3,339 research outputs found
Nonlinear Effects in the TGB_A Phase
We study the nonlinear interactions in the TGB_A phase by using a
rotationally invariant elastic free energy. By deforming a single grain
boundary so that the smectic layers undergo their rotation within a finite
interval, we construct a consistent three-dimensional structure. With this
structure we study the energetics and predict the ratio between the intragrain
and intergrain defect spacing, and compare our results with those from linear
elasticity and experiment.Comment: 4 pages, RevTeX, 2 included eps figure
Review of the Exponential and Cayley Map on SE(3) as relevant for Lie Group Integration of the Generalized Poisson Equation and Flexible Multibody Systems
The exponential and Cayley map on SE(3) are the prevailing coordinate maps
used in Lie group integration schemes for rigid body and flexible body systems.
Such geometric integrators are the Munthe-Kaas and generalized-alpha schemes,
which involve the differential and its directional derivative of the respective
coordinate map. Relevant closed form expressions, which were reported over the
last two decades, are scattered in the literature, and some are reported
without proof. This paper provides a reference summarizing all relevant closed
form relations along with the relevant proofs. including the right-trivialized
differential of the exponential and Cayley map and their directional
derivatives (resembling the Hessian). The latter gives rise to an implicit
generalized-alpha scheme for rigid/flexible multibody systems in terms of the
Cayley map with improved computational efficiency
A stiffness-based quality measure for compliant grasps and fixtures
This paper presents a systematic approach to quantifying the effectiveness of compliant grasps and fixtures of an object. The approach is physically motivated and applies to the grasping of two- and three-dimensional objects by any number of fingers. The approach is based on a characterization of the frame-invariant features of a grasp or fixture stiffness matrix. In particular, we define a set of frame-invariant characteristic stiffness parameters, and provide physical and geometric interpretation for these parameters. Using a physically meaningful scheme to make the rotational and translational stiffness parameters comparable, we define a frame-invariant quality measure, which we call the stiffness quality measure. An example of a frictional grasp illustrates the effectiveness of the quality measure. We then consider the optimal grasping of frictionless polygonal objects by three and four fingers. Such frictionless grasps are useful in high-load fixturing applications, and their relative simplicity allows an efficient computation of the globally optimal finger arrangement. We compute the optimal finger arrangement in several examples, and use these examples to discuss properties that characterize the stiffness quality measure
Relativistic Chasles' theorem and the conjugacy classes of the inhomogeneous Lorentz group
This work is devoted to the relativistic generalization of Chasles' theorem,
namely to the proof that every proper orthochronous isometry of Minkowski
spacetime, which sends some point to its chronological future, is generated
through the frame displacement of an observer which moves with constant
acceleration and constant angular velocity. The acceleration and angular
velocity can be chosen either aligned or perpendicular, and in the latter case
the angular velocity can be chosen equal or smaller than than the acceleration.
We start reviewing the classical Euler's and Chasles' theorems both in the Lie
algebra and group versions. We recall the relativistic generalization of
Euler's theorem and observe that every (infinitesimal) transformation can be
recovered from information of algebraic and geometric type, the former being
identified with the conjugacy class and the latter with some additional
geometric ingredients (the screw axis in the usual non-relativistic version).
Then the proper orthochronous inhomogeneous Lorentz Lie group is studied in
detail. We prove its exponentiality and identify a causal semigroup and the
corresponding Lie cone. Through the identification of new Ad-invariants we
classify the conjugacy classes, and show that those which admit a causal
representative have special physical significance. These results imply a
classification of the inequivalent Killing vector fields of Minkowski spacetime
which we express through simple representatives. Finally, we arrive at the
mentioned generalization of Chasles' theorem.Comment: Latex2e, 49 pages. v2: few typos correcte
A way of relating instantaneous and finite screws based on the screw triangle product
It has been a desire to unify the models for structural and parametric analyses and design in the field of robotic mechanisms. This requires a mathematical tool that enables analytical description, formulation and operation possible for both finite and instantaneous motions. This paper presents a method to investigate the algebraic structures of finite screws represented in a quasi-vector form and instantaneous screws represented in a vector form. By revisiting algebraic operations of screw compositions, this paper examines associativity and derivative properties of the screw triangle product of finite screws and produces a vigorous proof that a derivative of a screw triangle product can be expressed as a linear combination of instantaneous screws. It is proved that the entire set of finite screws forms an algebraic structure as Lie group under the screw triangle product and its time derivative at the initial pose forms the corresponding Lie algebra under the screw cross product, allowing the algebraic structures of finite screws in quasi-vector form and instantaneous screws in vector form to be revealed.
Triply-Periodic Smectics
Twist-grain-boundary phases in smectics are the geometrical analogs of the
Abrikosov flux lattice in superconductors. At large twist angles, the nonlinear
elasticity is important in evaluating their energetics. We analytically
construct the height function of a pi/2 twist-grain-boundary phase in smectic-A
liquid crystals, known as Schnerk's first surface. This construction, utilizing
elliptic functions, allows us to compute the energy of the structure
analytically. By identifying a set of heretofore unknown defects along the
pitch axis of the structure, we study the necessary topological structure of
grain boundaries at other angles, concluding that there exist a set of
privileged angles and that the \pi/2 and \pi/3 grain boundary structures are
particularly simple.Comment: 13 pages, 7 figure
On the Generation of Smooth Three-Dimensional Rigid Body Motions
This paper addresses the problem of generating smooth trajectories between an initial and final position and orientation in space. The main idea is to define a functional depending on velocity or its derivatives that measures the smoothness of a trajectory and find trajectories that minimize this functional. In order to ensure that the computed trajectories are independent of the parameterization of positions and orientations, we use the notions of Riemannian metric and covariant derivatives from differential geometry and formulate the problem as a variational problem on the Lie group of spatial rigid body displacements, SE (3). We show that by choosing an appropriate measure of smoothness, the trajectories can be made to satisfy boundary conditions on the velocities or higher order derivatives. Dynamically smooth trajectories can be obtained by incorporating the inertia of the system into the definition of the Riemannian metric. We state the necessary conditions for the shortest distance, minimum acceleration and minimum jerk trajectories. Analytical expressions for the smooth trajectories are derived for some special cases. We also provide several examples of the general case where the trajectories are computed numerically
Volterra Distortions, Spinning Strings, and Cosmic Defects
Cosmic strings, as topological spacetime defects, show striking resemblance
to defects in solid continua: distortions, which can be classified into
disclinations and dislocations, are line-like defects characterized by a delta
function-valued curvature and torsion distribution giving rise to rotational
and translational holonomy. We exploit this analogy and investigate how
distortions can be adapted in a systematic manner from solid state systems to
Einstein-Cartan gravity. As distortions are efficiently described within the
framework of a SO(3) {\rlap{\supset}\times}} T(3) gauge theory of solid
continua with line defects, we are led in a straightforward way to a Poincar\'e
gauge approach to gravity which is a natural framework for introducing the
notion of distorted spacetimes. Constructing all ten possible distorted
spacetimes, we recover, inter alia, the well-known exterior spacetime of a
spin-polarized cosmic string as a special case of such a geometry. In a second
step, we search for matter distributions which, in Einstein-Cartan gravity, act
as sources of distorted spacetimes. The resulting solutions, appropriately
matched to the distorted vacua, are cylindrically symmetric and are interpreted
as spin-polarized cosmic strings and cosmic dislocations.Comment: 24 pages, LaTeX, 9 eps figures; remarks on energy conditions added,
discussion extended, version to be published in Class. Quantum Gra
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