Rubens Sampaio | PUC-RIO - Academia.edu (original) (raw)

Papers by Rubens Sampaio

Journal of the Brazilian Society of Mechanical Sciences and Engineering, 2012

Technical systems are often modeled through systems of differential equations in which the parame... more Technical systems are often modeled through systems of differential equations in which the parameters and initial conditions are subject to uncertainties. Usually, special solutions of the differential equations like equilibrium positions and periodic orbits are of importance and frequently the corresponding equations are only set up with the intent to describe the behavior in the vicinity of a limit cycle or an equilibrium position. For the validity of the analysis it must therefore be assumed that the initial conditions lie indeed in the basins of attraction of the corresponding attractors. In order to estimate basins of attraction, Lyapunov functions can be used. However, there are no systematic approaches available for the construction of Lyapunov functions with the goal to achieve a good approximation of the basin of attraction. The present paper suggests a method for defining appropriate Lyapunov functions using insight from center manifold theory. With this approach, not only variations in the initial conditions, but also in the parameters can be studied. The results are used to calculate the likelihood for the system to reach a certain attractor assuming different random distributions for the initial conditions.

MATEC Web of Conferences, 2014

This paper deals with the theoretical study of how discrete elements attached to a continuous sto... more This paper deals with the theoretical study of how discrete elements attached to a continuous stochastic systems can affect their dynamical behavior. For this, it is studied the nonlinear longitudinal dynamics of an elastic bar, attached to springs and a lumped mass, with a random elastic modulus and subjected to a Gaussian white-noise distributed external force. Numerical simulations are conducted and their results are analyzed in function of the ratio between the masses of the discrete and the continuous parts of the system. This analysis reveals that the dynamic behavior of the bar is significantly altered when the lumped mass is varied, being more influenced by the randomness for small values of the lumped mass.

Journal of Guidance, Control, and Dynamics, 2002

It is well-known that flexible beams become stiffer when subjected to high speed rotations. This ... more It is well-known that flexible beams become stiffer when subjected to high speed rotations. This is due to the membrane-bending coupling resulting from the large displacements of the beam cross-section. This effect, often called geometric stiffening, has been largely discussed in the last two decades. Several methodologies have been proposed in the literature to account for the stiffening effect in the dynamics equations. However, considerable effort is generally done to derive linear models using steady-state assumptions and membrane-bending decoupling. This work aims first to present a brief review of the open literature on this subject. Then, a general non-linear model is formulated using a non-linear strain-displacement relation. This model is used to deeply analyze simplified models arising in the literature. In particular, the assumption of steady-state values for the centrifugal load is analyzed and its consequences are discussed. Thereafter, four finite element models are proposed, one based on non-linear theory and the others on simplified linear theories. These models are then applied to the study of a flexible beam undergoing prescribed high speed large rotations. The analyses show that one must account for the geometric stiffening effect to obtain realistic results. In addition, it is shown that models disregarding the axial displacement dynamics lead to erroneous results for the axial stress in the beam, which may be of main importance in structural integrity analysis. Hence, in the general case, geometric stiffening must be accounted for in association with the inclusion of full axialtransverse displacements coupling dynamics in the model.

Journal of Sound and Vibration

Proceeding Series of the Brazilian Society of Computational and Applied Mathematics, 2023

In this paper the inuence of asynchronous parametric excitation in stability maps of the simplest... more In this paper the inuence of asynchronous parametric excitation in stability maps of the simplest electromechanical system is analyzed. The system is composed by two interacting subsystems, a mechanical and an electromagnetic and it has the minimum number of elements necessary to be classied as an electromechanical system. The system does not have elements that can storage potential energies, neither mechanical nor electrical. The system dynamics is written in terms of 2 × 2 inertia matrix M and gyroscopic matrix G. Two parametric excitations terms are introduced in G. The terms have an amplitude ϵ, frequency ωand asynchrony with respect to each other θ. For dierent values of θ, stability maps, in terms of ϵ and ω, are constructed for the electromechanical system with the parametric excitation. In each map, it can be seen stability and instability regions of the trivial solution (system's equilibrium) of the system. The objective of the paper is to analyze how the value of θ aects these stability and instability regions. Palavras-chave. Electromechanical systems, asynchronous parametric excitation, stability maps.

