A Spatial Nine-Bar Linkage, Possible Configurations and Conditions for Paradoxical Mobility

Dominic R. Walter and Manfred L. Husty

ABSTRACT: This paper is about a linkage consisting of nine bars connected by spherical joints, where the bars form a spatial hexagon with its main diagonals. Additionally the supporting lines of the diagonals have one proper point in common. This linkage is rigid in general. Wunderlich [1] stated that, given the lengths of the bars, the computation of possible configurations (up to isometries) is a problem of degree 25. By direct computation using resultants it can be shown that the number of essentially different configurations is at most 21, and thatWunderlich’s result includes four solutions lying at infinity which are not interesting for practical purposes. Attempts to find a set of design parameters where all 21 solutions allow real assembling led without exception to at most 10 such solutions. Until now it is not clear if this is already the maximum. Concerning paradoxical mobilityWunderlich discussed two mechanisms, a general one and a special case of it, related to Bricard’s mechanism [2]. Systematic search for all mobile linkages showed that there is another mobile one which cannot be derived from the known mechanisms by specialization.

KEYWORDS: Spatial Nine-bar Linkage, Assembly Modes, Paradoxical Mobility, Bricard Mechanism

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