Nonlinear Dynamics

, Volume 67, Issue 1, pp 859–883 | Cite as

Nonlinear damping in a micromechanical oscillator

  • Stav Zaitsev
  • Oleg Shtempluck
  • Eyal Buks
  • Oded Gottlieb
Original Paper

Abstract

Nonlinear elastic effects play an important role in the dynamics of microelectromechanical systems (MEMS). A Duffing oscillator is widely used as an archetypical model of mechanical resonators with nonlinear elastic behavior. In contrast, nonlinear dissipation effects in micromechanical oscillators are often overlooked. In this work, we consider a doubly clamped micromechanical beam oscillator, which exhibits nonlinearity in both elastic and dissipative properties. The dynamics of the oscillator is measured in both frequency and time domains and compared to theoretical predictions based on a Duffing-like model with nonlinear dissipation. We especially focus on the behavior of the system near bifurcation points. The results show that nonlinear dissipation can have a significant impact on the dynamics of micromechanical systems. To account for the results, we have developed a continuous model of a geometrically nonlinear beam-string with a linear Voigt–Kelvin viscoelastic constitutive law, which shows a relation between linear and nonlinear damping. However, the experimental results suggest that this model alone cannot fully account for all the experimentally observed nonlinear dissipation, and that additional nonlinear dissipative processes exist in our devices.

Keywords

MEMS Duffing oscillator Nonlinear damping Saddle-node bifurcation Parameter identification Forced vibration 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Turner, K.L., Miller, S.A., Hartwell, P.G., MacDonald, N.C., Strogatz, S.H., Adams, S.G.: Five parametric resonances in a microelectromechanical system. Nature 396, 149–152 (1998) CrossRefGoogle Scholar
  2. 2.
    Roukes, M.: Nanoelectromechanical systems face the future. Phys. World 14, 25–25 (2001) Google Scholar
  3. 3.
    Roukes, M.: Nanomechanical systems. Technical digest of the 2000 solid state sensor and actuator workshop (2000) Google Scholar
  4. 4.
    Husain, A., Hone, J., Postma, H.W.C., Huang, X.M.H., Drake, T., Barbic, M., Scherer, A., Roukes, M.L.: Nanowire-based very-high-frequency electromechanical resonator. Appl. Phys. Lett. 83, 1240–1242 (2003) CrossRefGoogle Scholar
  5. 5.
    Sidles, J.A., Garbini, J.L., Bruland, K.J., Rugar, D., Zuger, O., Hoen, S., Yannoni, C.S.: Magnetic resonance force microscopy. Rev. Mod. Phys. 67(1), 249–265 (1995) CrossRefGoogle Scholar
  6. 6.
    Rugar, D., Budakian, R., Mamin, H.J., Chui, B.W.: Single spin detection by magnetic resonance force microscopy. Nature 430, 329–332 (2004) CrossRefGoogle Scholar
  7. 7.
    Ekinci, K.L., Yang, Y.T., Roukes, M.L.: Ultimate limits to inertial mass sensing based upon nanoelectromechanical systems. J. Appl. Phys. 95(5), 2682–2689 (2004) CrossRefGoogle Scholar
  8. 8.
    Ekinci, K.L., Huang, X.M.H., Roukes, M.L.: Ultrasensitive nanoelectromechanical mass detection. Appl. Phys. Lett. 84(22), 4469–4471 (2004) CrossRefGoogle Scholar
  9. 9.
    Ilic, B., Craighead, H.G., Krylov, S., Senaratne, W., Ober, C.: Attogram detection using nanoelectromechanical oscillators. J. Appl. Phys. 95, 3694 (2004) CrossRefGoogle Scholar
  10. 10.
    Nayfeh, A.H., Ouakad, H.M., Najar, F., Choura, S., Abdel-Rahman, E.M.: Nonlinear dynamics of a resonant gas sensor. Nonlinear Dyn. 59(4), 607–618 (2010) MATHCrossRefGoogle Scholar
  11. 11.
    Blencowe, M.: Quantum electromechanical systems. Phys. Rep. 395, 159–222 (2004) CrossRefGoogle Scholar
  12. 12.
