Self-organization into quantized eigenstates of a classical wave driven particle

A growing number of dynamical situations involve the coupling of particles or singularities with physical waves. In principle these situations are very far from the wave-particle duality at quantum scale where the wave is probabilistic by nature. Yet some dual characteristics were observed in a system where a macroscopic droplet is guided by a pilot-wave it generates. Here we investigate the behaviour of these entities when confined in a two-dimensional harmonic potential well. A discrete set of stable orbits is observed, in the shape of successive generalized Cassinian-like curves (circles, ovals, lemniscates, trefoils...). Along these specific trajectories, the droplet motion is characterized by a double quantization of the orbit spatial extent and of the angular momentum. We show that these trajectories are intertwined with the dynamical build-up of central wave-field modes. These dual self-organized modes form a basis of eigenstates on which more complex motions are naturally decomposed

Deformable orthogonal grids: lemniscates

AbstractIn this paper we describe a technique, based on complex polynomials, for creating plane regions with a hole and propose a new method to produce an orthogonal grid on it. The thickness of the grid can be easily controlled and the sizes of the cells can be automatically estimated. The grid is automatically adapted to the boundary of the region. We offer parameters for the control of the geometric shape of the region, which depend on the roots of the polynomial and its derivative

Movement Timing and Invariance Arise from Several Geometries

Human movements show several prominent features; movement duration is nearly independent of movement size (the isochrony principle), instantaneous speed depends on movement curvature (captured by the 2/3 power law), and complex movements are composed of simpler elements (movement compositionality). No existing theory can successfully account for all of these features, and the nature of the underlying motion primitives is still unknown. Also unknown is how the brain selects movement duration. Here we present a new theory of movement timing based on geometrical invariance. We propose that movement duration and compositionality arise from cooperation among Euclidian, equi-affine and full affine geometries. Each geometry posses a canonical measure of distance along curves, an invariant arc-length parameter. We suggest that for continuous movements, the actual movement duration reflects a particular tensorial mixture of these canonical parameters. Near geometrical singularities, specific combinations are selected to compensate for time expansion or compression in individual parameters. The theory was mathematically formulated using Cartan's moving frame method. Its predictions were tested on three data sets: drawings of elliptical curves, locomotion and drawing trajectories of complex figural forms (cloverleaves, lemniscates and limaçons, with varying ratios between the sizes of the large versus the small loops). Our theory accounted well for the kinematic and temporal features of these movements, in most cases better than the constrained Minimum Jerk model, even when taking into account the number of estimated free parameters. During both drawing and locomotion equi-affine geometry was the most dominant geometry, with affine geometry second most important during drawing; Euclidian geometry was second most important during locomotion. We further discuss the implications of this theory: the origin of the dominance of equi-affine geometry, the possibility that the brain uses different mixtures of these geometries to encode movement duration and speed, and the ontogeny of such representations

Approximating Generalised Cornu Spiral With Low Energy Curve

The computations of visual pleasing and mathematically fair curve are an ongoing process. In the earlier literature, researchers have been studying the smooth curve, or in a more technically term, the fair curve through the physical approach known as elastica, Elastica means a thin strip of elastic material. Daniel Bernoulli stated the elastica as variational problem in terms of strain energy and Malcom (1977) mentioned that the simplest way to characterize a spline mathematically is with the fact that a spline assumes a shape which minimizes its elastic strain energy

Re-experiencing Engineering Inventions within a Modern Virtual Environmen

This paper describes a personal journey in the world of Mathematics and Mechanics and the actual process of reinventing devices with focus on their mathematical and mechanical properties. A set of important accompanying skills have been identified along the path of constructing these devices

Evaluation of Quasi-Static Indentation Damage in Aluminum Honeycomb Core - Graphite/Epoxy Sandwich Structures

