Curvature Theory of Envelope Curve in Two-Dimensional Motion and Envelope Surface in Three-Dimensional Motion

2015 ◽  
Vol 7 (3) ◽  
Author(s):  
Wei Wang ◽  
Delun Wang
Keyword(s):  
Three Dimensional ◽  
Two Dimensional ◽  
Spatial Motion ◽  
Envelope Curve ◽  
Contact Conditions ◽  
Numerical Examples ◽  
Envelope Surface ◽  
Straight Line ◽  
Curvature Theory ◽  
Fixed Line

The curvature theories for envelope curve of a straight line in planar motion and envelope ruled surface of a plane in spatial motion are systematically presented in differential geometry language. Based on adjoint curve and adjoint surface methods as well as quasi-fixed line and quasi-fixed plane conditions, the centrode and axode are taken as two logical starting-points to study kinematic and geometric properties of the envelope curve of a line in two-dimensional motion and the envelope surface of a plane in three-dimensional motion. The analogical Euler–Savary equation of the line and the analogous infinitesimal Burmester theories of the plane are thoroughly revealed. The contact conditions of the plane-envelope and some common surfaces, such as circular and noncircular cylindrical surface, circular conical surface, and involute helicoid are also examined, and then the positions and dimensions of different osculating ruled surfaces are given. Two numerical examples are presented to demonstrate the curvature theories.

The Third Two-Dimensional Problem of Three-Dimensional Blade Systems of Hydraulic Machines—Part 2: Analytical Results

1981 ◽  
Vol 103 (1) ◽  
pp. 42-51
Author(s):  
P. K. Agarwal ◽  
G. V. Viktorov
Keyword(s):  
Three Dimensional ◽  
Two Dimensional ◽  
Dimensional Problem ◽  
Blade Profile ◽  
Flow Conditions ◽  
Numerical Examples ◽  
The Third ◽  
Hydraulic Machines ◽  
Method Of Solution ◽  
Axisymmetric Stream

This is the second part of a study of the “third” two-dimensional problem of three-dimensional blade systems of hydraulic machines. Part I described the formulation of the problem and the proposed method of solution to determine the velocity field on surfaces orthogonal to mean axisymmetric stream surfaces. Part II presents the numerical method of solving the integral equations; a few numerical examples for actual impellers/runners are also given. The results are presented in a series of figures and tables showing the distribution of the velocity component c2 along the blade profile on the surface q1 = const. The purpose of these numerical examples is to demonstrate the method and to help create a general understanding and awareness of the flow conditions existing in the runner passage.

scholarly journals Extrapolation Algorithm for Computing Multiple Cauchy Principal Value Integrals

2020 ◽  
Vol 2020 ◽  
pp. 1-15
Author(s):  
Jin Li ◽  
Yongling Cheng
Keyword(s):  
Asymptotic Expansion ◽  
Density Function ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Cauchy Principal ◽  
Numerical Examples ◽  
Extrapolation Algorithm ◽  
Expansion Formulae ◽  
Error Functional ◽  
Principal Value

In this paper, the computation of multiple (including two dimensional and three dimensional) Cauchy principal integral with generalized composite rectangle rule was discussed, with the density function approximated by the middle rectangle rule, while the singular kernel was analytically calculated. Based on the expansion of the density function, the asymptotic expansion formulae of error functional are obtained. A series is constructed to approach the singular point, then the extrapolation algorithm is presented, and the convergence rate is proved. At last, some numerical examples are presented to validate the theoretical analysis.

Three-dimensional waypoint following for fixed-wing unmanned aerial vehicles in obstacle-filled environments

2019 ◽  
Vol 234 (3) ◽  
pp. 640-654
Author(s):  
Nikhil Kumar Singh ◽  
Sikha Hota
Keyword(s):  
Unmanned Aerial Vehicles ◽  
Optimal Path ◽  
Three Dimensional ◽  
Computation Time ◽  
Two Dimensional ◽  
Aerial Vehicles ◽  
Optimal Paths ◽  
Straight Line ◽  
Different Types ◽  
Simulation Results

The paper computes optimal paths for fixed-wing unmanned aerial vehicles with bounded turn radii to follow a series of waypoints with specified directions in a three-dimensional obstacle-filled environment. In the existing literature, it was proved that the optimal path is of circular turn–straight line–circular turn (CSC) type for two consecutive waypoint configurations, when the points are sufficiently far apart and there is no obstacle in the field. The maximum of all minimum turn radii corresponding to all possible two-dimensional circular maneuvers was used for both the initial and final turns to develop the CSC-type paths. But, this paper considers the minimum turn radii for initial and final turns, corresponding to the maneuvering planes and which produces shorter CSC-type paths. In an obstacle-filled environment the shortest path may collide with obstacles, so a strategy is proposed to switch to the next best path that does not collide with obstacles. Using this technique, a series of waypoints is followed in the presence of obstacles of different types, for example, cylindrical, hemispherical, and spherical in shapes with different sizes. Finally, simulation results are presented to show the efficiency of the algorithm for obstacle avoidance. The computation time listed here indicates the potentiality of this algorithm for implementation in real time.

