1. Generalities

Uncertainties, even when reduced, can significantly change the prediction of the vibrational behavior of the structure ([bib13], [bib16]). It is therefore necessary to take them into account in a quantifiable and explicit manner in order to increase the realism and robustness of forecasts. In this context, a probabilistic uncertainty model contributes to the realism of the approach.

The classical probabilistic approach, called parametric, makes it possible to incorporate uncertainties in the data into mechanical analysis, for example parametric uncertainties in geometry, boundary conditions or material properties. In this approach, each parameter identified as a source of random uncertainties is modelled by a random variable. Since the input parameters of the model are thus characterized, probabilistic numerical methods seek to characterise in a probabilistic manner the result quantity (s) of the model. For complex structures, for which vibratory behavior depends on a large number of parameters, this type of probabilistic analysis is limited by the large amount of information required to characterize the input parameters and the difficulties in implementing the propagation of variability.

A new approach, called the non-parametric probabilistic approach to random uncertainties in structural dynamics, was recently proposed by C. Soize ([bib20] to [bib24]). This approach makes it possible to take into account model uncertainties (uncertainties in geometry for example) and modeling uncertainties (uncertainties in the kinematics of a beam or plate for example). It is based on the construction of random matrices of linear dynamic systems, after projection on a modal basis.

These two probabilistic approaches, one parametric and the other non-parametric, are complementary. Thus a mixed approach, parametric and non-parametric, can be developed (original method having resulted in publications ([bib6] and [bib11]). In particular, this mixed method is well suited to taking into account uncertainties in the analysis of a nonlinear dynamic system composed of a reduced linear structure on a modal basis and localized nonlinearities. Indeed, uncertainties in the linear structure can be treated naturally by the non-parametric approach and uncertainties in non-linearities can be treated naturally by the parametric approach.

The basic numerical model of the nonlinear dynamic system is a finite element model that will be called the « mean fine element model ». Whether it is the parametric or non-parametric approach, the laws of probability must be defined appropriately and as objectively as possible based on this average model. A Gaussian model of random matrices is not adapted to low frequency dynamics (negative natural frequencies). In order to build the corresponding probability distribution, we use Jayne’s principle of maximum entropy ([bib14], [bib15], [bib17]) as well as the available information (model, fine elements, average, algebraic properties of matrices, etc.)

In this document, we present the non-parametric approach for transitory or harmonic resolutions of the dynamic system. The parametric approach is more particularly presented in the case where it is combined with the non-parametric method.

Readers looking for the fundamental results of stochastic dynamics may refer to [18] and readers looking for the theoretical details of the probabilistic approach presented in this document may refer to [bib19]. Examples of uses of the approach are given in [bib5], and [bib7]. In [bib10], experimental trials made it possible to show the predictive nature of the approach.