Proceedings of Uncertainty 2023, 2023

In this paper, the spread of a general epidemic over time is modeled as a branching process. It i... more In this paper, the spread of a general epidemic over time is modeled as a branching process. It is a stochastic process sorted as an individual-based model, which records population growth over generations with uncertainties to its size. The source of randomness is inherently related to the individual behavior of each member in a population. In this context, the transmissibility of the disease, i.e., the contagion from an infected person to susceptible ones is the root. Therefore, a discrete random variable models the number of infections per infector and rules the branching process. Given the probabilistic model of the contagion, the objective of the paper is to compare three methodologies to evaluate the mass functions of further generations of the branching process: probability generating functions (pgf), Markov chains (MC) and Monte Carlo simulations (MCS). The former gives analytical expressions, that can be symbolic computed, to evaluate the probability of an arbitrary number of infected members for a desired generation, whereas MC is a semi-numerical methodology and the latter is indeed a numerical one. The comparison between all of them relies on computational cost (runtime and storage) and limitation of applicability in relation to the mass function of the contagion. One of the characteristics of interest in the analysis is the determination of which methodologies allow the calculation of the mass function of a further generation without computing the mass functions of previous ones. This feature is referred in here as not time-dependent. Another characteristic of interest is the determination of which methodologies allow the computation of just some values of the mass function of a generation, i.e., probabilities related to the same generation can be achieved independently from the others. This is so-called a local property.

Proceedings of Uncertainty 2023, 2023

To present ideas, a model problem consisting of a moving mass-belt system with random friction sh... more To present ideas, a model problem consisting of a moving mass-belt system with random friction showing the stick-slip phenomenon is treated. The dynamics is simulated. The objective of this work is to assess the behaviour of the computation cost in terms of the run-time, which is random, and its relationship with some of the output variables that define the dynamical behaviour of the mechanical system, such as the duration of the phases present in the simulation, sticks and slips, and the number of phases that occur in each realisation. All this is analysed from a stochastic perspective. However, the probabilistic model to analyse the distribution of a three-dimensional random vector, formed by the run-time, duration and number, belongs to R 4 , thus it is difficult to characterise and visualise. Hence, in this study, the use of random variable transformations to produce new independent variables is explored as an attempt to reduce the number of dimensions that need to be considered. Also, the change of variables is used to assess the link between the behaviour of the results and the chosen integration method. It is shown that the predictions obtained with the Monte Carlo method combined with a Multiple Scales analytical approximation are influenced by the number of transition phases rather than their durations.

This paper deals with a variational-based reduced-order model in dy-namic substructuring of two c... more This paper deals with a variational-based reduced-order model in dy-namic substructuring of two coupled structures through a physical dissipative flex-ible interface. We consider the linear elastodynamic of a dissipative structure com-posed of two main dissipative substructures perfectly connected through interfaces by a linking substructure. The linking substructure is flexible and is modeled in the context of the general linear viscoelasticity theory, yielding damping and stiffness operators depending on the frequency, while the two main dissipative substruc-tures are modeled in the context of linear elasticity with an additional classical viscous damping modeling which is assumed to be independent of the frequency. We present recent advances adapted to such a situation, which is positioned with respect to an appropriate review that we carry out on the different methods used in dynamic substructuring. It consists in constructing a reduced-order model using the free-interface elast...

This paper studies the transient dynamics of a linear dynamical system with elastic barriers exci... more This paper studies the transient dynamics of a linear dynamical system with elastic barriers excited by a deterministic transient force whose Fourier Transform has a bounded frequency narrow band. The system is then nonlinear. In order to measure the degree of non-linearity of the system, one looks for the mechanical energy transferred outside the frequency band of excitation as a function of the parameter η defined by ǫ a , in which ǫ is the size of the barrier gap and a is the amplitude of the excitation force. The mechanical energy transferred outside the frequency band of excitation can potentially be a source of excitation for other subsystems. Consequently, a quantification of this energy transfer is important for the understanding of the non-linear dynamical system. In addition, it is well known that this type of non-linear dynamical system is very sensitive to uncertainties. For this reason one studies the system as being deterministic, and also stochastic in order to take i...

