A treatise upon analytical mechanics : being the first book of the Mécanique céleste /

On spine: Laplace's mechanics.

Analytical mechanics for engineers,

Mode of access: Internet.

A treatise on infinitesimal calculus, containing differential and integral calculus, calculus of variations, applications to algebra and geometry, and analytical mechanics.

Vol. 3 and 4 form the author's Treatise on analytical mechanics.

Um esboço da história do conceito de trabalho virtual e suas aplicações

Neste trabalho abordamos a questão concernente à origem do princípio de trabalho virtual e sua consolidação como um dos conceitos fundamentais no estudo da mecânica analítica e, em particular, dos sistemas em equilíbrio estático. Ênfase foi dada às contribuições seminais de Stevin, Galileu e, sobretudo, as de d'Alembert e Lagrange, no tocante ao conceito de trabalho virtual. Além disso, faz-se um comentário geral sobre vínculos holônomos e deslocamento virtual. Alguns exemplos de emprego da equação de d'Alembert-Lagrange são apresentados, para mostrar como o princípio de trabalho virtual pode ser adequadamente aplicado.

A contribution for developing more efficient dynamic modelling algorithms of parallel robots

Parallel kinematic structures are considered very adequate architectures for positioning and orienti ng the tools of robotic mechanisms. However, developing dynamic models for this kind of systems is sometimes a difficult task. In fact, the direct application of traditional methods of robotics, for modelling and analysing such systems, usually does not lead to efficient and systematic algorithms. This work addre sses this issue: to present a modular approach to generate the dynamic model and through some convenient modifications, how we can make these methods more applicable to parallel structures as well. Kane’s formulati on to obtain the dynamic equations is shown to be one of the easiest ways to deal with redundant coordinates and kinematic constraints, so that a suitable c hoice of a set of coordinates allows the remaining of the modelling procedure to be computer aided. The advantages of this approach are discussed in the modelling of a 3-dof parallel asymmetric mechanisms.

A contribution on modeling methodologies for multibody systems.

Multibody System Dynamics has been responsible for revolutionizing Mechanical Engineering Design by using mathematical models to simulate and optimize the dynamic behavior of a wide range of mechanical systems. These mathematical models not only can provide valuable informations about a system that could otherwise be obtained only by experiments with prototypes, but also have been responsible for the development of many model-based control systems. This work represents a contribution for dynamic modeling of multibody mechanical systems by developing a novel recursive modular methodology that unifies the main contributions of several Classical Mechanics formalisms. The reason for proposing such a methodology is to motivate the implementation of computational routines for modeling complex multibody mechanical systems without being dependent on closed source software and, consequently, to contribute for the teaching of Multibody System Dynamics in undergraduate and graduate levels. All the theoretical developments are based on and motivated by a critical literature review, leading to a general matrix form of the dynamic equations of motion of a multibody mechanical system (that can be expressed in terms of any set of variables adopted for the description of motions performed by the system, even if such a set includes redundant variables) and to a general recursive methodology for obtaining mathematical models of complex systems given a set of equations describing the dynamics of each of its uncoupled subsystems and another set describing the constraints among these subsystems in the assembled system. This work also includes some discussions on the description of motion (using any possible set of motion variables and admitting any kind of constraint that can be expressed by an invariant), and on the conditions for solving forward and inverse dynamics problems given a mathematical model of a multibody system. Finally, some examples of computational packages based on the novel methodology, along with some case studies, are presented, highlighting the contributions that can be achieved by using the proposed methodology.

Comparison between analytical and 3D finite element solutions for borehole stresses in anisotropic elastic rock

We present a rigorous validation of the analytical Amadei solution for the stress concentration around an arbitrarily orientated borehole in general anisotropic elastic media. First, we revisit the theoretical framework of the Amadei solution and present analytical insights that show that the solution does indeed contain all special cases of symmetry, contrary to previous understanding, provided that the reduced strain coefficients b11 and b55 are not equal. It is shown from theoretical considerations and published experimental data that the b11 and b55 are not equal for realistic rocks. Second, we develop a 3D finite element elastic model within a hybrid analytical–numerical workflow that circumvents the need to rebuild and remesh the model for every borehole and material orientation. Third, we show that the borehole stresses computed from the numerical model and the analytical solution match almost perfectly for different borehole orientations (vertical, deviated and horizontal) and for several cases involving isotropic, transverse isotropic and orthorhombic symmetries. It is concluded that the analytical Amadei solution is valid with no restriction on the borehole orientation or the symmetry of the elastic anisotropy.

An analytical framework for quantifying aquifer response time scales associated with transient boundary conditions

A major challenge in studying coupled groundwater and surface-water interactions arises from the considerable difference in the response time scales of groundwater and surface-water systems affected by external forcings. Although coupled models representing the interaction of groundwater and surface-water systems have been studied for over a century, most have focused on groundwater quantity or quality issues rather than response time. In this study, we present an analytical framework, based on the concept of mean action time (MAT), to estimate the time scale required for groundwater systems to respond to changes in surface-water conditions. MAT can be used to estimate the transient response time scale by analyzing the governing mathematical model. This framework does not require any form of transient solution (either numerical or analytical) to the governing equation, yet it provides a closed form mathematical relationship for the response time as a function of the aquifer geometry, boundary conditions, and flow parameters. Our analysis indicates that aquifer systems have three fundamental time scales: (i) a time scale that depends on the intrinsic properties of the aquifer; (ii) a time scale that depends on the intrinsic properties of the boundary condition, and; (iii) a time scale that depends on the properties of the entire system. We discuss two practical scenarios where MAT estimates provide useful insights and we test the MAT predictions using new laboratory-scale experimental data sets.

