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Mathematical Problems in EngineeringVolume 2008 2008, Article ID 538725, 4 pages


Department of Statistics, Applied Mathematics, and Computation, State University of São Paulo UNESP at Rio Claro, 13500-230 Rio Claro, SP, Brazil

Civil Engineering Department, Pontifical Catholic University of Rio de Janeiro PUC-Rio, 22453-900 Rio de Janeiro, RJ, Brazil

Department of Structural and Geotechnical Engineering, University of São Paulo PEF-EPUSP-USP, 05508-900 São Paulo, SP, Brazil

Received 10 September 2008; Accepted 10 September 2008

Copyright © 2008 José Manoel Balthazar et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Nonlinear dynamical systems usuallydisplay high complexity. The last decades have seen a remarkable and fruitfuldevelopment of nonlinear dynamics, and a large number of papers have beenpublished in all branches of science. In modeling natural and man-made systems,it is assumed in general that the system is perfect and that all parameters ofthe system are known. However, real systems are usually imperfect, anduncertainties are present both in system parameters and in the modeling stage. Thisis associated with the lack of precise knowledge of the system parameters,random or noisy external loading, operating conditions, and variabilities inmanufacturing processes, among other things. In many situations, theseuncertainties are not important and may be overlooked in the mathematicalmodeling of the problem. However in several situations, the uncertainties canhave significant influence on the dynamic response and the stability of thesystem. Uncertainties may also be found in system response, even in cases whereall parameters are well established, such as systems exhibiting highsensitivity to initial conditions. This is particularly important in stronglynonlinear chaotic systems and those with fractal-basin boundaries. However, theinfluence of uncertainties in local and global bifurcations and basins ofattractions and on important engineering concepts such as reliability, safety,and robustness is not well studied inliterature. Even the definition of a random bifurcation is still an open problemin nonlinear dynamics. This is a rather broad topic in nonlinear dynamics.

So, the present special issue isdedicated to the influence of uncertainties in structural dynamics. In engineeringstructures, the main sources of uncertainties are: imperfections, uncertaintiesin system parameters mass, damping, and stiffness, uncertainties in theexternal load, such as random loads wind and earthquake, sensitivity toinitial conditions, and interaction between load and structure. These types ofuncertainties coupled to system nonlinearities may have a marked influence onthe structure’s response, particularly in a dynamic environment. So, it isuseful to study their influence on bifurcations, stability boundaries, andbasins of attraction. It is also interesting to discuss their influence onsafety factors, integrity measures, and confiability. These topics areessential for a safe design of structures and the development of mathematicallybased safe but not too conservative design codes and methodologies.

This issue comprises 16contributions from 8 different countries, which give a picture of the currentresearch on uncertainties in engineering problems. Several aspects areaddressed, including the following: influence of uncertainties on buckling and vibrationof structural elements: trusses, cables, piles, beams, plates, and shells, theconsideration of uncertainties in vibration control, stochastic systems, andnumerical algorithms for the analysis of systems with different types ofuncertainties.

The effect of geometricimperfections, unavoidable in real systems, and viscous damping on the type ofnonlinearity i.e., the hardening or softening behavior of circular plates andshallow spherical shells is investigated in the paper written by C. Touzé et al. The von Kármán large-deflection theory is used to derive thecontinuous models. Then, nonlinear normal modes NNMs are used for predicting thestructures’ nonlinear behavior.

D. S. Sophianopoulos et al. study the local instability of a 2 degree of freedom DOF weakly dampedsystems using the Liénard-Chipart stability criterion. The individual andcoupling effects ofmass and stiffness distribution on the dynamic asymptotic stability due tomainly infinitesimal damping areexamined. The validity of the theoretical findings presented herein is verifiedvia a nonlinear dynamic analysis.

The nonlinear modeling of cableswith flexural stiffness is discussed by W. Lacarbonara and A. Pacitti. A geometrically exact formulation of cables suffering axis stretchingand flexural curvature is presented. The dynamical formulation is based onnonlinearly viscoelastic constitutive laws for the tension and bending momentwith the additional constitutive nonlinearity accounting for the no-compressioncondition.

The paper by D. M. K. N. Kunitaki et al. uses probabilistic and fuzzy arithmetic approaches for the treatment of uncertaintiesin the installation of torpedo piles used in the foundations of mooring linesand risers of floating production systems for offshore oil exploitation. Methodologiesinvolving, respectively, the Monte Carlo method and concepts of fuzzy arithmetic are used to assess the sensitivity ofthe response to the variation of the uncertain parameters.

