Transition Curves Described Using Curvature Function

2017 ◽  
pp. 25-48
Author(s):  
Andrzej Kobryń
Keyword(s):  
Curvature Function ◽  
Transition Curves

scholarly journals Pansu–Wulff shapes in ℍ1

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Julián Pozuelo ◽  
Manuel Ritoré
Keyword(s):  
Mean Curvature ◽  
Isoperimetric Problem ◽  
Variational Formula ◽  
Geodesic Curvature ◽  
Vertical Axis ◽  
Singular Part ◽  
Tangent Plane ◽  
Curvature Function ◽  
Invariant Norm ◽  
The First Variation

Abstract We consider an asymmetric left-invariant norm ∥ ⋅ ∥ K {\|\cdot\|_{K}} in the first Heisenberg group ℍ 1 {\mathbb{H}^{1}} induced by a convex body K ⊂ ℝ 2 {K\subset\mathbb{R}^{2}} containing the origin in its interior. Associated to ∥ ⋅ ∥ K {\|\cdot\|_{K}} there is a perimeter functional, that coincides with the classical sub-Riemannian perimeter in case K is the closed unit disk centered at the origin of ℝ 2 {{\mathbb{R}}^{2}} . Under the assumption that K has C 2 {C^{2}} boundary with strictly positive geodesic curvature we compute the first variation formula of perimeter for sets with C 2 {C^{2}} boundary. The localization of the variational formula in the non-singular part of the boundary, composed of the points where the tangent plane is not horizontal, allows us to define a mean curvature function H K {H_{K}} out of the singular set. In the case of non-vanishing mean curvature, the condition that H K {H_{K}} be constant implies that the non-singular portion of the boundary is foliated by horizontal liftings of translations of ∂ ⁡ K {\partial K} dilated by a factor of 1 H K {\frac{1}{H_{K}}} . Based on this we can define a sphere 𝕊 K {\mathbb{S}_{K}} with constant mean curvature 1 by considering the union of all horizontal liftings of ∂ ⁡ K {\partial K} starting from ( 0 , 0 , 0 ) {(0,0,0)} until they meet again in a point of the vertical axis. We give some geometric properties of this sphere and, moreover, we prove that, up to non-homogeneous dilations and left-translations, they are the only solutions of the sub-Finsler isoperimetric problem in a restricted class of sets.

Stability of a Rigid Body With an Oscillating Particle: An Application of MACSYMA

1985 ◽  
Vol 52 (3) ◽  
pp. 686-692 ◽  
Author(s):  
L. A. Month ◽  
R. H. Rand
Keyword(s):  
Angular Velocity ◽  
Rigid Body ◽  
Classical Problem ◽  
Moment Of Inertia ◽  
Model Parameters ◽  
Stability Chart ◽  
The Stability ◽  
Transition Curves ◽  
Minimum Moment ◽  
Small Angular Velocity

This problem is a generalization of the classical problem of the stability of a spinning rigid body. We obtain the stability chart by using: (i) the computer algebra system MACSYMA in conjunction with a perturbation method, and (ii) numerical integration based on Floquet theory. We show that the form of the stability chart is different for each of the three cases in which the spin axis is the minimum, maximum, or middle principal moment of inertia axis. In particular, a rotation with arbitrarily small angular velocity about the maximum moment of inertia axis can be made unstable by appropriately choosing the model parameters. In contrast, a rotation about the minimum moment of inertia axis is always stable for a sufficiently small angular velocity. The MACSYMA program, which we used to obtain the transition curves, is included in the Appendix.

On the Planning and Construction of Railway Curved Track

2021 ◽  
Vol 13 (2) ◽  
Author(s):  
Sono Bhardawaj ◽  
Rakesh Chandmal Sharma ◽  
Sunil Kumar Sharma ◽  
Neeraj Sharma
Keyword(s):  
High Speed ◽  
Time Derivative ◽  
Railway Track ◽  
Lateral Acceleration ◽  
Transition Curve ◽  
Railway Vehicle ◽  
Geometric Conditions ◽  
Curve Track ◽  
Curved Track ◽  
Transition Curves

