Orientador: Masayoshi Tsuchida / Banca: José Manoel Balthazar / Banca: Elso Drigo Filho / Resumo: Neste trabalho estuda-se alguns sistemas dinâmicos utilizando um novo método para aproximar a matriz de transição de estados (STM) para sistemas periódicos no tempo. Este método é baseado na transformação de Lyapunov-Floquet (L-F), e utiliza a expansão polinomial de Chebyshev para aproximar o termo periódico. O método iterativo de Picard é usado para aproximar a STM. Os multiplicadores de Floquet, determinados através deste método, permitem construir o diagrama de estabilidade do sistema dinâmico. Esta técnica é aplicada para analisar a estabilidade e os pontos de bifurcação do sistema dinâmico formado por um pêndulo elástico com excitação vertical periódica no suporte. Além dessa aplicação, é analisada também a equação de Mathieu e a estabilidade do sistema dinâmico constituído por partículas carregadas e imersas em um campo magnético perturbado. / Abstract: In this work some dynamic systems are studied using a new method to approach state transition matrix (STM) for time-periodic systems. This method is based on Lyapunov- Floquet transformation (transformation L-F) and uses the Chebyshev polynomial expansion to approach the periodical term. The Picard iterative method is used to approach the STM. The Floquet multipliers determined through this method, allow to draw the stability diagram of the dynamic system. This technique is applied to analyze the stability and bifurcation points of the dynamic system formed by an elastic pendulum with periodic vertical excitation on support. Besides this application, the Mathieu equation is analyzed and also the stability of the dynamical system constituted by charged particle in a perturbed magnetic field is discussed. / Mestre

Made available in DSpace on 2014-06-11T19:27:08Z (GMT). No. of bitstreams: 0 Previous issue date: 2007-06-11Bitstream added on 2014-06-13T20:55:46Z : No. of bitstreams: 1 mesquita_ajn_me_sjrp.pdf: 655612 bytes, checksum: cb512103d01edb2f09f992e6cca22bdc (MD5) / Neste trabalho estuda-se alguns sistemas dinâmicos utilizando um novo método para aproximar a matriz de transição de estados (STM) para sistemas periódicos no tempo. Este método é baseado na transformação de Lyapunov-Floquet (L-F), e utiliza a expansão polinomial de Chebyshev para aproximar o termo periódico. O método iterativo de Picard é usado para aproximar a STM. Os multiplicadores de Floquet, determinados através deste método, permitem construir o diagrama de estabilidade do sistema dinâmico. Esta técnica é aplicada para analisar a estabilidade e os pontos de bifurcação do sistema dinâmico formado por um pêndulo elástico com excitação vertical periódica no suporte. Além dessa aplicação, é analisada também a equação de Mathieu e a estabilidade do sistema dinâmico constituído por partículas carregadas e imersas em um campo magnético perturbado. / In this work some dynamic systems are studied using a new method to approach state transition matrix (STM) for time-periodic systems. This method is based on Lyapunov- Floquet transformation (transformation L-F) and uses the Chebyshev polynomial expansion to approach the periodical term. The Picard iterative method is used to approach the STM. The Floquet multipliers determined through this method, allow to draw the stability diagram of the dynamic system. This technique is applied to analyze the stability and bifurcation points of the dynamic system formed by an elastic pendulum with periodic vertical excitation on support. Besides this application, the Mathieu equation is analyzed and also the stability of the dynamical system constituted by charged particle in a perturbed magnetic field is discussed.

abstract: It remains unquestionable that space-based technology is an indispensable component of modern daily lives. Success or failure of space missions is largely contingent upon the complex system analysis and design methodologies exerted in converting the initial idea into an elaborate functioning enterprise. It is for this reason that this dissertation seeks to contribute towards the search for simpler, efficacious and more reliable methodologies and tools that accurately model and analyze space systems dynamics. Inopportunely, despite the inimical physical hazards, space systems must endure a perturbing dynamical environment that persistently disorients spacecraft attitude, dislodges spacecraft from their designated orbital locations and compels spacecraft to follow undesired orbital trajectories. The ensuing dynamics’ analytical models are complexly structured, consisting of parametrically excited nonlinear systems with external periodic excitations–whose analysis and control is not a trivial task. Therefore, this dissertation’s objective is to overcome the limitations of traditional approaches (averaging and perturbation, linearization) commonly used to analyze and control such dynamics; and, further obtain more accurate closed-form analytical solutions in a lucid and broadly applicable manner. This dissertation hence implements a multi-faceted methodology that relies on Floquet theory, invariant center manifold reduction and normal forms simplification. At the heart of this approach is an intuitive system state augmentation technique that transforms non-autonomous nonlinear systems into autonomous ones. Two fitting representative types of space systems dynamics are investigated; i) attitude motion of a gravity gradient stabilized spacecraft in an eccentric orbit, ii) spacecraft motion in the vicinity of irregularly shaped small bodies. This investigation demonstrates how to analyze the motion stability, chaos, periodicity and resonance. Further, versal deformation of the normal forms scrutinizes the bifurcation behavior of the gravity gradient stabilized attitude motion. Control laws developed on transformed, more tractable analytical models show that; unlike linear control laws, nonlinear control strategies such as sliding mode control and bifurcation control stabilize the intricate, unwieldy astrodynamics. The pitch attitude dynamics are stabilized; and, a regular periodic orbit realized in the vicinity of small irregularly shaped bodies. Importantly, the outcomes obtained are unconventionally realized as closed-form analytical solutions obtained via the comprehensive approach introduced by this dissertation. / Dissertation/Thesis / Doctoral Dissertation Systems Engineering 2019