A Low Order Frequency Domain Algorithm for
Operational Modal Analysis
S. Chauhan, R. Martell, D.L. Brown, R.J. Allemang
Structural Dynamics Research Laboratory
University of Cincinnati, Cincinnati, OH USA
email: [email protected]
Abstract
Most of the algorithms for Operational Modal Analysis work in the time domain and there are very few
frequency domain based algorithms. One of the reasons for this is the poor numerical characteristics
associated with higher order frequency domain algorithms like Rational Fraction Polynomial (RFP). These
limitations can be improved by using methods such as frequency normalization and use of orthogonal
polynomials in the traditional experimental modal analysis set up. However estimating modal parameters
in the frequency domain using output-only response data still remains a challenge. In this paper a low
order frequency domain algorithms is proposed for Operational Modal Analysis. This algorithm is derived
using the Unified Matrix Polynomial Approach (UMPA) and is evaluated using a theoretical and an
experimental study.
1
Introduction
One of the earliest Operational Modal Analysis (OMA) algorithms to be developed was the Natural
Excitation Technique (NExT) [1] which utilized the cross-correlation functions between the measured
output responses. Soon many of the traditional Experimental Modal Analysis (EMA) time domain
algorithms such as Ibrahim Time Domain (ITD) [2-3], Eigensystem Realization Algorithm (ERA) [4-5],
Poly-reference Time Domain (PTD) [6-7], etc were reformulated to suit the operational modal analysis
framework. These algorithms also used correlation functions for modal parameter estimation. State space
model based modal parameter algorithms like Stochastic Subspace Identification (SSI) [8] algorithm were
also developed that worked in the time domain.
Brincker, Zhang and Andersen [9] proposed an algorithm named Frequency Domain Decomposition
(FDD) that involved frequency by frequency singular value decomposition of the output power spectra.
This algorithm, in principle, was similar to conventional CMIF algorithm [10] which is essentially a
spatial domain algorithm rather than a typical frequency domain algorithm. To completely identify a
particular mode, the FDD algorithm is followed by enhanced Frequency Domain Decomposition (eFDD)
[11] algorithm that determines the damping associated with the mode which wasn’t possible with FDD.
The modal parameter estimation portion of the eFDD algorithm again worked in the time domain.
Thus modal parameter identification in the case of OMA is mostly restricted to time domain algorithms.
One of the reasons for the lack of frequency domain algorithms for OMA is the poor numerical
conditioning associated with the high order frequency domain algorithms such as Rational Fraction
Polynomial (RFP) [12]. This problem is much more severe in the case of OMA as the order of the power
spectrum based model used in OMA is twice that of the frequency response function based model used in
EMA. Recently a method called PolyMAX [13-15] that builds upon the classical least squares complex
frequency domain estimator was proposed and shown to work in an OMA framework.
The work presented in this paper involves the utilization of the Unified Matrix Polynomial Approach
(UMPA) [16, 17] for developing a low order frequency domain algorithm (UMPA-LOFD) suited for the
output response based OMA framework. The algorithm is applied to an analytical system and also a
lightly damped circular plate. It is shown to have better numerical characteristics and results are
comparable to time domain based OMA algorithms.
2
2.1
Theoretical Background
Unified Matrix Polynomial Approach (UMPA)
This section describes briefly the theory behind the Unified Matrix Polynomial Approach and its
utilization for developing the low order frequency domain algorithm for OMA. To understand the UMPA
formulation, the polynomial model used for frequency response functions is considered.
n
H pq (ω i ) =
Xp
(ω ) ∑ β ( jω )
i
Fq (ω i )
=
k =0
m
∑ α ( jω )
k =0
k
k
k
(1)
k
Rewriting this model for a general multiple input, multiple output case and stating it in terms of frequency
response functions
m

k
∑ ( jω ) [α k ]
 k =0

n
[H (ω )] = ∑ ( jω )k [β k ]
 k =0

(2)
This model in the frequency domain is the AutoRegressive with eXogenous inputs (ARX(m,n)) model that
corresponds to the AutoRegressive (AR) model in time domain for the case of free decay or impulse
response data
∑ [α ]{h (t )} = 0
m
k =0
k
i+k
pq
(3)
The general matrix polynomial model concept recognizes that both time and frequency domain models
generate functionally similar matrix polynomial models. This model which describes both domains is thus
termed as Unified Matrix Polynomial Approach (UMPA) [16, 17].
2.2
Low Order Frequency Domain Algorithm
2.2.1
General UMPA Formulation
Lower order, frequency domain algorithms are basically UMPA based models that generate first or second
order matrix coefficient polynomials [18, 19]. Starting with the multiple input, multiple output frequency
response model of Eq. (2) and forming a second order matrix polynomial model.
m

 n

k
k
(
)
(
)
=
j
ω
[
α
]
[
H
ω
]
k 
∑
∑ ( jω ) [β k ]

 k =0
 k =0

for order m =2
[[α ]( jω )
2
i
2
]
+ [α1 ]( jωi ) + [α 0 ] [H (ωi )] = [β1 ]( jωi ) + [β 0 ]
(4)
This basic equation can be repeated for several frequencies and the matrix polynomial coefficients can be
obtained using either [α 2] or [α 0] normalization
[α2] Normalization
[[α 0 ] [α 1 ] [β 0 ]
( jω i )0 [H (ω i )]