Sandwich composites utilize a low density core and relatively stiff face sheets. These structures are ideal for applications that require high compressive strength, high bending stiffness, and very low weight such as aerospace vehicles. However, one problem with sandwich composites is their susceptibility to low velocity impact damage. Low velocity impacts result in both external damage, in the form of dents, and internal damage, in the form of core crushing, face sheet delaminations (two adjacent plies separating from one another), fiber fractures and matrix cracks. In general, it is assumed that visibly evident damage will be repaired. Barely visible impact damage (BVID) therefore represents a threshold, such that damage of this size or smaller must be considered to exist in flight structure, and structure must therefore be designed to tolerate this level of damage without a loss in performance. In order to design structures appropriately, it is necessary to understand the type and extent of internal damage present at or near the BVID threshold. Such damage assessments are then used as input for structural performance determinations. The purpose of this paper is to investigate how structural and impact parameters affect the nature and extent of damage in sandwich composites in the vicinity of BVID. The particular sandwich composites that were studied are comprised of an aluminum honeycomb core and face sheets made from multiple plies of unidirectional graphite fibers in an epoxy matrix. The plies in the face sheets have fibers oriented in the 0°, 90°, 45° and -45° directions. These plies are relatively stiff in the fiber direction and compliant in the perpendicular direction. Plies of different directions are stacked on top of each other to build face sheets that are quasi-isotropic, i.e., that have the same strength and stiffness in their in-plane directions. The parameters that are investigated in this paper are the core thickness, core density, face sheet stacking sequence (the sequence that the plies in various directions are placed on top of one another), load, and indenter diameter. To this end, specimens are indented using a quasi-static indentation test. In this test, load is applied monotonically using a fixed diameter indenter until the permanent dent becomes barely visible. This approach has been shown to produce essentially the same type of damage as low-velocity impact, but allows for more consistent and controllable levels of damage to be created. The damage was then evaluated non-destructively via ultrasonics and destructively via cross sectioning and microscopy. The results obtained by these two methods were then compared and synthesized to obtain an understanding of the internal state of damage as a function of those parameters studied. It was found that the two parameters that are most important are the face sheet stacking sequence and the core density. In terms of stacking sequence, delaminations are most prominent between plies with large differences in their fiber orientations. For adjacent plies with very different fiber directions (i.e., a 90° ply followed by a 0° ply), there is a large mismatch in stiffness and in coefficient of thermal expansion. This causes large shear stresses, which in turn lead to delamination. In addition, stiffer, higher density cores are observed to cause more delamination to occur than lower density, more compliant cores. It is expected that the data and trends collected in this study may be used to provide guidance for choosing structural geometries that optimize weight, cost, and impact resistance for practical structural applications

Doctor of Philosophy

dissertationMost humans have difficulty performing precision tasks, such as writing and painting, without additional physical support(s) to help steady or offload their arm's weight. To alleviate this problem, various passive and active devices have been developed. However, such devices often have a small workspace and lack scalable gravity compensation throughout the workspace and/or diversity in their applications. This dissertation describes the development of a Spatial Active Handrest (SAHR), a large-workspace manipulation aid, to offload the weight of the user's arm and increase user's accuracy over a large three-dimensional workspace. This device has four degrees-of-freedom and allows the user to perform dexterous tasks within a large workspace that matches the workspace of a human arm when performing daily tasks. Users can move this device to a desired position and orientation using force or position inputs, or a combination of both. The SAHR converts the given input(s) to desired velocit

Doctor of Philosophy

dissertationHumans generally have difficulty performing precision tasks with their unsupported hands. To compensate for this difficulty, people often seek to support or rest their hand and arm on a fixed surface. However, when the precision task needs to be performed over a workspace larger than what can be reached from a fixed position, a fixed support is no longer useful. This dissertation describes the development of the Active Handrest, a device that expands its user's dexterous workspace by providing ergonomic support and precise repositioning motions over a large workspace. The prototype Active Handrest is a planar computer-controlled support for the user's hand and arm. The device can be controlled through force input from the user, position input from a grasped tool, or a combination of inputs. The control algorithm of the Active Handrest converts the input(s) into device motions through admittance control where the device's desired velocity is calculated proportionally to the input force or its equivalent. A robotic 2-axis admittance device was constructed as the initial Planar Active Handrest, or PAHR, prototype. Experiments were conducted to optimize the device's control input strategies. Large workspace shape tracing experiments were used to compare the PAHR to unsupported, fixed support, and passive moveable support conditions. The Active Handrest was found to reduce task error and provide better speedaccuracy performance. Next, virtual fixture strategies were explored for the device. From the options considered, a virtual spring fixture strategy was chosen based on its effectiveness. An experiment was conducted to compare the PAHR with its virtual fixture strategy to traditional virtual fixture techniques for a grasped stylus. Virtual fixtures implemented on the Active Handrest were found to be as effective as fixtures implemented on a grasped tool. Finally, a higher degree-of-freedom Enhanced Planar Active Handrest, or E-PAHR, was constructed to provide support for large workspace precision tasks while more closely following the planar motions of the human arm. Experiments were conducted to investigate appropriate control strategies and device utility. The E-PAHR was found to provide a skill level equal to that of the PAHR with reduced user force input and lower perceived exertion

