Skip to main content

Introduction to Nonlinear Dynamics

  • Chapter
  • First Online:

Part of the book series: Microsystems ((MICT,volume 20))

Abstract

This chapter presents an introduction to fundamental concepts in nonlinear dynamics, which are essential for modeling and testing the mechanical behavior of MEMS devices. Because nonlinear dynamics is a wide topic, the focus here is on subjects that are most common and relevant to MEMS. Basic analytical techniques to analyze and tackle nonlinear systems are presented. Several applications on MEMS devices demonstrating phenomena at the microscale level are discussed. The aim of this chapter is to give MEMS researchers fundamental background on the subject. This should lead to proper understanding of the behavior of microsystems and enable the development of modeling methodologies that can capture the essence of the static and dynamic aspects of MEMS devices behaving nonlinearly.

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

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  1. Thompson J, Stewart H (2001) Nonlinear dynamics and chaos, Wiley, New York

    Google Scholar 

  2. Nayfeh A, Balachandran B (1995) Applied nonlinear dynamics, New York, Wiley

    Book  MATH  Google Scholar 

  3. Gilbert J R, Ananthasuresh G K, Senturia S D (1996) 3D Modeling of contact problems and hysteresis in coupled electro-mechanics, In Proceeding of the IEEE Annual International Conference on Microelectromechanical Systems (MEMS), 127–132

    Google Scholar 

  4. Mastrangelo C H, Hsu C H (1993) Mechanical stability and adhesion of microstructures undercapillary forces. I. Basic theory. Journal of Microelectromechanical Systems, 2:33–43

    Article  Google Scholar 

  5. Abdel-Rahman E M, Younis M I, Nayfeh A H (2004) Finite-amplitude motions of resonators and their stability. Journal of Computational and Theoretical Nanoscience, 4:385–391

    Article  Google Scholar 

  6. Nayfeh A H, Younis M I (2005) Dynamics of MEMS resonators under superharmonic and subharmonic excitations. Journal of Micromechanics and Microengineering, 15:1840–1847

    Article  Google Scholar 

  7. Krylov S, Harari I, Yaron C (2005) Stabilization of electrostatically actuated microstructures using parametric excitation. Journal of Micromechanics and Microengineering, 15:1188–1204.

    Article  Google Scholar 

  8. Mann B, Liu J, Hazra S (2006) Correcting measurement nonlinearity in dynamic nanoindentation, In Proceedings of 2006 ASME International Mechanical Engineering Congress and Exposition, IMECE2006, Chicago, Illinois, IMECE2006–15070.

    Google Scholar 

  9. Hirano T, Furuhata T, Gabriel K J, Fujita H (1992) Design, fabrica-tion, and operation of submicron gap comb-drive microactuators. Journal of Microelectromechanical Systems, 1(1):52–59.

    Article  Google Scholar 

  10. Pandey M, Aubin M, Zalalutdinov R B, Reichenbach A T, Zehnder, R H, Rand K, Craighead H G, Analysis of Frequency Locking in Optically Driven MEMS Resonators. Journal of Microelectromechanical systems, 15(6):1546–1554

    Google Scholar 

  11. Ananthasuresh G K, Gupta R K, Senturia S D (1996) An approach to macromodeling of MEMS for nonlinear dynamic simulation, In Proceeding of ASME International Conference of Mechanical Engineering Congress and Exposition (MEMS), Atlanta, GA, pp. 401–407

    Google Scholar 

  12. Krylov S, Maimon R (2004) Pull-in dynamics of an elastic beam actuated by continuously distributed electrostatic force,” ASME Journal of Vibrations and Acoustics, 126(3):332–342

    Article  Google Scholar 

  13. Krylov S (2008) Parametric excitation and stabilization of electrostatically actuated microstructures. International Journal of Multiscale Computational Engineering, 6(6):563–584

