On the non-linear stability theory of spinning solid bodies with liquid-filled cavities and its application to the columbus problem

In this paper, we first present a system of differential-integral equations for the largedisturbance to the general case that any arbitrarily shaped solid body with a cavity contain-ing viscous liquid rotates uniformly around the principal axis of inertia, and then develop aweakly non-linear stability theory by the Lyapunov direct approach. Applying this theoryto the Columbus problem, we have proved the consistency between the theory and Kelvin'sexperiments.

Fully Comprehensive Geometrically Non-Linear Dynamic Analysis of Multi-Body Beam Systems with Elastic Couplings

This paper is concerned with the dynamic analysis of flexible,non-linear multi-body beam systems. The focus is on problems where the strains within each elastic body (beam) remain small. Based on geometrically non-linear elasticity theory, the non-linear 3-D beam problem splits into either a linear or non-linear 2-D analysis of the beam cross-section and a non-linear 1-D analysis along the beam reference line. The splitting of the three-dimensional beam problem into two- and one-dimensional parts, called dimensional reduction,results in a tremendous savings of computational effort relative to the cost of three-dimensional finite element analysis,the only alternative for realistic beams. The analysis of beam-like structures made of laminated composite materials requires a much more complicated methodology. Hence, the analysis procedure based on Variational Asymptotic Method (VAM), a tool to carry out the dimensional reduction, is used here.The analysis methodology can be viewed as a 3-step procedure. First, the sectional properties of beams made of composite materials are determined either based on an asymptotic procedure that involves a 2-D finite element nonlinear analysis of the beam cross-section to capture trapeze effect or using strip-like beam analysis, starting from Classical Laminated Shell Theory (CLST). Second, the dynamic response of non-linear, flexible multi-body beam systems is simulated within the framework of energy-preserving and energy-decaying time integration schemes that provide unconditional stability for non-linear beam systems. Finally,local 3-D responses in the beams are recovered, based on the 1-D responses predicted in the second step. Numerical examples are presented and results from this analysis are compared with those available in the literature.

Geometrically non-linear dynamics of composite four-bar mechanisms

This work intends to demonstrate the importance of a geometrically nonlinear cross-sectional analysis of certain composite beam-based four-bar mechanisms in predicting system dynamic characteristics. All component bars of the mechanism are made of fiber reinforced laminates and have thin rectangular cross-sections. They could, in general, be pre-twisted and/or possess initial curvature, either by design or by defect. They are linked to each other by means of revolute joints. We restrict ourselves to linear materials with small strains within each elastic body (beam). Each component of the mechanism is modeled as a beam based on geometrically non-linear 3-D elasticity theory. The component problems are thus split into 2-D analyses of reference beam cross-sections and non-linear 1-D analyses along the three beam reference curves. For the thin rectangular cross-sections considered here, the 2-D cross-sectional non-linearity is also overwhelming. This can be perceived from the fact that such sections constitute a limiting case between thin-walled open and closed sections, thus inviting the non-linear phenomena observed in both. The strong elastic couplings of anisotropic composite laminates complicate the model further. However, a powerful mathematical tool called the Variational Asymptotic Method (VAM) not only enables such a dimensional reduction, but also provides asymptotically correct analytical solutions to the non-linear cross-sectional analysis. Such closed-form solutions are used here in conjunction with numerical techniques for the rest of the problem to predict multi-body dynamic responses more quickly and accurately than would otherwise be possible. The analysis methodology can be viewed as a three-step procedure: First, the cross-sectional properties of each bar of the mechanism is determined analytically based on an asymptotic procedure, starting from Classical Laminated Shell Theory (CLST) and taking advantage of its thin strip geometry. Second, the dynamic response of the non-linear, flexible four-bar mechanism is simulated by treating each bar as a 1-D beam, discretized using finite elements, and employing energy-preserving and -decaying time integration schemes for unconditional stability. Finally, local 3-D deformations and stresses in the entire system are recovered, based on the 1-D responses predicted in the previous step. With the model, tools and procedure in place, we identify and investigate a few four-bar mechanism problems where the cross-sectional non-linearities are significant in predicting better and critical system dynamic characteristics. This is carried out by varying stacking sequences (i.e. the arrangement of ply orientations within a laminate) and material properties, and speculating on the dominating diagonal and coupling terms in the closed-form non-linear beam stiffness matrix. A numerical example is presented which illustrates the importance of 2-D cross-sectional non-linearities and the behavior of the system is also observed by using commercial software (I-DEAS + NASTRAN + ADAMS). (C) 2012 Elsevier Ltd. All rights reserved.

