Research Papers: Design of Mechanisms and Robotic Systems

Programmable Multistable Mechanisms: Synthesis and Modeling

[+] Author and Article Information
Mohamed Zanaty

Institute of Microengineering,
Ecole Polytechnique Fédérale de
Lausanne (EPFL),
Rue de la Maladiere,
Neuchâtel 2000, Switzerland
e-mail: mohamed.zanaty@epfl.ch

Ilan Vardi

Institute of Microengineering,
Ecole Polytechnique Fédérale de
Lausanne (EPFL),
Rue de la Maladiere,
Neuchâtel 2000, Switzerland
e-mail: ilan.vardi@epfl.ch

Simon Henein

Institute of Microengineering,
Ecole Polytechnique Fédérale de
Lausanne (EPFL),
Rue de la Maladiere,
Neuchâtel 2000, Switzerland
e-mail: simon.henein@epfl.ch

1Corresponding author.

Contributed by the Mechanisms and Robotics Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received August 12, 2017; final manuscript received December 8, 2017; published online February 5, 2018. Assoc. Editor: Massimo Callegari.

J. Mech. Des 140(4), 042301 (Feb 05, 2018) (13 pages) Paper No: MD-17-1553; doi: 10.1115/1.4038926 History: Received August 12, 2017; Revised December 08, 2017

Compliant mechanisms can be classified according to the number of their stable states and are called multistable mechanisms if they have more than one stable state. We introduce a new family of mechanisms for which the number of stable states is modified by programming inputs. We call such mechanisms programmable multistable mechanisms (PMM). A complete qualitative analysis of a PMM, the T-mechanism, is provided including a description of its multistability as a function of the programming inputs. We give an exhaustive set of diagrams illustrating equilibrium states and their stiffness as one programming input varies while the other is fixed. Constant force behavior is also characterized. Our results use polynomial expressions for the reaction force derived from Euler–Bernoulli beam theory. Qualitative behavior follows from the evaluation of the zeros of the polynomial and its discriminant. These analytical results are validated by numerical finite element method simulations.

