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