TECHNICAL REVIEW High-resolution Fly-over Beamforming Clustering Approaches to Automatic Modal Parameter Estimation

TECHNICAL REVIEW High-resolution Fly-over Beamforming Clustering Approaches to Automatic Modal Parameter Estimation
TECHNICAL REVIEW
No. 1 – 2012
BV 0064 – 11
ISSN 0007 – 2621
ËBV-0064---5Î
High-resolution Fly-over Beamforming
Clustering Approaches to Automatic Modal Parameter Estimation
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Review
No. 1 – 2012
Contents
High-resolution Fly-over Beamforming Using a Small Practical Array ............... 1
Jorgen Hald, Yutaka Ishii, Tatsuya Ishii, Hideshi Oinuma, Kenichiro Nagai, Yuzuru Yokokawa and
Kazuomi Yamamoto
Clustering Approaches to Automatic Modal Parameter Estimation.................... 29
S. Chauhan and D. Tcherniak
Copyright © 2012, Brüel & Kjær Sound & Vibration Measurement A/S
All rights reserved. No part of this publication may be reproduced or distributed in any form, or by any
means, without prior written permission of the publishers. For details, contact:
Brüel & Kjær Sound & Vibration Measurement A/S, DK-2850 Nærum, Denmark.
Editor: Harry K. Zaveri
High-resolution Fly-over Beamforming
Using a Small Practical Array*
Jorgen Hald†, Yutaka Ishii‡, Tatsuya Ishii**, Hideshi Oinuma**,
Kenichiro Nagai**, Yuzuru Yokokawa** and Kazuomi Yamamoto**
Abstract
This paper describes a commercially available fly-over beamforming system
based on methodologies already published, but using an array that was designed
for quick and precise deployment on a concrete runway rather than for minimum
sidelobe level. Time domain tracking Delay And Sum (DAS) beamforming is the
first processing step, followed by Deconvolution in the frequency domain to
reduce sidelobes, enhance resolution, and get absolute scaling of the source maps.
The system has been used for a series of fly-over measurements on a Business Jet
type MU300 from Mitsubishi Heavy Industries. Results from a couple of these
measurements are presented: contribution spectra from selected areas on the
aircraft to the sound pressure level at the array are compared against the total
sound pressure spectrum measured by the array. One major aim of the paper is to
verify that the system performs well although the array was designed with quick
deployment as a main criterion. The results are very encouraging. A second aim is
to elaborate on the handling of the array-shading function in connection with the
calculation of the Point Spread Function (PSF) used in deconvolution. Recent
publications have used a simple formula to compensate for Doppler effects for the
case of flat broadband spectra. A more correct formula is derived in this paper,
also covering a Doppler correction to be made in the shading function, when that
function is used in the PSF calculation.
*
†
‡
**
First published in Proc. 33rd AIAA Aeroacoustics Conference, 2012
Brüel & Kjær, Denmark.
Brüel & Kjær Japan.
JAXA (Japan Aerospace Exploration Agency), Japan.
1
Résumé
Cet article concerne un système à formation de faisceau pour mesures acoustiques
de survols d'aéronefs, disponible dans le commerce et qui, intégrant des
méthodologies ayant déjà fait l'objet de publications, est doté d'une antenne
acoustique plus conçue au départ pour être déployée rapidement et avec précision
sur une piste en béton que pour minimiser les niveaux de lobe secondaire. La
technique de formation de faisceau, avec suivi des délais temporels et somme,
constitue le premier volet de traitement du signal, précédant une déconvolution
dans le domaine fréquentiel pour réduire les lobes latéraux, améliorer la résolution
et obtenir une imagerie des sources en échelle absolue. Ce système a été utilisé
pour les mesures de survol d'un avion à réaction MU300 du constructeur
Mitsubishi Heavy Industries. Les résultats de quelques-unes de ces mesures sont
présentés ici : les spectres associés à des zones spécifiques de l'aéronef et
contribuant au niveau de pression acoustique sont comparés au spectre de pression
acoustique total mesuré par l'antenne. Cet article vise avant tout à vérifier le
fonctionnement correct de l'antenne au vu et en dépit du principal critère ayant
présidé à sa conception, à savoir sa capacité à être déployée rapidement. Dans un
deuxième temps, il étudie la manière dont est gérée la fonction d'ombre acoustique
dans le calcul de la fonction PSF (dispersion de points) utilisée pour la
déconvolution. Des publications récentes avaient utilisé une formule simple pour
compenser les effets Doppler en cas de spectres bande large plats. Une formule
plus exacte est ici présentée, qui couvre également la correction Doppler à
appliquer à la fonction d'ombre acoustique lorsque cette fonction est utilisée pour
le calcul de la PSF.
Zusammenfassung
Dieser Artikel beschreibt ein kommerziell erhältliches Beamforming-System für
Überflugmessungen, das auf bereits veröffentlichter Methodik beruht, jedoch ein
Array verwendet, das in erster Linie für den schnellen und präzisen Einsatz auf
einer Landebahn aus Beton und weniger im Hinblick auf minimierte
Nebenmaxima entwickelt wurde. Der erste Verarbeitungsschritt ist Delay-AndSum-Beamforming im Zeitbereich, gefolgt von Dekonvolution (Entfaltung) im
Frequenzbereich, um Nebenmaxima zu reduzieren, die Auflösung zu verbessern
und eine absolute Skalierung der Quellenkarten zu erhalten. Das System wurde für
eine Reihe von Überflugmessungen mit einem Business-Jet vom Typ MU300 von
Mitsubishi Heavy Industries verwendet. Es werden Ergebnisse von einigen dieser
Messungen präsentiert: Teilspektren von ausgewählten Bereichen am Flugzeug,
2
die zum Schalldruckpegel am Array beitragen, werden mit dem vom Array
gemessenen Gesamtschalldruckspektrum verglichen. Ein wichtiges Anliegen des
Artikels ist die Verifizierung, dass das System einwandfrei arbeitet, obwohl das
Hauptkriterium bei der Konstruktion des Arrays seine schnelle Einsatzfähigkeit
war. Die Ergebnisse sind sehr ermutigend. Ein weiteres Anliegen ist die
detaillierte Behandlung der Array-Abschattungsfunktion in Verbindung mit der
Berechnung der zur Entfaltung verwendeten Point Spread-Funktion (PSF). In
neueren Veröffentlichungen wurde eine einfache Formel zur Kompensation der
Doppler-Effekte für flache Breitbandspektren verwendet. Im vorliegenden Artikel
wird eine korrektere Formel abgeleitet, die ebenfalls eine in der
Abschattungsfunktion erforderliche Doppler-Korrektur umfasst, wenn diese
Funktion bei der PSF-Berechnung verwendet wird.
Nomenclature
b(t)
B(ω)
Bij(ω)
c
DAS
Dfmi, Dfmj
f
Hij(ω)
i
I
j
k
κ
m
M
M0
pm(t)
p̂ m ( t )
= DAS beamformed time signal
= DAS beamformed frequency spectrum
= DAS beamformed spectrum at focus point j due to model source i
= Propagation speed of sound
= Delay And Sum
= Doppler frequency shift factor at microphone m for signal from
point i and j, respectively
= Frequency
= Element of Point Spread Function: From model source i to
focus point j
= Index of monopole point source in Deconvolution source model,
i = 1, 2, …, I
= Number of focus/source points in calculation mesh
= Index of focus position, j = 1, 2, …, I, or imaginary unit – 1
= Wavenumber (k = ω/c)
= Parameter defining steepness in radial cut-off of array
shading filters
= Microphone index, m = 1, 2, …, M
= Number of microphones
= Mach number
= Sound pressure time signal from microphone m
= Shaded time signal for microphone m
3
Pm(ω)
Pmi (ω)
PSF
Πmodel
Πmeasured
Qi ( ω)
rmj (t)
rmj
Rm
Rcoh(ω)
smi(t)
smi
s0i
Si(ω)
t
U
U
wm(τ )
Wm(ω)
ω
= Frequency spectrum from microphone m
= Frequency spectrum from microphone m due to model source i
= Point Spread Function (2D spatial power response to a
monopole point source)
= DAS beamformed pressure power (pressure squared) from
the point source model in deconvolution
= DAS beamformed pressure power from an actual measurement
= Amplitude spectrum of model point source i
= Distance from microphone m to moving focus point j
= Distance from microphone m to focus point j at the centre of
an averaging interval
= Distance of microphone m from array centre
= Frequency dependent radius of active sub-array
= Distance from microphone m to moving source point i
= Distance from microphone m to source point i at the centre of
an averaging interval
= Distance from array centre to source point i at the centre of
an averaging interval
= Power spectrum of model point source i
= Time
= Aircraft velocity vector
= Aircraft velocity, U ≡ |U|
= Delay domain shading function applied to microphone m
= Shading function in frequency domain
= Angular frequency (ω = 2πf )
Introduction
Beamforming has been widely used for noise source localization and
quantification on aircraft during fly-over for more than a decade [1 – 6]. The
standard Delay And Sum (DAS) beamforming algorithm, however, suffers from
poor low-frequency resolution, sidelobes producing ghost sources, and lack of
absolute scaling. A special scaling method was introduced in [2] to get absolute
contributions. During recent years, deconvolution has been introduced as a postprocessing step to scale the output contribution maps, but also improving both the
low-frequency resolution and the sidelobe suppression [3 – 8]. For a planar
distribution of incoherent monopole sources, which is a fairly good model for the
4
aerodynamic noise sources of an aircraft, the output of a DAS beamforming at a
given frequency will be approximately equal to the true source power distribution
convolved in 2D with a frequency-dependent spatial impulse response, which is
called the Point Spread Function (PSF). The PSF is defined entirely by the array
geometry and the relative positioning of the array and the source plane, so for
stationary sources it can be easily calculated and used in a deconvolution
algorithm to estimate the underlying real source distribution. A difficulty with the
use of deconvolution in connection with fly-over measurements is the fact that the
DAS beamforming algorithm must be implemented in the time domain in order to
track the aircraft, while deconvolution algorithms work only in the frequency
domain with the source at a fixed position relative to the array. Deconvolution in
its basic form therefore cannot take Doppler shifts into account. A method to do
that in an approximate and computationally efficient way was introduced in [3],
further developed in [4] and applied with actual fly-over measurements in [5]. The
method adapts the PSF to the output from a DAS measurement on a moving point
source, assuming flat broadband source spectra. Under that assumption, the
spectral shape will remain almost unchanged from the Doppler shifts. The method
is able to compensate for the change in lobe pattern caused by Doppler shifts.
Most of the published applications of microphone arrays for fly-over
measurement have been using rather large and complicated array geometries
requiring considerable time to deploy and to measure the exact microphone
positions. This paper describes an investigation of the possibility of building an
array system that can be quickly deployed on a runway and quickly taken down
again. The entire system including the array and the implemented processing
methodology is described in “Method and System Overview” on page 6, and its
performance is illustrated in “Application to MU300 Business-jet Fly-over” on
page 16 by results from a series of fly-over measurements on a business jet. The
calculation of the PSF is covered in some detail. A derivation of the Doppler
corrected PSF is given in the Appendix, and it turns out to have a slightly different
form than assumed in [3], [4] and [5], although it produces almost identical results
when a frequency-independent array-shading function is used. “Accuracy of the
Point Spread Function” on page 14 presents an investigation of the match between
the analytical frequency domain PSF and the DAS response to a moving point
source.
5
Method and System Overview
The applied method follows the same overall measurement and processing scheme
as the hybrid time-frequency approach described in [5]. Aircraft position during a
fly-over is measured with an onboard GPS system together with speed, roll, yaw
and pitch. Synchronization with array data is achieved by recording an IRIG-B
time-stamp signal together with the array data and also with the GPS data on the
aircraft. The beamforming calculation is performed with a standard tracking timedomain DAS algorithm [2]. For each focus point in the moving system, FFT and
averaging in short time intervals are then performed to obtain spectral noise source
maps representing the aircraft positions at the middle of the averaging intervals.
Diagonal Removal is implemented as described in [2], providing the capability of
suppressing the contributions to the averaged spectra from the wind noise in the
individual microphones. With sufficiently short averaging intervals, the array
beam pattern will remain almost constant during the corresponding sweep of each
focus point. This means that a deconvolution calculation can be performed for
each FFT frequency line and for each averaging interval in order to enhance
resolution, suppress sidelobes and scale the maps. “Beamforming and
Deconvolution Calculations” on page 8 elaborates on the compensation for
Doppler effects in the calculation of the PSF used for the deconvolution.
Compensation for wind was not implemented in the prototype software used for
data processing. Fortunately, there was almost no wind on the day when the
measurements used in this paper were taken. But the results presented in
“Application to MU300 Business-jet Fly-over” on page 16 reveal some small
source offsets which could probably be reduced through a wind correction to be
supported in released software. The prototype software also does not support
compensation for atmospheric losses. Such compensation will be needed to
correctly reconstruct the source levels on the aircraft, but is not necessary to
estimate the contributions from selected areas on the aircraft to the sound pressure
at the array.
Overall System Architecture
Fig. 1 shows the array geometry (left) and a picture of the array deployed on the
runway. The array design and the use of a frequency-dependent smooth arrayshading function are inspired by [2]. However, to support quick and precise
deployment on the runway, a simpler star-shaped array geometry was
implemented. The full array consists of nine identical line-arrays which are joined
together at a centre plate and with equal angular spacing controlled by aluminium
6
arcs. The 12 microphones on one line array of length 6 metres were clicked into an
aluminium tube, which was rotated in such a way around its axis that the ¼ inch
microphones were touching the runway. The surface geometry of the part of the
runway where the array was deployed was very smooth and regular, so it could be
characterized to a sufficient accuracy by just measuring a few slope parameters.
Measurement of individual microphone coordinates was therefore not necessary:
the vertical positions were automatically and accurately obtained from the known
microphone coordinates in the horizontal plane and the runway slopes.
Fig. 1. Array geometry and picture of the array on the runway. Each microphone is clicked
into position in the radial bars with the microphone tip touching the runway. “Half
windscreens” can be added. The array diameter is 12 m, and there are 9 radial line arrays
each with 12 microphones
Due to the turbulence-induced loss of coherence over distance, a smooth
shading function was used that focuses on a central sub-array, the radius of which
is inversely proportional to frequency [2]. At high frequencies only a small central
part of the array is therefore used, which must then have small microphone
spacing. To counteract the resolution loss at low-to-medium frequencies resulting
from the high microphone density at the centre, an additional weighting factor was
applied that ensured constant effective weight per unit area over the active part of
the array [2]. The effective frequency-dependent shading to be applied to each
microphone signal was implemented as a zero-phase FIR filter, which was applied
to the signal before the beamforming calculation.
7
An important reason for the use of a shading function that cuts away signals
from peripheral microphones at higher frequencies is to ensure that the PSF used
for deconvolution will approximate the beamformer response to a point source
measured under realistic conditions with air turbulence. The PSF is obtained
purely from a mathematical model, so it will not be affected by air turbulence. If
the PSF does not accurately model an actually measured point source response,
then the deconvolution process cannot accurately estimate the underlying real
source distribution that leads to the measured DAS map. The shading function
must guarantee that at every frequency we use only a central part of the array that
is not highly affected by air turbulence [2].
A different way of handling the problem of a limited coherence diameter would
be the use of nested arrays, where different sub-arrays are used in different
frequency bands [5]. An advantage of the array design and shading method chosen
in this paper is the possibility of changing the shading function, and thus the active
sub-array, continuously with frequency. As will be outlined below, Doppler
correction must be applied in the shading filter, when that filter is used in the PSF
calculation.
As the array target frequency range was from 500 Hz to 5 kHz, a sampling rate
of 16384 samples/second had to be used. The microphones were Brüel & Kjær
Type 4958 array microphones, and a Brüel & Kjær PULSE™ front-end was used
for the acquisition. In addition to the 108 array microphone signals, an IRIG-B
signal and a line camera trigger signal were also recorded. While GPS data from
the aircraft would be available through file transfer only after the measurement
campaign, the line camera signal provided immediate information about aircraft
passage time over the camera position within the time interval of the recorded
microphone signals.
Beamforming and Deconvolution Calculations
The implemented fly-over beamforming software supports two different kinds of
output map on the aircraft: Pressure Contribution Density and Sound Intensity.
Both quantities can be integrated over selected areas on the aircraft to obtain the
contributions from these areas to the sound pressure at the array and the radiated
sound power, respectively. This paper is concerned only with the first quantity.
As argued above, the estimation of the Pressure Contribution Density does not
require any compensation for losses during wave propagation in the atmosphere.
For the same reason, Doppler amplitude correction is not required either. The first
calculation step is to apply the shading filters Wm(ω) to the measured microphone
pressure signals pm(t), m = 1, 2, …, M being an index over the M microphones, ω
8
the temporal angular frequency and t the time. To achieve equal weight per area
over the active central sub-array, the shading filters were defined as [2]:
Rm

