OMA: From Research to Engineering Applications

Abstract

Ambient vibration modal identification, also known as Operational Modal Analysis (OMA), aims to identify the modal properties of a structure based on vibration data collected when the structure is under its operating conditions, i.e., when there is no initial excitation or known artificial excitation. This method for testing and/or monitoring historical buildings and civil structures, is particularly attractive for civil engineers concerned with the safety of complex historical structures. However, in practice, not only records of external force are missing, but uncertainties are involved to a significant extent. Hence, stochastic mechanics approaches are needed in combination with the identification methods to solve the problem. In this context, this paper’s contribution is to introduce an innovative ambient identification method based on the Hilbert Transform to obtain the analytical representation of the system response in terms of the correlation function. This approach opens the pathway for a monitoring system that is user friendly and can be used by people who have little to no knowledge of signal processing and stochastic analysis such as those who are responsible for the maintenance of a city’s historical buildings. In particular, this method operates in time domain only. Specifically, firstly the correlation functions matrix ${\mathbf{R}}_{X} \left( \tau \right)$ is determined based on the recorded time domain data. Next, performing a Singular Value Decomposition (SVD) on ${\mathbf{R}}_{X} \left( \tau \right)$ for $\tau = 0$ leads to an estimate of the modal matrix ${\varvec{\Phi}}$ containing all the modal shapes. In this manner, once ${\varvec{\Phi}}$ is known, the entire correlation functions matrix in modal space ${\mathbf{R}}_{Y} \left( \tau \right)$ is recovered. Further, the analytical signals of the auto-correlation functions in modal space are determined performing the sum of each auto-correlation function with its Hilbert transform. Moreover, since the analytical signal can be expressed in terms of amplitude and phase, then frequencies and damping ratios estimation is possible. Finally, in order to prove the reliability of the method several numerical examples and an experimental test are reported.

Appendix

The proposed algorithm, entirely reported in this appendix, requires as input only the output signals (X) and the time vector (time). It calculates automatically the frequencies (fid), the damping ratios (Z_ID_LOG) and the modal shapes both normalized with respect to the first component of each mode (PHI_IDNN) and not-normalized (PHI_ID). The entire developed MatLab function, called TD_ASM.m, is shown below.

In this code only two kind of interactions with the user are requested. The first one is the organization of the input data i.e. the time vector that is a row vector and the structural output process that is a three-dimensional array. The second one is a step of the function TimeStopSelection.m that is contained into TD_ASM.m and that requires the choice of the time interval to be used to perform the average in Eq. (16) and to identify the coefficient $c_{1}$ in Eq. (17). In order to simplify this step, TimeStopSelection.m has an interactive graphic interface that allows to choose, with few clicks, the aforementioned time intervals as reported in Fig. 5a, b respectively for frequency identification and damping ratios identification.

Fig. 5

Interactive graphic interface of TimeStopSelection.m: for frequency estimation a, for damping ratio estimation b