Research Papers

A Comprehensive Elliptic Integral Solution to the Large Deflection Problems of Thin Beams in Compliant Mechanisms

[+] Author and Article Information
Guimin Chen

e-mail: guimin.chen@gmail.com
Key Laboratory of Electronic Equipment
Structure Design of Ministry of Education,
School of Electro-Mechanical Engineering,
Xidian University,
Xi'an Shaanxi 710071, China

1Corresponding author.

Contributed by the Mechanisms and Robotics Committee of ASME for publication in the Journal of Mechanisms and Robotics. Manuscript received July 22, 2012; final manuscript received November 15, 2012; published online March 26, 2013. Assoc. Editor: Anupam Saxena.

J. Mechanisms Robotics 5(2), 021006 (Mar 26, 2013) (10 pages) Paper No: JMR-12-1105; doi: 10.1115/1.4023558 History: Received July 22, 2012; Revised November 15, 2012

The elliptic integral solution is often considered to be the most accurate method for analyzing large deflections of thin beams in compliant mechanisms. In this paper, a comprehensive solution based on the elliptic integrals is proposed for solving large deflection problems. By explicitly incorporating the number of inflection points and the sign of the end-moment load in the derivation, the comprehensive solution is capable of solving large deflections of thin beams with multiple inflection points and subject to any kinds of load cases. The comprehensive solution also extends the elliptic integral solutions to be suitable for any beam end angle. Deflected configurations of complex modes solved by the comprehensive solution are presented and discussed. The use of the comprehensive solution in analyzing compliant mechanisms is also demonstrated by examples.

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Fig. 1

Deflected configuration of a thin beam subject to combined force and moment loads (with the positive directions of P, nP and Mo shown)

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Fig. 2

Solutions of Eq. (12) for n=0.1, κ=0.1 and θo=0.1(λ=sin θo-n cos θo+κ=0.1003). The values of θ where λ=0.1003 represents the magnitude of the beam angle at the inflection points

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Fig. 3

The deflected curves with multiple inflection points

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Fig. 4

The angle of a deflected curve for m = 3 and Mo0<0

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Fig. 5

The deflected configurations corresponding to m = 2 and m = 4 when θo=0

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Fig. 6

The deflected curves for SM=1, n = 0, and θo=3π

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Fig. 7

The curves of m=1,SM=-1,κ=0.01,n=10, and θo=π

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Fig. 8

Comparison of the tip locus of the comprehensive solution with that of Ref. [1] for a beam subject to a pure moment

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Fig. 9

A partially compliant four-bar mechanism containing a flexible beam DQ

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Fig. 10

The free-body diagram of each link in the mechanism

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Fig. 11

The deflected shape calculated by the comprehensive solution

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Fig. 12

Plots of the moments and end forces of the flexible beam versus crank angle θ1

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Fig. 13

Plots of input moment Tin versus crank angle θ1

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Fig. 14

Fixed-guided compliant mechanism

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Fig. 15

Two possible deflection paths of the fixed-guided beam (β=30 deg) achieved by the comprehensive solution. (a) SM=1 when m = 2 and (b) SM=-1 when m = 2.

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Fig. 16

Comparison of the two deflected configurations (SM=1 and SM=-1) obtained by the comprehensive solution with the experimental results in Ref. [21]

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Fig. 17

Force Fv versus δ for different β

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Fig. 18

Circular-guided compliant mechanism containing a flexible beam OA

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Fig. 19

The deflected configurations of beam OA

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Fig. 20

Photographs of the circular-guided compliant mechanism. (a) β=β0, and OA is straight, (b) β=90 deg and the deflected curve of OA has two inflection points, (c) β=170 deg, and the deflected curve of OA has one inflection point

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Fig. 21

Plots of input moment Tin versus crank angle β

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Fig. 22

The coupler-curve guided mechanism in its initial position (solid lines) and a deflected position (dashed lines)

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Fig. 23

The free-body diagram of each link in the coupler-curve guided mechanism

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Fig. 24

Locus of Point E (the coupler curve) and the deflected configurations of beam OE

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Fig. 25

Photographs of the coupler-curve guided compliant mechanism. (a) OE has two inflection points (m = 2) and (b) OE has three inflection points (m = 3)

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Fig. 26

Plots of input moment Tin versus crank angle θ2




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