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Geometric analysis of Bennett's spherical 8-bar linkage and its spatial counterpart

Hellmuth Stachel

Abstract

We provide a geometric approach to two combinatorically symmmetric overconstrained spatial linkages. Both contain eight bodies and twelve revolute joints and collapse in aligned poses. The first one is spherical and the union of six spherical isograms. It is the spherical image of a Bricard octahedron of type~3 and was already analysed 1912 by Bennett. The second linkage is the dualized version and composed from six Bennett isograms. Our approach via line reflections discloses some symmetries at spatial poses.

Geometric analysis of Bennett's spherical 8-bar linkage and its spatial counterpart

Abstract

We provide a geometric approach to two combinatorically symmmetric overconstrained spatial linkages. Both contain eight bodies and twelve revolute joints and collapse in aligned poses. The first one is spherical and the union of six spherical isograms. It is the spherical image of a Bricard octahedron of type~3 and was already analysed 1912 by Bennett. The second linkage is the dualized version and composed from six Bennett isograms. Our approach via line reflections discloses some symmetries at spatial poses.
Paper Structure (4 sections, 4 theorems, 9 equations, 10 figures)

This paper contains 4 sections, 4 theorems, 9 equations, 10 figures.

Table of Contents

  1. Introduction
  2. The spherical 8-bar linkage
  3. The dualized version
  4. Notes and Comments

Key Result

lemma thmcounterlemma

For any two different oriented great circles $g_1, g_2$ on $\mathcal{S}^2$ there exists a unique pair of antipodal centers of symmetry $S, S^\ast \in\mathcal{S}^2$, i.e., spherical centers of a halfturn that maps $g_1$ to $g_2$, and vice versa. The centers are located on the common perpendicular of

Figures (10)

  • Figure 1: Bennett's composition of spherical isograms presented in Bennett is an overconstrained spherical linkage with 8 bars and 12 joints, arranged in six isograms.
  • Figure 2: At this linkgraph the links and joints of Bennet's 6-bar linkage (Figure \ref{['fig:spher_linkage1']}) and its spatial counterpart (Figure \ref{['fig:spatial_linkage_kmplt']}) are represented as knots and bars.
  • Figure 3: At a skew isogram $ABCD$ with side lengths $a, b$ the dihedral angles $\alpha,\beta$ of its convex hull match the proportion $a:b = \sin\alpha : \sin\beta$.
  • Figure 4: At a non-crossed spherical isogram (left) the common midpoint $S_1$ of both diagonal arcs is a center of symmetry. At a crossed spherical isogram (right) the axis $s_1$ of symmetry meets the two diagonal arcs orthogonally at there midpoints. If one side remains fixed, the coupler motion of a spherical isogram splits into two rational motions with bifurcations at the aligned poses.
  • Figure 5: This triple of spherical isograms is flexible and has aligned centers of symmetry (Lemma \ref{['lem:compound_isograms']}).
  • ...and 5 more figures

Theorems & Definitions (5)

  • definition thmcounterdefinition
  • lemma thmcounterlemma
  • lemma thmcounterlemma
  • theorem thmcountertheorem
  • theorem thmcountertheorem