UNIVERSITY OF CINCINNATI
12/27/2007
Date:___________________
Ravi Kumar Mantrala
I, _________________________________________________________,
hereby submit this work as part of the requirements for the degree of:
Master of Science
in:
Department of Mechanical Engineering
It is entitled:
Squeak and Rattle Detection : A Comparative Experimental Data Analysis
This work and its defense approved by:
Dr. Randall J Allemang
Chair: _______________________________
Dr. Allyn Philips
_______________________________
Dr. Jay Kim
_______________________________
_______________________________
_______________________________
Squeak and Rattle Detection: A Comparative Experimental Data Analysis
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 Engineering
of the College of Engineering
2007
by
Ravi Kumar Mantrala
B Tech (Mechanical Engineering), Jawaharlal Nehru Technological University (JNTU),
Hyderabad, India
Committee Chair: Dr. Randall J. Allemang
Committee Member: Dr. Allyn Phillips
Committee Member: Dr. Jay Kim
iv
Abstract
Squeak and rattle evaluation is a common problem faced by automotive OEM’s. With
increased importance to driving comfort and quality perception, squeak and rattle detection and
elimination in modern automotive systems has become much more important in recent years.
Many techniques involving time-frequency analysis, acoustics, digital signal processing, etc. are
used to understand and control this unpredictable and undesirable vibro-acoustic phenomenon.
An extensive literature survey was performed to gain understanding of the current state of the art.
Detailed progression from the Fourier transform to the modern wavelet transform is also
documented. This thesis is an attempt to perform a comparative analysis on experimental
acoustic data collected on four, completely trimmed vehicles. These cars were tested on the fouraxis, hydraulic road simulator at the Non-linear Testing Facility at the University of Cincinnati,
Structural Dynamics Research Lab (UC-SDRL). The data was then analyzed using timefrequency techniques such as short time Fourier transform (STFT) and also with the advanced,
complex Morlet wavelet technique. Both energy and amplitude normalization was performed on
the Morlet wavelet and effects studied on data. The objective of the analysis was to detect and
localize (in time and frequency) squeaks and rattles in automobiles from digital data recorded on
the microphones when the hydraulic actuators were exciting the vehicle with certain signals.
Understanding the applicability of the Morlet wavelets to the study of random bursts of acoustic
energy and to observe their effectiveness in comparison to the standard STFT was also an
important part of the study. Analysis of the data resulted in the conclusion that Morlet wavelets
might not offer great advantage over the standard STFT in the process of detection of squeaks
and rattles for automotive systems.
iii
iv
Acknowledgement
Firstly, I would like to thank Dr. Randall Allemang for his esteemed guidance and
encouragement through out the project. He was always there to help me out, be it understanding
concepts or guidance on experimental set up.
I would like to thank Dr. Allyn Phillips and Dr. Jay Kim for serving on the thesis
committee.
Dr. Phillips also guided and helped the team with troubleshooting the data
acquisition equipment, acquisition software and with in the wavelet code. His help and guidance
is greatly appreciated.
I thank the Ford project team members, who have become good friends over the period of
the project, Matt Allemang, Abbey Yee and Ryota Jinnai for all the great times during the long
set up and testing procedures. I would also like to thank Mr. Tim Mouch and Mr. Amiyo Basu of
Ford Motor Company for having funded the project through the Ford Foundation.
I would like to thank all my friends at the University of Cincinnati for having made every
moment of the two odd years at the university special and cherish able.
Last but not least, I am immensely indebted to my family back home. Nothing could have
been possible without their support.
v
Table of Contents
Abstract…………………………………………………………………………………………...iii
Acknowledgement………………………………………………………………………………...v
Table of Contents…………………………………………………………………………………vi
List of Figures……………………………………………………………………………………vii
List of Tables……………………………………………………………………………………...x
Chapter 1 - Introduction
1.1 Motivation………………………………………………………………………………...1
1.2 Problem Definition………………………………………………………………………..2
1.2.1 Squeak and Rattle - Theory and Background……………………………………2
1.2.2 Sound generation mechanism……………………………………………………3
1.3 Research Goals……………………………………………………………………………4
1.4 Thesis Organization………………………………………………………………………4
1.5 Squeak and Rattle – Early Developments and Literature………………………………...5
Chapter 2 - Time - Frequency analysis concepts
2.1 Digital Signal Processing considerations……………………………………………….13
2.1.1 Analog versus Digital Data……………………………………………………….13
2.1.2 Shannon’s Sampling Theorem……………………………………………………13
2.1.3 Rayleigh’s Criterion………………………………………………………………14
2.1.4 Digital Signal Processing Errors………………………………………………….14
2.2 Time Frequency Transform…………………………………………………………….15
2.3 Orthogonal and Non-Orthogonal Basis Functions……………………………………...15
2.4 Fourier Transform………………………………………………………………………16
2.5 Wavelet Transform Theory……………………………………………………………..22
2.5.1 Need for Wavelet Transform……………………………………………………..22
2.5.2 Wavelets…………………………………………………………………………..23
2.5.3 Continuous Wavelet Transform (CWT)………………………………………….23
2.5.4 Scaling and Translation…………………………………………………………..25
2.5.5 Computation of CWT…………………………………………………………….28
2.5.6 Computation of CWT using DFT………………………………………………...30
2.5.7 Wavelet Normalization…………………………………………………………...31
2.5.8 Redundancy in CWT……………………………………………………………...32
2.5.9 Discrete Wavelet Transform (DWT)……………………………………………..32
2.5.10 Wavelets as Band Pass Filters…………………………………………………...33
2.5.11 Concept of Sub-Band Coding…………………………………………………...34
2.5.12 Computation of DWT…………………………………………………………...36
2.6 Choice of Wavelet Properties…………………………………………………………..37
2.6.1 Discrete or Continuous Transform………………………………………………..37
2.6.2 Choice of Wavelet………………………………………………………………...37
vi
2.6.3 Complex Morlet Wavelet…………………………………………………………39
2.7 Parameter Discussion…………………………………………………………………...41
2.7.1 Wavenumber……………………………………………………………………...41
2.7.2 Wavelet Scale and Fourier Frequency …………………………………………...43
2.8 Edge Error………………………………………………………………………………44
2.9 Summary of Methods…………………………………………………………………...44
Chapter 3 - Testing and data acquisition
3.1 Testing ………………………………………………………………………………….46
3.1.1 Need for Testing…………………………………………………………………..46
3.1.2 Testing Methodologies……………………………………………………………46
3.2 Test Facility…………………………………………………………………………….47
3.2.1 Ideal Testing Conditions…………………………………………………………..47
3.2.2 Actual Testing Conditions ………………………………………………………..48
3.3 Test Set up………………………………………………………………………………50
3.3.1 General Configuration…………………………………………………………….50
3.3.2 Validation Configuration………………………………………………………….54
3.3.3 Sensor Calibration…………………………………………………………………56
3.3.4 Troubleshooting………………….………………………………………………..56
3.3.5 Data………………………………………………………………………………..57
Chapter 4- Analysis and Results
4.1 Choice of Scales………………………………………………………………………..58
4.2 MATLAB Code………………………………………………………………………..59
4.3 Display…………………………………………………………………………………59
4.4 Block and Overlap Processing…………………………………………………………60
4.5 Wavelet Filter Banks…………………………………………………………………...62
4.6 Code Validation ……………………………………………………………………….71
4.7 Results………………………………………………………………………………….81
4.8 Holder Exponent Analysis……………………………………………………………101
Chapter 5- Conclusion and Future Work
5.1 Conclusion…………………………………………………………………………….102
5.2 Retrospection………………………………………………………………………….102
5.3 Future Work…………………………………………………………………………...103
5.4 Possible Application in Qualitative Non-Stationary Sound Analysis…………………104
Bibliography
Appendix
A Wavelet analysis code
B Inner product, Orthogonality and Orthonormality
vii
List of Figures
Figure 2.1 Relation between time resolution and frequency domain……………………………………..13
Figure 2.2 Relation between frequency resolution and time domain……………………………………..14
Figure 2.3 (a): Time Frequency resolution in a STFT plot ……………………………………………….19
Figure 2.3 (b): Heisenberg representation of the Fourier Basis…………………………………………...19
Figure 2.3 (c): Idealized Heisenberg Time-frequency representation of STFT (Left) result from wider time
window (Right) result from shorter time window…………………………………………………...........20
Figure 2.4: Effect of scaling on a Morlet wavelet in the T-F plane……………………………………….26
Figure 2.5: Ideal TF plane decomposed with wavelets on a Dyadic grid…………………………………27
Figure 2.6(a): Cosine curve illustrating the concept of scale……………………………………………...27
Figure 2.7(a): CWT computation at a low scale…………………………………………………………. 29
Figure 2.7(b): CWT computation at a higher scale………………………………………………………..29
Figure 2.8: Band pass filter bank………………………………………………………………………….34
Figure 2.9: Figure illustrating the scaling function………………………………………………………..34
Figure 2.10: Illustration of sub-band coding………………………………………………………………35
Figure 2.11: Time and frequency domain representations of Morlet, Paul, DOG wavelets………………39
Figure 2.12(a): Complex Morlet in time domain………………………………………………………….40
Figure 2.12(b): Frequency domain equivalent of Complex Morlet……………………………………….40
Figure 2.13: Frequency and time domain representations of morlets of different k_0 values……………42
Figure 2.14: Effect of increasing wavenumber……………………………………………………………43
Figure 3.1 Test Set up……………………………………………………………………………………..49
Figure 3.2 (from L to R) Sensors Microphone with preamp, tri-axial accel, uni-axial accel …………….50
Figure 3.3 (a): Instrumentation………………………………………………………………………..53, 54
Figure 3.3 (b): (left) VXI system, (middle) wheel fastened to actuator pan, (right) dummies with
microphones (circles indicate microphone position)…………………………………………………….. 54
Figure 3.3 Speaker Set ……………………………………………………………………………………55
Figure 3.4 Speaker taped to foot well wall………………………………………………………………..55
Figure 4.1: Sample plot; Vehicle ‘E’, excitation (100%), ω 0 =50, δj =0.18, f = 100 – 5000Hz………….60
Figure 4.2: Illustration of flow of block processing for edge effect removal……………………………..61
Figure 4.3: Putting data together, averaging clear data and removal of affected data…………………….62
Figure 4.4(a): scale 2 = 0.0011……………………………………………………………………………63
Figure 4.4(b): scale 5 = 0.0017……………………………………………………………………………63
Figure 4.4(c): scale 10 = 0.0035…………………………………………………………………………..63
Figure 4.4(d): scale 20 = 0.0139…………………………………………………………………………..64
Figure 4.4(e): scale 30 = 0.0554…………………………………………………………………………..64
Figure 4.4(f): Filter bank concept with energy normalization (overlay of all scales)…………………….64
Figure 4.4(g): Filter bank concept with amplitude normalization (overlay of all scales)…………………65
Figure 4.5(a): Example of energy normalized wavelet plot with corresponding wavelet filter bank……..66
Figure 4.5(b): Example of wavelet plot and filter bank with amplitude normalized to 1…………………66
Figure 4.5(c): δj =0.25…………………………………………………………………………………….67
Figure 4.5(d): δj =0.3……………………………………………………………………………………..68
Figure 4.5(e): ω 0 =50……………………………………………………………………………………...68
Figure 4.5(f): ω 0 =10……………………………………………………………………………………...69
Figure 4.5(g): ω 0 =10, δj =0.2 f = 100 – 6000Hz…………………………………………………………70
Figure 4.5(h): Amplitude normalized, ω 0 =10, δj =0.2 f = 100 – 6000Hz………………………………..70
Figure 4.6 (a): Spectrogram of the ‘low frequency’ chirp; 200 – 1100Hz………………………………..71
viii
Figure 4.6(b): No shaker excitation (0%), ω 0 =50, t=1-5sec……………………………………………..72
Figure 4.6(c): Shaker excitation (100%), ω 0 =10, t = 15-20sec…………………………………………..74
Figure 4.6(d): Shaker excitation (100%), ω 0 =50, t=15-20sec……………………………………………74
Figure 4.6(e): Amplitude normalized, Shaker excitation (100%), ω 0 =10, t=15-20sec…………………..75
Figure 4.7(a): Spectrogram of the ‘medium frequency’ chirp; 3100 ~ 3700Hz…………………………..76
Figure 4.7(b): δj =0.2……………………………………………………………………………………..77
Figure 4.7(c): δj =0.1……………………………………………………………………………………...77
Figure 4.8(a): Spectrogram of the ‘high frequency’ chirp; 6100 ~ 8500Hz………………………………78
Figure 4.8(b): δj =0.25……………………………………………………………………………………79
Figure 4.8(c): δj =0.1……………………………………………………………………………………..79
Figure 4.8(d): Amplitude normalized, δj =0.1……………………………………………………………80
Fig 4.8 (e): AWT plot of the ‘high frequency’ chirp……………………………………………………...80
Figure 4.9(a): Shaker excitation (100%); ω 0 =50, t = 13-16sec…………………………………………..82
Figure 4.9(b): Shaker excitation (100%); ω 0 =10, t=13-16sec……………………………………………83
Fig 4.9 (c): AWT plot …………………………………………………………………………………….83
Figure 4.9(d): STFT, shaker excitation (100%), t= 13-16sec……………………………………………. 84
Figure 4.9(e): Shaker excitation (100%); ω 0 =35, t = 13-16sec…………………………………………..84
Figure 4.9(f): Shaker excitation (75%); ω 0 =35, t=13-16sec……………………………………………...85
Figure 4.9(g): Shaker excitation (50%); ω 0 =35, t =13-16sec…………………………………………….85
Figure 4.9(h): Shaker excitation (50%); ω 0 =30, t=16-21 sec…………………………………………….86
Figure 4.9(i): Shaker excitation (100%); ω 0 =30, t=16-21sec…………………………………………….87
Figure 4.9(j): STFT, shaker excitation (100%); t=16-21sec………………………………………………87
Figure 4.10(a): Shaker excitation (100%)…………………………………………………………………88
Figure 4.10(b): Shaker excitation (75%)…………………………………………………………………..89
Figure 4.10(c): Shaker excitation (50%)………………………………………………………………….90
Figure 4.10(d): STFT, shaker excitation (100%), t=11-16sec…………………………………………….91
Figure 4.11(a): Shaker excitation (100%); ω 0 =30, t=12-16sec…………………………………………..92
Figure 4.11(b): STFT, shaker excitation (100%), t = 12 – 16sec…………………………………………93
Figure 4.11(c): Shaker excitation (100%); ω 0 =10, t=11-16sec…………………………………………..93
Figure 4.11(d): Shaker excitation (100%); ω 0 =10, t=16-21sec…………………………………………..94
Figure 4.11(e): Shaker excitation (75%); ω 0 =10, t=11-16sec……………………………………………94
Figure 4.11(f): Shaker excitation (50%); ω 0 =10, t=11-16sec…………………………………………….95
Figure 4.11(g): STFT, shaker excitation (50%), t=11-16sec……………………………………………...96
Figure 4.12(a): Vehicle B, t=1-4sec……………………………………………………………………….97
Figure 4.12(b): Vehicle E, t=10-15sec…………………………………………………………………….98
Figure 4.12(c): Amplitude normalized, Vehicle B, t=1-4sec……………………………………………...99
Figure 4.12(d): STFT, Vehicle B, t=1-4sec……………………………………………………………….99
Figure 4.12(e): Amplitude normalized, Vehicle E, t=10-15sec………………………………………….100
Figure 4.12(f): STFT, Vehicle E, t=10-15sec……………………………………………………………100
Figure 4.13(a): Vehicle D, Random, shaker excitation (100%); ω 0 =50, δj =0.25, f = 1000 – 10000Hz,
t=33-34sec………………………………………………………………………………………………..103
ix
List of Tables
Table 3.1 Instrumentation Summary………………………………………………………………………50
Table 3.2 Sensor distribution……………………………………………………………………………...51
Table 3.3: Summary of data collected on vehicles A, B, C, D and E on MTS 320 road simulator……….58
Table 4.1: Noise ratings inside the vehicle………………………………………………………………..73
x
Chapter 1: Introduction
1.1 Motivation:
In recent years, the automotive industry has seen a gradual increase in the level of ride
comfort demanded by customers. The interpretation of ride comfort has also undergone huge
metamorphosis. Continuous improvements in technology coupled with increased awareness
levels about safety, economic viability and clean environment have taken the design and analysis
processes in the auto industry to a higher level. With this continuous improvement over the
years, automobiles have become more and more user and environment friendly.
All the factors considered above have given way to fierce competition amongst the US
and international automakers. Given this scenario, customer satisfaction in all fields is of utmost
importance for the growth of any corporation. In this context, noise, vibration and harshness
(NVH) problems like buzz, squeak and rattle are major concerns because any of these is directly
related to perceived quality of the vehicle as such. This is because of the effect that car interior
noise has on the perspective of the customer. In the recent past, with better engines, vehicle
structures and well designed interiors, chances of any noise being generated at all and, if
generated, the probability of those noises reaching the driver/passenger, has been reduced.
Therefore, any sound emanating from the interior stands out. There has been considerable
research and development activity concentrated around this problem by all manufacturers. NVH
groups have been an integral part of all automakers ever since the inception of the automotive
industry. The importance associated with the department has only increased over the years.
Detection and source localization of such noises is the backbone of current research in
this area. Different methods of experimentation and CAE analysis are being researched for an
1
effective solution. This thesis reviews and advances the state-of–the-art in squeak and rattle
detection methods.
1.2 Problem Description:
1.2.1 Squeak and Rattle (S&R) Issues – Theory and Background
Squeak and rattle are unexpected events, usually noises of short time duration, perceived
by a listener to stand out from the background (expected) noise [6].
Squeak is a friction induced noise caused by relative motion resulting from slip-stick
phenomenon between interacting surfaces. The elastic deformation of contact surfaces stores
energy that is released when static friction is overcome, producing audible squeak noise. Audible
squeaks are generally in the frequency range of 200 – 10000 Hz. The amplitude and frequency
content depends on a host of complex factors such as material constituents, coefficient of
friction, normal load and load history, sliding velocity, inertia and thermal effects, wear
characteristics, temperature and humidity conditions, etc.
