How Helical Can a Closed, Twisted Space Curve be?

2000 ◽  
Vol 107 (4) ◽  
pp. 327 ◽  
Author(s):  
Joel L. Weiner
Keyword(s):  
Space Curve ◽  
Twisted Space

Geometry Design and Contact Ratio Analysis for Circular Arc Parallel-Axis Helical Gears

2016 ◽  
Author(s):  
Z. Chen ◽  
B. Lei ◽  
Q. Zhao
Keyword(s):  
Rolling Process ◽  
Helical Gear ◽  
Space Curve ◽  
Impact Factors ◽  
Contact Ratio ◽  
Circular Arc ◽  
Practical Applications ◽  
Geometry Design ◽  
Gear Mechanism ◽  
The Impact

Based on space curve meshing theory, in this paper, we present a novel geometric design of a circular arc helical gear mechanism for parallel transmission with convex-concave circular arc profiles. The parameter equations describing the contact curves for both the driving gear and the driven gear were deduced from the space curve meshing equations, and parameter equations for calculating the convex-concave circular arc profiles were established both for internal meshing and external meshing. Furthermore, a formula for the contact ratio was deduced, and the impact factors influencing the contact ratio are discussed. Using the deduced equations, several numerical examples were considered to validate the contact ratio equation. The circular arc helical gear mechanism investigated in this study showed a high gear transmission performance when considering practical applications, such as a pure rolling process, a high contact ratio, and a large comprehensive strength.

Designing and Mapping Trimming Curves on Surfaces Using Orthogonal Projection

1990 ◽  
Author(s):  
Joseph Pegna ◽  
Franz-Erich Wolter
Keyword(s):  
Differential Equation ◽  
Orthogonal Projection ◽  
Broad Class ◽  
Design Tool ◽  
Space Curve ◽  
Computationally Efficient ◽  
Curves On Surfaces ◽  
Surface Representations ◽  
Differential Properties ◽  
Surface Curve

Abstract In the design and manufacturing of shell structures it is frequently necessary to construct trimming curves on surfaces. The novel method introduced in this paper was formulated to be coordinate independent and computationally efficient for a very general class of surfaces. Generality of the formulation is attained by solving a tensorial differential equation that is formulated in terms of local differential properties of the surface. In the method proposed here, a space curve is mapped onto the surface by tracing a surface curve whose points are connected to the space curve via surface normals. This surface curve is called to be an orthogonal projection of the space curve onto the surface. Tracing of the orthogonal projection is achieved by solving the aforementionned tensorial differential equation. For an implicitely represented surface, the differential equation is solved in three-space. For a parametric surface the tensorial differential equation is solved in the parametric space associated with the surface representation. This method has been tested on a broad class of examples including polynomials, splines, transcendental parametric and implicit surface representations. Orthogonal projection of a curve onto a surface was also developed in the context of surface blending. The orthogonal projection of a curve onto two surfaces to be blended provides not only a trimming curve design tool, but it was also used to construct smooth natural maps between trimming curves on different surfaces. This provides a coordinate and representation independent tool for constructing blend surfaces.

Uniqueness of the Geometric Representation in Large Rotation Finite Element Formulations

2010 ◽  
Vol 5 (4) ◽  
Author(s):  
Ahmed A. Shabana
Keyword(s):  
Finite Element ◽  
Finite Elements ◽  
Computer Aided Design ◽  
Curvature Vector ◽  
Space Curve ◽  
Body Motion ◽  
Large Rotation ◽  
Space Curves ◽  
Curvature Continuity ◽  
Finite Element Formulations

