A coupled parametric and nonparametric approach for modal analysis of a satellite

This study takes place in the context of dynamical prediction for satellite structures. Aims of such studies are to survey the dynamical response of satellite equipment and components, to check that requirements are correct and to give prediction of vibration levels, which are inputs for the experimental test validation according to the launcher specification. This prediction is done by modal analysis performed on a numerical model built by finite element method. Uncertainties on equipment and components properties lead to random frequency response function (FRF). This paper aims at understanding how modal approach can be adapted to probabilistic framework in order to calculate cumulative density function or, at least, some quantiles of a FRF. Starting point of deterministic modal analysis is the study of a single degree of freedom (DOF) system. C. Heinkelé analytically expresses the probability density function (PDF) of the FRF of an oscillator with a random natural pulsation following a uniform law. We have generalized this work to random natural pulsation following a law of finite variance. Then, the expression of a FRF between two DOF of the structure is a linear function of random oscillators FRF and DOF components of random eigenvectors. Assuming that random eigenvectors are close to their means, we have access to the characteristic function of the random FRF between two DOF as a multi-dimensional integral with respect to the joint PDF of the oscillators FRF. This paper mainly focuses on two major points which are calculation of oscillators joint PDF and inversion of characteristic functions. The first one is tackled by copulas theory. Dependence structure of random eigenvalues are identified and modeled by a copula. Then we apply results on copulas transformations to obtain joint PDF of oscillators FRF. Classical results concern monotonic transformations but we extended these ones to non-monotonic cases. Concerning inversion of characteristic function, several methods are studied to numerically compute the Gil-Pelaez formula. This approach allows to access some FRF quantiles by numerical integration which error can be controlled. Moreover, an interesting point is the flexibility in the identification of the random eigenvalues PDF. This is especially interesting in order to couple parametric identification with nonparametric one, when only few dispersion informations are given for equipments, housed in the satellite primary structure

A new adaptive response surface method for reliability analysis

Response surface method is a convenient tool to assess reliability for a wide range of structural mechanical problems. More specifically, adaptive schemes which consist in iteratively refine the experimental design close to the limit state have received much attention. However, it is generally difficult to take into account a lot of variables and to well handle approximation error. The method, proposed in this paper, addresses these points using sparse response surface and a relevant criterion for results accuracy. For this purpose, a response surface is built from an initial Latin Hypercube Sampling (LHS) where the most significant terms are chosen from statistical criteria and cross-validation method. At each step, LHS is refined in a region of interest defined with respect to an importance level on probability density in the design point. Two convergence criteria are used in the procedure: The first one concerns localization of the region and the second one the response surface quality. Finally, a bootstrap method is used to determine the influence of the response error on the estimated probability of failure. This method is applied to several examples and results are discussed

Determination of Bootstrap confidence intervals on sensitivity indices obtained by polynomial chaos expansion

L’analyse de sensibilité a pour but d’évaluer l’influence de la variabilité d’un ou plusieurs paramètres d’entrée d’un modèle sur la variabilité d’une ou plusieurs réponses. Parmi toutes les méthodes d’approximations, le développement sur une base de chaos polynômial est une des plus efficace pour le calcul des indices de sensibilité, car ils sont obtenus analytiquement grâce aux coefficients de la décomposition (Sudret (2008)). Les indices sont donc approximés et il est difficile d’évaluer l’erreur due à cette approximation. Afin d’évaluer la confiance que l’on peut leur accorder nous proposons de construire des intervalles de confiance par ré-échantillonnage Bootstrap (Efron, Tibshirani (1993)) sur le plan d’expérience utilisé pour construire l’approximation par chaos polynômial. L’utilisation de ces intervalles de confiance permet de trouver un plan d’expérience optimal garantissant le calcul des indices de sensibilité avec une précision donnée

Reliability approach in spacecraft structures

This paper presents an application of the probabilistic approach with reliability assessment on a spacecraft structure. The adopted strategy uses meta-modeling with first and second order polynomial functions. This method aims at minimizing computational time while giving relevant results. The first part focuses on computational tools employed in the strategy development. The second part presents a spacecraft application. The purpose is to highlight benefits of the probabilistic approach compared with the current deterministic one. From examples of reliability assessment we show some advantages which could be found in industrial applications

Synthèse modale probabiliste de systèmes à plusieurs degrés de liberté

Notre travail concerne les études dynamiques basse fréquence de satellites. Le but est d’étendre l’analyse modale en prenant en compte les incertitudes sur les paramètres d’entrée du modèle. Pour cela, l’approche probabiliste a été choisie. Les paramètres incertains du modèle sont donc définis par des variables aléatoires de lois connues. L’objectif de cette analyse est de déterminer la variabilité d’une fonction de réponse en fréquence (FRF) entre deux points de la structure. Nous supposons qu’il est possible d’identifier les lois de probabilité des valeurs propres du système et que l’amortissement modal est déterministe. Nous présentons une expression analytique des densités de probabilité de la FRF, ainsi qu’une méthodologie permettant de les calculer y compris dans le cas où les valeurs propres sont corrélées

