0
Research Papers

Analyzing the Stability Properties of Kaleidocycles

[+] Author and Article Information
C. Safsten, T. Fillmore, A. Logan, D. Halverson

Department of Mathematics,
Brigham Young University,
Provo, UT 84602

L. Howell

Department of Mechanical Engineering,
Brigham Young University,
Provo, UT 84602

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received September 2, 2015; final manuscript received January 21, 2016; published online February 10, 2016. Assoc. Editor: Shaoxing Qu.

J. Appl. Mech 83(5), 051001 (Feb 10, 2016) (13 pages) Paper No: JAM-15-1465; doi: 10.1115/1.4032572 History: Received September 02, 2015; Revised January 21, 2016

Kaleidocycles are continuously rotating n-jointed linkages. We consider a certain class of six-jointed kaleidocycles which have a spring at each joint. For this class of kaleidocycles, stored energy varies throughout the rotation process in a nonconstant, cyclic pattern. The purpose of this paper is to model and provide an analysis of the stored energy of a kaleidocycle throughout its motion. In particular, we will solve analytically for the number of stable equilibrium states for any kaleidocycle in this class.

FIGURES IN THIS ARTICLE
<>
Copyright © 2016 by ASME
Your Session has timed out. Please sign back in to continue.

References

Evans, T. A. , Rowberry, B. G. , Magleby, S. P. , and Howell, L. L. , 2015, “ Multistable Behavior of Compliant Kaleidocycles,” ASME Paper No. DETC2015-46637.
Schattschneider, D. , 1977, M. C. Escher Kaleidocycles, Ballantine Books, New York.
Dai, J. S. , and Jones, J. R. , 2002, “ Kinematics and Mobility Analysis of Carton Folds in Packing Manipulation Based on the Mechanism Equivalent,” Proc. Inst. Mech. Eng. C, 216(10), pp. 959–970. [CrossRef]
Howell, L. L. , and Midha, A. , 1994, “ A Method for the Design of Compliant Mechanisms With Small-Length Flexural Pivots,” ASME J. Mech. Des., 116(1), pp. 280–290. [CrossRef]
Baker, J. E. , 1980, “ An Analysis of the Bricard Linkages,” Mech. Mach. Theory, 15(4), pp. 267–286. [CrossRef]
Bricard, R. , 1897, “ Memoire sur la thorie de l'octadre articul,” J. Math. Pures Appl., 3, pp. 113–150.
You, Z. , and Chen, Y. , 2001, Motion Structures, Taylor and Francis, London.
Chen, Y. , You, Z. , and Tarnai, T. , 2005, “ Threefold-Symmetric Bricard Linkages for Deployable Structures,” Int. J. Solids Struct., 42(8), pp. 2287–2301. [CrossRef]
Song, C. Y. , Chen, Y. , and Chen, I. M. , 2014, “ Kinematic Study of the Original and Revised General Line-Symmetric Bricard 6R Linkages,” ASME J. Mech. Rob., 6(3), p. 031002. [CrossRef]
Wei, G. , and Dai, J. S. , 2014, “ Origami-Inspired Integrated Planar-Spherical Overconstrained Mechanisms,” ASME J. Mech. Des., 136(5), p. 051003. [CrossRef]
Chen, Y. , Peng, R. , and You, Z. , 2015, “ Origami of Thick Panels,” Science, 349(6246), pp. 396–400. [CrossRef] [PubMed]
Howell, L. L. , 2001, Compliant Mechanisms, Wiley, New York.
Howell, L. L. , Magleby, S. P. , and Olsen, B. O. , eds., 2013, Handbook of Compliant Mechanisms, Wiley, New York.
Vogtmann, D. E. , Gupta, S. K. , and Bergbreiter, S. , 2013, “ Characterization and Modeling of Elastomeric Joints in Miniature Compliant Mechanisms,” ASME J. Mech. Rob., 5(4), p. 041017. [CrossRef]
Venkiteswaran, V. K. , and Su, H. J. , 2015, “ A Parameter Optimization Framework for Determining the Pseudo-Rigid-Body Model of Cantilever-Beams,” Precis. Eng., 40, pp. 46–54. [CrossRef]
Dado, M. H. , 2001, “ Variable Parametric Pseudo-Rigid-Body Model for Large-Deflection Beams With End Loads,” Int. J. Nonlinear Mech., 36(7), pp. 1123–1133. [CrossRef]
Yu, Y. Q. , Feng, Z. L. , and Xu, Q. P. , 2012, “ A Pseudo-Rigid-Body 2R Model of Flexural Beam in Compliant Mechanisms,” Mech. Mach. Theory, 55, pp. 18–33. [CrossRef]
Dai, J. S. , and Cannella, F. , 2008, “ Stiffness Characteristics of Carton Folds for Packaging,” ASME J. Mech. Des., 130(2), p. 022305. [CrossRef]
Wang, D. A. , Chen, J. H. , and Pham, H. T. , 2014, “ A Tristable Compliant Micromechanism With Two Serially Connected Bistable Mechanisms,” Mech. Mach. Theory, 71, pp. 27–39. [CrossRef]
Chen, G. , Gou, Y. , and Zhang, A. , 2011, “ Synthesis of Compliant Multistable Mechanisms Through Use of a Single Bistable Mechanism,” ASME J. Mech. Des., 133(8), p. 081007. [CrossRef]
Williams, M. D. , van Keulen, F. , and Sheplak, M. , 2012, “ Modeling of Initially Curved Beam Structures for Design of Multistable MEMS,” ASME J. Appl. Mech., 79(1), p. 011006. [CrossRef]
Dellaert, D. , and Doutreloigne, J. , 2014, “ Design and Characterization of a Thermally Actuated Latching MEMS Switch for Telecommunication Applications,” J. Micromech. Microeng., 24(7), p. 075022. [CrossRef]
