scholarly journals A Note on the Acceleration and Jerk in Motion Along a Space Curve

2020 ◽  
Vol 28 (1) ◽  
pp. 151-164
Author(s):  
Kahraman Esen Özen ◽  
Mehmet Güner ◽  
Murat Tosun
Keyword(s):  
Time Derivative ◽  
Space Curve ◽  
Space Curves ◽  
Acceleration Vector ◽  
Orthogonal Frame

AbstractThe resolution of the acceleration vector of a particle moving along a space curve is well known thanks to Siacci [1]. This resolution comprises two special oblique components which lie in the osculating plane of the curve. The jerk is the time derivative of acceleration vector. For the jerk vector of the aforementioned particle, a similar resolution is presented as a new contribution to field [2]. It comprises three special oblique components which lie in the osculating and rectifying planes. In this paper, we have studied the Siacci’s resolution of the acceleration vector and aforementioned resolution of the jerk vector for the space curves which are equipped with the modified orthogonal frame. Moreover, we have given some illustrative examples to show how the our theorems work.

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 Evolution of space curves and the special ruled surfaces with modified orthogonal frame

2020 ◽  
Vol 5 (3) ◽  
pp. 2027-2039 ◽  
Author(s):  
Kemal Eren ◽  
◽  
Hidayet Huda Kosal
Keyword(s):  
Ruled Surfaces ◽  
Space Curves ◽  
Orthogonal Frame

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.

Accurate Representation of the Rail Geometry for Multibody System Applications

2009 ◽  
Vol 5 (1) ◽  
Author(s):  
Brian Marquis ◽  
Khaled E. Zaazaa ◽  
Tariq Sinokrot ◽  
Ahmed A. Shabana
Keyword(s):  
Space Curve ◽  
Contact Forces ◽  
Railway Track ◽  
Bank Angle ◽  
Space Curves ◽  
The Right ◽  
Yaw Angle ◽  
Horizontal Curvature ◽  
The Relationship

The objective of this study is to examine the geometric description of the spiral sections of railway track systems, in order to correctly define the relationship between the geometry of the right and left rails. The geometry of the space curves that define the rails are expressed in terms of the geometry of the space curve that defines the track center curve. Industry inputs such as the horizontal curvature, grade, and superelevation are used to define the track centerline space curve in terms of Euler angles. The analysis presented in this study shows that, in the general case of a spiral, the profile frames of the right and left rails that have zero yaw angles with respect to the track frame have different orientations. As a consequence, the longitudinal tangential creep forces acting on the right and left wheels, in the case of zero yaw angle, are not in the same direction. Nonetheless, the orientation difference between the profile frames of the right and left rails can be defined in terms of a single pitch angle. In the case of small bank angle that defines the superelevation of the track, one can show that this angle directly contributes to the track elevation. The results obtained in this study also show that the right and left rail longitudinal tangents can be parallel only in the case of a constant horizontal curvature. Since the spiral is used to connect track segments with different curvatures, the horizontal curvature cannot be assumed constant, and as a consequence, the right and left rail longitudinal tangents cannot be considered parallel in the spiral region. Numerical examples that demonstrate the effect of the errors that result from the assumption that the right and left rails in the spiral sections have the same geometry are presented. The numerical results obtained show that these errors can have a significant effect on the quality of the predicted creep contact forces.

scholarly journals Geometric Characterizations of Canal Surfaces in Minkowski 3-Space II

Mathematics ◽  
2019 ◽  
Vol 7 (8) ◽  
pp. 703 ◽  
Author(s):  
Jinhua Qian ◽  
Mengfei Su ◽  
Xueshan Fu ◽  
Seoung Dal Jung
Keyword(s):  
Gaussian Curvature ◽  
Mean Curvature ◽  
Space Curve ◽  
Timelike Curve ◽  
Geometric Properties ◽  
Spacelike Curve ◽  
Space Curves ◽  
Canal Surfaces ◽  
The Mean ◽  
The Relationship

Canal surfaces are defined and divided into nine types in Minkowski 3-space E 1 3 , which are obtained as the envelope of a family of pseudospheres S 1 2 , pseudohyperbolic spheres H 0 2 , or lightlike cones Q 2 , whose centers lie on a space curve (resp. spacelike curve, timelike curve, or null curve). This paper focuses on canal surfaces foliated by pseudohyperbolic spheres H 0 2 along three kinds of space curves in E 1 3 . The geometric properties of such surfaces are presented by classifying the linear Weingarten canal surfaces, especially the relationship between the Gaussian curvature and the mean curvature of canal surfaces. Last but not least, two examples are shown to illustrate the construction of such surfaces.

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.

scholarly journals Siacci's Theorem According to Darboux Frame

2017 ◽  
Vol 25 (3) ◽  
pp. 155-165
Author(s):  
Kahraman Esen Ozen ◽  
Murat Tosun ◽  
Mahmut Akyiğit
Keyword(s):  
Euclidean Space ◽  
Rigid Body ◽  
Space Curve ◽  
Dimensional Euclidean Space ◽  
3 Dimensional ◽  
Acceleration Vector ◽  
Darboux Frame

Abstract The resolution of the acceleration vector of rigid body moving along a space curve is well known thanks to Siacci [1]. In this resolution, the acceleration vector is stated as the sum of two special oblique components in the osculating plane of the point of the curve. In this paper, we have studied the Siacci’s theorem for the curves on regular surfaces in 3-dimensional Euclidean space. Also, an example is given for a helix lying on a cylinder.

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