Utilization of Blind Source Separation Techniques for Modal Analysis
B. Swaminathan+, B. Sharma+, S. Chauhan*
+
Structural Dynamics Research Lab,
University of Cincinnati, Cincinnati, OH, USA
*
Bruel & Kjaer Sound and Vibration Measurement A/S
Skodsborgvej 307, DK 2850, Naerum, Denmark
Email: [email protected], [email protected], [email protected]
Abstract
In past few years, there have been attempts at utilizing Blind Source Separation (BSS) and Independent
Component Analysis (ICA) techniques for modal analysis purposes. Most of the early work in this regard has
been promising, though restricted to application of these techniques to analytical and laboratory based
experimental structures. It is felt that in order to make these techniques applicable to more challenging
scenarios, they need to be modified keeping in view the demands of modal parameter estimation procedure.
This includes making them more robust and applicable to handle complex scenarios (for e.g. closely coupled
modes, heavily damped modes, low signal-to-noise ratio, etc.). This forms the motivation for this paper which
aims at tuning BSS / ICA methods for modal analysis purposes in an effective and efficient manner. Amongst
other methods, it is shown how to incorporate signal processing techniques, modify BSS techniques to
handle data in specified frequency ranges, extract modal parameters from limited output channels, etc. to
derive most benefits out of these algorithms.
1. Introduction
Blind Source Separation techniques emerged from the medical imaging and wireless communication fields
as image or signal processing techniques. Antoni [1] demonstrated how these techniques can be used for
blind separation of vibration components and laid down the associated principles. This was followed by
works that attempted at showing the application of these techniques for modal analysis [2-5]. These early
works aimed at establishing a connection between BSS techniques and Operational Modal Analysis (OMA).
These studies showed encouraging results, by means of application of BSS techniques to analytical and
simple laboratory structures and estimating modal parameters in the process.
As fundamental principles of application of BSS techniques for OMA became clear and understood, their
limitations in this regard also came into light. Mathematical formulation of BSS techniques shows that they
are more suitable to handle systems having no or negligible damping [5]. Early simulations, however,
suggest that this does not pose serious challenge as long as the modal coordinates [4] have distinct spectra
and are mutually uncorrelated. However, the verdict is still not clear and more rigourous study needs to be
conducted. Yet another limitation, on account of mathematical foundations of BSS techniques, is that they
find real modal vectors, which might not be the case for most real-life structures. A framework to extend BSS
techniques to take this limitation into account was recently suggested in [6]. Further, the effect of noise on
the performance of these techniques is not yet evaluated (for modal analysis applications). Typically, these
techniques work directly on the raw data without much signal pre-processing and thus their performance is
more sensitive to the quality of the data acquired. Current formulation of most BSS algorithms also poses a
limitation on account of the fact that one can estimate only as many modes as the number of output
responses being measured.
The work presented in this paper attempts at addressing some of these issues by means of utilizing Second
Order Blind Identification (SOBI) [7, 8] algorithm. In light of above discussion, a modified formulation of SOBI
algorithm is provided in the paper that aims at circumventing the limitation that only as many modes can be
found as the number of sensors measuring the output response. This modification also includes use of signal
processing techniques such as averaging and windowing to reduce the effect of noise. Suggested
modifications, thus improve the overall performance of SOBI algorithm and contributes towards making it
more suitable for OMA purposes and widely applicable to more realistic structures.
Section 2 presents SOBI in its original form and its modified form developed in this work, along with revisiting
the link between BSS and OMA. Section 3 demonstrates the positive effect these modifications have on
overall results, by means of studies conducted on a 15 degree-of-freedom analytical system and a lightly
damped circular plate. Finally, based on these results, conclusions are drawn in Section 4.
2. Second Order Blind Identification (SOBI) [7, 8]
2.1 Theoretical Background
Mathematically, an instantaneous BSS problem, in time domain, can be formulated as
x(t ) = As (t )
(01)
where x(t) is a column vector of m output observations representing an instantaneous linear mixture of
source signals s(t), which is a column vector of n sources at time instant t. A is an m X n matrix referred to as
“mixing system” or more commonly as “mixing matrix”.
