scholarly journals A Different View on Dynamics of Space Curves Geometry

2021 ◽  
Author(s):  
Yılmaz Tunçer
Keyword(s):  
Space Curve ◽  
Regular Space ◽  
Position Vector ◽  
Space Curves

AbstractIn this study, we define the X-torque curves, $$X-$$ X - equilibrium curves, X-moment conservative curves, $$X-$$ X - gyroscopic curves as new curves derived from a regular space curve by using the Frenet vectors of a space curve and its position vector, where $$X\in \left\{ T\left( s\right) , N\left( s\right) , B\left( s\right) \right\} $$ X ∈ T s , N s , B s and we examine these curves and we give their properties.

A pair of kinematically related space curves

2018 ◽  
Vol 15 (10) ◽  
pp. 1850180 ◽  
Author(s):  
Vishesh S. Bhat ◽  
R. Haribaskar
Keyword(s):  
Space Curve ◽  
Regular Space ◽  
Motion Trajectory ◽  
Space Curves ◽  
Related Space ◽  
Constant Pitch ◽  
Mannheim Curve ◽  
Special Case ◽  
Radius Function ◽  
Geometric Nature

We investigate the relation between two types of space curves, the Mannheim curves and constant-pitch curves and primarily explicate a method of deriving Mannheim curves and constant-pitch curves from each other by means of a suitable deformation of a space curve. We define a “radius” function and a “pitch” function for any arbitrary regular space curve and use these to characterize the two classes of curves. A few non-trivial examples of both Mannheim and constant pitch curves are discussed. The geometric nature of Mannheim curves is established by using the notion of osculating helices. The Frenet–Serret motion of a rigid body in theoretical kinematics is studied for the special case of a Mannheim curve and the axodes in this case are deduced. In particular, we show that the fixed axode is developable if and only if the motion trajectory is a Mannheim curve.

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.

Polyhedral Space Curve Meshing Reducer With Multiple Output Shafts

2012 ◽  
Author(s):  
Yangzhi Chen ◽  
Jiang Ding ◽  
Chuanghai Yao ◽  
Yueling Lv
Keyword(s):  
Design Method ◽  
Low Cost ◽  
Space Curve ◽  
Compact Structure ◽  
Input Shaft ◽  
Space Curves ◽  
Rotation Axes ◽  
Polyhedral Space ◽  
Driving Wheel ◽  
Single Input

In recent years, a gear named Space Curve Meshing Wheel (SCMW) has been invented based on the meshing theory of space curves instead of classic space surfaces. Well improved in many aspects after its invention, it has been applied within the Space Curve Meshing Reducer (SCMR). The design method of an invention named polyhedral SCMR is presented in this paper. With single input shaft and multiple output shafts, this SCMR has advantages like compact structure, flexible design and low cost. It is characterized by the application of the SCMW group containing one driving wheel and several driven wheels, whose rotation axes are concurrent at a point and radiate in polyhedral directions. A SCMW group can form a single-stage SCMR, while SCMW groups connected can form a multiple-stage SCMR. In this paper, geometric parameters of the polyhedral SCMR are defined, design formulas are derived, and an example is provided to illustrate the design process.

Integral closure of some co-ordinate rings

1975 ◽  
Vol 20 (1) ◽  
pp. 115-123
Author(s):  
David J. Smith
Keyword(s):  
Integral Closure ◽  
Principal Result ◽  
Space Curve ◽  
Coordinate Ring ◽  
Unique Factorization ◽  
Algebraic Space ◽  
Space Curves ◽  
Integrally Closed ◽  
Coordinate Rings ◽  
Special Case

In this paper, some methods are developed for obtaining explicitly a basis for the integral closure of a class of coordinate rings of algebraic space curves.The investigation of this problem was motivated by a need for examples of integrally closed rings with specified subrings with a view toward examining questions of unique factorization in them. The principal result, giving the elements to be adjoined to a ring of the form k[x1, …,xn] to obtain its integral closure, is limited to the rather special case of the coordinate ring of a space curve all of whose singularities are normal. But in numerous examples where the curve has nonnormal singularities, the same method, which is essentially a modification of the method of locally quadratic transformations, also gives the integral closure.

scholarly journals Evolutes and focal surfaces of framed immersions in the Euclidean space

2019 ◽  
Vol 150 (1) ◽  
pp. 497-516 ◽  
Author(s):  
Shun'ichi Honda ◽  
Masatomo Takahashi
Keyword(s):  
Euclidean Space ◽  
Smooth Curve ◽  
Singular Points ◽  
Moving Frame ◽  
Regular Space ◽  
Space Curves ◽  
Focal Surface

AbstractWe consider a smooth curve with singular points in the Euclidean space. As a smooth curve with singular points, we have introduced a framed curve or a framed immersion. A framed immersion is a smooth curve with a moving frame and the pair is an immersion. We define an evolute and a focal surface of a framed immersion in the Euclidean space. The evolutes and focal surfaces of framed immersions are generalizations of each object of regular space curves. We give relationships between singularities of the evolutes and of the focal surfaces. Moreover, we consider properties of the evolutes, focal surfaces and repeated evolutes.

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