Nonlinear damping in a micromechanical oscillator

Nonlinear Dynamics

Volumn 67, Issue 1, 2012, Pages 859-883

  Zaitsev, Stav ^a   Shtempluck, Oleg ^a   Buks, Eyal ^a   Gottlieb, Oded ^a  

^a TECHNION ISRAEL INSTITUTE OF TECHNOLOGY   (Israel)

Author keywords

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

Indexed keywords

BIFURCATION POINTS; CONSTITUTIVE LAW; CONTINUOUS MODELS; DISSIPATIVE PROCESS; DISSIPATIVE PROPERTIES; DUFFING OSCILLATOR; ELASTIC EFFECTS; FORCED VIBRATION; FREQUENCY AND TIME DOMAINS; GEOMETRICALLY NONLINEAR; MECHANICAL RESONATORS; MICRO-MECHANICAL; MICROMECHANICAL OSCILLATORS; MICROMECHANICAL SYSTEMS; NON-LINEAR ELASTIC BEHAVIOR; NON-LINEARITY; NONLINEAR DAMPING; NONLINEAR DISSIPATION; SADDLE NODE BIFURCATION; SIGNIFICANT IMPACTS; THEORETICAL PREDICTION;

BIFURCATION (MATHEMATICS); DAMPING; IDENTIFICATION (CONTROL SYSTEMS); MEMS; MICROELECTROMECHANICAL DEVICES; VIBRATIONS (MECHANICAL);

OSCILLATORS (MECHANICAL);

EID: 82255186528     PISSN: 0924090X     EISSN: None     Source Type: Journal    
DOI: 10.1007/s11071-011-0031-5     Document Type: Article

References (98)