The stochastic dynamics of a drill-string is analyzed, where the uncertainty is in the bitrock no... more The stochastic dynamics of a drill-string is analyzed, where the uncertainty is in the bitrock nonlinear interaction model. The Maximum Entropy Principle is used to construct a probabilistic model for the nonlinear operator related to the bit-rock interaction model. A numerical model is developed using the Timoshenko beam theory and it is discretized by means of the Finite Element Method. The nonlinear dynamics analyzed considers the main efforts that the column is subjected to as, for instance, imposed rotation at the top, fluid-structure interaction, impact between the column and the borehole, and finite strains (what couples axial, torsional and lateral vibrations).

This paper is devoted to the analysis of the efficiency of the reduced model constructed using th... more This paper is devoted to the analysis of the efficiency of the reduced model constructed using the POD or the Karhunen-Loeve, KL, method in nonlinear dynamics for continuous systems. We present a theory for continuous systems and we develop a numerical analysis based on the use of the finite element method applied to the weak formulation of different continuous problems. A nonlinear basis is constructed by the POD or the KL method. We prove the fundamental properties required to use such a basis of the admissible displacement field space in order to construct a reduced model. It is explained that if the nonlinear basis constructed with the POD or the KL method can be viewed as an optimal basis for representing the response of the system, it is not a priori an optimal nonlinear basis for reducing the model. A numerical solver of the generalized eigenvalue problem related to the POD or the KL method is proposed for large systems. Another solver is also proposed in introducing a reduce...

The aim of this paper is to use Bayesian statistics to update a probability density function (p.d... more The aim of this paper is to use Bayesian statistics to update a probability density function (p.d.f.) related to the tension parameter of the vocal folds, which is one of the main parameters responsible for the changing of the fundamental frequency of a voice signal, generated by a mechanical - mathematical model for producing voiced sounds. Three parameters are considered uncertain in the model used: the tension parameter, the neutral glottal area and the subglottal pressure. Random variables are associated to the uncertain parameters and their corresponding p.d.f.'s are constructed using the Maximum Entropy Principle. The Monte Carlo method is used to generate the voice signals, which are the outputs of the model. For each voice signal, the corresponding fundamental frequency is calculated and a p.d.f. for this random variable is constructed. Experimental values of the fundamental frequency are then used to update the p.d.f. of the fundamental frequency and, consequently, of t...

TEMA - Tendências em Matemática Aplicada e Computacional, 2006

The aim of this work is to discuss the dynamics of an one spacedimension structure without flexur... more The aim of this work is to discuss the dynamics of an one spacedimension structure without flexural stiffness. We model the behavior of a very light string and we present the problem, considering large deformations, considering that each particle has only vertical motion. We discuss the kind of constitutive equations needed in such motion. Using a perturbation method, we show that the D'Alembert's equation can be seen as a first approximation for a general model and, then, its limitations become clear.