Several finite element formulations used in the analysis of large rotation and large deformation problems employ independent interpolations for the displacement and rotation fields. As explained in this paper, three rotations defined as field variables can be sufficient to define a space curve that represents the element centerline. The frame defined by the rotations can differ from the Frenet frame of the space curve defined by the same rotation field and, therefore, such a rotation-based representation can provide measure of twist shear deformations and captures the rotation of the beam about its axis. However, the space curve defined using the rotation interpolation has a geometry that can significantly differ from the geometry defined by an independent displacement interpolation. Furthermore, the two different space curves defined by the two different interpolations can differ by a rigid body motion. Therefore, in these formulations, the uniqueness of the kinematic representation is an issue unless nonlinear algebraic constraint equations are used to establish relationships between the two independent displacement and rotation interpolations. Nonetheless, significant geometric and kinematic differences between two independent space curves cannot always be reduced by using restoring elastic forces. Because of the nonuniqueness of such a finite element representation, imposing continuity on higher derivatives such as the curvature vector is not straight forward as in the case of the absolute nodal coordinate formulation (ANCF) that defines unique displacement and rotation fields. ANCF finite elements allow for imposing curvature continuity without increasing the order of the interpolation or the number of nodal coordinates, as demonstrated in this paper. Furthermore, the relationship between ANCF finite elements and the B-spline representation used in computational geometry can be established, allowing for a straight forward integration of computer aided design and analysis.

scholarly journals The pencil of the 4th and 3rd order surfaces obtained as a harmonic equivalent of the pencil of quadrics through a 4th order space curve of the 1st category

2012 ◽  
Vol 10 (2) ◽  
pp. 193-207 ◽  
Author(s):  
Gordana Djukanovic ◽  
Marija Obradovic
Keyword(s):  
Cross Sections ◽  
Spatial Model ◽  
Spatial Models ◽  
Space Curve ◽  
Parabolic Cylinder ◽  
Triaxial Ellipsoid ◽  
Architectural Practice ◽  
Ordered Space ◽  
Pass Through ◽  
Frontal Projection

This paper shows the process of inverting the 4th ordered space curve of the first category with a self-intersecting point (with two planes of symmetry) and determining its harmonic equivalent. There are harmonic equivalents for five groups of surfaces obtained through the 4th order space curve of the 1st category. Mapping was done through a system of circular cross-sections. Both classical and relativistic geometry interpretations are presented. We also designed spatial models - a spatial model of the pencil of quadrics and a spatial model of the pencil of equivalent quadrics. Besides the boundary surfaces, one surface of the 3rd order, which is an equivalent to a triaxial ellipsoid, passes through this pencil of surface of the 4th order. The center of inversion is located on the contour of the ellipsoid. The parabolic cylinder is mapped into its equivalent, by mapping the contour parabola of the cylinder, in the frontal projection, in relation to the center and the sphere of inversion into a contour curve of the 4th order surface. The generating lines of the parabolic cylinder, which are in a projecting position and pass through the antipode, are mapped into circles (also in a projecting position) whose diameters are from the center of inversion to the contour line. The application of the 4th order surfaces in architectural practice is also presented.

scholarly journals A special interpretation of the concept constant breadth for a space curve

2019 ◽  
Vol 23 (Suppl. 1) ◽  
pp. 371-382
Author(s):  
Tuba Agirman-Aydin
Keyword(s):  
Variable Coefficients ◽  
Space Curve ◽  
Normal Vector ◽  
Exit Point ◽  
Constant Breadth ◽  
Normal Vectors ◽  
Linear Differential ◽  
Curve Of Constant Breadth ◽  
Definition Of ◽  
Curvature And Torsion

The definition of curve of constant breadth in the literature is made by using tangent vectors, which are parallel and opposite directions, at opposite points of the curve. In this study, normal vectors of the curve, which are parallel and opposite directions are placed at the exit point of the concept of curve of constant breadth. In this study, on the concept of curve of constant breadth according to normal vector is worked. At the conclusion of the study, is obtained a system of linear differential equations with variable coefficients characterizing space curves of constant breadth according to normal vector. The coefficients of this system of equations are functions depend on the curvature and torsion of the curve. Then is obtained an approximate solution of this system by using the Taylor matrix collocation method. In summary, in this study, a different interpretation is made for the concept of space curve of constant breadth, the first time. Then this interpretation is used to obtain a characterization. As a result, this characterization we?ve obtained is solved.

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