Skip to main content
SpringerLink
Log in

Two-to-one internal resonance in microscanners

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

To realize large scanning angles, torsional microscanners are normally excited at their natural frequencies. Usually, a bias DC voltage is also applied to scan around a desired nonzero tilt angle. As a result, a deep understanding of the mirror’s response to a DC-shifted primary resonance excitation is imperative. Along these lines, we use the method of multiple scales to obtain a second-order nonlinear approximate analytical solution of the mirror steady-state response. We show that the response of the mirror exhibits a softening-type behavior that increases as the magnitude of the DC component increases. For a given mirror, we can also identify a DC voltage range wherein the mirror exhibits a two-to-one internal resonance between the first two modes; that is, ω 2≈2ω 1. To analyze the mirror behavior within that range, we first treat the case where the excitation frequency is near the first-mode frequency; that is, Ω≈ω 1. Then we treat the case where the excitation frequency is near the second-mode frequency; that is, Ω≈ω 2. We analyze the stability of the response and compare the analytical results to numerical solutions obtained via long-time integration of the equations of motion. We show that, due to the internal resonance, the mirror exhibits complex dynamic behavior characterized by aperiodic responses to primary resonance excitations. This behavior results in undesirable oscillations that are detrimental to the mirror performance, namely bringing the target point in and out of focus and resulting in distorted images.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Van Kessel, P.F., Hornbeck, L.J., Meier, R.E., Douglass, M.R.: MEMS-based projection display. Proc. IEEE 86(8), 1687–1704 (1998)

    Article  Google Scholar 

  2. Ford, J.E., Aksyuk, V.A., Bishop, D.J., Walker, J.A.: Wavelength add-drop switching using tilting micromirrors. J. Lightwave Technol. 17(5), 904–911 (1999)

    Article  Google Scholar 

  3. Lin, L.Y., Goldstein, A.L., Tkach, R.W.: Free-space micromachined optical switches for optical networking. IEEE J. Sel. Top. Quantum Electron. 5(1) (1999)

  4. Collins, D.R., Sampsell, J.B., Hornbeck, L.J., Florence, P.A., Penz, P.A., Gately, M.T.: Application of improved deformable mirror array technology to neural network realization. Neural Netw. 1(1), 378–382 (1988)

    Google Scholar 

  5. Cohn, R.W., Sampsell, J.B.: Deformable mirror device uses in frequency excision and optical switching. Appl. Opt. 27, 937–940 (1988)

    Article  Google Scholar 

  6. Hornbeck, L.J.: 128 multiplied by 128 deformable mirror device. IEEE Trans. Electron Dev. 30(5), 539–545 (1983)

    Article  Google Scholar 

  7. Fan, L., Yu, M.C.: Two-dimensional optical scanners with large angular rotation realized by self-assembled micro-elevator. In: Proceedings of the IEEE LEOS Summer Topical Meeting on Optical MEMS, Monterey, CA pp. 107–108 (1998)

  8. Osterberg, P.M.: Electrostatically actuated microelectromechanical test structures for material property measurements. PhD thesis, MIT, Cambridge, MA (1995)

  9. Hornbeck, L.J.: Deformable-mirror spatial light modulators. SPIE Crit. Rev. Ser. 1150, 86–102 (1989)

    Google Scholar 

  10. Degani, O.B., Socher, E., Lipson, A., Leitner, T., Setter, D.J., Kaldor, S., Nemirovisky, Y.: Pull-in study of an electrostatic torsion microactuator. J. Microelectromech. Syst. 7(4), 373–379 (1998)

    Article  Google Scholar 

  11. Nemirovisky, Y., Degani, O.B.: A methodology and model for the pull-in parameters of electrostatic actuators. J. Microelectromech. Syst. 10(4), 601–614 (2001)

    Article  Google Scholar 

  12. Degani, O.B., Nemirovisky, Y.: Design considerations of rectangular electrostatic torsion actuators based on new analytical pull-in expressions. J. Microelectromech. Syst. 11(1), 20–26 (2002)

    Article  Google Scholar 

  13. Zhang, X.M., Chau, F.S., Quan, C., Liu, A.Q.: Modeling of the optical torsion micromirror. In: Proceedings of the SPIE Conference on Photonics Technology in the 21st Century: Semiconductors, Microstructures, and Nanostructures, Singapore, pp. 109–116 (1999)

