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.

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.

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.

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 SIMPLE STATIONARY CONFIGURATIONS OF SINGLE-LOOP SPATIAL N-REVOLUTE OVERCONSTRAINED LINKAGES

1996 ◽  
Vol 20 (1) ◽  
pp. 17-39 ◽  
Author(s):  
Chung-Ching Lee
Keyword(s):  
Computer Graphics ◽  
Differential Equations ◽  
Range Of Motion ◽  
Closed Form ◽  
Single Loop ◽  
Closed Form Solutions ◽  
Systematic Formulation ◽  
Overconstrained Linkages

Based on the derived matrix and its differential equations, a systematic formulation is presented to either identify the simple stationary configurations of movable spatial 4R, 5R and 6R overconstrained linkages or prove none of them occurs at all. Some examples are given to confirm the correctness and validity of the derived mathematical criterion. In addition, the closed-form solutions of linkage joint variables are well-established and with the help of computer graphics, geometrical meanings of linkage configurations are described. This approach can be used to provide a foundation for understanding the range of motion in overconstrained linkage application.

Span Limits for Elevated Temperature Piping

1999 ◽  
Vol 122 (2) ◽  
pp. 121-124 ◽  
Author(s):  
Charles Becht ◽  
Yaofeng Chen
Keyword(s):  
Elevated Temperature ◽  
Closed Form ◽  
Closed Form Solutions ◽  
Elastic Deflection ◽  
Pipe Deflection ◽  
Creep Regime

Pipe deflection due to self-weight quite often governs in the determination of the spacing between supports. Methods are readily available for calculation of elastic deflection of the pipe. However, such methods are not available for calculation of long-term deflection due to creep, which can be many times the initial elastic deflection for pipe operating in the creep regime of the material. Closed-form solutions for simple span conditions are presented which can be used by the analyst to develop charts for specific applications. These solutions provide insights into the problem of establishing span limits for elevated temperature pipe. [S0094-9930(00)01101-X]

scholarly journals Some triple trigonometrical series

1969 ◽  
Vol 10 (2) ◽  
pp. 121-125 ◽  
Author(s):  
C. J. Tranter
Keyword(s):  
Integral Equation ◽  
Closed Form ◽  
Fredholm Integral Equation ◽  
Jacobi Polynomials ◽  
Trigonometrical Series ◽  
Closed Form Solutions ◽  
General Series ◽  
Special Cases

This paper considers the determination of the coefficients in two sets of triple trigonometrical series and shows that these can be obtained in closed form. The series considered are special cases of some triple series in Jacobi polynomials studied by K. N. Srivastava [1]. Srivastava, however, shows that the problem for the more general series can be reduced to the solution of a Fredholm integral equation of the second kind and he does not discuss special cases which may lead to closed form solutions.

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