    Article  Google Scholar 

  14. Rebeiz G M (2003) RF MEMS: theory, design, and technology, Wiley, New York

    Book  Google Scholar 

  15. Varadan V M, Vinoy K J, Jose K A (2003) RF MEMS and their applications, Wiley New York

    Google Scholar 

  16. Krylov S, Maimon R (2004) Pull-in dynamics of an elastic beam actuated by continuously distributed electrostatic force. ASME Journal of Vibrations and Acoustics, 126(3):332–342

    Article  Google Scholar 

  17. Elata D, Bamberger H (2006) On the dynamic pull-in of electrostatic actuators with multiple degrees of freedom and multiple voltage sources, Journal of Microelectromechanical Systems, 15:131–140

    Article  Google Scholar 

  18. Nielson G N, Barbastathis G (2006) Dynamic Pull-in of Parallel-Plate and Torsional MEMS Actuators, Journal of Microelectromechanical Systems, 15:811–821

    Article  Google Scholar 

  19. Luo A C, Wang F Y (2004) Nonlinear dynamics of a Micro-electro-mechanical system with time-varying capacitors. Journal of Vibration and Acoustics, 126:77–83

    Article  Google Scholar 

  20. Sattler R, Plotz F, Fattinger G, Wachutka G (2002) Modeling of an electrostatic torsional actuator demonstrated with an RF MEMS switch. Sensors and Actuators A, 97–98:337–346

    Article  Google Scholar 

  21. Xiao Z, Sun Y, Li B, Lee S, Sidhu K S, Chin K K, Farmer K R (2004) Natural Frequency Shift of Rectangular Torsion Actuators due to Electrostatic Force, In Proceeding of NSTI-Nanotech 2004, pp. 1:9728422–7-6

    Google Scholar 

  22. Krylov S, Barnea D I (2005) Bouncing mode electrostatically actuated scanning micromirror for video applications,” Smart Material Structures, 14:1281–1296

    Article  Google Scholar 

  23. Guo J G, Zhao Y P (2004) Influence of van der Waals and Casimir Forces on Electrostatic Torsional Actuators, Journal of Microelectromechanical Systems, 13(6):1027–1035

    Article  MathSciNet  Google Scholar 

  24. Guo J G, Zhao Y P (2006) Dynamic stability of electrostatic torsional actuators with van der Waals effect. International Journal of Solids and Structures, 43:675–685

    Article  MATH  Google Scholar 

  25. Gusso A, Delben G J (2007) Influence of the Casimir force on the pull-in parameters of silicon based electrostatic torsional actuators. Sensors and Actuators A, 135:792–800

    Article  Google Scholar 

  26. Guo J G, Zhao Y P, Zhao Y P (2009) Instability analysis of torsional MEMS/NEMS actuators under capillary force. Journal of Colloid and Interface Science, 331:458–462

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  29. Zhao J P, Chen H L, Huang J M, Liu A Q (2005) A study of dynamic characteristics and simulation of MEMS torsional micromirrors. Sensors and Actuators A, 120:199–210

    Article  Google Scholar 

  30. Lee K B (2007) Closed-form expressions for pull-in parameters of two-degree-of-freedom torsional microactuators. Journal of Micromechanics and Microenginnering, 17:1853–1862.

    Article  Google Scholar 

  31. Nayfeh A, Mook D (1979) Nonlinear oscillations, Wiley, New York

    MATH  Google Scholar 

  32. Ayela F, Fournier T (1998) An experimental study of anharmonic micromachined silicon resonators. Measurement, Science and Technology, 9:1821–1830

    Article  Google Scholar 

  33. Gui C, Legtenberg R, Tilmans H A C, Fluitman J H J, Elwenspoek M (1998) Nonlinearity and hysteresis of resonant strain gauges. Journal of Microelectromechanical Systems, 7:122–127

    Article  Google Scholar 

  34. Wang Y C, Adams S G, Thorp J S, MacDonald N C, Hartwell P, Bertsch F (1998) Chaos in MEMS, parameter estimation and its potential application. IEEE Transactions on Circuits and Systems-I: Fundamental Theory and Applications, 45:1013–1020