SOBI is a BSS algorithm that separates the sources assuming that they have a temporal structure with
different autocorrelation functions (or power spectra) and are mutually uncorrelated (having zero crosscorrelation). It utilizes the concept of joint diagonalization for achieving this goal. Following are the steps
involved in this algorithm. Note that, number of sources are considered equal to number of sensors (or
observations) i.e. m = n.
1. First the covariance (mean removed correlation) matrix of the output observations is estimated
1
Rˆ x (0 ) =
N
N
∑ x(k )x (k )
T
(02)
k =1
where Rˆ x (0 ) is the covariance matrix at zero time lag and N is the total number of time samples
taken.
2. Compute EVD (or SVD) of Rˆ x (0 )
Rˆ x (0 ) = U x ∑ x VxT = Vx Λ xVxT = Vs Λ sVsT + VN Λ NVNT
(03)
where Vs is m X n matrix of eigenvectors associated with n principal eigenvalues of Λs = diag{λ1, λ2,
….., λn} in descending order. Vn is m X (m-n) matrix containing the (m-n) noise eigenvectors
associated with noise eigenvalues Λn = diag{λn+1, λn+2, ….., λm}. The number of sources n is thus
estimated based on the n most significant eigenvalues (or singular values in case of SVD).
3. Perform pre-whitening transformation
x (k ) = Λ s 2VsT x(k ) = Qx(k )
−1
4. Estimate the covariance matrix of the vector
other than τ =0.
1
Rˆ x (τ i ) =
N
(04)
x (k ) for a preselected set of time lags (τ 1 ,τ 2 ,......,τ L )
N
∑ x (k )x (k − τ )
T
k =1
i
(05)
5. Perform Joint Approximate Diagonalization on the above set of covariance matrices
Rx (τ i ) = U D (τ i )U T
(06)
to estimate the orthogonal matrix U that diagonalizes a set of covariance matrices. Several efficient
algorithms are available for this purpose including Jacobi techniques, Alternating Least Squares,
Parallel Factor Analysis etc. [9, 10].
6. The mixing matrix and source signals can be estimated as
1
Aˆ = Q +U x = Vs Λ s2U x
y (k ) = sˆ(k ) = U xT x (k )
It should be noted that
(07)
(08)
D(τ i ) is a diagonal matrix that has distinct diagonal entries. However, it is difficult to
determine a priori a single time lag τ at which the above criterion is satisfied. Joint diagonalizaton procedure
avoids this difficulty by providing an optimum solution considering a number of time lags.
2.2 BSS and OMA
The basic fundamental behind application of ICA / BSS techniques to modal analysis goes back to the
concept of expansion theorem [12] and modal filters [14]. According to the expansion theorem [15], the
response vector of a distributed parameter structure can be expressed as
x (t ) =
∞
∑φ η
r =1
r
r
(t )
(09)
where Φr are the modal vectors weighted by the modal coordinates ηr. For real systems, however, the
response of the system can be represented as a finite sum of modal vectors weighted by the modal
coordinates. In this manner, mathematically, expansion theorem yields similar formulation as expressed in
Eq. (01), with source vector s(t) and mixing matrix A being analogous to modal coordinate response vector
η(t) and modal vector matrix [Φ], and hence BSS techniques like SOBI can be applied to obtain modal
vectors and modal coordinates (which define the modal frequency and damping).
To obtain a particular modal coordinate ηi from response vector x, a modal filter vector ψi is required such
that
ψ iT φ i = 0,
for i ≠ j
(10)
and
ψ iT φ i ≠ 0,
for i = j
(11)
so that
ψ iT x(t ) = ψ iT ∑ φ rη r (t )
(12)
= ψ iT φiη i
(13)
N
r =1
or
Thus modal filter performs a coordinate transformation from physical to modal coordinates. Multiplying the
system response x with modal filter matrix ΨT results in uncoupling of the system response into single
degree of freedom (SDOF) modal coordinate responses (η).
