Abstract
A modal identification algorithm is developed, combining techniques from Second Order Blind Source Separation (SOBSS) and State Space Realization (SSR) theory. In this hybrid algorithm, a set of correlation matrices is generated using time-shifted, analytic data and assembled into several Hankel matrices. Dissimilar left and right matrices are found, which diagonalize the set of nonhermetian Hankel matrices. The complex-valued modal matrix is obtained from this decomposition. The modal responses, modal auto-correlation functions and discrete-time plant matrix (in state space modal form) are subsequently identified. System eigenvalues are computed from the plant matrix to obtain the natural frequencies and modal fractions of critical damping. Joint Approximate Diagonalization (JAD) of the Hankel matrices enables the under determined (more modes than sensors) problem to be effectively treated without restrictions on the number of sensors required. Because the analytic signal is used, the redundant complex conjugate pairs are eliminated, reducing the system order (number of modes) to be identified half. This enables smaller Hankel matrix sizes and reduced computational effort. The modal auto-correlation functions provide an expedient means of screening out spurious computational modes or modes corresponding to noise sources, eliminating the need for a consistency diagram. In addition, the reduction in the number of modes enables the modal responses to be identified when there are at least as many sensors as independent (not including conjugate pairs) modes. A further benefit of the algorithm is that identification of dissimilar left and right diagonalizers preclude the need for windowing of the analytic data. The effectiveness of the new modal identification method is demonstrated using vibration data from a 6 DOF simulation, 4-story building simulation and the Heritage court tower building.
Highlights
An introduction to blind source separation applied to modal identification is provided along with a review of previous developments in the literature.1.1
Joint Approximate Diagonalization (JAD) of the Hankel matrices enables the under determined problem to be effectively treated without restrictions on the number of sensors required
In Second Order Blind Identification (SOBI), it is assumed that the data correlation matrices, Ryy, are Hermetian for all τ and, the right diagonalizing matrix factor is the Hermetian transpose of the left
Summary
In the last several years research has evolved the application of Blind Source Separation (BSS) techniques to solve the Modal ID entification (MID) problem. BSS attempts to find source components, s j t C J , with prescribed properties embedded in measured data xi t C I. The techniques applicable for modal identification assume that the measured data is a linear mixture (as opposed to convolutive) of the components. The objective of BSS is to simultaneously estimate the mixing matrix, A, and the vector of components, s t , from the observed data, x t. The potential application of BSS on vibration data is fairly obvious. One might consider using BSS to estimate both the (inverse) modal matrix and the modal responses, q t Φ 1x t. Note that the modal matrix and modal responses are real valued if the topology of the damping matrix is restricted (e.g. proportional damping) and are otherwise complex-valued
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