Nonlinear Structural Behavior of Micro- and Nano-Actuators Using the Galerkin Discretization Technique
In this paper, the influence of van der Waals, as well as electrostatic forces on the structural behavior of MEMS and NEMS actuators, has been investigated using of a Euler-Bernoulli beam continuous model. In the proposed nonlinear model, the electrostatic fringing-fields and the mid-plane stretching (geometric nonlinearity) effects have been considered. The nonlinear integro-differential equation governing the static structural behavior of the actuator has been derived. An original Galerkin-based reduced-order model has been developed to avoid problems arising from the nonlinearities in the differential equation. The obtained reduced-order model equations have been solved numerically using the Newton-Raphson method. The basic design parameters such as the pull-in parameters (voltage and deflection at pull-in), as well as the detachment length due to the van der Waals force of some investigated micro- and nano-actuators have been calculated. The obtained numerical results have been compared with some other existing methods (finite-elements method and finite-difference method) and the comparison showed good agreement among all assumed numerical techniques.
[1] G. T. A. Kovacs, Micromachined Transducers Sourcebook. McGraw-Hill, NY, 1998
[2] H. C. Nathanson, and R. A. Wickstrom, “A resonant gate silicon surface transistor with high Q bandpass properties,” IEEE Appl. Phys. Lett., vol.7, 1965, pp. 84-86.
[3] F. D. Bannon, J. R. Clark, and C.T.C. Nguyen, “High-Q HF Microelectromechanical Filters,” J. Solid-State Circuits, vol. 35, 2000, pp. 512-526.
[4] T. P. Burg, A. R. Mirza, N. Milovic, C. H. Tsau, G. A. Popescu, J. S. Foster, and S. R. Manalis, “Vacuum-packaged suspended microchannel resonant mass sensor for biomolecular detection,” J. Microelectromech. Syst., vol. 15, 2006, pp. 1466-1476.
[5] W. T. Hsu, J. R. Clark, and C. T. C. Nguyen, “A resonant temperature sensor based on electrical spring softening”, Transducers 01, Eurosensors XV, The 11th International Conference on Solid-State Sensors and Actuators, Munich, Germany, 2001.
[6] B. Kim, M. A. Hopcroft, R. Melamud, C. M. Jha, M. Agarwal, S. A. Chandorkar and T. W. Kenny, “CMOS compatible wafer-scale encapsulation with MEMS resonators”, ASME InterPACK, Vancouver, Canada, 2007.
[7] T. Rueckes, K. Kim, E. Joselevich, G. Y. Tseng, C-L. Cheung, and Lieber C. M., “Carbon nanotube-based nonvolatile random access memory for molecular computing,” Science, vol. 289, 2000, pp.94–97.
[8] M. Dequesnes, S. Tang, and N.R. Aluru, “Static and Dynamic analysis of carbon nanotube-based switches”, Transactions of the ASME, vol. 126, 2004, pp. 230-237.
[9] P. G. Collins, K. B. Bradley, M. Ishigami, and A. Zettl, “Extreme oxygen sensitivity of electronic properties of carbon nanotubes,” Science, vol. 287, 2000, pp. 1801–1804.
[10] C. K. W. Adu, G. U. Sumanasekera, B. K. Pradhan, H. E. Romero, and P. C., Eklund, “Carbon nanotubes: a thermoelectric nano-nose,” Chem. Phys. Lett., vol. 337, 20001, pp. 31-35.
[11] H. M. Ouakad, and M. I., Younis, “Nonlinear dynamics of electrically actuated carbon nanotube resonators,” Journal of Computational and Nonlinear Dynamics, vol. 5, 2010, pp. 011009.
[12] A. F. Marquès, R.C. Castelló, and A.M. Shkel, “Modelling the Electrostatic Actuation of MEMS: State of the Art 2005”, Technical Report, IOC-DT, Vol 18, 2005, pp 1-24.
[13] H. Nathanson, W. Newell, R. Wickstrom, and J.R. Davis, “The Resonant Gate Transistor,” IEEE Transactions on Electronics devices, vol. 14, 1967, pp. 117-133.
[14] W. Newell, “Miniaturization of Tuning Forks,” Smithsonian/NASA Physics Query Form Science, vol. 161, 1986, pp. 1320-1326.
[15] D. J. Ijntema, and H. A. C. Tilmans, “Static and Dynamic Aspects of an Air-gap Capacitor”, Journal of Sensors and Actuators, vol. 35, 1992, pp. 121-128.
[16] P. M. Osterberg, S. D. Senturia, “M-TEST: a test chip for MEMS material property measurement using electrostatically actuated test structures,” J Microelectromech Syst, vol. 6, 1997, pp. 107-118.