2.3 Suggested Modifications
Mixing model shown in Eq. (1) does not take into consideration the additive noise, which is often present
while dealing with real life structures. From operational modal analysis application point of view, one of the
drawbacks of SOBI is in dealing with significantly noisy output response signals. In typical OMA scenario,
such as response measurements taken over a bridge or a building, SNR (Signal-to-Noise Ratio) might not be
good and this can deteriorate the performance of the algorithm. Apart from whitening (Eq. 4), SOBI doesn’t
involve any other signal pre-processing step and thus errors can creep in due to additive measurement and
process noise that is generally random in nature.
One of the ways, to overcome this issue and improve performance of SOBI, is by minimizing the errors in
measurement, i.e. effect of noise and bias, by means of techniques like averaging and use of windowing
functions. Thus, instead of working directly on raw data, SOBI algorithm can be modified to work with
correlation functions that are obtained after inverse Fourier transforming the averaged power spectra.
This means that in step 5, instead of applying joint diagonalization procedure to covariance matrices
calculated using complete data, it is performed on covariance matrices obtained after averaging procedure is
carried out for noise minimization. Welch Periodogram method [11] is one such method that can be used for
obtaining averaged output response power spectra. Along with averaging the power spectra, windowing and
overlapping can also be used to reduce the leakage (bias) errors.
This process of noise minimization, using power spectra averaging based technique, also provides a
mechanism to extend the effectiveness of SOBI algorithm by making it possible to apply this technique within
frequency band of interest. This is a significant improvement over SOBI in its original form, as it overcomes
the limitation of identifying only as many sources as number of measured responses. Thus step 5, in the
previous section, can be preceded by selecting averaged power spectra in the frequency band of interest,
say between ω1 and ω2,
Gˆ x (ω )ω1 ,ω2 . This is followed by inverse Fourier transforming the power spectra in
selected frequency range to obtain corresponding Covariance matrix.
(
~
Rˆ x (τ ) = ℑ−1 Gˆ x (ω )ω1 ,ω2
)
(14)
Step 5 then involves performing joint diagonalization of these covariance matrices, thus restricting SOBI to
estimate modes in the specified frequency range.
Advantages of these modifications are shown in the next section where modified version of SOBI is applied
to a 15 Degree-of-Freedom analytical system and a lightly damped circular plate.
3. Case Studies
3.1 15 Degree-of-Freedom Analytical System
A 15 DOF analytical M-C-K system is considered for analysis using the modified version of SOBI algorithm
(Figure 1). This system has some moderately damped (1-4 %) modes; with a pair of closely spaced modes
around 53.3 Hz and also some locally excited modes in 100-200 Hz frequency range.
Figure 1: Analytical 15 Degree of Freedom System
Theoretical modal frequency (in Hz) and damping (in % Critical) values for the system are shown in Table 1.
Response data is obtained at each DOF by exciting the system at these DOFs by uncorrelated random
forces. Simulated response data is sampled at a frequency of 1024 Hz and a total of 163840 response
samples are collected.
Complete Frequency Range Based Analysis
In this analysis, data is analyzed in complete frequency range from 0-512 Hz. Table 1 lists the estimates of
frequency and damping obtained using SOBI in its original form and its modified form. The estimates of
modal frequency and damping are obtained by applying SDOF frequency domain methods on the power
spectra of the obtained sources [12]. These are compared against the theoretical values of frequency and
damping. It should be noted that while using modified version of SOBI, frequency band of interest is chosen
to cover the complete frequency range. Thus, effectively this is similar to SOBI original, except that modified
version of SOBI works with covariance functions obtained after inverse Fourier transforming the averaged
complete power spectra, where as original form of SOBI works directly on the data without performing any
prior signal processing. Figure 2 shows the plots of power spectra of SDOF modal coordinates obtained
using original SOBI. All 15 modes are easily obtained using original SOBI. Similar plots are obtained for
modified version of SOBI as well.