1
(
jω i ) [H (ω i )]

[β1 ]]N0 ×4 Ni 
0
− ( jω i ) [I ] 


1
 − ( jω i ) [I ]  4 N
= −( jω i ) [H (ω i )]N 0 × Ni
(5)
= −( jω i ) [H (ω i )]N 0 × Ni
(6)
2
0 × Ni
[α 0] Normalization
[[α 1 ] [α 2 ] [β 0 ]
 ( jω i )1 [H (ω i )]


2
[β1 ]]N0 ×4 Ni ( jωi ) [H0(ωi )]
− ( jω i ) [I ]


1
 − ( jω i ) [I ]  4 N
0
0 × Ni
The coefficients are then used to form a companion matrix and eigenvalue decomposition can be applied
to estimate the modal parameters.
2.2.2
Extending UMPA to Operational Modal Analysis
The formulation as described above can be utilized to suit the operational modal analysis framework by
using power spectrums instead of frequency response functions. If {X(ω)} is the measured response and
{F(ω)} is the input force, the relationship between them in terms of frequency response function [H(ω)] is
given as follows [20]:
{X (ω )} = [H (ω )]{F (ω )}
(7)
{X (ω )}H = {F (ω )}H [H (ω )]H
(8)
Now multiplying Eq. (7) and Eq. (8)
{X (ω )}{X (ω )}H = [H (ω )]{F (ω )}{F (ω )}H [H (ω )]H
or with averaging,
[G XX (ω )] = [H (ω )] [G FF (ω )] [H (ω )]H
(9)
where [GXX(ω)] is the output response power spectra and [GFF(ω)] is the input force power spectra.
It should be noted that in OMA the power spectra of the input force is assumed to be broadband
and smooth. This means that the input power spectra is constant and has no poles or zeroes in the
frequency range of interest. The forcing is further assumed to be uniformly distributed spatially (Ni
approaching No, considering the response is being measured all over the structure). Thus the output
response power spectra [GXX(ω)] is proportional to the product [H(ω)][H(ω)]H and the order of output
response power spectrum is twice that of frequency response functions. This means the power spectrum
based UMPA model will be twice the order of a frequency response function based UMPA model.
Since [GFF(ω)] is constant, [GXX(ω)] can be expressed in terms of frequency response functions using the
polynomial model in Eq. (1).
[G XX (ω )] ∝ [H (ω )][I ][H (ω )]H
(10)
 n
 n
k 
k 
[
]
(
)
β
j
ω
∑
k

