The use of the Advanced Censored Closure (ACC) technique, recently proposed by the authors for predicting the peak response of linear structures vibrating under random processes, is extended to the case of non-linear oscillators driven by stationary white noise. The proposed approach requires the knowledge of mean upcrossing rate and spectral bandwidth of the response process, which in this paper are estimated through the Stochastic Averaging method. Numerical applications to oscillators with non-linear stiffness and damping are included, and the results are compared with those given by Monte Carlo Simulation and by other approximate formulations available in the literature.

no / The paper deals with the problem of predicting the maximum response statistics of Multi-Degree-of-Freedom (MDoF) linear structures subjected to non-stationary non-white noises. The extension of two different censored closures of Gumbel type, originally proposed by the authors for the response of Single-Degree-of-Freedom oscillators, it is presented. The improvement associated with the introduction in the closure of a consistent censorship factor, accounting for the response bandwidth, it is pointed out. Simple and effective step-by-step procedures are formulated and described in details. Numerical applications on a realistic 25-storey moment-resisting frame along with comparisons with classical approximations and Monte Carlo simulations are also included.

Much of the vibration in fast moving vehicles is caused bydistributed random excitation, such as turbulent flow and roadroughness. Piping systems transporting fast flowing fluid isanother example, where distributed random excitation will causeunwanted vibration. In order to reduce these vibrations andalso the noise they cause, it is important to have accurate andcomputationally efficient prediction methods available. The aim of this thesis is to present such a method. Thefirst step towards this end was to extend an existing spectralfinite element method (SFEM) to handle excitation of planetravelling pressure waves. Once the elementary response tothese waves is known, the response to arbitrary homogeneousrandom excitation can be found. One example of random excitation is turbulent boundary layer(TBL) excitation. From measurements a new modified Chase modelwas developed that allowed for a satisfactory prediction ofboth the measured wall pressure field and the vibrationresponse of a turbulence excited plate. In order to model morecomplicated structures, a new spectral super element method(SSEM) was formulated. It is based on a waveguide formulation,handles all kinds of boundaries and its elements are easily putinto an assembly with conventional finite elements. Finally, the work to model fluid-structure interaction withanother wave based method is presented. Similar to the previousmethods it seems to be computationally more efficient thanconventional finite elements. / <p>NR 20140805</p>

In this dissertation, certain problems of stochastic optimal control and relevant analysis of random vibrations are considered. Dynamic Programming approach is used to find an optimal control law for a linear single-degree-of-freedom system subjected to Gaussian white-noise excitation. To minimize a system's mean response energy, a bounded in magnitude control force is applied. This approach reduces the problem of finding the optimal control law to a problem of finding a solution to the Hamilton-Jacobi-Bellman (HJB) partial differential equation. A solution to this partial differential equation (PDE) is obtained by developed 'hybrid' solution method. The application of bounded in magnitude control law will always introduce a certain type of nonlinearity into the system's stochastic equation of motion. These systems may be analyzed by the Energy Balance method, which introduced and developed in this dissertation. Comparison of analytical results obtained by the Energy Balance method and by stochastic averaging method with numerical results is provided. The comparison of results indicates that the Energy Balance method is more accurate than the well-known stochastic averaging method.

<p>Much of the vibration in fast moving vehicles is caused bydistributed random excitation, such as turbulent flow and roadroughness. Piping systems transporting fast flowing fluid isanother example, where distributed random excitation will causeunwanted vibration. In order to reduce these vibrations andalso the noise they cause, it is important to have accurate andcomputationally efficient prediction methods available.</p><p>The aim of this thesis is to present such a method. Thefirst step towards this end was to extend an existing spectralfinite element method (SFEM) to handle excitation of planetravelling pressure waves. Once the elementary response tothese waves is known, the response to arbitrary homogeneousrandom excitation can be found.</p><p>One example of random excitation is turbulent boundary layer(TBL) excitation. From measurements a new modified Chase modelwas developed that allowed for a satisfactory prediction ofboth the measured wall pressure field and the vibrationresponse of a turbulence excited plate. In order to model morecomplicated structures, a new spectral super element method(SSEM) was formulated. It is based on a waveguide formulation,handles all kinds of boundaries and its elements are easily putinto an assembly with conventional finite elements.</p><p>Finally, the work to model fluid-structure interaction withanother wave based method is presented. Similar to the previousmethods it seems to be computationally more efficient thanconventional finite elements.</p>

