In this paper, the theory of solvable structures is applied to the falling cat problem, which concerns the ability of a cat (or any similar creature) to reorient its body in midair during a fall to land on its feet. Under some assumptions and simplifications in the falling cat problem, a coupled system formed by two second-order ordinary differential equations involving an arbitrary scalar function is obtained. In order to address the determination of exact solutions, the Lie algebra of point symmetries admitted by the system is computed, which turns out to be four-dimensional. However, the corresponding symmetry generators are not linearly independent pointwise, and therefore, the classical Lie-Bianchi method cannot be directly applied to solve the system. In order to overcome this difficulty, a solvable structure for the vector field associated with the system is computed, which is used to completely integrate the equations of motion. An alternative integration procedure based on a combination of the Noether theorem and the Lie-Bianchi method is also shown.

Os resíduos sólidos se referem aos materiais descartados que resultam das atividades humanas no meio social. A geração de resíduos tem aumentado devido ao crescimento populacional que acarreta o surgimento de diversos problemas socioambientais e econômicos. Este trabalho teve como objetivo realizar a prospecção tecnológica de patentes relacionadas à área de gestão de Resíduos Sólidos Urbanos (RSU), visando a identificar as tecnologias mais adequadas para implementação na Região de Integração do Carajás (RIC). A busca de patentes foi realizada na base de dados do Instituto Nacional da Propriedade Industrial, utilizando palavras-chave no título e resumo, assim como códigos do sistema de classificação internacional. A partir dos resultados, foram selecionadas patentes promissoras para o tratamento/reciclagem, transformação/digestão e gerenciamento de RSU e logística reversa que podem ser implementadas na gestão de RSU na RIC, principalmente por meio do consórcio público.

A gestão de informações acadêmicas da experiência profissional docente é inerente às Instituições de Ensino Superior que estimula a inovação de tecnologias facilitadoras dos processos gerenciais pelo desenvolvimento de software que favoreçam a construção de novas bases de dados sistematizadas. Nesse contexto, pretende-se analisar as tecnologias protegidas e utilizadas na gestão desse tipo de informações por meio da prospecção de tecnologias; da identificação de suas fontes de dados; a fim de viabilizar a criação de um software no âmbito da Universidade. Para tanto, foi realizada a pesquisa prospectiva de registros de programas de computador e de software públicos no Brasil, de patentes internacionais, e de bibliografias. De posse dos resultados analisados, não foram identificadas tecnologias existentes que atendam ao objetivo direto do estudo, o que oportuniza a proposição de construção de um conjunto de dados sistematizado mais confiável, menos vulnerável e que favoreça o processo de transferência de tecnologia.

In this paper, the general analytical solution of the mechanical system consisting of an axisymmetric body spinning on a horizontal surface is analyzed. The motion equations are given by a three-dimensional system in which one of the equations is of second order. The invariance of the system under time translation is applied to reduce the order of the system by means of the classical Lie reduction method. As a result, a reduced autonomous first-order system is obtained. It is also explained how to recover the general analytical solution of the original system from the general solution of the reduced motion equations. Finally, some particular situations are considered with the goal of developing further the expression of the analytical solution found. The case of a spinning polar spheroid is also addressed.

This work illustrates how one could apply Lie point symmetries for finding the analytical solution to first-order mechanical systems. Although the classical Lie method constitutes a powerful tool for solving differential equations, an obstacle appears in the case of systems of first-order equations because they admit an infinite number of symmetries, and it is not possible to compute them by following a systematic procedure. To overcome this difficulty, we follow the idea exposed in Nucci (J. Math. Phys. 37: 1772–1775, 1996), Nucci (Electr. J. Diff. Eqn. 12: 87–101, 2005), Nucci (J. Math. Phys. 42: 746, 2001) consisting of transforming the original system to an equivalent system in which one of the equations is of second–order. The presented approach is applied to Hathaway’s circular pursuit problem, leading to the analytical solution of the system expressed in terms of the general solution to an Abel equation.

In this article, it is studied the mechanical system formed by a pendulum with two reaction wheels in which the friction torque is assumed to follow a Coulomb law. A qualitative analysis of the system is performed for the damped case. Specifically, the equilibrium points for the unforced pendulum are analyzed. Also, in the forced case, the conditions for which there exist asymptotically stable solutions are determined. In order to study the exact analytical solution of the unforced pendulum, we also perform a Lie symmetry analysis. In this regard, it is shown that the exact general solution of the system for null motor torques can be expressed in terms of the general solution to an Abel equation. In the unforced and undamped case, the exact general solution is obtained in explicit form and expressed in terms of the Jacobi elliptic function by using the Lie symmetry approach.