Affine differential geometry analysis of human arm movements

Humans interact with their environment through sensory information and motor actions. These interactions may be understood via the underlying geometry of both perception and action. While the motor space is typically considered by default to be Euclidean, persistent behavioral observations point to a different underlying geometric structure. These observed regularities include the “two-thirds power law” which connects path curvature with velocity, and “local isochrony” which prescribes the relation between movement time and its extent. Starting with these empirical observations, we have developed a mathematical framework based on differential geometry, Lie group theory and Cartan’s moving frame method for the analysis of human hand trajectories. We also use this method to identify possible motion primitives, i.e., elementary building blocks from which more complicated movements are constructed. We show that a natural geometric description of continuous repetitive hand trajectories is not Euclidean but equi-affine. Specifically, equi-affine velocity is piecewise constant along movement segments, and movement execution time for a given segment is proportional to its equi-affine arc-length. Using this mathematical framework, we then analyze experimentally recorded drawing movements. To examine movement segmentation and classification, the two fundamental equi-affine differential invariants—equi-affine arc-length and curvature are calculated for the recorded movements. We also discuss the possible role of conic sections, i.e., curves with constant equi-affine curvature, as motor primitives and focus in more detail on parabolas, the equi-affine geodesics. Finally, we explore possible schemes for the internal neural coding of motor commands by showing that the equi-affine framework is compatible with the common model of population coding of the hand velocity vector when combined with a simple assumption on its dynamics. We then discuss several alternative explanations for the role that the equi-affine metric may play in internal representations of motion perception and production

Streamlined islands and the English Channel megaflood hypothesis

Recognising ice-age catastrophic megafloods is important because they had significant impact on large-scale drainage evolution and patterns of water and sediment movement to the oceans, and likely induced very rapid, short-term effects on climate. It has been previously proposed that a drainage system on the floor of the English Channel was initiated by catastrophic flooding in the Pleistocene but this suggestion has remained controversial. Here we examine this hypothesis through an analysis of key landform features. We use a new compilation of multi- and single-beam bathymetry together with sub-bottom profiler data to establish the internal structure, planform geometry and hence origin of a set of 36 mid-channel islands. Whilst there is evidence of modern-day surficial sediment processes, the majority of the islands can be clearly demonstrated to be formed of bedrock, and are hence erosional remnants rather than depositional features. The islands display classic lemniscate or tear-drop outlines, with elongated tips pointing downstream, typical of streamlined islands formed during high-magnitude water flow. The length-to-width ratio for the entire island population is 3.4 ± 1.3 and the degree-of-elongation or k-value is 3.7 ± 1.4. These values are comparable to streamlined islands in other proven Pleistocene catastrophic flood terrains and are distinctly different to values found in modern-day rivers. The island geometries show a correlation with bedrock type: with those carved from Upper Cretaceous chalk having larger length-to-width ratios (3.2 ± 1.3) than those carved into more mixed Paleogene terrigenous sandstones, siltstones and mudstones (3.0 ± 1.5). We attribute these differences to the former rock unit having a lower skin friction which allowed longer island growth to achieve minimum drag. The Paleogene islands, although less numerous than the Chalk islands, also assume more perfect lemniscate shapes. These lithologies therefore reached island equilibrium shape more quickly but were also susceptible to total erosion. Our observations support the hypothesis that the islands were initially carved by high-water volume flows via a unique catastrophic drainage of a pro-glacial lake in the southern North Sea at the Dover Strait rather than by fluvial erosion throughout the Pleistocene