The 3D nature of a real un-dismantled electrical contact interface

AbstractA 3D contact analysis and modeling suite of tools are developed and introduced in this work. The “3D Contact Map” of an electrical contact interface is presented demonstrating the 3D nature of the contact. It gives information on where the electrical contact spots in a 3D surface profile are located. An X-ray Computer Tomography (CT) technique is used to collect the 3D data to a resolution of around 5μm of a real un-dismantled contact interface for analysis. Previous work by Lalechos and Swingler presented “2D Contact Map” on a 2D contact profile from collected 3D data to a resolution of around 8μm. The main advantages of both 3D and 2D mapping techniques focus on the fact that they are non-destructive and there is no need to dismantle the component of interest. This current work focuses on the 3D mapping technique showing its advantages over the 2D mapping technique. For test purposes, a 16A rated AC single pole switch is scanned after two different current loading tests (0A and 16A). A comparison for the total mechanical area of contact, the number of contact spots and the total contact resistance is conducted using both the 2D and 3D mapping techniques to a resolution of around 5μm

Bounds on the hausdorff dimension of a renormalisation map arising from an excitable reaction-diffusion system on a fractal lattice

A renormalisation approach to investigate travelling wave solutions of an excitable reaction- diusion system on a deterministic fractal structure has recently been derived. The dynamics of a particular class of solutions which are governed by a two dimensional subspace of these renormalisation recursion relationships are discussed in this paper. The bifurcations of this mapping are discussed with reference to the discontinuities which arise at the singularities. The map is chaotic for a bounded region in parameter space and bounds on the Hausdor dimension of the associated invariant hyperbolic set are calculated

Analytical Study of the Julia Set of a Coupled Generalized Logistic Map

A coupled system of two generalized logistic maps is studied. In particular influence of the coupling to the behaviour of the Julia set in two dimensional complex space is analyzed both analytically and numerically. It is proved analytically that the Julia set disappears from the complex plane uniformly as a parameter interpolates from the chaotic phase to the integrable phase, if the coupling strength satisfies a certain condition.Comment: 30pages, 22figure

Conformal mapping methods for interfacial dynamics

The article provides a pedagogical review aimed at graduate students in materials science, physics, and applied mathematics, focusing on recent developments in the subject. Following a brief summary of concepts from complex analysis, the article begins with an overview of continuous conformal-map dynamics. This includes problems of interfacial motion driven by harmonic fields (such as viscous fingering and void electromigration), bi-harmonic fields (such as viscous sintering and elastic pore evolution), and non-harmonic, conformally invariant fields (such as growth by advection-diffusion and electro-deposition). The second part of the article is devoted to iterated conformal maps for analogous problems in stochastic interfacial dynamics (such as diffusion-limited aggregation, dielectric breakdown, brittle fracture, and advection-diffusion-limited aggregation). The third part notes that all of these models can be extended to curved surfaces by an auxilliary conformal mapping from the complex plane, such as stereographic projection to a sphere. The article concludes with an outlook for further research.Comment: 37 pages, 12 (mostly color) figure

Fractal dimension evolution and spatial replacement dynamics of urban growth

This paper presents a new perspective of looking at the relation between fractals and chaos by means of cities. Especially, a principle of space filling and spatial replacement is proposed to explain the fractal dimension of urban form. The fractal dimension evolution of urban growth can be empirically modeled with Boltzmann's equation. For the normalized data, Boltzmann's equation is equivalent to the logistic function. The logistic equation can be transformed into the well-known 1-dimensional logistic map, which is based on a 2-dimensional map suggesting spatial replacement dynamics of city development. The 2-dimensional recurrence relations can be employed to generate the nonlinear dynamical behaviors such as bifurcation and chaos. A discovery is made that, for the fractal dimension growth following the logistic curve, the normalized dimension value is the ratio of space filling. If the rate of spatial replacement (urban growth) is too high, the periodic oscillations and chaos will arise, and the city system will fall into disorder. The spatial replacement dynamics can be extended to general replacement dynamics, and bifurcation and chaos seem to be related with some kind of replacement process.Comment: 17 pages, 5 figures, 2 table