Helicoidal Surfaces Research Articles

Helicoidal surfaces with constant anisotropic mean curvature

We study surfaces which are in equilibrium for an anisotropic surface energy and which are invariant under a helicoidal motion. For anisotropic functionals with axially symmetric Wulff shapes, we generalize the recently developed twizzler representation [Perdomo, O., A dynamical interpretation of the profile curve of CMC twizzlers surfaces, e-print arXiv:1001.5198v1] to the anisotropic case and show how all helicoidal constant anisotropic mean curvature surfaces can be obtained by quadratures. When the functional is not axially symmetric, we produce a canonical critical point which is analogous to the classical helicoid.

Helicoidal surfaces in the three-dimensional Lorentz–Minkowski space satisfying Δ II r i = λ i r i

In this paper, we study helicoidal surfaces without parabolic points in the 3-dimensional Lorentz–Minkowski space under the condition Δ II r i = λ i r i where Δ II is the Laplace operator with respect to the second fundamental form and λ i is a real number. We prove that there are no helicoidal surfaces without parabolic points in the 3-dimensional Lorentz–Minkowski space satisfying that condition.

The Study of the Relationship between the Helix and the Individual Error in Gear Hob Measurements

As a tool of complex shape, the manufacturing error of hobs can be classed into three kinds: the datum level error, the tooth thickness error and the parameter error of helicoidal surface. The parameter error of helicoidal surface, which is composed of tooth error and helix error, is the most importance among them. For precision of hob and convenience of measurement, several individual errors are instead of the helix error. How to set reasonable individual errors to guarantee the helix error within tolerance, the relationship between them in gear hob measurements is analyzed, and the approach to decompose the helix error into individual error has been discussed in the paper. On the basis of above theory, a high precision mode can be easily established using the soft ware.

HELICOIDAL CDPC-SURFACES IN MINKOWSKI 3-SPACE

In this paper, we discuss the local existence and the construction of helicoidal CDPC-surfaces, i.e. the helicoidal surfaces with constant difference of the principal curvatures, in Minkowski 3-space [Formula: see text]. In general, there exist helicoidal surfaces with conjugate complex principal curvatures, which have no counterparts in R3. We prove this is still true for helicoidal CDPC-surfaces but not true for rotation surfaces in [Formula: see text].

HELICOIDAL SURFACES AND THEIR GAUSS MAP IN MINKOWSKI 3-SPACE

The helicoidal surface is a generalization of rotation surface in a Minkowski space. We study helicoidal surfaces in a Minkowski 3-space in terms of their Gauss map and provide some examples of new classes of helicoidal surfaces with constant mean curvature in a Minkowski 3-space.

Mean curvatures and Gauss maps of a pair of isometric helicoidal and rotation surfaces in Minkowski 3-space

It is proved that, in Minkowski 3-space, a CSM-helicoidal surface, i.e., a helicoidal surface under cubic screw motion is isometric to a rotation surface so that helices on the helicoidal surface correspond to parallel circles on the rotation surface. By distinguishing a CSM-helicoidal surface as three cases, that is, the case of type I, the case of type II with negative and positive pitch, the relations are discussed between the mean curvatures or Gauss maps of a pair of isometric helicoidal and rotation surface. A CSM-helicoidal surface of Case 1 or 2 and its isometric rotation surface with null axis have same mean curvatures (resp. Gauss maps) if and only if they are minimal. But each pair of isometric CSM-helicoidal surface of Case 3 and rotation surface with spacelike axis have different Gauss maps.

Modulated waves and helicoidal pseudospherical surfaces in nonlinear inhomogeneous elasticity

The nonlinear wave equation provides a Lagrangian description of one-dimensional stress propagation in a class of model inhomogeneous ideally hard elastic materials. It is associated with the geometry of pseudospherical surfaces and, as such, is integrable and admits an auto-Bäcklund transformation. Here, modulated waves associated with helicoidal pseudospherical surfaces are investigated.

Electronic states on the helicoidal surface

It is shown that free electrons on the helicoidal surface are affected by an effective potential, which vanishes in the s channel and gives rise to both metastable as well as stable bound states. The effective potential depends linearly on the “angular momentum” quantum number and the spin state resulting in a partially nondegenerate energy spectrum. The vanishing of the potential in the s channel illustrates a “miraculous” exact cancellation which is present for all quantum states. An attractive effective potential term arising from the spin connection cancels exactly a repulsive potential term, which is identical to the potential which affects relativistic bosons on the helicoidal surface.

