Bayesian Operational Modal Analysis

Bayesian Operational Modal Analysis (BAYOMA) adopts a Bayesian system identification approach for Operational Modal Analysis (OMA). Operational Modal Analysis (OMA) aims at identifying the modal properties (natural frequencies, damping ratios, mode shapes, etc.) of a constructed structure using only its (output) vibration response (e.g., velocity, acceleration) measured under operating conditions. The (input) excitations to the structure are not measured but are assumed to be 'ambient' ('broadband random'). In a Bayesian context, the set of modal parameters are viewed as uncertain parameters or random variables whose probability distribution is updated from the prior distribution (before data) to the posterior distribution (after data). The peak(s) of the posterior distribution represents the most probable value(s) (MPV) suggested by the data, while the spread of the distribution around the MPV reflects the remaining uncertainty of the parameters.

Pros and Cons

In the absence of (input) loading information, the identified modal properties from OMA often have significantly larger uncertainty (or variability) than their counterparts identified using free vibration or forced vibration (known input) tests. Quantifying and calculating the identification uncertainty of the modal parameters become relevant.

The advantage of a Bayesian approach for OMA is that it provides a fundamental means via the Bayes' Theorem to process the information in the data for making statistical inference on the modal properties in a manner consistent with modeling assumptions and probability logic.

The potential disadvantage of Bayesian approach is that the theoretical formulation can be more involved and less intuitive than their non-Bayesian counterparts. Algorithms are needed for efficient computation of the statistics (e.g., mean and variance) of the modal parameters from the posterior distribution.

Methods

Bayesian formulations have been developed for OMA in the time domain[1] and in the frequency domain using the spectral density matrix[2] and FFT (Fast Fourier Transform)[3] of ambient vibration data. Based on the formulation for FFT data, fast algorithms have been developed for computing the posterior statistics of modal parameters.[4] The fundamental precision limit of OMA has been investigated and presented as a set of uncertainty laws.[5][6]

Notes
  • See Jaynes[7] and Cox[8] for Bayesian inference in general.
  • See Beck[9] for Bayesian inference in structural dynamics (relevant for OMA)
  • The uncertainty of the modal parameters in OMA can also be quantified and calculated in a non-Bayesian manner. See Pintelon et al.[10]

See also

References
  1. Yuen, K.V.; Katafygiotis, L.S. (2001). "Bayesian time-domain approach for modal updating using ambient data". Probabilistic Engineering Mechanics. 16 (3): 219–231. doi:10.1016/S0266-8920(01)00004-2.
  2. Yuen, K.V.; Katafygiotis, L.S. (2001). "Bayesian spectral density approach for modal updating using ambient data". Earthquake Engineering and Structural Dynamics. 30 (8): 1103–1123. doi:10.1002/eqe.53.
  3. Yuen, K.V.; Katafygiotis, L.S. (2003). "Bayesian Fast Fourier Transform approach for modal updating using ambient data". Advances in Structural Engineering. 6 (2): 81–95. doi:10.1260/136943303769013183.
  4. Au, S.K.; Zhang, F.L.; Ni, Y.C. (2013). "Bayesian operational modal analysis: theory, computation, practice". Computers and Structures. 126: 3–14. doi:10.1016/j.compstruc.2012.12.015.
  5. Au, S.K. (2013). "Uncertainty law in ambient modal identification. Part I: theory". Mechanical Systems and Signal Processing. 48 (1–2): 15–33. doi:10.1016/j.ymssp.2013.07.016.
  6. Au, S.K. (2013). "Uncertainty law in ambient modal identification. Part II: implication and field verification". Mechanical Systems and Signal Processing. 48 (1–2): 34–48. doi:10.1016/j.ymssp.2013.07.017.
  7. Jaynes, E.T. (2003). Probability Theory: The Logic of Science. United Kingdom: Cambridge University Press.
  8. Cox, R.T. (1961). The Algebra of Probable Inference. Baltimore: Johns Hopkins University Press.
  9. Beck, J.L. (2010). "Bayesian system identification based on probability logic". Structural Control and Health Monitoring. 17 (7): 825–847. doi:10.1002/stc.424.
  10. Pintelon, R.; Guillaume, P.; Schoukens, J. (2007). "Uncertainty calculation in (operational) modal analysis". Mechanical Systems and Signal Processing. 21 (6): 2359–2373. Bibcode:2007MSSP...21.2359P. doi:10.1016/j.ymssp.2006.11.007.
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