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RESEARCH PAPERS

# A Polynomial Homotopy Formulation of the Inverse Static Analysis of Planar Compliant Mechanisms

[+] Author and Article Information
Hai-Jun Su1

Virtual Reality Application Center, 1620 Howe Hall, Iowa State University, Ames, IA 50011haijunsu@iastate.edu

J. Michael McCarthy

Robotics and Automation Laboratory, University of California, Irvine, Irvine, CA 92697jmmccart@uci.edu

1

Address all correspondence to this author.

J. Mech. Des 128(4), 776-786 (Oct 04, 2005) (11 pages) doi:10.1115/1.2202137 History: Received June 07, 2005; Revised October 04, 2005

## Abstract

This paper formulates the inverse static analysis of planar compliant mechanisms in polynomial form. The goal is to find the equilibrium configurations of the system in response to a known force/moment applied to the mechanism. The geometric constraint of the linkage defines a set of kinematics equations which are combined with equilibrium equations obtained from partial derivatives of the potential-energy function. In order to apply polynomial homotopy solver to these equations, we approximate the linear torsion spring torque at each joint by using sine and cosine functions. The results obtained from the homotopy solver are then refined using Newton-Raphson iteration. To demonstrate the analysis steps, we study two example planar compliant mechanisms, a four-bar linkage with two torsional springs, and a parallel platform supported by three linear springs. Numerical examples are provided together with plots of the potential energy during a movement between selected equilibrium positions.

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## Figures

Figure 1

A general three degree-of-freedom compliant platform mechanism and its rigid-body counterpart

Figure 2

Two typical flexural joints and their pseudo-rigid-body models

Figure 3

The linear function τ=kΔϕ and the approximating functions τ′(Δϕ). The maximum error is 1.5% for Δϕ∊[−π∕2,π∕2]

Figure 4

Figure 5

A compliant platform mechanism with three flexible limbs

Figure 6

Six equilibrium configurations of a compliant four-bar linkage. Configuarations (a), (b), (c), and (d) are on one assembly, while (e) and (f) are on the other assembly.

Figure 7

Energy curve and its derivative for the compliant four-bar linkage

Figure 8

Four equilibrium configurations of an equilateral compliant platform

Figure 9

Potential energy and moment required to move between the four equilibrium configurations of the compliant platform

Figure 10

Six equilibrium configurations of the alternate form of the equilateral compliant platform

Figure 11

Potential energy and external loading required to move between the six configurations

Figure 12

The other 15 equilibrium configurations of the equilateral compliant mechanism

Figure 13

The other three equilibrium configurations of the alternative form of the equilateral compliant mechanism

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