    Article  Google Scholar 

  35. Veijola T, Mattila T, Jaakkola O, KihamÄaki J, Lamminmäaki T, Oja A, Ruokonen K, SeppÄa H, SeppÄalÄa P, Tittonen I (2000) Large-displacement modeling and simulation of micromechanical electrostatically driven resonators using the harmonic balance method, In Proceedings of the International Microwave Symposium, 99–102

    Google Scholar 

  36. Younis M I, Nayfeh A H (2003) A study of the nonlinear response of a resonant microbeam to an electric actuation. Nonlinear Dynamics, 31:91–117

    Article  MATH  Google Scholar 

  37. Abdel-Rahman E M, Nayfeh A H (2003) Secondary resonances of electrically actuated resonant microsensors. Journal of Micromechanics and Microengineering, 13:1–11

    Article  Google Scholar 

  38. Najar F, Choura S, Abdel-Rahman E M, El-Borgi S, Nayfeh A (2006) Dynamic analysis of variable-geometry electrostatic microactuators. Journal of Micromechanics and Microengineering, 16:2449–2457

    Article  Google Scholar 

  39. Liu S, Davidson A, Lin Q (2004) Simulation studies on nonlinear dynamics and chaos in a MEMS cantilever control system. Journal of Micromechanics and Microengineering, 14:1064–1073

    Article  Google Scholar 

  40. De S K, Aluru N R (2004) Full-lagrangian schemes for dynamic analysis of electrostatic MEMS. Journal of Microelectromechanical Systems, 13:737–758

    Article  Google Scholar 

  41. Lenci S, Rega G (2006) Control of pull-In dynamics in a nonlinear thermoelasticelectrically actuated microbeam. Journal of Micromechanics and Microengineering, 16:390–401

    Article  Google Scholar 

  42. Liqin L, Gang T Y, Zhiqiang W (2006) Nonlinear dynamics of microelectromechanical systems. Journal of Vibration and Control, 12:57–65

    Article  MATH  Google Scholar 

  43. Elata D, Bamberger H (2006) On the dynamic pull-in of electrostatic actuators with multiple degrees of freedom and multiple voltage sources. Journal of Microelectromechanical Systems, 15:131–140

    Article  Google Scholar 

  44. Li H, Preidikman S, Balachandran B, Mote Jr C D (2006) Nonlinear free and forced oscillations of piezoelectric microresonators. Journal of Micromechanics and Microengineering, 16:356–367

    Article  Google Scholar 

  45. De S K, Aluru N R (2006) Complex Nonlinear oscillations in electrostatically actuated microstructures. Journal of Microelectromechanical Systems, 15:355–369

    Article  Google Scholar 

  46. Tadayon M A, Rajaei M, Sayyaadi H, Nakhaie, Jazar G, Alasty A (2006) Nonlinear dynamics of microresonators. International MEMS Conference, 34:961–966

    Google Scholar 

  47. Jazar G N (2006) Mathematical modeling and simulation of thermal effects in flexural microcantilever. Journal of Vibration and Control, 12(2):139–163

    Article  MATH  Google Scholar 

  48. Zhao J H, Bridges G E, Thomson D J (2006) Direct evidence of spring softening nonlinearity in micromachined mechanical resonator using optical beam deflection technique. Journal of Vacuum Science & Technology A, 24(3):732–736

    Article  Google Scholar 

  49. Zhang W-M, Meng G (2007) Nonlinear dynamic analysis of electrostatically actuated resonant MEMS sensors under parametric excitation. 7(3):370–380

    Google Scholar 

  50. Zhou J, Wang Z, Grots A, He X (2007) Electric field drives the nonlinear resonance of a piezoelectric nanowire. Solid State Communications, Elsevier, Solid State Communications. 144(3–4):118–123