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Harne, R. , and Wang, K. , 2013, “ A Review of the Recent Research on Vibration Energy Harvesting Via Bistable Systems,” Smart Mater. Struct., 22(2), p. 023001. [CrossRef]
Younesian, D. , and Alam, M.-R. , 2017, “ Multi-Stable Mechanisms for High-Efficiency and Broadband Ocean Wave Energy Harvesting,” Appl. Energy, 197, pp. 292–302. [CrossRef]
Oberhammer, J. , Tang, M. , Liu, A.-Q. , and Stemme, G. , 2006, “ Mechanically Tri-Stable, True Single-Pole-Double-Throw (SPDT) Switches,” J. Micromech. Microeng., 16(11), p. 2251. [CrossRef]
Receveur, R. A. , Marxer, C. R. , Woering, R. , Larik, V. C. , and de Rooij, N.-F. , 2005, “ Laterally Moving Bistable MEMS DC Switch for Biomedical Applications,” J. Microelectromech. Syst., 14(5), pp. 1089–1098. [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]
Chen, G. , Aten, Q. T. , Zirbel, S. , Jensen, B. D. , and Howell, L. L. , 2010, “ A Tristable Mechanism Configuration Employing Orthogonal Compliant Mechanisms,” ASME J. Mech. Rob., 2(1), p. 014501. [CrossRef]
Oh, Y. S. , and Kota, S. , 2009, “ Synthesis of Multistable Equilibrium Compliant Mechanisms Using Combinations of Bistable Mechanisms,” ASME J. Mech. Des., 131(2), p. 021002. [CrossRef]
Chen, G. , Liu, Y. , and Gou, Y. , 2012, “ A Compliant 5-Bar Tristable Mechanism Utilizing Metamorphic Transformation,” Advances in Reconfigurable Mechanisms and Robots I, Springer, London, pp. 233–242. [CrossRef]
Chen, G. , Zhang, S. , and Li, G. , 2013, “ Multistable Behaviors of Compliant Sarrus Mechanisms,” ASME J. Mech. Rob., 5(2), p. 021005. [CrossRef]
Chen, G. , and Du, Y. , 2012, “ Double-Young Tristable Mechanisms,” ASME J. Mech. Rob., 5(1), p. 011007. [CrossRef]
Halverson, P. A. , Howell, L. L. , and Magleby, S. P. , 2010, “ Tension-Based Multi-Stable Compliant Rolling-Contact Elements,” Mech. Mach. Theory, 45(2), pp. 147–156. [CrossRef]
Zhao, J. , Huang, Y. , Gao, R. , Chen, G. , Yang, Y. , Liu, S. , and Fan, K. , 2014, “ Novel Universal Multistable Mechanism Based on Magnetic—Mechanical—Inertial Coupling Effects,” IEEE Trans. Ind. Electron., 61(6), pp. 2714–2723. [CrossRef]
Howell, L. L. , Magleby, S. P. , and Olsen, B. M. , 2013, Handbook of Compliant Mechanisms, Wiley, Hoboken, NJ. [CrossRef]
Cosandier, F. , Henein, S. , Richard, M. , and Rubbert, L. , 2017, The Art of Exure Mechanism Design, EPFL Press, Lausanne, Switzerland.
Strogatz, S. H. , 2014, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering, Westview Press, Boulder, CO.
Saif, M. , 2000, “ On a Tunable Bistable MEMS-Theory and Experiment,” J. Microelectromech. Syst., 9(2), pp. 157–170. [CrossRef]
Cazottes, P. , Fernandes, A. , Pouget, J. , and Hafez, M. , 2009, “ Bistable Buckled Beam: Modeling of Actuating Force and Experimental Validations,” ASME J. Mech. Des., 131(10), p. 101001. [CrossRef]
Gerson, Y. , Krylov, S. , and Ilic, B. , 2010, “ Electrothermal Bistability Tuning in a Large Displacement Micro Actuator,” J. Micromech. Microeng., 20(11), p. 112001. [CrossRef]
Li, S. , and Wang, K. , 2015, “ Fluidic Origami With Embedded Pressure Dependent Multi-Stability: A Plant Inspired Innovation,” J. R. Soc. Interface, 12(111), p. 20150639.
Chen, G. , Wilcox, D. L. , and Howell, L. L. , 2009, “ Fully Compliant Double Tensural Tristable Micromechanisms (DTTM),” J. Micromech. Microeng., 19(2), p. 025011. [CrossRef]
Zanaty, M. , 2018, “Programmable Multistable Mechanisms,” Ph.D. thesis, Ecole Polytechnique Federale de Lausanne, EPFL, Lausanne, Switzerland.
Timoshenko, S. P. , and Gere, J. M. , 1961, Theory of Elastic Stability, McGrawHill-Kogakusha, Tokyo, Japan.
Birkhoff, G. , and Mac Lane, S. , 1966, A Survey of Modern Algebra, CRC Press, Boca Raton, FL.
Bensimhoun, M. , 2016, “Historical Account and Ultra-Simple Proofs of Descartes’s Rule of Signs, De Gua, Fourier, and Budan’s Rule,” e-print arXiv:1309.6664.
Howell, L. L. , 2001, Compliant Mechanisms, Wiley, Hoboken, NJ.
Herder, J. , 2001, “Free Energy System: Theory, Conception and Design of Statically Balanced Spring Mechanisms,” Ph.D. thesis, Delft University of Technology, Delft, The Netherlands.
Hao, G. , 2015, “ Extended Nonlinear Analytical Models of Compliant Parallelogram Mechanisms: Third-Order Models,” Trans. Can. Soc. Mech. Eng., 39(1), pp. 71–83.
Merkle, R. C. , 1993, “ Two Types of Mechanical Reversible Logic,” Nanotechnology, 4(2), p. 114. [CrossRef]
Hafiz, M. , Kosuru, L. , and Younis, M. I. , 2016, “ Microelectromechanical Reprogrammable Logic Device,” Nat. Commun., 7, p. 11137. [CrossRef] [PubMed]
Gomm, T. , Howell, L. L. , and Selfridge, R. H. , 2002, “ In-Plane Linear Displacement Bistable Microrelay,” J. Micromech. Microeng., 12(3), p. 257. [CrossRef]
Rafsanjani, A. , Akbarzadeh, A. , and Pasini, D. , 2015, “ Snapping Mechanical Metamaterials Under Tension,” Adv. Mater., 27(39), pp. 5931–5935. [CrossRef] [PubMed]
Zanaty, M. , Rogg, A. , Fussinger, T. , Lovera, A. , Baur, C. , Bellouard, Y. , and Henein, S. , 2017, “Safe Puncture Tool for Retinal Vein Cannulation,” Design of Medical Devices (DMD), Eindhoven, The Netherlands, Nov. 14–15.


