Keywords: system identification, proper orthogonal decomposition, Tikhonov regularisation, damping matrix identification, least squares estimation, constrained optimisation, Kronecker algebra.

The identification of damping plays a crucial role in the validation of numerical models of dynamical systems. In spite of extensive research, the knowledge regarding damping forces is least developed compared to the other forces acting on a structure. There are two basic reasons for this. Firstly, from a theoretical point of view, it is not in general clear which state variables are relevant to determine the damping forces. This fact makes it difficult to select what mathematical form of damping model should be used at the first place, let alone how to identify its parameters from the experimental measurements. Secondly, from experimental point of view, unlike the mass and stiffness properties, the damping properties can be identified only by a dynamic testing.

The most common method to model damping in a multiple-degree-of-freedom linear systems is to assume the so called viscous damping. Many researchers have proposed methods to identify viscous damping matrix from experimental measurements. These methods can be divided into two broad categories: (a) damping identification from modal testing and analysis and (b) direct damping identification from the forced response measurements in the frequency or time domain. The modal method entails estimating the modal parameters, such as natural frequencies, damping ratio and mode shapes, from the measured transfer functions, and consequently the damping matrix are reconstructed from the estimated modal parameters. Direct methods on the other hand bypass the modal parameter extraction step and fit the damping matrix to the measured response. The common issues regarding the identification of the damping matrix using conventional modal analysis are (a) the accuracy of the identified modal parameters, and consequently the system matrices, rely 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 [1]. The first problem is inherent to conventional modal analysis. If the peaks in the measured FRFs are not distinct or are closely spaced, the modal parameter extraction procedure is difficult to apply [2]. As a consequence, the identified system matrices using the extracted modal parameters become erroneous. It is therefore difficult to extend the modal identification procedure in the mid-frequency range, or for periodic systems (such as bladed disks in turbomachineries) inherently containing closely spaced modes. The second problem arises for systems with high damping such as a panel with viscoelastic damping. In this paper a new approach using proper orthogonal decomposition is proposed which can potentially circumvent, or at least alleviate, these difficulties.

The proper orthogonal decomposition (POD) method, also known as principle component analysis (PCA), entails the extraction of the dominant eigen-subspace of the response correlation matrix, namely proper orthogonal modes (POMs), over a given frequency band of interest. These dominant eigenvectors span the system response optimally for the prescribed frequency range of interest. Thus, the POD method has been used for the model reduction of linear as well as non-linear dynamical systems [3,4].

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 with a reasonable assumption. To achieve this objective, the POD method is adopted as a tool for a model reduction strategy to solve the inverse problem involving system identification over a frequency range of interest. A novel mathematical framework exploiting Kronecker Algebra is presented which allows for estimation of the POD reduced order damping matrix elements in the least-square sense. Furthermore, in order to deal with the ill-posedness of the problem, prior information concerning the system, such as symmetry of the damping matrix, is introduced as Tikhonov regularisation [5] in a constrained optimisation framework. Using computer simulation, it was shown that with only a few POD modes, the reconstructed FRF matches reasonably well with the original FRF.

1

S. Adhikari, "Optimal complex modes and an index of damping non-proportionality", Mechanical System and Signal Processing, 18(1), 1-27, 2004. doi:10.1016/S0888-3270(03)00048-7

2

D. J. Ewins, Modal Testing: Theory and Practice, Research Studies Press, Baldock, England, second edition, 2000.

3

A. Sarkar and R. Ghanem, "Mid-frequency structural dynamics with parameter uncertainty", Computer Methods in Applied Mechanics and Engineering, 191(47-48), 5499-5513, 2002. doi:10.1016/S0045-7825(02)00465-6

4

V. Lenaerts, G. Kerschen and J.-C. Golinval, "Identification of a continuous structure with a geometrical non-linearity. Part II: Proper orthogonal decomposition", Journal of Sound and Vibration, 262(4), 907-919, 2003. doi:10.1016/S0022-460X(02)01132-X

5

A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-Posed Problems, Winston-Wiley, Washington, 1977.

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