The objective of this thesis is to develop a novel methodology of fractional stochastic dynamics to study stochastic stability of viscoelastic systems under stochastic loadings. Numerous structures in civil engineering are driven by dynamic forces, such as seismic and wind loads, which can be described satisfactorily only by using probabilistic models, such as white noise processes, real noise processes, or bounded noise processes. Viscoelastic materials exhibit time-dependent stress relaxation and creep; it has been shown that fractional calculus provide a unique and powerful mathematical tool to model such a hereditary property. Investigation of stochastic stability of viscoelastic systems with fractional calculus frequently leads to a parametrized family of fractional stochastic differential equations of motion. Parametric excitation may cause parametric resonance or instability, which is more dangerous than ordinary resonance as it is characterized by exponential growth of the response amplitudes even in the presence of damping. The Lyapunov exponents and moment Lyapunov exponents provide not only the information about stability or instability of stochastic systems, but also how rapidly the response grows or diminishes with time. Lyapunov exponents characterizes sample stability or instability. However, this sample stability cannot assure the moment stability. Hence, to obtain a complete picture of the dynamic stability, it is important to study both the top Lyapunov exponent and the moment Lyapunov exponent. Unfortunately, it is very difficult to obtain the accurate values of theses two exponents. One has to resort to numerical and approximate approaches. The main contributions of this thesis are: (1) A new numerical simulation method is proposed to determine moment Lyapunov exponents of fractional stochastic systems, in which three steps are involved: discretization of fractional derivatives, numerical solution of the fractional equation, and an algorithm for calculating Lyapunov exponents from small data sets. (2) Higher-order stochastic averaging method is developed and applied to investigate stochastic stability of fractional viscoelastic single-degree-of-freedom structures under white noise, real noise, or bounded noise excitation. (3) For two-degree-of-freedom coupled non-gyroscopic and gyroscopic viscoelastic systems under random excitation, the Stratonovich equations of motion are set up, and then decoupled into four-dimensional Ito stochastic differential equations, by making use of the method of stochastic averaging for the non-viscoelastic terms and the method of Larionov for viscoelastic terms. An elegant scheme for formulating the eigenvalue problems is presented by using Khasminskii and Wedig’s mathematical transformations from the decoupled Ito equations. Moment Lyapunov exponents are approximately determined by solving the eigenvalue problems through Fourier series expansion. Stability boundaries, critical excitations, and stability index are obtained. The effects of various parameters on the stochastic stability of the system are discussed. Parametric resonances are studied in detail. Approximate analytical results are confirmed by numerical simulations.

[pt] Análise modal consiste na caracterização de um sistema através dos seus parâmetros modais. Quando a principal excitação é causada pelo ambiente em que o sistema está inserido, essa caracterização é definida como análise modal operacional (OMA). Nestes casos, os forçamentos não são conhecidos (mensuráveis) e apenas as respostas são monitoradas. Por terem natureza aleatória, esses sinais precisam ser incorporados ao modelo numérico através de processos estocásticos. O principal objetivo desta dissertação consiste em descrever as técnicas de identificação em OMA. Para isso, duas vertentes foram criadas, uma teórica e outra experimental. Na parte teórica, as hipóteses necessárias para a identificação de um sistema por OMA são apresentadas. Uma análise dos erros causados por sinais ruidosos também é feita, permitindo que a sensibilidade dos métodos seja avaliada. Além de contemplar os principais métodos de identificação, dois novos métodos são propostos. Ambos foram desenvolvidos a partir da Decomposição Ortogonal Própria (POD) e combinam uma eficiência computacional com a possibilidade de quantificar as incertezas dos parâmetros. Na vertente experimental, o objetivo é ilustrar e validar a identificação de estruturas. Para isso, três diferentes bancadas foram criadas: um prédio de dois andares, uma pá eólica e uma ponte suspensa. Após a construção, essas estruturas foram devidamente instrumentadas por diferentes sensores. Um sistema de aquisição dados foi montado através de hardwares comerciais e analisados através de uma interfase gráfica desenvolvida especialmente para OMA pelo Laboratório de Vibrações. / [en] Modal analysis consists in the characterization of a system through its modal parameters. When the main excitation source is the system s environment, this characterization is defined as operational modal analysis (OMA). On those cases, the forces are unknown (not measured) and only the responses are monitored. Because of there random nature, those signals are incorporated into the numerical model as stochastic processes. The main goal of this dissertation is to describe the identification techniques in OMA. Therefore, two different approaches were created: a theoretical one and an experimental one. In the theoretical part, the required hypotheses for system s identification with OMA are presented. An analysis of the errors caused by noisy signals are also performed, allowing the method s sensibility to be evaluated. Besides the standard identification methods, two new ones are proposed. They both has been developed as extension of the Proper Orthogonal Decomposition (POD) and combine an efficient computational process with the possibility of quantify the parameters uncertainties. In the experimental approach, the goal is to illustrate and validate the identification of structures. Therefore, three different test bench were created: a two floor building, a wind turbine blade and a cable-stayed bridge. After their construction, those structures were proper instrumented with different sensors. A data acquisition system were built using commercial hardwares and analyzed through a graphic interface specially made for OMA and developed in the vibration laboratory.