[17] L. Castañer, A. Rodriguez, J. Pons, and S. D. Senturia, “Pull- in Time-Energy Product of Electrostatic Actuators: Comparison of Experiments with Simulation,” Journal of Sensors and Actuators A, vol 27, 1999, pp. 263-269.
[18] O. Bochobza-Degani, and Y. Nemirovsky, “Modeling the pull-in parameters of electrostatic actuators with a novel lumped two degrees of freedom pull-in model,” Sensors and Actuators A, vol. 97, 2002, pp. 569-578.
[19] S. Krylov, and R. Maimon. “Pull-in Dynamics of an Elastic Beam Actuated by Continuously Distributed Electrostatic Force,” Transactions of the ASME: Journal of vibration and acoustics, vol. 126, 2004, pp. 332-342.
[20] A. H. Nayfeh, M. I. Younis, and E. M. Abdel-Rahman, “Dynamic Pull-in Phenomenon in MEMS Resonators,” Journal of Nonlinear Dynamics, vol. 48, 2007, pp. 153-163.
[21] M. Dequesnes, S. V. Rotkin, and N. R. Aluru, “Calculation of pull-in voltages for carbon-nanotube-based nanoelectromechanical switches,” Nanotechnology, vol. 13, 2002, pp. 120-131.
[22] T. Tang, A. Jagota, and C. Y., Hui, “Adhesion between single-walled carbon nanotubes,” Journal of Applied Physics, vol. 97, 2005, pp. 074304-07431.
[23] T. Tang, A. Jagota, C. Y., Hui, N. and J. Glassmaker, “Collapse of single-walled carbon nanotubes,” Journal of Applied Physics, vol. 97, 2005, pp. 074310-074311.
[24] W-H. Lin, and Y.P. Zhao, “Nonlinear behavior for nanoscale electrostatic actuators with Casimir force,” Chaos, Solitons and Fractals, vol. 23, 2005, pp. 1777-1785.
[25] G. W. Wang, Y. Zhang, and Y. P. Zhao, “Pull-in instability study of carbon nanotube tweezers under the influence of van der Waals forces,” Journal of Micromechanics and Microengineering, vol. 14, 2004, pp. 1119-1125.
[26] H. S. Elmer, and S. D. Senturia, “Generating efficient dynamical models for microelectromechanical systems from a few finite-element simulation runs,” Journal of Microelectromechanical Systems, vol. 8, 1999, pp. 280-289.
[27] F. Najar, S. Choura, E. M. Abdel-Rahman, S. El-Borgi, and A. H. Nayfeh, “Dynamic analysis of variable-geometry electrostatic microactuators,” Journal of Micromechanics and Microengineering, vol. 16, 2006, pp. 2449.
[28] A. H. Nayfeh, M. I. Younis, and E. M. Abdel-Rahman, “Characterization of the mechanical behavior of an electrically actuated microbeam,” Journal of Micromechanics and Microengineering, vol. 12.6, 2002, pp. 759-763.
[29] H. M. Ouakad, and M. I. Younis, “Modeling and Simulations of Collapse Instabilities of Microbeams due to Capillary Forces,” Hindawi Mathematical Problems in Engineering, vol. 2009, 2009, pp. 1-16.
[30] M. I. Younis, and A. H. Nayfeh, “A study of the nonlinear response of a resonant microbeam to an electric actuation,” Nonlinear Dynamics, vol. 31, 2003, pp. 91-117.
[31] G. J. Simitses, and D. H. Hodges, Fundamentals of Structural Stability, Elsevier, Amsterdam, 2006.
[32] A. H. Nayfeh, Nonlinear Interactions, Wiley Interscience, New York, 2000.
[33] R. K. Gupta, Electrostatic pull-in test structure design for in-situ mechanical property measurements of microelectromechanical systems, PhD Thesis, MIT, MA: Cambridge, 1997.
[34] J. M. Huang, K. M. Liew, C. H. Wong, S. Rajendran, M. J. Tan, and A. Q. Liu, “Mechanical design and optimization of capacitive micromachined switch,” Sensors and Actuators A, vol. 93, 2001, pp. 273-285.
[35] J. N. Israelachvili, “Intermolecular and surface forces,” Academic, London, 1992.
[36] H. M. Ouakad, “An electrostatically actuated MEMS arch band-pass filter,” Shock and Vibration, vol. 20, 2013, pp. 809-819.
[37] E. T. Whittaker, and G. Robinson, “The Newton-Raphson Method,” §44 in the Calculus of Observations: A Treatise on Numerical Mathematics: 4th edition, NY: Dover, pp. 84-87, 1967.