The aim of this work is to show how the moving frames method can be applied for reducing and solving two nonlinear mechanical systems: a bead on a rotating wire hoop and a spinning top. Once both problems are adequately formulated, we explicitly determine the corresponding moving frames associated to the symmetry group of transformations admitted by the systems. The knowledge of the moving frames for the action of the corresponding symmetry groups permits to perform order reductions. Furthermore, we are able to compute the general solutions to each problem from the general solutions of the corresponding reduced systems. Finally, we also discuss the connection of the presented approach with the classical method provided by the celebrated Noether's Theorem.

Given a Lie group of finite-dimensional transformations acting on a manifold, there is always an action known as a long-acting group action. This action describes the fundamental basis of the Lie theory connecting groups of symmetry in differential equations. Differential invariants emerge as constants of the action of the prolongation of a group. Élie Cartan extended this in the twentieth century involving the geometry of the action of this group, grounding the so-called moving frame theory. With this theory, various applications are possible and detailed in the literature, such as symmetries of variational problems, conservation laws, invariant differential forms, and group invariant solutions. In order to demonstrate the approach, two nonholonomic constrained mechanical systems are exemplified to obtain either the general closed-solution in explicit form, when possible, or an order reduction provided by the Lie symmetries via moving frames. The first example is a coin with mass m rolling without slipping and takes on an inclined plane (x, y) with angle $$\alpha $$ and nonlinear constraint. The second example is a Chetaev type described by a dog pursuing a man in a plane surface with a nonholonomic restriction. A full detailed analysis is addressed to define the Lie symmetries and the corresponding moving frames obtained in both examples.

The existence of Lie symmetries in differential equations can generate transformations in the dependent and independent variables and obtain new equations that may be easier to integrate. In particular, in some situations, one can reduce the order and it is possible to obtain first integrals. Thus, this article presents the application of the fundamental Lie theorem to obtain the complete solution of a classical nonlinear problem of the dynamics of mechanical systems: the bead on a rotating wire hoop. From the first integral obtained with the Lie symmetry generators, the exact solution can be found with the aid of the Jacobi elliptic functions.

A existência de simetrias em equações diferenciais pode gerar transformações em variáveis dependentes e independentes que facilitam a integração destas equações. Em especial, Sophus Lie desenvolveu no século XIX uma forma de extração de simetrias que podem ser usadas efetivamente para revelar as integrais primeiras, ou seja, as constantes de movimento, que muitas vezes podem estar escondidas. Estes invariantes podem em algumas situações ser identificados pelo teorema de Noether ou a partir de manipulações das próprias equações com transformações de Lie. Nos cursos iniciais de mecânica clássica, apesar de todo o formalismo em cima dos teoremas de conservação de energia e momento linear/angular, a relação disto com a existência de possíveis simetrias de Lie não é destacada de forma clara e objetiva. Neste sentido, o presente artigo busca apresentar uma introdução às simetrias de Lie usando uma linguagem acessível para um aluno de graduação de física, matemática ou engenharia com domínio básico em fundamentos de cálculo com várias variáveis. Para ilustrar a abordagem, considera-se um problema clássico de mecânica considerando um pião em regime de movimento com precessão estacionária. A partir das equações de movimento obtidas, as simetrias de Lie são identificadas e usadas na transformação para a redução da ordem. As integrais primeiras são obtidas a partir deste resultado com o Teorema de Noether, mostrando que neste exemplo e condição as simetrias de Lie também são simetrias de Noether. Por fim, a resolução das equações de movimento podem ser feitas usando funções elípticas de Jacobi para a obtenção dos ângulos de precessão, nutação e spin nas condições apresentadas.

Vibration energy harvesting has greatly expanded over the last few years. While linear vibration-based energy harvesting has received considerable attention in the literature, some current research is focused on the concept of purposeful inclusion of nonlinearities for broadband transduction. The energy harvesting process is performed in two steps: energy extraction and using this energy to feed electronic devices. Thus, this work discusses the use of an energy extraction device that is a chaotic non-linear mechanical device, coupled to a half-wave rectifier circuit to transform the alternating voltage generated by the piezoelectric ceramics into continuous voltage to power electronic devices. An analysis of the dynamic interaction between the two devices is done and it can be concluded that it is possible to use a mechanical device that operates in chaos coupled to a rectification circuit.

Common solution for energy harvester device is project design according harmonic excitation around natural frequency matching excitation source. Environment vibration is random and wide band causing short time of resonance in disagreement of project objectives. Control driven of energy harvesting to take advantage of the higher vibration range can results in greater energy converted. This study investigates a non-ideal excitation behavior and their efficiency in convert electricity via piezoelectric direct effect from the available system energy compared to harmonic excitation source. Numeric evaluation was performed based in bimorph piezoelectric beam in dimensionless consideration. Chaos behavior and harvest energy capability were compared from non-ideal to harmonic excitation. Results demonstrate considerable higher quantity of energy available for non-ideal approach compared to harmonic resonant design and encourages advance study to control and enhance energy from random and wide band vibration source.