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Fig. 1

Block diagram representation of a 3DOP PMM programmed to be (a) monostable and (b) bistable

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Fig. 3

Block diagram representation of (a) PBM and (b) 2DOP T-combination

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Fig. 4

Double parallelogram mechanism and its strain energy when programmed to be (a) monostable and (b) bistable

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Fig. 5

(a) Double parallelogram mechanism connection blocks and (b) 2DOP T-combination of DPMs

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Fig. 6

Stable states of the 2DOP T-mechanism programmed to be (a) monostable, (b) bistable, (c) tristable, and (d) quadrastable

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Fig. 7

Block diagram representation, example mechanism and DOS diagram of (a) T-connection, (b) serial connection, and (c) parallel connection

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Fig. 8

(a) Constructed T-mechanism, (b) top view, and (c) main components

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Fig. 9

Range of DOS for admissible η1, η2 for (a) α2=0.5, (b) α2=1, and (c) α2=1.5

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Fig. 10

Strain energy of the T-mechanism programmed to be (a) monostable at p̂1=0,p̂2=0, (b) bistable at p̂1=0,p̂2=0.12, (c) tristable at p̂1=0.0175,p̂2=0, and (d) quadrastable at p̂1=0.12,p̂2=0.0175

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Fig. 11

(a) Sign of the discriminant ΔΦ, (b) number of sign alternations nσ, and (c) DOS

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Fig. 12

Sign of (a) β0, (b) β1, (c) β2, and (d) β3

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Fig. 13

(a) Sign and zeros of β0 and (b) DOS with boundaries

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Fig. 14

Equilibrium and zero stiffness diagrams for the fixed values shown in (a): (b) p̂1=0.0, (c) p̂1=0.007, (d) p̂1=0.012, (e) p̂1=0.016, and (f) p̂1=0.02

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Fig. 15

Equilibrium and zero stiffness diagrams for the fixedvalues shown in (a): (b) p̂2=−0.06, (c) p̂2=−0.03, (d) p̂2=0.025, (e) p̂2=0.06, and (f) p̂2=0.12

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Fig. 16

Stiffness and sign of stiffness at equilibrium positions: (a) and (b) for q0, (c) and (d) for q1, (e) and (f) for q2, (g) and (h) for q3

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Fig. 17

(a) Selected values of p̂1, p̂2 leading to near zero force and near constant force regions: (b) zero force monostable mechanism at p̂1=0,p̂2=0.052, (c) constant force monostable mechanism at p̂1=0.012,p̂2=0, (d) zero force bistable mechanism at p̂1=0.017,p̂2=−0.045, (e) constant force bistable mechanism at p̂1=0.007,p̂2=0.092, and (f) zero force tristable mechanism at p̂1=0.017,p̂2=0.12

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Fig. 18

FEM rendering of T-mechanism deformation

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Fig. 19

(a) Values of p̂1, p̂2 for FEM simulation with T-mechanism programmed to be (b) monostable at p̂1=0,p̂2=0, (c) bistable at p̂1=0,p̂2=0.12, (d) tristable at p̂1=0.0175,p̂2=0, (e) quadrastable at p̂1=0.0175,p̂2=0.012, and (f) Present difference between analytical and numerical models




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