Nonlinear Structural Behavior of Micro- and Nano-Actuators Using the Galerkin Discretization Technique

Authors: Hassen M. Ouakad

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.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1123594

References:

[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.