Design Innovation Paper

A Lamina-Emergent Frustum Using a Bistable Collapsible Compliant Mechanism

[+] Author and Article Information
Rami Alfattani

Department of Mechanical Engineering,
University of South Florida,
4202 E. Fowler Avenue,
Tampa, FL 33620
e-mail: rami1@mail.usf.edu

Craig Lusk

Department of Mechanical Engineering,
University of South Florida,
4202 E. Fowler Avenue,
Tampa, FL 33620
e-mail: clusk2@usf.edu

Contributed by the Mechanisms and Robotics Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received October 17, 2016; final manuscript received August 4, 2017; published online September 18, 2018. Assoc. Editor: David Myszka.

J. Mech. Des 140(12), 125001 (Sep 18, 2018) (10 pages) Paper No: MD-16-1707; doi: 10.1115/1.4037621 History: Received October 17, 2016; Revised August 04, 2017

This paper presents a new bistable collapsible compliant mechanism (BCCM) that is utilized in a lamina-emergent frustum. The mechanism is based on transforming a polygon spiral into spatial frustum shape using a mechanism composed of compliant links and joints that exhibits a bistable behavior. A number of mechanism types (graphs) were considered to implement the shape-morphing spiral, including 4-bar, 6-bar, and 8-bar chains. Our design requirements permitted the selection of a particular 8-bar chain as the basis for the BCCM. The bistable behavior was added to the mechanism by introducing a snap-through bistability as the mechanism morphs. A parametric CAD was used to perform the dimensional synthesis. The design was successfully prototyped. We anticipate that the mechanism may be useful in commercial small animal enclosures or as a frame for a solar still.

