Dynamics of nonuniform deformable AFG viscoelastic microbeams

  • Mergen H. GhayeshEmail author
Technical Paper


This paper analyses the coupled dynamics of nonuniform deformable axially functionally graded (AFG) viscoelastic microbeams with special consideration to a Kelvin–Voigt type viscosity in the FGM system. When modelling AFG viscoelastic systems, linear assumptions are commonly utilised to simplify numerical or analytical calculations. Another important factor which is usually neglected in the literature on AFG systems is to ignore in-plane/axial displacements/inertia. Viscosity between infinitesimal elements of AFG systems is also usually neglected to simplify calculations. This paper is the first to analyse the viscosity on the dynamical behaviour of AFG microbeams with the help of the Kelvin–Voigt method of viscosity, and the Euler–Bernoulli beam theory. The size dependence in the model and numerical simulations is incorporated via the modified version of the couple stress theory. Viscous stress-components of the Kelvin–Voigt model incorporated via their negative work contribution. The use of an energy balance generates the continuous model of the AFG system in the longitudinal as well as transverse directions. There are both nonlinear and linear couplings between elastically generated and viscous related terms. The nonuniform shape of the microbeam is incorporated via an axial-coordinate-dependent width. A truncation/discretisation for the coupled nonlinear model is performed using Galerkin’s method. The couplings between the dynamics of the AFG viscoelastic microbeam is examined through analysing the influence of various microsystem parameters (e.g. gradient index, viscosity coefficient, and the taper ratio of the microbeam).



  1. Abouelregal AE (2018) Response of thermoelastic microbeams to a periodic external transverse excitation based on MCS theory. Microsyst Technol 24:1925–1933CrossRefGoogle Scholar
  2. Ahmed MS, Ghommem M, Abdelkefi A (2018) Nonlinear analysis and characteristics of electrically-coupled microbeams under mechanical shock. Microsyst Technol.
  3. Akgöz B, Civalek Ö (2013) A size-dependent shear deformation beam model based on the strain gradient elasticity theory. Int J Eng Sci 70:1–14MathSciNetCrossRefzbMATHGoogle Scholar
  4. Elwenspoek M, Jansen HV (2004) Silicon micromachining. Cambridge University Press, CambridgeGoogle Scholar
  5. Farajpour A, Farokhi H, Ghayesh MH et al (2018a) Nonlinear mechanics of nanotubes conveying fluid. Int J Eng Sci 133:132–143MathSciNetCrossRefzbMATHGoogle Scholar
  6. Farajpour A, Ghayesh MH, Farokhi H (2018b) A review on the mechanics of nanostructures. Int J Eng Sci 133:231–263MathSciNetCrossRefzbMATHGoogle Scholar
  7. Farokhi H, Ghayesh MH (2015a) Nonlinear dynamical behaviour of geometrically imperfect microplates based on modified couple stress theory. Int J Mech Sci 90:133–144Google Scholar
  8. Farokhi H, Ghayesh MH (2015b) Thermo-mechanical dynamics of perfect and imperfect Timoshenko microbeams. Int J Eng Sci 91:12–33Google Scholar
  9. Farokhi H, Ghayesh MH (2017) Nonlinear thermo-mechanical behaviour of MEMS resonators. Microsyst Technol 23:5303–5315CrossRefGoogle Scholar
  10. Farokhi H, Ghayesh MH (2018a) Nonlinear mechanics of electrically actuated microplates. Int J Eng Sci 123:197–213Google Scholar
  11. Farokhi H, Ghayesh MH (2018b) Supercritical nonlinear parametric dynamics of Timoshenko microbeams. Commun Nonlinear Sci Numer Simul 59:592–605Google Scholar