    Knobel, R.G., Cleland, A.N.: Nanometre-scale displacement sensing using a single electron transistor. Nature 424, 291–293 (2003) CrossRefGoogle Scholar
  13. 13.
    LaHaye, M.D., Buu, O., Camarota, B., Schwab, K.C.: Approaching the quantum limit of a nanomechanical resonator. Science 304, 74–77 (2004) CrossRefGoogle Scholar
  14. 14.
    Schwab, K., Henriksen, E.A., Worlock, J.M., Roukes, M.L.: Measurement of the quantum of thermal conductance. Nature 404, 974–977 (2000) CrossRefGoogle Scholar
  15. 15.
    Buks, E., Roukes, M.L.: Stiction, adhesion energy, and the Casimir effect in micromechanical systems. Phys. Rev. B 63, 33402 (2001) CrossRefGoogle Scholar
  16. 16.
    Buks, E., Roukes, M.L.: Metastability and the Casimir effect in micromechanical systems. Europhys. Lett. 54(2), 220–226 (2001) CrossRefGoogle Scholar
  17. 17.
    Schwab, K.C., Roukes, M.L.: Putting mechanics into quantum mechanics. Phys. Today 58, 36–42 (2005) CrossRefGoogle Scholar
  18. 18.
    Aspelmeyer, M., Schwab, K.: Focus on mechanical systems at the quantum limit. New J. Phys. 10(9), 095001 (2008) CrossRefGoogle Scholar
  19. 19.
    Kozinsky, I., Postma, H.W.C., Kogan, O., Husain, A., Roukes, M.L.: Basins of attraction of a nonlinear nanomechanical resonator. Phys. Rev. Lett. 99, 207201 (2007) CrossRefGoogle Scholar
  20. 20.
    Cross, M.C., Zumdieck, A., Lifshitz, R., Rogers, J.L.: Synchronization by nonlinear frequency pulling. Phys. Rev. Lett. 93, 224101 (2004) CrossRefGoogle Scholar
  21. 21.
    Erbe, A., Krömmer, H., Kraus, A., Blick, R.H., Corso, G., Richter, K.: Mechanical mixing in nonlinear nanomechanical resonators. Appl. Phys. Lett. 77, 3102–3104 (2000) CrossRefGoogle Scholar
  22. 22.
    Rhoads, J.F., Shaw, S.W., Turner, K.L., Baskaran, R.: Tunable microelectromechanical filters that exploit parametric resonance. J. Vib. Acoust. 127, 423–431 (2005) CrossRefGoogle Scholar
  23. 23.
    Reichenbach, R.B., Zalalutdinov, M., Aubin, K.L., Rand, R., Houston, B.H., Parpia, J.M., Craighead, H.G.: Third-order intermodulation in a micromechanical thermal mixer. J. Micro/Nanolithogr. MEMS MOEMS 14, 1244–1252 (2005) Google Scholar
  24. 24.
    Almog, R., Zaitsev, S., Shtempluck, O., Buks, E.: High intermodulation gain in a micromechanical Duffing resonator. Appl. Phys. Lett. 88, 213509 (2006) CrossRefGoogle Scholar
  25. 25.
    Almog, R., Zaitsev, S., Shtempluck, O., Buks, E.: Noise squeezing in a nanomechanical Duffing resonator. Phys. Rev. Lett. 98, 78103 (2007) CrossRefGoogle Scholar
  26. 26.
    Almog, R., Zaitsev, S., Shtempluck, O., Buks, E.: Signal amplification in a nanomechanical duffing resonator via stochastic resonance. Appl. Phys. Lett. 90, 13508 (2007) CrossRefGoogle Scholar
  27. 27.
    Zhang, W., Baskaran, R., Turner, K.L.: Nonlinear behavior of a parametric resonance-based mass sensor. In: Proc. IMECE2002, p. 33261 Nov (2002) Google Scholar
  28. 28.
    Buks, E., Yurke, B.: Mass detection with nonlinear nanomechanical resonator. Phys. Rev. E 74, 46619 (2006) CrossRefGoogle Scholar
  29. 29.
    Cleland, A.N., Roukes, M.L.: Noise processes in nanomechanical resonators. J. Appl. Phys. 92(5), 2758–2769 (2002) CrossRefGoogle Scholar
  30. 30.