Rattle, generally is an impact induced phenomenon that occurs when there is a relative
motion between components with a short loss of contact. It is generally caused by loose or overly
flexible elements under forced excitation. Impacts are caused when surfaces close to each other
move perpendicular to each other due to insufficient structural strength, forcing repeated
separation and contact. Possibilities of these actions increase when the vibration is excessive and
the tolerances are inadequate. Rattles generally occur in the frequency range of 200 – 2000 Hz.
Higher frequency rattles are referred to as “Buzz”.
All these sounds are audible only when the surface areas adjacent to the impact are
capable of radiating audible sound power levels [3].
2
The physical nature of S&R is well established because of the enormous amount of
research activity both at the industry and academic level. Research is now focused on how to
eradicate the problem by eliminating the root causes. Part modeling and subsequent analysis
using CAE software, vibration and modal testing of substructures, property study on materials
(plastics) used in the vehicle interior etc. are activities very widely performed in the industry. In
spite of a sustained effort in this regard, squeaks and rattles are observed when the fully trimmed
cars are tested on the proving grounds or off road testing methodologies based on multi axis
shaker tables/road simulators. So, research directed towards a robust car interior noise detection
scheme that can be used to test/analyze subsystems and also fully trimmed vehicles is under
study at the UC-SDRL. The techniques thus evolved are meant to replace the historic “find and
fix” approach which is associated with 47% of the concerns detected during vehicle launch, with
subsequent cost and durability issues [2].
1.2.2 Sound Generation Mechanism
Driving over road irregularities generates impacts on vehicle suspension. This excitation
causes free and forced vehicle vibration in the low frequency range according to suspension
design, with typical resonance frequencies between 2 to 15 Hz. Besides the rigid body vibration,
the whole chassis will respond to a wide frequency range, leading to deformation and relative
motion between components and impacts
[1]
. Almost all S&R can be attributed to structural
deficiencies, incompatible material pairs or poor geometric control. Newer fastening methods are
an example of poor geometrical control. Fast production has changed the older techniques of like
the use of screws on the instrumentation panel with plastic fasteners which may fail occasionally.
This can be a major cause of concern. S&R is generally caused due to relative motion exceeding
3
the threshold value. Relative motion might not always cause S&R but S&R is always caused by
relative motion
[3]
. In the current research project, the road surface is being simulated using the
MTS 320 hydraulic actuator system. When the vehicle is subjected to the forces of the hydraulic
actuator, gaps are filled momentarily, metal or plastic members rub nearest materials and loose
parts rattle. These are the primary mechanisms that generate the annoying squeak and rattle
sounds.
1.3 Research Goals
1. Understand the state-of-the-art in S&R testing and evaluation through extensive literature
research
2. Understand and evaluate the use of MTS 320 hydraulic shaker system with particular interest
in S&R evaluation
3. Understand various time - frequency quantization and localization techniques
4. Explore and apply analysis techniques suitable for test data
1.4 Thesis Organization
This thesis is divided into five chapters. The first chapter explains the ideology behind the
existing problem with reference to the automotive industry. It also includes a comprehensive
literature survey on the past and current research emphasis on this topic using various
experimental and analytical techniques.
The second chapter is the crux of all basic concepts related to signal processing and time
frequency transforms used in this thesis. It prepares the reader in order to understand the
4
technicalities of the analysis performed from an engineer’s point of view and directs to
exhaustive mathematical references for the initiated.
Chapter Three describes the testing facility, equipment, settings, software and the practices used
for successful and meaningful data acquisition.
It also describes the different equipment
configurations used for different kinds of data acquisition.
Chapter Four has the detailed description of methodologies used to validate code written in
MATLAB. It includes results obtained by applying continuous wavelet transforms and short time
fourier transforms to the current research and related analysis.
Chapter Five talks about the avenues for further improvement in the analysis technique
developed.
1.5 Squeak and Rattle – Early developments and literature
Market surveys as early as 1983 reported S&R as the third most important customer concern in
cars after 3 months of ownership [4]. Kavaranna and Rediers [3] gave a comprehensive listing and
elaboration about developments and literature from 1983-1999. Moving from the historic “find
and fix” approach to “do it right” design approach is stressed and systematic methodologies
based on improved structural integrity, use of good design principles, integrated design and
testing schemes for effective tackling of the problem are presented. Nolan, et al
[2]
conducted
experiments on the instrument panel, seat subsystem and proposed methods for squeak and rattle
target setting. Discussion related to disadvantages of using modal parameters to characterize
NVH characteristics of a vehicle is presented. The work proposes that high modal frequencies,
fewer coupled modes, lower amplitudes are desirable for good S&R performance. Basic spring
mass damper models simulating squeaks and rattles were created by Grenier [5]. Experimentation
5
based on these models was performed and results were analyzed using Zwicker loudness
[50]
methods. Grenier concluded that use of N10 percentile quantification in conjunction with higher
critical bands of Zwicker are enough for quantification of loudness but not discriminatory
enough to state that the loudness is because of the BSR phenomena. Soine, et al
[6]
studied
analytical models of coupled and uncoupled rattle mechanisms and compared them to
experiments conducted on a cantilever beam simulating the same. They were successful in
developing a methodology for benchmarking and experimental product development but, were
unable to relate S&R sounds to threshold levels of motion. Wyerman, et al.
[7]
wrote a
comprehensive paper which covers all aspects of NVH such as vehicle noise control, research by
OEM’s, importance of subsystem testing by lower tier manufacturers, absorptive and barrier
materials, testing methodologies, CAE approach etc. This report concludes that there is a higher
expectation from OEM’s by the consumer to optimize cost and weight of the NVH package onboard the vehicle. Consistent with these findings, they propose that NVH engineers be a part of
all major departments of vehicle development for effective use of available technology. Judek, et
al.
[8]
documented literature in the area of material pair interactions and evaluated various
experimental and computational solutions and methods. Also, the paper thoroughly reviews
mathematical simulations of model squeaks. Also, there were efforts in the direction of
developing a friction based theory for metal samples. Artificially induced squeaks were
experimentally investigated. The experiment helped in concluding that the content of squeak
frequency does not depend on exciting force frequency. Juneja, et al.
[9]
presented the theory on
which squeak phenomenon is based (stick slip) and dependence of squeaks on excitation
frequency, amplitude of excitation, geometry of contact, interference levels between material
pairs, normal loads, material properties of the two components, temperature and humidity. The
6
authors conducted a series of experiments on polyurethane and PVC elastomers and concluded
that materials with higher damping and stiffness values are better candidates for squeak
prevention. Trapp, et al. [10] evaluated the joint durability of virgin and Post Consumer Recyclate
(PCR) containing thermoplastic bosses. This study indicates that there is no major degradation in
performance of the attachment after the addition of PCR. Peterson, et al.
[11]
researched squeak
and rattle properties of polymeric materials like polypropylene, polycarbonate, etc. A tensile
tester was used to measure the friction force of the plastics sliding on one another. This paper
documented effects of the three important material properties (chemical composition, surface
topography, temperature) that affect adhesion of polymeric material surfaces. Experiments were
conducted on six materials. Hunt, et al.
[12]
contributed towards a comprehensive materials
database that is the result of a uniform experimental procedure to identify pertinent material
characteristics in order to understand and verify and reduce stick-slip in automotive applications.
This research also developed a test system to perform the experimentation and with the
experimentation results, concluded that by using the test system proposed, a standard for material
pair testing and analysis can be achieved.
There has been considerable emphasis on full vehicle/subsystem testing in order to
evaluate and improve NVH characteristics in the automotive industry. Beane, et al
[13]
.
investigated the automotive body panel structure to determine the contribution to vehicle sound
quality and NVH. The paper also describes usage of non-contacting laser based methodologies to
image the body panel modes at their specific resonant frequencies. Substructure testing has also
been a consideration during the development of the vehicle. With the test set up proposed, the
research group could experimentally validate the positive effect of application of a damping
material on the dash panel on powertrain noise reduction. Nolan, et al [14] discussed experimental
7
methodologies for squeak and rattle testing and analysis of instrumentation panels (IP). Modal
analysis and durability testing of the IP are also discussed. The authors claim that the research
has led to significant improvement in instrumentation panel structure design optimization.
Rusen, et al
[15]
in their paper on next generation techniques of S&R testing, have outlined a
structured way of testing in order to avoid excessive capital cost without compromising S&R
results. They also present feasibility studies performed on common S&R contenders. The authors
propose that individual component suppliers should also perform S&R evaluation on parts. Such
measurements may help in support developments and support sampling tests. Hurd [16] describes
a unique way of testing a portion of a vehicle (vehicle buck) with the IP mounted. This set up is
mounted on a multi axis shaker table and test data collected. Frequency transformed data is
analyzed in terms of signal energy. This paper proposes a new method of durability testing for
automotives. Byrd and Peterson
[17]
introduced new testing methodologies of S&R testing of
large modules of automobiles. This was an attempt to verify a new process replacing a legacy
system used earlier. The procedure to fixture parts and the intelligent use of shakers is outlined in
this paper. The project was also successful in an increase of vehicle module testing at the
company compared to the legacy systems used earlier. Liu, et al
[18]
presented an experimental
approach to study seat belt retractor vibration and rattle noises. An in-depth analysis was
performed from the design to the materials used for constructing the part itself. The study
showed that choice of right materials is of paramount importance for noise abatement. Also, a
successful correlation between objective measurement and subjective jury rating provided an
important bridge between in vehicle evaluation and component test. Malinow and Perkins
[19]
studied the behavior of the three main parts (sensor, spool and the lock pawl of the seat belt
retractor system) when a vehicle accelerates. Merits and demerits of frequency domain and time
8
domain shaker control methods are discussed in the paper. They conclude that the spool rattle is
heavily dependent on the pre-load generated by the webbing and that accelerations in the vertical
direction produces sensor rattle even at small magnitudes less than 1g. Frusti, et al
[20]
in their
paper on seat subsystem squeak and rattle, discussed the application of modal analysis to
examine seat dynamics and vibration behaviors. They were successful in relating the customer
S&R concerns and modal separation. Also, the group advises that seat designers should consider
seat dynamics as a priority issue.
Experimental techniques have always been coupled with analytical forms of assessment
to validate and sometimes quicken or focus analysis. Finite element analysis, a standard CAE
method, is a very powerful analysis tool used widely in today’s industrial scene. Hsieh, et al [21]
presented a finite element based method of improvement of rattle performance of vehicle
components. This paper concentrates on studying structural components in the glove
compartment latch and corner rubber bumpers. An analytical model for rattle is developed and
analysis is built around it. An important conclusion drawn is that the glove compartment rattle
performance can be improved by changing the frequency response characteristics of the
instrument panel or the cross-body beam. Kuo
[22]
in his paper on design to prevent squeak and
rattle is technically exhaustive and provides CAE figures for clear understanding. The paper
presents the correlation between squeak and rattle performance and diagonal distortions at body
closure openings and fastener openings in the IP. The author opines that the CAE model
projected squeak and rattle performance improvements correlated well with subjective
evaluations conducted on prototypes.
Analysis of noise signatures using time - frequency maps and Zwicker loudness
techniques has occupied an important space in the detection and analysis scheme of S&R sounds.
9
Lim, et al
[23]
studied temporal and spectral characteristics of crank shaft rumble noise in four
cylinder engine vehicles. Elaborate time - frequency and Zwicker loudness analysis was
performed on the experimental data. Analysis indicates direct correlation between modulation
frequency of the rumble and one-half order of the fundamental engine rotational speed.
Furthermore, the authors showed that the primary transmission path for crank rumble is structure
borne, except at higher frequency ranges where the path is air borne. Brines, et al [24] developed
an objective metric based on Zwicker loudness technique for a full vehicle S&R performance.
The experimental data is collected on a vehicle using the Direct Body Excitation
[51]
which uses
quiet electrodynamic shakers instead of hydraulic shakers. The paper presents a comparison of
the direct body excitation (DBE) method, 4 post shaker method and on road testing to evaluate
squeaks and rattles on full vehicles using approaches based on psychoacoustics. The authors
conclude that DBE, though it cannot simulate all the conditions that a four axis hydraulic road
simulator can, is highly effective in removing most of the back ground noise which masks low
frequency squeaks and rattles. Results were based on subjective evaluations and objective
measurements. Weisch, et al [1] created a noise quality index for car rattle by studying responses
of 70 test persons on 40 vehicles. The index was generated by means of multiple pair
comparisons and statistics on measurement data. Hamilton
[25]
performed a detailed study on
automotive door closing (impulsive) sounds. The author incorporated 4 general attributes which
quantify the character of an impulsive sound event. The paper includes recommendations for
values of these critical attributes. Also, the author feels that overslam bumpers should be an
integral part of the door system design. They help in softening the energy peak when a door is
slammed against the body. Thus there will be less high frequency modes that will be excited in
the door system. Feng and Hobelsberger
[26]
have used the concept of time domain filter banks
10
for analyzing squeak and rattle sounds. Recommendations for an analysis scheme for both on
and off road measurements are outlined. The authors also say that the percentile statistics need to
be used with care because of their inherent property of not taking into account loudness of S&R
events beyond a certain threshold. Uchida and Ueda
[27]
studied rattling noises in automobiles
using Wigner - Ville distribution. The authors were able to isolate rattles from the back ground
noise by formulating the power spectral density of the signal without the transient and divided
the earlier result with this PSD. Fuzzy logic is used to automate the process. The methodology
developed is proved to be as good as a skilled inspector. Brines and May
[28]
brought into light
the importance of an international standard for non-stationary loudness to be in place. This paper
outlines various loudness techniques in vogue at many OEM’s to quantify squeaks and rattles.
The authors prove that non stationary Zwicker loudness is a valuable tool for objective
quantification of squeak and rattle noise. Van Auken, et al [29] studied subjective ‘over the road’
noise discomfort ratings and objective measurements collected from 10 different driver subjects.
The results indicate that the driver noise discomfort ratings are influenced by the change in
overall loudness, sharpness and the floor acceleration as quantified by the time varying Zwicker
loudness and Aures’ sharpness metrics (Appendix A of [29]).
There is a huge collection of resources explaining the basics and applications of wavelet
theory both on the Internet written by eminent people and also in the form of books and scholarly
papers. Some papers which bear close relation to the work in this thesis have been referenced
here. Lin and Qu
[30]
describe the method of ‘soft thresholding’ to purge data of noise sources
using morlet wavelet. This is of interesting consequence if the signal to noise ratio of the signal
is low. Signals from roller bearings and gear boxes are presented. Scholl and Amman [31] devised
a method to study transient door closing events using orthogonal discrete wavelet transforms.
11
Scalograms produced by wavelets and traditional spectrograms are compared and contrasted.
Munoz-Najar, Hashemi
[32]
compared and contrasted the use of continuous and discrete wavelet
transforms in the analysis of acoustic emission signals. The advantage of CWT over DWT, in the
case of acoustic signal analysis is clearly outlined with examples. Ishimitsu, et al
[33]
introduced
the calculation of instantaneous correlation factor (ICF). The paper discusses the use of part of
the signal as a wavelet itself. Analysis was conducted on acoustic emissions from the interior of
a ship. The research resulted in useful selection of reference signal to improve active noise
control. Robertson, et al [34, 35] studied the use of Holder exponents in the use of structural health
monitoring. This journal paper shows a way of effective use of non-linearities associated with
discontinuities into dynamic response data due to damage typical to certain systems, as a damage
sensitive feature. This process of non-linearity detection is performed using the concept of
Holder exponents
[34]
. Application of this procedure is outlined in the conference paper on
damage detection for theme park rides. Holder exponent and other methodologies discussed in
the paper, showed some promise in understanding and analyzing the accelerometer signals.
Torrence and Compo
[36]
in their paper on the application of continuous wavelet transform to
study El Nino Southern Oscillation (ENSO). They present a clear way of using the wavelet
transform using the DFT to improve computational efficiency. Frenz, et al [37] certified the MTS
320 road simulator for the use of squeak and rattle studies over 800Hz in their study. The study
involved testing a luxury vehicle on the 4 poster system in different configurations. The paper
also talks about the method of determining how noise radiates from the hydraulic actuator. The
largest noise component, in the range of 0 – 500Hz, was determined to be the servo valve dither
and laid down guidelines to design a noise reduction enclosure for the actuators.
12
Chapter 2: Time - Frequency Analysis Concepts
2.1 Digital Signal Processing Considerations
2.1.1 Analog versus Digital Data
While most of today’s data acquisition starts with analog sensing, the analog signals thus
acquired are almost always followed by digitization of the signal at some point. This is clearly
because of the ease of data storage and computation given the capability of modern day digital
computers.
2.1.2 Shannon’s Sampling Theorem
This is one of the base principles on which signal processing is built upon. The theorem
explains the relation between time and frequency domains.
Fsamp = 1/∆t = Fnyq x 2
(2.1 a)
Fnyq ≥ Fsamp
(2.1 b)
Figure 2.1 Relation between time resolution and frequency domain
13
Fsamp is the digital sampling frequency, Fnyq is the corresponding Nyquist frequency and ∆t is the
time interval between each time sample.
2.1.3 Rayleigh’s Criterion
This states the relation between frequency resolution and time data. The relation can be
enumerated as
∆f = 1/T
(2.1 c)
Figure 2.2 Relation between frequency resolution and time domain
2.1.4 Digital Signal Processing Errors
Errors are always a part of data acquisition in any real system. These enter into the
measurement invariably due to a number of reasons. These errors can be classified broadly into
random and bias errors. Random errors can be easily dealt with by averaging the data. But, bias
errors need special attention. Aliasing and leakage errors are bias errors most commonly
encountered in a signal processing situation.
14
2.2 Time - Frequency Transform
A transform is basically a conversion from one domain to another without loss of
information. The time - frequency transform, therefore, is a mathematical relation between time
and frequency domains. This is possible today due to the introduction of the concept of Fourier
analysis. In order to represent general physical signals, it is necessary to analyze time as well as
frequency characteristics. This is known as time – frequency (T-F) representation [39]. Time frequency analysis has become a very common analysis scheme and has gained importance in
the science and engineering world ever since its inception.
