Advertisement

Simplified model and analysis of a pair of coupled thermo-optical MEMS oscillators

  • Richard H. RandEmail author
  • Alan T. Zehnder
  • B. Shayak
  • Aditya Bhaskar
Original paper
  • 23 Downloads

Abstract

Motivated by the dynamics of microscale oscillators with thermo-optical feedback, a simplified third-order model capturing the key features of these oscillators is developed, where each oscillator consists of a displacement variable coupled to a temperature variable. Further, the dynamics of a pair of such oscillators coupled via a linear spring is analyzed. The analytical procedures used are the variational equation method and the two-variable expansion method. It is shown that the analytical results are in agreement with the results of numerical integration. The bifurcation structure of the system is revealed through a bifurcation diagram.

Keywords

Micro-electromechanical systems Limit cycle oscillator Coupled oscillators Perturbation method Bifurcations 

Notes

Acknowledgements

This material is based upon work supported by the National Science Foundation under Grant Number CMMI-1634664. The authors wish to thank Professor John Guckenheimer for advising them on the bifurcations involved in this paper.

Funding

Funding has been received from NSF as acknowledged above. The entire research presented here is the authors’ own. No part of this article has been reproduced from other Articles or is under consideration elsewhere.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

Human and animal rights

Research on human and animal subjects was not necessary for this project.

Informed consent

All authors consent to submission of this article in its present form.

References

  1. 1.
    Aubin, K., Zalalutdinov, M., Alan, T., Reichenbach, R.B., Rand, R., Zehnder, A., Parpia, J., Craighead, H.: Limit cycle oscillations in cw laser-driven nems. J. Microelectromech. Syst. 13(6), 1018–1026 (2004).  https://doi.org/10.1109/JMEMS.2004.838360 CrossRefGoogle Scholar
  2. 2.
    Blocher, D., Rand, R.H., Zehnder, A.T.: Analysis of laser power threshold for self oscillation in thermo-optically excited doubly supported mems beams. Int. J. Non-Linear Mech. 57, 10–15 (2013).  https://doi.org/10.1016/j.ijnonlinmec.2013.06.010 CrossRefGoogle Scholar
  3. 3.
    Blocher, D., Zehnder, A.T., Rand, R.H., Mukerji, S.: Anchor deformations drive limit cycle oscillations in interferometrically transduced mems beams. Finite Elem. Anal. Des. 49(1), 52–57 (2012).  https://doi.org/10.1016/j.finel.2011.08.020 CrossRefGoogle Scholar
  4. 4.
    Chávez, J.P., Brzeski, P., Perlikowski, P.: Bifurcation analysis of non-linear oscillators interacting via soft impacts. Int. J. Non-Linear Mech. 92, 76–83 (2017).  https://doi.org/10.1016/j.ijnonlinmec.2017.02.018 CrossRefGoogle Scholar
  5. 5.
    Doedel, E.J., Champneys, A.R., Dercole, F., Fairgrieve, T., Kuznetsov, Y., Oldeman, B., Paffenroth, R., Sandstede, B., Wang, X., Zhang, C.: Auto-07p: continuation and bifurcation software for ordinary differential equations (2008). http://sourceforge.net/projects/auto-07p/. Accessed 12 Aug 2019
  6. 6.
    Fradkov, A.L., Andrievsky, B.: Synchronization and phase relations in the motion of two-pendulum system. Int. J. Non-Linear Mech. 42(6), 895–901 (2007).  https://doi.org/10.1016/j.ijnonlinmec.2007.03.016 CrossRefGoogle Scholar
  7. 7.
    Hennig, D.: Existence of nonlinear normal modes for coupled nonlinear oscillators. Nonlinear Dyn. 80(1), 937–944 (2015).  https://doi.org/10.1007/s11071-015-1918-3 MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Kuramoto, Y.: Self-entrainment of a population of coupled non-linear oscillators. In: International Symposium on Mathematical Problems in Theoretical Physics, pp. 420–422. Springer (1975)Google Scholar
  9. 9.
    Mirollo, R., Strogatz, S.: Synchronization of pulse-coupled biological oscillators. SIAM J. Appl. Math. 50(6), 1645–1662 (1990).  https://doi.org/10.1137/0150098 MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Nayfeh, A., Mook, D.: Nonlinear Oscillations. Pure and Applied Mathematics: A Wiley Series of Texts, Monographs and Tracts. Wiley, Hoboken (1979). https://books.google.com/books?id=SQq0QgAACAAJ
  11. 11.
    Nayfeh, A.H.: A perturbation method for treating nonlinear oscillation problems. J. Math. Phys. 44(1–4), 368–374 (1965).  https://doi.org/10.1002/sapm1965441368 MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Pandey, M., Aubin, K., Zalalutdinov, M., Reichenbach, R.B., Zehnder, A.T., Rand, R.H., Craighead, H.G.: Analysis of frequency locking in optically driven mems resonators. J. Microelectromech. Syst. 15(6), 1546–1554 (2006).  https://doi.org/10.1109/JMEMS.2006.879693 CrossRefGoogle Scholar
  13. 13.
    Pandey, M., Rand, R., Zehnder, A.: Perturbation analysis of entrainment in a micromechanical limit cycle oscillator. Commun. Nonlinear Sci. Numer. Simul. 12(7), 1291–1301 (2007).  https://doi.org/10.1016/j.cnsns.2006.01.017 CrossRefzbMATHGoogle Scholar
  14. 14.
    Rand, R.H.: Notes on Nonlinear Vibrations. Published on-line by The Internet-First University Press (2012). http://ecommons.library.cornell.edu/handle/1813/28989. Accessed 12 Aug 2019
  15. 15.
    Śliwa, I., Grygiel, K.: Periodic orbits, basins of attraction and chaotic beats in two coupled Kerr oscillators. Nonlinear Dyn. 67(1), 755–765 (2012).  https://doi.org/10.1007/s11071-011-0024-4 MathSciNetCrossRefGoogle Scholar
  16. 16.
    Stoker, J.: Nonlinear Vibrations (1966). https://books.google.com/books?id=KFZIJxZYoxgC. Accessed 12 Aug 2019
  17. 17.
    Storti, D., Rand, R.: Dynamics of two strongly coupled van der Pol oscillators. Int. J. Non-Linear Mech. 17(3), 143–152 (1982).  https://doi.org/10.1016/0020-7462(82)90014-2 MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Suchorsky, M.K., Rand, R.H.: A pair of van der pol oscillators coupled by fractional derivatives. Nonlinear Dyn. 69(1), 313–324 (2012).  https://doi.org/10.1007/s11071-011-0266-1 MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Valente, A.X.C., McClamroch, N., Mezić, I.: Hybrid dynamics of two coupled oscillators that can impact a fixed stop. Int. J. Non-linear Mech. 38(5), 677–689 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Younis, M.I.: MEMS Linear and Nonlinear Statics and Dynamics, vol. 20. Springer, Berlin (2011)CrossRefGoogle Scholar
  21. 21.
    Zehnder, A.T., Rand, R.H., Krylov, S.: Locking of electrostatically coupled thermo-optically driven mems limit cycle oscillators. Int. J. Non-Linear Mech. 102, 92–100 (2018).  https://doi.org/10.1016/j.ijnonlinmec.2018.03.009 CrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Theoretical and Applied Mechanics, Sibley School of Mechanical and Aerospace EngineeringCornell UniversityIthacaUSA
  2. 2.Department of MathematicsCornell UniversityIthacaUSA

Personalised recommendations