2 
W m ( ω ) = K ( ω )R m  1 – Erf κ  ------------------- – 1
R (ω)  


coh
(1)
where Rm is the distance from microphone m to the array centre, Erf is the Error
Function, κ is a factor that controls the steepness of the radial cut-off, Rcoh(ω) is
the assumed frequency-dependent coherence radius (i.e., the radius of the active
sub-array). Finally, K(ω) is a scaling factor ensuring that at every frequency the
sum of the microphone weights equals one. The filters Wm(ω) are applied to the
microphone signals as a set of FIR filters. Effectively, the microphone signals are
convolved by the impulse responses wm(τ) of the filters, providing the shaded
microphone signals p̂ m ( t ) :
p̂ m ( t ) = ( p m ⊗ w m ) ( t )
(2)
For each point in the calculation mesh following the aircraft, DAS beamforming is
then performed as:
M
bj ( t ) =
r mj ( t )
-
 p̂m  t + ------------c 
(3)
m=1
bj (t) being the beamformed time signal at a focus point with index j, c the
propagation speed of sound, and rmj (t) the distance from microphone m to the
selected focus point at time t. Eq. (3) must be calculated once for each desired
sample of the beamformed signal, typically with the same sampling frequency as
the measured microphone signals. With the applied rather low sampling rate in the
acquisition, sample interpolation had to be performed on the microphone signals
to accurately take into account the delays rmj (t)/c. Eq. (3) inherently performs a
de-dopplerization providing the frequency content at the source [2].
Once the beamformed time signals have been computed, averaging of
Autopower spectra (using FFT) is performed for each focus point in time intervals
that correspond to selected position intervals of the aircraft, typically of 10 m
9
length. With a flight speed of 60 m/s, 256 samples FFT record length,
16384 samples/second sampling rate, and 66.6% record overlap, the number of
averages will be around 30. For the subsequent deconvolution calculation, an
averaged spectrum is considered as belonging to a fixed position – the position of
the focus point at the middle of the averaging time interval.
To introduce deconvolution and derive the associated PSF in a simple way, the
case of non-moving source and focus points, i.e., with rmj (t) equal to a constant
distance rmj, shall be considered first. Using e jωt as the implicit complex time
factor, Eq. (3) is then easily transformed to the frequency domain:
M
Bj ( ω ) =

