Acta Mechanica Sinica

, Volume 34, Issue 3, pp 528–541 | Cite as

Anharmonic 1D actuator model including electrostatic and Casimir forces with fractional damping perturbed by an external force

Research Paper
  • 39 Downloads

Abstract

We modeled a one-dimensional actuator including the Casimir and electrostatic forces perturbed by an external force with fractional damping. The movable electrode was assumed to oscillate by an anharmonic elastic force originated from Murrell–Mottram or Lippincott potential. The nonlinear equations have been solved via the Adomian decomposition method. The behavior of the displacement of the electrode from equilibrium position, its velocity and acceleration were described versus time. Also, the changes of the displacement have been investigated according to the frequency of the external force and the voltage of the electrostatic force. The convergence of the Adomian method and the effect of the orders of expansion on the displacement versus time, frequency, and voltage were discussed. The pull-in parameter was obtained and compared with the other models in the literature. This parameter was described versus the equilibrium position and anharmonicity constant.

Keywords

Murrell–Mottram potential Lippincott potential Casimir force Fractional damping Pull-in parameter Adomian decomposition method 

References

  1. 1.
    Zhang, W.M., Yan, H., Pemg, Z.K., et al.: Electrostatic pull-in instability in MEMS/NEMS: a review. Sens. Actuators A Phys. 214, 187–218 (2014)CrossRefGoogle Scholar
  2. 2.
    Lin, W.H., Zhao, Y.P.: Pull-in instability of micro-switch actuators: model review. Int. J. Nonlinear Sci. Numer. Simul. 9, 175–183 (2008)CrossRefGoogle Scholar
  3. 3.
    Duan, J., Li, Z., Liu, J.: Pull-in instability analyses for NEMS actuators with quartic shape approximation. Appl. Math. Mech. 37, 303–314 (2016)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Dai, H.L., Wang, L.: Surface effect on the pull-in instability of cantilevered nano-switches based on a full nonlinear model. Physica E Low Dimens. Syst. Nanostruct. 73, 141–147 (2015)CrossRefGoogle Scholar
  5. 5.
    Ma, J.B., Jiang, L., Asokanthan, S.F.: Influence of surface effects on the pull-in instability of NEMS electrostatic switches. Nanotechnology 21, 505708–505717 (2010)CrossRefGoogle Scholar
  6. 6.
    Wang, K.F., Wang, B.L.: Influence of surface energy on the non-linear pull-in instability of nano-switches. Int. J. Nonlinear Mech. 59, 69–75 (2014)CrossRefGoogle Scholar
  7. 7.
    Kim, N., Aluru, N.R.: Effect of intermolecular force on the static/dynamic behaviour of M/NEM devices. Nanotechnology. 25, 485204–485216 (2014)CrossRefGoogle Scholar
  8. 8.
    Ramezani, A., Alasty, A., Akbari, J.: Closed-form solutions of the pull-in instability in nano-cantilevers under electrostatic and intermolecular forces. Int. J. Solids Struct. 44, 4925–4941 (2007)CrossRefMATHGoogle Scholar
  9. 9.
    Moeenfard, H., Ahmadian, M.T.: Analytical modeling of static behavior of electrostatically actuated nano/micromirrors considering van der Waals forces. Acta Mech. Sin. 28, 729–736 (2012)CrossRefGoogle Scholar
  10. 10.
    Lin, W.H., Zhao, Y.P.: Influence of damping on the dynamical behavior of the electrostatic parallel-plate and torsional actuators with intermolecular forces. Sensors 7, 3012–3026 (2007)CrossRefGoogle Scholar
  11. 11.
    Pritchard, R.H., Terentjev, E.M.: Oscillations and damping in the fractional Maxwell materials. J. Rheol. 61, 187–203 (2017)CrossRefGoogle Scholar
  12. 12.
    Ghanbari, M., Hossainpour, S., Rezazadeh, G.: Studying thin film damping in a micro-beam resonator based on non-classical theories. Acta Mech. Sin. 32, 369–379 (2016)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Casimir, H.B.G.: On the attraction between two perfectly conducting plates. Proc. Koninklijke Ned. Akad. Wet. 51, 793–796 (1948)MATHGoogle Scholar
  14. 14.
    Lin, W.H., Zhao, Y.P.: Casimir effect on the pull-in parameters of nanometer switches. Microsyst. Technol. 11, 80–85 (2005)CrossRefGoogle Scholar
  15. 15.
    Serry, F.M., Walliser, D., Maclay, G.J.: The role of the Casimir effect in the static deflection and stiction of membrane strips in Microelectromechanical systems MEMS. J. Appl. Phys. 84, 2501–2506 (1998)CrossRefGoogle Scholar
  16. 16.
    Ding, J.N., Wen, S.Z., Meng, Y.G.: Theoretical study of the sticking of a membrane strip in MEMS under the Casimir effect. J. Micromech. Microeng. 11, 202–208 (2001)CrossRefGoogle Scholar
  17. 17.
    Buks, E., Roukes, M.: Stiction, adhesion energy, and the Casimir effect in micromechanical systems. Phys. Rev. B 63, 033402-1–033402-4 (2001)CrossRefGoogle Scholar
  18. 18.
    Lin, W.H., Zhao, Y.P.: Nonlinear behavior for nanoscale electrostatic actuators with Casimir force Chaos. Solitons Fract. 23, 1777–1785 (2005)CrossRefMATHGoogle Scholar
  19. 19.
