Dynamic balancing of planar mechanisms using toric geometry

TL;DR

This paper introduces a novel algebraic approach using toric geometry and complex variables to determine the complete set of dynamically balanced planar four-bar mechanisms, providing necessary and sufficient conditions.

Contribution

It presents a new method employing Laurent polynomial factorization and toric polynomial division for dynamic balancing analysis of planar four-bar mechanisms.

Findings

Derived necessary and sufficient conditions for dynamic balancing.

Formulated the problem as Laurent polynomial factorization.

Applied toric polynomial division to mechanism analysis.

Abstract

In this paper, a new method to determine the complete set of dynamically balanced planar four-bar mechanims is presented. Using complex variables to model the kinematics of the mechanism, the dynamic balancing constraints are written as algebraic equations over complex variables and joint angular velocities. After elimination of the joint angular velocity variables, the problem is formulated as a problem of factorization of Laurent polynomials. Using toric polynomial division, necessary and sufficient conditions for dynamic balancing of planar four-bar mechanisms are derived.