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Technical Brief

Bi-BCM: A Closed-Form Solution for Fixed-Guided Beams in Compliant Mechanisms

[+] Author and Article Information
Fulei Ma

School of Electro-Mechanical Engineering,
Xidian University,
Xi'an 710071, Shaanxi, China

Guimin Chen

School of Electro-Mechanical Engineering,
Xidian University,
Xi'an 710071, Shaanxi, China
e-mail: guimin.chen@gmail.com

1Corresponding author.

Manuscript received March 10, 2016; final manuscript received October 4, 2016; published online November 23, 2016. Assoc. Editor: Venkat Krovi.

J. Mechanisms Robotics 9(1), 014501 (Nov 23, 2016) (8 pages) Paper No: JMR-16-1065; doi: 10.1115/1.4035084 History: Received March 10, 2016; Revised October 04, 2016

A fixed-guided beam, with one end is fixed while the other is guided in that the angle of that end does not change, is one of the most commonly used flexible segments in compliant mechanisms such as bistable mechanisms, compliant parallelogram mechanisms, compound compliant parallelogram mechanisms, and thermomechanical in-plane microactuators. In this paper, we split a fixed-guided beam into two elements, formulate each element using the beam constraint model (BCM) equations, and then assemble the two elements' equations to obtain the final solution for the load–deflection relations. Interestingly, the resulting load–deflection solution (referred to as Bi-BCM) is closed-form, in which the tip loads are expressed as functions of the tip deflections. The maximum allowable axial force of Bi-BCM is the quadruple of that of BCM. Bi-BCM also extends the capability of BCM for predicting the second mode bending of fixed-guided beams. Besides, the boundary line between the first and the second modes bending of fixed-guided beams can be easily obtained using a closed-form equation. Bi-BCM can be immediately used for quick design calculations of compliant mechanisms utilizing fixed-guided beams as their flexible segments (generally no iteration is required). Different examples are analyzed to illustrate the usage of Bi-BCM, and the results show the effectiveness of the closed-form solution.

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Figures

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Fig. 1

Illustration of different compliant mechanisms that employ fixed-guided beams: (a) a bistable mechanism, (b) a thermomechanical in-plane microactuator, (c) a compound compliant parallelogram mechanism, and (d) a double parallelogram mechanism

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Fig. 2

Schematic of a fixed-guided beam

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Fig. 3

Free body diagrams for the two element of a fixed-guided beam end remains zero

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Fig. 4

Illustration of two possible deflected configurations corresponding to the second mode bending (configuration I corresponding to solution (a) and configuration II corresponding to solution (b))

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Fig. 5

Schematic of a fixed-guided beam in a bistable mechanism

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Fig. 6

The force–deflection curves of the bistable mechanism

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Fig. 7

The boundary between first and second mode bending according to Eq. (42) for the fixed-guided beam. Several deflected beam configurations are also shown.

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Fig. 8

Comparison of the two deflected configurations obtained by Bi-BCM solution and the experimental results in Ref. [10]

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Fig. 9

Schematic of a fixed-guided beam taken from a compliant parallelogram mechanism

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Fig. 10

The force–deflection curves of the compliant parallelogram mechanism

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Fig. 11

Schematic of a deflected TIM leg

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Fig. 12

The force–displacement relationship of the thermomechanical in-plane microactuator

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