A closed linkage mechanism having the shape of a discrete M\"obius strip

TL;DR

This paper introduces a new family of closed linkage mechanisms modeled as discrete Möbius strips, exhibiting unique properties like 1-DOF and constant energy, with potential applications in mathematics and mechanical design.

Contribution

It extends Bricard's 6R linkage to a broader class of mechanisms with singular properties, linking mechanical design to mathematical structures.

Findings

Mechanisms have one-dimensional degree of freedom.

Energies remain constant across different states.

Properties confirmed mainly through numerical analysis.

Abstract

A closed linkage mechanism in three-dimensional space is an object comprising rigid bodies connected with hinges in a circular form like a rosary. Such linkages include Bricard6R and Bennett4R. To design such a closed linkage, it is necessary to solve a high-degree algebraic equation, which is generally difficult. In this lecture, the author proposes a new family of closed linkage mechanisms with an arbitrary number of hinges as an extension of a certain Bricard6R. They have singular properties, such as one-dimensional degree of freedom (1-DOF), and certain energies taking a constant value regardless of the state. These linkage mechanisms can be regarded as discrete M\"obius strips and may be of interest in the context of pure mathematics as well. However, many of the properties described here have been confirmed only numerically, with no rigorous mathematical proof, and should be interpreted with caution.