Nonlinear Dynamics

, Volume 92, Issue 3, pp 803–814 | Cite as

Nonlinear dynamics of doubly curved shallow microshells

  • Mergen H. Ghayesh
  • Hamed Farokhi
Original Paper


The nonlinear dynamical characteristics of a doubly curved shallow microshell are investigated thoroughly. A consistent nonlinear model for the microshell is developed on the basis of the modified couple stress theory (MCST) in an orthogonal curvilinear coordinate system. In particular, based on Donnell’s nonlinear theory, the expressions for the strain and the symmetric rotation gradient tensors are obtained in the framework of MCST, which are then used to derive the potential energy of the microshell. The analytical geometrically nonlinear equations of motion of the doubly microshell are obtained for in-plane displacements as well as the out-of-plane one. These equations of partial differential type are reduced to a large set of ordinary differential equations making use of a two-dimensional Galerkin scheme. Extensive numerical simulations are conducted to obtain the nonlinear resonant response of the system for various principal radii of curvature and to examine the effect of modal interactions and the length-scale parameter.


Nonlinear dynamics Microshells Modified couple stress theory Modal interactions Nonlinear resonant response 


  1. 1.
    Younis, M.I., Nayfeh, A.H.: A study of the nonlinear response of a resonant microbeam to an electric actuation. Nonlinear Dyn. 31, 91–117 (2003)CrossRefzbMATHGoogle Scholar
  2. 2.
    Younis, M.I., Abdel-Rahman, E.M., Nayfeh, A.: A reduced-order model for electrically actuated microbeam-based MEMS. J. Microelectromech. Syst. 12, 672–680 (2003)CrossRefGoogle Scholar
  3. 3.
    Younis, M.I., Abdel-Rahman, E.M., Nayfeh, A.H.: Dynamic simulations of a novel RF MEMS switch. In: Technical Proceedings of the Nsti Nanotech 2004, vol. 2, pp. 287–290 (2004)Google Scholar
  4. 4.
    Nayfeh, A.H., Younis, M.I.: Dynamics of MEMS resonators under superharmonic and subharmonic excitations. J. Micromech. Microeng. 15, 1840–1847 (2005)CrossRefGoogle Scholar
  5. 5.
    Younis, M.I., Alsaleem, F., Jordy, D.: The response of clamped-clamped microbeams under mechanical shock. Int. J. Non-Linear Mech. 42, 643–657 (2007)CrossRefGoogle Scholar
  6. 6.
    Ouakad, H.M., Younis, M.I.: The dynamic behavior of MEMS arch resonators actuated electrically. Int. J. Non-Linear Mech. 45, 704–713 (2010)CrossRefGoogle Scholar
  7. 7.
    Younis, M.I., Ouakad, H.M., Alsaleem, F.M., Miles, R., Cui, W.: Nonlinear dynamics of MEMS arches under harmonic electrostatic actuation. Microelectromech. Syst. 19, 647–656 (2010)CrossRefGoogle Scholar
  8. 8.
    Younis, M.I.: MEMS Linear and Nonlinear Statics and Dynamics. Springer, Berlin (2011)CrossRefGoogle Scholar
  9. 9.
    Ruzziconi, L., Younis, M.I., Lenci, S.: Parameter identification of an electrically actuated imperfect microbeam. Int. J. Non-Linear Mech 57, 208–219 (2013)CrossRefzbMATHGoogle Scholar
  10. 10.
    Laura, R., Ahmad, M.B., Mohammad, I.Y., Weili, C., Stefano, L.: Nonlinear dynamics of an electrically actuated imperfect microbeam resonator: experimental investigation and reduced-order modeling. J. Micromech. Microeng. 23, 075012 (2013)CrossRefGoogle Scholar
  11. 11.
    Ramini, A.H., Hennawi, Q.M., Younis, M.I.: Theoretical and experimental investigation of the nonlinear behavior of an electrostatically actuated in-plane MEMS arch. J. Microelectromech. Syst. 25, 570–578 (2016)CrossRefGoogle Scholar
  12. 12.
    Gutschmidt, S., Gottlieb, O.: Nonlinear dynamic behavior of a microbeam array subject to parametric actuation at low, medium and large DC-voltages. Nonlinear Dyn. 67, 1–36 (2012)CrossRefzbMATHGoogle Scholar
  13. 13.
    Ghayesh, M.H., Farokhi, H., Amabili, M.: Nonlinear behaviour of electrically actuated MEMS resonators. Int. J. Eng. Sci. 71, 137–155 (2013)CrossRefGoogle Scholar
  14. 14.
    Samaali, H., Najar, F., Choura, S., Nayfeh, A.H., Masmoudi, M.: A double microbeam MEMS ohmic switch for RF-applications with low actuation voltage. Nonlinear Dyn. 63, 719–734 (2011)CrossRefGoogle Scholar
  15. 15.
    LaRose Iii, R.P., Murphy, K.D.: Impact dynamics of MEMS switches. Nonlinear Dyn. 60, 327–339 (2010)CrossRefzbMATHGoogle Scholar