Analyse de la fiabilité d’une structure spatiale

L’analyse de la fiabilité des grandes structures est difficile car le nombre de variables aléatoires d’entrée et le nombre de réponses à traiter sont généralement grands. Lorsque le comportement mécanique de la structure n’est pas fortement non-linéaire, les surfaces de réponse présentent des outils intéressants pour évaluer la fiabilité. Ce papier expose une stratégie de résolution appliquée à une structure spatiale. Elle consiste à simplifier le problème initial dans le but de conserver l’information essentielle. Pour cela, la réduction s’appuie sur la sélection des variables les plus pertinentes. Un indicateur d’erreur est utilisé pour valider la précision des résultats. Des tirages de Monte Carlo viennent corroborer certains résultats

Etude numérique de l'influence de la structure de dépendance des valeurs propres en synthèse modale probabiliste

Ce travail a pour cadre la détermination de fonctions de réponse en fréquence (FRF) par synthèse modale. La modélisation probabiliste des paramètres d'entrée du modèle conduit à un problème aux valeurs propres aléatoires. Nous nous intéressons à la représentation de la structure de dépendance entre les valeurs propres et son influence sur la densité de probabilité de la FRF . Cette structure de dépendance est modélisée par une copule identifiée à partir de simulations de Monte-Carlo. En adaptant les travaux de C. Heinkelé au cas de l'amortissement critique, nous obtenons les expressions analytiques des densités de probabilité de la FRF d'un oscillateur harmonique. Nous utilisons ces résultats afin d'exprimer la densité jointe d'un vecteur de N oscillateurs connaissant la loi jointe des N premières valeurs propres du système

A coupled parametric and nonparametric approach for modal analysis of a satellite

International audienceThis study takes place in the context of dynamical prediction for satellite structures. Aims of such studies are to survey the dynamical response of satellite equipment and components, to check that requirements are correct and to give prediction of vibration levels, which are inputs for the experimental test validation according to the launcher specification. This prediction is done by modal analysis performed on a numerical model built by finite element method. Uncertainties on equipment and components properties lead to random frequency response function (FRF). This paper aims at understanding how modal approach can be adapted to probabilistic framework in order to calculate cumulative density function or, at least, some quantiles of a FRF. Starting point of deterministic modal analysis is the study of a single degree of freedom (DOF) system. C. Heinkelé analytically expresses the probability density function (PDF) of the FRF of an oscillator with a random natural pulsation following a uniform law. We have generalized this work to random natural pulsation following a law of finite variance. Then, the expression of a FRF between two DOF of the structure is a linear function of random oscillators FRF and DOF components of random eigenvectors. Assuming that random eigenvectors are close to their means, we have access to the characteristic function of the random FRF between two DOF as a multi-dimensional integral with respect to the joint PDF of the oscillators FRF. This paper mainly focuses on two major points which are calculation of oscillators joint PDF and inversion of characteristic functions. The first one is tackled by copulas theory. Dependence structure of random eigenvalues are identified and modeled by a copula. Then we apply results on copulas transformations to obtain joint PDF of oscillators FRF. Classical results concern monotonic transformations but we extended these ones to non-monotonic cases. Concerning inversion of characteristic function, several methods are studied to numerically compute the Gil-Pelaez formula. This approach allows to access some FRF quantiles by numerical integration which error can be controlled. Moreover, an interesting point is the flexibility in the identification of the random eigenvalues PDF. This is especially interesting in order to couple parametric identification with nonparametric one, when only few dispersion informations are given for equipments, housed in the satellite primary structure

Superposition modale probabiliste : identification de la loi jointe des valeurs propres et paramètres effectifs par la théorie des copules

Afin de prendre en compte les incertitudes de conception dans le calcul prédictif des fonctions de réponse en fréquence (FRF), une adaptation de la méthode de superposition modale au cadre probabiliste est proposée. L’identification de la loi de probabilité du vecteur composé des valeurs propres et paramètres effectifs aléatoires est réalisée en 3 étapes : sélection des modes prépondérants, discrimination des lois jointes à identifier, identification de ces lois de probabilité de grandes dimensions par décomposition Vine(copules bidimensionnelles)

Construction and identification of a D-Vine model applied to the probability distribution of modal parameters in structural dynamics

This study investigates the construction and identification of the probability distribution of random modal parameters (natural frequencies and effective parameters) in structural dynamics. As these parameters present various types of dependence structures, the retained approach is based on pair copula construction (PCC). A literature review leads us to choose a D-Vine model for the construction of modal parameters probability distributions. Identification of this model is based on likelihood maximization which makes it sensitive to the dimension of the distribution, namely the number of considered modes in our context. To this respect, a mode selection preprocessing step is proposed. It allows the selection of the relevant random modes for a given transfer function. The second point, addressed in this study, concerns the choice of the D-Vine model. Indeed, D-Vine model is not uniquely defined. Two strategies are proposed and compared. The first one is based on the context of the study whereas the second one is purely based on statistical considerations. Finally, the proposed approaches are numerically studied and compared with respect to their capabilities, first in the identification of the probability distribution of random modal parameters and second in the estimation of the 99% quantiles of some transfer functions