab code used in developing the CWT based
diagnostic tool
function varargout = CoeffPlotting(varargin)
% COEFFPLOTTING M-file for CoeffPlotting.fig
%
COEFFPLOTTING, by itself, creates a new COEFFPLOTTING or raises the
existing
%
singleton*.
93
%
%
H = COEFFPLOTTING returns the handle to a new COEFFPLOTTING or the
handle to
%
the existing singleton*.
%
%
COEFFPLOTTING('CALLBACK',hObject,eventData,handles,...) calls the local
%
function named CALLBACK in COEFFPLOTTING.M with the given input
arguments.
%
%
COEFFPLOTTING('Property','Value',...) creates a new COEFFPLOTTING or
raises the
%
existing singleton*. Starting from the left, property value pairs are
%
applied to the GUI before CoeffPlotting_OpeningFunction gets called. An
%
unrecognized property name or invalid value makes property application
%
stop. All inputs are passed to CoeffPlotting_OpeningFcn via varargin.
%
%
*See GUI Options on GUIDE's Tools menu. Choose "GUI allows only one
%
instance to run (singleton)".
%
% See also: GUIDE, GUIDATA, GUIHANDLES
gui_Singleton = 1;
gui_State = struct('gui_Name',
mfilename, ...
'gui_Singleton', gui_Singleton, ...
94
'gui_OpeningFcn', @CoeffPlotting_OpeningFcn, ...
'gui_OutputFcn', @CoeffPlotting_OutputFcn, ...
'gui_LayoutFcn', [] , ...
'gui_Callback', []);
if nargin && ischar(varargin{1})
gui_State.gui_Callback = str2func(varargin{1});
end
if nargout
[varargout{1:nargout}] = gui_mainfcn(gui_State, varargin{:});
else
gui_mainfcn(gui_State, varargin{:});
end
% End initialization code - DO NOT EDIT
% --- Executes just before CoeffPlotting is made visible.
function CoeffPlotting_OpeningFcn(hObject, eventdata, handles, varargin)
% This function has no output args, see OutputFcn.
% hObject
handle to figure
% Choose default command line output for CoeffPlotting
handles.output = hObject;
set(handles.pushbutton1,'enable','on');
95
handles.flag = 0;
% Update handles structure
guidata(hObject, handles);
set(handles.popupmenu1,'Enable','off');
set(handles.pushbutton1,'Enable','off');
set(handles.pushbutton2,'Enable','off');
set(handles.axes3,'Visible','off');
set(handles.axes4,'visible','off');
set(handles.popupmenu2,'Enable','off');
set(handles.pushbutton3,'Enable','off');
set(gcf,'Color',[0.925,0.914,0.847]);
% UIWAIT makes CoeffPlotting wait for user response (see UIRESUME)
% uiwait(handles.figure1);
% --- Outputs from this function are returned to the command line.
function varargout = CoeffPlotting_OutputFcn(hObject, eventdata, handles)
% varargout cell array for returning output args (see VARARGOUT);
% Get default command line output from handles structure
varargout{1} = handles.output;
% --- Executes on button press in pushbutton1.
function pushbutton1_Callback(hObject, eventdata, handles)
% hObject
handle to pushbutton1 (see GCBO)
% handles.counter = 30;
96
% guidata(hObject,handles);
i=1;
colormap(handles.axes2,'pink');
while(i<=handles.counter)
filename1 = sprintf('coeff_data_%03d.mat',i);
load(filename1,'coefs');
filename2 = sprintf('data_%03d',i);
load(filename2,'sig_Anal');
cdata = wcodemat(abs(coefs),128,'row',0);
axes(handles.axes2);
image('CData',cdata);
t_handle = title('Complex continuous wavelet spectrum of pinion vibration signal');
set(t_handle,'FontSize',14);
t_handle = xlabel('Time Points');
set(t_handle,'FontSize',14);
t_handle = ylabel('Scale');
set(t_handle,'FontSize',14);
axes(handles.axes1);
plot(sig_Anal);
t_handle = title('Plot of Pinion angular velocity');
set(t_handle,'FontSize',14);
t_handle = ylabel({'angular velocity','(rad/sec)'});
set(t_handle,'FontSize',14);
97
pause(0.001);
i=i+1;
end
% --- Executes on button press in pushbutton2.