Thin-Walled Structures, 2015

In this paper we perform a quantification of the uncertainty propagation of the dynamics of slend... more In this paper we perform a quantification of the uncertainty propagation of the dynamics of slender initially curved structures constructed with fiber reinforced composite materials. Depending on the manufacturing process, composite materials may have deviations with respect to the expected response, often called nominal response in a deterministic sense. The manufacturing aspects lead to uncertainty in the structural response associated with constituent proportions, material and/or geometric parameters among others. Another aspect of uncertainty that can be sensitive in composite structures is the mathematical model that represents the mechanics of the structural member, that is: the assumptions and type of hypotheses invoked reflect the most relevant aspects of the physics of a structure, however in some circumstances these hypotheses are not enough, and cannot represent properly the mechanics of the structure. Uncertainties should be considered in a structural system in order to improve the predictability of a given modeling scheme. There are two approaches to evaluate the propagation of uncertainties in structural models: the parametric probabilistic approach and the non-parametric probabilistic approach. In the parametric, one quantifies the uncertainty of given parameters (such as variation of the angles of fiber reinforcement and material constituents) by associating random variables to them. In the non-parametric, the propagation of uncertainty is quantified by considering uncertain the matrices of the whole system. In this study a shear deformable model of composite curved thinwalled beams is employed as the mean or expected model. The probabilistic model is constructed by adopting random variables for the uncertain entities (parameters or matrices) of the model. The probability density functions of the random entities are derived appealing to the maximum entropy principle under given constraints. Once the probabilistic model is discretized in the context of the finite element method, the Monte Carlo method is employed to perform the simulations. Then the statistics of the simulations is evaluated and the parametric and non-parametric approaches are compared.

In this paper a probabilistic model is proposed for the bit-rock interaction model of a drill-str... more In this paper a probabilistic model is proposed for the bit-rock interaction model of a drill-string system. A new strategy to take into account uncertainties in a local constitutive nonlinear equation using the nonparametric probabilistic approach is developed. The deterministic model considers the main forces that are applied to the column such as bit-rock interaction, fluid-structure interaction and impact forces. The nonlinear Timoshenko beam theory is applied and the system is discretized by means of the Finite Element Method.

This paper presents a finite element formulation for transient dynamic analysis of sandwich curve... more This paper presents a finite element formulation for transient dynamic analysis of sandwich curved beams with embedded viscoelastic material whose constitutive behavior is modeled by means of fractional derivative operators. The sandwich configuration is composed of a band as a viscoelastic core bonded to elastic metallic strips. The viscoelastic model used to describe the behavior of the core is a four-parameter fractional derivative model. The Grünwald definition of the fractional operator is used to implement the viscoelastic model into a finite element formulation. Then, discretized motion equations are solved with a direct time integration scheme based on the Newmark method. A useful aspect of the procedure is that only the anelastic displacements history is kept. This allows an important save of computational resources associated with the non-locality of the operators for fractional derivatives. Numerical studies are presented in order to validate the curved beam model with other approaches (frames of straight beams) as well as to analyze the influence of different parameters in the transient dynamics of naturally curved sandwich beams.

Journal of Vibration and Acoustics, 2014

For noise and vibration attenuation, various approaches can be employed depending on the frequenc... more For noise and vibration attenuation, various approaches can be employed depending on the frequency range to attenuate. Generally, active or passive piezoelectric techniques are effective in the low-frequency range, while dissipative materials, such as viscoelastic or porous treatments, are efficient for higher-frequency domain. In this work, a reduced-order model is developed for the approximation of a fully coupled electromechanical-acoustic system using modal projection techniques. The problem consists of an elastic structure with surface-mounted piezoelectric patches coupled with a compressible inviscid fluid. The piezoelectric elements, connected with resonant shunt circuits, are used for the vibration damping of the coupled system. Numerical examples are presented in order to illustrate the accuracy and the versatility of the proposed reduced-order model, especially in terms of prediction of attenuation.

Journal of the Brazilian Society of Mechanical Sciences and Engineering, 2010

A drill-string is a slender structure that turns and drills into the rock in search of oil. There... more A drill-string is a slender structure that turns and drills into the rock in search of oil. There are many sources of uncertainties in this complex dynamical system. However, this article is concerned only with uncertainties in the weight-on-hook, which is the supporting force exerted by the hook at the top. A probabilistic model is constructed for the random variable related to the weight-on-hook using the Maximum Entropy Principle, and the random response of the system is computed through Monte Carlo simulations. The idea is to understand how the performance of the system (which is measured by the rate of penetration) if aected by the uncertainties of the weight-onhook. The continuous system analyzed is discretized by means of the Finite Element Method and a computer code is developed to do the simulations.