2.3 Orthogonal and Non-Orthogonal Basis functions
In mathematics, a basis function is an integral element in a function space. Generally
speaking, basis functions are defining functions. They are a family of analyzing functions,
linearly independent of one another, that can represent any signal in that particular function
space. Any function in the function space can be expressed as a linear combination of these basis
functions. Some basis functions have additional properties, like orthogonality, which eases
reconstruction of the original signal. The constituents of the basis function are generally
represented in vector form. The number of constituents of this function talks about the dimension
of the function space. There can be more than one basis for a function space but all of them will
have same number of dimensions. This concept can be represented as follows:
f (t ) = ∑ µ k φ k (t )
(2.2)
k
where f(t) is the function being analyzed, k is the dimension of the basis, µ k is the
coefficient in the kth dimension and φ k is the corresponding basis component
15
An orthogonal basis system has the property of each of its constituent basis functions have an
inner product of zero (refer to Appendix B). This can be interpreted as a zero projection of each
basis element on any other constituent basis element. This makes the projection of the original
signal on each constituent basis unique and order independent. The major advantage derived is
an uncoupled situation in the transformed domain. Reconstructing the signal is thus a very
simple task. A non-orthogonal basis system, on the other hand, has no special properties. They
are just a set of linearly independent functions that can completely define the signal under
consideration. Projections of a constituent basis function on any other constituent may be non
zero. Projection of the signal on each constituent basis is not completely unique i.e. if projection
information on any one basis is missing, some information about many or all other projections on
the remaining basis elements is automatically lost. The projection talked about here is not the
usual normal projection since the basis is non-orthogonal. Therefore, reconstruction is trickier
but, is possible though, if the basis follows some predetermined admissibility conditions or
ordered evaluation. In fact, in a practical case, perfect reconstruction may not be possible at all.
The mathematical details of this discussion can be found in several applied linear algebra books.
Also, weighted orthogonality is a concept used frequently in the field of experimental modal
analysis. Inner product of different eigenvectors representing the normal modes of a structure
when weighted with the mass or the stiffness matrix is zero [53].
2.4 Fourier Transform (FT)
The Fourier transform is famously called “The Mathematical Prism” for its capability to
describe a time domain waveform in terms of both frequency domain magnitude and phase [40].
But, the FT is only a frequency transform and there is absolutely no way of relating time to the
16
values generated by this function i.e. it identifies the various frequency components in the signal
but not at what times they occur. There are two basic forms of Fourier transform and many
specific cases. The discrete Fourier transform is given by
( xˆ k ) = (1 / N )
n = N −1
∑x e
n =0
−i 2 Π kn / N
n
(2.3)
where N is the number of points in the time series, k =0 to N-1 is the frequency index. This form
is used when the signal is sampled and is discrete in nature. On the other hand, for continuous
signals, the summation is replaced by an integral over all time. This is also called the integral
transform.
∞
xˆ ( f ) =
∫ x(t )e
−i 2πft
dt
(2.4)
t = −∞
where t is continuous time, f is ordinary frequency
It is not difficult to see that the basis vector for the Fourier transform is [1, ejω, ej2ω, ej3ω,
…ejnω] which is an orthonormal set. Using knowledge about the inner product (Appendix B) the
integral transform is an inner product of the function and the basis vector. The digital version of
the integral transform is the Fast Fourier Transform (FFT). The FFT is readily available in coded
form in many mathematical packages. The drawback of the Fourier transform is its limited
capability to exactly identify only frequency components which are stationary or that are fully
observed in the window of analysis. This is because the Fourier series which is the origin of this
analysis tool is developed for waveforms periodic in time. If any of these conditions is not
satisfied there invariably exists an error called the truncation error or more commonly leakage.
Most real signals are non-stationary in an arbitrary observation time T and special
methodologies need to be developed for their analysis. The Fourier transform can be extended to
17
non-stationary waveform analysis by coupling it with a moving time window. This function is
called Short Time Fourier Transform (STFT). The STFT is an attempt to ‘time localize’ the
Fourier transform. STFT is essentially a repeated Fourier transform using a chosen time window.
The window chosen is generally rectangular and the width is defined by the user. The STFT in
the discrete sense can be calculated as follows
N −1
( xˆ f ,τ ) = (1 / N )∑ w(t − τ ) xn e −i 2πfn / N
(2.5)
n =0
where w(t) is the window function which localizes the waveform in time domain. Replacement
of the summation with an integral over all time gives the continuous equivalent. In the digital
domain, a window is equivalent to choosing N points in the time stream. While the classical
Fourier Transform compares the entire signal successively to infinite sines and cosines of
different frequencies, the STFT does the same process one segment (governed by the window) at
a time. The size of the window selected dictates the time and frequency resolution of analysis.
The time resolution (∆t) of any portion of the signal analyzed is unaltered. But, the frequency
resolution (∆f) depends on the window width. Therefore, there is a definite reduction in the
frequency resolution in the STFT when compared to the FFT where the window, theoretically, is
of infinite width. The Equations 2.1 (a) (b) and (c) are applicable with N equal to Fsamp. A
smaller window in time domain detects the higher frequencies better and vice versa
[42]
. But,
after reaching a certain resolution, it is not possible to attain better resolution by reducing the
window size further. This is based on the uncertainty principle proposed by Heisenberg. The
essence of this principle can be explained as one cannot measure the exact frequency and time
existing at a point in time. Unlike the traditional FT, STFT gives a T-F plot. Since the window
support remains the same through out the analysis, the T-F resolution is in the shape of a box
18
(some times called the time-frequency atom
[52]
) with constant area and shape on the T-F plot.
This is illustrated in the Fig (2.3 (a))
ω
ω2
2σ t
ω1
2σ ω
t1
t2
t
Figure 2.3 (a): Time Frequency resolution in a STFT plot [52]
Where, σ ω is the standard deviation of energy spectrum around the corresponding spectral
component ω1 , ω2 etc. σ t is the corresponding deviation in the temporal energy spectra
[48]
. A
thorough mathematical treatment of this concept is given in [30].
ω
t
Figure 2.3 (b): Heisenberg representation of the Fourier Basis [42]
19
ω
ω
t
t
Figure 2.3 (c): Idealized Heisenberg Time-frequency representation of STFT (Left) result from wider time
window (Right) result from shorter time window [42]
Summarizing the above discussion, the FT detects the frequency content of a time stream
by essentially fitting periodic sinusoids ranging through all frequency to the data. Depending on
the kind of analysis required, data may be averaged multiple times for better results. The results
thus obtained are completely devoid of any time information with or without averaging. This
indicates that that the frequencies detected have occurred all through the time stream analyzed
unless there is some prior information available on probable frequency distribution in the signal.
Since the analyzing functions used are infinite pure sinusoids, which have exactly the same
properties all through time, the result has to be interpreted this way. This way of interpretation
may or may not be true depending on the nature of the signal. An improvement to this method is
the STFT where, some amount of relative time localization is brought into play. This is done by
performing a Fourier analysis on blocks of the time stream using known windows. Since the
ending time of each block thus transformed is known, a relative time relation with respect to the
start time can be worked out. This again has issues observed in a FT stated earlier. Also, the
frequency resolution of the STFT is constant throughout the analysis due to a constant width of
20
the window. So a good low frequency resolution is equivalent to unnecessarily high frequency
resolution at higher frequencies. An important observation here is the usage of basic sinusoidal
orthogonal basis functions for analysis purposes. These are of good applicability if the signal
analyzed mainly contains combinations of sinusoidal data. With limited time localization
property provided by the STFT, this method is of substantial analysis applicability. But, consider
a situation where there are transients embedded in the data with arbitrary frequency content that
needs to be observed. Though an STFT offers some time localization as discussed, it
approximates the signal with sinusoids only. This is not a great way of perceiving the signal
especially when the full character of the transient needs to be captured. Since the transient burst
generally bears no resemblance to a common infinite sinusoid, the power changes in the signal
are not properly captured by the sinusoidal basis functions thus leaving the properties partially
uncovered. Consider a small wave having a definite shape and size, following some admissibility
conditions (which dictate its usability for analysis). This small wave is chosen to resemble the
transient itself. Now, let the small wave be used in an operation similar to STFT, but the length
of the window in an STFT, discussed earlier, is now dependent on the width of the small wave. It
is translated across the full time stream similar to the STFT. The width of the small wave, unlike
the infinite sinusoids of an STFT, is finite. This brings in the factor of finite time localization
into the analysis. Instantaneous comparison of this wave with the original signal all through time
results in instantaneous correlation factors. The procedure described is repeated for dilated
versions of the small wave. This action covers possibilities of occurrence of the small wave in
different shapes and sizes in the signal. The small wave, thus chosen, is not a single frequency
sinusoid any more as in the case of STFT. Dilations of the wave change the frequency properties
simultaneously and thus capture varied time and frequency contents of the signal. Now that the
21
transient buried in the signal resembles the analyzing wave, correlation factors calculated around
the vicinity of the transient are much higher compared to the others. Thus the full character of the
transient can be studied with a wise choice of the analyzing wave. This process is popularly
called Wavelet analysis and the various facets of this methodology are explained in the next
section.
2.5 Wavelet Transform Theory
2.5.1 Need for Wavelet Transform
The material discussed thus far brings to light requirement for more robust time
frequency analysis methods. As stated in the earlier section, the T-F resolution is constant in the
STFT and there exists a need for multi-resolution analysis (MRA) for better frequency domain
understanding of signals. The wavelet transform (WT) is an excellent avenue to achieve this
objective. T-F resolution problems are established to be physical limitations as stated by the
Heisenberg’s uncertainty principle. WTs are no exceptions. But, the capability to incorporate
MRA takes T-F analysis to a next level. The output of a WT is a time-scale plot and not exactly
the traditional T-F plot. But the scale is loosely related to frequency through relations that exist
separately for different wavelets. Therefore, the plot can be interpreted both ways.
The extent of possible applications of this analysis is quite large. Wavelet packages
(software) today are commonplace in analysis of data having transients embedded given its
desirable multi resolution properties. In particular, wavelets are well suited for this thesis, which
is an attempt to detect automotive squeaks and rattles, which are short time, high energy bursts,
that occur unpredictably (in time) with somewhat random frequency content.
22
2.5.2 Wavelets
Wavelets, as the name suggests, are small waves in the time domain. WT uses these small
waves which follow certain admissibility conditions. These will be dealt with in the following
sections. These small waves are localized in both time and frequency and are well suited for
transient signal analysis. Since this is an attempt to transform an existing time domain signal,
basis functions are associated with WT as in the FT and STFT. The basis functions are dilated
and compressed forms of the original wavelet called the mother wavelet. The scaling function
computes the dilated and compressed versions of the mother wavelet. As discussed earlier, basis
functions can either be orthogonal or non orthogonal depending on various factors of analysis
and the application itself. The Continuous (CWT) and Discrete (DWT) WTs are two different
ways of analyzing data. Orthogonal wavelets are special cases of DWT. These wavelets are
much more difficult to construct because of the properties the basis has to satisfy. The
orthogonality of the basis aids in development of faster algorithms [42]. Non orthogonal basis can
be used for both CWT and DWT computations.
2.5.3 Continuous Wavelet Transform (CWT)
The Continuous Wavelet Transform (CWT) was developed as an alternative approach to
STFT to overcome the resolution issue. The primary difference between the STFT and the WT is
the change of window support as the transform is calculated for each spectral component. The
general mathematical form of a wavelet is given as
ψ (τ , s ) (t ) =
1 (t − τ )
ψ
s
s
(2.6)
23
Where ψ (τ , s ) (t) is the mother wavelet, τ is the finite translation in time and s (s>0) is the scaling
factor (dilation parameter).
The CWT is defined as
∞
CWT ψ (τ , s ) (t ) = ∫ x (t )ψ (τ , s ) (t ) dt
(2.7)
−∞
A closer look at the equation shows that CWT is again a inner product of the signal x(t)
and the wavelet basis functionψ . For ψ to be an admissible window function, and to recover the
original signal x(t) from the transform, it has to satisfy the condition
∞
∫ψ (t )dt = 0
(2.8 (a))
−∞
It can be interpreted that the function should have some oscillations in the time domain to
satisfy the above condition. It is worthwhile to note that if there is no signal reconstruction
required, any wavelet can be used for analysis purposes as long as it follows Equation (2.8 (a)).
This is due to the fact that a signal subjected to CWT can be perfectly reconstructed only if the
wavelet follows the admissibility condition [45]. Equation 2.8 (b) is the admissibility condition for
wavelets.
2
ψˆ (ω )
Cψ = ∫
dω < ∞
ω
−∞
∞
(2.8 (b))
Where ψˆ ( w) represents the frequency transform of the waveletψ (t ) . The admissibility condition
impliesψˆ (0) = 0 . That is, the mother wavelet must be a band pass in the frequency domain.
Also, it can easily be interpreted that the STFT is just a simplified form of an orthogonal
WT with e
njωt
, n=0, 1, 2, 3, ….N as the basis function with scaling factor equal to 1 and each
translation is equal to the length of the window employed. This shows that the WT is just an
24
extension of the STFT to incorporate composite frequency analysis waveform for effective
transient detection and signal analysis purposes.
2.5.4 Scaling and Translation
Scaling and translation, as mentioned earlier are the principle components of a WT.
Scaling refers to the compression and dilation of the mother wavelet thus developing the basis
functions for the analysis. In the Equation (2.6) ‘s’ represents the scaling factor. This operates on
the wavelet function as a multiplication factor. Scale can also be related to the inverse of
frequency. A wide window detects low frequencies better and a smaller one, high frequencies.
Each of the scaled versions of the wavelet form an integral part of the basis function vector.
Choice of scales can be made in different ways depending on the application and frequency
range of interest. For an orthogonal basis system, one is limited to a set of discrete values [44]. For
a nonorthogonal system, one is free to choose arbitrary set of scales including non-integer values,
to build up a complete picture. Translation, on the other hand is represented by ‘τ’ in Equation
(2.6). ‘τ’ governs the lateral displacement, or time shift, of the wavelet window in the time
domain across the signal being analyzed. The values of s and τ are arbitrary for a CWT. This
gives the advantage of spanning the full time stream as per the analysis requirement.
25
Figure 2.4: Effect of scaling on a Morlet wavelet in the T-F plane
The Fig (2.4)
[48]
shows the idealized Heisenberg boxes for the WT. The wide windows
used to look at low frequencies covey precise frequency and relatively vague time information.
The narrow windows used to look at high frequencies, expectedly have opposite properties.
Another interesting fact is that, though the lengths and widths of the boxes change, the area of
each box is equal to others.
26
Figure 2.5: Ideal TF plane decomposed with wavelets on a Dyadic grid
Figure 2.6(a): Cosine curve illustrating the concept of scale
The Fig (2.6 a) illustrates the mathematical concept of scaling in a cosine wave. If a
function f(t) is scaled resulting in f(st), then f(st) is a dilated wave if s<1 and a contracted wave if
27
s>1. But, in the case of WT, the scaling parameter appears in the denominator. Therefore, the
scaling relation is reversed here.
2.5.5 Computation of a CWT
The interpretation of Equation (2.7) is discussed in this section. With data in hand, the
wavelet to be used for the analysis is chosen. Then, the next task is choosing a set of scales for
the analysis. Generally, computations do not need to be performed at all scales since practical
signals are normally bandlimited in frequency. Scales corresponding to frequency range of
interest can be isolated for improved computational efficiency.
The wavelet is placed at the beginning of the time signal (t=0). The signal is multiplied
by the wavelet function and then integrated over time. This value gives the continuous wavelet
coefficient for that time. The wavelet is then moved in time by ‘τ’ units. The process discussed
above is repeated. When the wavelet reaches the end of the signal the calculation of the first row
of coefficients is complete. The wavelet is scaled according to requirement and the second line of
coefficients is computed. This is continued till all the desired scales are exhausted. An illustrative
example is shown for clarity [43]. Fig (2.7 a) and Fig (2.7 b) are only indicative of the translation.
In these figures, the signal being analyzed is the transient curve (yellow curve). The filter in the
background, is representative of the wavelet. These figures show a subset of the possible
translations for explanatory purposes. The signal analyzed is non stationary in time and the
window of analysis is not a wavelet function but more of an illustration.
28
Figure 2.7(a): CWT computation at a low scale [43]
Figure 2.7(b): CWT computation at a higher scale [43]
If there is a spectral component in the signal equivalent to that of the window (scale
dependent) at a point in time, the CWT coefficient is high there. Also, a CWT is redundant at
higher frequencies. This is due to the overlap of windows. This property of redundancy comes
handy if the data being analyzed has sharp transients, but, with an increased requirement of time
and computer memory.
29
2.5.6 Computation of CWT Using DFT
It is very common to compute the CWT in the time domain. An equivalent computation
can be carried out in the frequency domain. This form is preferred sometimes due to its
significant speed compared to the former. This methodology has been discussed in detail in [44].
As a starting point to this method, the analytical frequency transform of the wavelet function
needs to be available.
The continuous wavelet transform of a discrete sequence xn is defined as the convolution
of xn with a scaled and translated version of ψ (η )
N −1
CWTψ n ( s ) = ∑ xn′ψ *[
n′ = 0
(n′ − n)δt
]
s
(2.9)
Where ψ * indicates the complex conjugate, N the number of discrete points in the time series xn ,
s is the scale and n is the translating variable.
It is evident from Equation (2.9) that, to approximate the CWT, the convolution needs to
be performed N times for each scale ‘s’. N convolutions are not a necessity. By choosing N
points, the convolution theorem allows us to do all N convolutions simultaneously in the Fourier
space using a DFT (Equation (2.3)). From the convolution theorem, the wavelet transform is the
inverse FT of the product
N −1
CWTψ n ( s ) = ∑ xˆ n′ψˆ * ( sω k )e iωk nδt
(2.10)
k =0
Where ψˆ is the analytical FT of the mother waveletψ (τ , s ) (t ) , x̂ n is the Fourier space equivalent
of the time stream x n . The angular frequency is defined as
30
N
⎧ 2πk
⎪ Nδt : k ≤ 2
⎪
⎪
ωk = ⎨
2πk
N
⎪−
:k >
2
⎪ Nδt
⎪
⎩
(2.11)
Using the above formulation, one can calculate the CWT using a frequency domain
calculation involving the numerically efficient FFT.
2.5.7 Wavelet Normalization
To ensure that the wavelet transforms (see Equations (2.6) and (2.10)) at each scale are
directly comparable to each other and to transforms of other time series, the wavelet function at
each scale s is normalized to have unit energy. This process is called energy normalization.
