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A unified position analysis of the Dixon and the generalized peaucellier linkages

journal contribution
posted on 2023-06-09, 00:27 authored by Nicolas Rojas, Aaron M Dollar, Federico Thomas
This paper shows how, using elementary Distance Geometry, a closure polynomial of degree 8 for the Dixon linkage can be derived without any trigonometric substitution, variable elimination, or artifice to collapse mirror configurations. The formulation permits the derivation of the geometric conditions required in order for each factor of the leading coefficient of this polynomial to vanish. These conditions either correspond to the case in which the quadrilateral defined by four joints is orthodiagonal, or to the case in which the center of the circle defined by three joints is on the line defined by two other joints. This latter condition remained concealed in previous formulations. Then, particular cases satisfying some of the mentioned geometric conditions are analyzed. Finally, the obtained polynomial is applied to derive the coupler curve of the generalized Peaucellier linkage, a linkage with the same topology as that of the celebrated Peaucellier straight-line linkage but with arbitrary link lengths. It is shown that this curve is 11-circular of degree 22 from which the bicircular quartic curve of the Cayley's scalene cell is derived as a particular case.

History

Publication status

  • Published

Journal

Mechanism and Machine Theory

ISSN

0094-114X

Publisher

Elsevier

Volume

94

Page range

28-40

Department affiliated with

  • Engineering and Design Publications

Full text available

  • No

Peer reviewed?

  • Yes

Legacy Posted Date

2016-03-07

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