Journal of Fluids and Structures, 2014

This paper deals with the problem of a pipe conveying fluid of interest in several engineering ap... more This paper deals with the problem of a pipe conveying fluid of interest in several engineering applications, such as micro-systems or drill-string dynamics. The deterministic stability analysis developed by Paidoussis and Issid (1974) is extended to the case for which there are model uncertainties induced by modeling errors in the computational model. The aim of this work is twofold: (1) to propose a probabilistic model for the fluid-structure interaction considering modeling errors and (2) to analyze the stability and reliability of the stochastic system. The Euler-Bernoulli beam model is used to model the pipe and the plug flow model is used to take into account the internal flow in the pipe. The resulting differential equation is discretized by means of the finite element method and a reduced-order model is constructed from some eigenmodes of the beam. A probabilistic approach is used to model uncertainties in the fluid-structure interaction. The proposed strategy takes into account global uncertainties related to the noninertial coupled fluid forces (related to damping and stiffness). The resulting random eigenvalue problem is used to analyze flutter and divergence unstable modes of the system for different values of the dimensionless flow speed. The numerical results show the random response of the system for different levels of uncertainty, and the reliability of the system for different dimensionless speeds and levels of uncertainty.

Journal of the Brazilian Society of Mechanical Sciences and Engineering, 2012

MATEC Web of Conferences, 2014

Journal of Guidance, Control, and Dynamics, 2002

Journal of Sound and Vibration

Proceeding Series of the Brazilian Society of Computational and Applied Mathematics, 2023

Proceedings of Uncertainty 2023, 2023

TEMA - Tendências em Matemática Aplicada e Computacional, 2006

Thin-Walled Structures, 2015

Journal of Vibration and Acoustics, 2014

Journal of the Brazilian Society of Mechanical Sciences and Engineering, 2010

Journal of Fluids and Structures, 2014

Discussion about quantification and propagation of uncertainties

Smooth Decomposition (SD) is a multivariate data or statistical analysis method to find normal mo... more Smooth Decomposition (SD) is a multivariate data or statistical analysis method to find normal modes and natural frequencies in an spatial data field. The projection used for this method is made such as it keeps the maximum variance possible for the displacement vector and also as it keeps the smoothest motions along time. From this method we can get the "energy" participation in the response of each normal mode during the simulation or the experimental test which can be a relevant information to validate results concerning the identification process. This method of identification can be used for linear and nonlinear systems and uses only output data given that the excitation satisfies some properties normally met by a well chosen random excitation, as a white noise, for example. The objective of this method is to identify systems from their displacement field under ambient excitation which, in many cases, can be hard to compute or to describe. As the method is only based on the covariance matrices of the displacement field and the corresponding velocity field, it is no needed further considerations and approximations. In this point the method is a great tool for modal analysis and system identification. In this paper, the presentation of the method is firstly done which will show us how we can interpret the results of SD for different systems and then the application of SD on simulated multi-DoF damped and undamped systems is performed and discussed to understand how SD can be a great tool for modal analysis. A discussion about the quality of the excitation is also performed.

The extraction of modal parameters from a real structure represents an important step in modal an... more The extraction of modal parameters from a real structure represents an important step in modal analysis. When only the output signal is available in an experiment, the system identification process is referred as operation modal analysis (OMA). Applications of those cases are fond for structures where the ambient excitation (wind, traffic, waves, nearby systems, etc.) can not be removed or is the only possible one. Once the input signals can not be measured, some assumptions in their random nature are needed together with a stochastic modeling of the system. Among several methods, the stochastic subspace identification (SSI) has been shown to be a consistent one and, therefore, was chosen to be used in this paper. Here, the modal analysis of a system under wind load is studied. The fluid-structure interaction force is usually not easy to be represented and its whiteness (assumption made in most of OMA methods) can not be easily conformed. In this way, a two floor building model is used for experimental validation, where different fluid-structure interaction were created. The paper begins with a presentation of the discrete state space model followed by the SSI theory. Two popular SSI algorithms are presented: covariance-driven and data-driven. A efficient way to select the correct parameters for the method is discussed together with a procedure to analyze the results. To exemplify the identification process, experimental results are shown and the identified parameters are listed. As conclusion, the wind has been shown to be a good excitation source for OMA once the system has been correct identified.

