A bifurcation and symmetry discussion of the Sommerfeld effect

The Arnold Sommerfeld effect is an intriguing resonance capture and release series of events originally demonstrated in 1902. A single event is studied using a two degree of freedom mathematical model of a motor with imbalance mounted to laterally restricted spring connected cart. For a certain power supplied, in general the motor rotates at a speed consistent with a motor on a rigid base. However at speeds close to the natural frequency of the cart, it seemingly takes on extra oscillations where for a single rotation it both speeds up and then slows down. Therefore in a standard experimental demonstration of the effect, as the supplied torque force is increased or decreased, this may give the illusion that the stable operation of the motor is losing and gaining stability. This is not strictly the case, instead small oscillations always present in the system solution are amplified near the resonant frequency. The imbalance in the motor causes a single resonance curve to fold back on itself forming two fold bifurcations which leads to hysteresis and an asymmetry between increasing and decreasing the motor speed. Although as outlined the basic mechanism is due the interplay between two stable and one unstable limit cycles, a more complicated bifurcation scenario is observed for higher imbalances in the motor. The presence of a Z2 phase space symmetry tempers the dynamics and bifurcation picture.Comment: 14 pages, 8 figure

Analysis of stationary random vibrating systems using smooth decomposition

International audienceA modified Karhunen-Loève Decomposition/Proper Orthogonal Decomposition method, named Smooth Decomposition (SD) (also named smooth Karhunen-Lo'eve decomposition), was recently introduced to analyze stationary random signal. It is based on a generalized eigenproblem defined from the covariance matrix of the random process and the covariance matrix of the associated time-derivative random process. The SD appears to be an interesting tool in terms of modal analysis. In this paper, the SD will be described in case of stationary random processes and extended also to stationary random fields. The main properties will be discussed and illustrated on a randomly excited clamped-free beam

Dynamics of curved thin-walled composite beams: uncertainty quantification due to randomly distributed thermal/hygroscopic aspects

In this paper we analyze the dynamic behavior of curved thin walled composite beams considering hygroscopic and thermal effects in the constitutive equations. A model of curved thin walled beam is employed as the basis for deterministic calculations which are performed in the context of finite element approaches. This model takes into account shear deformation due to bending and non-uniform torsion, also it incorporates the effect of hygro-thermal stresses and strains in the classical way, however considering them as uncertain due to the randomness associated with the material of the matrix resin (normally sensitive to the absorption of humidity) while the composite beam is constructed or while the structure is under service. The variability of the stiffness and mass properties of the composite beam is assumed as a random field along the structure taking into account the elastic coupling between bending, twisting, shear and axial motions together with the thermal and hygroscopics terms. The probabilistic model is constructed appealing to the Maximum Entropy Principle in order to derive the marginal probability density functions, according to increasing levels of entropy, i.e. with less number of constraints or less information. The analysis is performed in the frequency domain and the buckling loads by comparing the probabilistic models with different levels of information (i.e., given the mean and/or the bounds, etc.) with previously developed probabilistic approaches such as the ones with parametric uncertainty. Also the Entropy of the response is evaluated in order to quantify the propagation of uncertainty in the information of the model. A number of different hygroscopic sensitive composites are evaluated and the dynamic response of the structure constructed with them is compared with the homonymous case of a perfectly dry specimen of the same volumetric fraction of reinforcement.Publicado en: Mecánica Computacional vol. XXXV, no. 22Facultad de Ingenierí

Free vibrations of an uncertain energy pumping system

International audienceThe aim of this paper is to study the energy pumping (the irreversible energy transfer from one structure, linear, to another structure, nonlinear) robustness considering the uncertainties of the parameters of a two DOF mass-spring-damper, composed of two subsystems, coupled by a linear spring: one linear subsystem, the primary structure, and one nonlinear subsystem, the so-called NES (nonlinear energy sink). Three parameters of the system will be considered as uncertain: the nonlinear stiffness and the two dampers. Random variables are associated to the uncertain parameters and probability density functions are constructed for the random variables applying the Maximum Entropy Principle. A sensitivity analysis is then performed, considering different levels of dispersion, and conclusions are obtained about the influence of the uncertain parameters in the robustness of the system

O método do Lagrangeano aumentado no estudo de cabos umbilicais

O objetivo deste trabalho é apresentar um modelo numérico para cabos umbilicais hiperelásticos que experimentem grandes deslocamentos e grandes rotacoes. O modelo mecanico resulta num sistema náo-linear, que é resolvido por um método de decomposicáo-coordenacao via Lagrangeano Aumentado. Este tratamento posibilita a descricáo do acoplamento existente entre flexao e torcáo em cabos submetidos A grandes delocamentosPeer Reviewe

