This thesis deals with the study of two problems in the area of nonlinear dynamics using the method of multiple scales. Accordingly, it consists of two parts. In the first part of the thesis, we explore the asymptotic stability of a planar two-degree- of-freedom robot with two rotary (R) joints following a desired trajectory under feedback control. Although such robots have been extensively studied and there exists stability and other results for position control, there are no analytical results for asymptotic stability when the end of the robot or its joints are made to follow a time dependent trajectory. The nonlinear dynamics of a 2R planar robot, under a proportional plus derivative (PD) and a model based computed torque control, is studied. The method of multiple scales is applied to the two nonlinear second-order ordinary deferential equations which describes the dynamics of the feedback controlled 2R robot. Amplitude modulation equations, as a set of four first order equations, are derived. At a fixed point, the Routh-Hurwitz criterion is used to obtain positive values of proportional and derivative gains at which the controller is asymptotically stable or indeterminate. For the model based control, a parameter representing model mismatch is incorporated and again controller gains are obtained, for a chosen mismatch parameter value, where the controller results in asymptotic stability or is indeterminate. From numerical simulations with gain values in the indeterminate region, it is shown that for some values and ranges of the gains, the non- linear dynamical equations are chaotic and hence the 2R robot cannot follow the desired trajectory and be asymptotically stable. The second part of the thesis deals with the study of the nonlinear dynamics of a rotating flexible link, modeled as a one dimensional beam, undergoing large deformation and with geometric nonlinearities. The partial deferential equation of motion is discretized using a finite element approach to yield four nonlinear, non-autonomous and coupled ordinary deferential equations. The equations are non-dimensional zed using two characteristic velocities – the speed of sound in the material and a speed associated with the trans- verse bending vibration of the beam. The method of multiple scales is used to perform a detailed study of the system. A set of four autonomous equations of the first-order are derived considering primary resonance of the external excitation with one of the natural frequencies of the model and one-to-one internal resonance between two different natural frequencies of the model. Numerical simulations show that for certain ranges of values of these characteristic velocities, the slow flow equations can exhibit chaotic motions. The numerical simulations and the results are related to a rotating wind turbine blade and the approach can be used for the study of the nonlinear dynamics of a single link flexible manipulator. The second part of the thesis also deals with the synchronization of chaos in the equations of motion of the flexible beam. A nonlinear control scheme via active nonlinear control and Lyapunov stability theory is proposed to synchronize the chaotic system. The proposed controller ensures that the error between the controlled and the original system asymptotically go to zero. A numerical example using parameters of a rotating power generating wind turbine blade is used to illustrate the theoretical approach.

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Submitted by Cássia Santos (cassia.bcufg@gmail.com) on 2017-01-17T12:39:40Z No. of bitstreams: 3 Dissertação - Joaquim Orlando Parada (parte1) - 2013.pdf: 11591347 bytes, checksum: e970b2f0fffd5ccc2222bce05ea90d41 (MD5) Dissertação - Joaquim Orlando Parada (parte 2) - 2013.pdf: 18027973 bytes, checksum: 6bdbe04565ae04f1d810137fc59f37e2 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2017-01-18T10:31:58Z (GMT) No. of bitstreams: 3 Dissertação - Joaquim Orlando Parada (parte1) - 2013.pdf: 11591347 bytes, checksum: e970b2f0fffd5ccc2222bce05ea90d41 (MD5) Dissertação - Joaquim Orlando Parada (parte 2) - 2013.pdf: 18027973 bytes, checksum: 6bdbe04565ae04f1d810137fc59f37e2 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Made available in DSpace on 2017-01-18T10:31:58Z (GMT). No. of bitstreams: 3 Dissertação - Joaquim Orlando Parada (parte1) - 2013.pdf: 11591347 bytes, checksum: e970b2f0fffd5ccc2222bce05ea90d41 (MD5) Dissertação - Joaquim Orlando Parada (parte 2) - 2013.pdf: 18027973 bytes, checksum: 6bdbe04565ae04f1d810137fc59f37e2 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2013-06-21 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this work, the linear and nonlinear instability of pipes conveying static and pulsating fluid flow is analyzed. The dynamic equation of motion was derived for cantilevered and clamped-clamped pipes. For this purpose, the Euler Bernoulli beam theory and Hamilton’s principle were applied, resulting in a partial differential equation of second order in time. Thus, a model with four degrees of freedom, which satisfies the boundary condition, is used and, the Galekin method is applied to derive the set of coupled non linear ordinary equations of motion which are, in turn, solved by the fourth order Runge-Kutta method, and then some numerical results were obtained as Argand diagram, stability boudaries, time response, phase plane and, Poincaré section, through computational algorithms modeled in C++. These results revealed the importance of the nonlinear terms in the stability of the system, especially in the post-critical analysis, also revealed the existence of quasi-periodic motions, for the system subjected to a static flow and, chaotic motions for pulsating fluid flow / Nesta dissertação analisa-se a instabilidade linear e não linear de tubos com fluido interno em escoamento estático e pulsante. A equação de movimento dinâmico foi deduzida para tubos em balanço e biengastados. Para tanto, utilizou-se a teoria de vigas de Euler Bernoulli e o princípio variacional de Hamilton, resultado em uma equação diferencial parcial de segunda ordem no tempo. Tal equação foi discretizada, pelo método de Galerkin, em quatro equações diferenciais ordinárias, uma para cada grau de liberdade, em seguida transformadas em um conjunto de equações diferenciais de primeira ordem. Tais equações foram integradas pelo método de Runge-Kutta de quarta ordem e, posteriormente, foram obtidos alguns resultados numéricos como: diagrama de Argand, curvas de escape, diagrama de bifurcação, resposta no tempo, plano fase e, seção de Poincaré, através de algoritmos implementados computacionalmente na linguagem C++. Tais resultados revelaram a importância dos termos não lineares na estabilidade do sistema, especialmente na análise pós-crítica, revelaram também a existência de movimentos quase periódicos, para o sistema submetido a um fluxo estático e, caóticos para fluxo pulsante.