    Yasumura, K.Y., Stowe, T.D., Chow, E.M., Pfafman, T., Kenny, T.W., Stipe, B.C., Rugar, D.: Quality factors in micron- and submicron-thick cantilevers. J. Micromech. Syst. 9(1), 117–125 (2000) CrossRefGoogle Scholar
  31. 31.
    Ono, T., Wang, D.F., Esashi, M.: Time dependence of energy dissipation in resonating silicon cantilevers in ultrahigh vacuum. Appl. Phys. Lett. 83(10), 1950–1952 (2003) CrossRefGoogle Scholar
  32. 32.
    Liu, X., Thompson, E., White, B. Jr, Pohl, R.: Low-temperature internal friction in metal films and in plastically deformed bulk aluminum. Phys. Rev. B 59(18), 11767–11776 (1999) CrossRefGoogle Scholar
  33. 33.
    Harrington, D.A., Mohanty, P., Roukes, M.L.: Energy dissipation in suspended micromechanical resonators at low temperatures. Physica B 284–288, 2145–2146 (2000) CrossRefGoogle Scholar
  34. 34.
    Lifshitz, R., Roukes, M.L.: Thermoelastic damping in micro- and nanomechanical systems. Phys. Rev. B 61(8), 5600–5609 (2000) CrossRefGoogle Scholar
  35. 35.
    Houston, B.H., Photiadis, D.M., Marcus, M.H., Bucaro, J.A., Liu, X., Vignola, J.F.: Thermoelastic loss in microscale oscillators. Appl. Phys. Lett. 80(7), 1300–1302 (2002) CrossRefGoogle Scholar
  36. 36.
    Lifshitz, R.: Phonon-mediated dissipation in micro- and nano-mechanical systems. Physica B 316–317, 397–399 (2002) CrossRefGoogle Scholar
  37. 37.
    Wilson-Rae, I.: Intrinsic dissipation in nanomechanical resonators due to phonon tunneling. Phys. Rev. B 77, 245418 (2008) CrossRefGoogle Scholar
  38. 38.
    Remus, L.G., Blencowe, M.P., Tanaka, Y.: Damping and decoherence of a nanomechanical resonator due to a few two level systems. arXiv:0907.0431 [cond-mat] (2009)
  39. 39.
    Popovic, P., Nayfeh, A.H., Oh, K., Nayfeh, S.A.: An experimental investigation of energy transfer from a highfrequency mode to a low-frequency mode in a flexible structure. J. Vib. Control 1(1), 115–128 (1995) CrossRefGoogle Scholar
  40. 40.
    Hajj, M.R., Fung, J., Nayfeh, A.H., Fahey, S.O.: Damping identification using perturbation techniques and higher-order spectra. Nonlinear Dyn. 23(2), 189–203 (2000) MATHCrossRefGoogle Scholar
  41. 41.
    Jaksic, N., Boltezar, M.: An approach to parameter identification for a single-degree-of-freedom dynamical system based on short free acceleration response. J. Sound Vib. 250, 465–483 (2002) CrossRefGoogle Scholar
  42. 42.
    Zhang, W., Baskaran, R., Turner, K.L.: Effect of cubic nonlinearity on auto-parametrically amplified resonant MEMS mass sensor. Sens. Actuators A, Phys. 102, 139–150 (2002) CrossRefGoogle Scholar
  43. 43.
    Zhang, W., Baskaran, R., Turner, K.: Tuning the dynamic behavior of parametric resonance in a micromechanical oscillator. Appl. Phys. Lett. 82, 130–132 (2003) CrossRefGoogle Scholar
  44. 44.
    Krylov, S., Ilic, B.R., Schreiber, D., Seretensky, S., Craighead, H.: The pull-in behavior of electrostatically actuated bistable microstructures. J. Micromech. Microeng. 18(5), 055026 (2008) CrossRefGoogle Scholar
  45. 45.
    Nayfeh, A.H., Mook, D.T.: Nonlinear Oscillations. Wiley, New York (1995). Wiley Classics Library CrossRefGoogle Scholar
  46. 46.