W m ( ω )P m ( ω )e
jkr mj
(4)
m=1
where Bj is the beamformed spectrum, Wm is the shading filter applied to
microphone m, Pm is the spectrum measured by microphone m, and k = ω/c is the
wavenumber.
Consider now a source model in terms of a set of I incoherent monopole point
sources at each one of a grid of focus positions. Let i = 1, 2, …, I be an index over
the sources and j = 1, 2, …, I an index over the focus point. The sound pressure at
microphone m due to source i is then expressed in the following way:
– jks
mi
s 0i –jks mi
e
P mi ( ω ) ≡ Q i ( ω )s 0i ---------------- = Q i ( ω ) ------- e
s mi
s mi
(5)
where Qi is the source amplitude, smi is the distance from microphone m to source
number i, and s0i is the distance from the centre of the array to source number i.
With this definition, Qi is simply the amplitude of the sound pressure produced by
source number i at the centre of the array, which can be seen from Eq. (5) by
considering the array centre as microphone number 0. The beamformer output Bij
at position j due to source i is now obtained by use of Eq. (5) in Eq. (4):
M
B ij ( ω ) =
 Wm ( ω )Pmi ( ω )e
m=1
10
jkr mj
M
= Qi ( ω )
s 0i jk ( rmj – s mi )
-e
 Wm ( ω ) -----s mi
m=1
(6)
Based on Eq. (6) we define the power transfer function Hij (ω) from source i to
focus point j through the beamforming measurement and calculation process as
follows:
B ij ( ω )
H ij ( ω ) ≡ --------------Qi ( ω )
2
M
2
=

m=1
s 0i jk ( rmj – s mi )
W m ( ω ) ------- e
s mi
(7)
Since the sources are assumed incoherent, they contribute additively to the
power at focus position j, so defining the source power spectrum as Si ≡ ½|Qi|2, the
total power represented by the source model at focus point j is:
Π model, j ( ω ) =
 Hij ( ω )Si ( ω )
(8)
i
Deconvolution algorithms aim at identifying the non-negative point source
power values Si of the model such that the modelled power at all focus points
approximates as closely as possible the power values Πmeasured, j obtained at the
same points from use of the DAS beamformer Eq. (4) on the measured
microphone pressure data:
Solve
Π measured, j ( ω ) =
 Hij ( ω )Si ( ω )
with j = 1, 2, …, I, S i ≥ 0
(9)
i
The set of transfer functions Hij(ω) from a single source position i to all focus
points j constitutes the PSF for that source position, describing the response of the
beamformer to that point source:
PSF i ( ω ) ≡ { H ij ( ω ) } j = 1, 2, …, I
(10)
Once Eq. (9) has been (approximately) solved by a deconvolution algorithm, the
source strengths Si represent the sound pressure power of the ith model source at
the array centre. The pressure contribution density is therefore obtained just by
11
dividing Si by the area of the segment on the mapping plane represented by that
monopole source.
Several deconvolution algorithms have been developed for use in connection
with beamforming; see, for example, [7] for the DAMAS algorithm, introduced as
the first one, and [8] for an overview. The DAMAS algorithm is computationally
heavy, but supports arbitrary geometry of the focus/source grid, meaning that for a
fly-over application an irregular area covering only the fuselage and the wings can
be used [4, 5, 8]. Another advantage of DAMAS is that it can take into account the
full variation of the PSF with source position i. Algorithms like DAMAS2 and
FFT-NNLS are much faster, because they use 2D spatial FFT for the matrix-vector
multiplications to be calculated during the deconvolution iteration. This, however,
sets the restrictions that i) the focus/source grid must be regular rectangular, and
ii) the PSF must be assumed to be shift invariant to make the right-hand side in
Eq. (9) take the form of a convolution. The second requirement can be relaxed
through the use of nested iterations [8]. All results of this paper have been
obtained using an FFT-NNLS algorithm based on a single PSF with source
position at the centre of the mapping area. No nested algorithm was used.
Having introduced the concepts related to deconvolution in connection with
non-moving sources, we now return to the case of a moving source such as an
aircraft. In that case, the beamforming is performed in the time domain using
Eq. (3), followed by FFT and averaging in time intervals corresponding to selected
position intervals of the aircraft. As a result, we obtain for each averaging interval
a set of beamformed FFT Autopower spectra Πmeasured, j covering all focus point
indices j. Associating the spectra related to a specific averaging interval with the
focus grid position at the middle of the averaging interval, one might as a first
approximation just use the beamformed spectra in a stationary deconvolution
based on Eq. (9), i.e., with the PSF calculated using Eq. (7). This would, however,
not take into account the influence of Doppler shifts in the PSF calculation. The
following modification was suggested and used in [4] and [5]:
M
H ij ( ω ) =

m=1
2
s 0i jkDfmi ( rmj – s mi )
Wm ------- e
s mi
(11)
where Dfmi is the Doppler frequency shift factor of the signal from source i at
microphone m:
12
1
Dfmi = ---------------------------------------1 + M 0 cos ( ψ mi )
(12)
Here, M0 ≡ |U|/c is the Mach number, U being the source velocity vector, and ψmi
is the angle between the velocity vector U and a vector from microphone m to
point source number i. The inclusion of the Doppler shift factor in Eq. (11)
changes to wavenumber k to the wavenumber Dfmi k = (Dfmi ω)/c seen by
microphone m.
During testing of the beamforming software with simulated measurements, it
turned out that the shading filter needs to be taken into account when doing
Doppler corrections in the PSF calculation. Based on linear approximations in the
calculation of distances, when the calculation grid is near the centre of a selected
averaging interval, it is shown in the Appendix that Eq. (6) should be replaced by:
M
B ij ( ω ) ≅
s 0i Df mj jkDf mj ( r mj – s mi )
- ----------- e
 Wm ( ωDfmj ) -----s mi Df mi
m=1
Df mj
Q i  ω ----------- 
 Df mi 
(13)
when processing data taken near the centre of the interval. Here, Dfmj is the
Doppler frequency shift factor at microphone m associated with a point source at
focus position j. It is defined exactly as the factor for the source point i. Provided
the source spectra Qi (ω) are very flat, the Doppler correction factor on the
frequency in the argument of these spectra can be neglected, and Eq. (7) and
Eq. (13) then lead to the following formula for the elements Hij (ω) of the PSFs:
B ij ( ω ) 2
H ij ( ω ) ≡ --------------- ≅
Qi ( ω )
M