function pushbutton2_Callback(hObject, eventdata, handles)
% hObject
handle to pushbutton2 (see GCBO)
scale_min = str2double(get(handles.edit1,'string'));
scale_max = str2double(get(handles.edit2,'string'));
scale_inc = str2double(get(handles.edit3,'string'));
i=1;
delete coeff_*.mat;
handles.h = waitbar(0,'Starting Iteration');
guidata(hObject,handles);
while(1)
% End USER DATA
fig_no = cwimtool;
% set(fig_no,'visible','off');
filename1 = sprintf('data_%03d',i);
filename2 = sprintf('coeff_data_%03d.mat',i);
iteration_no = sprintf('Computing Iteration No: %d',i);
% message = sprintf('%d',i);
% set(
waitbar(i/handles.counter,handles.h,iteration_no);
98
try,
cw1dmngr('load',fig_no,filename1);
catch,
break,
end
%Setting Wavelets
%-----------------------[Txt_Data_NS,Edi_Data_NS,Pop_Wav_Fam,Pop_Wav_Num,Pop_Lev] =
utanapar('handles',fig_no);
if(get(handles.popupmenu1,'value')==1)
strf = 'cmor';
waveno = 4;
elseif(get(handles.popupmenu1,'value')==2)
strf = 'gear';
waveno = 5;
end
set(Pop_Wav_Fam,'Value',waveno);
% handles = guihandles(fig_no);
tab = wavemngr('fields',{'fsn',strf},'tabNums');
set(Pop_Wav_Num,'String',tab,'Value',1);
%-----------------------%Setting Scales
99
%-----------------------handles1 = wfigmngr('getValue',fig_no,['CW1D_handles']);
set(handles1.hdl_UIC.edi_min,'string',num2str(scale_min));
set(handles1.hdl_UIC.edi_max,'string',num2str(scale_max));
set(handles1.hdl_UIC.edi_stp,'string',num2str(scale_inc));
cw1dmngr('anal',fig_no);
try,
cw1dmngr('save',fig_no,filename2);
catch,
err = 1;
end
% close all;
wfigmngr('close',fig_no);
i=i+1;
end
delete(handles.h);
% handles.counter = i;
% guidata(hObject,handles);
set(handles.pushbutton1,'enable','on');
function edit1_Callback(hObject, eventdata, handles)
% hObject
handle to edit1 (see GCBO)
% eventdata reserved - to be defined in a future version of MATLAB
guidata(hObject,handles);
100
% --- Executes during object creation, after setting all properties.
function edit1_CreateFcn(hObject, eventdata, handles)
% hObject
handle to edit1 (see GCBO)
if ispc && isequal(get(hObject,'BackgroundColor'),
get(0,'defaultUicontrolBackgroundColor'))
set(hObject,'BackgroundColor','white');
end
function edit2_Callback(hObject, eventdata, handles)
% hObject
handle to edit2 (see GCBO)
guidata(hObject,handles);
% --- Executes during object creation, after setting all properties.
function edit2_CreateFcn(hObject, eventdata, handles)
% hObject
handle to edit2 (see GCBO)
if ispc && isequal(get(hObject,'BackgroundColor'),
get(0,'defaultUicontrolBackgroundColor'))
set(hObject,'BackgroundColor','white');
end
% --- Executes on selection change in popupmenu1.
function popupmenu1_Callback(hObject, eventdata, handles)
101
% hObject
handle to popupmenu1 (see GCBO)
% --- Executes during object creation, after setting all properties.
function popupmenu1_CreateFcn(hObject, eventdata, handles)
% hObject
handle to popupmenu1 (see GCBO)
if ispc && isequal(get(hObject,'BackgroundColor'),
get(0,'defaultUicontrolBackgroundColor'))
set(hObject,'BackgroundColor','white');
end
function exit_1_Callback(hObject, eventdata, handles)
% hObject
handle to exit_1 (see GCBO)
selection = questdlg('Quit the program',...
'Confirm',...
