Equilibrium Formulation of a 3 DOF Compliant Mechanism using Sylvester Dialytic Method of Elimination

TL;DR

This paper formulates the equilibrium states of a 3 DOF compliant mechanism using Sylvester's elimination method, deriving polynomial equations to identify all possible configurations under different spring conditions.

Contribution

It introduces a novel application of Sylvester's elimination method to analyze equilibrium configurations of a complex compliant mechanism with variable spring conditions.

Findings

Derived polynomial equations of degrees 4 and 48 for equilibrium analysis.

Validated the approach with numerical examples using Maple software.

Confirmed the method's effectiveness in determining all equilibrium configurations.

Abstract

This paper studies the equilibrium formulation of a three degree of freedom planar compliant platform mechanism, which is in contact with a solid body in its environment. The mechanism includes two platforms, which are connected in parallel by three linear springs. The capability of deformation by manipulating both platforms exceptionally complicates the problem. The analysis aims to determine all equilibrium configurations for two different cases: FIRST CASE all three springs have zero free lengths and SECOND CASE only two of the springs have zero free lengths. The proposed procedure calculates the pose of the top platform when it is not in contact with the surface, and then detects if the top platform is in contact to determine the equilibrium configurations. To solve the geometric equations of the mechanism, we use Sylvester method of elimination. The approach obtains 4th and 48th degree polynomial equations for the first and second cases, respectively. Numerical examples have been applied to verify the process of analysis. The results, which are numerically calculated by software Maple, prove the validity of the analysis.