PSI - Issue 64

Ge-Wei Chen et al. / Procedia Structural Integrity 64 (2024) 724–731 Ge-Wei Chen, Xinghua Chen, Piotr Omenzetter / Structural Integrity Procedia 00 (2019) 000–000

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̈ � ( ) = � ̈ � ( − 1) + ( )

(1) where ̈ � ( ) = [ ̈( ) � ̈( − 1) � … ̈( − + 1) � ] , ( ) = [ ( ) � �×� … �×� ] , and � is the system matrix made up of the model AR coefficients. The modal parameters of the discrete-time can be computed by the eigenvalue decomposition of the system matrix and then converted into their continuous-time counterparts (Alvin and Park, 1994). 4.2. Eigensystem realization with observer/Kalman filter method The observer/Kalman filter identification counteracts the lack of information on initial conditions and extends the applicability of ERA to general input/output systems (Juang et al, 1993). For output-only system identification, the inputs are unavailable and the stochastic subsystem, describing the influence of noise, is formulated as follows: ��� = � + � , � = � + � (2) where and are the discrete-time state and output matrices, respectively, � and � are the state and output vectors, respectively, and � and � are the process and measurement noise vectors, respectively. For systems experiencing unmeasured random inputs, it is assumed that the stochastic components control the system evolution and the outputs are observable. The observer gain, � , is introduced into the state equation, which then becomes: ��� = ( − ) � + � (3) where is an arbitrarily matrix that allows maximizing the system stability. The eigenvalues of matrix − can be set to any desired values for the specific case of observer gain , e.g., they can all be set to 0, such that ( − ) � = for ≥ . In this formulation, p is the number of independent Markov parameters for the stabilization to occur. The Markov parameters for the output-only system are � � = � ( = 0,1, … , − 1) . The Hankel matrix is then formed by overparameterizing the independent Markov parameters to compensate for the rank deficiency (Chang and Pakzad, 2013), and subsequently the general ERA method (Juang and Pappa, 1985) is utilized to determine the matrices , and and then the modal parameters. 5. System identification results Figure 3 shows typical HVT acceleration time histories from the middle of the longest span. After pre-processing, including trend removal, 5 th -order Butterworth high-pass filtering with the cut-off frequency of 0.5 Hz, and three-fold decimation, to eliminate low and high frequency noise, the vertical peak acceleration was 0.035 m/s 2 , which was more than twice that of the lateral peak acceleration of 0.013 m/s 2 . Both values are small, indicating weak excitation levels. Since in HVT an additional weak artificial excitation was applied, it is interesting to observe the difference in the vibration levels between the AVT and HVT. For a comprehensive comparison, the averaged power spectral densities (APSDs) (Felber, 1994) of all the channels were merged by averaging all PSDs across all channels. The APSDs are shown in Fig. 4, which demonstrates that both the vertical and lateral HVT vibrations were significantly stronger compared to the AVT, especially at higher frequencies. There were, however, some exceptions to this general observation, including the peak just below 4 Hz for the vertical direction and the very low frequency band around 4.5 Hz for lateral vibrations. Table 1 provides a comparison of the modal frequencies and damping ratios obtained using the AVT and HVT data and the numerical FEM. The modes are indicated by symbols V, V/T and L for the vertical, mixed vertical-torsional and lateral modes, respectively. The FEM results (Column 2) were taken as the reference. The mixed vertical-torsional mode V7/T1 was not identified by any of the three system identification algorithms, whereas vertical modes V3 to V6 and lateral mode L1 were also missed in the AVT (Columns 3 to 5). This was likely due to the relatively low participation of these modes in the measured dynamic responses. It can clearly be seen that four more vertical higher order modes, V3-V6, and the lateral fundamental mode L1 were recovered from the HVT compared to the AVT. This