[1] G. T. A. Kovacs, Micromachined Transducers Sourcebook. McGraw-Hill, NY, 1998
[2] H. C. Nathanson, and R. A. Wickstrom, “A resonant gate silicon surface transistor with high Q bandpass properties,” IEEE Appl. Phys. Lett., vol.7, 1965, pp. 84-86.
[3] F. D. Bannon, J. R. Clark, and C.T.C. Nguyen, “High-Q HF Microelectromechanical Filters,” J. Solid-State Circuits, vol. 35, 2000, pp. 512-526.
[4] T. P. Burg, A. R. Mirza, N. Milovic, C. H. Tsau, G. A. Popescu, J. S. Foster, and S. R. Manalis, “Vacuum-packaged suspended microchannel resonant mass sensor for biomolecular detection,” J. Microelectromech. Syst., vol. 15, 2006, pp. 1466-1476.
[5] W. T. Hsu, J. R. Clark, and C. T. C. Nguyen, “A resonant temperature sensor based on electrical spring softening”, Transducers 01, Eurosensors XV, The 11th International Conference on Solid-State Sensors and Actuators, Munich, Germany, 2001.
[6] B. Kim, M. A. Hopcroft, R. Melamud, C. M. Jha, M. Agarwal, S. A. Chandorkar and T. W. Kenny, “CMOS compatible wafer-scale encapsulation with MEMS resonators”, ASME InterPACK, Vancouver, Canada, 2007.
[7] T. Rueckes, K. Kim, E. Joselevich, G. Y. Tseng, C-L. Cheung, and Lieber C. M., “Carbon nanotube-based nonvolatile random access memory for molecular computing,” Science, vol. 289, 2000, pp.94–97.
[8] M. Dequesnes, S. Tang, and N.R. Aluru, “Static and Dynamic analysis of carbon nanotube-based switches”, Transactions of the ASME, vol. 126, 2004, pp. 230-237.
[9] P. G. Collins, K. B. Bradley, M. Ishigami, and A. Zettl, “Extreme oxygen sensitivity of electronic properties of carbon nanotubes,” Science, vol. 287, 2000, pp. 1801–1804.
[10] C. K. W. Adu, G. U. Sumanasekera, B. K. Pradhan, H. E. Romero, and P. C., Eklund, “Carbon nanotubes: a thermoelectric nano-nose,” Chem. Phys. Lett., vol. 337, 20001, pp. 31-35.
[11] H. M. Ouakad, and M. I., Younis, “Nonlinear dynamics of electrically actuated carbon nanotube resonators,” Journal of Computational and Nonlinear Dynamics, vol. 5, 2010, pp. 011009.
[12] A. F. Marquès, R.C. Castelló, and A.M. Shkel, “Modelling the Electrostatic Actuation of MEMS: State of the Art 2005”, Technical Report, IOC-DT, Vol 18, 2005, pp 1-24.
[13] H. Nathanson, W. Newell, R. Wickstrom, and J.R. Davis, “The Resonant Gate Transistor,” IEEE Transactions on Electronics devices, vol. 14, 1967, pp. 117-133.
[14] W. Newell, “Miniaturization of Tuning Forks,” Smithsonian/NASA Physics Query Form Science, vol. 161, 1986, pp. 1320-1326.
[15] D. J. Ijntema, and H. A. C. Tilmans, “Static and Dynamic Aspects of an Air-gap Capacitor”, Journal of Sensors and Actuators, vol. 35, 1992, pp. 121-128.
[16] P. M. Osterberg, S. D. Senturia, “M-TEST: a test chip for MEMS material property measurement using electrostatically actuated test structures,” J Microelectromech Syst, vol. 6, 1997, pp. 107-118.
[17] L. Castañer, A. Rodriguez, J. Pons, and S. D. Senturia, “Pull- in Time-Energy Product of Electrostatic Actuators: Comparison of Experiments with Simulation,” Journal of Sensors and Actuators A, vol 27, 1999, pp. 263-269.
[18] O. Bochobza-Degani, and Y. Nemirovsky, “Modeling the pull-in parameters of electrostatic actuators with a novel lumped two degrees of freedom pull-in model,” Sensors and Actuators A, vol. 97, 2002, pp. 569-578.
[19] S. Krylov, and R. Maimon. “Pull-in Dynamics of an Elastic Beam Actuated by Continuously Distributed Electrostatic Force,” Transactions of the ASME: Journal of vibration and acoustics, vol. 126, 2004, pp. 332-342.
[20] A. H. Nayfeh, M. I. Younis, and E. M. Abdel-Rahman, “Dynamic Pull-in Phenomenon in MEMS Resonators,” Journal of Nonlinear Dynamics, vol. 48, 2007, pp. 153-163.
[21] M. Dequesnes, S. V. Rotkin, and N. R. Aluru, “Calculation of pull-in voltages for carbon-nanotube-based nanoelectromechanical switches,” Nanotechnology, vol. 13, 2002, pp. 120-131.
