Solutions to the Vector Tetrahedron Equation

1965 ◽  
Vol 87 (2) ◽  
pp. 228-234 ◽  
Author(s):  
Milton A. Chace
Keyword(s):  
Closed Form ◽  
Digital Computer ◽  
Three Dimensional ◽  
Dimensional Vector ◽  
Spherical Coordinates ◽  
Tetrahedron Equation ◽  
Position Determination ◽  
Closed Form Solutions

A set of nine closed-form solutions are presented to the single, three-dimensional vector tetrahedron equation, sum of vectors equals zero. The set represents all possible combinations of unknown spherical coordinates among the vectors, assuming the coordinates are functionally independent. Optimum use is made of symmetry. The solutions are interpretable and can be evaluated reliably by digital computer in milliseconds. They have been successfully applied to position determination of many three-dimensional mechanisms.

scholarly journals Reconstruction of three-dimensional maps based on closed-form solutions of the variational problem of multisensor data registration

2019 ◽  
Vol 484 (6) ◽  
pp. 672-677
Author(s):  
A. V. Vokhmintcev ◽  
A. V. Melnikov ◽  
K. V. Mironov ◽  
V. V. Burlutskiy
Keyword(s):  
Closed Form ◽  
Large Scale ◽  
Three Dimensional ◽  
Dynamic Environment ◽  
Closed Form Solution ◽  
Point Clouds ◽  
Accurate Method ◽  
Form Solution ◽  
Closed Form Solutions ◽  
Reconstruction Methods

A closed-form solution is proposed for the problem of minimizing a functional consisting of two terms measuring mean-square distances for visually associated characteristic points on an image and meansquare distances for point clouds in terms of a point-to-plane metric. An accurate method for reconstructing three-dimensional dynamic environment is presented, and the properties of closed-form solutions are described. The proposed approach improves the accuracy and convergence of reconstruction methods for complex and large-scale scenes.

Kinematics of a Novel Single-Loop Under-Actuated Wrist

2012 ◽  
Author(s):  
Raffaele Di Gregorio
Keyword(s):  
Path Planning ◽  
Closed Form ◽  
Nonholonomic Constraint ◽  
Three Dimensional ◽  
Single Loop ◽  
Closed Form Solutions ◽  
Position Analysis ◽  
Tracking Tasks ◽  
Planning Algorithm ◽  
Path Planning Algorithm

A novel type of parallel wrist (PW) is proposed which, differently from previously presented PWs, features a single-loop architecture and only one nonholonomic constraint. Due to the presence of a nonholonomic constraint, the proposed PW type is under-actuated, that is, it is able to control the platform orientation in a three-dimensional workspace by employing only two actuated pairs, one prismatic (P) and the other revolute (R); and it cannot perform tracking tasks. Position analysis and path planning of this novel PW are studied. In particular, all the relevant position analysis problems are solved in closed form, and, based on these closed-form solutions, a path-planning algorithm is built.

scholarly journals Use of a Strongly Nonlinear Gambier Equation for the Construction of Exact Closed Form Solutions of Nonlinear ODEs

2007 ◽  
Vol 2007 ◽  
pp. 1-25
Author(s):  
M. P. Markakis
Keyword(s):  
Closed Form ◽  
Initial Conditions ◽  
Closed Form Solution ◽  
Form Solution ◽  
Nonlinear Odes ◽  
Analytical Technique ◽  
Closed Form Solutions ◽  
Strongly Nonlinear ◽  
Parameter Values

We establish an analytical method leading to a more general form of the exact solution of a nonlinear ODE of the second order due to Gambier. The treatment is based on the introduction and determination of a new function, by means of which the solution of the original equation is expressed. This treatment is applied to another nonlinear equation, subjected to the same general class as that of Gambier, by constructing step by step an appropriate analytical technique. The developed procedure yields a general exact closed form solution of this equation, valid for specific values of the parameters involved and containing two arbitrary (free) parameters evaluated by the relevant initial conditions. We finally verify this technique by applying it to two specific sets of parameter values of the equation under consideration.

Closed-Form Displacement Analysis of SDOF 8 Link Mechanisms

1996 ◽  
Author(s):  
Abdulaziz N. Almadi ◽  
Anoop K. Dhingra ◽  
Dilip Kohli
Keyword(s):  
Closed Form ◽  
Analysis Problem ◽  
Algebraic Form ◽  
Single Degree Of Freedom ◽  
Displacement Analysis ◽  
Kinematic Chains ◽  
Closed Form Solutions ◽  
Single Degree ◽  
Half Angle

Abstract This paper presents closed-form solutions to the displacement analysis problem of planar 8-link mechanisms with a single degree of freedom (SDOF). The degrees of I/O polynomials as well as the number of possible assembly configurations for all 71 8-link mechanisms resulting from 16 8-link kinematic chains are presented. Three numerical examples illustrating the applicability of the successive elimination procedure to the displacement analysis of 8-link mechanisms are presented. The first example deals with the determination of I/O polynomial for an 8-link mechanism containing no four-bar loops. The second and third examples, address in detail, some of the problems associated with the conversion of transcendental loop-closure equations into an algebraic form using tangent half-angle substitutions. These examples illustrate how extraneous roots can get introduced during the displacement analysis of mechanisms, and how one can derive an I/O polynomial devoid of the extraneous roots. Extensions of the proposed approach to the displacement analysis of SDOF spherical 8-link mechanisms is also presented.