m=1
2
s 0i Df mj jkDfmj ( rmj – s mi )
Wm ( ωDf mj ) ------- ----------- e
s mi Df mi
(14)
One very important difference between Eq. (11) and Eq. (14) is the addition of
the Doppler shift factor Dfmj on the frequency in the calculation of the arrayshading function Wm. The need for that factor comes from the fact that in the
tracking DAS algorithm the shading filters are applied to the measured
microphone signals, which include the Doppler shift, while the PSF is calculated
in the moving system, where Doppler correction has been made. If the array13
shading functions Wm are very flat over frequency intervals of length equal to the
maximum Doppler shift, the factors Dfmj are, of course, not needed. Another
difference between Eq. (11) and Eq. (14) is that Eq. (14) has the focus point
Doppler factor Dfmj in the exponential functions, whereas Eq. (11) uses the source
point Doppler factor Dfmi. However, experience from simulated measurements has
shown this difference to be of minor importance, since the two factors are very
similar. The influence of the amplitude factor Dfmj/Dfmi is also negligible.
Accuracy of the Point Spread Function
This section will investigate the agreement between the DAS response to a
moving point source and a corresponding PSF calculated using Eq. (14). The
geometry of the array described in “Overall System Architecture” on page 6 was
used, and the monopole point source was passing over at an altitude equal to 60 m
with a speed of 60 m/s, which is representative for the real fly-over measurements
to be described in “Application to MU300 Business-jet Fly-over” on page 16. A
pseudorandom type of source signal, totally flat from 0 Hz to 6400 Hz, was used,
consisting of 800 sine waves of equal amplitude, but with random phases. With
16384 samples/s in the simulated measurement and an FFT record length equal to
256 samples, the FFT line width became 64 Hz, so the source signal had eight
frequency lines for each FFT line. FFT and averaging was performed over 10 m
position intervals of the point source along the x-axis, so the averaging time was
1/6 of a second, which is comparable with the 1/8-second period length of the
source signal. The FFT and averaging used a Hanning window and 66% record
overlap.
For the array shading, the radius Rcoh(ω) of the active central sub-array must be
specified as a function of frequency, f. Reference [2] proposed the use of a radius
inversely proportional to frequency:
f 1 metre
R coh ( f ) = ---------------- ⋅ 1 m
f
(15)
f1 metre being the frequency with 1 metre radius of the central coherent area. Based
on actual measurements, a value around 4 kHz was proposed [2]. The actual
measurements presented in this paper were taken on a day with almost no wind,
and we have found a value of f1 metre = 6 kHz to provide good results, so that value
has also been used in the simulated measurements.
14
After shading, DAS beamforming was performed with a grid of 61 × 61 points
with 0.25 m spacing, covering an area of 15 m × 15 m with the point source at the
centre. The PSF was then calculated across the same grid of points, for a source
also at the centre. Diagonal Removal was not applied.
Fig. 2. Relative average deviation between PSF and DAS over a 15 m x 15 m area centred
at x = –30 m
a) Shading slope factor κ = 2.0
b) Shading slope factor κ = 4.0
Fig. 2 shows, as a function of frequency, the average relative deviation between
the two maps calculated as:
 ( Hij – Πmeasured, j )
j
2
Relative Deviation = ------------------------------------------------------------ ⋅ 100%
 ( Πmeasured, j )
(16)
2
j
where the PSF source position i is at the centre of the area, and Πmeasured, j is from
a simulated measurement on that source with unit amplitude. Fig. 2a shows the
result for a rather smooth radial cut-off in the shading function, κ = 2.0, while in
Fig. 2b a medium-steep cut-off κ = 4.0 was used. In both cases, three levels of
Doppler correction were used in the calculation of the PSF. The full red curve
represents the case of no Doppler correction made, meaning that the PSF was
calculated from Eq. (7). For the dotted red curve Eq. (14) was used, but without
the Doppler shift factor in the shading function Wm. The result is almost identical
with what would be obtained using Eq. (11). The full black spectrum is obtained
using Eq. (14). Clearly, use of the Doppler factor is needed in the calculation of
15
the shading function. The remaining error was highly dependent on the applied
signal and the averaging time, and no other important influencing factors were
identified. So this residual error seems to be caused by the very short averaging
performed in the tracking DAS beamformer. The change in shape of the deviation
spectra around 800 Hz occurs where the radial cut-off of the shading function sets
in.
Fig. 3 shows the deviation achieved through use of Eq. (14) for PSF calculation
at a set of x-coordinates. The deviation is seen to have approximately the same
level independent of position during the simulated fly-over, when Eq. (14) is used
for calculation of the PSF.
Fig. 3. Relative average deviation between PSF and DAS over a 15 m x 15 m area centred
at a set of different x-coordinates. κ = 4.0
Application to MU300 Business-jet Fly-over
The system was applied as a part of a fly-over test campaign in November 2010 at
Taiki Aerospace Research Field (Taiki, Hokkaido, Japan) under joint research
work between JAXA and Brüel & Kjær. JAXA was conducting the test campaign,
where fly-over noise source localization technologies, including their own
acoustic array, were developed. Around 120 measurements were taken on an
MU300 business jet from Mitsubishi Heavy Industries. Fig. 5 shows the MU300
aircraft, which has overall length and width equal to 14.8 m and 13.3 m,
respectively. It has two jet engines on the body, just behind and over the wings.
The nose of the aircraft is used as the reference in the position information
obtained from the onboard GPS system. As indicated in Fig. 4, the centre of the
global coordinate system is on the runway at the centre of the array.
The approach adopted for time-alignment of array recordings and aircraft
position information from the aircraft was described in “Overall System
16
Fig. 4. Taiki Aerospace Research Field
with indication of array position and global
coordinate system
Fig. 5. Picture of the MU300 business jet
Architecture” on page 6. The data file from the on-board GPS based positioning
system provided the following information (at 5 m intervals along the runway):
• Very accurate absolute time from the IRIG-B system
• Three position coordinates with accuracy between 5 cm and 30 cm
• Three speed coordinates with accuracy around 0.005 m/s
• Roll, pitch and yaw with approximate accuracy 0.005°
This information in combination with the IRIG-B signal recorded with the
microphone signals was used in all data processing for accurate reconstruction of
the aircraft position at every sample of the microphone signals.
Illustration of Processing Steps
The purpose of this section is just to illustrate the huge improvement in resolution
and dynamic range that is achieved through the combination of shading and
deconvolution. For this illustration a level flight was chosen with engine idle and
the aircraft in landing configuration. Altitude was 59 m, and the speed was 57 m/s.
Fig. 6 shows results for the 1 kHz octave band, averaged over a 15 m interval
centred where the nose of the aircraft is 5 m past the array centre, i.e., at x = 5 m.
The resulting FFT spectra were synthesized into full octave bands. The displayed
dynamic range is 20 dB, corresponding to 2 dB level difference between the
colours. Fig. 6a shows the DAS map obtained without shading, meaning that
resolution will be poor due to the concentration of microphones near the array
centre. Use of the shading function improves resolution considerably, as seen in
Fig. 6b, but it also amplifies the sidelobes due to the large microphone spacing
across the outer part of the active sub-array, where each microphone is also given a
large weight. Fortunately, the deconvolution process is able to significantly reduce
17
Fig. 6. Illustration of the improvements in resolution and dynamic range obtained through
the use of shading and deconvolution. The data are from a level flight with engine idle and
the aircraft in landing configuration. The display dynamic range is 20 dB, corresponding to 2
dB contour interval
a) DAS, no shading,
no diagonal removal
b) DAS, shading,
no diagonal removal
c) DAS + NNLS, shading,
diagonal removal
these sidelobes, as can be seen in Fig. 6c. Better sidelobe suppression could have
been achieved in DAS by the use of more optimized irregular array geometries
(e.g., multi-spiral), but here the focus has been on the ease of array deployment,
and deconvolution seems to compensate quite well.
All maps were calculated using a 16 m × 16 m grid with 0.25 m spacing,
leading to 65 × 65 = 4225 calculation points. In the following, 10 m averaging
intervals will be always used. The total calculation time for seven intervals,
including DAS and FFT-NNLS calculations, was approximately 5 min on a
standard Dell Latitude E6420 PC.
Contour Plots of Pressure Contribution Density
The results to be presented in this section and in “Pressure Contribution Spectra at
the Centre of the Array” on page 19 are all from a level flight at 63 m altitude, with
61 m/s speed, engine idle, and with the aircraft in clean configuration. All results
were obtained using shading, diagonal removal and FFT-NNLS deconvolution
based on a PSF with full Doppler correction as described in Eq. (14).
Fig. 7 contains contour plots of the pressure contribution density for the 1 kHz
octave band when the nose of the aircraft is at x = –30, –10, +10 and +30 m. A
20 dB fixed display range has been used to reveal source level changes during the
fly-over. At x = +10 m the engine nozzles are almost exactly over the centre of the
array. The strong nozzle sources are seen to shift a bit in the x-direction as the
aircraft moves past the array. This is at least partially because the engine is at a
slightly higher altitude than the mapping plane, which is at the level of the aircraft
nose, see Fig. 5. Based on aircraft geometry, the nozzle sources should shift
18
Fig. 7. Pressure contribution density plots for the 1 kHz octave at four positions during a
level flight with engine idle and the aircraft in clean configuration. The averaging intervals
were 10 m long and centred at the listed positions. The 20 dB colour scale from Fig. 6 is
reused. Threshold is constant across the four maps with full dynamic range used at
x = +10 m, where the engine nozzles are exactly over the array centre
a) x = –30 m
b) x = –10 m
c) x = +10 m
d) x = +30 m
approximately 1/3 of the engine length due to that phenomenon when the aircraft
moves from x = –30 m to x = +30 m. The two weaker sources in front of the
nozzles are probably the intakes, which are only partially visible from the array
because of the wings. Close to the wing tips, two more significant sources are seen
in this 1 kHz octave band. Probably these sources are the openings of two drain
tubes or small holes and gaps. The narrowband spectral results to be presented in
“Pressure Contribution Spectra at the Centre of the Array” on page 19 show that
these two sources are narrow-banded and concentrated near 1 kHz.
Fig. 8 contains contour plots similar to those of Fig. 7, but with the nose of the
aircraft at x = 0 m and covering the octave bands from 500 Hz to 4 kHz. Clearly,
the two sources some small distance from the wing tips exist only within the 1 kHz
octave band. The 500 Hz octave includes frequencies well below 500 Hz, where
the array is too small to make deconvolution work effectively, so here resolution is
poor. Notice that the system provides almost constant resolution across a fairly
wide frequency range. This is also true for the DAS maps, i.e., without
deconvolution, the explanation being that the diameter of the active sub-array is
inversely proportional to frequency above 1 kHz.
Pressure Contribution Spectra at the Centre of the Array
As mentioned in “Contour Plots of Pressure Contribution Density” on page 18, the
pressure contribution density maps – such as those in Fig. 7 – can be area
integrated to give estimates of the contributions from selected areas to the sound
pressure at the centre of the array. As a reference for these contributions, and for
validation purposes, it is desirable to compare them with the pressure measured
directly at the array. Since there was no microphone at the array centre, the
19
Fig. 8. Octave band pressure contribution density maps for the averaging interval at x = 0.
Again, the 20 dB colour scale from Fig. 6 is reused. For each map the threshold is adjusted
to show a 20 dB range
a) 500 Hz octave
b) 1 kHz octave
c) 2 kHz octave
d) 4 kHz octave
average pressure power across all microphones was used. The directly measured
spectra, however, contain Doppler shifts, whereas the area-integrated spectra are
based on maps of de-dopplerized data. To compare the spectral contents, the
Doppler shift status must be brought into line for the two spectra. Since it is
natural to have the Doppler shift included when dealing with the noise at the array,
a choice was made to “re-dopplerize” the pressure contribution density maps on
the aircraft. So this was actually done also for the maps in Fig. 7 and Fig. 8.
Fig. 9. Measured average pressure spectrum compared with pressure contributions
calculated by integrating over the full mapping area seen in Fig. 7d
The full black curve in Fig. 9 represents the directly measured array pressure
spectrum based on FFTs with 256 samples record length, Hanning window, and
averaging over a time interval corresponding to that used at the aircraft, but
delayed with the sound propagation time from the aircraft to the array. The three
pressure contribution spectra in the same figure were integrated over the full
20
mapping area for the averaging interval at x = +30 m, also represented in Fig. 7d.
As expected, the re-dopplerization shifts the spectral peaks in the contribution
spectra downwards to match very well with the peaks in the measured array
pressure spectrum. Without diagonal removal, the level of the calculated
contribution spectrum matches very well with the measured array pressure.
Diagonal removal leads to a small under-estimation amounting to approximately
1 dB, part of which is flow noise in the individual microphones.
In the derivation of the PSF in Eq. (14) we had to assume a flat spectrum to
proceed from Eq. (13). Clearly, the spectrum in Fig. 9 is not flat around the narrow
peak at 2.5 kHz. Eq. (14) was used anyway across the full frequency range, and
the spectral peaks seem well reproduced. But to ensure accurate handling of
Doppler effects around sharp spectral peaks (tones) in deconvolution, a special
handling should be implemented modelling the energy flow between frequency
lines [4]. This becomes an important issue, if the level difference between a peak
and the surrounding broadband spectrum approaches or even exceeds the dynamic
range (sidelobe suppression) of the array with DAS beamforming. This is not the
case here, but close.
By integrating the pressure contribution density over only partial areas, one can
estimate the contribution from these areas to the sound pressure at the array.
Fig. 10 shows a set of sub-areas to be considered beyond the full mapping area:
i) the engine nozzles; ii) the central area, covering engine intake, wheel wells, and
the inner part of the wings; iii) the outer part, approximately 1/3 of both wings.
The colours of the areas will be re-used in the contribution spectra.
Fig. 11 contains the contribution spectra for the same aircraft positions as
represented by the contour maps in Fig. 7. In the present clean configuration of the
aircraft, the engine nozzles are seen to have by far the dominating noise
contribution, even though the engine is in idle condition. A small exception is a
narrow frequency band near 1 kHz, where the sources near the wing tips are
dominant. Except for very few exceptions, the full area contribution is within 1 dB
from the measured average sound pressure over the array. Part of this difference is
due to flow noise in the individual microphones. So the underestimation on the
full-area contribution due to the use of diagonal removal is very small, but may
cause low-level secondary sources in Fig. 7 and Fig. 8 to become invisible. The
30 dB dynamic range shown in the spectra of Fig. 11 is probably a bit too large to
say that all visible details are “real”.
21
Fig. 10. Sub-areas used for integration of pressure contribution. The sources at the inner
front edge of the wings are probably two fins
Fig. 11. Pressure contributions from the areas of Fig. 10 to the sound pressure at the array
centre
22
a) x = –30 m
b) x = –10 m
c) x = +10 m
d) x = +30 m
Conclusion
The paper has described a system and a methodology for performing highresolution fly-over beamforming using an array designed for fast and precise
deployment on a runway. Due to this requirement, a rather simple array geometry
not optimized for best sidelobe suppression was used. Our hope was that the use of
deconvolution could compensate for that. The system was designed to cover the
frequency range from 500 Hz to 5 kHz, and it proved to provide very good
resolution and dynamic range across these frequencies, except perhaps just around
500 Hz. Results have been presented in the paper from a couple of measurements
out of approximately 120 recordings taken on an MU300 business jet at Taiki
Aerospace Research Field, Taiki, Hokkaido, Japan, in November 2010. The results
are very encouraging.
A special focus has been on the use of an array-shading function that changes
continuously with frequency, and in particular on the implications of that in
connection with deconvolution. It was shown that Doppler shifts have to be taken
into account in the use of the shading function in connection with calculation of
the Point Spread Function used for deconvolution.
23
Appendix
This appendix presents a derivation of Eq. (13). We consider an arbitrarily
selected averaging interval, and we choose also arbitrarily a single model point
source with index i. For convenience we measure time relative to the centre of the
selected averaging interval.
The first problem is to derive a linear approximation for the time tm where the
source signal radiated at time ts reaches microphone m, valid for |ts| << 1. To do
that, the distance from point source i to microphone m is approximated as:
s mi ( t s ) ≅ s mi + cos ( ψ mi ) Ut s
for t s << 1
(17)
where U ≡ |U| is the aircraft speed, and ψ mi is the angle between the velocity
vector U and a vector from microphone m to the point source, both at the centre of
the averaging interval. In the same way we get for the distance from focus point j
to the microphone:
r mj ( t ) ≅ r mj + cos ( ψ mj ) Ut
for t << 1
(18)
The signal radiated at time ts arrives at microphone m at time tm given as:
s mi ( t s )
t m = t s + --------------c
(19)
Using Eq. (17) and the expression in Eq. (12) for the Doppler shift factor, we
can rewrite Eq. (19) as:
ts
s mi
t m ≅ ------ + ---------c
Df mi
(20)
This equation is easily solved for ts with the result:
s mi
t s ≅ Df mi ⋅  t m – -----
c 
which is then valid for |tm – smi /c| << 1.
24
(21)
For the microphone signals from point source i we need an expression
equivalent with Eq. (5), just in the time domain and for the moving source. As
argued in “Beamforming and Deconvolution Calculations” on page 8, the Doppler
amplitude factor can be neglected, when estimating Pressure Contributions. Doing
that, the microphone pressure gets the following simple form [2]:
s mi ( t s )
qi ( ts )
p mi  t s + ----------------  = s 0i ---------------