Figure 2: Power Spectra of SDOF Modal Coordinates (Original SOBI)
Table 1 shows that the comparison on the basis of modal frequency and damping estimates is quite
satisfactory though damping values obtained using SOBI (both algorithms) are slightly overestimated. Both
SOBI approaches, original and modified, show fairly similar results.
Table 1: Comparison of Modal Parameter Estimates for the 15 DOF System
Theoretical
Original SOBI
Modified SOBI
Freq (Hz)
Damping (%)
Freq (Hz)
Damping (%)
Freq (Hz)
Damping (%)
15.985
1.004
15.990
1.360
15.990
1.360
30.858
1.937
30.818
2.238
30.818
2.238
43.600
2.735
43.604
2.884
43.604
2.891
46.444
2.912
46.486
3.107
46.485
3.112
53.317
3.338
53.291
3.560
53.344
3.492
53.391
3.345
53.471
3.246
53.439
3.341
59.413
3.715
59.413
3.819
59.421
3.850
61.624
3.858
61.662
4.055
61.660
4.070
68.811
4.298
68.795
4.192
68.798
4.220
73.630
4.593
73.532
4.668
73.522
4.697
128.84
2.609
128.83
2.669
128.83
2.670
136.55
2.455
136.56
2.492
136.56
2.494
143.86
2.329
143.87
2.379
143.87
2.382
150.83
2.221
150.82
2.254
150.82
2.259
157.47
2.122
157.47
2.168
157.47
2.169
The mode shapes of this system are complex. Except for the first mode at 15.98 Hz, which has fairly real
mode shape, other modes have complex mode shapes. Since, BSS techniques (such as SOBI in its original
form) are typically suited to give real mode shapes (although modified SOBI does produce complex mode
shapes), it is expected that results might not be as accurate as expected. However, as shown by Table 2,
when mode shapes obtained using the two algorithms are compared with theoretical mode shapes, the
comparison between theoretical mode shapes and SOBI Original mode shapes is pretty good. MAC (Modal
Assurance Criterion) [12] comparison for the mode shapes corresponding to the two closely spaced modes
(around 53.3 Hz) is not as good as other modes, which highlights that BSS techniques such as SOBI find it
difficult to handle effectively the cases involving closely spaced modes.
Modified SOBI’s performance, on the other hand, is inferior in comparison (Highlighted in Table 2). One of
the possible reasons can be the fact that the output response data obtained for this analytical system is free
from noise and use of signal processing techniques such as averaging, windowing and overlapping
invariably introduces some errors into the processed power spectra. Further, the modal vectors obtained
using modified SOBI are complex (due to discrete Fourier transformation) and it is observed that if only the
real part of the modal vectors is compared with the theoretical modes, MAC is significantly improved.
Table 2: MAC Comparison
Theoretical
Modes
Theoretical vs. Original
SOBI
Theoretical vs. Modified
SOBI
Theo. vs. Mod. SOBI Mode
Shapes (Real Part Only)
15.985
1.00
1.00
1.00
30.858
1.00
1.00
1.00
43.600
1.00
1.00
1.00
46.444
1.00
1.00
1.00
53.317
0.91
0.52
0.98
53.391
0.88
0.48
0.96
59.413
1.00
0.60
0.99
61.624
1.00
0.39
0.98
68.811
1.00
0.99
0.99
73.630
0.99
0.99
0.99
128.84
1.00
1.00
1.00
136.55
1.00
1.00
1.00
143.86
1.00
1.00
1.00
150.83
1.00
1.00
1.00
157.47
1.00
1.00
1.00
Analysis in a Limited Frequency Band
Table 3 highlights the real advantage of Modified SOBI which is not apparent when it is applied to complete
frequency range like Original SOBI. In this case Modified SOBI is applied to the data in three different
frequency ranges of 10-40 Hz, 35-85 Hz and 100-200 Hz. MAC values (highlighted in Table 3) for the two
closely spaced modes at 53.3 Hz and also 59.4 Hz and 61.6 Hz mode show significant improvements in
comparison to results obtained using Modified SOBI in complete range. MAC number for these four modes
improves from 0.4-0.6 range to around 1 (around 0.95 for the closely spaced modes). In fact, the results are
even superior to those obtained using Original SOBI in which case MAC is 0.91 and 0.88 (see Table 2). A
typical plot of power spectra of SDOF modal coordinates (for a chosen frequency range of 10-40 Hz) is
shown in Figure 3. There are only two proper estimates of modal coordinates in the selected frequency
range (as indicated in Figure 3) and these estimates correspond to the modes at 15.99 Hz and 30.819 Hz.
