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

Mustafa M. Mustafa, Carl D. Crane, Ibrahim Hamarash

TL;DR

This work addresses the equilibrium configurations of a 3 DOF planar compliant parallel mechanism that interacts with a stiff environment through three springs. It employs the Sylvester dialytic elimination method to transform the geometric and statics constraints into univariate polynomials, obtaining a $4^{\text{th}}$-degree polynomial for the zero free-length case and a $48^{\text{th}}$-degree polynomial for the non-zero free-length case, with validation against Maple-produced solutions. The approach includes a contact-detection step based on a plane-face test and a detailed derivation of equilibrium conditions via force projections and moments about the contact point. The results reveal multiple real equilibria, organized into distinct groups relative to the surface, and highlight the presence of extraneous solutions in the higher-degree case, informing future design and optimization of compliant mechanisms in contact-rich environments.

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.

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

TL;DR

This work addresses the equilibrium configurations of a 3 DOF planar compliant parallel mechanism that interacts with a stiff environment through three springs. It employs the Sylvester dialytic elimination method to transform the geometric and statics constraints into univariate polynomials, obtaining a -degree polynomial for the zero free-length case and a -degree polynomial for the non-zero free-length case, with validation against Maple-produced solutions. The approach includes a contact-detection step based on a plane-face test and a detailed derivation of equilibrium conditions via force projections and moments about the contact point. The results reveal multiple real equilibria, organized into distinct groups relative to the surface, and highlight the presence of extraneous solutions in the higher-degree case, informing future design and optimization of compliant mechanisms in contact-rich environments.

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.
Paper Structure (9 sections, 53 equations, 4 figures, 2 tables)

This paper contains 9 sections, 53 equations, 4 figures, 2 tables.

Table of Contents

  1. INTRODUCTION
  2. Problem Statement
  3. Solution Approach
  4. Calculating the Pose of the Top Platform
  5. Determining if the Top Platform is in Contact
  6. Finding the Equilibrium Pose when the Top Platform is in Contact
  7. Case i: $\mathit{L_{01}=\mathit{L_{02}}=L_{03}=0}$
  8. Case ii: $\mathit{L_{01}\neq0,}$ $\mathit{L_{02}}=L_{03}=0$
  9. Conclusions

Figures (4)

  • Figure 1: The compliant planar platform mechanism
  • Figure 2: Definition of point $E$, distance $L$, and angle $\beta$
  • Figure 3: Two real solutions for zero free lengths case
  • Figure 4: Eight real solutions for one non-zero free length Case