1. K.L. Turner S.A. Miller P.G. Hartwell N.C. MacDonald S.H. Strogatz S.G. Adams 1998 Five parametric resonances in a microelectromechanical system Nature 396 149 152 10.1038/24122 (Pubitemid 28523612)
2. M. Roukes 2001 Nanoelectromechanical systems face the future Phys. World 14 25 25
3. Roukes, M.: Nanomechanical systems. Technical digest of the 2000 solid state sensor and actuator workshop (2000)
4. A. Husain J. Hone H.W.C. Postma X.M.H. Huang T. Drake M. Barbic A. Scherer M.L. Roukes 2003 Nanowire-based very-high-frequency electromechanical resonator Appl. Phys. Lett. 83 1240 1242 10.1063/1.1601311
5. J.A. Sidles J.L. Garbini K.J. Bruland D. Rugar O. Zuger S. Hoen C.S. Yannoni 1995 Magnetic resonance force microscopy Rev. Mod. Phys. 67 1 249 265 10.1103/RevModPhys.67.249
6. D. Rugar R. Budakian H.J. Mamin B.W. Chui 2004 Single spin detection by magnetic resonance force microscopy Nature 430 329 332 10.1038/nature02658 (Pubitemid 38937736)
7. K.L. Ekinci Y.T. Yang M.L. Roukes 2004 Ultimate limits to inertial mass sensing based upon nanoelectromechanical systems J. Appl. Phys. 95 5 2682 2689 10.1063/1.1642738
8. K.L. Ekinci X.M.H. Huang M.L. Roukes 2004 Ultrasensitive nanoelectromechanical mass detection Appl. Phys. Lett. 84 22 4469 4471 10.1063/1.1755417
9. B. Ilic H.G. Craighead S. Krylov W. Senaratne C. Ober 2004 Attogram detection using nanoelectromechanical oscillators J. Appl. Phys. 95 3694 10.1063/1.1650542
10. A.H. Nayfeh H.M. Ouakad F. Najar S. Choura E.M. Abdel-Rahman 2010 Nonlinear dynamics of a resonant gas sensor Nonlinear Dyn. 59 4 607 618 1189.78046 10.1007/s11071-009-9567-z
11. M. Blencowe 2004 Quantum electromechanical systems Phys. Rep. 395 159 222 10.1016/j.physrep.2003.12.005 (Pubitemid 38491780)
12. R.G. Knobel A.N. Cleland 2003 Nanometre-scale displacement sensing using a single electron transistor Nature 424 291 293 10.1038/nature01773 (Pubitemid 36899366)
13. M.D. LaHaye O. Buu B. Camarota K.C. Schwab 2004 Approaching the quantum limit of a nanomechanical resonator Science 304 74 77 10.1126/science.1094419 (Pubitemid 38451458)
14. K. Schwab E.A. Henriksen J.M. Worlock M.L. Roukes 2000 Measurement of the quantum of thermal conductance Nature 404 974 977 10.1038/35010065 (Pubitemid 30243586)
15. E. Buks M.L. Roukes 2001 Stiction, adhesion energy, and the Casimir effect in micromechanical systems Phys. Rev. B 63 33402 10.1103/PhysRevB.63. 033402
16. E. Buks M.L. Roukes 2001 Metastability and the Casimir effect in micromechanical systems Europhys. Lett. 54 2 220 226 10.1209/epl/i2001-00298-x
17. K.C. Schwab M.L. Roukes 2005 Putting mechanics into quantum mechanics Phys. Today 58 36 42 10.1063/1.2012461 (Pubitemid 41148701)
18. M. Aspelmeyer K. Schwab 2008 Focus on mechanical systems at the quantum limit New J. Phys. 10 9 095001 10.1088/1367-2630/10/9/095001
19. I. Kozinsky H.W.C. Postma O. Kogan A. Husain M.L. Roukes 2007 Basins of attraction of a nonlinear nanomechanical resonator Phys. Rev. Lett. 99 207201 10.1103/PhysRevLett.99.207201 (Pubitemid 350113505)
20. M.C. Cross A. Zumdieck R. Lifshitz J.L. Rogers 2004 Synchronization by nonlinear frequency pulling Phys. Rev. Lett. 93 224101 10.1103/PhysRevLett.93. 224101 (Pubitemid 40072792)
21. A. Erbe H. Krömmer A. Kraus R.H. Blick G. Corso K. Richter 2000 Mechanical mixing in nonlinear nanomechanical resonators Appl. Phys. Lett. 77 3102 3104 10.1063/1.1324721
22. J.F. Rhoads S.W. Shaw K.L. Turner R. Baskaran 2005 Tunable microelectromechanical filters that exploit parametric resonance J. Vib. Acoust. 127 423 431 10.1115/1.2013301 (Pubitemid 41581525)
23. R.B. Reichenbach M. Zalalutdinov K.L. Aubin R. Rand B.H. Houston J.M. Parpia H.G. Craighead 2005 Third-order intermodulation in a micromechanical thermal mixer J. Micro/Nanolithogr. MEMS MOEMS 14 1244 1252 (Pubitemid 41830199)
