Statistical modal identification using ambient or strong wind response data

The problem of identification of the modal parameters of a structural model using measured ambient or strong wind response time histories is addressed. A Bayesian probabilistic approach is followed to obtain not only the most probable (optimal) values but also the probability distribution of the updated modal parameters. This is very important when one plans to use these estimates for further processing, such as for updating the theoretical finite-element model of the structure, because it provides a rational basis for weighting differently the errors of the various modal parameters, the errors being the differences between the theoretical and identified values of these parameters. The approach is introduced for a SDOF system and it can be extended to general MDOF systems. The statistical properties of an estimator of the spectral density are presented. Based on these statistical results expressions for the updated probability density function (PDF) of the modal parameters are derived. The updated PDF is well approximated by a Gaussian distribution centered at the optimal parameters at which the updated PDF is maximized. Numerical examples using simulated data are presented to illustrate the proposed method

A methodology for treating non-identifiability in model updating

The problem of updating a structural model and its associated uncertainties by utilizing structural response data is addressed. The present paper focuses on the problem of model updating in the general nonidentifiable case for which certain simplifying assumptions available for identifiable cases are not valid. It is shown that in this case, the PDF is distributed in the neighborhood of an extended and extremely complicated manifold of the parameter space. The computational difficulties associated with calculating the highly complex posterior PDF are discussed and an algorithm for an efficient approximate representation of the above manifold and the posterior PDF is presented. Using this approximation, expressions for calculating the uncertain predictive response are established. A numerical example involving noisy data is presented to demonstrate the proposed method

An optimal sensor location methodology for designing modal experiments

A methodology is proposed for making decisions regarding the optimal location of sensors for modal identification. Uncertainties are quantified using probability distributions, and a Bayesian methodology is proposed for deriving appropriate expressions for the updated probability density function (PDF) of the modal parameters based on measured ambient response time histories. The optimal sensor configuration is selected as the one that minimizes the information entropy which is a unique measure of the uncertainty in the modal parameters. The information entropy measure is also extended to handle large uncertainties expected in the pre-test nominal modal model of a structure. Genetic algorithms are well-suited for solving the resulting discrete optimization problem. In experimental design, the proposed information entropy can be used to design cost-effective modal experiments by exploring, comparing and evaluating the benefits from placing additional sensors in the structure in relation to the improvement in the quality of the modal predictions

Probabilistic model updating using dynamic data

The problem of updating a structural model and its associated uncertainties by utilizing structural response data is addressed. Using a Bayesian probabilistic formulation, the posterior probability density function (PDF) of the uncertain model parameters for given measured data can be obtained. Previous results provide asymptotic approximations to the posterior PDF under the assumption of relatively small model error and measurement noise, relatively small number of model parameters to be identified, and large number of data. Using such results one can approximate-the posterior PDF as a weighted sum of Gaussian distributions centered at some optimal values of the parameters at which some positive measure-of-fit function is minimized. The present paper addresses the problem of model updating in the general case for which the assumptions for the previous asymptotic approximations are not satisfied. In this case the posterior PDF is distributed in the neighborhood of an extended and extremely complicated manifold of the parameter space. The computational difficulties associated with calculating the highly complex posterior PDF are discussed and an algorithm for an efficient approximate representation of the above manifold and the posterior PDF is presented. Using this approximation, expressions for calculating the uncertain predictive response are established. A numerical example involving noisy data is presented to demonstrate the proposed method

A probabilistic framework for model updating and structural health monitoring

This paper addresses the problem of model updating and structural health monitoring utilizing measured dynamic data. In order to explicitly treat the uncertainties arising from measurement noise, modeling error, and an inherent nonuniqueness of model updating, the proposed methodology follows a Bayesian framework for model updating. A set of damage index parameters are introduced to measure the possible damage within the structure. The proposed health monitoring methodology aims in calculating the joint probability density function (PDF) of these damage indices utilizing the posterior PDF of the stiffness parameters for both the "healthy" and "damaged" structure. This approach allows for a probabilistic assessment of damage based on the measured data and any prior information. The proposed health monitoring methodology is illustrated with a numerical example

Optimal control of structures: A reliability-based approach

A probability-based optimal control methodology for use in the design of active tuned-mass dampers is presented. The probabilistic approach provides a rational basis for accounting for both loading and structural modeling uncertainties. A reliability-based performance index is considered which accounts for structural safety considerations in the design of the active tuned-mass damper. This index is directly related to the structural failure probability which is formulated in terms of multi-dimensional integrals over the space of uncertain parameters. A new asymptotic expansion is used to compute approximately these reliability-type integrals. Accuracy issues are addressed by comparing asymptotic results with the ones obtained using numerical integration. A numerical study is carried out to investigate the effects of structural uncertainties on the optimal design and performance of the active mass dampers. It is found that consideration of structural modelling uncertainties improves substantially the performance and robustness of the ATMD design

On stability of linear dynamical systems with small Markov perturbations

The mean-square stability analysis of linear equations with coefficients perturbed by small bounded functions of Markov processes is studied. The stability is determined by the sign of the mean-square Lyapunov exponent. This Lyapunov index depends on a small parameter epsilon related to the strength of the perturbations. A method is proposed for calculating the Lyapunov index from the largest real eigenvalue of a specially constructed matrix expanded as, power series in epsilon. An algorithms presented for calculating the terms in the expansion as well as the number of terms needed to be included in the expansion for the purpose of determining the system stability. The foundation of the algorithm is based on well-known results of the Kato perturbation theory for closed operators. For sufficiently small epsilon, the methodology gives the necessary and sufficient conditions for mean-square stability The method and the algorithm are illustrated by analyzing the stability of MDOF linear-systems. Examples include the parametric resonance of an oscillator with low damping, as well as the stability of a bridge deck subject to wind excitation

Simulation of homogeneous and partially isotropic random fields

Four different approaches for the simulation of random fields, one of which is introduced herein, are examined. They are based on the spectral representation method, formulated to represent homogeneous and partially isotropic random fields, and generate simulations with random variability in both their amplitudes and phases. The four methods are compared in terms of the variability of the amplitudes and phases of the simulated processes. It is shown that, although all of them reproduce well the prescribed auto-spectral and cross-spectral density functions, some of them preserve spectral characteristics such as homogeneity and amplitude variability better than others. The four approaches are utilized in the simulation of spatially variable ground motions experiencing loss of coherence