15 Quantitative Study on the Modal Parameters Estimated Using the PLSCF. . . 163 1. Choose a modal parameter estimation method capable of producing poles and mode shape vectors at varying model orders 2. Estimate poles at mode shape vectors at various model orders 3. Initiate the statistical representation of the pole estimates 4. Introduce an occurrence threshold and temporarily exclude modal parameter estimates below said threshold 5. Compute the MAC for each remaining bin and remove modal parameter estimates whose MAC value is poor in comparison to the MAC values of the majority of poles in a given bin 6. Combine adjacent bins whose MAC values are similar 7. Add any adjacent modal parameters that did not pass step (4) to a bin if the MAC values are similar 8. Compute mean values and standard deviation for each remaining bin The AMA algorithm has shown successful in detecting structural modes from datasets on several real structures, e.g. the Little Belt Suspension Bridge, the Heritage Court Tower and a Ro-Lo ship [23]. The modal parameter estimates from the experimental dataset using the AMA algorithm, were then used to simulated 300 s forced response sequences at a sampling frequency of 5000 Hz using superposition and digital filter theory [24]. Gaussian white noise processes were used as input in all 35 DOFs with a noise level corresponding to 0.01% of the standard deviation of the forced response. The simulated data were input into the PLSCF algorithm and the MITD algorithm to estimate modal parameters. A model order of 100 was used for both algorithm, to ensure that many modal parameters were output. For the PLSCF algorithm three of the four corner points were used as references of the 35 responses. The correlation function estimates used to compute the spectral densities were postmultiplied by an exponential window so that the correlation function value at the end of the measurement time was reduced to 0.01%. This is a common procedure used to reduce the bias error added from truncations in the time domain [25]. A total of 100 averages were used to compute the spectral density estimates. The damping ratio added from applying an exponential window was subtracted from the estimated damping ratios output by the PLSCF algorithm. For the MITD algorithm an unbiased Welch estimator with a blocksize of 512 samples, was used to compute the correlation functions for all 35 measurement channels. The first 15 time lag values were removed from all correlation functions to suppress measurement noise [26]. Exactly 90 time lag values from each of the 1225 correlation functions were used as input into the MITD algorithm. At this point the correlation functions had almost fully decayed. For the modal parameter estimates output by both algorithm, only poles originating from an under-damped system were considered, i.e. having low damping and positive frequency. Furthermore, damping ratio estimates above 10% were discarded since the damping ratio of the plexiglass plate does not exceed 3.5% for any of the modes of interest. 15.4 Results The modal parameters estimated using the two algorithms are presented in the following. The stabilization diagrams are seen in Figs. 15.1 and 15.2. In the present work, when using the PLSCF algorithm, it was crucial that a representative number of references were chosen and that the number of references were limited to only a few. The most clear stabilization diagram were obtained by choosing three of the four corner points as references, which was also mentioned in Sect. 15.3. By choosing more references, the number of spurious poles present in the stabilization diagrams would rise. It was attempted to use 35 references, which rendered the stabilization diagram completely indecipherable for the AMA algorithm to interpret. This observation is related to the fact that the number of eigenvalues estimated equals mNi, where m is the number of frequency lines and Ni denotes the number of references. By increasing the number of references more eigenvalues are estimated as a multiple of the number of frequency lines. Therefore, if too many references (e.g. SVD) are used, equation condensation should be employed, while, when only a few references are used equation condensation should not be required. For the PLSCF algorithm nine modes were identified, which correspond to the number of modes used to simulate the data. It is quite clear that the frequency component of the pole estimates stabilize for increasing model order and that barely any spurious pole estimates are present. This is in line with previous observations, that very clear stabilization diagrams are output, when using the PLSCF algorithm, that was presented in Sect. 15.2. Upon zooming onto the first two and closely spaced modes in the same plot, it is seen, up to a model order of 20, that the model parameters are not stabilizing well on the frequency axis. Above a model order of 50 it appears that the estimates are strongly frequency stable. In the intermediate rage, model order ranging from 20 to 50, the frequency component of the model parameter estimates are slightly skewed to the right, yielding higher frequency estimates. It should be clarified that in Fig. 15.1, the horizontal dashed black line corresponds to the occurrence threshold that was described in step (4), see Sect. 15.2. Modal parameter estimates marked
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