ture (λ,φλ) results from det(F−λ I)=0, F ϕλ =λ ϕλ, φλ =Hϕλ, (6) where λ ranges over the set of eigenvalues of F. In practice, the truncation order of the SVD is increased from 1 to the maximal system order in Equation (4) to get a stabilization diagram of the obtained modes vs model order. This gives results for successive different but redundant models and modes that are common to many successive models can be distinguished from the spurious modes. There are many papers on the used identification techniques. A complete description can be found in [2, 6, 7, 3], and the related references. A proof of non-stationary consistency of these subspace methods can be found in [3]. 3 Confidence Interval Computation The statistical uncertainty of the obtained modal parameters is necessary to assess the confidence one can have in these values, e.g. when comparing the modal parameters of different states of a structure. Modal parameters with little confidence (and hence large confidence intervals) are little useful for comparing structural states. The uncertainties of the modal parameters at a chosen system order can be computed from the uncertainty of the subspace matrix by doing a sensitivity analysis, and the covariance of the subspace matrixΣH can be evaluated by cutting the sensor data into blocks on which instances of the subspace matrix are computed. It holds Δfj =Jf j Δ(vecH), Δdj =Jdj Δ(vecH), Δϕj =Jϕj Δ(vecH), with the frequencies fj, damping ratios dj and mode shapes ϕj, and their sensitivities Jwith respect to vecH. It follows covfj =Jf j ΣHJT f j , covdj =Jdj ΣHJT dj , covϕj =Jϕj ΣHJT ϕj . This offers a possibility to compute confidence intervals on the modal parameters at a certain system order without repeating the system identification. In [9] this algorithm was described in detail for the covariance-driven SSI. In this paper, three extensions of the confidence interval computation of [9] are used: • As the mode shapes are defined up to a complex constant, the confidence interval computation on them requires an additional constraint. In [9], the confidence intervals are computed with respect to one point of the mode shape that is normalized to value one, which results in a confidence interval of size zero of this point. In [4], the confidence intervals of the mode shapes are computed with respect to the maximal amplitude of deflection, which is applied in this paper. Confidence Intervals of Modal Parameters during Progressive Damage Test 241
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