Densing Sets

AbstractLet H be a family of "large" (in various senses, e.g., of positive Hausdorff dimension or Lebesgue measure) subsets of R. We study sets D of real numbers which are H-densing, namely have the property that, given any set H ∈ H and ϵ > 0, there exist an a ∈ D for which the set aH is ϵ-dense modulo 1. In the special case, where H consists of all subsets of R having a finite accumulations point, H-densing sets are simply Glasner sets, studied earlier

Regularity of patterns in the factorization of n!

AbstractConsider the multiplicities ep1(n),ep2(n),…,epk(n) in which the primes p1,p2,…,pk appear in the factorization of n!. We show that these multiplicities are jointly uniformly distributed modulo (m1,m2,…,mk) for any fixed integers m1,m2,…,mk, thus improving a result of Luca and Stănică [F. Luca, P. Stănică, On the prime power factorization of n!, J. Number Theory 102 (2003) 298–305]. To prove the theorem, we obtain a result regarding the joint distribution of several completely q-additive functions, which seems to be of independent interest

Kinematic and stochastic surface topography of machined tial6v4-parts by means of ball nose end milling

Ball nose end mills are usually applied during 5-axes machining of high functional parts especially in the aerospace industry. The systematical study of the relationship between process forces and kinematics, surface topography and subsurface properties is fundamental to ensure a high surface integrity. This paper deals with the topography of machined surfaces of TiAl6V4 parts by means of ball nose end milling. The machined surface has been analyzed and the kinematic topography, influenced by the process parameters and the geometry of the cutting tool, has been computed. By subtracting the surface measurements from the computed topography, the stochastic topography of the machined surface, e.g. roughness and cracks, can be determined. Furthermore, an approach is given for predicting the stochastic topography based on the process forces during machining of TiAl6V4. © 2011 Published by Elsevier Ltd.DFG/CRC/87

Contact zone analysis based on multidexel workpiece model and detailed tool geometry representation

A new method for analyzing the tool-workpiece-contact area in cutting processes is presented. To gain enhanced knowledge about tool-workpiece interaction, determination of chip thickness, contact length and resulting cross-section area of the undeformed chip is of major interest. Compared to common simulation approaches, where rotation-symmetrically constructed tool geometry is used, the new method uses a detailed three dimensional tool shape model for an extended and more accurate contact zone analysis. As a corresponding representation of the workpiece and its time dependent shape-changes a multidexel model is used. To prepare the geometric tool model, the contained BREP topology is built up within the simulation system using data from a STEP-file. First of all functional parts of the tool like rake and flank faces and cutting edges are labeled for further processing. In a second step the identified NURBS-faces are discretized for the application in material-removal calculation. This way a mesh is built-up based on triangle elements which maps the geometry of each cutting edge into a 2D parametric representation. In relation to rake face, each node is described by its position on the cutting edge and its perpendicular distance to this edge. To perform contact zone analysis each cutting geometry and a multidexel model are intersected in discrete time steps corresponding to a tool rotation of about three degrees. The intersection point of each dexel and the cutting geometry is calculated. Parametric cutting geometry allows for a direct computation of local cutting depth and contact length for each involved point. Based on the local values of contact length and cross section area of the undeformed chip the characteristic values for the entire contact zone are calculated and used to predict mechanical as well as thermal loads caused by the cutting process. To demonstrate the application of the novel approach, prediction of forces in slot milling of 1.1191 steel is presented.DFG/PP/148

Analysis of airplane boarding via space-time geometry and random matrix theory

We show that airplane boarding can be asymptotically modeled by 2-dimensional Lorentzian geometry. Boarding time is given by the maximal proper time among curves in the model. Discrepancies between the model and simulation results are closely related to random matrix theory. We then show how such models can be used to explain why some commonly practiced airline boarding policies are ineffective and even detrimental.Comment: 4 page

Influence of Machining Parameters on Heat Generation during Milling of Aluminum Alloys