scholarly journals On rigid origami I: piecewise-planar paper with straight-line creases

2019 ◽  
Vol 475 (2232) ◽  
pp. 20190215 ◽  
Author(s):  
Zeyuan He ◽  
Simon D. Guest
Keyword(s):  
Configuration Space ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Deployable Structures ◽  
Straight Line ◽  
Paper Folding ◽  
Two Dimensional Materials ◽  
Hinge Model ◽  
Special Case ◽  
Important Branch

Origami (paper folding) is an effective tool for transforming two-dimensional materials into three-dimensional structures, and has been widely applied to robots, deployable structures, metamaterials, etc. Rigid origami is an important branch of origami where the facets are rigid, focusing on the kinematics of a panel-hinge model. Here, we develop a theoretical framework for rigid origami, and show how this framework can be used to connect rigid origami and its cognate areas, such as the rigidity theory, graph theory, linkage folding and computer science. First, we give definitions regarding fundamental aspects of rigid origami, then focus on how to describe the configuration space of a creased paper. The shape and 0-connectedness of the configuration space are analysed using algebraic, geometric and numeric methods. In the algebraic part, we study the tangent space and generic rigid-foldability based on the polynomial nature of constraints for a panel-hinge system. In the geometric part, we analyse corresponding spherical linkage folding and discuss the special case when there is no cycle in the interior of a crease pattern. In the numeric part, we review methods to trace folding motion and avoid self-intersection. Our results will be instructive for the mathematical and engineering design of origami structures.

Reflection from Curved Mirrors in a Plane

2019 ◽  
Vol 7 (1) ◽  
pp. 46-54 ◽  
Author(s):  
Л. Жихарев ◽  
L. Zhikharev
Keyword(s):  
Dimensional Space ◽  
Spherical Surface ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Interesting Phenomenon ◽  
One Dimensional ◽  
Straight Line ◽  
The Family ◽  
Dimensional Object ◽  
Three Dimensional Space

Reflection from a certain mirror is one of the main types of transformations in geometry. On a plane a mirror represents a straight line. When reflecting, we obtain an object, each point of which is symmetric with respect to this straight line. In this paper have been considered examples of reflection from a circle – a general case of a straight line, if the latter is defined through a circle of infinite radius. While analyzing a simple reflection and generalization of this process to the cases of such curvature of the mirror, an interesting phenomenon was found – an increase in the reflection dimension by one, that is, under reflection of a one-dimensional object from the circle, a two-dimensional curve is obtained. Thus, under reflection of a point from the circle was obtained the family of Pascal's snails. The main cases, related to reflection from a circular mirror the simplest two-dimensional objects – a segment and a circle at their various arrangement, were also considered. In these examples, the reflections are two-dimensional objects – areas of bizarre shape, bounded by sections of curves – Pascal snails. The most interesting is the reflection of two-dimensional objects on a plane, because the reflection is too informative to fit in the appropriate space. To represent the models of obtained reflections, it was proposed to move into three-dimensional space, and also developed a general algorithm allowing obtain the object reflection from the curved mirror in the space of any dimension. Threedimensional models of the reflections obtained by this algorithm have been presented. This paper reveals the prospects for further research related to transition to three-dimensional space and reflection of objects from a spherical surface (possibility to obtain four-dimensional and five-dimensional reflections), as well as studies of reflections from geometric curves in the plane, and more complex surfaces in space.

A Three-Dimensional Deterministic Model for Rough Surface Line-Contact EHL Problems

2007 ◽  
Author(s):  
Ning Ren ◽  
W. Wayne Chen ◽  
Dong Zhu ◽  
Yuchuan Liu ◽  
Q. Jane Wang
Keyword(s):  
Rough Surface ◽  
Present Model ◽  
Deterministic Model ◽  
Three Dimensional ◽  
Line Contact ◽  
Two Dimensional ◽  
Numerical Examples ◽  
Asperity Contacts ◽  
Line Contacts ◽  
2D Model

This paper reports the development of a novel three-dimensional (3D) deterministic model for rough surface line-contact mixed-EHL problems. This model is of great importance because line contacts are found in many mechanical components. The macro aspects of a line-contact problem can be simplified into a two-dimensional (2D) model, but the topography of contacting rough surfaces, micro asperity contacts, and lubricant flows around asperities are often 3D. The present model is based on Hu and Zhu’s unified mixed EHL model [1] and the mixed FFT-based approach formulated by Chen et al [2]. It is numerically verified through comparisons with results from conventional 2D line-contact EHL theories. Numerical examples involving sinusoidal roughness and digitized 3D machined surfaces are analyzed.

