Aalborg Universitet
Modal Identification of Output-Only Systems using Frequency Domain Decomposition
Brincker, Rune; Zhang, L.; Andersen, P.
Published in:
Proceedings of the European COST F3 Conference on System Identification & Structural Health Monitoring, 6-9
June, 2000, Universidad Politécnica de Madrid, Spain
Publication date:
2000
Document Version
Publisher's PDF, also known as Version of record
Link to publication from Aalborg University
Citation for published version (APA):
Brincker, R., Zhang, L., & Andersen, P. (2000). Modal Identification of Output-Only Systems using Frequency
Domain Decomposition. In J. A. Güemes (Ed.), Proceedings of the European COST F3 Conference on System
Identification & Structural Health Monitoring, 6-9 June, 2000, Universidad Politécnica de Madrid, Spain (pp. 273-
282). Universidad Politécnica de Madrid.
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Modal Identification
of
Output-Only Systems
using Frequency Domain Decomposition
Rune Brincker'l, Lingmi Zhang
2
l and Palle Andersen
3
l
IJ
Department
of
Building Technology
and
Structural Engineering
Aalborg University, Sonhgaardsholmsvej
57,
DK
9000, Aalborg
lJ
Institute
of
Vibration Engineering,
Nanjing University
of
Aeronautics and Astronautics, Nanjing, 210016, China, P.R.
J)Structural Vibration Solutions
ApS
NOVI
Science Park,Niels Jernes
Vej
10,
DK
9220 Aalborg East, Denmark
SUMMARY: In this paper a new frequency domain technique
is
introduced for the modal
identification
of
output-only systems, i.e. in the case where the modal parameters must be
estimated without knowing the input exciting the system. By its user friendliness the
technique
is
closely related to the classical approach where the modal parameters are
estimated by simple peak picking. However, by introducing a decomposition
of
the spectral
density function matrix, the response spectra can be separated into a set
of
single degree
of
freedom systems, each corresponding
to
an individual mode. By using this decomposition
technique close modes can be identified with high accuracy even in the case
of
strong noise
contamination
of
the signals. Also, the technique clearly indicates harmonic components in
the response signals.
INTRODUCTION
Modal identification
of
output-only systems is normally associated with the identification
of
modal parameters from the natural responses of civil engineering structures, space structures
and large mechanical structures. Normally, in these cases the loads are unknown, and thus,
the modal identification has
to
be carried out based on the responses only. Real case
examples on some civil engineering structures can be found in Ventura and Horyna
[1]
or
Andersen et
al.
[2].
The present paper deals with a new way
of
identifying the modal parameters
of
a structure
from the responses only when the structure
is
loaded by a broad-banded excitation.
The technique presented in this paper
is
an extension
of
the classical frequency domain
approach often referred
to
as
the Basic Frequency Domain (BFD) technique, or the Peak
Picking technique. The classical approach is based on simple signal processing using the
Discrete Fourier Transform, and
is
using the fact that well separated modes can be estimated
273
directly from the power spectral density matrix at the peak, Bendat and Piersol [3]. Other
implementations
of
the technique make use
of
the coherence between channels, Felber [
4]
.
The classical technique gives reasonable estimates
of
natural frequencies and mode shapes
if
the modes are well separated. However, in the case
of
close modes, it can be difficult
to
detect the close modes, and even in the case where close modes are detected, estimates
becomes heavily biased. Further, the frequency estimates are limited by the frequency
resolution
of
the spectral density estimate, and in all cases, damping estimation is uncertain or
impossible.
Even though the classical approach has limitations concerning accuracy in the identification
process, the classical approach has important advantages when compared to other
approaches.
It
is natural to compare with classical two-stage time domain approaches such
as
the Polyreference technique, V old et al [5], the Ibrahim Time Domain tehcnique, Ibrahim and
Milkulcik [6], and the Eigensystem Realization Algorithm, Juang and
Papa [7], or to compare
with the new one-stage time domain identification tehcniques know
as
the Stochastic
Subspace
Identification algorithms, Van Overschee and De Moor [8]. The main advantages
compared to these other techniques is that the classical approach is much more user-friendly,
it is faster, simpler to use, and gives the user a "feeling"
of
the data he or she is dealing with.
The fact that the user works directly with the spectral density functions helps the user in
figuring out what is structural just by looking at the spectral density functions. This reinforces
the users understanding
of
the physics and thus provides a valuable basis for a meaningful
identification.
The technique presented in this paper is a Frequency Domain Decomposition (FDD)
technique. It removes all the disadvantages associated with the classical approach, but keeps
the important features
of
user-friendliness and even improves the physical understanding by
dealing directly with the spectral density function. Further, the technique gives a clear
indication
of
harmonic components in the response signals.
In this paper it is shown that taking the Singular Value Decomposition (SVD)
of
the spectral
matrix, the spectral matrix is decomposed into a set
of
auto spectral density functions, each
corresponding to a single degree
of
freedom (SDOF) system. This result is exact in the case
where the loading is white noise, the structure is lightly damped, and when the mode shapes
of
close modes are geometrically orthogonal.