    Google Scholar 

  51. Sahaia T, Rustom B, Bhiladvalab C, Zehnder A T (2007) Thermomechanical transitions in doubly-clamped micro-oscillators. International Journal of Non-Linear Mechanics, 42(4):596–607

    Article  Google Scholar 

  52. Ai S, Pelesko J A (2007) Dynamics of a canonical electrostatic MEMS/NEMS system. Journal of Dynamics and Differential Equation, 20:609–641

    Article  MathSciNet  Google Scholar 

  53. Trusov A A, Shkel A M (2007) The effect of high order non-linearities on sub-harmonics excitation with parallel plate capacitive actuators. In Proceedings of the ASME 2007 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference IDETC/CIE 2007, Las Vegas

    Google Scholar 

  54. Mestrom R M, Fey R H, Phan K L, van Beek J T, Nijmeijer H (2008) Modeling the dynamics of a MEMS resonator: simulations and experiments. Sensors & Actuators A, 142:306–315

    Article  Google Scholar 

  55. Batra R C, Porfiri M, Spinello D (2008) Vibrations of narrow microbeams predeformed by an electric field. Journal of Sound and Vibration, 309(3–5): 600–612

    Article  Google Scholar 

  56. Zamanian M, Khadem S E, Mahmoodi S N (2008) The effect of a piezoelectric layer on the mechanical behavior of an electrostatic actuated microbeam. Smart Materials and Structures, 17:065024(15pp)

    Article  Google Scholar 

  57. Alsaleem F, Younis M I, Ouakad H (2009) On the nonlinear resonances and dynamic pull-in of electrostatically actuated resonators. Journal of Micromechanics and Microengineering, 19:045013(1–14)

    Google Scholar 

  58. Nayfeh A (1981) Introduction to perturbation techniques, Wiley, New York

    MATH  Google Scholar 

  59. Younis M I, Alsaleem F M (2009) Exploration of new concepts for mass detection in electrostatically-actuated structures based on nonlinear phenomena. Journal of Computational and Nonlinear Dynamics, 4(2):1–15

    Article  Google Scholar 

  60. Alsaleem F M, Younis M I, Ruzziconi L (2010) An experimental and theoretical investigation of dynamic pull-in in MEMS resonators actuated electrostatically. Journal of Microelectromechanical Systems, 19(4):794–806

    Article  Google Scholar 

  61. Mahmoodi S N, Jalili N, Daqaq M F (2008) Modeling, nonlinear dynamics, and identification of a piezoelectrically actuated micro-cantilever sensor. IEEE/ASME Transactions on Mechatronics, 13(1):58–65

    Article  Google Scholar 

  62. Veijola T, Kuisma H, Lahdenpera J, Ryhanen T (1995) Equivalent-circuit model of the squeezed gas film in a silicon accelerometer. Sensors and Actuators A, 48:239–248

    Article  Google Scholar 

  63. Rugar D, Grutter P (1991) Mechanical parametric amplification and thermomechanical noise squeezing. Physical Review Letters, 67(6):699–702.

    Article  Google Scholar 

  64. Turner K L, Miller S A, Hartwell P G, MacDonald N C, Strogatz S H and Adams S G (1998) Five parametric resonances in a micromechanical system Nature, 36:149–152.

    Article  Google Scholar 

  65. Zhang W, Baskaran R and Turner K L (2002) Effect of cubic nonlinearity on auto-parametrically amplified resonant MEMS mass sensor. Sensors Actuators A, 102:139–150

    Article  Google Scholar 

  66. Zhang W, Turner K L (2005) Application of parametric resonance amplification in a single-crystal silicon micro-oscillator based mass sensor. Sensors and Actuators A, 122:23–30

    Article  Google Scholar 

  67. Rhoads J, Shaw S, Turner K L, Baskaran R (2005) Tunable microelectromechanical filters that exploit parametric resonance. Journal of Sound and Acoustics, 127:423–430

