Optimization of Curvature and Torsion of the Curve Based on the Basic Vector of the Space Curve

2017 ◽  
Author(s):  
Liu Donghai ◽  
Lu liyu
Keyword(s):  
Space Curve ◽  
Basic Vector ◽  
Curvature And Torsion

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.

Noncommutativity of Finite Rotations and Definitions of Curvature and Torsion

2019 ◽  
Vol 14 (9) ◽  
Author(s):  
Ahmed A. Shabana ◽  
Hao Ling
Keyword(s):  
Euler Angle ◽  
Rigid Body Dynamics ◽  
Space Curve ◽  
Euler Angles ◽  
Arc Length ◽  
Important Conclusion ◽  
Finite Rotations ◽  
Definition Of ◽  
Curvature And Torsion ◽  
Curve Geometry

The geometry of a space curve, including its curvature and torsion, can be uniquely defined in terms of only one parameter which can be the arc length parameter. Using the differential geometry equations, the Frenet frame of the space curve is completely defined using the curve equation and the arc length parameter only. Therefore, when Euler angles are used to describe the curve geometry, these angles are no longer independent and can be expressed in terms of one parameter as field variables. The relationships between Euler angles used in the definition of the curve geometry are developed in a closed-differential form expressed in terms of the curve curvature and torsion. While the curvature and torsion of a space curve are unique, the Euler-angle representation of the space curve is not unique because of the noncommutative nature of the finite rotations. Depending on the sequence of Euler angles used, different expressions for the curvature and torsion can be obtained in terms of Euler angles, despite the fact that only one Euler angle can be treated as an independent variable, and such an independent angle can be used as the curve parameter instead of its arc length, as discussed in this paper. The curve differential equations developed in this paper demonstrate that the curvature and torsion expressed in terms of Euler angles do not depend on the sequence of rotations only in the case of infinitesimal rotations. This important conclusion is consistent with the definition of Euler angles as generalized coordinates in rigid body dynamics. This paper generalizes this definition by demonstrating that finite rotations cannot be directly associated with physical geometric properties or deformation modes except in the cases when infinitesimal-rotation assumptions are used.

The form of the tangent-developable at points of zero torsion on space curves

1980 ◽  
Vol 88 (3) ◽  
pp. 403-407 ◽  
Author(s):  
J. P. Cleave
Keyword(s):  
Power Series ◽  
Space Curve ◽  
Arc Length ◽  
Space Curves ◽  
Normal Plane ◽  
Tangent Developable ◽  
Cuspidal Edge ◽  
Curvature And Torsion ◽  
Image Position ◽  
Classical Derivation

A tangent-developable is a surface generated by the tangent lines of a space curve. The intersection of a tangent-developable with the normal plane at a point P of the curve generally has a cusp at that point. Thus the tangent-developable of a space curve has a cuspidal edge along the curve. The classical derivation of this phenomenon takes the trihedron (t, n, b) at P as coordinate axes to which the curve is referred. Then the intersection of the part of the tangent-developable generated by tangent lines at points close to P with the normal plane at P (i.e. the plane through P containing n and b) is given parametrically by power serieswhere K, T are the curvature and torsion, respectively, of the curve at P and s is arc-length measured from P ((2) p. 68). It is tacitly understood in this analysis that curvature and torsion are both defined and non-zero.

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.

COMPUTATIONAL DESCRIPTION OF THE GEOMETRY OF ALIGNED CARBON NANOTUBES IN POLYMER NANOCOMPOSITES

2021 ◽  
Author(s):  
STEPAN V. LOMOV ◽  
JEONYOON LEEJEONYOON LEE ◽  
BRIAN L. WARDLE ◽  
NIKITA A. GUDKOV ◽  
ISKANDER S. AKHATOV ◽  
...  
Keyword(s):  
Carbon Nanotubes ◽  
Polymer Nanocomposites ◽  
Finite Element Modelling ◽  
Volume Fraction ◽  
Voronoi Tessellation ◽  
Geometrical Model ◽  
Limiting Factor ◽  
Full Paper ◽  
Aligned Carbon Nanotubes ◽  
Curvature And Torsion

The paper considers nanocomposites, reinforced with aligned carbon nanotubes (A- CNTs). Nominally aligned, the CNTs in the forest are wavy, which has important consequences in downgraded mechanical properties, and influences electric and thermal performance. The most detailed geometrical model of A-CNTs was proposed by Stein and Wardle (Nanotechnology, 27:035701, 2015). It creates a centerline trajectory of a CNT in steps, each step defining a section of the CNT, growing in the alignment direction with certain deviations. The paper, starting from this framework, formulates a model of the CNT geometry, which is based on the concept of correlation length of the CNT waviness and maximum admissible CNT curvature and torsion. The value of the maximum curvature can be linked to the buckling criteria for CNTs, or derived from ab initio and finite element modelling. It is used as a limiting factor for the growth, defining the waviness and tortuosity of the CNTs. The CNTs in the forest are placed in a random non-regular way, using Voronoi tessellation. The full paper includes investigation of the proposed algorithm for several values of the CNT volume fraction (in the range 0.5%…8%), the dependency of the modelled geometry on the curvature, and the apparent twist of the CNT centerlines. The modelling results are compared with experimental observations in 3D TEM imaging.

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