@phdthesis{480,
type = "Thesis",
year = "1996",
title = "Statistiques du comportement de systèmes dynamiques non linéaires - Application à la tenue à la mer de navires rapides",
journal = "",
editor = "",
volume = "",
number = "",
pages = "",
author = "Monbet Valerie",
url = "https://archimer.ifremer.fr/doc/00000/480/",
organization = "",
address = "",
school = "Université de Rennes 1",
abstract = "In this thesis, we propose a method to estimate the mean number of level upcrossings for the degrees of freedom (and/or their first and second derivatives) of a structure submitted to random excitation such as wave forces. Although this method could be applied to several types of systems, we are especially interested here in the study of the behaviour of an high speed ship for the comfort of passengers. The non linear modelization of the system and the representation of sea waves as a Gaussian stationnary process allows to have a different approach of problems like seakeeping than purely hydrodynamic and deterministic methods. Criteria of comfort may be given by statistics of the number of level upcrossings and a statistical analysis allows, by describing the sea surface with random processes, to estimate statistics of the upcrossing number.
We propose in a first part a bibliography and present several methods amoung the most frequently applied to approximate, without simulation, the joint probability density function of the response of a non linear oscillator under random excitations. The joint probability density function integrated in the Rice formula gives the expected number of upcrossings. We conclude this synthesis by a table which gives the more important conditions under which each method is available and the obtained results.
Then, for a mixing stationnary process with smooth trajectories, we estimate the mean number of upcrossings with two non parametrical estimators (an empirical one and an adaptative one) from discretized observations and we compare these estimators to a regression estimator. The adaptative estimator is construct such that the joint probability density function of the process and its derivative is approximate by a polynomial perturbation of the product of the marginal densities. We evaluate the asymptotic quadratic risk of both estimators with respects to the number of observed points, the discretization time step and the dimension of the projection space for the adaptative estimator. In the particular case of Gaussian processes, the empirical estimator is compared to the parametrical one.
At last we tackle the problem of the behaviour of an high speed ship on real sea waves. We explain how we construct an approximate non linear model for the motions for the hull, by adding to the calculated linear model some non linear terms justified by the diffraction-radiation problem. Then, we justify the choice of the method which is applied to estimate the upcrossing frequency of the pitch acceleration. The accelerations and not only the displacements of the system are studied and this characteristic is really important for the choice of the method. Finally we compare several approximations of the mean number of upcrossings of the pitch acceleration for relatively short samples. We conclude this work by giving a quite general methodology to solve problems of non linear systems submitted to random excitations.
",
key = ""
}