  16. 16.
    Farokhi, H., Ghayesh, M.H.: Nonlinear mechanics of electrically actuated microplates. Int. J. Eng. Sci. 123, 197–213 (2018)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Farokhi, H., Ghayesh, M.H.: Supercritical nonlinear parametric dynamics of Timoshenko microbeams. Commun. Nonlinear Sci. Numer. Simul. 59, 592–605 (2018)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Tahani, M., Askari, A.R., Mohandes, Y., Hassani, B.: Size-dependent free vibration analysis of electrostatically pre-deformed rectangular micro-plates based on the modified couple stress theory. Int. J. Mech. Sci. 94–95, 185–198 (2015)CrossRefGoogle Scholar
  19. 19.
    Farokhi, H., Ghayesh, M.H.: Size-dependent parametric dynamics of imperfect microbeams. Int. J. Eng. Sci. 99, 39–55 (2016)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Farokhi, H., Ghayesh, M.H., Hussain, S.: Large-amplitude dynamical behaviour of microcantilevers. Int. J. Eng. Sci. 106, 29–41 (2016)CrossRefGoogle Scholar
  21. 21.
    Yu, Y., Wu, B., Lim, C.W.: Numerical and analytical approximations to large post-buckling deformation of MEMS. Int. J. Mech. Sci. 55, 95–103 (2012)CrossRefGoogle Scholar
  22. 22.
    Li, Y., Meguid, S.A., Fu, Y., Xu, D.: Nonlinear analysis of thermally and electrically actuated functionally graded material microbeam. Proc. R. Soc. Lond. A Math. Phys. Eng. Sci. 470, 20130473 (2013)CrossRefzbMATHGoogle Scholar
  23. 23.
    Ghayesh, M.H., Farokhi, H., Alici, G.: Size-dependent performance of microgyroscopes. Int. J. Eng. Sci. 100, 99–111 (2016)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Ghayesh, M.H., Farokhi, H., Hussain, S.: Viscoelastically coupled size-dependent dynamics of microbeams. Int. J. Eng. Sci. 109, 243–255 (2016)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Farokhi, H., Ghayesh, M.H.: Thermo-mechanical dynamics of perfect and imperfect Timoshenko microbeams. Int. J. Eng. Sci. 91, 12–33 (2015)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Ghayesh, M.H., Farokhi, H.: Chaotic motion of a parametrically excited microbeam. Int. J. Eng. Sci. 96, 34–45 (2015)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Ghayesh, M.H., Farokhi, H.: Nonlinear dynamics of microplates. Int. J. Eng. Sci. 86, 60–73 (2015)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Ghayesh, M.H., Farokhi, H., Gholipour, A., Tavallaeinejad, M.: Nonlinear oscillations of functionally graded microplates. Int. J. Eng. Sci. 122, 56–72 (2018)CrossRefGoogle Scholar
  29. 29.
    Fleck, N.A., Muller, G.M., Ashby, M.F., Hutchinson, J.W.: Strain gradient plasticity: theory and experiment. Acta Metallurgica et Materialia 42, 475–487 (1994)CrossRefGoogle Scholar
  30. 30.
    McFarland, A.W., Colton, J.S.: Role of material microstructure in plate stiffness with relevance to microcantilever sensors. J. Micromech. Microeng. 15, 1060 (2005)CrossRefGoogle Scholar
  31. 31.
    Haque, M.A., Saif, M.T.A.: Strain gradient effect in nanoscale thin films. Acta Materialia 51, 3053–3061 (2003)CrossRefGoogle Scholar
  32. 32.
    Hosseini-Hashemi, S., Sharifpour, F., Ilkhani, M.R.: On the free vibrations of size-dependent closed micro/nano-spherical shell based on the modified couple stress theory. Int. J. Mech. Sci. 115–116, 501–515 (2016)CrossRefGoogle Scholar
  33. 33.
    Farokhi, H., Ghayesh, M.H., Amabili, M.: Nonlinear dynamics of a geometrically imperfect microbeam based on the modified couple stress theory. Int. J. Eng. Sci. 68, 11–23 (2013)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Rahaeifard, M., Kahrobaiyan, M.H., Ahmadian, M.T., Firoozbakhsh, K.: Size-dependent pull-in phenomena in nonlinear microbridges. Int. J. Mech. Sci. 54, 306–310 (2012)CrossRefGoogle Scholar
  35. 35.
    Ghayesh, M.H., Amabili, M., Farokhi, H.: Nonlinear forced vibrations of a microbeam based on the strain gradient elasticity theory. Int. J. Eng. Sci. 63, 52–60 (2013)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Ramezani, S.: A shear deformation micro-plate model based on the most general form of strain gradient elasticity. Int. J. Mech. Sci. 57, 34–42 (2012)CrossRefGoogle Scholar
  37. 37.
