150.00 160.00 s -0.50 0.50 g 86:STRL:DRVR:+Z (a) 0.00 0.50 s -4.20e-3 4.20e-3 g 2 (b) 0.00 50.00 Hz 100e-9 100e-6 g 2 (c) 0.00 50.00 Hz -180.00 180.00 ° (d) Fig.5 A time record (a), (b) a typical cross-correlation function (in green the unwindowed one), calculated with respect to a certain reference output signal with its cross-power spectrum, amplitude (c) and phase (d) An exponential window reduces the effect of leakage and the influence of the higher time lags, which have a larger variance. Moreover, the application of an exponential window to correlations is compatible with the modal model and the pole estimates can be corrected. A thorough discussion and a comparison between the correlogram and periodogram estimates can be found in [9-10]. The weighted correlogram approach to estimate half spectra is illustrated in Fig.5. An exponential window of 10% has been used and its effect on the correlation (Fig.5b) and spectrum (Figures 5c and 5d) data is clearly visible. 3.3 Operational modal parameter estimation It can be shown that the l l× half spectrum matrix can be modally decomposed as [9,10] ( ) { } { }* * * 1 j j j n i i i i yy i i i v g v g S ω ω λ ω λ + = 〈 〉 〈 〉 = + − − ∑ (4) which replace the modal participation factors in the case of output-only data and iλ are the poles, occurring in complexconjugated pairs and related to the eigenfrequencies iω and damping ratios iξ by the relation * 2 , j 1 i i i i i i λ λ ξω ω ξ =− ± − . (5) By using an operational modal identification technique, as the recalled output-only data version of the Stochastic Subspace Identification in the time or frequency domain [11] or of the Polyreference Least Square Complex Exponential in the time domain or PolyMAX (Polyreference Least Square Complex Frequency Domain) in the frequency domain [12,13], it is possible to build up a stabilization diagram (Fig.6a), assuming subsequently an increasing number of poles. The stabilization diagram gives a strong indication of the number of present physical modes and allows the selection of the best estimates for physical poles and, hence, for eigenfrequencies and damping ratios and for the said operational reference factors. where n is the number of modes, { } l iv ∈ℂ are the mode shapes, l ig〈 〉 ∈ℂ are the so-called operational reference factors, 318
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