1/ 2
⎛ 2πs ⎞ ˆ
ψˆ ( sωk ) = ⎜
⎟ ψ 0 ( sω k )
⎝ δt ⎠
(2.12)
Where ψˆ 0 is the unscaled mother wavelet.
N −1
∑ ψˆ (sω )
k =0
k
2
=N
(2.13)
Where N is the number of points in the DFT.
The 1
s factor in Equation (2.6) is the normalization factor if the time domain
convolution approach is used to compute the CWT. Concisely, the normalization ensures that all
wavelets at all scales have the same area in the frequency domain and thus have same energy.
This is a common practice in wavelet analysis. But energy normalization results in unequal
weighting of frequencies in the Fourier domain. Requirements for different practical applications
31
may require frequencies to be weighted equally unlike the standard procedure. Amplitude
normalization is performed in such situations. The energy normalization factor is equated to
some constant value in such cases. The normalization is not scale dependent any more. A
graphical treatment of this case is included in Section (4.5).
2.5.8 Redundancy in CWT
This is a characteristic property of a CWT. The very fact that a 1-D time signal is
transformed to 2-Dimensions (time and scale) shows the repetition of data. It is gauged as a
disadvantage for applications where data compression and image capture/storage are required.
But for applications such as fault/damage detection, acoustic emission study or analysis of any
short (in time) transient event, all involving feature extraction, CWT is used widely. This
property is a burden on the computer’s memory and processor, but the translation from one scale
to another allows the transients in the signal to be more dissected thoroughly. The CWT is
generally computed using non-orthogonal basis system for these applications. The primary
reason for this is that wavelets generated through an orthogonal basis often do not look similar to
the signal being analyzed. Secondly, creating new orthogonal wavelets is a mathematically
cumbersome task. As discussed earlier, the crux of this technique is evaluation of correlation
between the signal and the scaled wavelet. With years of research experience, it has been
established that some wavelets are suited for certain types of signals.
2.5.9 Discrete Wavelet Transform (DWT)
This form of the transform is popularly used for data compression applications. The
concept of the transform essentially remains the same as the CWT. The one difference between
32
the CWT and DWT is the choice of scale and translation. In the Equation (2.6), the values of s
and τ are not arbitrary as in the CWT case.
s = 2j; τ = k where j, k∈ Z.
(2.14)
A special case of Equation (2.14) is choosing s0 = 2, t0 = 1, s = 2j and τ = k*2j. This
arrangement is called the dyadic grid and is very commonly used to represent WT data. Fast
algorithms are shown to be built using this configuration by Mallat given that the wavelet uses an
orthogonal basis
[30]
. The DWT was developed with the intention of reducing the memory
resources and improving computing efficiency. It is developed on the lines of sub-band coding
principles. The aim of this process is to pass the signal to be analyzed through a filter bank to
break it up into high and low frequency bands at every stage. To make the process simpler,
wavelets with an orthogonal basis are generally preferred for these operations. They are of great
use in compact representation of data as discussed in the earlier sections. Many wavelets used for
the DWT, like the Daubechies wavelets (dbn)
[54]
, cannot be represented in an equation form.
Unlike the Morlet and Shannon wavelets, they are compactly supported in time domain. They
have properties of a band pass filter in the frequency domain.
2.5.10 Wavelets as Band Pass Filters
If there is a question posed about the number of discrete wavelets needed to encode
information in a signal without redundancy, the actual answer is infinite. In an attempt to make
data manageable, wavelets, which act as band pass filters can be created. Recall from the Fourier
transform theory that a compressed version of a signal in time represents a stretched frequency
band with the center frequency shifting up by the factor of multiplication.
F ( f (at )) =
1 ˆ ω
f( )
a
a
(2.15)
33
Using this property, wavelets can be designed such that the band pass filter bank (Fig 2.8)
can be formulated. The ratio of the center frequency of the band to the bandwidth is a constant.
This factor is referred to as the fidelity factor ‘Q’ of the wavelet [45].
Figure 2.8: Band pass filter bank
It is evident from Fig 2.8 that this filter bank needs to be infinite in length to cover the
full frequency spectrum. In order to circumvent this problem, a scaling function which is also a
wavelet in the time domain is developed depending on the requirement. The scaling function is a
low pass filter which fits in at the lowest band. The scale at which this filter comes into picture is
defined by the user. It is interesting to note that not all wavelets need a scaling function. This
process is also termed as Multi-Resolution Analysis (MRA) if the scaling function follows
certain conditions, one of which is orthogonality. These conditions are explained thoroughly in
many MRA or wavelet related mathematical texts.
Scaling Function Φ
Φ
Ψ4
Ψ3
Ψ2
Ψ1
f
Figure 2.9: Figure illustrating the scaling function
2.5.11 Concept of Sub-band Coding
Creating a sub-band is the process of splitting up the frequency band into smaller
manageable frequency bands using band pass filters.
34
Figure 2.10: Illustration of sub-band coding [56]
This process can be done in two different ways. The first is to choose various band pass
filters to analyze data in those bands and cover the entire frequency band. The problem with this
method is the creation of band limited filters. The other way to look at the problem is by splitting
the frequency spectrum into two at the middle. The upper half can be classified as the high
frequency band and the lower half, low frequency band. In wavelet jargon, the high frequency
band is called ‘detail’ and the low frequency is called ‘approximation’. Splitting the existing low
frequency band in the middle will result in two sub bands each of which can again be classified
as high and low frequency band respectively. This sub division is continued till the desired
accuracy is obtained.
35
2.5.12 Computation of DWT
The DWT is an implementation of the idea of sub-band coding using orthogonal
wavelets. A close look at the frequency band subdivision will bring into light another advantage.
Now, the bandwidths have essentially been halved by the filtering process. By Nyquist-Shannon
sampling theorem, only half the number of discrete samples that existed earlier will be needed to
completely define the frequencies in the low frequency band. To improve the memory efficiency,
both chunks of data can now be down sampled by 2 and still not lose any valid information. This
is possible because of the knowledge of high pass and low pass bandwidth. The down sampled
signal now has half the frequency resolution compared to the original signal. Mathematically, the
filtering and down sampling on a sequence x[n] can be expressed as follows.
yhigh [k ] = ∑ x[n]g[− n + 2k ]
(2.16)
ylow [k ] = ∑ x[n]h[− n + 2k ]
(2.17),
n
g[n] is the high pass filter,
n
h[n] is the low pass filter.
Reconstruction of the original signal is possible since orthonormal filters are used. The
process followed thus far is reversed for reconstruction. Down sampling is replaced by the up
sampling operation which basically is placing a zero in between every two existing samples. For
every layer, the reconstruction formula can be written as follows.
∞
x[ n] = ∑ ( yhigh .g[− n + 2k ] + ylow .h[− n + 2k ])
(2.18)
−∞
36
2.6 Choice of Wavelet Properties
This section deals with understanding the effects of various parameters integral to a
wavelet transform. The theory required was presented earlier in this chapter. The reasons behind
the choice of a particular wavelet and its governing parameters are presented henceforth.
2.6.1 Discrete or Continuous Transform
Orthogonal wavelets are computationally quicker due to well evolved fast algorithms.
But they have limitations on the scale and translation, as described earlier, if one wants to take
advantage of their special properties. Orthogonal wavelets produce shift variant transforms [45, 46]
due to properties inherent in them. This makes the analysis position dependent, i.e. a transient in
time is interpreted differently at different time instants. The dyadic grid, as described in Section
2.5.9, is very sparse to clearly understand the characteristics of a transient like signal. Also,
feature components cannot be separated from the irrelevant components by these sparse grids [34].
Nuances of the signal can easily be missed in such arrangements. This is of absolute concern,
especially in the case of pattern recognition and feature extraction. Thus, dyadic discrete wavelet
transforms are not suited for feature extraction. A continuous wavelet transform (CWT) appears
to be most appropriate for all signal analysis purposes in this project.
2.6.2 Choice of Wavelet
Wavelet analysis as discussed earlier has best results if the function used resembles the
analyzed signal when dilated and translated. It is not an easy task to build new wavelets though
researchers have done that in the past. There are a host of functions in the wavelet literature that
one can choose from and still reach valid conclusions because of the built-in ‘zoom in, zoom out’
37
property that the WT offers. Some commonly used orthogonal wavelets are Haar, Daubechies,
Coiflets, Symlets and Meyer
Mexican Hat
[54]
[54]
used for data compression, signal de-noising, etc. Morlet and
wavelets are some common non-orthogonal wavelets used for transient
identification. The complex Morlet wavelet is chosen for further analysis because of its wide
spread use in the area of feature extraction and damage detection. This wavelet shares a lot in
common with its real counterpart, the real Morlet wavelet except that real wavelets can capture
only amplitude data and no phase data compared to the complex Morlet wavelet. The complex
Morlet offers good balance between time and frequency resolution unlike the Haar and the
Shannon wavelets which have lopsided properties
[47]
. Some common wavelets and their
frequency transforms are shown in Fig (2.11). ‘m’ represents the wavenumber (discussed later in
this chapter) of the particular wavelet. A comprehensive collection of common wavelets, their
construction details and their frequency transforms can be found in Goswami, et al [54].
38
Figure 2.11: Time and frequency domain representations of Morlet, Paul, DOG wavelets [44]
2.6.3 Complex Morlet Wavelet
Properties of the complex Morlet are studied closely in the following sections due to its
extensive use in the current research. The Morlet wavelet is in essence a modulated Gaussian
function. It can be expressed in closed form as follows:
ψ 0 (t ) = π −1/ 4 eiω t e − t
0
2
/2
(2.19)
Where ω 0 is called the wavenumber or the center frequency (radians/sec). The wavenumber is
roughly equal to the number of ripples in a Morlet.
The analytical FT of this function is given as follows:
ψˆ 0 ( sω ) = π −1/ 4 e − ( sω −ω
0)
2
2
(2.20)
39
The maximum value of the FT of the unscaledψˆ 0 occurs at the center frequency ω 0 [44, 47].
4
real
imag
3
2
1
0
-1
-2
-3
-4
0
50
100
150
200
250
300
Figure 2.12(a): Complex Morlet in time domain
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-0.1
0
50
100
150
200
250
300
Figure 2.12(b): Frequency domain equivalent of Complex Morlet
A Morlet wavelet dilated by a factor ‘s’ will have the center frequency at ω 0 / s . Both the
increase and decrease of amplitude is shown in Fig (2.12 (a)). But for analysis purposes, only
40
one half of the signal is used. The other half is a mirror image and corresponds to negative
frequency equivalent in the Fourier domain.
2.7 Parameter Discussion
Understanding the effects of various parameters that define the wavelet is of utmost
importance to draw any conclusions from the results. This is because changes in these
parameters alter the output greatly at times. Inputs such as the frequency range selection,
wavenumber of the wavelet, number of scales are discussed in this section. The code used for the
analysis is included in the Appendix A. For visibility purposes, the imaginary part of the wavelet
is not displayed in the illustrative plots shown in this section.
2.7.1 Wavenumber
Wavenumber, as mentioned earlier is the center frequency of the wavelet and determines
the number of significant oscillations of the complex sinusoid within the Gaussian window
[48]
.
Wavenumber has units of radians/sec. For a given scale, increasing the wavenumber shifts the
FT of the wavelet to the right. For very high wavenumbers, the wavelet appears like noise and
thus correlates high with noise in the data. This is not desired. So a balance needs to be
maintained for effective use of the wavelet technique.
41
Time Domain
wavenumber = 10
Frequency Domain
Wavenumber = 10
4
15
2
10
0
5
-2
-4
0
500
1000
1500
2000
0
0
Time Domain
wavenumber = 20
50
100
Frequency Domain
wavenumber = 20
4
15
2
10
0
5
-2
-4
0
500
1000
1500
2000
0
0
50
100
Figure 2.13: Frequency and time domain representations of morlets of different k_0 values
From Fig (2.13) it can be observed that the number of ripples increase as the wavenumber
increases. Note that the frequency domain figure has been zoomed in to show the curve. The
effect of varying ω 0 as illustrated in [48] can be explained as follows:
42
Figure 2.14: Effect of increasing wavenumber
ω 0 [48]
With no change in the Gaussian envelope, an increase in wavenumber shifts the
Heisenberg box up in frequency. Since the FT of the wavelet retains the same shape, the
dimensions of the box remain constant with this operation. Higher wavenumber is indicative of
increase in the number of oscillations and thus allows for a better resolution of frequency
components
. It is interesting to note that with higher ω 0 , the grid in the T-F plane in Fig
[48]
(2.14) will move up in frequency.
2.7.2 Wavelet Scale and Fourier Frequency
The wavelet plot generally is shown as a time-scale (T-S) plot. But, relations between
Fourier frequency (ω) and scale (s) exist for some wavelets such as Morlet, Paul, Deivative of
43
Gauss (DOG), etc [44]. Equivalent relations can be found for the above wavelets in Torrence and
Compo [44]. This is possible because of presence of dominant periodic components in the wavelet
due to which the frequency domain equivalent looks like a band pass filter. The relation for a
Morlet is presented in Addison, et al
[48]
and in Meyers, et al
[49]
. Such relations are not very
meaningful for wavelets without any dominant periodic components. For a Morlet,
ω=
ω0 + ω0 2 + 2
(2.21)
2s
2.8 Edge Error
Errors arising from processing truncated time data are well understood. An error similar
to the truncation error (leakage in the FT) is seen with the digitized form of wavelet analysis.
Errors occur at the beginning and end of the wavelet power spectrum. The easiest way to
improve the computation is by zero padding the end of time series. Since these values enter the
transform, they should be removed from the final estimation. Padding with zeros introduces
discontinuities at the end points which results in the area called Cone of Influence
[44]
. This is
observed to be higher at lower frequencies (higher scale values). Amplitudes of wavelet
coefficients in this area are not accurately estimated.
2.9 Summary of Methods
A comprehensive discussion on various time-frequency localization techniques has been
presented in this chapter. The FT detects completely observed transients or sinusoidal frequency
components effectively but, with no relation to time. The STFT, takes a step further and adds the
time localization character to the analysis. As the STFT window width is retained constant
through out a particular analysis, the time and frequency resolution remains constant too. The
44
WT uses a characteristic wave for analysis unlike the sinusoids in STFT. Dilates of this wave,
when used, change the time-frequency characteristics of the analysis instantaneously. Energy and
amplitude normalization, the two independent facets of the WT, process data differently and
thus, result in different graphical interpretations. A comparative data analysis with STFT and
WT using Morlet wavelet is presented in Chapter Four. The time and frequency localization
capabilities of these techniques are compared and contrasted. STFT is computationally extremely
efficient. On the other hand, WT offers critical advantages with multi resolution capabilities.
But, it has issues associated with memory and time. The analysis capability comparison of a
Morlet wavelet to the STFT is interesting due to the Morlet shape itself.
45
Chapter 3: Testing and Data Acquisition
3.1 Testing
3.1.1 Need for Testing
Increasingly, testing of final products is becoming a validation methodology. Analysis has
become the primary consideration with the extensive use of analysis software and extremely
powerful computers. It is a known fact that the cost, expertise and technical manpower required
to perform large scale testing is too expensive for most automotive manufacturers (OEMs).
Squeak and rattle, however, is still a problem that is being handled by testing and experimental
data processing due to poor predictability of S&R problems with respect to practical
manufacturing issues. Moreover, the capability to reproduce actual noises during testing, as
heard by customers during operation, will aid in better understanding and probable development
of early corrective methodology for future projects.
3.1.2 Testing Methodologies
Squeak and rattle testing and data acquisition is commonly done in two ways.
1. Operational data collection on moving cars
2. Testing of the structure full or subsystems (such as IP cutaway) on a simulation platform
like the MTS 320 road simulator, multi axis shaker table or using electrodynamic shakers
[51]
.
There are pros and cons to the methods mentioned above. A very prominent problem especially
with noise measurements conducted on a road simulator is the noise emanating from the
hydraulic shakers itself. These are squeaks and squeals from the valves and seals. These tend to
be in the lower frequency range (10 – 800Hz). With these noises in the background, actual
46
squeaks, in that frequency range, get masked. Operational data on the other hand does not need
shakers for excitation but is acquired when the vehicle is moving on a road/track. Excitation due
to the actual road input cannot be easily repeated.
Also, instrumentation of a car when being tested on a simulator is much easier compared to the
operational approach. Limited power to support the instrumentation on board a moving car and
noise from the instrumentation are also important considerations. A simulator also gives us the
advantage of manually controlling the input (file information) to the system and reproducing
inputs.
The current research has been carried out on the MTS 320 road simulator primarily because of its
availability at UC-SDRL. As part of the project, the team studied the accelerometer and
microphone responses at various levels of excitation of the shaker input.
3.2 Test Facility
3.2.1 Ideal Test Condition
Ideally, any noise evaluation system should be isolated from all non-contributing noise
producing sources. For the current system, the hydraulic pump and the two distribution towers
should be located away from the testing facility. The shakers should be shrouded in a noise
abating enclosure. The pans should have a damper coating to reduce any acoustic emissions due
to natural frequencies.
3.2.1 Actual Test Conditions
47
The MTS 320 road simulator system is located in the UC-SDRL high bay lab area. The control
room, pump room and the distribution towers are located in close vicinity of the four poster (see
Figure 3.1). The service lift, the hydraulic actuators and the distribution towers are all mounted
on an isolation mass (poured 25x14x9 inch concrete block). The pump room, containing the
hydraulic pump, is about 8x8x10 feet with the internal wall spaces filled with sand, all openings
are sealed to create as much acoustic isolation as possible and the control room is about 8x20x10
feet. The hydraulic shakers (Model 248.03) can be individually moved according to the
requirement using an air floatation device. The force rating of the actuators are 5.5 kips, 2.25
inch2 piston area, 3.15 inch diameter rod and 6 inch stroke length[15]. A service lift is used to
raise the car. Actuators are maneuvered using the floatation device. The actuators are then firmly
bolted down to the steel plates embedded in the concrete block. The wheels of the vehicle are
then firmly strapped down to the pans using nylon ratchet straps.