24. R. Almog S. Zaitsev O. Shtempluck E. Buks 2006 High intermodulation gain in a micromechanical Duffing resonator Appl. Phys. Lett. 88 213509 10.1063/1.2207490 (Pubitemid 43814872)
25. R. Almog S. Zaitsev O. Shtempluck E. Buks 2007 Noise squeezing in a nanomechanical Duffing resonator Phys. Rev. Lett. 98 78103 10.1103/PhysRevLett. 98.078103
26. R. Almog S. Zaitsev O. Shtempluck E. Buks 2007 Signal amplification in a nanomechanical duffing resonator via stochastic resonance Appl. Phys. Lett. 90 13508 10.1063/1.2430689 (Pubitemid 46068379)
27. Zhang, W., Baskaran, R., Turner, K.L.: Nonlinear behavior of a parametric resonance-based mass sensor. In: Proc. IMECE2002, p. 33261 Nov (2002)
28. E. Buks B. Yurke 2006 Mass detection with nonlinear nanomechanical resonator Phys. Rev. E 74 46619 10.1103/PhysRevE.74.046619 (Pubitemid 44656329)
29. A.N. Cleland M.L. Roukes 2002 Noise processes in nanomechanical resonators J. Appl. Phys. 92 5 2758 2769 10.1063/1.1499745 (Pubitemid 35037891)
30. K.Y. Yasumura T.D. Stowe E.M. Chow T. Pfafman T.W. Kenny B.C. Stipe D. Rugar 2000 Quality factors in micron- and submicron-thick cantilevers J. Micromech. Syst. 9 1 117 125 10.1109/84.825786 (Pubitemid 30581703)
31. T. Ono D.F. Wang M. Esashi 2003 Time dependence of energy dissipation in resonating silicon cantilevers in ultrahigh vacuum Appl. Phys. Lett. 83 10 1950 1952 10.1063/1.1608485
32. X. Liu E. Thompson B. White Jr R. Pohl 1999 Low-temperature internal friction in metal films and in plastically deformed bulk aluminum Phys. Rev. B 59 18 11767 11776 10.1103/PhysRevB.59.11767
33. D.A. Harrington P. Mohanty M.L. Roukes 2000 Energy dissipation in suspended micromechanical resonators at low temperatures Physica B 284-288 2145 2146 10.1016/S0921-4526(99)02998-1
34. R. Lifshitz M.L. Roukes 2000 Thermoelastic damping in micro- and nanomechanical systems Phys. Rev. B 61 8 5600 5609 10.1103/PhysRevB.61.5600
35. B.H. Houston D.M. Photiadis M.H. Marcus J.A. Bucaro X. Liu J.F. Vignola 2002 Thermoelastic loss in microscale oscillators Appl. Phys. Lett. 80 7 1300 1302 10.1063/1.1449534 (Pubitemid 34180558)
36. R. Lifshitz 2002 Phonon-mediated dissipation in micro- and nano-mechanical systems Physica B 316-317 397 399 10.1016/S0921-4526(02)00524-0 (Pubitemid 34544576)
37. I. Wilson-Rae 2008 Intrinsic dissipation in nanomechanical resonators due to phonon tunneling Phys. Rev. B 77 245418 10.1103/PhysRevB.77.245418
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. P. Popovic A.H. Nayfeh K. Oh S.A. Nayfeh 1995 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 10.1177/ 107754639500100108
40. M.R. Hajj J. Fung A.H. Nayfeh S.O. Fahey 2000 Damping identification using perturbation techniques and higher-order spectra Nonlinear Dyn. 23 2 189 203 0981.70018 10.1023/A:1008335522973
41. N. Jaksic M. Boltezar 2002 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 10.1006/jsvi.2001.3953 (Pubitemid 35262828)
42. W. Zhang R. Baskaran K.L. Turner 2002 Effect of cubic nonlinearity on auto-parametrically amplified resonant MEMS mass sensor Sens. Actuators A, Phys. 102 139 150 10.1016/S0924-4247(02)00299-6 (Pubitemid 35422086)
43. W. Zhang R. Baskaran K. Turner 2003 Tuning the dynamic behavior of parametric resonance in a micromechanical oscillator Appl. Phys. Lett. 82 130 132 10.1063/1.1534615
44. S. Krylov B.R. Ilic D. Schreiber S. Seretensky H. Craighead 2008 The pull-in behavior of electrostatically actuated bistable microstructures J. Micromech. Microeng. 18 5 055026 10.1088/0960-1317/18/5/055026 (Pubitemid 351606163)
45. Nayfeh, A.H., Mook, D.T.: Nonlinear Oscillations. Wiley, New York (1995). Wiley Classics Library
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)
47. Landau, L.D., Lifshitz, E.M.: Mechanics, 3rd edn. Pergamon, New York (1976)
48. Nayfeh, A.H.: Introduction to Perturbation Techniques. Wiley, New York (1981)
