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Bistable Quad-Nets Composed of Four-Bar Linkages

Gudrun Szewieczek, Daniel Huczala, Martin Pfurner, Hans-Peter Schröcker

Abstract

We study mechanical structures composed of spatial four-bar linkages that are bistable, that is, they allow for two distinct configurations. They have an interpretation as quad nets in the Study quadric which can be used to prove existence of arbitrarily large structures of this type. We propose a purely geometric construction of such examples, starting from infinitesimally flexible quad nets in Euclidean space and applying Whiteley de-averaging. This point of view situates the problem within the broader framework of discrete differential geometry and enables the construction of bistable structures from well-known classes of quad nets, such as discrete minimal surfaces. The proposed construction does not rely on numerical optimization and allows control over axis positions and snap angles.

Bistable Quad-Nets Composed of Four-Bar Linkages

Abstract

We study mechanical structures composed of spatial four-bar linkages that are bistable, that is, they allow for two distinct configurations. They have an interpretation as quad nets in the Study quadric which can be used to prove existence of arbitrarily large structures of this type. We propose a purely geometric construction of such examples, starting from infinitesimally flexible quad nets in Euclidean space and applying Whiteley de-averaging. This point of view situates the problem within the broader framework of discrete differential geometry and enables the construction of bistable structures from well-known classes of quad nets, such as discrete minimal surfaces. The proposed construction does not rely on numerical optimization and allows control over axis positions and snap angles.

Paper Structure

This paper contains 16 sections, 3 theorems, 38 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Snapping Four-Bars
  3. Constructions of Snapping Four-Bars
  4. Snapping Four-Bars and Rotation Quadrilaterals
  5. Rotation Quadrilaterals and Dual Quaternions
  6. Snapping Four-Bar Nets
  7. Mechanisms on Quad Graphs
  8. Quad Nets in Quadrics
  9. Surface Design
  10. Rolling of Discrete Surfaces with Congruent Faces
  11. Isometric Quad Nets from Infinitesimally Flexible Nets
  12. Discrete Koenigs Nets
  13. Revolute Axes and Snapping Angles From Koenigs Nets
  14. A Snapping Cube
  15. 3D Printed Prototypes
  16. ...and 1 more sections

Key Result

Proposition 1

For every snapping four-bar with axes $(R_i, R_j, R_k, R_l)$, there exists a quadrilateral with vertices $(g_i, g_j, g_k, g_l)$ located on the respective axes that is congruent to the corresponding quadrilateral $(h_i, h_j, h_k, h_l)$ in the second configuration $(R'_i, R'_j, R'_k, R'_l)$ of the sna

Figures (10)

  • Figure 1: Two configurations of a snapping four-bar (center), in poses where two axes are aligned (a, c, e, g) and in intermediate poses (b, d, f, h).
  • Figure 2: Schematic sketch of a planar scissor linkage. Its linkgraph consists of two quads. Note that graph vertices (black dots) correspond to links while edges correspond to revolute joints (white dots).
  • Figure 3: Construction of a $\mathcal{Q}$-net on $\mathbb{Z}^2$. We prescribe the origin (left), then compute the values along the two coordinate axes (center) before consecutively determining the remaining values (right).
  • Figure 4: Top: A graph with $\mathbb{Z}^2$-combinatorics with black and white coloring of the faces and the related dual graph. Bottom: The quad net $h$ rolls on the quad net $g$. Both $g$ and $h$ correspond to the original graph while the discrete rolling motion corresponds to the dual graph (right).
  • Figure 5: Initial data and iterative construction for Theorem \ref{['th:1']}.
  • ...and 5 more figures

Theorems & Definitions (21)

  • Example 1
  • Definition 1
  • Definition 2
  • Remark
  • Definition 3
  • Definition 4
  • Remark 1
  • Proposition 1
  • proof : Proof of Proposition \ref{['prop:congr_quad']}
  • Remark 2
  • ...and 11 more