This work proposes an application of uncertainties analysis in a rotating system. As stochastic m... more This work proposes an application of uncertainties analysis in a rotating system. As stochastic modelling is not so common in rotordynamics field, it was chosen a simple model, which can qualitatively represent the main behavior of o rotating machinery. Another advantage of this choice is the processing time of model simulation. As Monte Carlo simulation is used for the stochastic model processing, a lower processing time is desirable. The model takes into account a rotor with two flexible bearing and an outboard disc. It is considered as random parameters the shaft elasticity modulus and the disc mass. In order to analyze the stochastic rotating system response, Campbell diagram is accomplished. In that way, it is possible to verify how the natural frequencies vary with the rotating speed. The rotating speed that is coincident to the natural frequency is called critical speed. Random parameters were adopted and a stochastic Campbell diagram were obtained. Consequently, the histograms of the critical speed were also obtained. Besides, a sensitivity analysis was accomplished, considering the influence of the coefficient of variation on the critical speeds. The is noted an important variation on the critical speeds skewness for high coefficient of variation.

A dynamic study of timber footbridges with uncertain mechanical properties under the action of st... more A dynamic study of timber footbridges with uncertain mechanical properties under the action of stochastic walking loads is presented in this paper. These structural systems made of timber are increasingly employed due to the high relation stiffness/weight that wood exhibits in relation to others structural materials. More, the development and implementation of laminated beams permits larger spans. These features can lead to lightweight structural systems in which the acceleration levels can exceed the human comfort limits. The sources of uncertainty of this structural model are the timber mechanical and physical properties, Modulus of Elasticity (MOE) and mass density. Also, the geometrical design of the boards that compose the laminated timber beams supporting the floor involves variability in the distances between finger joints. Probability Density Functions (PDFs) of the timber properties are formulated from the Principle of Maximum Entropy (PME). The finger joints distance generates the lengthwise variability of the MOE and mass density functions in each board of the laminated beams. The influence of these stochastic variables in the structural response on a forced vibration problem that includes a stochastic model of the load induced by the human walking is assessed. Pedestrians arrive to the footbridge under a Poisson distribution. The arrival velocity is such that a medium/low transit density is achieved in accordance with the footbridge dimension. The PDFs of the natural frequencies of the structure, the mode shapes and the structural response are numerically obtained through the Finite Element Method (FEM) and Monte Carlo Simulations (MCS). Values of peak accelerations produced at the mean span of the footbridge are evaluated in relation to the footbridge occupancy at each instant. The present stochastic model contributes to obtain a more realistic description of the response of this type of structures.

The study of the nonlinear dynamic response of a guyed mast considering the uncertainty of the gu... more The study of the nonlinear dynamic response of a guyed mast considering the uncertainty of the guys pre-tension is reported in this work. A computational model is constructed with the mast represented by an equivalent beam-column and the three guys at one level, by cables with an initial pretension and only having tensile capacity. The system of partial differential equations is discretized using the finite element method considering hermite elements for the mast (Bernoulli beam theory) and nonlinear cable elements for the guys. Also, the second order effect due to the axial loads on the mast is taken into account. A lateral dynamic load is applied on the mast. An ad hoc software developed by the first author is employed to solve the problem. Once the deterministic problem is stated, an uncertainty quantification is carried out. In this case, the stochastic variable is the pretension. Since the design value can be modified at the construction stage and during the service life, the pretension force is modeled as a random variable with a probability density function (PDF) derived from the Principle of Maximum Entropy (PME). The model herein presented contributes to attain a more realistic description of the structure, mainly regarding the three-dimensional representation and the sensibility to the variability of the guys pretensions.

