Identification of vibration sources from measurements of transverse displacement, from a Bayesian point of view.

The objective of this work is to identify vibratory loads applied on a structure using an inverse method and taking into account different types of uncertainty. In the case of partially known structures such as composite materials, errors on model parameters yield biased results in addition to randomness provided by measurement noise. The RIFF (Résolution Inverse Filtrée Fenêtrée, in french) method is used for the identification of vibratory sources and, under certain conditions, for the identification of bias on model parameters. This local method is based on the estimation of discrete spatial and temporal derivative of the equation of motion of the structure and has the advantage of using a minimum of prior information. Unlike global methods that require knowledge of boundary conditions, the RIFF method only requires the knowledge of the operator of the structure and a small number of measurement points around a specific grid node to determine the force which is applied on it. Combined with a finite element model, the RIFF method is particularly suited to the study of substructures. Inverse problems are inherently unstable due to measurement noise, a regularization step is then necessary to force the solution of the method to have a better physical behavior. Low-pass filtering, Tikhonov regularization or Bayesian methods based on conditional probabilities are all methods of regularization. In this work, the choice is to use Bayesian methods because they offer two advantages: First of all, a priori information such as the noise level or type of source can be formalized and taken into account by the Bayesian formalism. Second, the data of the problem are seen as random variables. Therefore, it is possible to estimate the probability density function of each unknown parameter, including the most probable value, the variance and the shape of its distribution. In other words, Bayesian methods consider both the information arising from theory, numerical simulation or past experiences and also give access to the error bars on the estimation of the various parameters of the model. When the number of parameters increases, the probability density functions are not numerically tractable anymore. One should then use random sampling methods, such as MCMC methods, and especially the Gibbs sampling. This one uses conditional probabilities of parameters which are organized into a tree-like structure, a so-called hierarchical model, to improve convergence of Markov chains. Finally, this algorithm allows the estimation of the probability density of parameters of interest from the histograms of data sampled.