    Bordag, M., Mohideen, U., Mostepanenko, V.M.: New developments in the Casimir effect. Phys. Rep. 353, 1–205 (2001)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Guo, J.G., Zhao, Y.P.: Influence of van der Waals and Casimir forces on electrostatic torsional actuators. J. Microelectromech. Syst. 13, 1027–1035 (2004)CrossRefGoogle Scholar
  21. 21.
    Mojahedi, M., Ahmadian, M.T., Firoozbakhsh, K.: The oscillatory behavior, static and dynamic analyses of a micro/nano gyroscope considering geometric nonlinearities and intermolecular forces. Acta Mech. Sin. 29, 851–863 (2013)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Draganescu, G.E., Bereteu, L., Ercuta, A., et al.: Anharmonic vibrations of a nano-sized oscillator with fractional damping. Commun. Nonlinear Sci. Numer. Simul. 15, 922–926 (2010)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Wang, K.F., Wang, B.L.: A general model for nano-cantilever switches with consideration of surface effects and nonlinear curvature. Physica E Low Dimens. Syst. Nanostruct. 66, 197–208 (2015)CrossRefGoogle Scholar
  24. 24.
    Zhang, Y., Liu, Y., Murphy, K.D.: Nonlinear dynamic response of beam and its application in nanomechanical resonator. Acta Mech. Sin. 28, 190–200 (2012)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Mansoori Kermani, M., Dehestani, M.: Solving the nonlinear equations for one-dimensional nano-sized model including Rydberg and Varshni potentials and Casimir force using the decomposition method. Appl. Math. Model. 37, 3399–3406 (2013)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Adomian, G.A.: Review of decomposition method in applied mathematics. J. Math. Anal. Appl. 138, 501–544 (1988)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Kuang, J.H., Chen, C.J.: Adomian decomposition method used for solving nonlinear pull-in behavior in electrostatic micro-actuators. Math. Comput. Model. 41, 1479–1491 (2005)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Chen, W., Lu, Z.: An algorithm for Adomian decomposition method. Appl. Math. Comput. 159, 221–235 (2004)MathSciNetMATHGoogle Scholar
  29. 29.
    Wazwaz, A.M.: A comparison between Adomian decomposition method and Taylor series method in the series solution. Appl. Math. Comput. 97, 37–44 (1998)MathSciNetMATHGoogle Scholar
  30. 30.
    Abbasbandy, S.: A numerical solution of Blasius equation by Adomian’s decomposition method and comparison with homotopy perturbation method. Chaos Solition. Fract. 31, 257–260 (2007)CrossRefGoogle Scholar
  31. 31.
    Lavoie, J.L., Osler, T.J., Tremblay, R.: Fractional derivatives and special functions. SIAM Rev. 18, 240–268 (1976)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Abramowits, M., Stegun, I.A.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards, Washington, DC (1964)Google Scholar
  33. 33.
    Gómez-Aguilar, J.F., Yépez-Martínez, H., Calderón-Ramón, C., et al.: Modeling of a Mass–Spring–Damper system by fractional derivatives with and without a singular kernel. Entropy 17, 6289–6303 (2015)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Basto, M., Semiao, V., Calheiros, F.L.: Numerical study of modified Adomian’s method applied to Burgers equation. J. Comput. Appl. Math. 206, 927–949 (2007)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Ramana, P.V., Raghu Prasad, B.K.: Modified adomian decomposition method for Van der Pol equations. Int. J. Nonlinear Mech. 65, 121–132 (2014)CrossRefGoogle Scholar
  36. 36.
    Lippincott, E.R., Steele, D., Caldwell, P.: General relation between potential energy and internuclear distance for diatomic molecules. III. Excited states. J. Chem. Phys. 35, 123–141 (1961)CrossRefGoogle Scholar
  37. 37.
    Schlegel, H.B., Wolfe, S., Bernardi, F.: Ab initio computation of force constants. II. The estimation of dissociation energies from ab initio SCF calculations. Can. J. Chem. 53, 3599–3601 (1975)CrossRefGoogle Scholar
  38. 38.
    Lim, T.C.: Relationship between Morse and Murrell-Mottram potentials at long range. J. Math. Chem. 36, 139–145 (2004)MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Lim, T.C.: Relationship between the 2-body energy of the Biswas-Hamann and the Murrell-Mottram potential functions. Z. Naturforsch. 59a, 116–118 (2004)Google Scholar
  40. 40.
    Cherruault, Y., Adomian, G.: Decomposition method: a new proof of convergence. Math. Comput. Model. 18, 103–106 (1993)MathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    Lin, W.H., Zhao, Y.P.: Dynamics behavior of nanoscale electrostatic actuators. Chin. Phys. Lett. 20, 2070–2073 (2003)CrossRefGoogle Scholar
  42. 42.
    Wolfram Research, Inc. Mathematica, v. 9; Wolfram Research, Inc.: 1988–2015Google Scholar

Copyright information

© The Chinese Society of Theoretical and Applied Mechanics; Institute of Mechanics, Chinese Academy of Sciences and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of ChemistryShahid Bahonar University of KermanKermanIran

Personalised recommendations