Slides de um minicurso dado como parte do Programa de Verão 2017 do LNCC sobre fundamentos de pro... more Slides de um minicurso dado como parte do Programa de Verão 2017 do LNCC sobre fundamentos de probabilidade e sobre algumas quastões relacionadas a incerteza e sua quantificação.

In this paper a parametric analysis of a sample of responses of a dry-friction oscillator is perf... more In this paper a parametric analysis of a sample of responses of a dry-friction oscillator is performed in order to construct a statistical model. The system consists of a simple oscillator (mass-spring) moving on a base with a rough surface. Due to this roughness, the mass is subject to a dry-frictional force modeled as a Coulomb friction. It is considered that the base has an imposed stochastic bang-bang motion which excites the system in a stochastic way. The base velocity is modeled by a Poisson process for which a probabilistic model is fully specified. The non-smooth behavior of the dry-frictional force associated with the non-smooth stochastic base motion induces in the system stochastic stick-slip oscillations. The system response is composed by a random sequence alternating stick and slip-modes. With realizations of the system, a statistical model is constructed for this sequence. In this statistical model, the variables of interest of the sequence are modeled as random variables, as for example, the number of time intervals in which stick or slip occur, the instants at which they begin, and their duration. Samples of the system response are computed by integration of the dynamic equation of the system using independent samples of the base motion. Statistics and histograms of the random variables which characterize the stick-slip process are estimated for the generated samples. The objective of the paper is to analyze how these estimated statistics and histograms vary with the system parameters, i.e., to make a parametric analysis of the statistical model of the stick-slip process.

Proper ways to measure uncertainties are discussed. It is shown that an envelop with mean and st... more Proper ways to measure uncertainties are discussed. It is shown that an envelop with mean and standard deviation cannot describe uncertainty.

Stick-slip is modeled using Coulomb's friction and through a stochastic treatment the proportion ... more Stick-slip is modeled using Coulomb's friction and through a stochastic treatment the proportion of stick versus slip duration is described.

Segunda parte dos slides de um minicurso sobre Modelagem Estocástica da dinâmica do stick-slip da... more Segunda parte dos slides de um minicurso sobre Modelagem Estocástica da dinâmica do stick-slip dado durante o Uncertainties 2016, realizado em Maresias de 15 a 19/fevereiro/2016.

This work deals with the design of a suspension device, idealized as a spring-mass-damper sys... more This work deals with the design of a suspension device, idealized as a
spring-mass-damper system. The amplitude of a nominal system is
constrained to satisfy certain limitations in a given frequency band
and the design is to be done as a reliability;based optimization. This constitutes a major difficulty since the constraint becomes a random process. To concentrate in the main ideas, only the stiffness of the system will be considered random. The stiffness is characterized by a uniform random variable, and its mean and standard deviation are
the optimization parameters. The design problem is stated as a two;
objective optimization. They are the mean and the standard devia;
tion of the stiffness: one searches for the lowest stiffness and the greatest standard deviation, while the amplitude response must be
within the acceptable domain of vibration, which is prescribed. To
generate the Pareto front, the Normal Boundary Intersection (NBI)
method is used in the RFNM algorithm. Results show that a not;
connected Pareto curve can be obtained for some choice of constraint. Hence, in this simple example, one shows that difficult situations can occur in the design of dynamic systems when prescribing an amplitude;response hull. Despite the simplicity of the example treated here, chosen to highlight the main ideas without distraction, the strategy proposed here can be generalized for more complex cases and give valuable results, able to help designers to choose for the best compromise between the mean and the standard deviation in reliability;based designs.

First, of two parts of slides corresponding of a short course given during the Uncertainty 2016, ... more First, of two parts of slides corresponding of a short course given during the Uncertainty 2016, Maresias, 16--19/Mars/2016.

