application of frequency-domain system identification techniques in ...

26 Chapter 2. Operational Modal Analysis1995; Maia and Silva, 1997). The multiple coherence mγ 2 o in response point o atcircular frequency ω is defined byN imγo 2 (ω) = ∑ ∑N iH [o,s] (ω) · ŜF F [s,t](ω) · (H[o,t] (ω) ) ∗Ŝ XX[o,o] (ω)s=1 t=1(2.44)This quantity is a measure of the correlation between the response signal and allof the input forces. A unity value expresses a perfect linear relationship betweenthe considered signals. A coherence value less than unity, between 0 and 1, can bedue to one or a combination of the following causes• presence of uncorrelated noise on the response and/or force measurements• presence of leakage in the measurements• non-linear behavior of the test structureIn practice, averaging techniques are used in order to reduce the effect of measurementnoise. It should be noted that the values in the coherence function aredependent on the number of averages that were used to obtain the data. Sincethe coherence function is a measure for the accuracy of the estimate, the functioncan be used for the computation of the uncertainty on the FRFs (see e.g.,(Verboven, 2002)).Parameter confidence intervalsGiven the uncertainty on the FRF measurements, a good approximation of thecovariance matrix of the ML estimate ˆθ ML is obtained by inverting the Fisherinformation matrix (Pintelon and Schoukens, 2001)}cov{ˆθML ≃ [ 2Re ( Jp H J −1p)](2.45)with J p the Jacobian matrix evaluated in the last iteration step of the Gauss-Newton optimization. As one is often mainly interested in the uncertainty on themodal frequencies and damping ratios, only the covariance matrix of the denominatorcoefficients is required. Starting from (2.45), it can be shown (Verboven, 2002)that this matrix is given bycov{ˆα ML } ≃[2N∑oN ik=1(Tk − Sk T R −1kS ) ] −1k(2.46)Hence, it is not necessary to invert the full matrix occurring in (2.45). From (2.46)it is possible to compute the uncertainty on the modal frequency and damping ratios(Guillaume et al., 1989).