Thin-walled components, i.e. fuselage frames of airplanes, are prone to an unstable process behavior during milling. Therefore, tools with a chamfer between the cutting edge and the flank face are often used for such machining tasks. During milling, the chamfered area comes into contact with the just cut surface. This contact leads to process damping forces and the induced heat into the workpiece in this contact zone is increased. Furthermore, the amount of induced heat depends on the process parameters. At certain spots on the machined surface this may lead to a local overheating, which can reduce stiffness significantly. When this occurs during milling of a thin-walled component, the component is often regarded as reject. In this paper, the influence of chamfers and process parameters on the induced heat into the workpiece is investigated experimentally. In addition, a simulation which predict the temperature in the workpiece in dependence of the process parameters is presented.Ministry of Economics, Labour and Transport of Lower Saxony/ZW3-80134969DFG/DE 447/90-

Modeling a thermomechanical NC-simulation

This paper presents a method for a NC-Simulation based prediction of shape errors caused by thermal expansions in machining of complex workpieces. In the first part of the paper the basic approach of modeling a thermomechanical NC-Simulation for a faster and more precise process simulation is shown. Therefore, a fast dexel based material removal simulation including process models for calculation of localized heat flux and forces is linked to a FE model for simulation of thermal conduction in the workpiece. Interdependencies of thermal process and workpiece conditions are considered by a closed simulation loop. In the second part of the paper the modeling of each component is explained. To consider thermomechanical effects in material removal simulation the dexel based workpiece model is extended by additional information like temperature and deformation in every dexel. An inverse projection of the workpiece deformation on a triangulated tool model allows consideration this effect by deformation of the tool model. Thereby, a realistic shape of the workpiece can be simulated. In addition, the current cutting conditions like area of undeformed chip-thickness or contact length are changed. This results in diversified cutting forces and heat fluxes. For a realistic simulation of the thermal conduction the dimensions of the FE model have to be adapted by a time dependent virtual domain method. In the last part of the paper, results of the simulation are compared to measured data. The comparison shows that process temperatures in different workpiece areas are predicted accurately

Prediction of the Principal Stress Direction for 5-axis Ball End Milling

While regenerating damaged components, e.g. compressor blades, the removal of excess weld material called re-contouring often determines the surface integrity including the residual stress state. A load-specific residual stress state is beneficial for lifetime. This leads to the necessity to predict the resulting residual stress state after machining. The paper describes two models, which predict the principal stress direction as a residual stress characteristic for 5-axis ball nose end milling of Ti-6Al-4 V. One model uses process force components, the other is based on the microtopography of the workpiece, which is influenced by the kinematics of the process.DFG/Collaborative Research Centre/87

Approaches for improving cutting processes and machine too in re-contouring

Re-contouring in the repair process of aircraft engine blades and vanes is a crucial task. Highest demands are made on the geometrical accuracy as well as on the machined surface of the part. Complexity rises even more due to the unique part characteristic originating from the operation and repair history. This requires well-designed processes and machine tool technologies. In this paper, approaches for coping with these challenges and improving the re-contouring process are described and discussed. This includes an advanced process simulation with its capabilities to accurately depict different material areas and predict process forces. Beyond, experimental investigations on workpiece-tooldeflection are presented. Finally, a machine tool prototype with a novel electromagnetic guiding system is introduced and the benefits of this technology in the field of repair are outlined.DFG/CRC/87

Discrete charging of metallic grains: Statistics of addition spectra

We analyze the statistics of electrostatic energies (and their differences) for a quantum dot system composed of a finite number $K$ of electron islands (metallic grains) with random capacitance-inductance matrix $C$, for which the total charge is discrete, $Q=Ne$ (where $e$ is the charge of an electron and $N$ is an integer). The analysis is based on a generalized charging model, where the electrons are distributed among the grains such that the electrostatic energy E(N) is minimal. Its second difference (inverse compressibility) $\chi_{N}=E(N+1)-2 E(N)+E(N-1)$ represents the spacing between adjacent Coulomb blockade peaks appearing when the conductance of the quantum dot is plotted against gate voltage. The statistics of this quantity has been the focus of experimental and theoretical investigations during the last two decades. We provide an algorithm for calculating the distribution function corresponding to $\chi_{N}$ and show that this function is piecewise polynomial.Comment: 21 pages, no figures, mathematical nomenclature (except for Abstract and Introduction