On flexes associated with higher-order flexible bar-joint frameworks

TL;DR

This paper redefines higher-order flexes in bar-joint frameworks using algebraic curves and Puiseux series, addressing previous ambiguities and incorporating real-world considerations.

Contribution

It introduces a new algebraic method to define and compute higher-order flexes in flexible frameworks, resolving open questions in the field.

Findings

Defined higher-order flexes via algebraic curves

Demonstrated computation using Puiseux series

Accounted for real-world constraints in flexibility analysis

Abstract

The famous example of the double-Watt mechanism given by Connelly and Servatius raises some problems concerning the classical definitions of higher-order flexibility and rigidity, respectively. Recently, the author was able to give a proper redefinition of the flexion/rigidity order for bar-joint frameworks, but the question for the flexes associated with higher-order flexible structures remained open. In this paper we properly define these flexes based on the theory of algebraic curves and demonstrate their computation by means of Puiseux series. The presented algebraic approach also allows to take reality issues into account.