Research Papers

A Kinetostatic Formulation for Load-Flow Visualization in Compliant Mechanisms

[+] Author and Article Information
Girish Krishnan

Mechanical Engineering,
University of Michigan,
Ann Arbor, MI 48105
e-mail: gikrishn@umich.edu

Charles Kim

Associate Professor
Mechanical Engineering,
Bucknell University,
Lewisburg, PA 17837
e-mail: charles.kim@bucknell.edu

Sridhar Kota

Department of
Mechanical Engineering,
University of Michigan,
Ann Arbor, MI 48105
e-mail: kota@umich.edu

Contributed by the Mechanisms and Robotics Committee of ASME for publication in the JOURNAL OF MECHANISMS AND ROBOTICS. Manuscript received August 17, 2011; final manuscript received November 7, 2012; published online April 12, 2013. Assoc. Editor: Anupam Saxena.

J. Mechanisms Robotics 5(2), 021007 (Apr 12, 2013) (9 pages) Paper No: JMR-11-1092; doi: 10.1115/1.4023872 History: Received August 17, 2011; Revised November 07, 2012

Load flow visualization, which is an important step in structural and machine assembly design may aid in the analysis and eventual synthesis of compliant mechanisms. In this paper, we present a kineto-static formulation to visualize load flow in compliant mechanisms. This formulation uses the concept of transferred forces to quantify load flow from input to the output of a compliant mechanism. The magnitude and direction of load flow in the constituent members enables functional decomposition of the compliant mechanism into (i) Constraints (C): members that are constrained to deform in a particular direction and (ii) Transmitters (T): members that transmit load to the output. Furthermore, it is shown that a constraint member and an adjacent transmitter member can be grouped together to constitute a fundamental building block known as an CT set whose load flow behavior is maximally decoupled from the rest of the mechanism. We can thereby explain the deformation behavior of a number of compliant mechanisms from literature by visualizing load flow, and identifying building blocks.

Copyright © 2013 by ASME
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Fig. 1

Definition of Load Flow using the concept of transferred load (a) Force applied at i producing a deformation uj at point j. (b) The transferred force fjtr is an applied force at j that produces the same displacement uj at j. (c) Assuming small displacement assumptions, the transferred load is equal and opposite to the reaction load at j if this point were fixed.

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Fig. 2

Transferred-force evaluation and load-flow visualization for a cantilever beam with a transverse end applied load. (a) End load at i produces displacement at j, (b) transferred load at j produces the same displacement as in the previous case, (c) load-flow visualization entails evaluation of the transferred load at multiple points along the length of the beam, and (d) load-flow visualization for axial loads.

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Fig. 3

Load flow visualization in a MBB structure whose (a) geometry is modeled as beams, and (b) transferred forces and moments plotted at various locations

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Fig. 4

(a) A parallelogram flexure consisting of two beams in parallel. Load flow is plotted for (b) transverse applied load, (c) applied in-plane moment, and (d) axial applied load.

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Fig. 5

Load-flow visualization in a compliant gripper [15]. Constraints are marked C, while transmitters are marked T.

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Fig. 6

Load-flow visualization in a displacement amplifying compliant mechanism [16]

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Fig. 7

Relating load flow to the direction of deformation in (a) a beam constraint, and (b) a flexural hinge with a rigid body

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Fig. 8

A general constraint-transmitter (CT) set consisting of input and output suspensions and a transmitter element

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Fig. 9

A CT set with a general suspension as the constraint and a rigid transmitter

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Fig. 10

A compliant dyad as a CT set

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Fig. 11

A CT set with a rigid member in between relatively short beam flexures

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Fig. 12

A compliant mechanism with two load transfer stages. The input force fi is applied at point A. fjtr and fktr are transferred forces at points j and k, respectively.

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Fig. 13

Decomposition of a mechanism composed of series combination of CT sets. (a)The mechanism of Fig. 5 is (b) broken down into building blocks based on the principle of series combination of CT sets. (c) The direction of displacement (Dod) at each point is determined by projecting the direction of the transferred load (ftr) on the direction of freedom (Dof) line.

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Fig. 14

Mechanisms with multiple load flow paths (a) Mechanism with input and output. Mechanism is divided to separate two distinct load paths (b) Load flow in one load path, and (c) Load flow in the other load path.

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Fig. 15

Decomposition of mechanism shown in Fig. 6 composed of a parallel combination of CT sets. (a) CT sets for the first load path (b) CT sets for the second load path, and (c) The direction of displacement (Dod) at each point, and (d) defining the direction of motion of the rigid body.

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Fig. 16

Understanding the working of a lumbar support mechanism [17] (a) Transferring the user's weight to deform the lumbar support (b) base structure consisting of the lumbar support surface and an additional beam support, and (c) CT sets that constitute the mechanism topology




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