Resumo A forma mais comum de definir matematicamente o momento angular é através da operação de produto vetorial como definido por Josiah Willard Gibbs e ao mesmo tempo por Oliver Heaviside. Neste sentido, a contribuição deste artigo é mostrar que a definição do momento angular também pode ser feita usando outros sistemas algébricos existentes e menos conhecidos da maioria dos estudantes, como a álgebra de Hamilton, a álgebra de Grassmann e a álgebra geométrica de Clifford. Por fim, realizamos uma comparação crítica entre estas definições e o seus significados.

Abstract The most common way to mathematically define the angular moment is through the vector product operation as defined by Josiah Willard Gibbs and at the same time by Oliver Heaviside. This way, the contribution of this article is to show that the definition of angular moment can also be made using other algebraic systems existing and less known to most students, such as Hamilton algebra, Grassmann algebra and Clifford geometric algebra. Finally, we make a critical comparison between these definitions and their meanings.

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.

The existence of symmetries in differential equations can generate transformations of dependent and independent variables that facilitate the integration of these equations. In the nineteenth century, Sophus Lie developed a method of extracting symmetries that can be used effectively to reveal first integrals of a differential equation. These invariants can in some situations be identified by the Noether theorem or from manipulating the equations themselves with Lie transformations. Despite the formalism over conservation theorems for energy and linear/angular momentum, initial courses in classical mechanics often do not clearly or objectively highlight the relationship between conservation laws and the existence of possible Lie symmetries. For this reason, we seek to present an introduction to Lie symmetries using a language accessible to a graduate student in physics, mathematics, or engineering with basic mastery of classical mechanics in several variables. In order to illustrate the approach, we consider the classical problem of a spinning top with stationary precession. From the equations of motion, the Lie symmetries are identified and used in a transformation that results in an order reduction. The first integrals are obtained from this result using the Noether theorem, and we illustrate that the Lie symmetries for this problem are also Noether symmetries. Finally, the solution to the equations of motion are written using Jacobi elliptic functions, and from this we obtain the precession, nutation and spin angles under the presented conditions.

Abstract In this study, the discrete-time Volterra series are used to update parameters in a nonlinear finite element model. The main idea of the Volterra series is to describe the discrete-time output of a nonlinear system using multidimensional convolutions between the Volterra kernels represented in a Kautz orthogonal basis and the excitations. A metric based on the residue between the experimental and the numerical Volterra kernels is used to identify the parameters of the numerical model. First, the identification of the linear parameters is performed using a metric based only on the first order Volterra kernels. Then the nonlinear parameters are identified through a metric based on the higher-order kernels. The originality of this nonlinear updating method stems from the decoupling of linear and nonlinear parameters and the use of global nonlinear model. In order to put in light the applicability of this technique, this work focus on the identification of the parameters in a nonlinear finite element model of a beam that was preloaded by compression mechanism. This work shows that the updated numerical model was able to represent the behaviour observed in the experimental measurements.

A parametrização da velocidade angular instantânea de um corpo rígido é usualmente descrita nos cursos de mecânica clássica com o auxílio dos ângulos de Euler. No entanto, para evitar problemas de singularidade na integração numérica das equações de movimento de rotação, outras formas de parametrização são mais interessantes. Dentre as diversas opções, os quatérnions são uma escolha bastante popular e com forte apelo prático na parametrização e integração das equações de Euler. Neste sentido, o presente artigo contribui com uma apresentação didática e comparação da integração numérica das equações de Euler para descrição do movimento de um giroscópio com a cinemática inversa solucionada via ângulos de Euler e por quatérnions, que não são comumente descritas em livros didáticos clássicos em um primeiro curso de mecânica. Busca-se mostrar que além deste trunfo numérico, a integração das equações de movimento com parametrização via quatérnions também pode ser usada com um fim didático mais nobre: introduzir e motivar os estudantes de um curso de mecânica, tanto físicos como engenheiros, no uso de outros sistemas algébricos mais avançados, como a Álgebra Geométrica (Álgebra de Clifford).

The angular velocity of a rigid body is usually described in classical mechanics courses by using Euler angles. However, in order to avoid singularity problems on the numerical integration of the motion equations, another ways are more interesting. Among them, the quaternions are a very popular choice and with a strong practical appeal on the parameterization and integration of the Euler's equations. In this sense, this paper proposes a comparison between the numerical integration of the Euler equations of a gyroscope with the inverse kinematics solved through Euler angles and by quaternions, that are not commonly described in classical mechanics textbooks on a first mechanical course. The goal is to show that apart the numerical asset, the integration of the motion equations with the quaternions parameterization also can be used with a noblest didactical way: to introduce and motivate the students with the use of others more advanced algebraic systems, as Geometric Algebra (Clifford's Algebra).