Rest of the 13 estimates can be neglected.
Figure 3: Power Spectra of SDOF Modal Coordinates (Modified SOBI 10-40 Hz)
Table 3: Frequency Band based Analysis using Modified SOBI
Theoretical Modes
Modified SOBI (Applied in Freq. Band)
MAC
Frequency (Hz)
Damping (%)
Frequency (Hz)
Damping (%)
15.985
1.004
15.99
1.3604
1.00
30.858
1.937
30.819
2.2371
1.00
43.600
2.735
43.611
2.8753
1.00
46.444
2.912
46.486
3.1092
1.00
53.317
3.338
53.345
3.4503
0.95
53.391
3.345
53.427
3.3519
0.94
59.413
3.715
59.414
3.8261
1.00
61.624
3.858
61.672
4.0557
1.00
68.811
4.298
68.805
4.1962
1.00
73.630
4.593
73.542
4.6779
1.00
128.84
2.609
128.83
2.6703
1.00
136.55
2.455
136.56
2.4930
1.00
143.86
2.329
143.88
2.3839
1.00
150.83
2.221
150.82
2.2603
1.00
157.47
2.122
157.47
2.1683
1.00
Analysis conducted on 15 DOF analytical system verifies that, when applied in frequency bands, Modified
SOBI is as effective (and in certain cases better) than original SOBI algorithm and compares well with the
theoretical modal parameters.
This study is now followed by analysis on an experimental structure to assess the utility of Modified SOBI
further, especially in terms of assessing its performance in limited response sensors scenario (Number of
sensors measuring output response is less than number of modes of interest).
3.2 Lightly Damped Circular Plate
A lightly damped aluminum circular plate (Figure 4) is considered for experimental validation of Modified
SOBI method. 30 responses are taken on the plate in a configuration shown in Figure 5. Plate is randomly
excited by tapping it with fingers all over its surface. Data is acquired for a period of 5 mins. at a sampling
rate of 1600 Hz, thus providing 4,80,000 samples.
Figure 4: Experimental Set Up for Lightly Damped Circular Plate
Figure 5: Lightly Damped Circular Plate (Sensor Locations)
Since there are 30 responses observed over the plate, due to algorithmic limitations one can at most identify
30 modes in the frequency range of interest (0-700 Hz) using original SOBI. For comparison purposes, an
EMA test is also performed by exciting the plate by means of random excitation at three locations using
electrodynamic shakers. Using EMA algorithms, such as Polyreference Time Domain (PTD) [13], a total of
21 modes are identified. This is in agreement with the corresponding Complex Mode Indicator Function
(CMIF) plot [12] (Figure 6) which also indicates the presence of 21 modes. These modes are listed in Table 4
and are considered the reference against which performance of both SOBI algorithms (original and modified
forms) will be evaluated.
Figure 6: CMIF Plot for EMA Case
Figure 7: Cross MAC Plot (EMA modes vs. Modified SOBI modes)
Table 4 lists the modal parameters obtained using the three approaches; EMA, original SOBI and modified
SOBI. It should be noted that in case of modified SOBI, modal parameters are obtained by performing
analysis in various frequency bands where as original SOBI estimates the modes by analyzing complete
frequency range (0-800 Hz). Further, modal frequency and damping, in case of SOBI algorithms, is obtained
by applying SDOF frequency domain methods to power spectra of obtained sources (as done in case of 15
DOF system).
Advantage of modified SOBI is apparent in this analysis as it is able to identify all the expected modes in
comparison to original SOBI which is able to identify only 19 modes, though overall estimates for these
modes are in good agreement with the values obtained using EMA.