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)
50. Strogatz, S.H.: Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering. Perseus Books, Readings (1994)
51. H.B. Chan M. Dykman C. Stambaugh 2008 Paths of fluctuation induced switching Phys. Rev. Lett. 100 130602 10.1103/PhysRevLett.100.130602 (Pubitemid 351490851)
52. M.I. Dykman B. Golding D. Ryvkine 2004 Critical exponent crossovers in escape near a bifurcation point Phys. Rev. Lett. 92 8 080602 10.1103/PhysRevLett.92.080602
53. B. Yurke E. Buks 2006 Performance of cavity-parametric amplifiers, employing Kerr nonlinearites, in the presence of two-photon loss J. Lightwave Technol. 24 12 5054 5066 10.1109/JLT.2006.884490 (Pubitemid 46350966)
54. E. Buks B. Yurke 2006 Dephasing due to intermode coupling in superconducting stripline resonators Phys. Rev. A 73 23815 10.1103/PhysRevA.73. 023815 (Pubitemid 43270216)
55. B. Ravindra A.K. Mallik 1994 Role of nonlinear dissipation in soft Duffing oscillators Phys. Rev. E 49 6 4950 4953 10.1103/PhysRevE.49.4950
56. B. Ravindra A.K. Mallik 1994 Stability analysis of a non-linearly damped Duffing oscillator J. Sound Vib. 171 5 708 716 1273313 1064.70512 10.1006/jsvi.1994.1153
57. J.L. Trueba J. Rams M.A.F. Sanjuan 2000 Analytical estimates of the effect of nonlinear damping in some nonlinear oscillators Int. J. Bifurc. Chaos 10 9 2257 2267 1793826 0999.70021 10.1142/S0218127400001419
58. J.P. Baltanas J.L. Trueba M.A.F. Sanjuan 2001 Energy dissipation in a nonlinearly damped Duffing oscillator Physica D 159 22 34 0994.37009 10.1016/S0167-2789(01)00329-3 (Pubitemid 32975773)
59. M.A.F. Sanjuan 1999 The effect of nonlinear damping on the universal escape oscillator Int. J. Bifurc. Chaos 9 4 735 744 0977.34041 10.1142/S0218127499000523 (Pubitemid 129487674)
60. S. Krylov R. Maimon 2004 Pull-in dynamics of an elastic beam actuated by continuously distributed electrostatic force J. Vib. Acoust. 126 332 343 10.1115/1.1760559
61. X.J. Jing Z.Q. Lang 2009 Frequency domain analysis of a dimensionless cubic nonlinear damping system subject to harmonic input Nonlinear Dyn. 58 3 469 485 2562943 1183.70042 10.1007/s11071-009-9493-0
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)
63. S. Gutschmidt O. Gottlieb 2010 Internal resonances and bifurcations of a microbeam array below the first pull-in instability Int. J. Bifurc. Chaos 20 3 605 618 1193.74076 10.1142/S0218127410025910
64. R. Lifshitz M.C. Cross 2003 Response of parametrically driven nonlinear coupled oscillators with application to micromechanical and nanomechanical resonator arrays Phys. Rev. B 67 134302 10.1103/PhysRevB.67.134302
65. M. Bikdash B. Balachandran A. Nayfeh 1994 Melnikov analysis for a ship with a general roll-damping model Nonlinear Dyn. 6 101 124
66. O. Gottlieb M. Feldman 1997 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 10.1115/1.2829102
67. A.J. Dick B. Balachandran D.L. DeVoe C.D. Mote Jr. 2006 Parametric identification of piezoelectric microscale resonators J. Micromech. Microeng. 16 1593 1601 10.1088/0960-1317/16/8/021 (Pubitemid 44102308)
68. W.Q. Zhu Y.J. Wu 2003 First-passage time of duffing oscillator under combined harmonic and white-noise excitations Nonlinear Dyn. 32 3 291 305 1062.70605 10.1023/A:1024414020813
69. J. Aldridge A. Cleland 2005 Noise-enabled precision measurements of a Duffing nanomechanical resonator Phys. Rev. Lett. 94 156403 10.1103/PhysRevLett. 94.156403 (Pubitemid 40620382)
70. M.I. Younis A.H. Nayfeh 2003 A study of the nonlinear response of a resonant microbeam to an electric actuation Nonlinear Dyn. 31 1 91 117 2157181 1047.74027 10.1023/A:1022103118330
71. E. Buks M.L. Roukes 2002 Electrically tunable collective response in a coupled micromechanical array J. Micromech. Syst. 11 6 802 807 10.1109/JMEMS.2002.805056