c
s mi ( t s )
(22)
containing the propagation delay smi (ts)/c and the inverse distance decay in
connection with the source signal qi. In Eq. (22) we approximate the time-varying
inverse distance decay by its value for ts = 0, and we use the delay approximation
of Eq. (21):
s 0i
s mi 
p mi ( t m ) ≅ ------- q i  Df mi ⋅  t m – -----

s mi
c 
s mi
for t m – ------ << 1
c
(23)
The first step in the DAS calculation is application of the individual shading
filters to the microphone signals. In the time domain this can be expressed as
convolution with the impulse responses wm(τ) of these filters, see Eq. (2):
∞
p̂ mi ( t m ) = ( p mi ⊗ w m ) ( t m ) =
 pmi ( τ )wm ( tm – τ ) dτ
–∞
s 0i
≅ ------s mi
∞
s mi
 qi  Dfmi ⋅  τ – ------c -  wm ( tm – τ )dτ
(24)
–∞
Here the approximation of Eq. (23) has been inserted. An apparent conflict in this
context is the integral going from –∞ to +∞ while at the same time we use an
approximation valid for only small values of the integration variable. Both here
and in the later Fourier integrals we have to think of using windowed source
signals, meaning that the integrals will have contributions only from time
segments close to time zero.
25
The shaded microphone signals are now used in DAS beamforming as
expressed in Eq. (3):
M
b ij ( t ) =
r mj ( t )
p̂ mi  t + -------------- 

c 

(25)
m=1
To proceed, we need to use the linear approximation of Eq. (18) for the
distances rmj(t) between microphones and focus points. As a result we obtain an
approximation similar to the one in Eq. (20):
r mj ( t ) r mj
t
t + ------------- ≅ ------- + ----------c
Df mj
c
(26)
for t << 1
Use of Eqs. (26) and (24) in Eq. (25) leads to:
M
b ij ( t ) =

m=1
M
≅

m=1
r mj ( t )
p̂ mi  t + ------------- ≅

c 
s 0i
-----s mi
∞
M
r mj
t
----------- 
 p̂mi  ------c - + Df

mj
m=1
s mi
r mj
t
----------- – τ dτ
 qi  Dfmi ⋅  τ – ------c -  wm  ------c - + Df
mj
(27)
–∞
The final step to obtain the frequency domain response Bij(ω) is to Fourier
transform the signal bij(t). To do that we notice first that in Eq. (27) the time
variable t occurs only in the argument of the shading impulse response function
wm. The argument of wm has the form of a linear function of t. To work out the
Fourier integral of bij(t) we therefore need the following formula:
∞
 w ( at + b ) e
–∞
26
– jαωt
– 1 jωαb ⁄ a
dt = a e
α
W  --- ω
a 
(28)
with α =1, a = (Dfmj)–1, and b = (rmj/c) – τ. W is the Fourier transform of w.
Eq. (28) can be easily verified by substituting a new variable for (at+b) in the
integral. From use of Eq. (28) in Eq. (27) we get:
∞
 bij ( t )e
B ij ( ω ) ≡
– j ωt
(29)
dt
–∞
M
≅
r
 mj

jDf mj ω ------s 0i
c
W m ( Df mi ω ) ------- Df mj e
s mi
m=1

–∞

jDf mj ω -------- – τ
s mi 
 c

W m ( Df mj ω ) dτ
- Df mj e
q i  Df mi ⋅  τ – -----

c 

m=1
M
=
∞
s 0i
-----s mi
r mj
∞

–∞
s mi –j Df mj ωτ
q i  Df mi ⋅  τ – -----dτ
- e


c 
where the factors are just re-arranged in the last line. The remaining integral in
Eq. (29) has also the form of Eq. (28), only with α = Dfmj, a = Dfmi, and
b = –Dfmismi /c. Use of Eq. (28) with these parameter values in Eq. (29) leads to:
M
B ij ( ω ) ≅