Cette thèse présente une étude expérimentale et numérique d'essais de chute libre et de vibrations aléatoires d'emballages en carton ondulé contenant différents calages en mousse. Un modèle d'homogénéisation efficace pour le carton ondulé a été développé. Dans ce modèle homogénéisé, le carton ondulé est représenté par une plaque 2D. Au lieu d'utiliser une loi de comportement locale (reliant les déformations aux contraintes) à chaque point d'intégration, l'homogénéisation conduit aux rigidités globales (reliant les déformations généralisées aux forces résultantes) pour la plaque équivalente homogène. Ce modèle a été implémenté dans le logiciel Abaqus. Le comportement de la mousse a été déterminé expérimentalement et modélisé avec un modèle de mousse déformable d'Abaqus. Les emballages sont soumis à des tests de chute libre d'une hauteur donnée sur un sol rigide et des essais de vibrations aléatoires type bruit blanc sur table vibrante. Les résultats numériques obtenus en utilisant la simulation avec le modèle homogénéisé concordent bien avec les résultats expérimentaux. Nous avons également montré que la contribution de la caisse en carton ondulé à la réponse au choc et à la vibration aléatoire ne pouvait pas être négligée dans la conception des calages. / This thesis presents experimental and numerical studies of drop tests random vibrations of corrugated cardboard packaging containing different foam cushions. An efficient homogenization model for the corrugated cardboard has been developed. In this homogenized model, the corrugated cardboard is represented by a 2D plate. Instead of using a local constitutive law (relating the strains to the stresses) at each material point, the homogenization leads to global rigidities (relating the generalized strains to the resultant forces) for the equivalent homogeneous plate. This model was implemented into the FE software Abaqus. The foam behaviour was experimentally determined and modelled using a crushable foam model of Abaqus. The packages are tested in free fall from a given height on a rigid floor and submitted to white noise random vibrations. The numerical results obtained using the FE simulation with the homogenized model agree well with the experimental results. We have also shown that the contribution of the corrugated cardboard box to the shock and random vibration responses could not be neglected in the design of cushioning package.

Cette recherche porte sur le développement d’un modèle de structure de géométrie complexe équipée de composants à comportement non linéaire viscoélastique dans le but de simuler sa réponse à des excitations définies par des densités spectrales de puissance (DSP). L’application industrielle concerne l’isolation vibratoire d’un module de refroidissement automobile monté sur plot de suspension en élastomère. Une revue du comportement des élastomères en fonction de leurs conditions d’environnement et de sollicitations identifie les paramètres des différents modèles analysés. Des essais préliminaires ont été menés pour définir les intervalles des niveaux de sollicitations et quantifier l’échauffement des plots. La caractérisation expérimentale de la suspension porte sur des plots en élastomère munis de leurs interfaces afin d’agréger dans un seul modèle les non linéarités du comportement viscoélastique, des glissements et des frottements. Les boucles effort-déflexion axiales et radiales mesurées sont traitées avec un système expert développé spécialement pour caler les paramètres du modèle retenu. Il s’agit du modèle de Dahl généralisé qu’il a fallu étendre aux aspects viscoélastiques. Ce processus de calage automatique a été codé avec un logiciel développé sous Octave/Matlab. Des méthodes d’interpolations et extrapolations rendent opérationnel ce modèle sur toute la gamme fonctionnement de l’application définie dans l’espace fréquence-déflexion. Ce processus a été codé dans le module UserSubroutine pour Abaqus. Soumettre le système mécanique non linéaire à des excitations définies par une DSP nécessite de s’appesantir sur le traitement des vibrations aléatoires. En effet il faut, pour calculer les réponses, considérer le passage fréquence-temps et inversement pour les comparer éventuellement aux exigences des normes. De plus, la taille et la complexité du modèle EF de la structure industrielle rendent impossible une résolution temporelle sur l’ensemble de ses degrés de liberté. Il s’agit alors de faire appel à des techniques d’homogénéisation et de condensation dynamique afin de prévoir la réponse aux excitations à large bande fréquentielle dans le but d’analyser les performances de l’isolation vibratoire. / This research work regards the development of a complex structure model with non-linear viscoelastic components. The purpose of this study is to simulate the response of this structure submitted to a random vibration excitation based on a power spectral density definition (PSD). The industrial applicative case is the vibratory insulation of a automotive engine cooling module supported by elastomer mounts. A brief review of elastomers behavior depending on solicitations types enables to identify the parameters of the different investigated models. Preliminary tests have been conducted to define the range of amplitudes of excitations and evaluate the internal warming of rubbers during the full structure validation test. The experimental characterization of the suspension is based on rubbers mounts and their interfaces with the cooling module, in order to take into account in a unique model all nonlinearities due to the viscoelastic behavior, the slidings, and the friction. Measured force-deflection hysteretic cycles in axial and radial direction are post-processed with an expert system developed to obtain the parameters of the retained model: the modified Dahl’s model, generalized to viscoleastic aspect. This process has been developed with Octave/Matlab code. Interpolation and extrapolation methods enable to obtain a good model response on the global operating range. These methods have been coded in an Abaqus UserSubroutine. Imposing random vibration excitation of a non linear mechanical system based on PSD imposes to take into account signal processing aspects. To evaluate response levels versus norms requirements, it’s mandatory to consider the time-frequency transfer. In addition, the size and the complexity of the total finite element model of the industrial structure don’t allow a global resolution in the time domain for all the degrees of freedom. Homogenization and dynamic reduction techniques are used to evaluate the response of the system submitted to large frequency range excitations, and to analyse the behavior of the suspension.

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