72. P. Ullersma 1966 An exactly solvable model for Brownian motion: I. derivation of the Langevin equation Physica 32 27 55 198905 10.1016/0031- 8914(66)90102-9
73. P. Ullersma 1966 An exactly solvable model for Brownian motion: II. derivation of the Fokker-Planck equation and the master equation Physica 32 56 73 198906 10.1016/0031-8914(66)90103-0
74. A.O. Caldeira A.J. Leggett 1983 Path integral approach to quantum Brownian motion Physica A 121 587 616 726154 0585.60082 10.1016/0378-4371(83) 90013-4
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)
76. P. Mohanty D.A. Harrington K.L. Ekinci Y.T. Yang M.J. Murphy M.L. Roukes 2002 Intrinsic dissipation in high-frequency micromechanical resonators Phys. Rev. B 66 85416 10.1103/PhysRevB.66.085416
77. Zener, C.: Elasticity and Anelasticity of Metals. The University of Chicago Press, Chicago (1948)
78. T.H. Stievater W.S. Rabinovich N.A. Papanicolaou R. Bass J.B. Boos 2007 Measured limits of detection based on thermal-mechanical frequency noise in micromechanical sensors Appl. Phys. Lett. 90 051114 10.1063/1.2450643
79. T. Ke 1947 Stress relaxation across grain boundaries in metals Phys. Rev. 72 1 41 46 10.1103/PhysRev.72.41
80. Ono, T., Esashi, M.: Effect of ion attachment on mechanical dissipation of a resonator. Appl. Phys. Lett. 87(44105) (2005)
81. G. Zolfagharkhani A. Gaidarzhy S. Shim R.L. Badzey P. Mohanty 2005 Quantum friction in nanomechanical oscillators at millikelvin temperatures Phys. Rev. B 72 224101 10.1103/PhysRevB.72.224101 (Pubitemid 43022638)
82. Geller, M.R., Varley, J.B.: Friction in nanoelectromechanical systems: Clamping loss in the GHz regime. arXiv:cond-mat/0512710 (2005)
83. M. Cross R. Lifshitz 2001 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 10.1103/PhysRevB.64.085324
84. P. Hänggi G.L. Ingold 2005 Fundamental aspects of quantum Brownian motion Chaos 15 2 026105 2150233 10.1063/1.1853631 (Pubitemid 43127629)
85. Landau, L.D., Lifshitz, E.M.: Statistical Physics, Part 1, 3rd edn. Pergamon, New York (1980)
86. R. Kubo 1966 The fluctuation-dissipation theorem Rep. Prog. Phys. 29 255 284 10.1088/0034-4885/29/1/306
87. S. Chandrasekhar 1943 Stochastic problems in physics and astronomy Rev. Mod. Phys. 15 1 1 89 8000 0061.46403 10.1103/RevModPhys.15.1
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)
89. S. Habib H. Kandrup 1992 Nonlinear noise in cosmology Phys. Rev. D 46 5303 5314 1196884 10.1103/PhysRevD.46.5303
90. B. Yurke D.S. Greywall A.N. Pargellis P.A. Busch 1995 Theory of amplifier-noise evasion in an oscillator employing nonlinear resonator Phys. Rev. A 51 5 4211 4229 10.1103/PhysRevA.51.4211
91. D. Rugar P. Grüetter 1991 Mechanical parametric amplification and thermomechanical noise squeezing Phys. Rev. Lett. 67 699 702 10.1103/PhysRevLett.67.699
92. H.A. Kramers 1940 Brownian motion in a field of force and the diffusion model of chemical reactions Physica 7 284 304 2962 0061.46405 10.1016/S0031-8914(40)90098-2
93. P. Hänggi P. Talkner M. Borkovec 1990 Reaction-rate theory: fifty years after Kramers Rev. Mod. Phys. 62 251 342 10.1103/RevModPhys.62.251
94. O. Kogan 2007 Controlling transitions in a Duffing oscillator by sweeping parameters in time Phys. Rev. E 76 037203 10.1103/PhysRevE.76.037203 (Pubitemid 47507683)
95. M.J. Leamy O. Gottlieb 2000 Internal resonances in whirling strings involving longitudinal dynamics and material non-linearities J. Sound Vib. 236 683 703 10.1006/jsvi.2000.3039
96. M.J. Leamy O. Gottlieb 2001 Nonlinear dynamics of a taut string with material nonlinearities J. Vib. Acoust. 123 53 60 10.1115/1.1325411 (Pubitemid 33716015)
97. Meirovitch, L.: Principles and Techniques of Vibrations. Prentice-Hall, New York (1997)
98. Mintz, T.: Nonlinear dynamics and stability of a microbeam array subject to parametric excitation. Master's thesis, Technion - Israel Institute of Technology (2009)