If
these assumptions are not satisfied, the
decomposition into
SDOF
systems is approximate, but still the results are significantly more
accurate than the results
of
the classical approach.
THEORETICAL BACKGROUND OF FREQUENCY DOMAIN DECOMPOSTION
The relationship between the unknown inputs
x(t)
and the measured responses
y(t)
can be
expressed a
s,
Bendat & Piersol
[9]
:
(1)
WhereGxx(jw)is
the r x r Power Spectral Density (PSD) matrix
of
the input,
ris
the
number
of
inputs, Gyy(Jw)is the mx m PSD matrix
of
the responses, m is the number
of
274
[3]. Other
lber [4).
. e shapes
if
difficult to
, estimates
frequency
ncertain or
ntification
to other
es
such as
rahim and
>compare
)tochastic
ivantages
·-friendly,
ling with.
e user in
-einforces
:!aningful
1 (FDD)
1ut
keeps
'lding by
a clear
spectral
ns, each
the case
:s
hapes
ied, the
ly
more
[
ON
can be
(1)
is the
ber
of
responses,
H(jm)
is the
mxr
Frequency Response Function (FRF) matrix, and
"-"and
superscript T denote complex conjugate and transpose, respectively .
The FRF can be written in prutial fraction, i.e. pole/residue form
n R R
H(jm)
= L k + k
k=I
jm-
J.k
jm-
J.k
(2)
where n is the number
of
modes,
J.k
is the pole and
Rk
is the residue
(3)
where
iflk,
1 k is the mode shape vector and the modal participation vector, respectively.
Suppose the input is white noise, i.e. its PSD is a constant matrix, i.e. G xx
(jm)
=
C,
then
Equation
(1) becomes
(4)
where superscript H denotes complex conjugate and transpose. Multiplying the two partial
fraction factors and making use
of
the Heaviside partial fraction theorem, after some
mathematical manipulations, the output
PSD can be reduced to a pole/residue form
as
follows
n A A B B
G (jm) = L k + k + k + k
YY
k=t
}w
-
J.k
}w-:tk
-
}w
-:tk
-
}w
-
:tk
(5)
where
Ak
is
the k th residue matrix
of
the output PSD. As the output PSD itself the residue
matrix is an
mxm
hermitian matrix and is given by
(6)
The contribution to the residue from the k th mode is given by
(7)
whereak
is minus the real part
of
the pole
J.k
= - ak + jmk . As it appears, this term becomes
dominating when the damping is light, and thus, is case
of
light damping, the residue
becomes propmtional to the mode shape vector
- T T
Ak
oc RkCRk =
(PkYk
Cykiflk
=
dkiflk<Pl
(8)
275
where d k is a scalar constant. At a certain frequency m only a limited number
of
modes will
contribute significantly, typically one or two modes. Let this set
of
modes be denoted by
Sub(m).
Thus, in the case
of
a lightly damped structure, the response spectral density can
always be written
(9)
This is a modal decomposition
of
the spectral matrix. The expression is similar to the results
one would get directly from Equation
(1) under the assumption
of
independent white noise
input, i.e. a diagonal spectral input matrix.
IDENTIFICATION ALGORITHM
In the Frequency Domain Decomposition (FDD) identification, the first step is to estimate the
power spectral density matrix. The estimate
of
the output PSD
GyyCiOJ)
known at discrete
frequencies
m=
mi
is then decomposed by taking the Singular Value Decomposition (SVD)
of
the matrix
(10)
where the matrix
Ui
= [uil,ui
2
,K
,uim] is a unitary matrix holding the singular vectors uiJ,
and Si is a diagonal matrix holding the scalar singular values siJ. Near a peak corresponding
to the k th mode in the spectrum this mode or may be a possible close mode will be
dominating.
If
only the k th mode is dominating there will only be one term in Equation (9).
Thus, in this case, the first singular vector u
il
is an estimate
of
the mode shape
(11)
and the corresponding singular value is the auto power spectral density function
of
the
corresponding single degree
of
freedom system, refer to Equation (9). This power spectral
density function is identified around the peak by comparing the mode shape estimate
~
with
the singular vectors for the frequency lines around the peak. As long as a singular vector is
found that has high MAC value with
~
the corresponding singular value belongs to the
SDOF density function.
From the piece
of
the SDOF density function obtained around the peak
of
the PSD, the
natural frequency and the damping can be obtained. In this paper the piece
of
the
SDOF
PSD
was taken back to time domain by inverse FFT, and the frequency and the damping was
simply estimated from the crossing times and the logarithmic decrement
of
the corresponding
SDOF auto correlation function.
In the case two modes are dominating, the first singular vector will always be a good estimate
of
the mode shape
of
the strongest mode. However, in case the two modes are orthogonal, the
first two singular vectors are unbiased estimates
of
the corresponding mode shape vectors. In
276