Nonlinear Spatial Equilibria and Stability of Cables Under Uni-axial Torque and Thrust

1994 ◽  
Vol 61 (4) ◽  
pp. 879-886 ◽  
Author(s):  
C.-L. Lu ◽  
N. C. Perkins
Keyword(s):  
Closed Form ◽  
Elliptic Integral ◽  
Broad Class ◽  
Local Equilibrium ◽  
Three Dimensional ◽  
Equilibrium States ◽  
Form Complex ◽  
Closed Form Solutions ◽  
The Stability ◽  
Low Tension

Low tension cables subject to torque may form complex three-dimensional (spatial) equilibria. The resulting nonlinear static deformations, which are dominated by cable flexure and torsion, may produce interior loops or kinks that can seriously degrade the performance of the cable. Using Kirchhoffrod assumptions, a theoretical model governing cable flexure and torsion is derived herein and used to analyze (1) globally large equilibrium states, and (2) local equilibrium stability. For the broad class of problems described by pure boundary loading, the equilibrium boundary value problem is integrable and admits closed-form elliptic integral solutions. Attention is focused on the example problem of a cable subject to uni-axial torque and thrust. Closed-form solutions are presented for the complex three-dimensional equilibrium states which, heretofore, were analyzed using purely numerical methods. Moreover, the stability of these equilibrium states is assessed and new and important stability conclusions are drawn.

Closed-Form Solutions for Code Case 2286 Allowable Compressive Stresses Analogous to Vacuum Chart Method

2012 ◽  
Author(s):  
Harsh Kumar Baid ◽  
Donald LaBounty ◽  
Amiya Chatterjee
Keyword(s):  
Closed Form ◽  
Compressive Stress ◽  
Pressure Vessels ◽  
Closed Form Solutions ◽  
Compressive Stresses ◽  
Division 1 ◽  
Section Viii ◽  
Allowable Compressive Stress ◽  
User Friendly

The allowable compressive stresses in pressure vessels can be calculated either from ASME Section VIII Division 1, Paragraph UG-28 vacuum chart method [2] or Code Case 2286 [1]. Code Case 2286 has been incorporated into ASME Section VIII Division 2, Part 5. For Division 1 vessels, the vacuum chart method is a user-friendly tool for determining allowable compressive stress. In this paper, the authors present the development of allowable compressive stress data based on closed-form solutions of Code Case 2286. These closed-form solutions yield exact allowable compressive stress values which are not influenced by any kind of sensitivity. The development presented in the paper is also user-friendly, similar to the vacuum chart, for the determination of allowable compressive stresses. These designs, based on Code Case 2286, are economical without any compromise in the safety of the pressure vessel. Examples are included to demonstrate the results.

On the Efficiency of Analyzing 3D Anisotropic, Transversely Isotropic, and Isotropic Bodies in BEM

2010 ◽  
Vol 26 (4) ◽  
pp. 483-491
Author(s):  
Y. C. Shiah ◽  
W. X. Sun
Keyword(s):  
Closed Form ◽  
Computational Efficiency ◽  
Transverse Isotropy ◽  
Three Dimensional ◽  
Transversely Isotropic ◽  
Fundamental Solutions ◽  
Engineering Practice ◽  
Closed Form Solutions ◽  
Anisotropic Bodies ◽  
Isotropic Bodies

ABSTRACTDue to a lack of closed-form solutions for three dimensional anisotropic bodies, the computational burden of evaluating the fundamental solutions in the boundary element method (BEM) has been a research focus over the years. In engineering practice, transversely isotropic material has gained popularity in the use of composites. As a degenerate case of the generally anisotropic material, transverse isotropy still needs to be treated separately to ease the computations. This paper aims to investigate the computational efficiency of the BEM implementations for 3D anisotropic, transversely isotropic, and isotropic bodies. For evaluating the fundamental solutions of 3D anisotropy, the explicit formulations reported in [1,2] are implemented. For treating transversely isotropic materials, numerous closed form solutions have been reported in the literature. For the present study, the formulations presented by Pan and Chou [3] are particularly employed. At the end, a numerical example is presented to compare the computational efficiency of the three cases and to demonstrate how the CPU time varies with the number of meshes.

Automatic Closed-Form Kinematics-Solutions for Recursive Single-Loop Chains

1992 ◽  
Author(s):  
Andrés Kecskeméthy ◽  
Manfred Hiller
Keyword(s):  
Closed Form ◽  
Isotropy Group ◽  
Projection Operators ◽  
Closure Condition ◽  
Single Loop ◽  
Closed Form Solutions ◽  
Point Line ◽  
Geometric Elements ◽  
Scalar Equations

Abstract Described in this paper is a simplified method for the automatic detection and formulation of closed-form solutions for a special class of recursively solvable single-loop mechanisms. The objective is to generate a cascade of scalar equations from the closure condition of the loop, each containing exactly one unknown more than the predecessors, and each being maximally second-order in this unknown. In the proposed method this problem is reduced to the repeated determination of two subchains which are members of the isotropy group of either of the geometric elements point, line and plane and containing as much current unknowns as possible. The scalar equations then arise from unique projection operators applied to unique equation partitionings of the closure condition. This yields a general but easy-to-implement algorithm. The concepts are illustrated with some examples processed with an implementation in Mathematica based on the geometric elements point and plane.

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