  12. Farokhi H, Ghayesh MH, Amabili M (2013a) Nonlinear dynamics of a geometrically imperfect microbeam based on the modified couple stress theory. Int J Eng Sci 68:11–23Google Scholar
  13. Farokhi H, Ghayesh MH, Amabili M (2013b) Nonlinear resonant behavior of microbeams over the buckled state. Appl Physics A Mat Sci Process 113:297–307 Google Scholar
  14. Farokhi H, Ghayesh MH, Hussain Sh (2016) Large-amplitude dynamical behaviour of microcantilevers. Int J Eng Sci 106:29–41Google Scholar
  15. Farokhi H, Ghayesh MH, Gholipour, Hussain Sh (2017) Motion characteristics of bilayered extensible Timoshenko microbeams. Int J Eng Sci 112:1–17Google Scholar
  16. Ghayesh MH (2012) Subharmonic dynamics of an axially accelerating beam. Arch Appl Mech 82(9):1169–1181Google Scholar
  17. Ghayesh MH (2017) Nonlinear dynamics of multilayered microplates. J Comput Nonlinear Dyn 13:021006–021012CrossRefGoogle Scholar
  18. Ghayesh MH (2018a) Dynamics of functionally graded viscoelastic microbeams. Int J Eng Sci 124:115–131Google Scholar
  19. Ghayesh MH (2018b) Nonlinear vibration analysis of axially functionally graded shear-deformable tapered beams. Appl Math Model 59:583–596Google Scholar
  20. Ghayesh MH (2018c) Functionally graded microbeams: simultaneous presence of imperfection and viscoelasticity. Int J Mech Sci 140:339–350Google Scholar
  21. Ghayesh MH, Moradian N (2011) Nonlinear dynamic response of axially moving, stretched viscoelastic strings. Arch Appl Mech 81:781–799 Google Scholar
  22. Ghayesh MH, Farokhi H (2015a) Chaotic motion of a parametrically excited microbeam. Int J Eng Sci 96:34–45MathSciNetCrossRefzbMATHGoogle Scholar
  23. Ghayesh MH, Farokhi H (2015b) Nonlinear dynamics of microplates. Int J Eng Sci 86:60–73Google Scholar
  24. Ghayesh MH, Yourdkhani M, Balar S, Reid T (2010) Vibrations and stability of axially traveling laminated beams. Appl Math Comput 217:545–556 Google Scholar
  25. Ghayesh MH, Kazemirad S, Darabi MA (2011) A general solution procedure for vibrations of systems with cubic nonlinearities and nonlinear/time-dependent internal boundary conditions. J Sound Vib 330:5382–5400Google Scholar
  26. Ghayesh MH, Kazemirad S, Reid T (2012) Nonlinear vibrations and stability of parametrically exited systems with cubic nonlinearities and internal boundary conditions: A general solution procedure. Appl Math Model 36(7):3299–3311Google Scholar
  27. Ghayesh MH, Amabili M, Farokhi H (2013a) Three-dimensional nonlinear size-dependent behaviour of Timoshenko microbeams. Int J Eng Sci 71:1–14MathSciNetCrossRefzbMATHGoogle Scholar
  28. Ghayesh MH, Amabili M, Farokhi H (2013b) Coupled global dynamics of an axially moving viscoelastic beam. Int J Non-linear Mech 51:54–74Google Scholar
  29. Ghayesh MH, Amabili M, Farokhi H (2013c) Nonlinear forced vibrations of a microbeam based on the strain gradient elasticity theory. Int J Eng Sci 63:52–60 Google Scholar
  30. Ghayesh MH, Farokhi H, Amabili M (2013d) Nonlinear dynamics of a microscale beam based on the modified couple stress theory. Compos B Eng 50:318–324CrossRefzbMATHGoogle Scholar
  31. Ghayesh MH, Farokhi H, Amabili M (2013e) Nonlinear behaviour of electrically actuated MEMS resonators. Int J Eng Sci 71:137–155CrossRefzbMATHGoogle Scholar
  32. Ghayesh MH, Paidoussis MP, Amabili M (2013f) Nonlinear dynamics of cantilevered extensible pipes conveying fluid. J Sound Vib 332:6405–6418Google Scholar