  ∑ [β k ]( jω ) 
 ×  km=0

G XX (ω i ) =  km=0
 [α ]( jω )k   [α ]( jω )k 
k
k
 ∑
  ∑

k =0
k =0
H
(11)
Since (n < m), a partial fraction model can be formed for the output power spectrum. This partial fraction
model for a particular response location p and reference location q is given by
N
R pqk
k =1
jω − λ k
G pq (ω ) = ∑
+
R ∗pqk
+
jω − λ∗k
S pqk
jω * − λ k
+
S ∗pqk
(12-a)
jω * − λ∗k
and can be more conveniently written as [15]
N
R pqk
k =1
jω − λ k
G pq (ω ) = ∑
N
R pqk
k =1
jω − λ k
G pq (ω ) = ∑
+
+
R ∗pqk
jω − λ∗k
R ∗pqk
jω − λ∗k
+
+
S pqk
− jω − λ k
S pqk
jω − (− λ k )
+
+
S ∗pqk
(12-b)
− jω − λ∗k
S ∗pqk
(
jω − − λ∗k
)
(12-c)
where S pqk and S ∗pqk are redefined to incorporate (-1).
Note that λk is the pole and Rpqk and Spqk are the kth mathematical residues. These residues are different
from the residue obtained using a frequency response function based partial fraction model since they do
not contain modal scaling factor (as no force is measured). The form of Eq. (12-c) clearly indicates that
the roots that will be found from the power spectrum data will be λk , λ∗k , − λ k and − λ∗k for each model
order 1 to N.
The high order of the power spectrum based model in comparison to FRF based model causes various
disadvantages like numerical conditioning, effect of noise etc which makes it more difficult for the
frequency domain based algorithm to give good results as they inherently suffer from numerical
conditioning problems. However this problem can be tackled by using positive power spectra [21]. In this
approach first power spectrums are calculated from measured output time responses and then respective
correlation functions are obtained by inverse Fourier transformation. After this step, the negative lag
portion of the correlation functions is set out to zero and the positive power spectra are obtained by
Fourier transforming this resultant data. This is mathematically represented as
G
+
pq
N
R pqk
k =1
jω − λ k
(ω ) = ∑
+
R ∗pqk
jω − λ∗k
(13)
It is clear from this equation that not only the order of positive power spectrum is the same as that of the
frequency response functions, but also they contain all the necessary system information. Now Eq. 4-6 can
be formed based on [GXX(ω)]+ data and second order UMPA model can be utilized for the purpose of
modal parameter estimation.
3
3.1
Experimental Studies
Analytical 15 Degree of Freedom System
The low order frequency domain algorithm UMPA-LOFD is applied to an analytical 15 degree of freedom
system as shown in Figure 1. The system is excited by a white random uncorrelated input at all 15 degrees
of freedom. Output response power spectrums are obtained from the raw time data using the correlogram
method [22, 23] and these power spectrums are further processed to obtain positive power spectrums as
explained in previous section.
Figure 1: Analytical 15 Degree of Freedom System
Figure 2 shows the auto power spectrum and positive power spectrum for the degree of freedom number 1
or driving point 1 (GXX and GXX +). A complex mode indicator function (CMIF) plot based on power
11
11
spectrums as shown in Figure 3 indicates clearly the presence of all 15 modes including a repeated mode
around 53.3 Hz.
Figure 2: Auto power spectrum and positive power spectrum for the first degree of freedom
10
10
-11
10
-12
10
-13
10
-14
Magnitude
10
-15
10
-16
10
-17
10
-18
10
-19
10
-20
10
20
40
60
80
100
120
Frequency (Hz)
140
160
180
200
Figure 3: Complex Mode Indicator Function (CMIF) based on power spectrum
Modal parameters obtained from UMPA-LOFD algorithm are listed in Table along with those obtained
from time domain algorithms like ERA, PTD. It should be noted that though these algorithms are referred
by the name through which they are known popularly in the conventional frequency response function
based experimental modal analysis framework, in this study they are essentially operational modal
analysis algorithms i.e. working on output-only data. As can be seen, the results of the UMPA-LOFD
algorithm compare very well with the time domain algorithms. Though the damping is slightly off (This is
the case with time domain algorithms as well), the error is relatively insignificant as damping is given in
terms of percentage critical.
Table 1: Analytical System Modal Parameter Comparison
True Modes
UMPA-LOFD
UMPA-ERA
UMPA-PTD
(Low Order,
Frequency Domain)
(Low Order, Time
Domain)
(High Order, Time
Domain)
Damp
Freq
Damp
Freq
Damp
Freq
Damp
Freq
1.0042
15.985
2.338
15.963
2.286
15.904
2.261
15.917
1.9372
30.858
2.517
30.863
2.478
30.691
2.445
30.731
2.7347
43.6
3.043
43.680
3.059
43.435
3.022
43.451
2.9122
46.444
3.431
46.437
3.394
46.179
3.399
46.178
3.3375
53.317
3.932
53.209
3.634
53.015
3.682
52.992
3.3454
53.391
3.296
53.306
3.385
53.148
3.390
53.102
3.7145
59.413
4.430
59.116
4.075
59.058
3.998
59.087
3.858
61.624
4.180
61.133
4.373
61.055
4.469
60.923
4.2978
68.811
4.291
69.237
4.397
68.633
4.800
68.462
4.5925
73.63
4.812
73.253
4.963
73.264
4.753
73.576
2.6093
128.84
2.712
128.848
2.672
128.909
2.432
128.587
2.4548
136.55
2.563
136.547
2.419
136.637
2.531
136.254
2.3288
143.86
2.426
143.869
2.360
143.973
2.591
143.325
2.221
150.83
2.314
150.799
2.390
150.606
2.486
150.576
2.122
157.47
2.216
157.444
2.155
157.510
2.573
157.123