Beharic, J. , Lucas, T. M. , and Harnett, C. K. , 2014, “ Analysis of a Compressed Bistable Buckled Beam on a Flexible Support,” ASME J. Appl. Mech., 81(8), p. 081011. [CrossRef]
Gomm, T. , Howell, L. L. , and Selfridge, R. H. , 2002, “ In-Plane Linear-Displacement Bistable Microrelay,” J. Micromech. Microeng., 12(3), pp. 257–264. [CrossRef]
Pucheta, M. A. , and Cardona, A. , 2010, “ Design of Bistable Compliant Mechanisms Using Precision-Position and Rigid-Body Replacement Methods,” Mech. Mach. Theory, 45(2), pp. 304–326. [CrossRef]
Tanner, J. D. , and Jensen, B. D. , 2013, “ Power-Free Bistable Threshold Accelerometer Made From a Carbon Nanotube Framework,” J. Mech. Sci., 4(2), pp. 397–405. [CrossRef]
Silverberg, J. , Na, J. , Evans, A. , Liu, B. , Hull, T. , Santangelo, C. , Lang, R. , Hayward, R. , and Cohen, I. , 2015, “ Origami Structures With a Critical Transition to Bistability Arising From Hidden Degrees of Freedom,” Nat. Mater., 14(4), pp. 389–393. [CrossRef] [PubMed]
Hanna, B. H. , Magleby, S. P. , Lang, R. J. , and Howell, L. L. , 2015, “ Force-Deflection Modeling for Generalized Origami Waterbomb-Base Mechanisms,” ASME J. Appl. Mech., 82(8), p. 081001. [CrossRef]
Waitukaitis, S. , Menaut, R. , Chen, B. G. , and van Hecke, M. , 2015, “ Origami Multistability: From Single Vertices to Metasheets,” Phys. Rev. Lett., 114(5), p. 055503. [CrossRef] [PubMed]
Yasuda, H. , and Yang, J. , 2015, “ Reentrant Origami-Based Metamaterials With Negative Poisson's Ratio and Bistability,” Phys. Rev. Lett., 114(18), p. 185502. [CrossRef] [PubMed]
Lang, R. J. , 2011, Origami Design Secrets: Mathematical Methods for an Ancient Art, CRC Press, Boca Raton, FL.
Greenberg, H. C. , Gong, M. L. , Howell, L. L. , and Magleby, S. P. , 2011, “ Identifying Links Between Origami and Compliant Mechanisms,” Mech. Sci., 2(2), pp. 217–225. [CrossRef]
Winder, B. G. , Magleby, S. P. , and Howell, L. L. , 2009, “ Kinematic Representations of Pop-Up Paper Mechanisms,” ASME J. Mech. Rob., 1(2), pp. 217–225. [CrossRef]
Silverberg, J. L. , Evans, A. A. , McLeod, L. , Hayward, R. C. , Hull, T. , Santangelo, C. D. , and Cohen, I. , 2014, “ Using Origami Design Principles to Fold Reprogrammable Mechanical Metamaterials,” Science, 345(6197), pp. 647–650. [CrossRef] [PubMed]
Schenk, M. , and Guest, S. D. , 2012, “ Geometry of Miura-Folded Metamaterials,” Proc. Natl. Acad. Sci. U.S.A., 110(9), pp. 3276–3281. [CrossRef]
Miura, K. , 1985, “ Method of Packaging and Deployment of Large Membranes in Space,” The Institute of Space and Astronautical Science, Tokyo, Tech. Report 618.
Zirbel, S. A. , Lang, R. J. , Magleby, S. P. , Thomson, M. W. , Sigel, D. A. , Walkemeyer, P. E. , Trease, B. P. , and Howell, L. L. , 2013, “ Accommodating Thickness in Origami-Based Deployable Arrays,” ASME J. Mech. Des., 135(11), p. 111005. [CrossRef]
Ma, J. , and You, Z. , 2013, “ Energy Absorption of Thin-Walled Square Tubes With a Prefolded Origami Pattern—Part I: Geometry and Numerical Simulation,” ASME J. Appl. Mech., 81(1), p. 011003. [CrossRef]
Guest, S. D. , and Pellegrino, S. , 1994, “ The Folding of Triangulated Cylinders—Part I: Geometric Considerations,” ASME J. Appl. Mech., 61(4), pp. 773–777. [CrossRef]
Hawkes, E. , An, B. , Benbernou, N. M. , Tanaka, H. , Kim, S. , Demaine, E. D. , Rus, D. , and Wood, R. J. , 2010, “ Programmable Matter by Folding,” Proc. Natl. Acad. Sci. U.S.A., 107(28), pp. 12441–12445. [CrossRef] [PubMed]
Felton, S. , Tolley, M. , Demaine, E. , Rus, D. , and Wood, R. , 2014, “ A Method for Self-Folding Machines,” Science, 345(6197), pp. 644–646. [CrossRef] [PubMed]
Quaglia, C. P. , Dascanio, A. J. , and Thrall, A. P. , 2014, “ Bascule Shelters: A Novel Erection Strategy for Origami-Inspired Deployable Structures,” Eng. Struct., 75, pp. 276–287. [CrossRef]
Francis, K. C. , Blanch, J. E. , Magleby, S. P. , and Howell, L. L. , 2013, “ Origami-Like Creases in Sheet Materials for Compliant Mechanism Design,” Mech. Sci., 4(2), pp. 371–380. [CrossRef]
Delimont, I. L. , Magleby, S. P. , and Howell, L. L. , 2015, “ A Family of Dual-Segment Compliant Joints Suitable for Use as Surrogate Folds,” ASME J. Mech. Des., 137(9), p. 092302. [CrossRef]
Evans, T. , 2015, “ Deployable and Foldable Arrays of Spatial Mechanisms,” M.S. thesis, Brigham Young University, Provo, UT.
Rowberry, B. G. , 2013, “ Stability of n = 6 Normal and Right-Angled Kaleidocycles Under the Influence of Energy Elements,” B.S. honors thesis, Brigham Young University, Provo, UT.