Este estudo tem por objetivo identificar o efeito do armazenamento térmico na manutenção da qualidade, bem como, no controle do escurecimento da polpa durante o armazenamento refrigerado de cultivares de pêssegos tardios. Os frutos foram colhidos fisiologicamente maturos. Os frutos foram submetidos às temperaturas de 20 ± 0,5ºC e 75 ± 3% de UR, em câmara frigorífica, durante 0, 24 ou 48. Após, os frutos foram mantidos a 0 ± 0,5ºC e 92 ± 3% de UR, por 28 dias. Ao final do experimento, os frutos das cultivares de polpa amarela, independentemente de armazenamento refrigerado, não apresentaram sinais de escurecimento interno, apresentaram qualidade adequada para comercialização. Quando não submetidos ao armazenamento refrigerado, após 14 dia de armazenamento, mais dois sem refrigeração, os frutos de polpa branca apresentaram escurecimento da polpa. A partir dos resultados observados na análise de sólidos solúveis, acidez titulável e desenvolvimento de podridões, esses frutos apresentaram um maior nível de amadurecimento. O condicionamento térmico de 48 horas, nas duas cultivares, resultou em um amadurecimento maior e apodrecimento dos frutos. Armazenamento refrigerado por 24 horas foi eficiente no controle de escurecimento da polpa e na manutenção da qualidade das cultivares de pêssego de polpa branca.

The purpose of this study was to identify the effect of delayed storage on the maintenance of the quality and on the control of pulp browning during cold storage of late peach cultivars. The fruit was harvested at the mature-green stage. All cultivars were exposed to temperatures of 20 ± 0.5ºC and 75 ± 3% of moisture in a cold room for 0.24, or 48 hours. Afterwards, the peaches were kept at 0 ± 0.5ºC and 92 ± 3% of R.H. for 28 days. In the end, the yellow pulp cultivars, regardless of temperature and length of delayed storage, did not present signs of internal browning and were of adequate quality for commercialization. When they were not subjected to delayed storage, the white pulp peaches presented pulp browning after 14 days of cold storage plus two days out of the cold room. From the results observed in the analysis of soluble solids, titratable acidity and decay development, these peaches showed an increased level of ripening. In both cultivars, delayed storage for 48 hours resulted in increased ripening and decay. Delayed storage for 24 hours was effective in controlling pulp browning and in maintaining the quality of the white pulp peach cultivars.

The aim of this paper is to perform a comparative study between different distance measures or metrics for use in nonlinear model updating using vibration test data. Four metrics derived from both frequency and time domain updating approaches are studied, including the harmonic balance method, the constitutive equation error, the restoring force surface and the Karhunen-Loève decomposition. In the first section, a benchmark model with local nonlinear stiffness is defined in order to illustrate each method. Secondly, each nonlinear updating metric is succinctly reviewed. Finally, the relative performances of the different metrics are investigated based on numerical simulations. These results allow us to characterize the applicability and limitations of the different approaches.

Structural health monitoring (SHM) is related to the ability of monitoring the state and deciding the level of damage or deterioration within aerospace, civil and mechanical systems. In this sense, this paper deals with the application of a two-step auto-regressive and auto-regressive with exogenous inputs (AR-ARX) model for linear prediction of damage diagnosis in structural systems. This damage detection algorithm is based on the monitoring of residual error as damage-sensitive indexes, obtained through vibration response measurements. In complex structures there are many positions under observation and a large amount of data to be handed, making difficult the visualization of the signals. This paper also investigates data compression by using principal component analysis. In order to establish a threshold value, a fuzzy c-means clustering is taken to quantify the damage-sensitive index in an unsupervised learning mode. Tests are made in a benchmark problem, as proposed by IASC-ASCE with different damage patterns. The diagnosis that was obtained showed high correlation with the actual integrity state of the structure.

Flutter is an in-flight vibration of flexible structures caused by energy in the airstream absorbed by the lifting surface. This aeroelastic phenomenon is a problem of considerable interest in the aeronautic industry, because flutter is a potentially destructive instability resulting from an interaction between aerodynamic, inertial, and elastic forces. To overcome this effect, it is possible to use passive or active methodologies, but passive control adds mass to the structure and it is, therefore, undesirable. Thus, in this paper, the goal is to use linear matrix inequalities (LMIs) techniques to design an active state-feedback control to suppress flutter. Due to unmeasurable aerodynamic-lag states, one needs to use a dynamic observer. So, LMIs also were applied to design a state-estimator. The simulated model consists of a classical flat plate in a two-dimensional flow. Two regulators were designed, the first one is a non-robust design for parametric variation and the second one is a robust control design, both designed by using LMIs. The parametric uncertainties are modeled through polytopic uncertainties. The paper concludes with numerical simulations for each controller. The open-loop and closed-loop responses are also compared and the results show the flutter suppression. The perfomance for both controllers are compared and discussed.

The study of algorithms for active vibrations control in flexible structures became an area of enormous interest, mainly due to the countless demands of an optimal performance of mechanical systems as aircraft and aerospace structures. Smart structures, formed by a structure base, coupled with piezoelectric actuators and sensor are capable to guarantee the conditions demanded through the application of several types of controllers. This article shows some steps that should be followed in the design of a smart structure. It is discussed: the optimal placement of actuators, the model reduction and the controller design through techniques involving linear matrix inequalities (LMI). It is considered as constraints in LMI: the decay rate, voltage input limitation in the actuators and bounded output peak (output energy). Two controllers robust to parametric variation are designed: the first one considers the actuator in non-optimal location and the second one the actuator is put in an optimal placement. The performance are compared and discussed. The simulations to illustrate the methodology are made with a cantilever beam with bonded piezoelectric actuators.