48
Distribution towers
Actuators
Pump Room
Control Room
Figure 3.1 Test Set up
Given the description, there are some differences between how the system should be configured
compared to how it actually is configured. The ratchets are metal ended and have some loosely
connected parts. When the system is vibrated, they tend to make some noise. The high bay itself
has a lot of noise sources. The building HVAC, fans, adjacent civil engineering facility which
may have some effect on the data collected. Care was taken to test primarily at times when these
problems could be eliminated. Based on a prior acoustics project carried out at SDRL, the
primary noise source was determined to be the servo-valve on each actuator. Secondary sources
include the distribution towers and oil hoses [38].
49
3.3 Test Set up
3.3.1 General configuration
Five fully trimmed vehicles were tested on the MTS 320 hydraulic road simulator system. The
cars will be referred to as Vehicles ‘A’, ‘B’, ‘C’, ‘D’ and ‘E’ in the order of being tested, with
‘A’ being the first to be tested and ‘E’ the last, through out this thesis. Vehicles ‘A’ and ‘D’ are
exactly the same cars and ‘E’, a best in class vehicle. Each car was instrumented with 40+ triaxial accelerometers, 8 uni-axial accelerometers and 8 microphones.
Type
Sensitivity(nominal)
ICP powered
Model
LVDT
3.34 volts/inch
No
Tri axial
1 volt/g
Yes
PCB XT356B18
1 volt/g
Yes
PCB UT333M07
20 milli volt/Pascal
Yes
PCB 130A10/P10
accelerometer
Uni axial
accelerometer
Microphone
Table 3.1 Instrumentation Summary
Figure 3.2 (from L to R) Sensors Microphone with preamp, tri-axial accel, uni-axial accel
50
Sensor
Uni - axial accelerometers
Uni - axial accelerometers
Tri - axial accelerometers
Tri - axial accelerometers
Tri - axial accelerometers
Tri - axial accelerometers
Tri - axial accelerometers
Microphones
Tri - axial accelerometers
Part of car
Four poster pans
Wheel spindles
A – pillar
B – pillar
Bulk head / fire wall
Cross body beam and
HVAC
Seat rails
Front seats
Instrumentation panel and
steering wheel
Numbering scheme
10 series
20 series
100/200 series
300/400 series
500 series
600 series
700 series
800 series
900 series
Table 3.2 Sensor distribution
Accelerometers and microphones were positioned at locations very comparable to one another
across different cars tested. This is to ensure the comparability of data across vehicles. Data from
the sensors is collected using the required cabling. 32 sensor channels were connected to the 8
channel break out boxes which directly hook into data cards (HP E1432 A) of a VXI mainframe
(HP E8400 A). The mainframe uses a controller card (HP E8491 A) to connect to a personal
computer. Remaining channels were connected to 16 channel break out boxes which are in turn
hooked to the data acquisition cards through a signal conditioner (PCB Model 441A101) using
ribbon cables. All the accelerometers and microphones used were ICP devices. Each actuator is
equipped with an internal Linear Variable Differential Transformer (LVDT) and pressure sensors
for instantaneous displacement and pressure measurements. The four channels of the road
simulator stroke (LVDT) signals were also recorded. Unlike the accelerometer and microphone
channels, the stroke signals are voltage devices. In all, 150+ channels of acquisition were used
for all data collected in this configuration. The mainframe is interfaced with the personal
computer using Firewire (IEEE 1394).
51
Signal input into the hydraulic shaker is computationally generated using a software program,
SimTest(R). SimTest(R) is a commercially available MATLAB(R) based function developed by
Simulation Techniques Inc. which generates an input file (road profile file for the shakers) from
an existing response file. In the current case, the response files were provided by an auto
manufacturer. MTS Flex Test(R), commercially available controller software for the MTS 320
road simulator, is used to control the excitation level, frequency input for random signals, RMS
peak values of signal input, as well as to play the road profile files generated by SimTest, etc.
Road profile files and uncorrelated random signal files were used as inputs to suit the
requirement of the research objectives. The excitation was 50%, 75% or 100% of the original
file. Excitation level here means scaled version of the original input file (100% excitation). 50%
excitation would mean 0.5 times amplitude of the original input file. 100% excitation can be
physically interpreted as ± 0.8 to ± 0.9 inches of LVDT stroke for both the random and road
files. Different levels of excitation were evaluated in the hope of exciting as many squeaks or
rattles as possible. This is because some body parts may respond at lower inputs and others at
higher. The frequency range in the Flex Test software was set at 0-200Hz. But, the probability of
only the lower frequencies entering the system is higher due to fact that the road simulator
cannot faithfully reproduce frequencies above 60 – 70 Hz due to the oil column resonance of the
actuators [38]. However, the output frequency (of squeaks and rattles) is much higher. The process
is highly non-linear because squeaks are often initiated when arbitrary material rubs together or
loose plastic/sheet metal parts hitting one another. Moreover, the ideology behind the use of the
road simulator for this thesis is to best replicate the events that generate S&R and not to
introduce certain frequency ranges into the test subject. For the road profile files, the responses at
the wheels were matched to the files provided using SimTest.
52
Accelerometer
on ‘A’ pillar
Accelerometers
on ‘A’ pillar
Accelerometer
on cross body
Accelerometers on HVAC
Accelerometer on
wheel spindle
Accelerometers on IP
53
Accelerometer
on HVAC
Accelerometers on Bulkhead
Figure 3.3 (a): Instrumentation
Figure 3.3 (b): (left) VXI system, (middle) wheel fastened to actuator pan, (right) dummies with microphones
(circles indicate microphone position)
3.3.2 Validation Configuration
The validation configuration is a special configuration used for the validation of the Wavelet
code. A small personal computer speaker was taped down securely to the walls of foot well of
Vehicle ‘E’. Only the eight microphones, four uni – axial accelerometers and four stroke
(LVDT) channels were set ON in the DAC Express acquisition software.
54
Figure 3.3 Speaker Set
Figure 3.4 Speaker taped to foot well wall
The speaker was driven using a chirp signal generated using a MATLAB code on the same
personal computer which was used for the acquisition. The chirp signal was generated at three
different frequency ranges low (200-1100Hz), medium (3100-3700Hz) and high (6100-8500Hz).
The four poster had random signal input (generated by Flex Test) during this testing
configuration. Each frequency range was run at three excitation levels (see Section 3.4) and each
excitation level was measured at three different volumes (25%, 50% and 75%) of the speakers.
Speaker volume was controlled through the volume control GUI of the Windows XP based
computer which ran the chirp code. For example, 25% volume indicates that the cursor was at
one fourth height of the vertical volume control bar. Data was taken at different volume levels to
establish the capability of the code to discover low volume signals and check performance over
different volumes. The analog volume control on the speaker was fixed during all testing (see Fig
(3.3)). Note that this configuration was used only on vehicle ‘E’. A subjective noise
perceptibility chart is included in Section (4.6) of Chapter 4.
55
3.3.3 Sensor Calibration
Nominal calibration values from the calibration sheet given by the manufacturer were used for
all the accelerometers, both tri - axial and uni – axial. The microphones used were calibrated at
The Modal Shop(R) based in Cincinnati. However, three microphones used in the testing purpose
did not have calibration values recorded. So the nominal values have been used for calculation
purposes.
3.3.4 Troubleshooting
With 160 channels of acquisition, sensor cabling can become a major problem. It is very
laborious to track down a sensor if a proper tracking system is not in place. Elaborate
arrangements were made to ensure minimum time for replacement of a cable or an accelerometer
or the patch panel, should a problem be detected. Numbering schemes for accelerometers
depending on the position on the body of the vehicle was developed. Each cable connecting the
accelerometer to the patch panel was numbered on both sides using printed tape. Each patch
panel had a label indicating the channels entering it. It also had a number corresponding to its
position in the tracking list generated in Microsoft Excel. This helps in assuring that the ribbon
cables connecting these boxes to the signal conditioner are in the right order. This arrangement is
double checked when the vehicle is stripped of the instrumentation and errors are noted and
incorporated in the data. This way a sensor in the wrong position and/or direction and/or location
can be avoided. In addition to the tracking system, when the instrumentation is first mounted on
the vehicle, a short, time domain data record is collected. This is processed to check if all the
sensors are generating valid output using a MATLAB code. If there is a problem, the sensor is
56
tracked down and the channel is thoroughly checked to understand the issue. Data for analysis is
collected only after ensuring proper connections.
3.4 Data
With all the instrumentation and cabling in place, the data is acquired and stored real time on the
VXI SCSI disk, initially, and then transferred onto the personal computer and backed up on
DVD’s. DAC Express and X-Acquisition are two Windows based acquisition software used
through out the breadth of the project. DAC Express has various capabilities such as acquiring
time data, playing back user selected parts of data, trigger acquisition, multiple sampling rate
selections, etc. Data thus acquired, was used primarily for squeak and rattle detection. This data
is stored in the Standard Data Format
[55]
on the computer. X-Acquisition on the other hand, is
used to acquire averaged transmissibility data to be used for structural analysis. X-Acquisition
stores data in the Universal File Format
[57]
. Not all data was taken for all vehicles based upon
vehicle availability.
Data was acquired at 3 different sampling frequencies (25 KHz, 2.5 KHz and 200 Hz) and 3
excitation levels (50%, 75% and 100%).
Though data was acquired at different frequency sampling rates, only the 25 KHz sampling data
files were used for all the analysis related to this thesis.
57
Sampling Frequency = 25600Hz
Vehicle
A
Road Files
Cobble Stone
Belgian Block
NVH 1
Gravel
Harshness
NVH 2
Actuator excitaion(%)
100, 75, 50
100, 75, 50
100, 75, 50
100, 75, 50
100, 75, 50
100, 75, 50
B
Cobble Stone
Belgian Block
NVH 1
Gravel
Harshness
NVH 2
Ditch Twist
100, 75, 50
100, 75, 50
100, 75, 50
100, 75, 50
100, 75, 50
100, 75, 50
100, 75, 50
Random
100, 75, 50
Cobble Stone
Belgian Block
Gravel
Harshness
NVH 2
100, 75, 50
100, 75, 50
100, 75, 50
100, 75, 50
100, 75, 50
Random
100, 75, 50
Cobble Stone
Belgian Block
NVH 1
Gravel
Harshness
NVH 2
NVH 3
100, 75, 50
100, 75, 50
100, 75, 50
100, 75, 50
100, 75, 50
100, 75, 50
100, 75, 50
Random
100, 75, 50
Cobble Stone
Belgian Block
NVH 1
Gravel
Harshness
NVH 2
NVH 3
Ditch Twist
100, 75, 50
100, 75, 50
100, 75, 50
100, 75, 50
100, 75, 50
100, 75, 50
100, 75, 50
100, 75, 50
Random
100, 75, 50
C
D
E
Note: vehicle A and D are the same
Table 3.3: Summary of data collected on vehicles A, B, C, D and E on MTS 320 road simulator
58
Chapter 4: Data Analysis and Results
4.1 Choice of Scales
Recalling discussion from Chapter 3, data used for analysis is sampled at 25 KHz. Theoretically,
the maximum frequency that can be clearly reconstructed digitally is about 12.5 KHz. To
maintain a safely factor, we analyze data in the range of 0 – 11 KHz. The wavelet code allows
choice of frequency instead of the traditional scale values for ease of understanding. Using
Equation (2.21) the least scale (s0) value corresponding to the maximum frequency (in the
specified range) is calculated. Number of scales (J) and the scale size (resolution δj ) are
calculated by using the following equations. The scales can be written in the form of fractional
powers of 2.
s j = s0 2 j*δj
j = 0, 1, 2,……J
(4.1)
If fmax and fmin correspond to the maximum and minimum frequencies in the range of analysis,
J = δj −1 log 2 (
f max
)
f min
(4.2)
A grid of dyadic scales can be built by substituting δj =1. A smaller resolution value will
increase the number scales at which CWT is calculated and therefore increases the computations
performed. It has been observed that MATLAB cannot handle very low δj values and quickly
runs out of memory. A reasonable value of δj =0.2-0.25 is chosen when a signal of 6 – 8 seconds
length is analyzed. However, for short signals, a lower δj can be used for a finer plot. Data
records collected at 25 KHz sampling on the vehicles are of different time lengths (20 seconds 12 minutes long). Given the limitation stated above, blind processing consumes a lot of time and
can be a problem.
59
4.2 MATLAB Code
A MATLAB code incorporating all the discussed parameters was built. This used parts of
analysis code developed by Torrence and Compo [44] for oceanographic data analysis. The current
code is set to read Standard Data File format. But, it can easily be rewritten to read other
standard formats.
4.3 Display
As mentioned in Chapter I, squeak and rattle sounds can be interpreted as a sudden increase in
energy of the microphone signal in comparison to the background. Considering this reasoning,
wavelet power spectrum is used as the user display. This also allows the integration of energy
distributed between the real and imaginary parts of the wavelet. The magnitudes of complex
CWT coefficients obtained by following the procedure described in Chapter 2 are squared
(square of magnitude of complex coefficients) to obtain the power spectrum. A general 3D color
surface plot can be formulated using time, frequency (calculated using Equation (2.21) from
respective scales) and log of the power spectrum on the X, Y and Z axes respectively. It is easier
to notice the subtle shade changes with the Z axis pointing at the viewer. Therefore, a 2D version
of the 3D plot is shown in all the results discussed. A color bar in the side indicates intensity.
Though a wavelet plot is traditionally represented with scale as one of the describing
components, frequency was used in its place throughout this report. This form of representation
is used by many authors due to the ease of interpretation [35, 48]. The title of the plot may include
some or all details of the file analyzed such as vehicle tested (A, B, C, D or E), the wavenumber
( ω 0 ), the scale resolution ( δj ), selected frequency range (f) and the starting and ending time of
data selected (time frame (t)). Additionally, wavelet plots in Section (4.6) include the volume set
60
on the computer from which the signal was fed into the speakers. The interpretation of this
convention was described in Section (3.3.2) of Chapter 3. All wavelet and filter bank plots
presented correspond to the energy normalization configuration unless specified.
Figure 4.1: Sample plot; Vehicle ‘E’, excitation (100%),
ω 0 =50, δj =0.18, f = 100 – 5000Hz
4.4 Block and Overlap Processing
Results obtained, which will be showcased henceforth, were generated from microphone signals
sampled at 25 KHz. It was observed that MATLAB was running out of virtual memory with 6-7
seconds long blocks of data. Blocks of 2048 discrete points were found to be comfortable to
work with and also aided in improving the memory usage. The edge error discussed in Section
2.8 now repeats in every block. This makes the display nearly unreadable for analysis. To
overcome this issue, a form of overlap processing was developed and applied. An illustration of
this processing is shown for clarity in Fig (4.2). Each block shown is 2048 points long.
61
Methodology discussed in Section (2.5.6) is performed to obtain the CWT coefficients. The next
block chosen has 75% overlap with the earlier block. This process is repeated. The hatched
portion, 256 point long, was found to have the effect of the edge error after close inspection.
Coefficients lying in these portions are deleted from the processed blocks. The part of blocks
which overlap, are averaged together. This is pictorially shown in Fig (4.3). It is easy to see that
this process conserves the continuity in time. All data, except for the first and last three blocks,
gets averaged 3 times.
2048
Affected data
1
Clear data
512
2
3
4
256
5
6
7
8
Figure 4.2: Illustration of flow of block processing for edge effect removal
62
1
Affected data
2
Clear data
averaged
Continuous data
3
averaged
Continuous Time
Figure 4.3: Putting data together, averaging clear data and removal of affected data
4.5 Wavelet Filter Banks
This section explains the effect on frequency transforms of wavelet filter banks with change in
wavenumber, frequency range selection, scale resolution and normalization. This will aid in
deeper understanding of the analysis presented in later sections.
The frequency transform of the Morlet wavelet looks like a band pass filter. The properties of
this filter, evidently, are dependent on the parameters that control the shape of the wavelet. The
following Figs 4.4(a) (b) (c) (d) and (e), show the time and frequency domain equivalents of an
63
energy normalized Morlet when the analysis frequency was f = 100 – 8000Hz, ω 0 = 50 and
δj =0.2. The number of digital time points (discretization) used is retained for all the scales as is
the case in the working code. The choice of scales was taken in a logarithmic fashion as
described by Equations (4.1) and (4.2). The value of scale (s) is indicated in the title. This value
is derived using the process described in the Section 4.1. The overlap of all scales, 32 in this
case, to describe the given frequency range, is indicated in Fig 4.4 (f).
Figure 4.4(a): scale 2 = 0.0011
Figure 4.4(b): scale 5 = 0.0017
Figure 4.4(c): scale 10 = 0.0035
64
Figure 4.4(d): scale 20 = 0.0139
Figure 4.4(e): scale 30 = 0.0554
ω 0 =50, δj =0.2, f = 100 – 8000Hz (scale value decreasing from L to R) in Figs 4.4 (f) and (g)
Figure 4.4(f): Filter bank concept with energy normalization (overlay of all scales)
65
Amplitude normalization applied on the wavelet instead of the common energy normalization is
shown in Fig 4.4(g). Filter banks corresponding to STFT will look similar to the amplitude
normalized wavelet bank but without the change in the frequency spread indicated in Fig 4.4 (g).
Figure 4.4(g): Filter bank concept with amplitude normalization (overlay of all scales)
The frequency domain wavelet filter bank is shown on the right hand side of an example wavelet
plot in Fig 4.5 (a). The parameters have been retained from the previous discussion. The
frequency transform of the wavelet is wider at higher frequencies (smaller scales) i.e. has a
bigger frequency spread. This is reflected in the wavelet plot as wide lobes at higher frequency
and narrow ones at lower frequency. The increase in the lobe width can be observed to be
logarithmic. This is due to the choice of wavelet scales in the frequency range of analysis (see
section on choice of scales). Also, in Fig 4.5 (a) it can be seen that the amplitude of the filters
reduces, from left to right but all the filters have equal area under the curve. This property
evidently enhances (weights higher) the lower frequencies compared to the higher ones. Energy
normalization, a character imparted to this analysis, as described in Section (2.5.7), is the reason
behind this observation.
66
Figure 4.5(a): Example of energy normalized wavelet plot with corresponding wavelet filter bank
Figure 4.5(b): Example of wavelet plot and filter bank with amplitude normalized to 1
67
The amplitude normalization does not affect the width of any filter but only the amplitudes. All
frequencies are now weighted equally but the scale energy now is different at different scales.
