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.

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Fig. 1

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

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Fig. 2

An n = 6 kaleidocycle

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Fig. 3

The core segments, with hinge lines indicated

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Fig. 4

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

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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.

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Fig. 6

The relation g[α](θ,ϕ)=0

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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).

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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±.

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Fig. 9

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

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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.

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Fig. 11

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

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Fig. 12

The parameterized curve S[π/2,1]

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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.



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