[22] T. Tang, A. Jagota, and C. Y., Hui, “Adhesion between single-walled carbon nanotubes,” Journal of Applied Physics, vol. 97, 2005, pp. 074304-07431.
[23] T. Tang, A. Jagota, C. Y., Hui, N. and J. Glassmaker, “Collapse of single-walled carbon nanotubes,” Journal of Applied Physics, vol. 97, 2005, pp. 074310-074311.
[24] W-H. Lin, and Y.P. Zhao, “Nonlinear behavior for nanoscale electrostatic actuators with Casimir force,” Chaos, Solitons and Fractals, vol. 23, 2005, pp. 1777-1785.
[25] G. W. Wang, Y. Zhang, and Y. P. Zhao, “Pull-in instability study of carbon nanotube tweezers under the influence of van der Waals forces,” Journal of Micromechanics and Microengineering, vol. 14, 2004, pp. 1119-1125.
[26] H. S. Elmer, and S. D. Senturia, “Generating efficient dynamical models for microelectromechanical systems from a few finite-element simulation runs,” Journal of Microelectromechanical Systems, vol. 8, 1999, pp. 280-289.
[27] F. Najar, S. Choura, E. M. Abdel-Rahman, S. El-Borgi, and A. H. Nayfeh, “Dynamic analysis of variable-geometry electrostatic microactuators,” Journal of Micromechanics and Microengineering, vol. 16, 2006, pp. 2449.
[28] A. H. Nayfeh, M. I. Younis, and E. M. Abdel-Rahman, “Characterization of the mechanical behavior of an electrically actuated microbeam,” Journal of Micromechanics and Microengineering, vol. 12.6, 2002, pp. 759-763.
[29] H. M. Ouakad, and M. I. Younis, “Modeling and Simulations of Collapse Instabilities of Microbeams due to Capillary Forces,” Hindawi Mathematical Problems in Engineering, vol. 2009, 2009, pp. 1-16.
[30] M. I. Younis, and A. H. Nayfeh, “A study of the nonlinear response of a resonant microbeam to an electric actuation,” Nonlinear Dynamics, vol. 31, 2003, pp. 91-117.
[31] G. J. Simitses, and D. H. Hodges, Fundamentals of Structural Stability, Elsevier, Amsterdam, 2006.
[32] A. H. Nayfeh, Nonlinear Interactions, Wiley Interscience, New York, 2000.
[33] R. K. Gupta, Electrostatic pull-in test structure design for in-situ mechanical property measurements of microelectromechanical systems, PhD Thesis, MIT, MA: Cambridge, 1997.
[34] J. M. Huang, K. M. Liew, C. H. Wong, S. Rajendran, M. J. Tan, and A. Q. Liu, “Mechanical design and optimization of capacitive micromachined switch,” Sensors and Actuators A, vol. 93, 2001, pp. 273-285.
[35] J. N. Israelachvili, “Intermolecular and surface forces,” Academic, London, 1992.
[36] H. M. Ouakad, “An electrostatically actuated MEMS arch band-pass filter,” Shock and Vibration, vol. 20, 2013, pp. 809-819.
[37] E. T. Whittaker, and G. Robinson, “The Newton-Raphson Method,” §44 in the Calculus of Observations: A Treatise on Numerical Mathematics: 4th edition, NY: Dover, pp. 84-87, 1967.
@article{"International Journal of Mechanical, Industrial and Aerospace Sciences:72558", author = "Hassen M. Ouakad", title = "Nonlinear Structural Behavior of Micro- and Nano-Actuators Using the Galerkin Discretization Technique", abstract = "In this paper, the influence of van der Waals, as well as electrostatic forces on the structural behavior of MEMS and NEMS actuators, has been investigated using of a Euler-Bernoulli beam continuous model. In the proposed nonlinear model, the electrostatic fringing-fields and the mid-plane stretching (geometric nonlinearity) effects have been considered. The nonlinear integro-differential equation governing the static structural behavior of the actuator has been derived. An original Galerkin-based reduced-order model has been developed to avoid problems arising from the nonlinearities in the differential equation. The obtained reduced-order model equations have been solved numerically using the Newton-Raphson method. The basic design parameters such as the pull-in parameters (voltage and deflection at pull-in), as well as the detachment length due to the van der Waals force of some investigated micro- and nano-actuators have been calculated. The obtained numerical results have been compared with some other existing methods (finite-elements method and finite-difference method) and the comparison showed good agreement among all assumed numerical techniques.
", keywords = "MEMS, NEMS, fringing-fields, mid-plane stretching, Galerkin method. ", volume = "10", number = "4", pages = "666-7", }