    Lei, J., He, Y., Zhang, B., Liu, D., Shen, L., Guo, S.: A size-dependent FG micro-plate model incorporating higher-order shear and normal deformation effects based on a modified couple stress theory. Int. J. Mech. Sci. 104, 8–23 (2015)CrossRefGoogle Scholar
  38. 38.
    Farokhi, H., Ghayesh, M.H.: Nonlinear dynamical behaviour of geometrically imperfect microplates based on modified couple stress theory. Int. J. Mech. Sci. 90, 133–144 (2015)CrossRefGoogle Scholar
  39. 39.
    Yang, F., Chong, A.C.M., Lam, D.C.C., Tong, P.: Couple stress based strain gradient theory for elasticity. Int. J. Solids Struct. 39, 2731–2743 (2002)CrossRefzbMATHGoogle Scholar
  40. 40.
    Ghayesh, M.H., Farokhi, H., Amabili, M.: Nonlinear dynamics of a microscale beam based on the modified couple stress theory. Compos. Part B Eng. 50, 318–324 (2013)CrossRefzbMATHGoogle Scholar
  41. 41.
    Gholipour, A., Farokhi, H., Ghayesh, M.H.: In-plane and out-of-plane nonlinear size-dependent dynamics of microplates. Nonlinear Dyn. 79, 1771–1785 (2015)CrossRefGoogle Scholar
  42. 42.
    Sahmani, S., Ansari, R., Gholami, R., Darvizeh, A.: Dynamic stability analysis of functionally graded higher-order shear deformable microshells based on the modified couple stress elasticity theory. Compos. Part B Eng. 51, 44–53 (2013)CrossRefzbMATHGoogle Scholar
  43. 43.
    Lou, J., He, L., Wu, H., Du, J.: Pre-buckling and buckling analyses of functionally graded microshells under axial and radial loads based on the modified couple stress theory. Compos. Struct. 142, 226–237 (2016)CrossRefGoogle Scholar
  44. 44.
    Beni, Y.T., Mehralian, F., Razavi, H.: Free vibration analysis of size-dependent shear deformable functionally graded cylindrical shell on the basis of modified couple stress theory. Compos. Struct. 120, 65–78 (2015)CrossRefGoogle Scholar
  45. 45.
    Mehralian, F., Beni, Y.T.: Size-dependent torsional buckling analysis of functionally graded cylindrical shell. Compos. Part B. Eng. 94, 11–25 (2016)CrossRefGoogle Scholar
  46. 46.
    Fadaee, M., Ilkhani, M.R.: Closed-form solution for freely vibrating functionally graded thick doubly curved panel-a new generic approach. Lat. Am. J. Solids Struct. 12, 1748–1770 (2015)CrossRefGoogle Scholar
  47. 47.
    Fadaee, M., Ilkhani, M.R., Hosseini-Hashemi, S.: A new generic exact solution for free vibration of functionally graded moderately thick doubly curved shallow shell panel. J. Vib. Control 22, 3355–3367 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Donnell, L.H.: A new theory for the buckling of thin cylinders under axial compression and bending. Trans. Asme 56, 795–806 (1934)Google Scholar
  49. 49.
    Eringen, A.C.: Mechanics of Continua. Wiley, New York (1967)zbMATHGoogle Scholar
  50. 50.
    Ghayesh, M.H., Farokhi, H., Gholipour, A.: Oscillations of functionally graded microbeams. Int. J. Eng. Sci. 110, 35–53 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Ghayesh, M.H., Farokhi, H., Gholipour, A.: Vibration analysis of geometrically imperfect three-layered shear-deformable microbeams. Int. J. Mech. Sci. 122, 370–383 (2017)CrossRefGoogle Scholar
  52. 52.
    Ghayesh, M.H., Amabili, M., Farokhi, H.: Three-dimensional nonlinear size-dependent behaviour of Timoshenko microbeams. Int. J. Eng. Sci. 71, 1–14 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  53. 53.
    Ghayesh, M.H., Farokhi, H., Amabili, M.: In-plane and out-of-plane motion characteristics of microbeams with modal interactions. Compos. Part B Eng. 60, 423–439 (2014)CrossRefGoogle Scholar
  54. 54.
    Farokhi, H., Ghayesh, M.H.: Nonlinear resonant response of imperfect extensible Timoshenko microbeams. Int. J. Mech. Mater. Des. 13, 43–55 (2017)CrossRefGoogle Scholar
  55. 55.
    Farokhi, H., Ghayesh, M.H., Gholipour, A., Hussain, S.: Motion characteristics of bilayered extensible Timoshenko microbeams. Int. J. Eng. Sci. 112, 1–17 (2017)MathSciNetCrossRefGoogle Scholar
  56. 56.
    Chen, S.H., Feng, B.: Size effect in micro-scale cantilever beam bending. Acta Mechanica 219, 291–307 (2011)CrossRefzbMATHGoogle Scholar

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© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mechanical EngineeringUniversity of AdelaideAdelaideAustralia
  2. 2.Department of AeronauticsImperial College LondonLondonUK

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