A one-step non-iterative solution algorithm for mechanisms—II. Application and closed-form solutions for planar mechanisms

Abstract In this paper an inversion approach is developed for the analysis of planar mechanisms using closed-form equations. The vector loop equation approach is used, and the occurrence matrices of the variables in the position equations are obtained. After the loop equations are formed, dependency checking of the unknowns is performed to determine if it is possible to solve for any two equations in two unknowns. For the cases where the closed-form solutions cannot be implemented directly, possible inversions of the mechanism are studied. If the vector loop equations for an inversion can be solved in closed-form, they are identified and solved, and the solutions are transformed back to the original linkage. The method developed in this paper eliminates the uncertainties involved, and the large number of computations required in solving the equations by iterative methods.

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A one-step non-iterative solution algorithm for mechanisms—Part I. General method

Mechanism and Machine Theory ◽

10.1016/0094-114x(74)90011-1 ◽

1974 ◽

Vol 9 (1) ◽

pp. 107-114 ◽

Cited By ~ 3

Author(s):

M.A Townsend

Keyword(s):

Iterative Solution ◽

Solution Algorithm ◽

One Step ◽

General Method

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A Hybrid Approach to Solving the Position Equations for Planar Mechanisms

Journal of Mechanical Design ◽

10.1115/1.2826731 ◽

1995 ◽

Vol 117 (4) ◽

pp. 627-632 ◽

Cited By ~ 2

Author(s):

S. Bawab ◽

G. L. Kinzel

Keyword(s):

Closed Form ◽

Hybrid Approach ◽

Closed Form Solution ◽

Iterative Solution ◽

Form Solution ◽

Planar Mechanisms ◽

Solution Approach ◽

Newton Raphson ◽

Raphson Method ◽

Straightforward Approach

In this paper, a straightforward approach is developed to solve the nonlinear position equations for a linkage when a closed-form solution to some of the equations can be obtained. This is done with the aid of dependency checking concepts that organizes a system 2n equations and 2n unknowns (variables) into smaller sets of equations. When a set of two equations and two unknowns is obtained, the variables are analyzed using a closed-form (non-iterative) solution approach. Otherwise, an iterative approach such as the Newton-Raphson method is used for the analysis.

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Finitely and Multiply Separated Synthesis of Link and Geared Mechanisms Using Symbolic Computing

Journal of Mechanical Design ◽

10.1115/1.2919226 ◽

1993 ◽

Vol 115 (3) ◽

pp. 560-567 ◽

Cited By ~ 4

Author(s):

A. K. Dhingra ◽

N. K. Mani

Keyword(s):

Closed Form ◽

Closed Form Solution ◽

Iterative Solution ◽

Form Solution ◽

Motion Generation ◽

Closed Form Solutions ◽

Symbolic Computing ◽

Manipulation Language ◽

Solution Techniques

A computer amenable symbolic computing approach for the synthesis of six different link and geared mechanisms is presented. Burmester theory, complex number algebra, and loop closure equations are employed to develop governing equations for the mechanism to be synthesized. Closed-form and iterative solution techniques have been developed which permit synthesis of six-link Watt and Stephenson chains for function, path, and motion generation tasks with up to eleven precision points. Closed-form solution techniques have also been developed for the synthesis of geared five-bar, six-bar, and five-link cycloidal crack mechanisms, for synthesis tasks with up to six finitely and multiply separated precision points. The symbolic manipulation language MACSYMA is used to simplify the resulting synthesis equations and obtain closed-form solutions. A design methodology which demonstrates the feasibility and versatility of symbolic computing in computer-aided mechanisms design is outlined. A computer program which incorporates these synthesis procedures is developed. Two examples are presented to illustrate the role of symbolic computing in an automated mechanism design process.

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