    Google Scholar 

  68. DeMartini B E, Rhoads J F, Turner K L, Shaw S W, Moehlis J (2007) Linear and nonlinear tuning of parametrically excited MEMS oscillators. Journal of Microelectromechanical Systems, 16(2): 310–318

    Article  Google Scholar 

  69. Requa M, Turner K (2006) Electromechanically driven and sensed parametric resonance in silicon microcantilevers. Applied Physics Letters, 88:263508(1–3)

    Article  Google Scholar 

  70. Rhoads J F, Shaw S W, Turner K L (2006) The nonlinear response of resonant microbeam systems with purely-parametric electrostatic actuation. Journal of Micromechanics and Microengineering, 16:890–899.

    Article  Google Scholar 

  71. Zalalutdinov M, Olkhovets A, Zehnder A T, Ilic B, Czaplewski D, Craighead H G, Parpia J M (2001) Optically pumped parametric amplification for micromechanical oscillators. Applied Physical Letters, 78:3142–3144

    Article  Google Scholar 

  72. Carr D W, Evoy S, Sekaric L, Craighead H G, Parpia J M (2000) Parametric amplification in a torsional microresonator. Applied Physics Letters, 77(10)

    Google Scholar 

  73. Zook D J, Burns D W, Herb W R, Guckel H, Kang J W, and Ahn Y (1996) Optically excited self-resonant microbeams. Sensors and Actuators A, 52:92–98

    Article  Google Scholar 

  74. Pandey M, Rand R H, Zehnder A T (2008) Frequency locking in a forced Mathieu–van der Pol–Duffing system. Nonlinear Dynamics, 54:3–12

    Article  MathSciNet  MATH  Google Scholar 

  75. Aubin K, Zalalutdinov M, Alan T, Reichenbach R B, Rand R, Zehnder A, Parpia J, Craighead H (2004) Limit cycle oscillations in CW laser-driven NEMS. Journal of Microelectromechanical Systems, 13(6):1018–1026

    Article  Google Scholar 

  76. Albrecht T R, Grtitter P, Horne D, Rugar D (1991) Frequency modulation detection using high Q cantilevers for enhanced force microscope sensitivity. Applied Physical Letters, 69(2):668–673

    Google Scholar 

  77. Yabuno H, Kaneko H, Kuroda M, Kobayashi T (2008) Van der Pol type self-excited micro-cantilever probe of atomic force microscopy. Nonlinear Dynamics. 54:137–149

    Article  MATH  Google Scholar 

  78. Ouakad H, Nayfeh A H, Choura S, Abdel-Rahman E M, Najar F, Hamad B (2008) Nonlinear feedback control and dynamics of an electrostatically actuated microbeam filter. In Proceeding of the 2008 ASME International Mechanical Engineering Congress and Exposition, IMECE2008, Boston, MA

    Google Scholar 

  79. Rand R H (2005) Lecture Notes on Nonlinear Vibrations. version 52, 2005, http://audiophile.tam.cornell.edu/randdocs/nlvibe52.pd

    Google Scholar 

  80. Hayashi C (1964) Nonlinear Oscillations in Physical Systems, McGraw Hill, New York

    MATH  Google Scholar 

  81. Nayfeh A (2000) Nonlinear Interactions: Analytical, Computational, and Experimental Methods, Wiley Interscience, New York

    MATH  Google Scholar 

  82. Arafat H N, Nayfeh A H, Abdel-Rahman E M (2008) Modal interactions in contact-mode atomic force microscopes. 54(1–2):151–166

    Google Scholar 

  83. Daqaq M F, Abdel-Rahman E M, Nayfeh A H (2009) Two-to-one internal resonance in microscanners. 57(1–2):231–251

    Google Scholar 

  84. Vyas A, Peroulis D, Bajaj A K (2008) Dynamics of a nonlinear microresonator based on resonantly interacting flexural-torsional modes. Nonlinear Dynamics 54(1–2):31–52