'Yes','No','Yes');
switch selection,
case 'Yes',
delete(handles.figure1)
case 'No'
return
end
function interrupt_callback(handles)
delete(handles.h);
return;
102
function edit3_Callback(hObject, eventdata, handles)
% hObject
handle to edit3 (see GCBO)
% --- Executes during object creation, after setting all properties.
function edit3_CreateFcn(hObject, eventdata, handles)
% hObject
handle to edit3 (see GCBO)
if ispc && isequal(get(hObject,'BackgroundColor'),
get(0,'defaultUicontrolBackgroundColor'))
set(hObject,'BackgroundColor','white');
end
% -------------------------------------------------------------------function LoadData_Callback(hObject, eventdata, handles)
% hObject
handle to LoadData (see GCBO)
try,
delete data_*.mat;
delete coeff_*.mat;
No_of_pt = 16384;
% load('data','Velocity');
[filename,pathname] = uigetfile({'*.mat','Matlab Data File';'*.*','All Files';},'Load Data
File');
load(fullfile(pathname,filename));
figure(999);
103
set(999,'NumberTitle','off');
set(999,'Name','Graphical Input');
set(999,'WindowStyle','modal');
plot(Velocity(:,2));
xlabel('Time (Sec)');
ylabel('Pinion Angular Velocity (rad/sec)');
title('Graphical Input to choose data analysis range');
[X,Y] = ginput(2);
close 999;
signal = Velocity(floor(X(1)):floor(X(2)),2);
signal = signal-mean(signal);
No_of_blocks = floor(length(signal)/No_of_pt);
signal = signal(1:No_of_blocks*No_of_pt);
for i=1:No_of_blocks
sig_Anal = signal((i-1)*(No_of_pt)+1:i*No_of_pt);
filename = sprintf('data_%03d.mat',i);
save(filename,'sig_Anal');
end
catch,
errordlg('Error in Data Loading Operation','Error');
return;
end
handles.counter = No_of_blocks;
104
handles.flag = 1;
guidata(hObject,handles);
set(handles.popupmenu1,'Enable','on');
set(handles.pushbutton1,'Enable','on');
set(handles.pushbutton2,'Enable','on');
% --- Executes on selection change in popupmenu2.
function popupmenu2_Callback(hObject, eventdata, handles)
% hObject
handle to popupmenu2 (see GCBO)
% --- Executes during object creation, after setting all properties.
function popupmenu2_CreateFcn(hObject, eventdata, handles)
% hObject
handle to popupmenu2 (see GCBO)
if ispc
set(hObject,'BackgroundColor','white');
else
set(hObject,'BackgroundColor',get(0,'defaultUicontrolBackgroundColor'));
end
% --- Executes on button press in checkbox1.
function checkbox1_Callback(hObject, eventdata, handles)
% hObject
handle to checkbox1 (see GCBO)
% Hint: get(hObject,'Value') returns toggle state of checkbox1
if (get(hObject,'Value') == get(hObject,'Max'))
105
% then checkbox is checked-take approriate action
set(handles.popupmenu2,'Enable','on');
set(handles.popupmenu1,'Enable','off');
set(handles.pushbutton1,'Enable','off');
set(handles.pushbutton2,'Enable','off');
set(handles.pushbutton3,'Enable','on');
set(handles.axes1,'visible','off');cla(handles.axes1);
set(handles.axes2,'visible','off');cla(handles.axes2);
set(handles.axes3,'visible','on');
set(handles.axes4,'visible','on');
else
% checkbox is not checked-take approriate action
if(handles.flag)
set(handles.pushbutton1,'Enable','on');
set(handles.pushbutton2,'Enable','on');
else
set(handles.pushbutton1,'Enable','off');
set(handles.pushbutton2,'Enable','off');
end
set(handles.pushbutton3,'Enable','off');
set(handles.popupmenu2,'Enable','off');
set(handles.popupmenu1,'Enable','on');
set(handles.axes1,'visible','on');
106
set(handles.axes2,'visible','on');
set(handles.axes3,'visible','off');cla(handles.axes3);
set(handles.axes4,'visible','off');cla(handles.axes4);
end
% --- Executes on button press in pushbutton3.
function pushbutton3_Callback(hObject, eventdata, handles)
% hObject
handle to pushbutton3 (see GCBO)
[Filename,Pathname] = uigetfile({'*.mat','signal Data File (*.mat)'},'Load signal');
signal = load(fullfile(Pathname,Filename));
[Filename,Pathname] = uigetfile({'*.mat','Coefficients Data File (*.mat)'},'Load signal');
coeff = load(fullfile(Pathname,Filename));
axes(handles.axes3);
plot(signal.sig_Anal);
t_handle = title('Pinion Angular Velocity');
set(t_handle,'FontSize',14);
t_handle = xlabel('Time Pts');
set(t_handle,'FontSize',14);
switch(get(handles.popupmenu2,'Value'))
case 1,
axes(handles.axes4);
plot(kurtosis(abs(coeff.coefs).'));
t_handle = title('Kurtosis');
set(t_handle,'FontSize',14);
107
t_handle = xlabel('Scale');
set(t_handle,'FontSize',14);
case 2,
axes(handles.axes4);
plot(mean(abs(coeff.coefs),2).');
t_handle = title('Time averaged wavelet spectrum');
set(t_handle,'FontSize',14);
t_handle = xlabel('Scale');
set(t_handle,'FontSize',14);
case 3,
axes(handles.axes4);
plot(var(abs(coeff.coefs),0,2).');
t_handle = title('Wavelet Variance');
set(t_handle,'FontSize',14);
t_handle = xlabel('Scale');
set(t_handle,'FontSize',14);
end
108
109
A.2 – SIMULINK model used to solve the non-linear system of equations
110
A.3 – MATLAB function simulating the varying system
parameters
% damping calculator
% Raghavendran Raghunathan
% Function requird to run simulation
function damping_vel = stiffness_calc(disp_vel_para);
Qc1 = 1000;
Qc2 = 56;
Qmb = 10000;
rp = 0.122;
rg = (93/23)*0.122;
Q = [ Qc1 -Qc1 0 0;
-Qc1 Qc1+rp^2*Qmb -rp*rg*Qmb 0;
0 -rp*rg*Qmb Qc2+rg^2*Qmb -Qc2;
0 0 -Qc2 Qc2];
damping_vel = Q*disp_vel_para(5:8);
return;
% Error Simulation code
% Raghavedran Raghunathan
% This code is required to run the simulation
function err_count = Error1(disp)
a = 0.5;
111
E = 20e-6;
E1 = 20e-6;
N = 23;
r = 0.3;
global VALUE_REAL;
global INDEX_INTEGER;
theta_shaft1 = mod(disp(2)/(2*pi),1);
% if(abs(disp(1)/(2*pi/N) - INDEX_INTEGER) < 1e-5)
%
VALUE_REAL = rand(1);
%
INDEX_INTEGER = INDEX_INTEGER +1;
% end
if((mod(disp(1),(2*pi/N)) - 0) < 1e-3)
VALUE_REAL = rand(1);
end
amp = (1 - r*(1-VALUE_REAL));
theta = mod(disp(2)*N/(2*pi),1);
if(theta<0)
err_count = 0;
elseif(theta>=0 && theta<=a)
err_count = amp*E/a*theta;
elseif(theta>a && theta<=1)
err_count = amp*E/(1-a)*(1-theta);
end
112
if(disp(5)>13&&disp(5)<=13)
err_count = err_count + E1*sin(2*pi*theta_shaft1);
else
err_count = err_count;
end
% err_count = 0;
return;
% force calculator
% Raghavendran Raghunathan
% Code required to run simulation
function force = force_calc(disp_vel_para);
Mn = 6150;
% Mn = 100;
N = 23;
rp = .122;
rg = (93/23)*0.122;
Qmb = 10000;
% ln = 300e-6;
ln = 0;
pom1 = mod(disp_vel_para(2)*N/(2*pi),1);
113
if(disp_vel_para(5)<=pi*98/3)
MSpeed = 2*Mn;
elseif(disp_vel_para(5)>pi*98/3 && disp_vel_para(5)<=pi*100/3)
MSpeed = 2*Mn*(pi*100/3 - disp_vel_para(5))*3/(2*pi);
else
MSpeed = 0;
end
% if(disp_vel_para(5)<=pi*98/3)
%
MSpeed = 2*Mn;
% else
%
MSpeed = 2*Mn*(pi*100/3 - disp_vel_para(5))*3/(2*pi);
% end
% if(disp_vel_para(5)<=56)
%
MSpeed = 2*Mn;
% else
%
MSpeed = 2*Mn*(58.4 - disp_vel_para(5))*3/(2*pi);
% end
% Force_temp1 = disp_vel_para(9)*max((rp*disp_vel_para(2)rg*disp_vel_para(3)+disp_vel_para(10)-ln),...
%
min( (rp*disp_vel_para(2)-rg*disp_vel_para(3)+disp_vel_para(10)+ln),0));
Force_temp1 = disp_vel_para(9)*(rp*disp_vel_para(2)rg*disp_vel_para(3)+disp_vel_para(10));
Force_temp = Force_temp1 + Qmb*(rp*disp_vel_para(6) - rg*disp_vel_para(7));
114
force = [MSpeed,-disp_vel_para(11)*abs(Force_temp)rp*disp_vel_para(9)*disp_vel_para(10),...
+disp_vel_para(12)*abs(Force_temp)+rp*disp_vel_para(9)*disp_vel_para(10),disp_vel_para(13)];
return;
% Moment due to friction calculator
% Raghavendran Raghunathan
% Code required to run simulation
function [moment] = FricMoment(disp)
i = 2.69;
N = 23;
rp = .122;
okr = 2*pi*rp/N;
mu = 0.02;
% okr = 0;
theta1 = mod(disp(2)*N/(2*pi),1);
if(theta1<0)
momentp = 0;
elseif(theta1>=0 && theta1<=1/i)
momentp = okr*mu*theta1;
elseif(theta1>1/i && theta1<=1)
momentp = -okr*mu*theta1;
115
end
if(theta1<0)
momentg = 0;
elseif(theta1>=0 && theta1<=1/i)
momentg = okr*mu*(1-theta1);
elseif(theta1>1/i && theta1<=1)
momentg = -okr*mu*(1-theta1);
end
moment = [momentp;momentg];
return;
%Load Calculator
% Raghavendran Raghunathan
% Code required to run simulation
function tn = load1(t);
if(t<=1)
tn = 0;
elseif(t>1 && t<=5)
tn = (24867/4)*(t-1);
else
tn = 24867;
end
return;
116
%mass calculator
% Raghavendran Raghunathan
% Code required to run simulation
function mass_acc = mass_calc(input)
IMotor = 12.2;
ILoad = 100;
IPinion = 1;
IGear = 200;
if(input(5)<=1)
mass = [ IMotor 0 0 0;
0 IPinion 0 0;
0 0 IGear 0;
0 0 0 ILoad];
else
mass = [ IMotor 0 0 0;
0 IPinion 0 0;
0 0 IGear 0;
0 0 0 86010];
end
mass_acc = inv(mass)*input(1:4);
return;
117
% Stiffness calculator\
% Raghavendran Raghunathan
% Code required to run simulation
function stiffness_disp = stiffness_calc(disp_vel_para);
Kc1 = 3e6;
Kc2 = 3.5e7;
Kmb = disp_vel_para(9);
rp = 0.122;
rg = (93/23)*0.122;
K = [ Kc1 -Kc1 0 0;
-Kc1 Kc1+rp^2*Kmb -rp*rg*Kmb 0;
0 -rp*rg*Kmb Kc2+rg^2*Kmb -Kc2;
0 0 -Kc2 Kc2];
stiffness_disp = K*disp_vel_para(1:4);
return;
% Meshing stiffness variation calculator
% Raghavendran Raghunathan
% Code required to run simulation
function stiff = Stiffness(input)
cs = 2.4e9; a = 0.53; b = 0.65; c = 0.92;
N = 23;
theta = mod(input(2)*N/(2*pi),1);
118
theta_shaft1 = mod(input(2)/(2*pi),1);
theta_shaft2 = mod(input(3)/(2*pi),1);
g1 = 0;g2=0;
if(input(5)>7&&input(5)<=9)
if(theta_shaft1>=0 && theta_shaft1<=1/30)
g1 = 0.06;
else
g1 = 0.0;
end
elseif(input(5)>9&&input(5)<=12)
if(theta_shaft1>=0 && theta_shaft1<=1/30)
g1 = 0.06;
else
g1 = 0.0;
end
if(theta_shaft2>=5/93 && theta_shaft2<=5/93+0.01)
g2 = 0.06;
else
g2 = 0.0;
end
end
g = 0.06+g1+g2;
if(theta<0)
119
stiff = 0;
elseif(theta>=0 && theta<=a)
stiff = cs*(1-g);
elseif(theta>a && theta<=b)
stiff = cs*(1+g*(theta-b)/(b-a));
elseif(theta>b && theta<=c)
stiff = cs;
elseif(theta>c && theta<=1)
stiff = cs*(1-g*(theta-c)/(1-c));
end
% stiff = cs;
return;
120