The effects of uncertainties in nonlineardamping coefficients on the parametric vibration of a cantilever beam with a lumpedmass are investigated by D. G. Silva and P. S. Varoto. The effects of a turbulent frictional dampingforce on the dynamic behavior of the flexible structure are studied numericallyand experimentally. The results indicate that variations on the dampingcoefficient significantly alter the dynamics of the structure under investigation.

The influence of uncertainties onthe dynamic buckling loads of structures liable to asymmetric postbuckling behavioris studied by P. B. Gonçalves and D. M. Santee. A parametric analysis illustratesthe influence of uncertainties in system parameters and random perturbations ofthe forcing on the dynamic buckling load. A lower bound for the buckling loads,obtained by the application of the Melnikov criterion, is proposed, whichcompare well with the scatter of buckling loads obtained numerically.

The paper by J. Lew presents anapproach to model validation for structures with significant parametervariations. Model uncertainty of the structural dynamics is quantified with theuse of a singular value decomposition technique to extract the principal componentsof parameter change, and an interval model is generated to represent the systemwith parameter uncertainty. A beam structure with an attached subsystem, whichhas significant parameter uncertainty, is used to demonstrate the proposedapproach.

J. Zhu et al. investigate a robustKalman filtering design for continuous-time Markovian jump nonlinear systemswith uncertain noise. The statistical characteristics of system noise andobservation noise are time-varying or unmeasurable instead of stationary. Byview of robust estimation, maximum admissible upper bound of the uncertainty tonoise covariance matrix is given based on system state estimation performance. Therobustness of the Kalman filter against noise uncertainty and stability ofdynamic systems is studied by Game theory.

The “Ga-basedfuzzy sliding mode controller for nonlinear systems” is studied by P. C. Chen et al. First, they approximate anddescribe an uncertain and nonlinear plant for the tracking of a referencetrajectory via a fuzzy model incorporating fuzzy logic control rules. Next, theinitial values of the consequent parameter vector are decided via a geneticalgorithm. After this, an adaptive fuzzy sliding model controller, designed tosimultaneously stabilize and control the system, is derived. The stability ofthe nonlinear system is ensured by the derivation of the stability criterionbased upon Lyapunov-s direct method.

The robust active vibration controlof flexible structures considering uncertainties in system parameters is addressedby D. D. Bueno et al. The paper proposes an experimental methodology forvibration control in a 3D truss structure using PZT wafer stacks and a robust controlalgorithm solved by linear matrix inequalities.

The paper written by D. Chen and W. Zhang is concerned with the sliding mode control for uncertain stochasticneutral systems with multiple delays. A switching surface is adopted first. Then,by means of linear matrix inequalities, a sufficient condition is derived toensure the global stochastic stability of the stochastic system in the slidingmode for all admissible uncertainties. The synthesized sliding mode controllerguarantees the existence of the sliding mode.

W. T. M. Lima and L. M. Bezerra’spaper presents an implicit time integration scheme for transient responsesolution of structures under large deformations and long-time durations. Theinfluence of different substep sizes on the numerical dissipation of the methodis studied throughout three practical examples. The method shows goodperformance and may be considered good for nonlinear transient response ofstructures.

C. Soize and A. Batou study the identification of stochastic loads applied to a nonlineardynamical system for which a few experimental responses are available using anuncertain computational model. A nonparametric probabilistic approach of bothparameter uncertainties and model uncertainties is implemented to take intoaccount uncertainties, and the level of uncertainties is identified using themaximum likelihood method. The identified stochastic simplified computationalmodel which is obtained is then used to perform the identification of thestochastic loads applied to the real nonlinear dynamical system.

A homotopy perturbation method forsolving reaction-diffusion equations is proposed by L. Mo et al. In thismethod, the trial function initial solution is chosen with some unknownparameters, which are identified using the method of weighted residuals. Someexamples are given, and the obtained results are compared with the exactsolutions.

Z. H. Wang presents, on the basisof Lambert W function, an iterative algorithm for the calculation of therightmost roots of the neutral delay differential equations so that thestability of the delay equations can be determined directly. The application ofthe method is illustrated with two examples.

Finally, dynamical models for computer viruses propagation are proposedby J. R. C. Piqueira and F. B. Cesar. Data from three different viruses arecollected in the Internet, and two different identification techniques,autoregressive and Fourier analyses, are applied showing that it is possible toforecast the dynamics of a new virus propagation by using the data collectedfrom other viruses that formerly infected the network.

José Manoel BalthazarPaulo Batista GonçalvesReyolando M. R. L. F. Brasil

Author: José Manoel Balthazar, Paulo Batista Gonçalves, and Reyolando M. R. L. F. Brasil

Source: https://www.hindawi.com/


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