Increasing demand for railway vehicle speed has pushed the railway track designers to develop high-quality track. An important measure of track quality is the character of the transition curve track connecting different intersecting straight tracks. A good transition curve track must be able to negotiate the intermittent stresses and dynamic effects caused by changes in lateral acceleration at high speed. This paper presents the constructional methods for planning transition curves considering the dynamics of movement. These methods consider the non-compensated lateral acceleration, deviation in lateral acceleration and its higher time derivatives. This paper discusses the laying methods of circular, vertical and transition curves. Key aspects in laying a curved track e.g. widening of gauge on curves are discussed in this paper. This paper also suggests a transition curve which is effective not only from a dynamic point of view considering lateral acceleration and its higher time derivative but also consider the geometric conditions along with the required deflection angle.

scholarly journals Mesomorphism Dependence on Cis-Trans Configuration and Varying Terminal/Lateral Group

2016 ◽  
Vol 65 ◽  
pp. 71-79
Author(s):  
G.N. Bhola ◽  
Umed C. Bhoya
Keyword(s):  
Thermal Stability ◽  
Homologous Series ◽  
Vinyl Ester ◽  
Molecular Structures ◽  
Transition Curve ◽  
Group Efficiency ◽  
Lateral Group ◽  
Normal Manner ◽  
Thermotropic Liquid Crystals ◽  
Transition Curves

A chalconyl trans vinyl ester homologous series of thermotropic liquid crystals RO-C6H4-CH=CH-COO-C6H4-CH=CH-CO-C6H3(OCH3)2meta-para is synthesized and studied with a view to understand and establish the relation between molecular structure and mesomorphism or mesomorphic behaviors. Novel homologous series consists of thirteen homologues (C1to C18). First five (C1to C5) homologues are nonmesogenic. Rest of the homologues (C6to C18) homologues is mesogenic. Monotropically smectogenic character appears from C6to C10homologue and C12to C18homologues are enantiotropically smectogenic. Nematogenic property is totally absent throughout a series. Textures of smectic phase are of the type smectic A or C. Transition curves of a phase diagram behaved in normal manner. Analytical, spectral and thermal data supports the molecular structures of homologues. Odd-even effect is observed for Sm-I/I-Sm transition curve. Thermal stability for smectic is 93.7 °C and the mesomorphic phase lengths from enantiotropic homologues for smectic from minimum to maximum is 6.0 °C to 28.0 °C at the C12and C18homologue respectively. Series is smectogenic without exhibition of nematic property. The group efficiency order on the basis of thermal stability is derived; from comparative study of present series and analogous series.

Effect of Neutron Irradiation on the Mechanical Properties of an A508 CL2 and 15Kh2NMFA Irradiated in the NOMAD_3 Rig in the BR2 Cooling Water

2021 ◽  
Author(s):  
Inge Uytdenhouwen ◽  
Rachid Chaouadi
Keyword(s):  
Neutron Irradiation ◽  
Nuclear Reactor ◽  
Cooling Water ◽  
Charpy Impact ◽  
Tensile Tests ◽  
Significant Loss ◽  
Irradiation Temperature ◽  
Irradiation Damage ◽  
Transition Curves ◽  
Reactor Pressure

Abstract The typical operating temperatures of a nuclear reactor pressure vessel in a PWR are between 290°C and 300°C. However, many BWRs and some PWRs operate at slightly lower temperatures down to 260°C. Most of the literature and neutron irradiation damage is therefore focused on those irradiation temperatures. It is well-known that the lower the irradiation temperature, the more neutron irradiation damage occurs, because no appreciable annealing happens below approximately 230°C. The NOMAD_3 irradiation consisted in total of 24 Charpy sized samples from an A508 Cl.2 forging and a 15Kh2NMFA material. They were irradiated to three various fluences between 1.55 and 7.90 × 1019 n/cm2 (E > 1MeV) at approximately 100°C. The hardening of the A508 Cl.2 was between 260 and 400 MPa which was much higher than the NOMAD_0 properties which were irradiated at approximately 280°C. The tensile tests of irradiated materials are all characterized by a significant loss of work hardening capacity leading to plastic flow localization promptly after the yield strength is reached. This affects also the shape of the Charpy impact transition curves. The radiation embrittlement derived from Charpy impact tests, ΔT41J, is up to 156°C for the highest fluence. For this irradiation, the embrittlement to hardening ratio was also around 0.43 +/−0.2°C/MPa as it was found in the previous campaign NOMAD_0. This paper discusses the tensile, hardness and impact properties of the NOMAD_3 irradiation campaign. It is compared to the NOMAD_0 with respect to effect of irradiation temperature and annealing recovery.

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