Figures

Grahic Jump Location
Fig. 1

A compliant-mechanism kaleidocycle that uses polypropylene “surrogate folds” to create the function of the rotating joints and the torsion springs

Grahic Jump Location
Fig. 2

An n = 6 kaleidocycle

Grahic Jump Location
Fig. 3

The core segments, with hinge lines indicated

Grahic Jump Location
Fig. 4

How θ, ϕ, and α are defined from the core segments and hinge lines

Grahic Jump Location
Fig. 5

A visualization of the geometric construction of the tetrahedron. The lines L1 and L2 are skew. The segments S1=a1b1¯ and S2=a2b2¯ lay on L1 and L2, respectively. The join of S1 and S2 is the tetrahedron τ. The dashed segment σ is the core segment, whose length is the shortest path between L1 and L2.

Grahic Jump Location
Fig. 6

The relation g[α](θ,ϕ)=0

Grahic Jump Location
Fig. 7

This is the region plot generated numerically for all kaleidocycles K[π/2,1](θ0,ϕ0). The plot shows the number of stable states based on θ0 (x-axis) and ϕ0 (y-axis).

Grahic Jump Location
Fig. 8

A plot of Ξ±[π/20](θ) for K[π/2,1](θ0,π/20). The kaleidocycle K[π/2,1](π/20,π/20) appears to have eight equilibria. The two plots correspond to the two different branches of Ξ±.

Grahic Jump Location
Fig. 9

Energy as a function of θ over the full rotation for K[π/2,1](π/30,π/30)

Grahic Jump Location
Fig. 10

Ξ±[ϕ0](θ) for K[π/2,1](θ0,π/20). Only the parts of the graph corresponding to stable equilibria are displayed. The bar indicates four stable equilibria.

Grahic Jump Location
Fig. 11

The bar representing K[π/2,1](θ0,π/20) now intersects the plot at a boundary point

Grahic Jump Location
Fig. 12

The parameterized curve S[π/2,1]

Grahic Jump Location
Fig. 13

These are the plots of S[α,r] for various values of r and α as described in Table 1. Different colored regions represent regions with different numbers of stable states, as follows: Gray—monostable, blue—bistable, yellow—tristable, and red—quadstable.

Tables

Errata

Discussions

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