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Patel, J. , and Ananthasuresh, G. , 2007, “ A Kinematic Theory for Radially Foldable Planar Linkages,” Int. J. Solids Struct., 44(18–19), pp. 6279–6298. [CrossRef]
Hartmann, A. , and Hoffert, B. , 1999, “ SPACE-Saving Furniture Projects for the Home,” Libr. J., 124(8), p. 107.
Frensemeier, M. , Schirra, D. , Weinmann, M. , Weber, O. , and Kroner, E. , 2016, “ Shape-Memory Topographies on Nickel–Titanium Alloys Trained by Embossing and Pulse Electrochemical Machining,” Advanced Engineering Materials, Wiley-VCH Verlag, Weinheim, Germany, pp. 1388–1395.
Howell, L. L. , 2001, Complaint Mechanisms, Wiley, New York.
Qiu, J. , Lang, J. H. , and Slocum, A. H. , 2004, “ A Curved-Beam Bistable Mechanism,” J. Microelectromech. Syst., 13(2), pp. 137–146. [CrossRef]
Shouyin, Z. , and Guimin, C. , 2011, “ Design of Compliant Bistable Mechanism for Rear Trunk Lid of Cars,” ICIRA Fourth International Conference, Aachen, Germany, Dec. 6–8. pp. 291–299.
Manan, D. , and Dibakar, S. , 2014, “ Design of Double Toggle Switching Mechanisms,” J. Mech. Mach. Theory, 71, pp. 163–190. [CrossRef]
Guimin, C. , Shouyin, Z. , and Geng, L. , 2013, “ Multistable Behaviors of Compliant Sarrus Mechanisms,” ASME J. Mech. Rob., 5(2), p. 021005. [CrossRef]
Jacobsen, J. O. , Winder, B. G. , Howell, S. P. , and Magleby, S. P. , 2010, “ Lamina Emergent Mechanisms and Their Basic Elements,” ASME J. Mech. Rob., 2(1), p. 011003.
Wilding, S. E. , Howell, L. L. , and Magleby, S. P. , 2012, “ Spherical Lamina Emergent Mechanisms,” Mech. Mach. Theory, 49, pp. 187–197. [CrossRef]
Norton, R. L. , 2012, Design of Machinery, McGraw-Hill, Worcester, UK.
Howell, L. L. , Midha, A. , and Norton, T. W. , 1996, “ Evaluation of Equivalent Spring Stiffness for Use in a Pseudo-Rigid-Body Model of Large-Deflection Compliant Mechanisms,” ASME J. Mech. Des., 118(1), pp. 126–131. [CrossRef]
Xilun Ding, X. L. , 2015, “ Design of a Type of Deployable/Retractable Mechanism Using,” Mech. Mach. Theory, 92, pp. 273–288. [CrossRef]
Pucheta, M. A. , Butti, A. , Tamellini, V. , Cardona, A. , and Ghezzi, L. , 2012, “ Topological Synthesis of Planar Metamorphic Mechanisms for Low-Voltage Circuit Breakers,” Mech. Based Des. Struct. Mach., 40(4), pp. 453–468. [CrossRef]
Liu, T. S. , and Chou, C. C. , 1993, “ Type Synthesis of Vehicle Planner Suspension Mechanism Using Graph Theory,” ASME J. Mech. Des., 115(3), pp. 652–657. [CrossRef]
Feng, C. M. , and Liu, T. S. , 2013, “ A Graph-Theory Approach to Designing Deployable Mechanism of Reflector Antenna,” Acta Astronaut., 87, pp. 40–47. [CrossRef]
Tsai, L.-W. , 1999, Mechanism Design: Enumeration of Kinematic Structures according to Function, CRC Press, Boca Raton, FL.
Su, H. , and McCarthy, L. M. , 2007, “ Synthesis of Bistable Compliant Four-Bar Mechanisms Using Polynomial Homotopy,” ASME J. Mech. Des., 129(10), pp. 1094–1098. [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]
Wiebe, R. , Virgin, L. , Stanciulescu, I. , Spottswood, S. , and Eason, T. , 2013, “ Characterizing Dynamic Transitions Associated With Snap-Through: A Discrete System,” ASME J. Comput. Nonlinear Dyn., 8(1), p. 011010. [CrossRef]
Qun Chen, L. , and Li, K. , 2016, “ Equilibriums and Their Stabilities of the Snap-Through Mechanism,” Arch. Appl. Mech., 86(3), pp. 403–410. [CrossRef]
Alqasimi, A. , Lusk, C. , and Chimento, J. , 2014, “ Design of a Linear Bi-stable Compliant Crank-Slider-Mechanism (LBCCSM),” Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Buffalo, NY, Aug. 17–20.
Alqasimi, A. , and Lusk, C. , 2015, “ Shape-Morphing Space Frame (SMSF) Using Linear Bistable Elements,” Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Boston, MA, Aug. 2–5.
Lusk, C. , and Montalbano, P. , 2011, “ Design Concepts for Shape-Shifting Surfaces,” Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Washington, DC, Aug. 28–31.
Gattas, J. M. , and You, Z. , 2015, “ Geometric Assembly of Rigid-Foldable Morphing Sandwich Structures,” Eng. Struct., 94, pp. 149–159. [CrossRef]
Wheeler, C. M. , and Culpepper, M. L. , 2016, “ Soft Origami: Classification, Constraint, and Actuation of Highly Compliant Origami Structures,” ASME J. Mech. Rob., 8(5), p. 051012. [CrossRef]
Peraza Hernandez, E. , Hartl, D. , and Lagoudas, D. , 2016, “ Kinematics of Origami Structures With Smooth Folds,” ASME J. Mech. Rob., 8(6), p. 0161019. [CrossRef]
Solomon, J. , Vouga, E. , Wardetzky, M. , and Grinspun, E. , 2012, “ Flexible Developable Surfaces,” Comput. Graph. Forum, 31(5), p. 1567. [CrossRef]
Wen-Tung, C. , Chen-Chou, L. , and Long-Iong, W. , 2005, “ A Note on Grashof's Theorem,” J. Mar. Sci. Technol., 13(4), pp. 239–248. http://jmst.ntou.edu.tw/marine/13-4/239-248.pdf
Vladimir, P. , and Ekaterina, E. , 2015, “ Number Structural Synthesis and Enumeration Process of All Possible Sets of Multiple Joints for 1-DOF up to 5-Loop 12-Link,” Mech. Mach. Theory, 90, pp. 108–127. [CrossRef]
Kinzel, E. C. , Schmiedeler, J. P. , and Pennock, G. R. , 2006, “ Kinematic Synthesis for Finitely Separated Positions Using Geometric Constraint Programming,” ASME Mech. Mach. Theory, 128(5), pp. 1070–1079 [CrossRef]
Jacobsen, J. , Chen, G. , Howell, L. , and Magleby, S. , 2009, “ Lamina Emergent Torsion (LET) Joint,” ASME Mech. Mach. Theory, 44(11), pp. 2098–2109. [CrossRef]


