Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Synthesis of multistable equilibrium linkage systems using an optimization approach


This paper introduces a methodology for designing multistable equilibrium (MSE) systems. The methodology derives design criteria that relate system equilibrium characteristics to a potential energy curve. These design criteria are then used in a performance index that guides a candidate’s potential energy to approach the desired potential energy curve. As an example, a four-bar linkage with linear translational springs attached demonstrates the design methodology and the difficulties inherent in synthesizing MSE systems. To solve for the unknown design variables of the example problem a genetic algorithm is used.

This is a preview of subscription content, log in to check access.


  1. 1.

    Capanu M, Boyd JG, Hesketh PJ (2000) Design, fabrication, and testing of a bistable electromagnetically actuated microvalve. J Microelectromech Syst 9(2):181–189

  2. 2.

    Chironis NP (1991) Mechanisms and mechanical devices sourcebook. McGraw-Hill, New York

  3. 3.

    Erdman AG, Sandor GN, Kota S (2001) Mechanism design: analysis and synthesis, Vol. 1. 4th edn. Prentice-Hall, NJ

  4. 4.

    Greenwood DT (1988) Principles of dynamics. 2nd end. Prentice-Hall, New Jersey,

  5. 5.

    Hafez M, Lichter MD, Dubowsky S (2002) Optimized binary modular reconfigurable robotic devices. Proceedings of the 2002 IEEE International Conference on Robotics and Automation, Washington D.C.

  6. 6.

    Hoffman M, Kopka P, Voges E (1999) Bistable micromechanical fiber-optic switches on silicon with thermal actuators. Sens and Actuators 78:28–35

  7. 7.

    Houck CR, Joines, JA, Kay MG (1995) A genetic algorithm for function optimization: a matlab implementation. North Carolina State University

  8. 8.

    Howell LL, Rao SS, Midha A (1994) Reliability-based optimal design of a bistable compliant mechanism. J Mech Des 116:1115–1121

  9. 9.

    Ingber AL (1989) Very fast simulated re-annealing. J Mathl Comput Model 12:967–973

  10. 10.

    Jensen BD, Howell LL (2000) Identification of compliant pseudo-rigid-body mechanism configurations resulting in bistable behavior. The Proceedings of the 2000 ASME Design Engineering Technical Conference, Baltimore, Maryland, Paper No. DETC2000/MECH-14147

  11. 11.

    Jensen BD, Howell LL, Salmon LG (1998) Introduction of two-link, in-plane, bistable compliant MEMS. The Proceedings of the 1998 ASME Design Engineering Technical Conference, Atlanta, Georgia, Paper No. DETC98/MECH-5837

  12. 12.

    Opdahl PG, Jensen BD, Howell LL (1998) An investigation into compliant bistable mechanisms. The Proceedings of the 1998 ASME Design Engineering Technical Conference, Atlanta, Georgia, Paper No. DETC98/MECH-5914

  13. 13.

    Papalambros P, Wilde D (2000) Principles of optimal design: modeling and computation. 2nd edn. Cambridge University Press

  14. 14.

    Pfaffenberger RC, Patterson JH (1987) Statistical methods: for business and economics, 3rd edn. Richard D. Irwin

  15. 15.

    Saif M, Taher A (2000) On a tunable bistable MEMS — theory and experiment. J Microelectricalmech Syst 9(2):157–170

  16. 16.

    Schomburg WK, Goll C (1998) Design optimization of bistable microdiaphragm valves. Sens Actuators A 64:259–264

  17. 17.

    Schulze EF (1955) Designing snap-action toggles. Prod Eng 26(1):168–170

  18. 18.

    Stützle T, Hoos HH (1999) Analyzing the run-time behaviour of iterated local search for the TSP. Proceedings of the 1999 Metaheuristics International Conference, Angra dos Reis, Brazil, July 19–22

  19. 19.

    Vangbo M, Bäcklund Y (1998) A lateral symmetrically bistable buckled beam. J Micromech Microeng 8:29–32

Download references

Author information

Correspondence to C.W. King.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

King, C., Campbell, M., Beaman, J. et al. Synthesis of multistable equilibrium linkage systems using an optimization approach. Struct Multidisc Optim 29, 477–487 (2005).

Download citation


  • Equilibrium
  • Four-bar linkage
  • Multistable