Figure 4 through Figure 7 show the consistency diagrams obtained using various algorithms (ERA, PTD,
UMPA-LOFD and RFP). Except for RFP, the consistency diagrams obtained using other algorithms are
very clear and show good stability of the modes. Note that the diamonds (◊) in the consistency diagram
represent stable pole and vector. The poor quality of consistency diagram obtained using RFP algorithm
(Figure 7) makes it useless to be used for modal parameter estimation process. It is also shown by means
of Figure 8 that processing the positive power spectrums rather than normal power spectrums result in
much clear and definitive consistency diagrams.
Figure 4: Consistency diagram for Polyreference Time Domain (PTD) algorithm (Analytical system)
Figure 5: Consistency diagram for Eigensystem Realization Algorithm (ERA) (Analytical system)
Figure 6: Consistency diagram for Low Order Frequency Domain (UMPA-LOFD) algorithm
(Analytical system)
Figure 7: Consistency diagram for Rational Fraction Polynomial (RFP) algorithm (Analytical
system)
Frequency (Hz)
Figure 8: Consistency diagram for Low Order Frequency Domain (UMPA-LOFD) algorithm based
on complete power spectrum (Analytical system)
3.2
Lightly Damped Circular Plate
A lightly damped circular plate made of aluminum is randomly across the entire surface using an impact
hammer. The response is measured by placing accelerometers at 30 locations on the plate (Figure 9). A
two shaker test is also conducted to compare the OMA results with EMA results. It should be noted that
for the shaker test the plate is excited at two locations by an ergodic, stationary, broad-band pure random
signal.
Figure 10 shows the CMIF plot based on power spectrums and clearly indicates the presence of the
various modes present.
Figure 9: Experimental set up for the lightly damped circular plate
Frequency (Hz)
Figure 10: CMIF plot based on complete power spectrums obtained when plate is excited
sufficiently over its surface
Table 2: Lightly Damped Circular Plate Modal Parameter Comparison
System Modes
UMPA-LOFD
UMPA-ERA
UMPA-PTD
Using EMA
(Low Order,
Frequency Domain)
(Low Order,
Time Domain)
(High Order, Time
Domain)
Damp
Freq
Damp
Freq
Damp
Freq
Damp
Freq
0.258
56.591
0.612
56.478
0.611
56.436
0.611
56.439
0.285
57.194
0.621
57.253
0.619
57.197
0.632
57.191
0.312
96.577
0.636
96.665
0.631
96.561
0.636
96.571
0.412
132.101
0.342
131.830
0.338
131.705
0.351
131.702
0.147
132.650
0.285
132.760
0.310
132.601
0.304
132.589
0.243
219.582
0.300
219.375
0.301
219.092
0.302
219.094
0.216
220.952
0.364
221.358
0.370
221.088
0.371
221.075
0.214
231.172
0.256
230.851
0.260
230.553
0.252
230.545
0.137
232.077
0.225
232.394
0.220
232.095
0.212
232.102
0.089
352.997
0.147
351.677
0.144
351.180
0.152
351.214
0.174
355.509
0.219
355.773
0.222
355.283
0.224
355.303
0.180
374.554
0.268
373.933
0.271
373.382
0.273
373.424
0.176
377.569
0.236
377.505
0.242
377.013
0.239
376.990
0.313
412.414
0.241
411.727
0.238
411.138
0.245
411.168
0.209
486.801
0.219
485.405
0.220
484.627
0.220
484.720
The modal parameters estimated using various algorithms including the UMPA-LOFD algorithm are listed
in Table and show good agreement among them. Figures 11-14 show the consistency diagrams for the
different algorithms. The consistency diagram for UMPA-LOFD algorithm is much clear than the other
frequency domain algorithm RFP. It is also comparable to consistency diagrams of the time domain
algorithms.
Frequency (Hz)
Figure 11: Consistency diagram for Polyreference Time Domain (PTD) algorithm (Circular plate)
Frequency (Hz)
Figure 12: Consistency diagram for Eigensystem Realization Algorithm (ERA) algorithm (Circular
plate)
Frequency (Hz)
Figure 1311: Consistency diagram for Lower Order Frequency Domain (UMPA-LOFD) algorithm
(Circular plate)
Frequency (Hz)
Figure 14: Consistency diagram for Rational Fraction Polynomial (RFP) algorithm (Circular plate)
Further the independence of the various estimated modes is checked by the means of modal assurance
criterion (MAC) plot as shown in Figure 15. It is evident that all the 15 modes, most of which are closely
spaced modes, are independent and represent different modes of the system. The mode shapes obtained for
the circular plate are shown in Figure 16. The mode shapes are of similar nature to the ones obtained
through experimental modal analysis, except that they are not scaled.
Figure 125: MAC plot for Low Order Frequency Domain Algorithm
1st Mode (56.5 Hz) (1st Bending mode )
3rd Mode (96.6 Hz) (1st Umbrella mode)
4th Mode (131.8 Hz) (1st Torsion mode)
12th Mode (373.9 Hz) (Torsion + Bending mode)
Figure 136: Mode Shapes
Conclusions
A frequency domain algorithm based on power spectrum data is developed using Unified Matrix
Polynomial Approach (UMPA). This algorithm, UMPA-LOFD, is a low order algorithm and offers a
frequency domain alternative for operational modal analysis purposes. The algorithm is shown to have
comparable results as the time domain algorithms and has better numerical characteristics than the high
order frequency domain algorithms like the Rational Fraction Polynomial algorithm. The good results
obtained using the algorithm makes it a good addition to the family of operational modal analysis
algorithms.
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