HELICOIDAL SURFACES AND THEIR GAUSS MAP IN MINKOWSKI 3-SPACE II

The helicoidal surface is a generalization of rotation surface in a Minkowski space. We study helicoidal surfaces in a Minkowski 3-space in terms of their Gauss map and provide some examples of new classes of helicoidal surfaces with constant mean curvature in a Minkowski 3-space.

HELICOIDAL SURFACES WITH POINTWISE 1-TYPE GAUSS MAP

The helicoidal surfaces with pointwise 1-type or harmonic gauss map in Euclidean 3-space are studied. The notion of pointwise 1- type Gauss map is a generalization of usual sense of 1-type Gauss map. In particular, we prove that an ordinary helicoid is the only genuine he- licoidal surface of polynomial kind with pointwise 1-type Gauss map of the flrst kind and a right cone is the only rational helicoidal surface with pointwise 1-type Gauss map of the second kind. Also, we give a charac- terization of rational helicoidal surface with harmonic or pointwise 1-type Gauss map.

Three dimensional modeling and finite element simulation of a generic end mill

The geometry of cutting flutes and the surfaces of end mills is one of the crucial parameters affecting the quality of the machining in the case of end milling. These are usually represented by two-dimensional models. This paper describes in detail the methodology to model the geometry of a flat end mill in terms of three-dimensional parameters. The geometric definition of the end mill is developed in terms of surface patches; flutes as helicoidal surfaces, the shank as a surface of revolution and the blending surfaces as bicubic Bezier and biparametric sweep surfaces. The proposed model defines the end mill in terms of three-dimensional rotational angles rather than the conventional two dimensional angles. To validate the methodology, the flat end milling cutter is directly rendered in OpenGL environment in terms of three-dimensional parameters. Further, an interface is developed that directly pulls the proposed three-dimensional model defined with the help of parametric equations into a commercial CAD modeling environment. This facilitates a wide range of downstream technological applications. The modeled tool is used for finite element simulations to study the cutting flutes under static and transient dynamic load conditions. The results of stress distribution (von mises stress), translational displacement and deformation are presented for static and transient dynamic analysis for the end mill cutter flute and its body. The method described in this paper offers a simple and intuitive way of generating high-quality end mill models for use in machining process simulations. It can be easily extended to generate other tools without relying on analytical or numerical formulations.

S1101-3-4 歯車の歯当たりとかみ合い評価法について : 可展円筒歯車歯面の図式展開について(伝動装置の基礎と応用セッション3)

Tooth bearing patterns give us very useful information about gear engagement situations. However, the detail analysis of gear engagement using actual contact patterns is not enough up to now, because the actual patterns are normally captured by a still camera as projected images. In the case of the involute helicoid surface, it is possible to develop it onto the plane geometrically. It is more convenient to compare the actual patterns with simulated ones on the plane. In this paper, geometrical developable surfaces are classified in the first place, and the involute helicoid surface is developed onto the plane geometrically with the help of the geodesic curvatures of tooth profiles and tooth leads. It is found that the lead circular helices are developed as the concentric circles on a plane, and the involute tooth profile is developed as an involute curve which has a certain diameter of a base circle.

The affine Cauchy problem

The aim of this paper is to solve the Cauchy problem for locally strongly convex surfaces which are extremal for the equiaffine area functional. These surfaces are called affine maximal surfaces and here, we give a new complex representation which let us describe the solution to the corresponding Cauchy problem. As applications, we obtain a generalized symmetry principle, characterize when a curve in R 3 can be a geodesic or pre-geodesic of a such surface and study the helicoidal affine maximal surfaces. Finally, we investigate the existence and uniqueness of affine maximal surfaces with a given analytic curve in its singular set.

Toroidal grinding method for curved cutting edge twist drills

In this paper, a new grinding procedure for the curved cutting edge twist drills is presented, i.e., the toroidal grinding procedure. Within this procedure, the clearance surface of the twist drill is a cylindrical helicoid surface with a constant pitch. Based on the principle of distance minimization and envelope theory, a numerical method of calculating the grinding wheel profile is developed. This grinding method allows the profile of the main cutting edge to be continuous and obtained directly during the grinding procedure. The feasibility of the toroidal grinding method is verified by drill grinding experiment and drilling measurements.

On The Mean, Gauss, The Second Gaussian and The Second Mean Curvature of The Helicoidal Surfaces with Light-Like Axis in $\mathbb{R}^{3}$

In this paper, the second Gaussian and the second mean curvature of the helicoidal surfaces with light-like axis of type $IV^{+}$ is obtained in Minkowski 3-space. In addition, some relations between the mean, Gauss, the second Gaussian and the second mean curvature of the helicoidal surfaces with light-like axis of type $IV^{+}$ are given in Minkowski 3-space.

