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
Your Session has timed out. Please sign back in to continue.


Howell, L. L., 2001, Compliant Mechanisms, Wiley, New York.
Saxena, A., and Ananthasuresh, G. K., 2003, “A Computational Approach to the Number of Synthesis of Linkages,” J. Mech. Des., 125(1), pp. 110–118. [CrossRef]
Krishnan, G., and Ananthasuresh, G. K., 2008, “Evaluation and Design of Displacement-Amplifying Compliant Mechanisms for Sensor Applications,” J. Mech. Des., 130(10), p. 102304. [CrossRef]
Hegde, S., and Ananthasuresh, G. K., 2010, “Design of Single-Input-Single-Output Compliant Mechanisms for Practical Applications Using Selection Maps,” J. Mech. Des., 132(8), p. 081007. [CrossRef]
Skakoon, J. G., 2008, The Elements of Mechanical Design, ASME, New York.
Ullman, D. G., 2003, The Mechanical Design Process, McGraw-Hill, Boston.
Suzuki, K., and Kikuchi, N., 1991, “A Homogenization Method for Shape and Topology Optimization,” Comput. Methods Appl. Mech. Eng., 93(3), pp. 291–318. [CrossRef]
Lu, K.-J., and Kota, S., 2003, “Synthesis of Shape Morphing Compliant Mechanisms Using a Load Path Representation Method,” Smart Mater. Struct., SPIE-5049, pp. 337–348. [CrossRef]
Juvinall, R. C., 1983, Fundamentals of Machine Component Design, Wiley, New York.
Chong, K. P., and Boresi, A., 2000, Elasticity in Engineering Mechanics, Wiley, New York.
Kelly, D. W., and Tosh, M. W., 2000, “Interpreting Load Paths and Stress Trajectories in Elasticity,” Eng. Comput., 17(2), pp. 117–135. [CrossRef]
Harasaki, H., and Arora, J. S., 2001, “New Concepts of Transferred and Potential Transferred Forces in Structures,” Comput. Methods Appl. Mech. Eng., 191(3–5), pp. 385–406. [CrossRef]
Marhadi, K., and Vekataraman, S., 2009, “Comparison of Quantitative and Qualitative Information Provided by Different Structural Load Path Definitions,” Int. J. Nonlinear Sci. Numer. Simul., 3, pp. 384–400. [CrossRef]
Blanding, D. K., 1999, Exact Constraint: Machine Design Using Kinematic Principles, ASME, New York.
Lu, K. J., and Kota, S., 2006, “Topology and Dimensional Synthesis of Compliant Mechanisms Using Discrete Optimization,” J. Mech. Design, 128(5), pp. 1080–1091. [CrossRef]
Saxena, A., and Ananthasuresh, G. K., 2000, “On an Optimal Property of Compliant Topologies,” Struct. Multidiscip. Optim., 19, pp 36–49. [CrossRef]
Lu, K.-J., and Kota, S., 2005, “An Effective Method of Synthesizing Compliant Adaptive Structures Using Load Path Representation,” J. Intell. Mater. Syst. Struct., 16(4), pp. 307–317. [CrossRef]
Krishnan, G., Kim, C., and Kota, S., 2010, “Load-Transmitter Constraint Sets: Part II-a Building Block Method for the Synthesis of Compliant Mechanisms,” Proceedings of 2010 ASME International Design Engineering Technical Conferences and Computers and Information in Engineering Conferences, August, Montreal, CA.


Grahic Jump Location
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.

Grahic Jump Location
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

Grahic Jump Location
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.

Grahic Jump Location
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.

Grahic Jump Location
Fig. 5

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

Grahic Jump Location
Fig. 6

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

Grahic Jump Location
Fig. 7

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

Grahic Jump Location
Fig. 10

A compliant dyad as a CT set

Grahic Jump Location
Fig. 8

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

Grahic Jump Location
Fig. 9

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

Grahic Jump Location
Fig. 11

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

Grahic Jump Location
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.

Grahic Jump Location
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.

Grahic Jump Location
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.

Grahic Jump Location
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.

Grahic Jump Location
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




Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In