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A Unified Matrix Polynomial Approach To Modal Identification
Journal of Sound and Vibration (1998) 211(3), 301322
A UNIFIED MATRIX POLYNOMIAL APPROACH
TO MODAL IDENTIFICATION
R. J. A
LLEMANG AND D. L. BROWN
Structural Dynamics Research Laboratory, Department of Mechanical, Industrial and
Nuclear Engineering, University of Cincinnati, Cincinnati, Ohio 45221-0072, U.S.A.
(Received 20 June 1996, and in nal form 4 March 1997)
One important current focus of modal identication is a reformulation of modal
parameter estimation algorithms into a single, consistent mathematical formulation with
a corresponding set of denitions and unifying concepts. Particularly, a matrix poly-
nomial approach is used to unify the presentation with respect to current algorithms such
as the least-squares complex exponential (LSCE), the polyreference time domain (PTD),
Ibrahim time domain (ITD), eigensystem realization algorithm (ERA), rational fraction
polynomial (RFP), polyreference frequency domain (PFD) and the complex mode indi-
cation function (CMIF) methods. Using this unied matrix polynomial approach
(UMPA) allows a discussion of the similarities and dierences of the commonly used
methods. The use of least squares (LS), total least squares (TLS), double least squares
(DLS) and singular value decomposition (SVD) methods is discussed in order to take
advantage of redundant measurement data. Eigenvalue and SVD transformation methods
are utilized to reduce the eective size of the resulting eigenvalueeigenvector problem as
well.
7 1998 Academic Press Limited
1. INTRODUCTION
Modal parameter estimation is a special case of system identication where the a priori
model of the system is known to be in the form of modal parameters. Over the past
twenty years, a number of algorithms have been developed to estimate modal parameters
from measured frequency or impulse response function data. While most of these
individual algorithms, summarized in Table 1, are well understood, the comparison of
one algorithm to another has become one of the thrusts of current research in this area.
Comparison of the dierent algorithms is possible when the algorithms are reformulated
using a common mathematical structure.
This reformulation attempts to characterize dierent classes of modal parameter
estimation techniques in terms of the structure of the underlying matrix polynomials
rather than the physically based models used historically. Since the modal parameter
estimation process involves a greatly over-determined problem (more data than indepen-
dent equations), this reformulation is helpful in understanding the dierent numerical
characteristics of each algorithm and, therefore, the slightly dierent estimates of modal
parameters that each algorithm yields. As a part of this reformulation of the algorithms,
the development of a conceptual understanding of modal parameter estimation technol-
ogy has emerged. This understanding involves the ability to visualize the measured data
in terms of the concept of characteristic space, the data domain (time, frequency, spatial),
the evaluation of the model (polynomial) order of the problem, the condensation of the
0022460X/97/130301 + 22 $25.00/0/sv971321
7 1998 Academic Press Limited R. J. ALLEMANG AND D. L. BROWN
302
T
ABLE 1
Acronymsmodal parameter estimation algorithms
Modal parameter estimation algorithms
CEA
Complex exponential algorithm [1, 2]
LSCE
Least squares complex exponential [1, 2]
PTD
Polyreference time domain [3, 4]
ITD
Ibrahim time domain [5, 6]
MRITD
Multiple reference Ibrahim time domain [5]
ERA
Eigensystem realization algorithm [7, 8]
PFD
Polyreference frequency domain [9, 1012, 13, 14]
SFD
Simultaneous frequency domain [15]
MRFD
Multi-reference frequency domain [16]
RFP
Rational fraction polynomial [17]
OP
Orthogonal polynomial [1820]
CMIF
Complex mode indication function [19, 21]
data, and a common parameter estimation theory that can serve as the basis for
devloping any of the algorithms in use today. The following sections review these
concepts as applied to the current modal parameter estimation methodology.
1.1.