Modified SOBI, on the other hand, provides satisfactory estimates for all the modes except for the
discrepancy in frequency estimates which can be explained in terms of different frequency resolution used in
the two approaches. MAC plot between modes obtained through EMA and those obtained using modified
SOBI, shown in Figure 7, indicates that mode shapes corresponding to the two closely spaced around 133
Hz are not matching well. However, if real part of mode shapes obtained from modified SOBI is considered
and compared with EMA, the MAC improves significantly. These MAC values are listed in Table 4. This
behaviour is similar to that observed while analyzing closely spaced modes in 15 DOF system. Some higher
modes (above 650 Hz) show similarity to some of the other lower modes. This is, perhaps, due to limited
spatial resolution due to which these modes appear to have similar mode shape as the lower modes.
Table 4: Comparison of Modal Parameter Estimates for the Circular Plate
EMA Modes
Original SOBI Modes
Modified SOBI Modes
Freq (Hz)
Damp (%)
Freq (Hz)
Damp (%)
MAC
Freq (Hz)
Damp (%)
MAC (Real
Part)
57.476
0.149
57.987
0.353
0.99
57.844
0.400
0.98
58.257
0.265
56.076
0.405
0.96
56.938
0.347
0.96
96.811
0.663
96.637
0.762
1.00
96.425
0.810
1.00
133.595
0.045
132.74
0.176
0.89
132.73
0.167
0.99
133.865
0.064
133.00
0.111
0.99
132.96
0.135
0.94
221.766
0.158
219.70
0.194
0.97
219.67
0.195
0.97
223.183
0.098
222.92
0.151
1.00
222.91
0.136
1.00
233.448
0.053
231.96
0.136
0.98
231.93
0.125
0.98
234.212
0.050
233.28
0.106
0.97
233.27
0.086
0.95
355.934
0.217
350.99
0.153
0.98
350.97
0.143
0.97
358.990
0.258
358.23
0.068
0.98
358.23
0.059
0.98
378.200
0.130
377.24
0.130
0.98
377.21
0.116
0.98
381.017
0.109
379.16
0.167
1.00
379.11
0.139
1.00
413.485
0.511
412.59
0.469
0.99
412.45
0.470
0.99
495.306
0.120
488.48
0.126
1.00
488.47
0.116
1.00
571.080
0.119
567.56
0.144
0.97
567.52
0.125
0.97
572.242
0.419
569.93
0.250
0.93
569.99
0.206
0.92
643.368
0.123
639.04
0.170
0.98
639.13
0.174
0.97
647.299
0.105
646.69
0.125
0.98
646.08
0.128
0.97
672.637
0.071
-
-
-
664.68
0.108
0.99
676.281
0.088
-
-
-
674.44
0.081
0.98
Limited Sensor Based Analysis
One of the main contributions of this paper is to demonstrate improved ability of modified SOBI algorithm to
deal with situations where number of modes of interest exceeds the sensors measuring the response. This
analysis brings to fore this advantage of modified version of SOBI. As mentioned before, one of the
limitations of SOBI is that the number of modes that can be identified is at most equal to the number of
sensors measuring the response. Following analysis shows that this limitation can be overcome by applying
the proposed modified version of SOBI in a number of frequency bands, instead of complete frequency
range as is the case with original SOBI.
For this case, eighteen channels (out of thirty) are selected; other channels are not considered (See Figure
8). Goal is to identify all twenty one modes, given the response information corresponding to these eighteen
channels only.
Figure 8: Selected Channels
Figure 9: Power Spectra of SDOF Modal Coordinates (Original SOBI Limited Channel Study)
When original SOBI is applied on this limited dataset, it is known beforehand that at most eighteen modes
could be identified due to algorithmic limitations. However, power spectra of estimated modal coordinates
(Figure 9); indicate that only nine modes are properly identified. This underlines the drawback on part of
original SOBI and severely affects its utilization for OMA purposes. Modified SOBI takes care of this
limitation effectively since it can be applied in limited frequency ranges.
