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Research Papers

A Symbolic Formulation for Analytical Compliance Analysis and Synthesis of Flexure Mechanisms

[+] Author and Article Information
Hai-Jun Su1

Department of Mechanical and Aerospace Engineering,  The Ohio State University, Columbus, OH 43210su.298@osu.eduRobotics Institute,  Beihang University, Beijing 100083, P. R. Chinasu.298@osu.edu

Hongliang Shi

Department of Mechanical and Aerospace Engineering,  The Ohio State University, Columbus, OH 43210shi.347@osu.eduRobotics Institute,  Beihang University, Beijing 100083, P. R. Chinashi.347@osu.edu

JingJun Yu

Department of Mechanical and Aerospace Engineering,  The Ohio State University, Columbus, OH 43210jjyu@buaa.edu.cnRobotics Institute,  Beihang University, Beijing 100083, P. R. Chinajjyu@buaa.edu.cn

1

Corresponding author.

J. Mech. Des 134(5), 051009 (Apr 25, 2012) (9 pages) doi:10.1115/1.4006441 History: Received August 18, 2011; Revised March 06, 2012; Published April 24, 2012; Online April 25, 2012

This paper presents a symbolic formulation for analytical compliance analysis and synthesis of flexure mechanisms with serial, parallel, or hybrid topologies. Our approach is based on the screw theory that characterizes flexure deformations with motion twists and loadings with force wrenches. In this work, we first derive a symbolic formulation of the compliance and stiffness matrices for commonly used flexure elements, flexure joints, and simple chains. Elements of these matrices are all explicit functions of flexure parameters. To analyze a general flexure mechanism, we subdivide it into multiple structural modules, which we identify as serial, parallel, or hybrid chains. We then analyze each module with the known flexure structures in the library. At last, we use a bottom-up approach to obtain the compliance/stiffness matrix for the overall mechanism. This is done by taking appropriate coordinate transformation of twists and wrenches in space. Four practical examples are provided to demonstrate the approach. A numerical example is employed to compare analytical compliance models against a finite element model. The results show that the errors are sufficiently small (2%, compared with finite element (FE) model), if the range of motion is limited to linear deformations. This work provides a systematical approach for compliance analysis and synthesis of general flexure mechanisms. The symbolic formulation enables subsequent design tasks, such as compliance synthesis or sensitivity analysis.

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Copyright © 2012 by American Society of Mechanical Engineers
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Figures

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Figure 1

A flexure mechanism is deformed by T̂ under a general loading Ŵ

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Figure 2

A typical blade flexure with a rectangular cross section. The thickness is much smaller than the length, i.e., t≪l.

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Figure 3

The ratio β versus t/w

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Figure 4

A long wire flexure of diameter d and length l with d≪l

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Figure 5

A serial chain of two identical wires joined by a rigid rod of length L. Each wire has a length l and a diameter d.

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Figure 6

A parallel flexure mechanism formed by two parallel ideal blade flexures

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Figure 7

The error of instant center Δry /l versus ψ ∈ [0,3π/4] with η = 0.01 and ρ ∈ [0.25,3.75]

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Figure 8

(a) A typical cross-strip flexure pivot and (b) a parallelogram linear spring

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Figure 9

The change of rational compliance versus ψ ∈ [0,3π/4] with η = 0.01 and ρ ∈ [0.5,4]

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Figure 10

A flexure platform mechanism with three identical chains of two wire flexures

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Figure 11

The rotational compliance c25 versus α ∈ [15 deg,75 deg] with Poisson’s ratio ν = 0.3 and ξ = r/L ∈ [0,2]

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Figure 12

The rotational compliance c25 versus α ∈ [15 deg,75 deg] with Poisson’s ratio ν = 0.3 and ξ = r/L ∈ [0,2]

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Figure 13

The rotational compliance c25 versus ξ = r/L ∈ [0,2] with Poisson’s ratio ν = 0.3 and α = 0

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