DEFINITION OF MODAL PARAMETERS
Modal identication involves estimating the modal parameters of a structural system
from measured inputoutput data. Most current modal parameter estimation is based
upon the measured data being the frequency response function or the equivalent impulse
response function, typically found by inverse Fourier transforming the frequency re-
sponse function. Modal parameters include the complex-valued modal frequencies
l
r
,
modal vectors
{c
r
} and modal scaling (modal mass or modal A). Additionally, most
current algorithms estimate modal participation vectors
{L
r
} and residue vectors {A
r
} as
part of the overall process. Modal participation vectors are a result of multiple reference
modal parameter estimation algorithms and relate how well each modal vector is excited
from each of the reference locations included in the measured data. The combination of
the modal participation vector
{L
r
} and the modal vector {c
r
} for a given mode give the
residue matrix [A
r
] for that mode.
In general, modal parameters are considered to be global properties of the system. The
concept of global modal parameters simply means that there is only one answer for each
modal parameter and that the modal parameter estimation solution procedure enforces
this constraint. Most of the current modal parameter estimation algorighms estimate the
modal frequencies and damping in a global sense but very few estimate the modal vectors
in a global sense.
2. SIMILARITIES IN MODAL PARAMETER ESTIMATION ALGORITHMS
The similarities in modal parameter estimation algorithms arise from the common
theoretical basis of the algorithms. Fundamentally, each algorithm starts with a system
that can be represented by a second order, linear, constant coecient matrix equation.
This fundamental equation depends upon several assumptions: linearity, time invariance,
observability and reciprocity. Rather than working with this matrix equation directly, UNIFIED MATRIX POLYNOMIAL
303
Figure 1. MDOFsuperposition of SDOF (positive frequency poles).
most modal parameter estimation algorithms utilize measured frequency response func-
tions (or the time domain equivalent, the impusle response functions) as the experimental
database for the algorithm.
The current approach in modal identication involves using numerical techniques to
separate the contributions of individual modes of vibration in measurements such as
frequency response functions. The concept involves estimating the individual single-de-
gree-of-freedom contributions (SDOF) to the multiple-degree-of-freedom measurement
(MDOF):
[H(
v)]
N
o
× N
i
=
s
N
r = 1
[A
r
]
N
o
× N
1
j
v l
r
+ [A
r
]
N
o
× N
i
j
v l*
r
.
(1)
This concept is represented mathematically in equation (1) and graphically in Figure 1.
Equation (1) represents a mathematical problem that, at rst observation, is non-linear
in terms of the unknown modal parameters. Once the modal frequencies
l
r
are known,
the mathematical problem is linear with respect to the remaining unknown modal
parameters [A
r
]. For this reason, the numerical approach in many algorithms involves
two or more linear stages. Typically, the modal frequencies and modal participation
vectors are found in a rst stage and residues, modal vectors and modal scaling are
determined in a second stage. This general approach is discussed in the sections 2.3
and 2.4. R. J. ALLEMANG AND D. L. BROWN
304
2.1.
DATA DOMAIN
Modal parameters can be estimated from a variety of dierent measurements that exist
as discrete data in dierent data domains (time and/or frequency). These measurements
can include free-decays, forced responses, frequency response functions (FRFs) or
impulse response functions (IRFs). These measurements can be processed one at a time
or in partial or complete sets simultaneously. The measurements can be generated with
no measured inputs, a single measured input, or multiple measured inputs. The data can
be measured individually or simultaneously. There is a tremendous variation in the types
of measurements and in the types of constraints that can be placed upon the testing
procedures used to acquire these data. For most measurement situations, FRFs are
utilized in the frequency domain and IRFs are utilized in the time domain.
2.2.
CHARACTERISTIC SPACE
From a conceptual viewpoint, the measurement space of a modal identication
problem can be visualized as occupying a volume with the co-ordinate axes dened in
terms of the three sets of characteristics. Two axes of the conceptual volume correspond
to spat