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Fig. 1

The-ball-on-a-hill analogy for bistable mechanisms. Adapted from Ref. [4].

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Fig. 2

A compliant “snap-through” mechanism that is used to illustrate the type of bistable behavior used in the BCCM design

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Fig. 3

The shape change process: (a) a polygonal spiral in its planar position and (b) a frustum shape after straightening the spiral lines

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Fig. 4

Polygon spiral calculation and ratio with k = 4 segments and n = 8 sectors

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Fig. 5

A slice of the polygon frustum to measure the height

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Fig. 6

The height (H) and the sector width (b) as a function of the number of sectors n for a unit radius R

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Fig. 7

Sector calculation for n = 8 sectors

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Fig. 8

A single sector of the design (when n = 4) that shows a repeatable quadrilateral element with constant size ratio in the two stable configurations

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Fig. 9

(a) Limited collapsing motion is possible in 6-bar mechanisms and (b) better collapsing motion is possible in 8-bar mechanisms

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Fig. 10

Three kinematic categories when Q = 0, 1, and 2 are obtained from Eqs. (12) and (13). For category 1, (Q = 0) leads to T = 4, B = 4 and can represent nine linkage chains. For category 2, (Q = 1) leads to T = 2, B = 5 and can represent five linkage chains. For category 3, (Q = 2) leads to T = 0, B = 6 and can represent two linkage chains.

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Fig. 11

All 16 possible 1DoF sets result from Eqs. (12) and (13) [11]. Category 1 represents nine configurations of Q = 0, T = 4, and B = 4. Category 2 represents five configurations of Q = 1, T = 2, and B = 5. Category 3 represents two configurations of Q = 2, T = 0, and B = 6.

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Fig. 12

Two 1DoF mechanisms are acquired and pass all design criteria. Mechanism 1 is composed of Q = 2 and B = 4. Mechanism 2 is composed of T = 4 and B = 4.

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Fig. 13

Graph representations of the two feasible mechanisms

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Fig. 14

Repeating (polymerizing) the structure of the two mechanisms in Fig. 11(a) graph representation and (b) mechanism representation

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Fig. 15

The dimensional synthesis process to create the sector mechanism from the selected 8-bar kinematic chain. Parametric CAD was used to apply the exterior constraints as shown in part (a) and to implement the interior scheme of the selected 8-bar kinematic chain (shown in part (b)), which achieves the final two designs shown in part (c) which solidworks produced after equality constraints were set between equivalent links in the two positions.

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Fig. 16

BCCM bistable configurations

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Fig. 17

(a) A stable position forming the lamina polygon spiral shape. The BCCM design: part (b) a stable position forming the frustum shape.

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Fig. 18

BCCM prototype made of 1/8 in thick polypropylene material

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Fig. 19

The torsion-bar that connects the BCCM to the base [32]

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Fig. 20

The top is made of a 1/16 in (mm) thick polypropylene material

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Fig. 21

Lamina-emergent frustum in both stable positions, the planar position and frustum position (top and side views)



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