    Dykman, M., Krivoglaz, M.: Theory of nonlinear oscillator interacting with a medium. In: Khalatnikov, I.M. (ed.) Soviet Scientific Reviews, Section A, Physics Reviews, vol. 5, pp. 265–441. Harwood Academic, Reading (1984) Google Scholar
  47. 47.
    Landau, L.D., Lifshitz, E.M.: Mechanics, 3rd edn. Pergamon, New York (1976) Google Scholar
  48. 48.
    Nayfeh, A.H.: Introduction to Perturbation Techniques. Wiley, New York (1981) MATHGoogle Scholar
  49. 49.
    Arnold, V.I.: Geometrical methods in the theory of ordinary differential equations, Grundlehren der mathematischen Wissenschaften, vol. 250, 2nd edn. Springer, New York (1988) CrossRefGoogle Scholar
  50. 50.
    Strogatz, S.H.: Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering. Perseus Books, Readings (1994) Google Scholar
  51. 51.
    Chan, H.B., Dykman, M., Stambaugh, C.: Paths of fluctuation induced switching. Phys. Rev. Lett. 100, 130602 (2008) CrossRefGoogle Scholar
  52. 52.
    Dykman, M.I., Golding, B., Ryvkine, D.: Critical exponent crossovers in escape near a bifurcation point. Phys. Rev. Lett. 92(8), 080602 (2004) CrossRefGoogle Scholar
  53. 53.
    Yurke, B., Buks, E.: Performance of cavity-parametric amplifiers, employing Kerr nonlinearites, in the presence of two-photon loss. J. Lightwave Technol. 24(12), 5054–5066 (2006) CrossRefGoogle Scholar
  54. 54.
    Buks, E., Yurke, B.: Dephasing due to intermode coupling in superconducting stripline resonators. Phys. Rev. A 73, 23815 (2006) CrossRefGoogle Scholar
  55. 55.
    Ravindra, B., Mallik, A.K.: Role of nonlinear dissipation in soft Duffing oscillators. Phys. Rev. E 49(6), 4950–4953 (1994) CrossRefGoogle Scholar
  56. 56.
    Ravindra, B., Mallik, A.K.: Stability analysis of a non-linearly damped Duffing oscillator. J. Sound Vib. 171(5), 708–716 (1994) MathSciNetMATHCrossRefGoogle Scholar
  57. 57.
    Trueba, J.L., Rams, J., Sanjuan, M.A.F.: Analytical estimates of the effect of nonlinear damping in some nonlinear oscillators. Int. J. Bifurc. Chaos 10(9), 2257–2267 (2000) MathSciNetMATHCrossRefGoogle Scholar
  58. 58.
    Baltanas, J.P., Trueba, J.L., Sanjuan, M.A.F.: Energy dissipation in a nonlinearly damped Duffing oscillator. Physica D 159, 22–34 (2001) MATHCrossRefGoogle Scholar
  59. 59.
    Sanjuan, M.A.F.: The effect of nonlinear damping on the universal escape oscillator. Int. J. Bifurc. Chaos 9(4), 735–744 (1999) MATHCrossRefGoogle Scholar
  60. 60.
    Krylov, S., Maimon, R.: Pull-in dynamics of an elastic beam actuated by continuously distributed electrostatic force. J. Vib. Acoust. 126, 332–343 (2004) CrossRefGoogle Scholar
  61. 61.
    Jing, X.J., Lang, Z.Q.: Frequency domain analysis of a dimensionless cubic nonlinear damping system subject to harmonic input. Nonlinear Dyn. 58(3), 469–485 (2009) MathSciNetMATHCrossRefGoogle Scholar
  62. 62.
    Lifshitz, R., Cross, M.: Nonlinear dynamics of nanomechanical and micromechanical resonators. In: Schuster, H.G. (ed.) Reviews of nonlinear dynamics and complexity, vol. 1, pp. 1–48. Wiley-VCH, New York (2008) CrossRefGoogle Scholar
  63. 63.
    Gutschmidt, S., Gottlieb, O.: Internal resonances and bifurcations of a microbeam array below the first pull-in instability. Int. J. Bifurc. Chaos 20(3), 605–618 (2010) MATHCrossRefGoogle Scholar
  64. 64.
