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.

Copyright © 2018 by ASME
Your Session has timed out. Please sign back in to continue.


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.


Grahic Jump Location
Fig. 1

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

Grahic Jump Location
Fig. 3

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

Grahic Jump Location
Fig. 4

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

Grahic Jump Location
Fig. 5

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

Grahic Jump Location
Fig. 6

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

Grahic Jump Location
Fig. 7

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

Grahic Jump Location
Fig. 8

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

Grahic Jump Location
Fig. 9

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

Grahic Jump Location
Fig. 11

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

Grahic Jump Location
Fig. 12

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

Grahic Jump Location
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

Grahic Jump Location
Fig. 13

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

Grahic Jump Location
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

Grahic Jump Location
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

Grahic Jump Location
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

Grahic Jump Location
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

Grahic Jump Location
Fig. 18

FEM rendering of T-mechanism deformation

Grahic Jump Location
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



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