Identification of Damping Using Proper Orthogonal Decomposition
Khalil, M., Adhikari, S. and Sarkar, A.,
The Eighth International Conference on Computational Structures Technology,
Las Palmas de Gran Canaria, Spain, 2006.
System identification plays a crucial role in the validation of
numerical models. In the context of damped
multiple-degree-of-freedom linear dynamic systems, the process of
system identification involves identification of the mass,
stiffness and damping matrices. Traditionally this has been done
using experimental modal analysis[1,2,3].
There are two broad approaches for identifying the system matrices
- (a) system identification from modal data, and (b) direct
identification from the forced response measurements. The modal
testing and analysis method seeks to determine the modal
parameters, such as natural frequencies, damping ratio and mode
shapes, from the measured transfer functions, and then reconstruct
the system matrices from these data. In one of the earliest works,
Lancaster[4] had shown that the mass, damping and
stiffness matrices can be obtained from the measured complex modes
and frequencies. Ibrahim[5] used the higher order
analytical modes together with the experimental set of complex
modes to compute improved mass, stiffness and damping matrices.
Later, Adhikari[6] proposed a method in which the system
matrices are identified using the residues and poles of the
measured transfer functions. In the second category, Roemer and
Mook[7] have developed methods in the time domain for
simultaneous identification of the mass, damping and stiffness
matrices. Chen et al.[8] have proposed a direct
frequency domain technique for identification of the system
matrices.
Each of the above methods have their own advantages and
disadvantages. The common issues regarding the identification of
the system matrices using conventional modal analysis are (a) the
accuracy of the identified modal parameters, and consequently the
system matrices, relies on the number of distinct `peaks' in the
measured frequency response functions (FRFs), and (b) if the
damping is non-proportional, the identification of complex modes
poses a serious challenge[9,10]. The first problem
is inherent to conventional modal analysis. If the peaks in the
measured FRFs are are not distinctive or are closely spaced, the
modal parameter extraction procedure itself becomes inaccurate
[1]. As a consequence, the identified system matrices
using the extracted modal parameters become erroneous. For this
reason it is difficult to extend the modal identification
procedure in the mid-frequency range, or for periodic systems with
closely spaced modes such as bladed disks in turbomachineries. The
second problem arises for systems with high damping materials such
as a panel with viscoelastic damping. In this paper a new approach
using proper orthogonal decomposition (POD) is proposed which can
potentially circumvent, or at least alleviate, these difficulties.
The POD method entails the extraction of the dominant eigenspace
of the response correlation matrix, also known as the
proper orthogonal modes, over a given frequency band of
interest. These dominant eigenvectors span the system response
optimally on the prescribed frequency range of interest.
This permits the construction of an optimal and adaptive
reduced-order model of the dynamical system. The POD method has
been extensively used for the model reduction of linear as well as
non-linear dynamical systems, for example see
references[11,12,13]. POD has also been used for
the identification of a non-linear dynamical system whereby the
non-linear stiffness parameter was identified with a reduced-order
model[14].
The primary objective of this investigation is the identification
of the damping matrix of a general linear dynamical system in the
medium frequency range, with the mass and stiffness matrices being
known a priori. To achieve this objective, the POD method is used
as a tool for model reduction strategy to solve the inverse
problem involving large scale system identification over a
frequency range of interest.
Acknowledgments
The first author acknowledges the support of the National
Sciences and Engineering Research Council of Canada. The second
author acknowledges the support of the UK Engineering and Physical
Sciences Research Council (EPSRC). The third author acknowledges
the support of a Discovery Grant from National Sciences and
Engineering Research Council of Canada and the Canada Research
Chair Program.
References
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BiBTeX Entry
@INPROCEEDINGS{cp23,
AUTHOR={M. Khalil and S. Adhikari and A. Sarkar},
TITLE = {Identification of Damping Using Proper Orthogonal Decomposition},
BOOKTITLE = {Proceedings of the Eighth International Conference on Computational Structures Technology},
EDITOR = {Topping, B. H. V. and Montero, G. and Montenegro, R.},
PUBLISHER = {Civil-Comp Press},
ADDRESS = {Stirlingshire, United Kingdom},
NOTE = {paper 73},
YEAR = 2006
}
by Sondipon Adhikari