    Lifshitz, R., Cross, M.C.: Response of parametrically driven nonlinear coupled oscillators with application to micromechanical and nanomechanical resonator arrays. Phys. Rev. B 67, 134302 (2003) CrossRefGoogle Scholar
  65. 65.
    Bikdash, M., Balachandran, B., Nayfeh, A.: Melnikov analysis for a ship with a general roll-damping model.Nonlinear Dyn. 6, 101–124 (1994) Google Scholar
  66. 66.
    Gottlieb, O., Feldman, M.: Application of a Hilbert transform-based algorithm for parameter estimation of a nonlinear ocean system roll model. J. Offshore Mech. Arct. Eng. 119, 239–243 (1997) CrossRefGoogle Scholar
  67. 67.
    Dick, A.J., Balachandran, B., DeVoe, D.L., Mote, C.D. Jr.: Parametric identification of piezoelectric microscale resonators. J. Micromech. Microeng. 16, 1593–1601 (2006) CrossRefGoogle Scholar
  68. 68.
    Zhu, W.Q., Wu, Y.J.: First-passage time of duffing oscillator under combined harmonic and white-noise excitations. Nonlinear Dyn. 32(3), 291–305 (2003) MATHCrossRefGoogle Scholar
  69. 69.
    Aldridge, J., Cleland, A.: Noise-enabled precision measurements of a Duffing nanomechanical resonator. Phys. Rev. Lett. 94, 156403 (2005) CrossRefGoogle Scholar
  70. 70.
    Younis, M.I., Nayfeh, A.H.: A study of the nonlinear response of a resonant microbeam to an electric actuation. Nonlinear Dyn. 31(1), 91–117 (2003) MathSciNetMATHCrossRefGoogle Scholar
  71. 71.
    Buks, E., Roukes, M.L.: Electrically tunable collective response in a coupled micromechanical array. J. Micromech. Syst. 11(6), 802–807 (2002) CrossRefGoogle Scholar
  72. 72.
    Ullersma, P.: An exactly solvable model for Brownian motion: I. derivation of the Langevin equation. Physica 32, 27–55 (1966) MathSciNetCrossRefGoogle Scholar
  73. 73.
    Ullersma, P.: An exactly solvable model for Brownian motion: II. derivation of the Fokker-Planck equation and the master equation. Physica 32, 56–73 (1966) MathSciNetCrossRefGoogle Scholar
  74. 74.
    Caldeira, A.O., Leggett, A.J.: Path integral approach to quantum Brownian motion. Physica A 121, 587–616 (1983) MathSciNetMATHCrossRefGoogle Scholar
  75. 75.
    Hänggi, P.: Generalized Langevin equations: A useful tool for the perplexed modeller of nonequilibrium fluctuations? In: Stochastic Dynamics. Lecture Notes in Physics, vol. 484, pp. 15–22. Springer, Berlin (1997) CrossRefGoogle Scholar
  76. 76.
    Mohanty, P., Harrington, D.A., Ekinci, K.L., Yang, Y.T., Murphy, M.J., Roukes, M.L.: Intrinsic dissipation in high-frequency micromechanical resonators. Phys. Rev. B 66, 85416 (2002) CrossRefGoogle Scholar
  77. 77.
    Zener, C.: Elasticity and Anelasticity of Metals. The University of Chicago Press, Chicago (1948) Google Scholar
  78. 78.
    Stievater, T.H., Rabinovich, W.S., Papanicolaou, N.A., Bass, R., Boos, J.B.: Measured limits of detection based on thermal-mechanical frequency noise in micromechanical sensors. Appl. Phys. Lett. 90, 051114 (2007) CrossRefGoogle Scholar
  79. 79.
    Ke, T.: Stress relaxation across grain boundaries in metals. Phys. Rev. 72(1), 41–46 (1947) CrossRefGoogle Scholar
  80. 80.
    Ono, T., Esashi, M.: Effect of ion attachment on mechanical dissipation of a resonator. Appl. Phys. Lett. 87(44105) (2005) Google Scholar
  81. 81.
