Polynomial Solution of the Spatial Burmester Problem

Author:

Innocenti C.1

Affiliation:

1. Dipartimento di Ingegneria delle Costruzioni Meccaniche, University of Bologna, Viale Risorgimento, 2-40136, Bologna, Italy

Abstract

This paper presents a new method for the dimensional synthesis of the spatial guidance linkage that features the guided body connected to the base by the interposition of five rods having spherical joints at both extremities. The linkage is required to index the guided body through seven arbitrarily-chosen rigid-body positions. Core of the proposed method is an original algebraic elimination procedure that allows five unknowns to be dropped from a set of six second-order algebraic equations in six unknowns. As a result, a final univariate polynomial equation of twentieth order is obtained whose twenty roots, in the complex domain, represent as many possible placements for a connecting rod. A numerical example is reported.

Publisher

ASME International

Subject

Computer Graphics and Computer-Aided Design,Computer Science Applications,Mechanical Engineering,Mechanics of Materials

Reference11 articles.

1. Burmester L. , 1876, “Geradfu¨hrung durch das Kurbelgetriebe,” Civilingenieur, Vol. 22, pp. 597–606.

2. Burmester L. , 1877, “Geradfu¨hrung durch das Kurbelgetriebe,” Civilingenieur, Vol. 23, pp. 227–250.

3. Dijksman E. A. , 1969, “Calculation and Construction of the Burmester Points for Five Positions of a Moving Plane,” ASME Journal of Engineering for Industry, Vol. 91B-1, pp. 66–74.

4. Freudenstein F. , and RothB., 1963, “Numerical Solution of Systems of Nonlinear Equations,” Journal of the Association of Computing Machinery, Vol. 10, pp. 550–556.

5. Hunt, K. H., 1978, Kinematic Geometry of Mechanisms, Clarendon Press, Oxford.

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