Figs 4.4(f), 4.5 (c) and (d) indicate the decrease in the density of the bank with higher scale
resolution ( δj ) for same ω 0 (= 50) and f (= 100 – 8000Hz). A lower δj improves readability of
the plot. This property will be discussed further in the next section in terms of wavelet plots (see
Figs 4.7(b) and (c)). But, with lower δj , there is an apparent increase in computations. Another
important observation is that at low frequencies (higher scale values), peaks in the Fourier
domain are not well defined at higher wavenumbers. This is expected because of the incomplete
sampling of the Morlet in the time domain. This is not of a major consequence because the
frequencies in that range are already smeared due to noise. Better results can be expected at low
frequencies at low wavenumbers. This is demonstrated later in this section (see Fig 4.14 (f)).
Figure 4.5(c):
δj =0.25
68
Figure 4.5(d):
δj =0.3
Figs 4.5 (e) and (f) show the filter bank with smaller frequency ranges at different wavenumbers.
δj =0.2 f = 2000 – 6000Hz in Figs 4.5 (e) and (f)
Figure 4.5(e):
ω 0 =50
69
Comparing Figs 4.4 (f) and 4.5 (g), it can be noticed that the amplitude of the bank as a whole
decreases relatively with decrease in the wavenumber ( ω 0 ). The overall weighting, especially the
low frequency, is less comparatively. This is the reason for the low frequency components to be
recognized better at low wavenumbers in the energy normalization configuration. Discussion on
this is presented in the following sections.
Figure 4.5(f):
ω 0 =10
Fig 4.5 (g) and (h) have δj =0.2 f = 100 – 6000Hz as parameters. Fig 4.5 (h) is amplitude
normalized.
70
Figure 4.5(g):
ω 0 =10, δj =0.2 f = 100 – 6000Hz
Figure 4.5(h): Amplitude normalized, ω 0 =10, δj =0.2 f = 100 – 6000Hz
The properties discussed in this section are valid for both energy and amplitude normalized
wavelets.
71
4.6 Code Verification
A continuous wavelet analysis code incorporated with the above discussed parameters was
implemented on the data. A separate set of data was acquired in the validation configuration
discussed in Section (3.3.2). The road simulator system was run at 0, 50, 75 and 100%
excitations. Data was collected with the speakers in the foot well emanating the chirp sound of
known frequency and time content using the DAC Express software. This data was analyzed
using the code developed for the analysis. Since the content of the signal is known, it is easy to
interpret the plot and check for errors. This process was repeated for three different frequency
ranges as mentioned in Section (3.3.2). MATLAB spectrograms of the chirp signals and the plots
generated by the code are compared. Note that all the wavelet plots presented are generated from
data measured in the validation configuration on Vehicle ‘E’ only. Data analyzed using an
established Matlab code employing analytic wavelet transform
[52, 58]
is also presented for some
cases as a cross check. This code was developed by Dr. Jay H. Kim [58] and his graduate students
at the University of Cincinnati.
Figure 4.6 (a): Spectrogram of the ‘low frequency’ chirp; 200 – 1100Hz
72
The spectrogram Fig 4.6 (a) shows the frequency content of 200 ~ 1100Hz (This plot was
generated using the spectrogram() command in MATLAB). Fig 4.6 (b) is the wavelet plot of the
chirp signal recorded by the microphones placed inside the vehicle when there was no excitation
imparted to the shakers (0% excitation). The presence of low frequency ambient noise is clearly
seen from the plot. The color bar indicates the energy in the signal. The high energy areas
indicate presence of a disturbance with the backdrop of the minimal background noise. Since this
is a zero excitation (0%) condition, the chirp and its time frequency characteristics are easily
recognizable. In fact, they match the spectrogram results closely. This indicates the T-F tracking
capability of the code. In Figs 4.6 (b), (c), (d) and (e) the low frequency chirp is plotted at
Volume = 25, f = 100 – 5000Hz, δj =0.2.
Figure 4.6(b): No shaker excitation (0%),
ω 0 =50, t=1-5sec
A noise perceptibility chart was also created in this configuration (see Table 4.1). The author,
seated in the driver’s seat, rated the chirp sounds as heard by an amateur squeak and rattle
73
technician. The rating was on a scale of 1 – 10 with 1 meaning ‘not heard’ and 10 meaning ‘loud
and easily heard’. It is the opinion of the author that the chirp in the frequency range 31003700Hz, Volume 25, was the closest to a real squeak sound. The plot in Fig 4.6(b) pertains to
Volume 25. Figs 4.6(c) and (d) are plots of the same chirp signal but with the hydraulic actuators
now at 100% excitation.
Chirp Frequency range 200 ~ 1100 Hz (low frequency)
Chirp time = 0.3 seconds
Shaker Excitation
100%
75%
50%
Volume of Speaker
25
5
7
8
50
9
10
10
100
10
10
10
Chirp Frequency range 3100 ~ 3700 Hz (medium frequency)
Chirp time = 0.4 seconds
Shaker Excitation
100%
75%
50%
Volume of Speaker
25
7
8
9
50
8
9
10
100
10
10
10
Chirp Frequency range 6100 ~ 8500 Hz (high frequency)
Chirp time = 0.3 seconds
Shaker Excitation
100%
75%
50%
Volume of Speaker
25
6
7
8
50
9
10
10
100
10
10
10
Table 4.1: Noise ratings inside the vehicle
74
Figure 4.6(c): Shaker excitation (100%),
ω 0 =10, t = 15-20sec
Figure 4.6(d): Shaker excitation (100%),
ω 0 =50, t=15-20sec
75
A low value of wavenumber ω 0 is used here to effectively characterize the low frequency
components (see Fig 4.6(c)). The signature of the chirp is clearly seen in the wavelet plot. Also,
random patches of high energy are also noticeable. This is a strong indication of an existence of
multiple squeaks or rattles. Similar cases will be dealt with in detail in the results section. Heavy
low frequency background noise is also detected. This noise is thought to be induced into the
vehicle interior due to the vehicle’s suspension itself. The low frequency rattle signature and the
chirp are clearly identified in spite of the background noise. This proves the robustness of the
technique applied for analysis. Fig 4.6(d) shows how low frequency information appears
smeared with higher wavenumber value. This is because the wavelet now has more oscillations
and low frequency information is smeared as described in [48].
Figure 4.6(e): Amplitude normalized, Shaker excitation (100%),
ω 0 =10, t=15-20sec
Comparing Figs 4.6 (c) and (e), it is observed that the amplitude normalized plot weights all
frequencies equally as expected. The plot does not provide any new information.
76
Figure 4.7(a): Spectrogram of the ‘medium frequency’ chirp; 3100 ~ 3700Hz
In order to present an overall picture, similar analysis was applied to the mid frequency and high
frequency chirps too. The effect of scale resolution increase (lower δj ) can be seen between Figs
4.7(b) and 4.7(c) over the frequency range f = 500-6000Hz with ω 0 =50, t = 3-7sec, Volume = 25
and shakers excited at 100%. These figures correspond to analysis performed on the signal
bearing the mid-frequency chirp signal. The same frequency range of 500-6000Hz is analyzed
closer with higher number of scales in fig 4.7(c). The continuity of the chirp is clearly improved
with additional scales.
Note that 8 microphones are used for effective capturing of any disturbance as described in
Sections 3.3.1 and 3.3.2. Wavelet plots of all the microphones are not included for every sample
plot. The author confirmed that CWT performed on all the microphone channels for an arbitrary
77
signal input chosen from the data bank at disposal, resulted in plots similar in appearance.
Therefore, an arbitrary microphone channel is chosen to be presented in this report.
Figure 4.7(b):
δj =0.2
Figure 4.7(c):
δj =0.1
78
Figure 4.8(a): Spectrogram of the ‘high frequency’ chirp; 6100 ~ 8500Hz
A closer look at Figs 4.8(b) and (c) will show the resolution improvement due to lower δj value
as observed earlier. It is also suggested that a wavenumber of about 50 is found to have a good
balance of frequency and time sharpness. It will be of interest to note that all the data sets shown
in this section had the chirp played at Volume = 25 and 100% actuator excitation. It can be said
with confidence that, in all these cases, the chirp is located with excellent clarity. It is only
obvious that in files with higher chirp volume or lower actuator excitation or both, representation
of the chirp in the wavelet plot will be extremely clear. Figs 4.8 (b), (c) and (d) are processed
plots of the high frequency chirp signal with the actuators at 100% excitation, ω 0 = 50, Volume =
25, f= 1000-10000Hz and t = 2-5 sec.
79
Figure 4.8(b): δj =0.25
Figure 4.8(c): δj =0.1
80
Figure 4.8(d): Amplitude normalized, δj =0.1
Fig 4.8 (e): AWT plot of the ‘high frequency’ chirp
Fig 4.8(e) is the plot of the same data analyzed in this section. The chirp signature is not
very evident unlike the wavelet transform.
81
4.7 Results
A few successful detections using the CWT code are discussed in this section. Again, as
mentioned in the previous section, data only from an arbitrarily chosen microphone is presented.
STFT plots (generated using MATLAB spectrogram function) of the same time frames are also
presented to showcase a comparative study. Hanned, 256 point windows with zero overlap were
applied on the data for obtaining the STFT plots.
Case 1
A very distinct rattle is observed on vehicle ‘D’ when the actuators were driven with the road
profile file Harshness at 100% excitation. Fig 4.9 (a) exhibits disturbances short in time and
sweeping a frequency range of 0-8000 Hz (at least). This is the exact characteristic which is
typical of a rattle. The result was cross checked on all 8 microphones. Each showed similar
pattern in time. Looking at the characteristics of this disturbance, it can be said that this is caused
by a loose part set into vibration by the simulated road input.
82
Figure 4.9(a): Shaker excitation (100%);
ω 0 =50, t = 13-16sec
The low frequency noise smears the plot till about 1700Hz due to the use of a higher
wavenumber ω 0 =50. Analysis on the same time frame at a lower wavenumber ω 0 =10 is shown
in Fig 4.9 (b). The low frequency content of the rattle is seen clearly. All wavelet plots
corresponding to Case 1 have f = 100 – 8000Hz and δj = 0.2. Fig 4.9 (c) is the AWT plot of the
same data processed for Fig 4.9 (a). The disturbance seen in Fig 4.1 (a) is seen here too.
83
Figure 4.9(b): Shaker excitation (100%);
ω 0 =10, t=13-16sec
The code allows the choice of frequency range. Frequencies above 1000Hz are shown in Fig
4.9(d).
Fig 4.9 (c): AWT plot
84
Figure 4.9(d): STFT, shaker excitation (100%), t= 13-16sec
Figure 4.9(e): Shaker excitation (100%);
ω 0 =35, t = 13-16sec
85
Figure 4.9(f): Shaker excitation (75%);
ω 0 =35, t=13-16sec
Figure 4.9(g): Shaker excitation (50%);
ω 0 =35, t =13-16sec
86
Fig 4.9 (f) and (g) are plots of the same file but at 75% and 50% excitations respectively. It is
easy to see that the intensity of the rattle at lower excitations, decreases. This is an expected
result if the threshold of the rattle is cleared at lower excitations values itself. Figs 4.9 (h) and (i)
correspond to the 50% and 100% excitations of the following three seconds. It is observed that
the disturbance continues in time. This is prominently seen in the plot corresponding to full
excitation. At 50% the rattle seems to persist but with comparatively lesser strength. Also, data
from a different channel was processed to reiterate that the disturbance is perceived by different
microphones.
Figure 4.9(h): Shaker excitation (50%);
ω 0 =30, t=16-21 sec
87
Figure 4.9(i): Shaker excitation (100%);
ω 0 =30, t=16-21sec
Figure 4.9(j): STFT, shaker excitation (100%); t=16-21sec
88
Comparing STFT plots Figs 4.9 (d) and (j) to their wavelet counterparts, Figs 4.9 (b) and (i), it is
clearly observed that the STFT itself can detect the presence of disturbance in the time stream.
The frequency content estimated at various times by both techniques matches to a large extent.
Case 2
A sudden strong rattle is noticed in data from Vehicle ‘E’. The hydraulic actuators were being
fed the Cobblestone road profile file at full excitation (100%). All wavelet plots included to
describe this case use ω 0 =30, δj =0.25 f = 100-8000Hz, t = 11-16sec.
Figure 4.10(a): Shaker excitation (100%)
This disturbance is not a continuous one like Case 1. This is a case where a body parts come in
contact with one another and produce intermittent noise. A medium wavenumber of 30 is chosen
to perform the analysis. There is a clear existence of an intermittent disturbance at about 11.6-
89
11.7 seconds. The frequency sweep can be observed from the plot. There is also indication of
low intensity disturbances at approximately 13 and 13.7 seconds.
Figure 4.10(b): Shaker excitation (75%)
90
Figure 4.10(c): Shaker excitation (50%)
91
Fig 4.10 (b) and (c) are the low excitation cases. The disturbance disappears as the excitation
decreases.
Figure 4.10(d): STFT, shaker excitation (100%), t=11-16sec
Fig 4.10 (d) is the STFT of the same short time history used for plotting Fig 4.10 (a). There is a
striking resemblance between the two plots when compared.
Case 3
A low frequency disturbance is observed in the test data recorded for the Vehicle B. Actuators
were on full excitation on the NVH 2 road file. All wavelet plots in this case use the frequency
range f = 100-8000Hz and δj =0.25.
92
Figure 4.11(a): Shaker excitation (100%);
ω 0 =30, t=12-16sec
The Fig 4.11 (a) shows clear indications of low frequency disturbance all through time. Since the
active frequency content appears to be in the low frequency range, a smaller wavenumber is
employed in an attempt to see the content clearly. Fig 4.11 (c) shows the same part of the file
analyzed at with a wavelet containing lesser number of oscillations. As expected, a cleaner low
frequency band is obtained.
Figs 4.11 (a) and (b) are wavelet and STFT plots of the same signal. As noticed earlier, the STFT
detects the disturbances, thus matching the capability of the Morlet wavelet.
93
Figure 4.11(b): STFT, shaker excitation (100%), t = 12 – 16sec
Figure 4.11(c): Shaker excitation (100%);
ω 0 =10, t=11-16sec
94
Figure 4.11(d): Shaker excitation (100%);
Figure 4.11(e): Shaker excitation (75%);
ω 0 =10, t=16-21sec
ω 0 =10, t=11-16sec
95
Figure 4.11(f): Shaker excitation (50%);
ω 0 =10, t=11-16sec
The analysis of the plots of the three different actuator excitations, shows that this is a continuing
disturbance and that it is present at appreciable intensities at the two other fractional excitations
also. Also, comparing Fig 4.11 (f) and (g), it can be observed that the low intensities are also
appreciably encoded by the STFT.
96
Figure 4.11(g): STFT, shaker excitation (50%), t=11-16sec
Case 4
This is an example which can be interpreted as an impact sound. Analysis was conducted on
microphone data when the Vehicle ‘B’ was being subjected to a Ditch Twist file through the
actuators at full excitation. This is very different when compared to the earlier cases. The near
complete blue background indicates no presence of any other disturbances. This was later found
to be noise related to suspension twist. All the wavelet plots in this case use ω 0 =30, δj =0.25, f =
100 – 8000Hz as parameters. The actuators were at 100% excitation for all files considered here.
97
Figure 4.12(a): Vehicle B, t=1-4sec
This particular finding is an anomaly with respect to the vehicle tested (Vehicle ‘B’). A twist test
file was unavailable for all the vehicles tested. Vehicle ‘E’ was the only other car which came
with a twist file and analysis on data pertaining to Vehicle ‘E’ did not reveal this anomalous
behavior. This is illustrated in Fig 4.12(b). An arbitrary time frame is presented for generality.
98
Figure 4.12(b): Vehicle E, t=10-15sec
A closer observation reveals the existence of a constant band across time in Fig 4.12 (a). This is a
frequency related to the MTS 320 hydraulic pump. This was discussed in the evaluation carried
out earlier at UC-SDRL on the actuator system [38].
99
Figure 4.12(c): Amplitude normalized, Vehicle B, t=1-4sec
Figure 4.12(d): STFT, Vehicle B, t=1-4sec
100
Figure 4.12(e): Amplitude normalized, Vehicle E, t=10-15sec
Figure 4.12(f): STFT, Vehicle E, t=10-15sec
101
The amplitude normalized analysis detects the disturbance with the same precision as the energy
normalized wavelet analysis. As discussed in the theory section, all frequencies are weighted
equally. This property was found to be of no additional importance in the process of transient
detection when Morlets are used as analyzing wavelets. Fig 4.12 (e) is the AWT plot of the data
staring at t = 1 to t = 5. Comparing it with Figs 4.12 (a) and (d), we can see that the
The STFT results in all the four cases examined in this section show that the Morlet is not
superior to the STFT in the case of detecting a transient. Moreover, the computations involved in
a STFT are much less demanding. This result is due to the fact that the Morlet is basically a
sinusoid modulated by a Gaussian curve which can be compared to an STFT, of comparable
window width, performed with a Hanned window. The Morlet still gives the advantage of MRA
which is not possible in the STFT. Other wavelets may improve the detection process owing to
their frequency domain characteristics. However, wavelet analysis may still be the popular
choice when signature characterization of S&R is attempted. This is due to the additional ‘zoom
in zoom out’ property of wavelet analysis.
4.8 Holder Exponent Analysis
As already discussed, the wavelet analysis code has issues with processing long time histories.
The Holder exponent method was applied to the data in an attempt to verify its use to solve the
current problem since, blind processing of short time frames in long time histories results in
unwarranted use of resources. The Holder exponent is a measure of regularity of a signal.
Regularity is related to the smoothness of the signal. If a signal is continuous, it has a large
Holder exponent value. On the contrary, a discontinuous signal will have a low value. This
concept, in conjunction with CWT may be useful for quicker detection of S&R in long time
102
streams. The Holder exponents, from wavelet coefficients, can be extracted the following way [34,
35]
.
If the signal f(t) has a holder exponent α over [a,b], then there exists A>0 such that:
WTψ n ( s ) ≤ Asα + 0.5
∀(u, s ) ∈ [a, b]
(4.3)
Where WT (u , s ) is the wavelet transform modulus of f(t). The exponent α can be found at a
specific time point by finding the slope of the log of the modulus at that time versus the log of
the scale vector s
log WTψ n ( s ) = log( A) + (α + 0.5) log( s )
m=
The procedure enumerated
[34]
log WTψ n ( s )
log( s )
− 0 .5 = α
(4.4)
(4.5)
was applied to some of the data sets in hand. Since the process
involves extraction of an exponent corresponding to each time point, (a sample four second
window has 25600*4 time points) excessive memory is required to compute all exponents.
