A Complex Number Approach to the Generation of the Cubic of Stationary Curvature

1995 ◽  
Author(s):  
L. G. Johnson ◽  
T. R. Chase
Keyword(s):  
Closed Form ◽  
Complex Number ◽  
Closed Form Solution ◽  
Form Solution ◽  
New Approach ◽  
Natural Sequence

Abstract A complex number approach to generating the cubic of stationary curvature (CSC) has been developed. The approach provides a closed form solution for generating points on the curve. The new approach eliminates the need for considering the Euler-Savary equation and centrode curvature as intermediate steps for obtaining points on the curve. Furthermore, the method guarantees that the points will be generated in their natural sequence and it simultaneously produces points on the centerpoint and circlepoint curves. The method can be applied to analyze an existing linkage or to synthesize a linkage to produce a coupler curve with specified stationary curvature at one position. Two analysis and one synthesis examples are provided.

Rupture Instability of Plane Membranes and Solids

1960 ◽  
Vol 27 (3) ◽  
pp. 501-504
Author(s):  
S. F. Borg
Keyword(s):  
Closed Form ◽  
Continuum Mechanics ◽  
Closed Form Solution ◽  
Practical Interest ◽  
Form Solution ◽  
Compatibility Conditions ◽  
New Approach ◽  
Conservation Equations ◽  
Dynamic Phenomena

A fundamentally new approach to the rupture-fracture problem is presented. Because of the particular type of dynamic phenomena being investigated, the formulation is given in terms of the conservation equations of continuum mechanics instead of in the usual elasticity-plasticity relations. The introduction of a similarity co-ordinate permits a complete closed-form solution to a particular problem of practical interest subject to certain compatibility conditions which depend upon the specific properties of the material under consideration.

Structural Error Synthesis of a Four-Bar Function Generator Using the Reliability Concept

1984 ◽  
Vol 198 (2) ◽  
pp. 87-90 ◽  
Author(s):  
A K Khare ◽  
A C Rao
Keyword(s):  
Closed Form ◽  
Closed Form Solution ◽  
Form Solution ◽  
Optimization Techniques ◽  
Function Generator ◽  
New Approach ◽  
Numerical Example ◽  
Structural Error ◽  
Synthesis Of Mechanisms

Structural error synthesis of mechanisms is usually carried out either by the precision point approach or by using optimization techniques. A new approach for such problems using the reliability concept is presented in this paper. Besides being simple, this approach leads to a closed form solution and the mechanism can be designed to perform with any desired reliability. Its application is illustrated by means of a numerical example and the results are compared with those available.

A Complex Number Approach for Absolute Precision Position Synthesis for Three Precision Positions

1992 ◽  
Author(s):  
John A. Mirth
Keyword(s):  
Closed Form ◽  
Complex Number ◽  
Closed Form Solution ◽  
Form Solution ◽  
Path Generation ◽  
Motion Generation ◽  
The Absolute ◽  
Position Problem ◽  
Position Synthesis

Abstract Dyads can be synthesized by prescribing the precision point coordinates and the absolute planar orientations of one dyad vector at each of three precision positions. This differs from traditional complex number methods wherein the vector orientations are described relative to one another. Absolute precision position synthesis can be performed for both motion generation, and path generation with prescribed timing. The method presented uses vector loop equations and complex number notation to produce a closed form solution for the three absolute precision position problem. Absolute precision position synthesis is applicable to cases that require specific coupler geometries. The synthesis of flat-folding mechanisms is an example of one such application.

On a Closed-Form Solution of Prandtl's System of Equations

2013 ◽  
Vol 40 (2) ◽  
pp. 106-114
Author(s):  
J. Venetis ◽  
Aimilios (Preferred name Emilios) Sideridis
Keyword(s):  
Closed Form ◽  
Closed Form Solution ◽  
Form Solution ◽  
System Of Equations

scholarly journals A Closed-Form Solution to Planar Feature-Based Registration of LiDAR Point Clouds

2021 ◽  
Vol 10 (7) ◽  
pp. 435
Author(s):  
Yongbo Wang ◽  
Nanshan Zheng ◽  
Zhengfu Bian
Keyword(s):  
Closed Form ◽  
Closed Form Solution ◽  
Simulated Data ◽  
Real Data ◽  
Point Clouds ◽  
Form Solution ◽  
Spatial Transformation ◽  
Dual Quaternions ◽  
Feature Based ◽  
Planar Feature

Since pairwise registration is a necessary step for the seamless fusion of point clouds from neighboring stations, a closed-form solution to planar feature-based registration of LiDAR (Light Detection and Ranging) point clouds is proposed in this paper. Based on the Plücker coordinate-based representation of linear features in three-dimensional space, a quad tuple-based representation of planar features is introduced, which makes it possible to directly determine the difference between any two planar features. Dual quaternions are employed to represent spatial transformation and operations between dual quaternions and the quad tuple-based representation of planar features are given, with which an error norm is constructed. Based on L2-norm-minimization, detailed derivations of the proposed solution are explained step by step. Two experiments were designed in which simulated data and real data were both used to verify the correctness and the feasibility of the proposed solution. With the simulated data, the calculated registration results were consistent with the pre-established parameters, which verifies the correctness of the presented solution. With the real data, the calculated registration results were consistent with the results calculated by iterative methods. Conclusions can be drawn from the two experiments: (1) The proposed solution does not require any initial estimates of the unknown parameters in advance, which assures the stability and robustness of the solution; (2) Using dual quaternions to represent spatial transformation greatly reduces the additional constraints in the estimation process.

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