AComparison of Experimental, Operational, and Combined
Experimental-Operational Parameter Estimation Techniques
T. Lauwagie∗, R. Van Assche, J. Van der Straeten, W. Heylen
K.U.Leuven, Department of Mechanical Engineering,
Celestijnenlaan 300 B, B-3001, Heverlee, Belgium
e-mail: ward.heylen@mech.kuleuven.be
Abstract
A modal analysis aims at the identification of the modal parameters of a test structure from the measured vi-
bratory behaviour. Traditionally, both the input forces and the resulting responses are measured. However, in
many applications it is not possible to measure (all) the input forces. During the last decade, two new classes
of modal parameter estimation techniques have been developed to overcome this problem: the operational
techniques and the combined experimental-operational techniques. Operational modal analysis techniques
can identify the modal parameters from the responses of the structure; they do not require the input forces.
The combined experimental-operational techniques, only require a part of the input forces to estimate the
modal parameters.
The work presented in this paper is part of the evaluation process of these new modal parameters estima-
tion techniques; it compares the modal parameters provided by the operational and combined experimental-
operational modal analysis techniques with the modal parameters obtained with the experimental modal
analysis technique.
Nomenclature
∗ Complex conjugate
T Transpose
† Pseudo-inverse
{} Column vector
〈〉 Row vector
[] Matrix
h(t) Impulse response function
H Frequency response function (FRF)
j Imaginary unit
L Modal participation factor
P Projection operator
Q Modal scaling factor
λ Eigenvalue
ψ Mode shape
γ Discrete eigenvalue
Ψ Discrete eigenvector
1 Introduction
Modal analysis techniques allow to measure the modal parameters, i.e. resonant frequencies, damping ratios,
mode shape vectors and modal participation factors, of a mechanical structure. With the experimental modal
analysis (EMA) approach [1, 2], the structure under investigation is placed in a test set-up where a number
of controlled input forces are applied and measured. The response of the structure is measured in a grid
of test locations. Although experimental modal analysis is a very versatile technique, it cannot be applied
in two important cases. First of all, it cannot be used to analyse a system in operational conditions, e.g. a
∗Now at IMEC, Kapeldreef 75, B-3001, Heverlee, Belgium.
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flying aircraft. Furthermore, it is not straightforward to apply an experimental modal analysis on massive
structures like highway bridges or oil rigs. Here, the main problem is the power of the excitation device,
which is usually insufficient to excite the structure with the required magnitude.
The operational modal analysis (OMA) techniques [3, 4] were developed to overcome the two main short-
comings of the experimental modal analysis approach. The operational techniques only require the response
signals of the investigated structure, they do not require the signals of the input forces. Because of this, it is
possible to use ambient forces as excitation forces. Typical examples of ambient excitation forces are wind
or traffic in the case of a bridge, and atmospheric turbulences in the case of a flying aircraft. Of course,
the operational modal analysis approach has a number of limitations. The input forces need to have a uni-
form spectrum. In case the input spectrum is not flat, the predominant excitation frequencies can appear as
system poles in the parameter estimation step, although they are not. Furthermore, the operational modal
analysis techniques can identify resonant frequencies, damping ratios and mode shape vectors; the modal
participation factors, however, can only be identified if the structure is measured in at least two different test
configurations [5].
Currently, a third class of modal analysis techniques is being developed and evaluated [6, 7], namely the
combined experimental-operational techniques (OMAX). The goal of these techniques is to combine the
advantages of the two previous approaches, by considering two classes of input forces: ambient forces that
cannot be measured and artificially applied forces that are measured. In a simplified way, one could state
that the ambient excitation forces are used to excite test structure with a sufficiently high amplitude, while
the artificially applied forces are used to identify the modal participation factors and, in the case the spectra
of the ambient forces are not uniform, to distinguish the actual modes from the spurious modes that are
generated by the frequency contents of the ambient input forces.
The work that is described here is part of the evaluation processes of the combined experimental-operational
modal analysis techniques. In this paper, the modal parameters obtained with the three modal analysis
approaches are critically compared and discussed.
2 Parameter Estimation Methods
In this text, the following three parameters estimation methods are used to identify the modal parameters:
- Poly-reference least-squares complex exponential method (EMA)
- Data driven stochastic subspace identification method (OMA)
- Combined least-squares frequency method (OMAX)
The goal of this section is to provide a summary of the theoretical background of these methods. For each
method, references with detailed information are provided.
2.1 Poly-reference Least-Squares Complex Exponential (PLSCE)
In the case of an experimental modal analysis, both the input and response signals are measured. From
the measured time signals, the averages of the frequency response functions (FRF) between every input
and output location can be computed. There is whole range of methods to extract modal parameters from
measured FRFs, however, in this work only the most commonly used method is considered, i.e. the poly-
reference least-squares complex exponential method [8]. This method is based on the re-transformation of
the FRFs to the time domain which provides an averaged version of the impulse response functions. These
impulse response functions can be grouped into an impulse response matrix that is related to the modal
parameters as
[
h(t)
]
=
N∑
r=1
(
Qr{ψ}r{ψ}
T
re
λrt +Q∗r{ψ}
∗
r{ψ}
∗
r
Teλ
∗
rt
)
(1)
2998 PROCEEDINGS OF ISMA2006