However, presence of noise all through time was also a hindrance in interpreting results
obtained. An example is presented in Figs 4.13 (a) and (b). Note that only one second of data
was used lower memory usage and to keep MATLAB functioning. In order to remove the
interference of low frequency noise, the frequency range was fixed at 1000 – 10000 Hz. The
Holder exponent plot (Fig 4.13 (b)) is inconclusive. It shows low exponent values all over the
time stream though majority of noise components have been removed. Moreover, this would
have been a useful tool if it did not use the wavelet coefficients as input. The computation of
wavelet coefficients and then the exponent values is not economical. This methodology may be
applicable at lower sampling frequencies with more statistical post processing as was the case in
[34].
103
Figure 4.13(a): Vehicle D, Random, shaker excitation (100%);
ω 0 =50, δj =0.25, f = 1000 – 10000Hz, t=33-
34sec
Figure 4.13(b): Holder exponents corresponding data used in fig 4.13(a)
104
Chapter 5: Conclusion and Future Work
5.1 Conclusion
The data acquisition methodology, MATLAB code and the analysis presented shows promise in
the area of automotive S&R detection and quantification. Several examples have been presented
of successful detections. The code developed can be used a robust analysis tool for both
laboratory based road simulator data and on road operational data. Both energy and amplitude
normalized complex Morlet wavelet analysis was compared to STFT analysis. Independent
wavelet parameter control incorporated in the code helps in varied forms of analysis depending
on the interest of the researcher and application. Though not tested, the code should be able to
work in higher sampling frequencies covering full spectrum of human hearing. Similar wavelets
like the Mexican hat [44] can replace the Morlet wavelet in the existing code with relative ease.
5.2 Retrospection
The code thus developed has proved to be a handy tool to analyze signatures of transient events
occurring in time streams. The frequency range selection, scale resolution variation, etc. can
easily be used by anyone with basic understanding of wavelet analysis. The use, though, is
subjected to certain constraining factors such as computer virtual memory and manual time.
Processing 6 – 7 second long time streams sampled at 25 KHz itself is found to be extremely
heavy on normal computer memory and MATLAB is observed to be unable to handle this
amount of data. Therefore, analysis has to be repeated several times to cover the full time stream.
The code does not have the capability to discern between good and bad data. Meaning, should
there be discontinuities in the acquisition process for any reason, the transform will still be
performed on the data available regardless of the error (gaps in data). Therefore, time domain
105
data needs to be checked for workability manually in order to get meaningful results. The
analysis is primarily quantitative. The human element is not considered inside the scope of the
current project. But the possibility of application is discussed later in this chapter.
Though four different vehicles were tested, only one vehicle of a certain model was tested due to
limited resources. Therefore, reaching a well researched conclusion on vehicle quality based on
statistical number of samples is not possible.
5.3 Future Work
1. Data acquisition on more vehicles belonging to same model should be performed for
better comparability and all rounded conclusions on quality of particular vehicles.
2. Signature extraction of squeaks and rattles can help in creation of a new wavelet with
particular application to S&R analysis. A wavelet and its dilates which are similar in
character to squeak or rattle in time domain, can be used effectively in the detection
process.
3. Robust methodology for indicating existence of disturbances in long time streams aiding
quicker analysis times.
4. Neural networks are commonly used in the fault detection schemes and are proven to
yield good results. Coupling the existing methodology with neural networks to automate
the process can be very interesting for automotive majors. Quick and reliable results can
be obtained with a well trained neural network.
5. Development of a comprehensive road profile file to be run on a road simulator by
making use of the signature knowledge base. This will help in reducing testing and
acquisition time but, yielding good results.
106
5.4 Possible Application in Qualitative Non-Stationary Sound Analysis
Research in the direction of characterizing S&R qualitatively has been ongoing in the automotive
field for a long time. Loudness and sharpness of sound is well understood in the stationary sense.
International standards exit and serve as a basis for a common understanding. One major setback
in this field of research is the non existence of a standard for measurement of non-stationary
signal loudness measurement. S&R, as was reiterated in this research are intermittent
disturbances and will require robust non-stationary methodologies to qualitatively understand
them. There exist individually developed methodologies that are used in the industry. The
procedure is described in literature
[26, 28]
. The author believes that the filter bank property of
orthogonal wavelet transforms can be used advantageously to split frequency data into critical
bands as described in ISO 532 - B (international loudness standard). Also, the CWT based
technique described can be coupled with non-stationary loudness measurements to come up with
relations between squeak/rattle signatures and corresponding loudness ranges. This will help in
formulating a squeak/ rattle database both in the quantitative and qualitative sense.
107
Bibliography
1. Weisch, G., Stucklschwaiger, W., Mendonca, A. A., Monteiro, N. T. S., Santos, A. A.
(1997), “The creation of car interior noise quality index for the evaluation of rattle
phenomena”, Proceedings of the 1997 SAE Noise and Vibration Conference, SAE paper
972018
2. Nolan, S. A., Rediers, B. E., Loftus, H. M., Leist, T. (1996), “Vehicle Squeak and Rattle
Benchmarking”, Proceedings of the IMAC – XIV, SEM, pp.483-489
3. Kavarana, F., Rediers, B. (1999), “Squeak and Rattle – State of the art and beyond”,
Proceedings of the 1999 Noise and Vibration Conference, SAE paper 1999-01-1728
4. Nolan, S. A., Sammut, J. P. (1992), “Automotive Squeak and Rattle Prevention”,
Proceedings of the 1992 SAE Noise and Vibration Conference, SAE paper 921065, pp.
355-363
5. Grenier, G. C. (2003), “The Rattle Trap”, Proceedings of the 2003 SAE Noise and
Vibration Conference, SAE paper 2003-01-1525
6. Soine, D. E., Evensen, H. A., Van Carsen, C. D. (1999), “A Design Assessment Tool For
Squeak and Rattle Performance”, Proc. SPIE Vol. 3727, Proceedings of the IMAC-XVII,
p.1428-1432
7. Wyrman, B., Baker, B., Dinsmore, M., Carey, A., Saha, P., Hadi, R. (2003), “Automotive
noise and vibration practices in the new millennium”, Proceedings of the 2003 SAE Noise
and Vibration Conference, SAE paper 2003-01-1589
8. Judek, T., Eiss, N., Trapp, M., Loftus, H. (1996), “Material Friction Pair Testing To
Reduce Vehicle Squeaks”, Proceedings of the IMAC-XIV, SEM, pp. 502-510
108
9. Juneja, V., Rediers, B., Kavarana, F. (1999), “Squeak Studies on Material Pairs”, SAE
paper 1999-01-1727, Proceedings of the 1999 SAE Noise and Vibration Conference
10. Trapp, M. A., Rozmus, R., Dapoz, A. (1997), “Joint Performance of Injection Molded
Thermoplastic Bosses Containing Post Consumer Recyclate: Possible Squeak and Rattle
Implications” SAE 972055, Proceedings of the 1997 SAE Noise and Vibration
Conference
11. Peterson, C., Wieslander, C., Eiss, N. S. (1999), “Squeak and Rattle Properties of
Polymeric Materials”, SAE paper 1999-01-1860, Proceedings of the 1999 SAE Noise and
Vibration Conference
12. Hunt, K., Rediers, B., Brines, R., McCormick, R., Leist, T., Artale, R. (2001), “Towards
a Standard for Material Friction Pair Testing to Reduce Automotive Squeaks”,
Proceedings of the 2001 SAE Noise and Vibration Conference, SAE 2001-01-1547
13. Beane, S. M., Marchi, M. M., Synder, D. S. (1995), “Utilizing Optimized Panel Damping
Treatments to Improve Powertrain Induced NVH and Sound Quality”, Technical Note,
Applied Acoustics, 45 (1995) pp181-187
14. Nolan, S. A., Yao, Y. X., Tran, V., Weber, W. F., Heard, G. S. (1996), “Instrument Panel
Squeak and Rattle Testing and Requirements”, Proceedings of the IMAC-XIV, SEM, pp.
490-494
15. Rusen, W., Peterson, E. L., McCormick, R., Byrd, R. (1997), “Next Generation Means
for Detecting Squeaks and Rattles in Instrument Panels”, Proceedings of the 1997 SAE
Noise and Vibration Conference, SAE 972061
109
16. Hurd, A. (1991), “Combining Accelerated Laboratory durability with Squeak and Rattle
Evaluation”, SAE paper 911051 pp107-118, Proceedings of the 1991 SAE Noise and
Vibration Conference
17. Byrd, R., Peterson, E. L. (1999), “A comparison of Different Squeak and Rattle Test
Methods for Large Modules and Subsystems”, Proceedings of the 1999 SAE Noise and
Vibration conference, SAE paper 99-0693
18. Liu, K. J., DeVilbiss, T., Freeman, M. (1999), “Experimental Analysis of Rattle Noise
Abatement in Seatbelt Retractor Assembly”, Proceedings of the SAE Noise and Vibration
Conference, SAE paper 1999-01-1723
19. Malinow, I. G., Perkins, N. C. (1999), “Seat Belt Retractor Rattle: Understanding Root
Causes and Testing Methods”, Proceedings of the 1999 SAE Noise and Vibration
Conference, SAE paper 1999-01-1729
20. Frusti, T. M., Fletcher, J. N., Grinn, J. G., Gu, Y. (1996), “Vehicle Seat Sub system
Squeak and Rattle Testing and Requirements”, Proceedings of the IMAC-XIV, SEM, pp.
497-501
21. Hsieh, S., Borowski, V. J., Yuan Her, J. H., Shaw, S. W. (1997), “A CAE Methodology
for Reducing Rattle in Structural Components”, Proceedings of the 1997 SAE Noise and
Vibration Conference, SAE paper 972057
22. Kuo, E. Y. (2003), “Up-Front Body Structural Designs for Squeak and Rattle
Prevention”, Proceedings of the 2003 SAE Noise and Vibration Conference, SAE paper
203-01-1523
23. Lim, T. C., Witer, A. J. (2000), “ Experimental Characterization of Engine Crankshaft
Rumble Noise Signature”, Journal of Applied Acoustics, 60 (2000) pp 45-62
110
24. Brines, R. S., Weiss, L., Stanley, G. (2002), “ Objective Metric for Evaluation of Full
Vehicle Squeak and Rattle Performance”, Proceedings of the International Congress and
Exposition on Noise Control Engineering, Inter-Noise 2002
25. Hamilton, D. (1999), “Sound Quality of Impulsive Noises: An Applied Study of
Automotive Door Closing Sounds”, Proceedings of the 1999 SAE Noise and Vibration
Conference, SAE paper 1999-01-1684
26. Feng, J., Hobelsberger, J. (1999), “ Detection and scaling of Squeak and Rattle Sounds”,
Proceedings of the 1999 SAE Sound and Vibration Conference, SAE paper 1999-01-1722
27. Uchida, H., Ueda, K. (1998), “Detection of Transient Noise of Car Interior Using NonStationary Signal Analysis”, Proceedings of the 1998 SAE Sound and Vibration
Conference, SAE paper 980589
28. Brines, R. S., May, P. (1997), “Subject: Application of Zwicker Loudness Measurements
to Squeak and Rattle”, source unknown
29. Van Auken, M., Zellner, J. W., Kunkel, D. T. (1998), “Correlation of Zwicker’s
Loudness and Other Noise Metrics with Drivers’ Over the Road Transient Noise
Discomfort”, Proceedings of the 1998 SAE Noise and Vibration Conference, SAE paper
980585, 1998
30. Lin, J., Qu, L. (2000), “Feature Extraction Based on Morlet Wavelet and its Application
for Mechanical Fault Diagnosis”, Journal of Sound and Vibration, 234(1), pp135-148
31. Scholl, D., Amman, S. (1999), “A New Wavelet Tecnique for Transient Sound
Visualization and Application to Automotive Door Closing Events”, Proceedings of the
1999 SAE Noise and Vibration Conference, SAE paper 1999-01-1682
111
32. Munoz-Najar, A., Hashemi, J. (1999), “Continuous Wavelet Transform Analysis of
Acoustic Emission Signals”, Proceedings of the IMAC XVII, pp99-102
33. Ishimitsu, S., Kitagawa, H., Horihata, S. (2000), “Active control of the Room Noise of
Ship by the Correlation analysis of Measured Wavelet”, Proceedings of the International
Congress and Exposition on Noise Control engineering, Inter-Noise 2000, paper
IN2000/552
34. Robertson, A. N., Farrar, C. R., Sohn, H. (2003), “Singularity Detection for Structural
Health Monitoring Using Holder Exponents”, Mechanical Systems and Signal
Processing, Vol 17, No (6), pp1163-1184
35. Robertson, A. N., Sohn, H., Farrar, C. R. (2004), “ Damage Detection Using Wavelet
Transforms for Theme Park Rides”, Proceedings of the IMAC XXII, 2004
36. Torrence, C., Compo, G. P. (1998), “A Practical Guide to Wavelet Analysis”, Bulletin of
the American Meteorological Society, Vol. 79, No. 1, pp. 61–78
37. Frenz, E., Dumbacher, S., Allemang, R. (2001), “Acoustic intensity Measurement of
MTS 320 Road Simulator Hydraulic Actuator”, Proceedings of the IMAC XIX, pp11731179
38. Frenz, E. R. (2004), “ Noise source reduction of a hydraulic road simulator for use as a
BSR evaluation platform”, Masters Thesis, University of Cincinnati
39. Ishimitsu, S. (2004), “Noise Source Contribution Analysis of Accelerating Cars by
Wavelets”, Proceedings of the International Congress and Exposition on Noise Control
engineering, Inter-Noise 2004
40. Ramirez, R.W. (1985), “The FFT, Fundamentals and Concepts”, Prentice Hall Inc., ISBN
0-13-314386-4
112
41. Allemang, R.J., Mechanical Vibrations III course notes, UC-SDRL
http://www.sdrl.uc.edu/academic-course-info/vibrations-iii-20-263-663
42. Hubbard, B. B., “The World According to Wavelets”, A K Peters, ISBN 1-56881-047-4
43. Polikar, R. (2001), “The Engineer’s Ultimate Guide to Wavelet Analysis, The Wavelet
Tutorial”, http://users.rowan.edu/~polikar/WAVELETS/WTtutorial.html, web resource.
44. Torrence, C., Compo, G. P. (1998), “A Practical Guide to Wavelet Analysis”, Bulletin of
American Meteorological Society, Vol 79, Issue 1, 1998, pp 61-78
The CWT code is available on the world wide web at:
http://paos.colorado.edu/research/wavelets/
45. Quian, S. (2002), “Introduction to Time Frequency and Wavelet Transforms”, Prentice
Hall PTR, ISBN 0-1303030-7
46. Ma, K., Tang, X. (2001), “Translation-Invariant Face Feature Estimation Using Discrete
Wavelet Transform”, Wavelet Analysis and Its Applications: Proceedings of Second
International Conference, WAA 2001, Volume 2251/2001
47. Teolis, A. (1998), “Computational Signal Processing with Wavelets”, Birkhauser, ISBN
0-8176-3909-8
48. Addison, P. S, Watson, J. N., Feng, T. (2002), “Low-Oscillation Complex Wavelets”,
Journal of Sound and Vibration, Volume 254, Issue 4, 2002, pp 733-762,
49. Meyers, S. D., Kelly, G. G., O’Brien, J. J. (1993), “An Introduction to Wavelet Analysis
in Oceanography and Meteorology: With Application to the Yanai Waves”, Monthly
Weather Review, Volume 121, pp2858-2866
50. Zwicker, E., Fastl, H. (1990), “Psychoacoustics: Facts and Models”, Springer-Verlag,
Berlin, 1990
113
51. Brines, R.S., Weiss, L. G., Peterson, E. L. (2001), “The Application of Direct Body
Excitation Toward Developing a Full Vehicle Objective Squeak and Rattle Metric”,
Proceedings of the 2001 SAE Noise and Vibration Conference, SAE paper 2001-01-1554
52. Mallat, S. (1998), “ A Wavelet Tour of Signal Processing”, Academic Press, ISBN 0-12466605-1
53. Allemang, R. J. (2004), Mechanical Vibrations II course notes, UC-SDRL
http://www.sdrl.uc.edu/academic-course-info/vibrations-ii-20-293-663/
54. Goswami, J. C., Chan, A. K. (1999), “Fundamentals of Wavelets: Theory, Algorithms
and Applications”, John Wiley & Sons Inc., ISBN 0-471-19748-3
55. Hall, M. L. (1993), “A Standard Data Format for Instrument Data Interchange - HP’s
SDF for Analysers – Technical”, Hewlett – Packard Journal
56. Valens, C. (1999), “A Really Friendly Guide to Wavelets”,
http://pagesperso-orange.fr/polyvalens/clemens/wavelets/wavelets.html#section5, web
resource
57. IDEAS Core Utility User’s Guide, SDRC, 1990
58. Zhu, X., & Kim, J. (2006), “Application of analytic wavelet transform to analysis of
highly impulsive noises” Journal of Sound and Vibration, 294(4-5), 841-855
114
Appendix – A
Following is the Matlab code used for analysis purposes. This is a set of 3 pieces of code. Name
the second wavelet75_256.m and the third as wave_bases.m. Copy all the files in one folder.
%***************************************************************
%******Wavelet analysis input code
%
INPUTS:
%
%
dj = The spacing between discrete scales. A smaller
%
# will give better scale (frequency) resolution,
%
but be slower to plot.
%
%
w0 = wavenumber, The mother wavelet parameter. refer to report for
more details
%
For 'MORLET' this is k0 (wavenumber), default is 6.
%
%
t0 = Sets the start time of the time axis.
%
%
Fs = The sample rate of filechoice.
%
%
Freqmin = The minimum frequency to analyze.
%
%
Freqmax = The maximum frequency to analyze.