Finite element updating of a bridge model using operational modal analysis

Operational Modal Analysis (OMA) consist in finding the dynamic characteristic of a structure through its modal parameters using output-only signals. Differently from the classical approach of Experimental Modal Analysis (EMA), where the input signal are also measured, OMA only uses the stochastic nature of the inputs, assumed to be random due the ambient conditions. An important application of this technique appears in the model validation, where numerical and experimental results are compared (Brincker and Ventura, 2015).Because of the large size of bridges, the identification of this kind of structures are restricted to operational modal analysis. Output-only methods are necessary since a controlled input is usually hard and expensive to apply. Also, the ambient forces such as wind, waves, traffic and ground motion can notbe eliminated. The advantage of OMA when compared to EMA is that those forces do not need to be measured and quantified.In this paper, the stochastic subspace identification method is used to characterize the dynamic behavior of a small bridge model under wind load (Overschee and Moor, 1996)(Wagner et al., 2017). The identified natural frequencies and mode shapes are used to validate its finite element model, specially regarding the imposed boundary conditions (clamp-clamp). In reality, those conditions are uncertain and need to be taken into account to improve the predictability of the model (Ritto et al., 2008)(Ritto et al.,2016). The clamp condition is changed into a free condition with displacements and torsional linear springs, where the uncertainties parameters are the respective stiffnesses.Publicado en: Mecánica Computacional vol. XXXV, no. 22Facultad de Ingenierí

Modal Features and Dynamic Behavior of a Nonlinear 3D Guyed Mast with Uncertain Guys Pretension

The study of the nonlinear dynamic characteristics and response of a guyed mast, considering the uncertainty of the guys pretension 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. Starting from the energy formulation of beams and nonlinear cables, the continuous equations are discretized using finite element techniques, considering Hermite elements for the mast (Bernoulli beam theory) and quadratic elements for the nonlinear guys. Also, the second order effect due to the axial loads on the mast is taken into account. An ad hoc software, developed by the first author, is employed here to explore natural frequencies and modes of the structure considering the uncertainty propagation of the stochastic guys pretension. Since the guys design value can be modified at the construction stage and more, 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. The results here presented (natural frequencies and modes) obtained through an uncertainty quantification analysis, improve the understanding of the real dynamic properties and behavior of slender and flexible guyed structures.Publicado en: Mecánica Computacional vol. XXXV, no. 21Facultad de Ingenierí

The robust smooth orthogonal decomposition for system identification: a new way to quantify the modal parameters uncertainties

Recently, the proper orthogonal decomposition (POD) has generated a family of methods that allow system identification using output-only data. They all have been developed to overcome some of the POD limitations in the field of linear modal analysis. Two important achievement was accomplish by the smooth orthogonal decomposition (SOD) (Bellizzi and Sampaio, 2015) (Chelidze andZhou, 2006) (Farooq and Feeny, 2008): first, the method eliminates the need of a priori knowledge of the inertia matrix to relate the proper orthogonal modes (POMs) to the linear normal modes (LNMs). Second, the method allows a direct estimation of the system´s natural frequencies. Although this powerful tool has provided good predictions, experimental tests have shown inconsistent results when significant noise levels are present in the signal. Compared with other operational modal analysis identification techniques, the so far proposed SOD has shown to be the one with more noise sensitivity (Brincker and Ventura, 2015). The reason can be shown through an analysis of the noise distortion in the correlation estimation of the measured data. In this article, two new robust versions of the SOD are presented. They solve the problem of the noise sensibility and also have new important features. The robust versions of the SOD allow the identification of the modal parameters and their uncertainties, that the SOD could not do (Wagner et al., 2017). Thanks to the method simplicity, efficiency implementations can be use to perform real-time identification (duringthe data acquisition phase). An application shows how the methods are used.Publicado en: Mecánica Computacional vol. XXXV, no. 22Facultad de Ingenierí

Modal Features and Dynamic Behavior of a Nonlinear 3D Guyed Mast with Uncertain Guys Pretension

The study of the nonlinear dynamic characteristics and response of a guyed mast, considering the uncertainty of the guys pretension 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. Starting from the energy formulation of beams and nonlinear cables, the continuous equations are discretized using finite element techniques, considering Hermite elements for the mast (Bernoulli beam theory) and quadratic elements for the nonlinear guys. Also, the second order effect due to the axial loads on the mast is taken into account. An ad hoc software, developed by the first author, is employed here to explore natural frequencies and modes of the structure considering the uncertainty propagation of the stochastic guys pretension. Since the guys design value can be modified at the construction stage and more, 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. The results here presented (natural frequencies and modes) obtained through an uncertainty quantification analysis, improve the understanding of the real dynamic properties and behavior of slender and flexible guyed structures.Publicado en: Mecánica Computacional vol. XXXV, no. 21Facultad de Ingenierí