    Zolfagharkhani, G., Gaidarzhy, A., Shim, S., Badzey, R.L., Mohanty, P.: Quantum friction in nanomechanical oscillators at millikelvin temperatures. Phys. Rev. B 72, 224101 (2005) CrossRefGoogle Scholar
  82. 82.
    Geller, M.R., Varley, J.B.: Friction in nanoelectromechanical systems: Clamping loss in the GHz regime. arXiv:cond-mat/0512710 (2005)
  83. 83.
    Cross, M., Lifshitz, R.: Elastic wave transmission at an abrupt junction in a thin plate with application to heat transport and vibrations in mesoscopic systems. Phys. Rev. B 64, 85324 (2001) CrossRefGoogle Scholar
  84. 84.
    Hänggi, P., Ingold, G.L.: Fundamental aspects of quantum Brownian motion. Chaos 15(2), 026105 (2005) MathSciNetCrossRefGoogle Scholar
  85. 85.
    Landau, L.D., Lifshitz, E.M.: Statistical Physics, Part 1, 3rd edn. Pergamon, New York (1980) Google Scholar
  86. 86.
    Kubo, R.: The fluctuation-dissipation theorem. Rep. Prog. Phys. 29, 255–284 (1966) CrossRefGoogle Scholar
  87. 87.
    Chandrasekhar, S.: Stochastic problems in physics and astronomy. Rev. Mod. Phys. 15(1), 1–89 (1943) MathSciNetMATHCrossRefGoogle Scholar
  88. 88.
    Klimontovich, Y.L.: Statistical Theory of Open Systems: Volume 1: A Unified Approach to Kinetic Description of Processes in Active Systems. Kluwer Academic, Norwell (1995) MATHGoogle Scholar
  89. 89.
    Habib, S., Kandrup, H.: Nonlinear noise in cosmology. Phys. Rev. D 46, 5303–5314 (1992) MathSciNetCrossRefGoogle Scholar
  90. 90.
    Yurke, B., Greywall, D.S., Pargellis, A.N., Busch, P.A.: Theory of amplifier-noise evasion in an oscillator employing nonlinear resonator. Phys. Rev. A 51(5), 4211–4229 (1995) CrossRefGoogle Scholar
  91. 91.
    Rugar, D., Grüetter, P.: Mechanical parametric amplification and thermomechanical noise squeezing. Phys. Rev. Lett. 67, 699–702 (1991) CrossRefGoogle Scholar
  92. 92.
    Kramers, H.A.: Brownian motion in a field of force and the diffusion model of chemical reactions. Physica 7, 284–304 (1940) MathSciNetMATHCrossRefGoogle Scholar
  93. 93.
    Hänggi, P., Talkner, P., Borkovec, M.: Reaction-rate theory: fifty years after Kramers. Rev. Mod. Phys. 62, 251–342 (1990) CrossRefGoogle Scholar
  94. 94.
    Kogan, O.: Controlling transitions in a Duffing oscillator by sweeping parameters in time. Phys. Rev. E 76, 037203 (2007) CrossRefGoogle Scholar
  95. 95.
    Leamy, M.J., Gottlieb, O.: Internal resonances in whirling strings involving longitudinal dynamics and material non-linearities. J. Sound Vib. 236, 683–703 (2000) CrossRefGoogle Scholar
  96. 96.
    Leamy, M.J., Gottlieb, O.: Nonlinear dynamics of a taut string with material nonlinearities. J. Vib. Acoust. 123, 53–60 (2001) CrossRefGoogle Scholar
  97. 97.
    Meirovitch, L.: Principles and Techniques of Vibrations. Prentice-Hall, New York (1997) Google Scholar
  98. 98.
    Mintz, T.: Nonlinear dynamics and stability of a microbeam array subject to parametric excitation. Master’s thesis, Technion – Israel Institute of Technology (2009) Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  • Stav Zaitsev
    • 1
  • Oleg Shtempluck
    • 1
  • Eyal Buks
    • 1
  • Oded Gottlieb
    • 2
  1. 1.Department of Electrical EngineeringTechnion–Israel Institute of TechnologyHaifaIsrael
  2. 2.Department of Mechanical EngineeringTechnion–Israel Institute of TechnologyHaifaIsrael

Personalised recommendations