m=1
M
=
r mj

s mi

jDf mj ω -------jω – Df mj -------s 0i

c 1
c   Df mj 
----------- e
W m ( Df mj ω ) ------- Df mj e
Q i ----------- ω
 Dfmi 
Df mi
s mi
s 0i Df mj jDf mj k ( r mj – smi )
- ----------- e
 Wm ( Dfmj ω ) -----s mi Df mi
m=1
Df mj
Q i  ----------- ω
 Df mi 
(30)
which is the expression given in Eq. (13). Q.E.D.
Acknowledgments
The authors would like to thank Diamond Air Service Incorporation for their
support in conducting the fly-over tests.
27
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algorithms for the mapping of acoustic sources,” AIAA Paper 2006-2711.
28
Clustering Approaches to Automatic Modal
Parameter Estimation*
S. Chauhan and D. Tcherniak
Abstract
Most modal parameter estimation techniques utilize Stabilization/Consistency
Diagrams as a tool for distinguishing between physical system modes and
mathematical modes. However, this process results in giving several estimates of
the same mode and the task of choosing one estimate over the others is left to the
user. This task is highly judgmental, with user expertise playing a big role as to
which estimate is selected, and can be very tedious, especially in situations when
the data is difficult to analyze (low signal-to-noise ratio, closely spaced modes,
heavily damped modes, etc.). One of the ways to get around this issue is to
incorporate smart selection of estimates in the algorithm itself, so as to avoid the
user interaction which, as stated previously, can be very subjective.
In this paper, two clustering-based approaches are suggested for the purpose of
automatic mode selection. These methods build upon the existing Stabilization
Diagram tool; differing in the manner in which the stabilization diagram is
constructed and clusters are being formed. Both approaches utilize a Euclidian
distance-based approach to automatically select the best estimate.
Résumé
La plupart des méthodes d'estimation des paramètres modaux font intervenir des
diagrammes de stabilisation/cohérence, cet outil servant à faire la distinction entre
modes de systèmes réels et modes mathématiques. Toutefois, ces processus
résultent en plusieurs estimations d'un même mode, laissant à l'opérateur la
responsabilité de choisir l'une d'entre elles. C'est une tâche souvent très
fastidieuse, surtout lorsque les données sont très difficiles à analyser (faible
rapport signal/bruit, modes peu espacés, modes très amortis, etc.) et qui fait
* First published in Proc. 27th Conference and Exposition on Structural Dynamics 2009
(IMAC XXVII)
29
fortement appel à son jugement et à son expérience. Un des moyens de contourner
cette difficulté est d'incorporer à l'algorithme une sélection judicieuse
d'estimations afin d'éviter de trop faire appel à la subjectivité de l'opérateur.
Cet article suggère deux approches sectorielles pour la sélection automatique des
modes. Ces méthodes s'appuient sur l'outil diagramme de stabilisation existant,
mais diffèrent dans la manière de construire ce dernier et de grouper les
diagrammes. Toutes deux font appel à une approche basée sur la distance
euclidienne pour sélectionner automatiquement la meilleure estimation.
Zusammenfassung
Die meisten Techniken zur Modalparameterbestimmung verwenden Stabilitäts/
Konsistenz-Diagramme als Werkzeug zur Unterscheidung zwischen
physikalischen Systemmoden und mathematischen Moden. Dieser Prozess ergibt
jedoch mehrere Abschätzungen für dieselbe Mode und die Aufgabe, eine davon
auszuwählen und den anderen vorzuziehen, wird dem Anwender überlassen. Es
handelt sich weitgehend um eine Ermessensfrage, wobei die Sachkenntnis des
Anwenders beim Auswählen der Abschätzung eine große Rolle spielt, und die
Aufgabe kann schwerfallen, besonders bei Daten, die schwierig zu analysieren
sind (geringer Signal-Rauschabstand, dicht benachbarte Moden, stark gedämpfte
Moden usw.). Eine Möglichkeit zur Umgehung dieses Problems besteht darin,
eine intelligente Auswahl in den Algorithmus einzubauen, um die
Benutzerinteraktion zu vermeiden, die wie erwähnt sehr subjektiv sein kann.
In diesem Artikel werden zwei auf Clustering basierte Ansätze für eine
automatische Modenauswahl vorgeschlagen. Diese Methoden bauen auf dem
vorhandenen Stabilitäts-Diagramm auf und unterscheiden sich darin, auf welche
Weise das Stabilitäts-Diagramm konstruiert wird und Cluster gebildet werden.
Beide Ansätze verwenden eine euklidische abstandsbasierte Methode, um
automatisch die beste Abschätzung auszuwählen.
Nomenclature
ω
H(*)
h(t)
30
Frequency
Frequency Response Function Matrix
Impulse Response Function
Nref
No
m
[α]
[β]
ERA
UMPA
EMA
OMA
Number of input references
Number of output responses
Model order
Denominator matrix polynomial coefficient
Numerator matrix polynomial coefficient
Eigensystem Realization Algorithm
Unified Matrix Polynomial Approach
Experimental Modal Analysis
Operational Modal Analysis
Introduction
The Stabilization Diagram (or Consistency Diagram) is an important tool often
utilized by the user for obtaining correct modal parameters. Most modal parameter
estimation algorithms use this tool for distinguishing physical system modes from
mathematical or computational modes. A stabilization diagram involves tracking
of the modal parameters as a function of increasing model order (or change in data
subsets, modal parameter estimation algorithms, etc.).
Once the stabilization diagram is prepared, the user is left with the task of
choosing one estimate (of a mode) amongst the many estimates obtained at
various iterations (as a function of model order or different solution set or even
different parameter estimation method). This final step often poses the question
“Which estimate should I choose?” The need to facilitate this process and avoid
the uncertainty involved in this process of selecting an estimate based on user
judgment has resulted in researchers working on approaches to automate the
process of mode selection. There have been two common approaches to this; the
first being a Logical Rules based approach and the second utilizing data-clustering
techniques, which have been traditionally popular in fields such as pattern
recognition, machine learning, data mining and bioinformatics.
In general, there are two issues that analysts face while dealing with modal
parameter estimation. The first issue involves separating physical modes from
computational modes and the second involves choosing an estimate, amongst
several others, that best represents a mode. These also form the core of any
technique that tries to automate the modal parameter estimation procedure. Pappa,
James, et al., proposed a four-step logical rule based approach to automatic mode
selection [1]. This approach utilized modal parameter estimation using ERA in the
first step. The poles obtained are then passed through a threshold step that
31
eliminates the mathematical modes. Once modal parameters from the first
iteration are identified and filtered using the previously mentioned steps, a logical
rules based engine is used to judge whether the estimate from the next iteration is
that of a new mode or an already existing one. If it turns out to be yet another
estimate of an already existing mode from the previous iteration, it is found
whether it is a better estimate than the already existing one (in which case it
replaces the current estimate) or not. Here, it is important to note that this
approach does not compare a mode in an iteration only to those from the previous
iteration, as in the traditional stabilization diagram approach, but compares it with
the best mode amongst those found in all previous iterations. The algorithm
continues until all the iterations are completed. This approach uses a consistentmode indicator (CMI) [2] as a measure of selecting the better of the two estimates.
This approach was later improved to include the Genetic Algorithm based
supervisor that required initial estimates of the modes to be searched [3]. It also
included a dynamic fitness function to capture the modes that were not provided as
initial estimates to be searched but might still exist as true modes of the structure.
A similar rule based approach was employed in [4]. This approach also utilizes,
amongst other measures, tolerances in estimates of frequency, damping and modal
vector to check whether or not a mode is stable with respect to that in the previous
iteration. An important point worth noticing in this approach in comparison to the
approach that Pappa, James, et al., suggested is that the estimate is only compared
with modes obtained in the previous iteration. Further, MAC between the modal
vectors is not used as a criterion for stability of modes but as a criterion to
distinguish between double or closely spaced modes. In this manner, several
clusters or groups of modes are obtained after all the iterations, with each cluster
containing various estimates of the same mode. Finally, one mode in each cluster
is selected as a best mode representing that cluster, the selection procedure being
based on damping ratio.
In [4, 5], a methodology based on energy analysis of the modes is proposed to
distinguish between system modes and spurious non-physical modes. It utilizes
techniques such as model reduction, balanced truncation and pole/zero
cancellation to assess the overall influence of removing a mode; true modes
influencing the model greatly, spurious modes having minimal effect.
There have also been attempts to use classical clustering techniques for the
purpose of automatic mode selection. These include utilizing techniques like
Fuzzy C-Means Clustering, Support Vector Machine, etc. A Maximum Likelihood
estimator based procedure that employs a similar multistage rule based approach
32
was suggested in [6]. In [7], this approach is complimented with Fuzzy Clustering,
according to which the modes are classified into physical and computational
modes. It is suggested in [8] that this methodology can also be used for structural
health monitoring purposes.
Unlike the above approaches to utilize Fuzzy clustering to classify a mode as
physical or computational, in [9] Fuzzy clustering is used not only to classify a
mode as physical or computational, but also to find the final system modes.
However, Fuzzy C-Means Clustering requires that the number of clusters (or in
other words modes) is known beforehand. It also requires initial guesses of the
cluster centres. In this work, a Genetic Algorithm based methodology was also
suggested that can be used to make initial guesses about cluster centres.
In a recent paper, Carden and Brownjohn used a similar Fuzzy Clustering
technique for structural health monitoring [10]. Instead of forming clusters based
on frequency and damping estimates, they formed clusters based on real and
imaginary parts of the obtained poles. The main purpose of this study was not
automatic mode selection but structural health monitoring. The technique was
applied to the Z24 bridge and Republic Plaza Office tower, a high rise building in
Singapore.
In addition to the methods discussed above, some other approaches utilizing
clustering techniques for automatic mode selection and better understanding of the
stability diagram are found in the literature. Goethals, Vanluyten, et al., [11] used a
K-Means clustering technique along with a self-learning classification algorithm,
Least Squares Support Vector Machines, for separating spurious poles and real
poles. This approach requires a training set for the Support Vector machine which
is obtained by utilizing the clustering technique in the first stage implemented
along with some rules/criteria.
One thing common to various approaches listed previously in this section is that
the process of automatically selecting the modal parameters starts once the