Revisiting borehole stresses in anisotropic elastic media : comparison of analytical versus numerical solutions

We present a rigorous validation of the analyticalAmadei solution for the stress concentration around arbitrarily orientated borehole in general anisotropic elastic media. First, we revisit the theoretical framework of the Amadei solution and present analytical insights that show that the solution does indeed contain all special cases of symmetry, contrary to previous understanding, provided that the reduced strain coefficients β11 and β55 are not equal. It is shown from theoretical considerations and published experimental data that the β11 and β55 are not equal for realistic rocks. Second, we develop a 3D finite-element elastic model within a hybrid analyticalnumerical workflow that circumvents the need to rebuild and remesh the model for every borehole and material orientation. Third, we show that the borehole stresses computed from the numerical model and the analytical solution match almost perfectly for different borehole orientations (vertical, deviated and horizontal) and for several cases involving isotropic and transverse isotropic symmetries. It is concluded that the analytical Amadei solution is valid with no restrictions on the borehole orientation or elastic anisotropy symmetry.

Analytical estimation of stress intensity factors in patched cracked plates

The fatigue and fracture performance of a cracked plate can be substantially improved by providing patches as reinforcements. The effectiveness of the patches is related to the reduction they cause in the stress intensity factor (SIF) of the crack. So, for reliable design, one needs an accurate evaluation of the SIF in terms of the crack, patch and adhesive parameters. In this investigation, a centrally cracked large plate with a pair of symmetric bonded narrow patches, oriented normally to the crack line, is analysed by a continuum approach. The narrow patches are treated as transversely flexible line members. The formulation leads to an integral equation which is solved numerically using point collocation. The convergence is rapid. It is found that substantial reductions in SIF are possible with practicable patch dimensions and locations. The patch is more effective when placed on the crack than ahead of the crack. The present analysis indicates that a little distance inwards of the crack tip, not the crack tip itself, is the ideal location, for the patch.

Prediction of masonry compressive behaviour using a damage mechanics inspired modelling method

Masonry under compression is affected by the properties of its constituents and their interfaces. In spite of extensive investigations of the behaviour of masonry under compression, the information in the literature cannot be regarded as comprehensive due to ongoing inventions of new generation products – for example, polymer modified thin layer mortared masonry and drystack masonry. As comprehensive experimental studies are very expensive, an analytical model inspired by damage mechanics is developed and applied to the prediction of the compressive behaviour of masonry in this paper. The model incorporates a parabolic progressively softening stress-strain curve for the units and a progressively stiffening stress-strain curve until a threshold strain for the combined mortar and the unit-mortar interfaces is reached. The model simulates the mutual constraints imposed by each of these constituents through their respective tensile and compressive behaviour and volumetric changes. The advantage of the model is that it requires only the properties of the constituents and considers masonry as a continuum and computes the average properties of the composite masonry prisms/wallettes; it does not require discretisation of prism or wallette similar to the finite element methods. The capability of the model in capturing the phenomenological behaviour of masonry with appropriate elastic response, stiffness degradation and post peak softening is presented through numerical examples. The fitting of the experimental data to the model parameters is demonstrated through calibration of some selected test data on units and mortar from the literature; the calibrated model is shown to predict the responses of the experimentally determined masonry built using the corresponding units and mortar quite well. Through a series of sensitivity studies, the model is also shown to predict the masonry strength appropriately for changes to the properties of the units and mortar, the mortar joint thickness and the ratio of the height of unit to mortar joint thickness. The unit strength is shown to affect the masonry strength significantly. Although the mortar strength has only a marginal effect, reduction in mortar joint thickness is shown to have a profound effect on the masonry strength. The results obtained from the model are compared with the various provisions in the Australian Masonry Structures Standard AS3700 (2011) and Eurocode 6.

Motion of a bore over a sloping beach: an approximate analytical approach

The motion of a bore over a sloping beach, earlier considered numerically by Keller, Levine & Whitham (1960), is studied by an approximate analytic technique. This technique is an extension of Whitham's (1958) approach for the propagation of shocks into a non-uniform medium. It gives the entire flow behind the bore and is shown to be equivalent to the theory of modulated simple waves of Varley, Ventakaraman & Cumberbatch (1971).

A semi-analytical particle filter for identification of nonlinear oscillators

Particle filters find important applications in the problems of state and parameter estimations of dynamical systems of engineering interest. Since a typical filtering algorithm involves Monte Carlo simulations of the process equations, sample variance of the estimator is inversely proportional to the number of particles. The sample variance may be reduced if one uses a Rao-Blackwell marginalization of states and performs analytical computations as much as possible. In this work, we propose a semi-analytical particle filter, requiring no Rao-Blackwell marginalization, for state and parameter estimations of nonlinear dynamical systems with additively Gaussian process/observation noises. Through local linearizations of the nonlinear drift fields in the process/observation equations via explicit Ito-Taylor expansions, the given nonlinear system is transformed into an ensemble of locally linearized systems. Using the most recent observation, conditionally Gaussian posterior density functions of the linearized systems are analytically obtained through the Kalman filter. This information is further exploited within the particle filter algorithm for obtaining samples from the optimal posterior density of the states. The potential of the method in state/parameter estimations is demonstrated through numerical illustrations for a few nonlinear oscillators. The proposed filter is found to yield estimates with reduced sample variance and improved accuracy vis-a-vis results from a form of sequential importance sampling filter.