A dynamic study of timber footbridges with stochastic mechanical properties is presented in this ... more A dynamic study of timber footbridges with stochastic mechanical properties is presented in this paper. These structural systems made of timber are increasingly employed due to the high relation stiffness/weight that this material exhibits. More, the development and implementation of laminated beams permits larger spans. The sources of uncertainty of this structural model are the timber mechanical and physical properties. Also, the geometrical design of the laminates that compose the laminated timber beams supporting the floor involves variability in the distances between finger joints. Probability Density Functions (PDFs) of the timber properties are formulated from the Principle of Maximum Entropy (PME) and visual surveys of structural size Eucalyptus grandis laminated beams are used to obtain statistical parameters. The influence of these stochastic variables in the structural response on a forced vibration problem that includes a deterministic model of the load induced by the human walking is assessed. The PDFs of the natural frequencies of the structure, the mode shapes and the structural response are numerically obtained through the Finite Element Method (FEM) and Monte Carlo Simulations (MCS). In order to carry out this analysis, plate elements and laminated beam elements derived starting from the First Order Shear Deformation Theory (FSDT) are employed. The present stochastic model contributes to obtain a more realistic description of the response of this type of structures applicable for the study of the human comfort conditions.

Talk during the CNMAC 2016; Gramado, Rio Grande do Sul, Brazil; September, 5 to 9th.

Opening Plenary conference in CNMAC 2016, Gramado September 5 to 9th.

Uma introdução aos modos normais não-lineares de sistemas mecânicos, 2023

Modos normais para sistemas lineares se referem a uma situação em que apenas a inércia e a rigide... more Modos normais para sistemas lineares se referem a uma situação em que apenas a inércia e a rigidez atuam em um sistema. Nesse caso o problema de achar soluções de um sistema linear se reduz a resolver um problema de autovalores. Com os autovalores e os autovetores se determinam famílias de soluções periódicas que dependem apenas das características do sistema linear e suas características independem de energia. Para o caso não-linear a situação muda muito. Não é mais possível determinar as soluções a partir de um problema de autovalores e as características das soluções periódicas variam com a energia.
Rosenberg, nos anos 60, observou que a determinação de frequência e modos pode ser feita de maneira alternativa ao problema de autova- lor, resolvendo um problema com condições periódicas. O período da solução encontrada é relacionado à frequência natural do sistema e a razão entre os deslocamentos é utilizada para definir os modos. Porém, adotando essa estratégia, que é equivalente à fornecida pelo problema de autovalor no caso linear, no caso não-linear surgem novidades. Uma delas é que os modos dependem da energia do sistema, que no caso de sistemas lineares normais é conservada.
Uma característica de sistemas lineares é a superposição de soluções, uma combinação linear de soluções também é uma solução do sistema. Esse fato implica que as soluções do problema de autovalores fornecem uma base para o espaço de soluções de um problema linear. No caso não-linear esse resultado não é válido e, aparentemente, não há sentido em calcular modos não-lineares, pois o espaço de soluções não é um espaço vetorial. Como se mostrará nesse livro, a perda da superposição não leva a perda de importância. Os modos não-lineares estão dire- tamente associados a dinâmica do sistema e são fundamentais para se entender como as soluções mudam com a energia do sistema e como no- vas soluções podem surgir. Uma série de situações, como localização da energia, aparentemente inexplicáveis, são facilmente explicáveis através dos modos não-lineares. Situações como enrijecimento, aumento da frequência com o aumento da energia de um sistema, e relaxamento (também chamado de amolecimento), o caso contrário, são também explicáveis com os modos não-lineares. Um fenômeno interessante de
transmissão irreversível de energia entre diferentes graus de liberdade do sistema (também chamado de bombeamento de energia) pode ser melhor compreendido quando se conhece os modos não-lineares.
Modos não-lineares são ligados a dinâmica de sistemas e ajudam no seu entendimento. Nesse livro, que mostra uma primeira visão do assunto, por motivos didáticos, se tratará apenas de sistemas discretos conservativos. Se deixará de lado sistemas contínuos e dissipativos. O enfoque é na parte computacional — balanço harmônico, método do tiro, continuação — e com dois capítulos dissertando sobre modos não-lineares e sua importância.
O assunto é atual e o número de publicações vem aumentando con- sideravelmente. Porém, esse é o primeiro livro em português sobre esse tema!