Bayesian Spectral Density Approach

Keywords ambient vibration correlation function Duffing oscillator hydraulic jump information entropy modal identification optimal sensor placement spectral density structural health monitoring Wishart distribution 

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Chapters 3 and 4 introduce two recently developed Bayesian methods for updating the mathematical models of dynamical systems. Chapter 3 presents the Bayesian spectral density approach. The spectral density estimator is defined to take into account of the aliasing and leakage effect. The statistical properties of the spectral density estimator are examined and... 

In this section, Bayesian analysis is performed to identify the uncertain coefficients of the quadratic function. The effective temperature T is assumed to be different from the measured values since there is measurement noise in the data acquisition process and the temperature in different parts of the building could also be non-uniform. The difference is assumed Gaussian with zero mean and variance a. In this study, this standard deviation is taken to be aj = 0.5°C since the difference between the indoor and outdoor temperature measurement was around 1 °C and the average value was used as the measured temperature T for the wth day. On the other hand, the squared fundamental frequency is identified by the Bayesian spectral density approach to be presented in Chapter 3. Therefore, the uncertain parameters include the coefficients of thequadratic function and the effective temperatures 0 = [l>o, b, b2, T, Ti,7 ], where iV is the number of data points. The data include the measurements of the temperature and the identified squared fundamental frequencies X> = [li,. .., j, 2> >... 

The Bayesian spectral density approach for parametric identification and model updating regression analysis are applied. During the monitoring period, four typhoons flitted over Macao. The structural behavior under such violent wind excitation is treated as discordance and the measurements obtained under these events are not taken into account for the analysis. By excluding these fifteen days of measurements, there are 168 pairs of identified squared fundamental frequency and measured temperature in the data set. Figure 2.28(a) shows the variation of the identified squared fundamental frequencies with their associated uncertainties represented by a confidence interval that is bounded by the plus or minus three standard derivations from the estimated values. It is noticed that this confidence interval contains 99.7% of the probability. Since the confidence intervals are narrow compared with the variation... 

In this chapter, the Bayesian spectral density approach, which is a frequency-domain approach, for modal/model updating using wide-band response data is presented. It utilizes the statistical properties of the spectral density estimator to obtain not only the optimal values of model parameters but also their associated uncertainty by means of the updated probability distribution of the uncertain parameters. Uncertainty quantification is important for many applications, such as damage detection and reliability analysis. 

Another case is investigated with a very short duration of measurement, namely T = 60 s, so it contains roughly 38 fundamental periods of the oscillator. The Bayesian spectral density approach is used for its identification with the frequency index set /C = 1,2,..., 45. Figure 3.13 shows the conditional updated PDFs of and with all other parameters fixed at their optimal values. It is obvious that the conditional PDFs are non-Gaussian so the Gaussian... 

Let = [S2, f, 5/0, 5eo] denote the vector of the uncertain model parameters to be identified. The most probable values of the parameters and their associated uncertainty are updated by the Bayesian spectral density approach. 

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