Relating Crevassing to Non-Linear Strain in the Floating Part of Jakobshavn Isbrae, West Greenland

Jakobshavn Isbrae is a major ice stream that drains the west-central Greenland ice sheet and becomes afloat in Jakobshavn Isfiord (69degreesN, 49degreesW), where it has maintained the world's fastest-known sustained velocity and calving rate (7 km a(-1)) for at least four decades. The floating portion is approximately 12 km long and 6 km wide. Surface elevations and motion vectors were determined photogrammetrically for about 500 crevasses on the floating ice, and adjacent grounded ice, using aerial photographs obtained 2 weeks apart in July 1985. Surface strain rates were computed from a mesh of 399 quadrilateral elements having velocity measurements at each corner. It is shown that heavy crevassing of floating ice invalidates the assumptions of linear strain theory that (i) surface strain in the floating ice is homogeneous in both space and time, (ii) the squares and products of strain components are nil, and (iii) first- and second-order rotation components are small compared to strain components. Therefore, strain rates and rotation rates were also computed using non-linear strain theory. The percentage difference between computed linear and non-linear second invariants of strain rate per element were greatest (mostly in the range 40-70%) where crevassing is greatest. Isopleths of strain rate parallel and transverse to flow and elevation isopleths relate crevassing to known and inferred pinning points.

Simulation de la nage anguilliforme

Ce document traite premièrement des diverses tentatives de modélisation et de simulation de la nage anguilliforme puis élabore une nouvelle technique, basée sur la méthode de la frontière immergée généralisée et la théorie des poutres de Reissner-Simo. Cette dernière, comme les équations des fluides polaires, est dérivée de la mécanique des milieux continus puis les équations obtenues sont discrétisées afin de les amener à une résolution numérique. Pour la première fois, la théorie des schémas de Runge-Kutta additifs est combinée à celle des schémas de Runge-Kutta-Munthe-Kaas pour engendrer une méthode d’ordre de convergence formel arbitraire. De plus, les opérations d’interpolation et d’étalement sont traitées d’un nouveau point de vue qui suggère l’usage des splines interpolatoires nodales en lieu et place des fonctions d’étalement traditionnelles. Enfin, de nombreuses vérifications numériques sont faites avant de considérer les simulations de la nage.

Theory of Non-linear Waves in Multi-dimensions: with Special Reference to Surface Water Waves

The surface water waves are "modal" waves in which the "physical space" (t, x, y, z) is the product of a propagation space (t, x, y) and a cross space, the z-axis in the vertical direction. We have derived a new set of equations for the long waves in shallow water in the propagation space. When the ratio of the amplitude of the disturbance to the depth of the water is small, these equations reduce to the equations derived by Whitham (1967) by the variational principle. Then we have derived a single equation in (t, x, y)-space which is a generalization of the fourth order Boussinesq equation for one-dimensional waves. In the neighbourhood of a wave froat, this equation reduces to the multidimensional generalization of the KdV equation derived by Shen & Keller (1973). We have also included a systematic discussion of the orders of the various non-dimensional parameters. This is followed by a presentation of a general theory of approximating a system of quasi-linear equations following one of the modes. When we apply this general method to the surface water wave equations in the propagation space, we get the Shen-Keller equation.

Non-linear theory of a positive-column in a magnetic-field

A nonlinear theory of an intermediate pressure discharge column in a magnetic field is presented. Motion of the neutral gas is considered. The continuity and momentum transfer equations for charged particles and neutral particles are solved by numerical methods. The main result obtained is that the rotating velocities of ionic gas and neutral gas are approximately equal. Bohm's criterion and potential inversion in the presence of neutral gas motion are also discussed.

Non-linear electronic transport in Pt nanowires deposited by focused ion beam

Electrical transport and structural properties of platinum nanowires, deposited using the focussed ion beam method have been investigated. Energy dispersive X-ray spectroscopy reveals metal-rich grains (atomic composition 31% Pt and 50% Ga) in a largely non-metallic matrix of C, O and Si. Resistivity measurements (15-300 K) reveal a negative temperature coefficient with the room-temperature resistivity 80-300 times higher than that of bulk Pt. Temperature dependent current-voltage characteristics exhibit non-linear behaviour in the entire range investigated. The conductance spectra indicate increasing non-linearity with decreasing temperature, reaching 4% at 15 K. The observed electrical behaviour is explained in terms of a model for inter-grain tunnelling in disordered media, a mechanism that is consistent with the strongly disordered nature of the nanowires observed in the structure and composition analysis.

K-Theory of non-linear projective toric varieties

We define a category of quasi-coherent sheaves of topological spaces on projective toric varieties and prove a splitting result for its algebraic K-theory, generalising earlier results for projective spaces. The splitting is expressed in terms of the number of interior lattice points of dilations of a polytope associated to the variety. The proof uses combinatorial and geometrical results on polytopal complexes. The same methods also give an elementary explicit calculation of the cohomology groups of a projective toric variety over any commutative ring.