    Article  MATH  Google Scholar 

  85. Strogatz S H (1994) Nonlinear dynamics and chaos, Westview Press, Cambridge, MA

    Google Scholar 

  86. Moon F C (2004) Chaotic Vibrations: An Introduction for Applied Scientists and Engineers, Wiley Interscience, New Jersey

    Book  Google Scholar 

  87. Wang Y, Adams S, Thorp J, MacDonald N, Hartwell P, Bertsch F (1998) Chaos in MEMS, parameter estimation and its potential application. IEEE Trans. Circuits Syst. I, Fundamental Theory Application, 45(10):1013–1020

    Google Scholar 

  88. DeMartini B E, Butterfield H, Moehlis J, Turner K L (2007) Chaos for a Microelectromechanical oscillator governed by the nonlinear Mathieu equation. Journal of Microelectromechanical Systems, 16(6):1314–1323

    Article  Google Scholar 

  89. Haghighi H S, Markazi A H D (2009) Chaos prediction and control in MEMS resonators. Communication Nonlinear Sciences Numerical Simulatation, doi:10.1016/j.cnsns.2009.10.002

    Google Scholar 

  90. De S K, Aluru N (2005) Complex oscillations and chaos in electrostatic microelectromechanical systems under superharmonic excitations. Physical Review Letters, 94(20):204 101.1–204 101.4

    Google Scholar 

  91. Park K, Chen Q, Lai Y C (2008) Energy enhancement and chaos control in microelectromehcanical systems. Physical Review E, 77:026210

    Article  Google Scholar 

  92. Liu S, Davidson A, Lin Q (2004) Simulation studies on nonlinear dynamics and chaos in a MEMS cantilever control system. Journal of Micromechanics and Microengineering, 14(7):1064–1073

    Article  Google Scholar 

  93. Chavarette F R, Balthazar J M, Felix J L P, Rafikov M (2009) A reducing of a chaotic movement to a periodic orbit, of a micro-electro-mechanical system, by using an optimal linear control design. Communication Nonlinear Sciences Numerical Simulation, 14:1844–1853

    Article  Google Scholar 

  94. Stulemeijer J, Herfst R W, Bielen J A (2009) Chaos in electrostatically actuated RF-MEMS measured and modeled. Proceeding of 22nd International Conference on Micro Electro Mechanical Systems MEMS 2009, Sorrento, Italy, pp. 920–922

    Google Scholar 

  95. Ashhab M, Salapaka M, Dahleh M, Mezi´c I (1999) Melnikov-based dynamical analysis of microcantilevers in scanning probe microscopy. Nonlinear Dynamics, 20(3):197–220

    Article  MathSciNet  MATH  Google Scholar 

  96. Hu S, Raman A (2006) Chaos in atomic force microscopy. Physical Review Letters, 96(3):036–107

    Article  Google Scholar 

  97. Jamitzky F, Stark M, Bunk W, Heckl W, Stark R (2006) Chaos in dynamic atomic force microscopy. Nanotechnology, 17(7):S213–S220

    Article  Google Scholar 

  98. Liu M, Chelidze D (2008) A new type of atomic force microscope based on chaotic motions. International Journal of Non-Linear Mechanics, 43:521–526

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Mohammad I. Younis Ph.D. .

Rights and permissions

Reprints and permissions

Copyright information

© 2011 Springer Science+Business Media, LLC

About this chapter

Cite this chapter

Younis, M.I. (2011). Introduction to Nonlinear Dynamics. In: MEMS Linear and Nonlinear Statics and Dynamics. Microsystems, vol 20. Springer, Boston, MA. https://doi.org/10.1007/978-1-4419-6020-7_5

Download citation

  • DOI: https://doi.org/10.1007/978-1-4419-6020-7_5

  • Published:

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4419-6019-1

  • Online ISBN: 978-1-4419-6020-7

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics