Modal updating

In the special case of modal updating, assume that only the lowest Nm modes contribute significantly to the model response and only the modal parameters of these modes are to be identified. Then, the uncertain structural parameters include the modal fi-equencies the modal damping ratios and the components of the measured DOFs of the mode shapes except the ones which are used for normalization, m = 1, 2,..., Nm, thus, there are a total number of Nm(No + 1) unknown modal parameters in modal identification. 

Katafygiotis, L. S. and Yuen, K.-V. Bayesian spectral density approach for modal updating using ambient data. Earthquake Engineering and Structural Dynamics 30(8) (2001), 1103-1123. 

Yuen, K.-V. and Katafygiotis, L. S. Bayesian modal updating using complete input and incomplete response noisy measurements. Journal of Engineering Mechanics (ASCE) 128(3) (2002), 340-350. 

Yuen KV, Katafygiotis LS (2001a) Bayesian time-domain approach for modal updating using ambient data. Probab Eng Mech 16(3) 219-231... 

Individual samples of isotopically heavier granitic rocks have also been reported from the Canadian Shield (5T i = +7.3 to +11.3 new and updated data from Bottomley et al. 2003) and from the Isle of Skye, Scotland (8 Li = +8.1 Pistiner and Henderson 2003). These are all rocks with long near-surface histories and modal chlorite or epidote, suggesting their isotopic... 

Tins text is designed to provide students witli a clear, concise introduction to pharmacology. The basic explanations presented in tliis text are not intended to surest tliat pharmacology is an easy subject. Drug therapy is one of tlie most important and complicated treatment modalities in modem healtli carei Because of its importance and complexity and the frequent additions and changes in the field of pharmacology, it is imperative tliat health care professionals constantly review and update tlieir knowledge. 

The adaptive MCMC simulation method is applied to updating the robust reliability for a two-story structural frame, depicted in Figure 2.21. The bay width and story height are 5.0 m and 2.5 m, respectively. The Young s modulus and mass density are taken to be 200 GPa and 7800 kg/m, respectively. The beams have a cross-sectional area of 0.01 vn5 and a moment of inertia of 6.0 X 10 m but they are 0.02 m and 1.5 x 10 m" for the columns. As a result, the structure has modal frequencies of 5.20 and 15.4 Hz. The structure is assumed to have 1.0% of critical damping for all modes. A simple model with two degrees of freedom is used in the system identification in order to estimate the inter-story stiffnesses and to assess the reliability of the structure. Specifically, the stiffness matrix is given by ... 

The prior PDF p(0 C) is taken to be independent log-normal PDFs with means of 0.9 and 1.2 and unit variance. Using the modal data V, the updated PDF for the stiffness parameter vector 0 is formulated as ... 

Figure 2.27 shows the updated failure probability of the frame. Assessment of the impact of damage on the reliability of the structure can be performed even though there are infinitely many most probable parameter estimates in this case and it is also seen from the similarity of Figures 2.24 and 2.27. The loss of information about the second modal frequency does not have much effect on the updated failure probability because the first mode dominates the response of this frame. Even the individual values of 0i and 02 cannot be identified, different combinations... 

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. 

Figure 4.14 Contours of the marginal updated joint PDF of the first two modal frequencies...

Keywords Bayesian inference damage detection eigenvalue problem iterative algorithm modal data mode shape expansion model updating structural health monitoring substmcture system identification... 

In this chapter, a Bayesian model updating method using incomplete modal data is presented with applications to structural health monitoring. As reported in the literature [18,51,52,267], the realistic assumption is made that only the modal frequencies and partial mode shapes of some modes are measured system mode shapes are also introduced, which avoid mode matching between the measured modes and those of the dynamical model. The novel feature... 

In the next section, the proposed updating approach is presented which provides estimates of the system modal frequencies and system mode shapes, as well as estimates of the stiffness model parameters, based on incomplete modal data. Examples with a twelve-story building and a three-dimensional braced frame wiU be used to demonstrate the method with applications to structural health monitoring. 

The mode shapes are usually measured with incomplete components, i.e., with missing DOFs but the modal frequencies are measured with relatively high accuracy. Therefore, the sequence of optimization starts from computing the missing components of the mode shapes. First set the updated model parameters at their nominal values ... 

The three optimization problems are coupled. In other words, the expressions of the optimal vectors depend on the optimal values of other parameters. In order to search for the overall optimal parameters efficiently, the proposed methodology updates the system modal frequencies, system mode shapes and stiffness scaling parameters in an iterative manner by using successively the optimization results of Section 5.3. The iterative procedure consists of the following steps ... 

In this example, the Bayesian model updating method is applied to update the finite-element model of a three-dimensional six-story braced frame, which is based on a model of an actual laboratory test structure. It is square in plan with width a = 5 m. There are four columns for each floor, one at each comer. Each of them have interstory stiffnesses of 10 MN/m and 15 MN/m in the x and y directions, respectively. Furthermore, each face in each floor is stiffened by a brace and its stiffness is taken to be 20 MN/m. As a result, the interstory stiffness is 80 MN/m and 100 MN/m in the x and y directions, respectively. The floor mass is taken to be 10 metric tons for each floor. As a result, the first five modal frequencies of the stmcture are 3.432, 3.837, 6.305, 10.10 and 11.29 Hz. 

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