188 M. Chang and S.N. Pakzad noise input, it can be assumed that the stochastic response governs the entire system and the output responses are represented as a function of [A,C]. The observer gainMyi is introduced to Eq. 18.1 and the output response is observed as: yi DCAi x0 C i 1 X D0 b Y yi 1 (18.7) In Eq. 18.7, b Y D CA M D C.ACMC/ M, and Ai approaches zero for i p where p is the number of independent Markov parameters. The Markov parameters for ERA-OKID-OO are determined as Y DCA M, which are calculated using the recursive relationship of auto-regressive coefficients b Y . 18.3 Physical Contribution Ratio The Physical Contribution Ratio (PCR) is introduced to quantify the participation of modal vibration when modal parameters are estimated from the impulse response [1]. ERA based SID methods commonly formulate the Hankel matrix composed of sequentially arranged Markov parameters (impulse responses) and estimate the equivalent numerical model using Singular Value Decomposition (SVD). Assuming that the Markov parameters include noise contribution, the effect is directly shown in discretized state-space matrices. Markov parameters (yi DCAi 1B) in Eq. 18.2 are transformed into the modal coordinates from physical coordinates using the eigenvalue decomposition of state matrix as: yi DC ƒi 1 1B (18.8) In Eq. 18.8, and ƒare the matrices of eigenvectors and eigenvalues of A. Accordingly, the mode shape and modal amplitude matrixes are determined as: C D 1 2 m and 1B D b1 T b2 T bm T T (18.9) In Eq. 18.9, 1 to m are the column vectors of identified mode shapes and b1 to bm are the row vectors of transformed impulse at each node; mdenotes the number of identified modes including noise parameters. Due to the overparameterization, the results of SID normally involve large amount of spurious modes that can be removed using stabilization diagram and expert judgments [15]. In this study, the number of identified modes mis conservatively determined as ten times larger than true structural modes in order not to lose true modes with small PCR values. Expanding the estimated impulse response, the PCR of the jth mode in the kth diagonal component is defined as [1]. PCRkj D Xn kjbjk j n 1 Xm Xm iD1 ki bik i n 1 (18.10) In Eq. 18.10, n is the time index and j is the jth eigenvalue of A. The PCR quantifies the contribution of true modes over time, indicating a measure of the signal power in true structural modes amongst estimated impulse responses. Since the measurement noise is decentralized with reference to all the modes, the increase of noise contamination result in the decrease in PCR values. 18.4 Validation Using Measured Data Two examples are used to validate the performance of PCR depending on the identification methods: (1) numerically simulated output data from a 4-DOF simply supported beam model and (2) ambient vibration data from Golden Gate Bridge (GGB). Based on the exact modal information in numerically simulated model, the relationship between estimated PCR values and errors in identified modal parameters is investigated. The GGB example validates the use of PCR for practical implementation to achieve accurate modal identification.
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