%
%
%***************************************************************
clear all
clc
%****************************************************************
%
SDF FILE INPUT:
%
%***************************************************************
[filename, pathname] = uigetfile({'*.sdf','SDF Files (*.sdf)'},'Select SDF
Data File');
if(filename==0), return, end;
clear options
options.offset = 0;
options.length = 1024;
options.chans = []; % null ... all channels
sdf = readDXopen([pathname filename],options);
sdf = readDXdata(sdf,options);
sdf = readDXclose(sdf,options);
nTimePts = double(sdf.dataHdr.num_of_points * sdf.scanBig.num_of_scan);
nChans
= length(sdf.chanNum);
Tmin=0;
SampleRate=1/(sdf.xdata(2)-sdf.xdata(1));
Tmax=nTimePts/SampleRate;
Fmin=0;
Fmax=0.8*SampleRate/2;
2
disp(['SDF File Information: ' ]);
disp([' ' ]);
disp(['Number of time points: ' num2str(nTimePts)]);
disp(['Number of channels: ' num2str(nChans)]);
disp(['Sample rate: ' num2str(SampleRate) ' Samples/Second']);
disp(['Starting Time: ' num2str(Tmin)]);
disp(['Ending Time: ' num2str(Tmax) ' Seconds']);
disp(['Minimum Frequency: ' num2str(Fmin) ' Hertz']);
disp(['Maximum Frequency: ' num2str(Fmax) ' Hertz']);
disp([' ' ]);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%
Freqmin = input('Enter in the minimum Frequency to be analyzed: ');
Freqmax = input('Enter in the maximum Frequency to be analyzed: ');
choice=2;
if (choice==2)
displayfunc = 2;
end
map_chan=input('which channel? ' );
map_chan=map_chan-1;
BS=2048;Tstart=0;Tend=8;
Tstart=input(['Starting Time: (' num2str(Tstart) '): ']);
if isempty(Tstart),Tstart = 0;end;
Tend=input(['Ending Time: (' num2str(Tend) '):']);
if isempty(Tend),Tend = 8;end;
clear options
% options.chans = 0:2;
options.offset = fix(Tstart)*12.5*BS;
options.length = (fix(Tend)-fix(Tstart))*12.5*BS;
% options.length = fix(((Tend+.1)-fix(Tstart))*12.5*BS);
options.chans = map_chan; % null ... all channels
sdf = readDXopen([pathname filename],options);
sdf = readDXdata(sdf,options);
x = sdf.ydata(:);
x = x-mean(x);
Fs=1/(sdf.xdata(2)-sdf.xdata(1));
dt = 1/Fs;t0=Tstart;
t = t0 + (0:length(x)-1)/Fs;
t = t';
l=length(t);
sigsize = l;
3
sdf = readDXclose(sdf,options);
%***************************************************************
%
END INPUT SECTION
%***************************************************************
%***************************************************************
%
WAVELET ANALYSIS SECTION
%***************************************************************
% 1 = pad the time series with zeroes
pad = 1;
critical_wavenumber = 1.43*Freqmin;
w0 = input('Enter in a wavenumber (6 is default): ');
s0 = (w0 + sqrt(2+w0^2))/(4*pi)*Freqmax^-1;
signalsize = length(x)
if displayfunc == 1
j1max = (2.442*10^7)/(signalsize^1.095);
end
if displayfunc == 2
j1max = (3.735*10^6)/(signalsize^.9696);
end
dj_min = (log2(Freqmax/Freqmin)/j1max)
dj = input('Enter in a resolution: ');
j1 = fix(log2(Freqmax/Freqmin)/dj);
%***************************************************************
%
TRANSFORM GENERATION SECTION
%***************************************************************
%Call the wavelet.m file to calculate the wavelet transform
[W,period,scale,coi,daughter]=wavelet75_256(x,dt,pad,dj,s0,j1,mother,w0);
%Calculate the wavelet power spectrum
wps=abs(W).^2;
wps = 10*log10(wps/4E-10); %convert Pa to dB
clear W;
%***************************************************************
%
WAVELET OUTPUT SECTION
%***************************************************************
frequency = 1 ./period;
%
****************************************
%
activate if u want to plot the frequency equivalents of the morlets
used at
%
different scales
%
pt=1;
4
%
%
%
%
%
%
for pt=1:length(scale)
plot(daughter(pt,:))
hold on;
pt=pt+1;
end
***********************************************
figure
if displayfunc == 1
contourf(t, frequency ,wps,10);drawnow;
end
if displayfunc == 2
surf(t, frequency ,wps, 'EdgeColor', 'none');
axis([t(1), t(end) frequency(j1+1) Freqmax])
% activate if you wan tscale to be displayed instead of frequency
%
axis([t(1), t(end) scale(1) scale(end)])
set(gca, 'layer', 'top', 'xgrid', 'off', 'ygrid',
'off','YTick',[],'YTickMode','manual',...
'YTickLabelMode','manual','YTick',fliplr(frequency),'YTickLabel',(fliplr(fix(
frequency)))');
xlabel('Time (sec)');
ylabel('Frequency (Hz)');
v = [0 100];
caxis(v);
colorbar;
W0 = num2str(w0);
DJ = num2str(dj);
ch = num2str(map_chan+1);
description = [mother, ', ', 'w0= ', W0, ', ', 'dj= ', DJ, ',
',filename, ' ,', ch];
title(description)
end
drawnow;
clear wps t x frequency
%***************************************************************
%
END WAVELET OUTPUT SECTION
%***************************************************************
% activate if u want to listen to the data you are analysing
%
%
%
%
%
%
q = input(' listen to channel? 1 yes ; 0 no
if (q==1)
x = sdf.ydata(:) / max(abs(sdf.ydata));
len = round(10 * Fs); % 10 seconds
wavplay(x(1:len),Fs);
end
')
5
Wavelet75_256:
%WAVELET 1D Wavelet transform with optional singificance testing
%
%
[WAVE,PERIOD,SCALE,COI] = wavelet(Y,DT,PAD,DJ,S0,J1,MOTHER,PARAM)
%
%
Computes the wavelet transform of the vector Y (length N),
%
with sampling rate DT.
%
%
By default, the Morlet wavelet (k0=6) is used.
%
The wavelet basis is normalized to have total energy=1 at all scales.
%
%
% INPUTS:
%
%
Y = the time series of length N.
%
DT = amount of time between each Y value, i.e. the sampling time.
%
% OUTPUTS:
%
%
WAVE is the WAVELET transform of Y. This is a complex array
%
of dimensions (N,J1+1). FLOAT(WAVE) gives the WAVELET amplitude,
%
ATAN(IMAGINARY(WAVE),FLOAT(WAVE) gives the WAVELET phase.
%
The WAVELET power spectrum is ABS(WAVE)^2.
%
Its units are sigma^2 (the time series variance).
%
%
% OPTIONAL INPUTS:
%
% *** Note *** setting any of the following to -1 will cause the default
%
value to be used.
%
%
PAD = if set to 1 (default is 0), pad time series with enough zeroes to
get
%
N up to the next higher power of 2. This prevents wraparound
%
from the end of the time series to the beginning, and also
%
speeds up the FFT's used to do the wavelet transform.
%
This will not eliminate all edge effects (see COI below).
%
%
DJ = the spacing between discrete scales. Default is 0.25.
%
A smaller # will give better scale resolution, but be slower to
plot.
%
%
S0 = the smallest scale of the wavelet. Default is 2*DT.
%
%
J1 = the # of scales minus one. Scales range from S0 up to S0*2^(J1*DJ),
%
to give a total of (J1+1) scales. Default is J1 = (LOG2(N DT/S0))/DJ.
%
%
MOTHER = the mother wavelet function.
%
The choices are 'MORLET', 'PAUL', or 'DOG'
%
%
PARAM = the mother wavelet parameter.
%
For 'MORLET' this is k0 (wavenumber), default is 6.
%
For 'PAUL' this is m (order), default is 4.
%
For 'DOG' this is m (m-th derivative), default is 2.
%
6
%
% OPTIONAL OUTPUTS:
%
%
PERIOD = the vector of "Fourier" periods (in time units) that
corresponds
%
to the SCALEs.
%
%
SCALE = the vector of scale indices, given by S0*2^(j*DJ), j=0...J1
%
where J1+1 is the total # of scales.
%
%
COI = if specified, then return the Cone-of-Influence, which is a vector
%
of N points that contains the maximum period of useful information
%
at that particular time.
%
Periods greater than this are subject to edge effects.
%
This can be used to plot COI lines on a contour plot by doing:
%
%
contour(time,log(period),log(power))
%
plot(time,log(coi),'k')
%
%---------------------------------------------------------------------------%
Copyright (C) 1995-2004, Christopher Torrence and Gilbert P. Compo
%
%
This software may be used, copied, or redistributed as long as it is not
%
sold and this copyright notice is reproduced on each copy made. This
%
routine is provided as is without any express or implied warranties
%
whatsoever.
%
% Notice: Please acknowledge the use of the above software in any
publications:
%
``Wavelet software was provided by C. Torrence and G. Compo,
%
and is available at URL: http://paos.colorado.edu/research/wavelets/''.
%
% Reference: Torrence, C. and G. P. Compo, 1998: A Practical Guide to
%
Wavelet Analysis. <I>Bull. Amer. Meteor. Soc.</I>, 79, 61-78.
%
% Please send a copy of such publications to either C. Torrence or G. Compo:
% Dr. Christopher Torrence
Dr. Gilbert P. Compo
% Research Systems, Inc.
Climate Diagnostics Center
% 4990 Pearl East Circle
325 Broadway R/CDC1
% Boulder, CO 80301, USA
Boulder, CO 80305-3328, USA
% E-mail: chris[AT]rsinc[DOT]com
E-mail: compo[AT]colorado[DOT]edu
%---------------------------------------------------------------------------% 25 %(256 point) overlap of each following block. shift for acquiring blocks
frm time
% stream is 512;
function [wavenew,period,scale,coi,daug] = ...
wavelet75_256(Y,dt,pad,dj,s0,J1,mother,param);
if (nargin < 8), param = -1;, end
if (nargin < 7), mother = -1;, end
if (nargin < 6), J1 = -1;, end
if (nargin < 5), s0 = -1;, end
if (nargin < 4), dj = -1;, end
if (nargin < 3), pad = 0;, end
if (nargin < 2)
error('Must input a vector Y and sampling time DT')
end
7
n1
if
if
if
if
= length(Y);
(s0 == -1), s0=2*dt;, end
(dj == -1), dj = 1./4.;, end
(J1 == -1), J1=fix((log(n1*dt/s0)/log(2))/dj);, end
(mother == -1), mother = 'MORLET';, end
fourier_factor = 0;
%....construct time series to analyze, pad if necessary
% x(1:n1) = Y - mean(Y);
x(1:n1) = Y;
%....construct SCALE array & empty PERIOD & WAVE arrays
scale = s0*2.^((0:J1)*dj);
period = scale;
BS1 = 2048;
shift = 512;
ctr = 1;
wavenew = zeros(J1+1,n1); % define the wavelet array
wavenew = wavenew + i*wavenew; % make it complex
wave = zeros(J1+1,n1*5); % define the wavelet array
wave = wave + i*wave; % make it complex
wavetemp = zeros(J1+1,BS1*2); % define the wavelet array
wavetemp = wavetemp + i*wavetemp; % make it complex
wavetemp1 = zeros(J1+1,BS1); % define the wavelet array
wavetemp1 = wavetemp1 + i*wavetemp1; % make it complex
wavetemp2 = zeros(J1+1,BS1); % define the wavelet array
wavetemp2 = wavetemp2 + i*wavetemp2; % make it complex
w = zeros(J1+1,1024);
w = w + i*w;
for ct = 1:shift:((fix(n1/BS1)-1)*BS1)
%ct=1:2048 then 2049:4096.... processing the data in blocks of 2048
xtemp = [x(ct:ct+BS1-1),zeros(1,BS1)];
ktemp = 1:BS1;
ktemp = ktemp.*((2.*pi)/(length(xtemp)*dt));
%calculation of angular
freq eq(5)
k = [ktemp, -ktemp(fix((length(xtemp))/2):-1:1)];
f = fft(xtemp);
% loop through all scales and compute transform
for a1 = 1:J1+1
[daughter,fourier_factor,coi,dofmin]=wave_bases(mother,k,scale(a1),param);
wavetemp(a1,:) = ifft(f.*daughter); % wavelet transform[Eqn(4)]
daug(a1,:)=daughter;
end
wave(:,(ctr-1)*BS1+1:ctr*BS1) = wavetemp(:,1:BS1); %getting rid of the
zero padded area
ctr = ctr+1;
end
8
disp('size of wave')
size(wave)
wavetemp1(:,:) = wave(:,1:BS1);
wavetemp2(:,:) = wave(:,BS1+1:2*BS1);
%overlap processing of the first 2 blocks
% wavenew(:,1:2*BS1-512) = [wavetemp1(:,1:2048512+128),(wavetemp1(:,BS1+128-512+1:BS1-128)+wavetemp2(:,129:128+256))./2, ...
%
wavetemp2(:,512-128+1:BS1)]; %size of wavenew =
[J1+1 * 3584 ] for BS1=2048
wavenew(:,1:BS1+512) = [wavetemp1(:,1:512+256),(wavetemp1(:,512+256+1:BS1256)+wavetemp2(:,257:256+1024))./2, ...
wavetemp2(:,1024+256+1:BS1)];
for ct=3:1:ctr-1
wavetemp1(:,:) = wave(:,(ct)*BS1+1:(ct+1)*BS1);
w(1:J1+1,1:1024) = (wavenew(:,1024+256+((ct-3)*512)+1:1024+256+((ct3)*512)+1024)+wavetemp1(:,257:256+1024))./2;
%3584 = 4096-512; 1536 = 2048-512;384 = 512-128;
wavenew(:,1024+256+((ct-3)*512)+1:1024+256+((ct-3)*512)+1792) =
[w(:,:),wavetemp1(:,1281:BS1)];
end
period = fourier_factor*scale;
disp('scale')
scale
% fourier wavelength calculation
return
Wave_bases.m:
%WAVE_BASES 1D Wavelet functions Morlet, Paul, or DOG
%
% [DAUGHTER,FOURIER_FACTOR,COI,DOFMIN] = ...
%
wave_bases(MOTHER,K,SCALE,PARAM);
%
%
Computes the wavelet function as a function of Fourier frequency,
%
used for the wavelet transform in Fourier space.
%
(This program is called automatically by WAVELET)
%
% INPUTS:
%
%
MOTHER = a string, equal to 'MORLET' or 'PAUL' or 'DOG'
%
K = a vector, the Fourier frequencies at which to calculate the wavelet
%
SCALE = a number, the wavelet scale
%
PARAM = the nondimensional parameter for the wavelet function
%
% OUTPUTS:
%
%
DAUGHTER = a vector, the wavelet function
%
FOURIER_FACTOR = the ratio of Fourier period to scale
%
COI = a number, the cone-of-influence size at the scale
%
DOFMIN = a number, degrees of freedom for each point in the wavelet
power
9
%
(either 2 for Morlet and Paul, or 1 for the DOG)
%
%---------------------------------------------------------------------------%
Copyright (C) 1995-1998, Christopher Torrence and Gilbert P. Compo
%
University of Colorado, Program in Atmospheric and Oceanic Sciences.
%
This software may be used, copied, or redistributed as long as it is not
%
sold and this copyright notice is reproduced on each copy made. This
%
routine is provided as is without any express or implied warranties
%
whatsoever.
%---------------------------------------------------------------------------function [daughter,fourier_factor,coi,dofmin] = ...
wave_bases(mother,k,scale,param);
mother = upper(mother);
n = length(k);
if (strcmp(mother,'MORLET')) %----------------------------------if (param == -1), param = 6.;, end
k0 = param;
expnt = -(scale.*k - k0).^2/2.*(k>0);
norm =(pi^(-0.25))*sqrt(n)*sqrt(scale*k(2)) ;
[Eqn(7)]
%
norm=(pi^(-0.25))*sqrt(n)*scale ;
daughter = norm*exp(expnt);
daughter = daughter.*(k > 0.);
Morlet
% total energy=N
% Heaviside step function
fourier_factor = (4*pi)/(k0 + sqrt(2 + k0^2)); % Scale-->Fourier [Sec.3h]
coi = fourier_factor/sqrt(2);
% Cone-of-influence
[Sec.3g]
dofmin = 2;
% Degrees of freedom
elseif (strcmp(mother,'PAUL')) %-------------------------------- Paul
if (param == -1), param = 4.;, end
m = param;
expnt = -(scale.*k).*(k > 0.);
norm = sqrt(scale*k(2))*(2^m/sqrt(m*prod(2:(2*m-1))))*sqrt(n);
daughter = norm*((scale.*k).^m).*exp(expnt);
daughter = daughter.*(k > 0.);
% Heaviside step function
fourier_factor = 4*pi/(2*m+1);
coi = fourier_factor*sqrt(2);
dofmin = 2;
elseif (strcmp(mother,'DOG')) %-------------------------------- DOG
if (param == -1), param = 2.;, end
m = param;
expnt = -(scale.*k).^2 ./ 2.0;
norm = sqrt(scale*k(2)/gamma(m+0.5))*sqrt(n);
daughter = -norm*(i^m)*((scale.*k).^m).*exp(expnt);
fourier_factor = 2*pi*sqrt(2./(2*m+1));
coi = fourier_factor/sqrt(2);
dofmin = 1;
else
error('Mother must be one of MORLET,PAUL,DOG')
end
return
10
Appendix B
Inner product and Orthogonality
Let f(t) and g(t) be two functions in L2 [a, b] (where L2 [a, b] represent the set of square
integrable function in [a, b]). Then the inner product is defined as follows
b
f (t ), g (t ) = ∫ f (t ).g * (t )dt
a
Two vectors are said to be orthogonal if their inner product is zero.
v, w = ∑ vn w* n = 0
n
Similarly, two functions f(t), g(t) are said to be orthogonal if their inner product is zero.
b
f (t ), g (t ) = ∫ f (t ).g * (t )dt =0
a
A set of vectors (v1, v2, …..vn) are said to be orhtonormal if they are pairwise orthogonal to each
other and have a length of one.
vm , vn = δ mn
Where δ mn is the Kronecker delta function
⎧1ifm = n
⎩0ifm ≠ n
δ mn = ⎨
Similarly a set of functions {φ k (t )}, k = 1,2,3.... is said to be orthonormal if
b
φ k (t ), φl (t ) = ∫ φ k (t ).φl (t )dt = δ kl ;
a
*
∫ {φ (t ) }dt = 1
b
2
k
a
11