Skill Acquisition in Multi-Dimensional Manipulator Control

1967 ◽  
Vol 9 (2) ◽  
pp. 133-139 ◽  
Author(s):  
Gershon Weltman ◽  
Aharon Nachson ◽  
Hilde Groth
Keyword(s):  
Skill Acquisition ◽  
Movement Time ◽  
Three Dimensional ◽  
Dimensional Case ◽  
Maximum Deviation ◽  
Two Dimensional ◽  
Vertical Movements ◽  
Straight Line ◽  
Line Path ◽  
Manipulator Control

Movements of a three-jointed electrically-powered manipulator were controlled by vertical movements of the second, third and fourth fingers of the subject's hand. Both two-dimensional and three-dimensional movement problems were examined. In the two-dimensional case, subjects were shown a silhouette of the manipulator with a lit endpoint or the manipulator endpoint alone. The manipulator was always fully visible in the three-dimensional case. Recordings were made of time-to-target, maximum deviation from a straight line path, and the percent of time that various numbers of controls were activated simultaneously. The results indicated that with practice the subjects tended to approach targets on a straight line course in both situations. They also increased the percentage of time that several controls were activated together. Being able to see the manipulator improved control coordination and decreased movement time in the two-dimensional case, but did not affect movement accuracy.

Simple but Efficient Method for Integrating 1/r Singularities

2013 ◽  
Vol 444-445 ◽  
pp. 615-620
Author(s):  
Feng Liu ◽  
Hong Zheng ◽  
Chun Guang Li
Keyword(s):  
Efficient Method ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Numerical Examples ◽  
Integration Schemes ◽  
Integration Techniques ◽  
Elastic Fracture ◽  
Duffy Transformation

New integration schemes are presented for integrands with singularity of 1/r. We partition the element with a singular center into several triangles sharing the center. Then, a transformation between a standard square and each of the triangles is conducted. We prove such a transformation itself brings about the Jacobian with the factor r, leading to no need to introduce any other transformation. Both two-dimensional and three-dimensional cases are considered. Compared to the Duffy transformation, the proposed methods enjoy more excellent numerical properties. Numerical examples in elastic fracture are also presented to illustrate the performance of the new integration techniques.

Capacitance Calculation for Offset Via Structures Using an Integral Approximation Approach Based on Finite Element Method

2010 ◽  
Vol 2010 (1) ◽  
pp. 000408-000412
Author(s):  
Hanfeng Wang ◽  
Yaojiang Zhang ◽  
James L. Drewniak ◽  
Jun Fan ◽  
Bruce Archambeault
Keyword(s):  
Finite Element Method ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Combined Method ◽  
Integral Approximation ◽  
Approximation Approach ◽  
Numerical Examples ◽  
Cpu Time ◽  
Fem Method ◽  
Similar Accuracy

A simple yet efficient approach is presented to extract the via-plane capacitances for an offset via structure. According to the integral approximation approach, the geometry of offset via is first divided into several segments with equally distributed angles from the origin. The two-dimensional FEM method for the concentric case is used for each segment based on its pad-stack parameters. Then, the final offset via-plane capacitance is approximated as the average of these ‘segmental’ capacitance values. Numerical examples demonstrated that the combined method has similar accuracy with a three-dimensional solver but it has much higher efficiency in both CPU time and memory cost.

scholarly journals Design of three-dimensional origami structures based on a vertex approach

2015 ◽  
Vol 471 (2181) ◽  
pp. 20150407 ◽  
Author(s):  
Xiang Zhou ◽  
Hai Wang ◽  
Zhong You
Keyword(s):  
Coordinate System ◽  
Three Dimensional ◽  
New Method ◽  
Cartesian Coordinate System ◽  
Engineering Applications ◽  
Line Segments ◽  
Numerical Examples ◽  
Input Point ◽  
Straight Line ◽  
Cartesian Coordinate

Origami geometric design is fundamental to many engineering applications of origami structures. This paper presents a new method for the design of three-dimensional (3D) origami structures suitable for engineering use. Using input point sets specified, respectively, in the x − z and y − z planes of a Cartesian coordinate system, the proposed method generates the coordinates of the vertices of a folded origami structure, whose fold lines are then defined by straight line segments each connecting two adjacent vertices. It is mathematically guaranteed that the origami structures obtained by this method are developable. Moreover, an algorithm to simulate the unfolding process from designed 3D configurations to planar crease patterns is provided. The validity and versatility of the proposed method are demonstrated through several numerical examples ranging from Miura-Ori to cylinder and curved-crease designs. Furthermore, it is shown that the proposed method can be used to design origami structures to support two given surfaces.

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