  14. Zhang, X.M., Chau, F.S., Quan, C., Lam, Y.L., Liu, A.Q.: Study of the static characteristics of a torsional micromirror. J. Microelectromech. Syst. 90, 73–81 (2001)

    Google Scholar 

  15. Degani, O.B., Nemirovisky, Y.: Modeling the pull-in parameters of electrostatic actuators with a novel lumped two degrees of freedom pull-in model. Sens. Actuators A 97–98, 569–578 (2002)

    Article  Google Scholar 

  16. Huang, J.M., Liu, A.Q., Deng, Z.L., Zhang, Q.X., Ahn, J., Asundi, A.: An approach to the coupling effect between torsion and bending for electrostatic torsional micromirrors. Sens. Actuators A 115, 159–167 (2004)

    Article  Google Scholar 

  17. Wetzel, G.C., Strozewski, K.J.: Dynamical model of microscale electromechanical spatial light modulator. J. Appl. Phys. 73(11), 7120–7124 (1993)

    Article  Google Scholar 

  18. Sattler, R., Florian, P., Fattinger, G., Wachutka, G.: Modeling of an electrostatic torsional actuator: Demonstrated with an RF MEMS switch. Sens. Actuators A 97, 337–346 (2002)

    Article  Google Scholar 

  19. Sane, H.S., Yazdi, N., Carlos, H.M.: Application of sliding mode control to electrostatically actuated two-axis gimbaled micromirrors. In: Proceedings of the American Control Conference, Denver, CO, pp. 3726–3731 (2003)

  20. Ataman, C., Urey, H.: Modeling and characterization of comb-actuated microscanners. J. Micromech. Microeng. 16, 9–16 (2006)

    Article  Google Scholar 

  21. Camon, H., Lanaudi, F.: Fabrication, simulation, and experiment of a rotating electrostatic silicon mirror with large angular deflections. In: Proceedings of the IEEE Micro Electro Mechanical Systems (MEMS), Japan, pp. 645–650 (2000)

  22. Zhao, J.P., Chen, H.L., Huang, J.M., Liu, A.Q.: A study of dynamic characteristics and simulations of MEMS torsional micromirror. Sens. Actuators A 120, 199–210 (2005)

    Article  Google Scholar 

  23. Daqaq, M.F.: Adaptation of nontraditional control techniques to nonlinear micro and macro mechanical systems. PhD thesis, Virginia Tech, Blacksburg, VA (2006)

  24. Timoshenko, S.: Theory of Plates and Shells. McGraw-Hill, New York (1987)

    Google Scholar 

  25. Nayfeh, A.H.: Introduction to Perturbation Techniques. Wiley-Interscience, New York (1981)

    MATH  Google Scholar 

  26. Daqaq, M.F., Abdel-Rahman, E.M., Nayfeh, A.H.: Towards a stable low-voltage torsional microscanner. Microsyst. Technol. 14, 725–737 (2008)

    Article  Google Scholar 

  27. Nayfeh, A.H.: Resolving controversies in the application of the method of multiple scales and the generalized method of averaging. Nonlinear Dyn. 40, 61–102 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  28. Arafat, H.N.: Nonlinear response of cantilever beams. PhD thesis, Virginia Tech, Blacksburg, VA (1999)

Download references

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, Clemson University, Clemson, SC, 29634, USA

    Mohammed F. Daqaq

  2. Systems Design Engineering, University of Waterloo, Waterloo, Canada

    Eihab M. Abdel-Rahman

  3. Department of Engineering Science and Mechanics, Virginia Tech, Blacksburg, VA, 24061, USA

    Ali H. Nayfeh

Authors
  1. Mohammed F. Daqaq
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Eihab M. Abdel-Rahman
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ali H. Nayfeh
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mohammed F. Daqaq.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Daqaq, M.F., Abdel-Rahman, E.M. & Nayfeh, A.H. Two-to-one internal resonance in microscanners. Nonlinear Dyn 57, 231–251 (2009). https://doi.org/10.1007/s11071-008-9435-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-008-9435-2

Keywords

Search

Navigation