Before discussing the results obtained using modified SOBI algorithm, effect of reducing the number of
sensors from original thirty to eighteen on the estimation of modes is discussed. One of the potential dangers
of reducing the number of sensors is that some of the modes might not be observable due to limited spatial
resolution. This is also indicated by the auto MAC plot for the EMA modes shapes defined by the selected
points, as shown in Figure 10. Indeed, some of the modes appear to be identical to some of the other
modes, for e.g. closely spaced modes around 133 Hz are very identical, which is also the case for closely
spaced modes around 571 Hz. On inspecting the mode shapes for these two pair of modes, the reasons for
this observation becomes even clearer. It turns out that these modes are torsional modes and each closely
spaced mode differs from the other only with respect to relative motion between the points, torsional motion
being shifted by 45 deg. in the two cases. Removing immediately adjacent points to the ones selected has
resulted in affecting the observability of these modes and thus it is expected that these modes might not be
estimated. It is important to understand that, in this analysis, EMA mode shapes for comparison purposes
are obtained by truncating the original EMA mode shapes (mode shapes obtained during analysis in
previous section where complete data from all 30 sensors is used). It is observed that even EMA algorithms
are not able to estimate these modes if data corresponding to the selected 18 channels is used. CMIF plot
based on power spectra of the 18 selected channels also supports this (Figure 11). Some of the other modes
also show similarity to other modes due to observability issues (Modes around 57 Hz and 233 Hz, 355 Hz
and 670 Hz). The observability can be improved by altering the selection of channels on the physical
structure, and results for a second set of 18 response points on the structure (discussed in Appendix A)
indicate improved estimation (all the modes are identifiable) and better resolution between some of the
closely lying modes.
Figure 10: Auto MAC Plot (EMA Mode Shapes defined by Selected Channels)
Figure 11: CMIF Plot for Limited Channel Configuration
There are still, however, nineteen valid modes that need to be identified by means of information available
from the eighteen channels. Estimates of these modes using modified SOBI algorithm along with
corresponding MAC values (compared with EMA mode shapes defined by selected points) are shown in
Table 5. All the estimates compare well with the EMA estimates, underlying the improved performance of
SOBI algorithm with proposed modifications. Cross MAC plot between EMA and modified SOBI estimates,
shown in Figure 12, is very similar to the auto MAC plot for EMA estimates (Figure 10), i.e. certain EMA
modes that looked similar to each other due to limited spatial resolution, look similar to corresponding modes
obtained using modified SOBI as well (for e.g. modes around 232 Hz and 57 Hz, and 355 Hz and 670 Hz).
Table 5: Modal Parameter Comparison for Limited Sensor Case
EMA Modes
Modified SOBI Modes
Freq (Hz)
Damp (%)
Freq (Hz)
Damp (%)
MAC
57.476
0.149
56.921
0.327
0.99
58.257
0.265
57.672
0.412
0.99
96.811
0.663
96.419
0.799
1.00
133.595
0.045
-
-
-
133.865
0.064
132.90
0.161
1.00
221.766
0.158
219.67
0.194
0.95
223.183
0.098
222.92
0.145
0.99
233.448
0.053
231.95
0.129
1.00
234.212
0.050
233.27
0.101
0.96
355.934
0.217
350.98
0.148
0.98
358.990
0.258
358.23
0.065
0.98
378.200
0.130
377.23
0.124
0.98
381.017
0.109
379.14
0.158
1.00
413.485
0.511
412.56
0.438
0.99
495.306
0.120
488.46
0.121
1.00
571.080
0.119
-
-
-
572.242
0.419
569.85
0.208
0.99
643.368
0.123
639.08
0.175
0.98
647.299
0.105
646.10
0.126
0.98
672.637
0.071
664.60
0.108
0.98
676.281
0.088
674.42
0.080
0.98
Figure 12: Cross MAC (EMA vs. Modified SOBI) Limited Channels Configuration
6. Conclusions
It has been shown, by means of work presented in this paper, how modifying the original Second Order Blind
Identification algorithm can improve its effectiveness and utility for Operational Modal Analysis purposes.