Geometric Modeling of Fluted Cutters

Geometries of cutting tools are usually represented by two-dimensional models. This paper outlines the construction of detailed computer aided design models for a variety of fluted cutters that includes slab mills, end mills, and drills; and establishes a new three-dimensional definition for the geometry of fluted cutters in terms of biparametric surface patches. This work presents unified models of both plain and helical slab mills, end mills (with different end geometries), and drills (with a variety of point styles). The surfaces meant for cutting operations, known as flutes, are modeled as helicoidal surfaces. To model the flutes, sectional geometry of tip-to-tip profile is developed and then it is swept according to a sweeping rule, determined by the type of the cutter under consideration. The slab mill consists of flutes and two planar end surfaces, while the end mills and drills have shanks and end geometries also (in addition to the flutes). The geometric models of shank and end geometries are separately developed. The transitional surfaces of these cutters are modeled as bicubic Bézier surfaces or biparametric sweep surfaces. The proposed models employ a novel nomenclature that defines the form of cutting tools in terms of three-dimensional rotational angles. The relations required to map the proposed three-dimensional angles to conventional angles (forward mapping) and their reverse relations (inverse mapping) are also developed for all three types of fluted cutters. The new paradigm offers immense technological advantages in terms of numerous downstream applications.

Contra‐rotating fans noise prediction for jet engine performance optimization

This work fits within the framework of the European research project VITAL, which is dedicated to the development of technologies aimed at reducing the environmental impact of jet engines. In this respect, a new concept of ducted contra‐rotating fans is investigated. The present contribution to this project is to elaborate a noise prediction tool for performances optimization. On the basis of the helicoidal surface theory, Hanson previously developed a harmonic formulation for the propfan rotor/rotor interaction. This method is here extended in order to take into account the duct effects and the noise radiated by the rear struts. A computer program has been written to predict the pressure distribution in the near field and to estimate the noise spectra in the far field. The results are validated with computational fluid dynamics simulations and compared with available public data. Those predictions are then included in a global noise computation with the other engine acoustic sources (coaxial jet, combustor and so on), and a multidisciplinary optimizer is used for the engine noise minimization.

Modeling of twist drills in terms of 3D angles

Traditional nomenclatures of specifying cutting tool geometries are two-dimensional (2D) in nature. The present work presents a paradigm to model the geometries of a variety of twist drills in terms of three-dimensional (3D) parameters. The work outlines the construction of a detailed computer-aided design (CAD) model for a fluted twist drill and establishes a new 3D definition for the geometry of drill in terms of biparametric surface patches. The flutes of the drill are modeled as helicoidal surfaces. For this, sectional geometry of tip-to-tip profile is developed and then swept. The geometric model of the shank is developed separately. The transitional surfaces are modeled as bicubic Bezier surfaces. With this methodology, we propose a new 3D nomenclature for drill geometries in terms of 3D rotational angles. The relations necessary to map the proposed three-dimensional angles to two-dimensional conventional angles, known as forward mapping and their reverse relations (inverse mapping) are also developed. The new paradigm offers immense technological advantages in terms of numerous downstream applications.

Examples and structure of CMC surfaces in some Riemannian and Lorentzian homogeneous spaces

It is proved that the holomorphic quadratic differential associated to CMC surfaces in Riemannian products S × R and H × R discovered by U. Abresch and H. Rosenberg could be obtained as a linear combination of usual Hopf differentials. Using this fact, we are able to extend it for Lorentzian products. Families of examples of helicoidal CMC surfaces on these spaces are explicitly described. We also present some characterizations of CMC rotationally invariant discs and spheres.

Helicoidal vortex model for wind turbine aeroelastic simulation

The vortex method has been extended to account for blade flexibility, which is a potential source of unsteadiness in the flow past a wind turbine rotor. The code has been validated previously under the assumption of rigid blades. The aerodynamics method is based on the Goldstein model, which distributes the flow vorticity on rigid helicoidal surfaces defined uniquely by the flow parameters (tip speed ratio and average power extracted by the rotor) and the blade geometry (maximum radius and root lengths). The structure is treated as a beam with degrees of freedom in bending and torsion. The high twist of the wind turbine blades is responsible for induced velocities in the plane of the rotor as well as out of plane. A modal decomposition has been shown to be the most accurate and efficient approach for an implicit coupling of the structural and aerodynamics equations. Results for a homogeneous blade are presented for a low speed of 5 m/s and yaw angles of 0°, 5° and 10° and compared with rigid blade results and experiments of the National Renewable Energy Laboratory (NREL). The nonhomogeneous NREL blade has also been modeled and results are presented for V = 8 and 10 m/s at zero yaw that include the effect of the tower on the blade loading.