parameter estimation algorithm has provided the modes. The automation process
is restricted to the task of filtering physical system modes and computational
modes and selecting one out of many estimates of a single mode. It should be
noted that it is very possible that this process might still not identify all the modes
in the given frequency range. This is due to the fact that the performance of a
modal parameter estimation algorithm depends considerably on several factors
including the chosen frequency range, the choice of inputs selected as references
and, most importantly, the choice of parameter estimation algorithm itself. Thus,
the task of choosing an algorithm and setting its initial parameters (frequency
33
range of interest, number and choice of references, etc.) still hold the key to good
results and require intelligent selection. In this manner, performance of automatic
mode selection is also influenced by these factors.
The next section presents the conceptual background to the two approaches to
automatic mode selection suggested in this paper. One of these approaches, named
Best Mode based approach, presents a new way of constructing the stability
diagram that is more informative in comparison with the traditional approach of
constructing the stability diagram. Theoretical background is followed by results
of studies conducted on an analytical rotor dataset in the third section where the
performance of the two approaches is evaluated. The fact that these approaches
result in formation of clusters having several estimates of the same mode also
provides a means to calculate useful statistical data about the estimated modes.
Finally, conclusions are made and the scope for future research is presented.
Theoretical Background
Historically, the concept of stability was applied to high-order algorithms like
Least Squares Complex Exponential (LSCE) [12], Polyreference Time Domain
(PTD) [13, 14], Rational Fraction Polynomial (RFP) [15], etc. This can be
understood through Unified Matrix Polynomial Approach (UMPA) [16 – 18]
which is a mathematical concept that helps in understanding various modal
parameter estimation algorithms by developing these algorithms from a common
framework. UMPA based equations for high-order algorithms in the time and
frequency domains are given as:
High-order time domain
[ h ( ti + 1 ) ]
[ [ α 1 ] [ α 2 ]… [ α m ] ]
[ h ( ti + 2 ) ]
N ref × mN ref
…
[ h ( t i + m ) ] mN × N
ref
o
34
= –[ h ( ti + 0 ) ]
N ref × N o
(1)
High-order frequency domain
1
( jω i ) [ H ( ω i ) ]
2
( jω i ) [ H ( ω i ) ]
…
m
[ [ α 1 ] [ α 2 ]… [ α m ] [ β 1 ] [ β 2 ]… [ β n ] ]
( jω i ) [ H ( ω i ) ]
N ref × mN ref + ( n + 1 )N o
1
– ( jω i ) [ I ]
2
– ( jω i ) [ I ]
…
n
– ( jω i ) [ I ]
0
= – ( jω i ) [ H ( ω i ) ]
N ref × N o
mN ref + ( n + 1 )N o × N o
(2)
Nref and No refer to the number of inputs (or number of reference outputs in the
case of OMA) and the number of outputs. m refers to the model order and the
number of roots of the equation (or poles) is m × Nref , out of which 2N are true
modes of the system and the rest are non-physical or computational modes (in
general m × Nref >> 2N). The stabilization diagram involved tracking the
estimates of modal frequency, damping and mode shapes as a function of
increasing model order m. Stability is generally defined in terms of tolerance
percentage set for each of the three parameters: frequency, damping and mode
shape. This approach is well documented and details can be obtained in [18, 28].
It should be noted that in equations (1) and (2), it is the [α0] coefficient (or the
lower order coefficient) that has been normalized. Similar equations can be
formed by choosing different coefficient normalization. This forms yet another
way of looking at the stability diagram where instead of varying the order, one can
vary the coefficient being normalized (for a fixed order) and then track the
stability of obtained modes as a function of coefficients [19].
As stated earlier, the concept of stabilization is not just restricted to high-order
algorithms but can also be extended to other algorithms. In [20], this concept of
stability was extended to other parameter estimation algorithms, in which case a
diagram similar to the stability diagram, called the Consistency Diagram, is
drawn. This approach is commonly applied to lower-order algorithms
(Eigensystem Realization Algorithm (ERA) [21, 22], Ibrahim Time Domain (ITD)
[23, 24], Polyreference Frequency Domain (PFD) [25-27]).
35
In a recent paper, Phillips and Allemang [28] provide several methods for
obtaining clear stabilization/consistency diagrams. These methods include the
utilization of normal mode criteria, using long vector comparison, both high- and
low-order coefficient normalization, different frequency normalization, etc., and it
is shown that effective use of these methods can greatly help in making the
stabilization/consistency diagram a better tool.
Fig. 1. A typical stability diagram: + New Mode,× Stable Frequency, * Stable Frequency and
Damping, ∇ Stable Frequency and Vector, ◊ Stable Mode with Frequency, Damping and
Vector all stable
16
14
12
Iteration
10
8
6
4
2
0
400
500
600
700
800
900
Frequency
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A typical stabilization diagram is shown in Fig. 1 where stabilized poles are
represented by ◊. This means that if a pole stabilizes, within the user-specified
tolerances of frequency, damping and vector, with respect to a mode in the
previous iteration, it is represented by ◊. It should be noted that comparing a mode
in an iteration to that in the previous iteration is the most common way of
constructing the stability diagram. In this manner several estimates of a mode are
estimated corresponding to a solution performed at various iterations (for
example, in Fig. 1, 12 estimates of the mode around 720 Hz are found that are
classified as stable). The task of choosing one of these twelve estimates is left to
the user and, as mentioned before, this can be very tedious in cases where the user
36
is inexperienced, the data quality is not very good, or the structure is complex with
closely coupled modes, heavily damped modes, etc. This creates a need for
automating this selection process, thus avoiding user judgment by formulating a
more reliable and intelligent process.
From the literature review provided in the previous section, it is apparent that
various approaches towards automating this procedure involve forming of groups
or clusters of valid poles (true modes of the system) obtained after the application
of the modal parameter estimation technique, with each cluster representing a
different mode and containing several estimates of the same mode. This procedure
normally involves a logical rules based approach which is often coupled with the
use of clustering algorithms like Fuzzy C-Means, K-Means, etc., to form and
validate a cluster and select one estimate in a cluster as the best estimate of that
particular mode.
The approaches presented in this paper are based on the fundamentals of good
selection of rules and clustering technique in order to achieve the ultimate goal of
automatic mode selection. For the purpose of this study, modal parameter
estimation based on varying model order (suitable for high-order algorithms like
PTD, RFP, etc.) is considered. It should however be noted that this procedure is
extensible to other cases as well (different solution set, varying coefficient
normalization, etc.). Further, in this paper only traditional FRF based experimental
modal analysis studies are conducted to illustrate these techniques. However, these
are equally adaptable to Operational Modal Analysis (OMA) as well.
First Approach (Traditional Stability Diagram Based)
Comparing modes from the current iteration to those in the previous iteration is a
common approach to developing a stability diagram. Since this approach to
automatic mode selection is based on comparing modes to those in the previous
iteration only (as is the most common practice), it is referred to as Traditional
Stability Diagram based approach.
Assume that a total of N iterations of modal parameter estimation are
performed. At each iteration, different numbers of poles (equal to m × Nref, m
being the model order and Nref the number of references) are obtained as model
order m is varied. Out of these poles, some are true system modes and others are
computational modes that are to be filtered. The approach is summarised as:
1) Select a mode in i + 1th iteration and perform a stability test for this mode
by comparing it with modes from ith iteration.
2) The stability test may include several criteria like frequency, damping,
mode shape tolerances, modal phase colinearity, modal phase deviation,
37
maximum and minimum criteria, presence of conjugate poles, positive
damping, etc.
3) If the mode passes the stability criterion, it is either assigned to an already
existing cluster having other estimates of the same mode or a new cluster is
initiated if this is the first stable estimate of a mode.
4) The procedure is repeated for all the modes obtained in i+1th iteration, by
comparing them to the modes obtained in the previous iteration (ith
iteration), before moving to the next iteration.
5) Steps 1 – 4 are repeated for all the iterations.
6) At the conclusion of step 5, clusters of various modes have been formed.
These clusters are passed through a filtering stage to filter out those clusters
that do not contain a sufficient number of estimates. This is yet another rule
integrated in the process.
7) Finally, for every cluster, an estimate is selected as the best estimate
representing that cluster, based on the calculation of a Euclidian distance
measure. This best estimate is also the automatically selected mode.
The process flowchart is illustrated in Fig. 2.
Fig. 2. Flowchart for Traditional Stability Diagram based approach
New mode
Next iteration
No
Compare with
modes in previous
iterations
Stability
criterion
satisfied with
any mode?
No
All modes
in this iteration
compared?
Yes
No
Yes
Last iteration?
Yes
The first
stable estimate
of a mode?
Yes
No
Assign to an
existing cluster
Filter the cluster to
obtain valid clusters
Initiate a
new cluster
Calculate the best
mode of the cluster
based on Euclidean
distance measurement
120663
38
Second Approach (Best Mode Based)
It is worth noting that in the first approach the stability diagram is still being
developed in the classical manner of comparing modes from an iteration to those
obtained in the previous iteration. In this sense, one ignores the information
provided in earlier iterations, using only information provided in the current and
the iteration previous to the current one.
In the proposed second approach, each mode, whether computational or
physical, is assigned to a cluster. For each cluster a best mode is selected amongst
all the estimates that form the cluster. Unlike the first approach, every new mode is
now compared to these best modes representing various clusters. Depending on
whether the comparison criterion is satisfied, a mode is assigned to an existing
cluster or assigned to an entirely new cluster. Further, every time a mode is
assigned to a cluster, the best mode of that cluster is recalculated. When modes
from all the iterations have been considered, cluster filtering is done to separate
computational and physical modes.
The flowchart for this approach is shown in Fig. 3.
Fig. 3. Flowchart for Best Mode based approach
New mode
No
Compare with
best mode of the
clusters
Stability
criterion
satisfied with
any cluster?
No Initiate a new cluster
with this mode
Yes
Assign the mode
to that cluster
All modes
compared?
Yes
Recalculate the
best mode for