These modifications take into consideration more elaborate signal processing techniques, such as
averaging, windowing etc., to reduce the effect of noise that is generally present while acquiring response
data on real life structures, there by improving performance of SOBI. One of the main advantages of the
suggested modification is that it enables SOBI to be applied even to situations where number of sensors
measuring the response is lesser than the number of modes to be estimated. It is, however, important to
note that this does not avoid observability related issues that might still be inherently present due to limited
spatial resolution.
In a more practical scenario (experimental analysis of the circular plate), it is shown that the modified form of
SOBI outperforms its original version. This is attributed to incorporation of better noise reduction signal
processing capabilities in the algorithm and extending its applicability to limited frequency ranges.
Results presented in this paper are encouraging and form good foundation for future research in this area.
One of the obvious research direction is to assess the performance of modified SOBI on real life structures
that are typical OMA applications, like buildings, bridges etc.
SOBI utilizes the concept of modal expansion theorem, according to which response of a system can be
decomposed into several single degree of freedom systems (defined by resonant frequencies of the system)
by means of modal vectors of the system which act as modal filters. Typically, modal expansion theorem is
valid for systems having distinct modes and negligible or very light damping. Damping constraints also imply
that expansion theorem is defined for systems with real modal vectors. Although one should not draw
conclusions only on the basis of work presented in this paper, yet it is interesting to note that SOBI can
identify heavily damped modes as well as modes with complex modal vectors (as shown in section 3.1, 15
DOF system). However, estimation of closely spaced modes does pose some challenges. In view of this
discussion, future work in this research should concentrate on improving SOBI further. This requires making
SOBI capable of handling closely spaced modes, and assessing its performance more rigourously for
practical structures and for systems with heavily damped and complex modes.
Acknowledgements
We would like to thank Dr. Randall J. Allemang, SDRL, University of Cincinnati, for his expert guidance and
feedback which were critical for completion of the paper.
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Appendix A
This section details results from a limited-sensors based study for an altered combination of response points
on the lightly damped circular plate. This study is aimed at evaluating the effect of the selection points on the
structure, on estimation of all the modes using the Modified SOBI methods. Eighteen points are chosen out
of the available thirty responses, with the selection shown in Figure A.1.
Figure A.1: Selected Channels
Modal estimates for this set of channels are listed in Table A.1. It is observed that this combination of
channels is able to resolve the torsion modes around 133 Hz with better distinction. However, this selection
of points too is unable to distinctly observe the closely lying modes around 571 Hz, although results are
better in comparison to configuration shown in Figure 8, Section 3.2. This indicates the importance of
selecting proper measurement locations in order to observe and estimate all modes of interest.
Table A.1: Modal Parameter Comparison for Limited Sensor Case
EMA Modes
Modified SOBI modes
Freq (Hz)
Damp (%)
Freq (Hz)
Damp (%)
MAC
57.476
0.149
57.028
0.413
0.98
58.257
0.265
57.987
0.318
0.96
96.811
0.663
96.576
0.917
0.97
133.595
0.045
132.760
0.184
0.99
133.865
0.064
132.940
0.151
0.95
221.766
0.158
219.680
0.197
0.98
223.183
0.098
222.970
0.151
0.99
233.448
0.053
231.930
0.121
0.98
234.212
0.050
233.270
0.100
0.93
355.934
0.217
350.950
0.146
0.95
358.990
0.258
358.230
0.079
0.96
378.200
0.130
377.270
0.126
0.92
381.017
0.109
379.090
0.150
0.98
413.485
0.511
412.440
0.430
0.99
495.306
0.120
488.510
0.107
0.94
571.080
0.119
567.720
0.150
0.68
572.242
0.419
569.640
0.181
0.83
643.368
0.123
639.110
0.165
0.99
647.299
0.105
646.070
0.122
0.96
672.637
0.071
664.560
0.108
0.99
676.281
0.088
674.370
0.077
0.99