the cluster
Filter the clusters
to separate physical
and computational
modes
120664
The best modes representing the valid clusters obtained after the filtering stage
are also the automatically selected modes. It should be noted that the process of
determining the best modes is based on a Euclidian distance measure, in the same
manner as the previously discussed first approach. The advantage of this approach
39
over the first approach is that it utilizes the complete information obtained from
various iterations while calculating the stability of a mode and not just the
previous iteration. This results in stability diagrams that are more informative in
comparison. This aspect is further illustrated by means of examples in the next
section.
Studies on Analytical Dataset
Fig. 4 shows a stability diagram obtained for an analytical rotor dataset. The
stability diagram is prepared in the traditional manner. The symbols in the stability
diagram represent:
+ New Mode
× Stable Frequency
* Stable Frequency and Damping
∇ Stable Frequency and Vector
◊ Stable Mode with Frequency, Damping and Vector all stable
Fig. 4. Stability diagram (0 – 100 Hz frequency range)
35
30
Iteration
25
20
15
10
5
0
0
10
20
30
40
50
60
70
80
Frequency
40
90
100
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Damping and frequency values for the various modes in the frequency range of
interest are shown in Table 1. Modes indicated in gray are repeated or closely
spaced modes. It is worth noting that this is a pretty complex system with plenty of
modes that are not only repeated modes but also very similar in terms of damping.
Table 1. Modal Parameters of the Analytical System
Analytical
Traditional Stability
Diagram based approach
Freq (Hz)
Damping
(%)
8.3549
0.0526
Best Mode based
approach
Freq (Hz)
Damping
(%)
Freq (Hz)
Damping
(%)
2.4780
0.0156
8.3549
0.0525
10.1817
0.0640
10.1818
0.0640
15.3281
0.0963
15.3281
16.7506
0.1052
16.7506
16.7554
0.1053
16.7555
17.4015
0.1093
17.4014
0.1100
17.4016
0.1093
17.4015
0.1079
21.3338
0.1340
21.3351
0.1330
21.3338
0.1340
22.3986
0.1407
22.3986
0.1409
22.3986
0.1409
22.4139
0.1408
22.4140
0.1410
22.4140
0.1410
22.9526
0.1442
22.9526
0.1445
22.9526
0.1445
26.5336
0.1667
26.5326
0.1664
26.5337
0.1667
26.5337
0.1667
28.0756
0.1764
28.0755
0.1765
28.0755
0.1765
65.6950
0.4128
65.6950
0.4128
65.6950
0.4128
75.6227
0.4752
75.6227
0.4752
75.6227
0.4752
75.8303
0.4765
75.8303
0.4765
75.8303
0.4765
8.3549
0.0521
10.1816
0.0641
0.0965
15.3281
0.0965
0.1052
16.7506
0.1052
0.1054
16.7554
0.1052
Once the stability diagram is prepared, the user has to select various modes
based on this stability diagram. This procedure might be easy in case of certain
modes (for example modes at 65.6, 75.6 and 75.8 Hz) while extremely difficult in
others (repeated roots at 16.75 and 22.39 Hz; see Fig. 5, which shows the zoomed
in portion of stability diagram between 16.5 – 23 Hz). The user has to rely
significant amount of experience and knowledge in order to recover these modes.
41
Fig. 5. Stability diagram (16.5 – 23 Hz frequency range)
30
Iteration
25
20
15
10
17
18
19
20
21
22
Frequency
23
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Fig. 6 shows the results of the automatic mode selection done using the
traditional stability diagram based approach. Using this approach, a total of 11
modes are obtained which are listed in Table 1 and indicated as solid red diamonds
with arrows in Fig. 6. Nine modes are not identified, all of them being repeated
modes (except the mode at 2.47 Hz). On observing the stability diagram,
particularly for these modes, it is evident that in the case of these modes the
stability is not very good. In such a case, the user will not generally pick these
modes at all or will try to improve his estimation.
Fig. 7 shows the stability diagram prepared using the Best Mode based
approach. A total of 17 modes are identified using this approach, which signifies
an improvement over previous approach. It should be noted that similar stability
criteria are used for both approaches. The advantage of the second approach of
being able to identify more modes than the previous approach comes from the fact
that it takes into consideration information available from all previous iterations.
Thus, there is less chance of missing a good mode in this approach as stability
level is decided with respect to the best mode of the cluster, not with respect to the
immediately previous mode (which can be an outlier). This can be illustrated in
this case by the fact that stability levels for certain modes (like repeated modes at
10.18, 17.4, 26.53 Hz, etc.) is improved in comparison to when stabilities are
42
Fig. 6. Traditional stability diagram approach based automatic mode selection
35
30
Iteration
25
20
15
10
5
0
0
10
20
30
40
50
60
70
80
Frequency
90
100
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Fig. 7. Best mode based stability diagram and automatic mode selection
35
30
Iteration
25
20
15
10
5
0
0
10
20
30
40
50
60
70
80
Frequency
90
100
120671
43
calculated in a traditional manner on an iteration by iteration basis. This results in
a better stabilization diagram and ultimately more modes being identified. Yet
another aspect of the Best Mode based approach is illustrated by observing
stabilization levels around 2.47 Hz using both approaches. Although, this mode is
not identified using either approaches, it is clear that the Best Mode based
approach is more indicative of the presence of this mode than the previous
approach. On the basis of this knowledge the user might carry out modal
parameter estimation with different algorithmic parameters in order to obtain this
mode.
Cluster based approaches, as suggested in the paper, also provide a way to
obtain statistical data about the automatically chosen modal parameter estimates.
Since each cluster contain various estimates of the same mode, one can easily
obtain statistical measures such as mean and standard deviation. These statistics
provide useful additional information about the estimated parameters that can also
aid the user in making a better decision regarding parameter estimation. Table 2
and Table 3 list the mean and standard deviation for the various estimates obtained
using the two approaches.
Table 2. Cluster Statistics for Traditional Stability Diagram based approach
Traditional Stability Diagram Based Approach
Frequency (Hz)
Automatically
Selected
Mode
44
Mean
Damping (% Critical)
Standard
Deviation
Automatically
Selected
Mode
Mean
Standard
Deviation
8.3549
8.3548
2.0833e-04
0.0526
0.0521
0.0017
15.3281
15.3280
1.8492e-04
0.0965
0.0961
8.1028e-04
16.7506
16.7506
3.6995e-04
0.1052
0.1053
0.0011
16.7555
16.7554
2.3726e-04
0.1054
0.1058
0.0012
22.3986
22.3987
1.8765e-04
0.1409
0.1412
0.0013
22.4140
22.4140
2.0279e-04
0.1410
0.1411
8.9666e-04
22.9526
22.9527
2.2956e-04
0.1445
0.1435
0.0016
28.0755
28.0756
4.3830e-04
0.1765
0.1765
0.0016
65.6950
65.6950
3.6534e-004
0.4128
0.4127
6.5969e-004
75.6227
75.6227
3.3838e-004
0.4752
0.4752
6.8348e-004
75.8303
75.8305
0.0013
0.4765
0.4763
6.6210e-004
Table 3. Cluster Statistics for Best Mode based approach
Best Mode Based Approach
Frequency (Hz)
Automatically
Selected
Mode
Mean
Damping (% Critical)
Standard
Deviation
Automatically
Selected
Mode
Mean
Standard
Deviation
8.3549
8.3548
1.9935e-04
0.0521
0.0518
0.0019
10.1818
10.1817
4.3937e-004
0.0654
0.0650
0.0015
15.3281
15.3281
3.8048e-004
0.0965
0.0967
0.0014
16.7506
16.7505
1.9212e-004
0.1052
0.1055
8.6664e-004
16.7554
16.7554
2.2717e-004
0.1052
0.1058
0.0012
17.4014
17.4018
3.9135e-004
0.1100
0.1094
0.0012
17.4015
17.4017
1.4785e-004
0.1079
0.1086
0.0018
21.3351
21.3338
1.7139e-004
0.1330
0.1344
6.6254e-004
22.3986
22.3986
2.1752e-004
0.1409
0.1414
0.0017
22.4140
22.4141
7.2579e-004
0.1410
0.1416
0.0024
22.9526
22.9527
2.3311e-004
0.1445
0.1435
0.0016
26.5336
26.5336
3.6793e-004
0.1664
0.1666
0.0021
26.5337
26.5336
7.0399e-004
0.1667
0.1667
2.4026e-004
28.0755
28.0756
5.5423e-004
0.1765
0.1768
0.0018
65.6950
65.6949
3.9947e-004
0.4128
0.4128
6.9889e-004
75.6227
75.6227
4.6055e-004
0.4752
0.4752
6.7244e-004
75.8303
75.8306
0.0014
0.4765
0.4763
6.5138e-004
Conclusions
The stability diagram is an important tool in the modal parameter estimation phase
whose understanding and utilization is highly subjective, depending on expertise
and user know-how. In this paper, two methods (Traditional Stability Diagram
based and Best Mode based) for automating the modal parameter estimation
process are suggested. Both methods involve the formation of clusters on a logical
rule based approach. However, these methods differ in the manner of formulating
the clusters. The Best Mode based method utilizes complete information available
from all previous iterations while preparing the stability diagram, unlike the
Traditional Stability Diagram based method which uses information only from the
immediately previous iteration. On formation of clusters, a Euclidian distance
based automatic mode selection process is carried out to identify the best mode
representing the cluster.
45
It is shown by means of studies performed on an analytical dataset that the Best
Mode based method results in a stability diagram which is more informative in
comparison to the one obtained with the traditional approach. This is indicative in
terms of improved stability of certain modes which do not stabilize sufficiently
enough while using the traditional approach and hence the risk of not choosing
these modes is lessened. The automatic mode selection procedure also works
satisfactorily. It is further observed that even in cases where a mode might not
have been automatically selected using either approach the Best Mode method is
more indicative of its presence than the Traditional Stability Diagram method.
Automatic mode selection is a desired tool in the modal analysis community as
it can not only help in reducing the overall analysis and estimation time but also
lends itself very well to online monitoring, which holds the key to many
applications including Structural Health Monitoring, Flutter Analysis, etc. The
studies conducted in this paper suggest positively that proposed methods are
capable of automating the modal parameter estimation process. It does not,
however, rule out user know-how completely, because there are still several
factors that require user intervention; for example, choice of parameter estimation
algorithm, frequency range of interest, choice of input references, etc.
Nonetheless, this is a valuable tool as once the initial parameters (like frequency
range, number of references, etc.) for the estimation algorithm are chosen, the
automatic mode selection procedure makes the rest of the process faster, easier,
reliable and independent of user interaction.
Although the two methods gave satisfactory results on a complicated dataset,
the need to further test them with real-life data cannot be overlooked. It is
important to note that since these methods are based on the implementation of
certain logical rules, their success depends on the robustness of these rules under
various conditions. Thus, it is important to test these methods more rigorously by
means of several real-life datasets so as to make them fulfil the original goals of
automatic mode selection, which include being robust, reliable and involving no
user interaction.
46
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49
50
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(Continued from cover page 2)
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Frequency Response for Measurement Microphones – a Question of
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High-resolution Fly-over Beamforming
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