  33. Ghayesh MH, Farokhi H, Amabili M (2014) In-plane and out-of-plane motion characteristics of microbeams with modal interactions. Compos B Eng 60:423–439CrossRefGoogle Scholar
  34. Ghayesh MH, Farokhi H, Alici G (2016a) Size-dependent performance of microgyroscopes. Int J Eng Sci 100:99–111MathSciNetCrossRefzbMATHGoogle Scholar
  35. Ghayesh MH, Farokhi H, Hussain Sh (2016b) Viscoelastically coupled size-dependent dynamics of microbeams. Int J Eng Sci 109:243–255 Google Scholar
  36. Ghayesh MH, Farokhi H, Gholipour A (2017a) Vibration analysis of geometrically imperfect three-layered shear-deformable microbeams. Int J Mech Sci 122:370–383Google Scholar
  37. Ghayesh MH, Farokhi H, Gholipour A (2017b) Oscillations of functionally graded microbeams. Int J Eng Sci 110:35–53Google Scholar
  38. Ghayesh MH, Farokhi H, Gholipour A et al (2018) Nonlinear oscillations of functionally graded microplates. Int J Eng Sci 122:56–72CrossRefzbMATHGoogle Scholar
  39. Gholipour A, Farokhi H, Ghayesh MH (2015) In-plane and out-of-plane nonlinear size-dependent dynamics of microplates. Nonlinear Dyn 79:1771–1785CrossRefGoogle Scholar
  40. Gholipour A, Ghayesh MH, Zander A (2018a) Nonlinear biomechanics of bifurcated atherosclerotic coronary arteries. Int J Eng Sci 133:60–83Google Scholar
  41. Gholipour A, Ghayesh MH, Zander A, Mahajan R (2018b) Three-dimensional biomechanics of coronary arteries. Int J Eng Sci 130:93–114 Google Scholar
  42. Hari K, Verma SK, Praveen Krishna IR et al (2018) Out-of-plane dual flexure MEMS piezoresistive accelerometer with low cross axis sensitivity. Microsyst Technol 24(5):2437–2444CrossRefGoogle Scholar
  43. He M-X, Sun J-Q (2018) Multi-objective structural-acoustic optimization of beams made of functionally graded materials. Compos Struct 185:221–228CrossRefGoogle Scholar
  44. Liu F, Gao S, Niu S et al (2018) Optimal design of high-g MEMS piezoresistive accelerometer based on Timoshenko beam theory. Microsyst Technol 24(2):855–867CrossRefGoogle Scholar
  45. Rajaei A, Vahidi-Moghaddam A, Ayati M et al (2018) Integral sliding mode control for nonlinear damped model of arch microbeams. Microsyst Technol 25(1):57–68Google Scholar
  46. Saxena S, Sharma R, Pant BD (2017) Dynamic characterization of fabricated guided two beam and four beam cantilever type MEMS based piezoelectric energy harvester having pyramidal shape seismic mass. Microsyst Technol 23:5947–5958CrossRefGoogle Scholar
  47. Searle T, Yildirim T, Ghayesh MH et al (2018) Design, fabrication, and test of a coupled parametric-transverse nonlinearly broadband energy harvester. IEEE Trans Energy Convers 33:457–464CrossRefGoogle Scholar
  48. Shafiei N, Kazemi M, Ghadiri M (2016) Nonlinear vibration of axially functionally graded tapered microbeams. Int J Eng Sci 102:12–26MathSciNetCrossRefzbMATHGoogle Scholar
  49. Şimşek M (2015) Size dependent nonlinear free vibration of an axially functionally graded (AFG) microbeam using He’s variational method. Compos Struct 131:207–214CrossRefGoogle Scholar
  50. Tang Y, Lv X, Yang T (2019) Bi-directional functionally graded beams: asymmetric modes and nonlinear free vibration. Compos B Eng 156:319–331CrossRefGoogle Scholar
  51. Teh KS, Lin L (1999) Time-dependent buckling phenomena of polysilicon micro beams. Microelectron J 30:1169–1172CrossRefGoogle Scholar
  52. Tran N, Ghayesh MH, Arjomandi M (2018) Ambient vibration energy harvesters: a review on nonlinear techniques for performance enhancement. Int J Eng Sci 127:162–185MathSciNetCrossRefzbMATHGoogle Scholar
  53. Tuck K, Jungen A, Geisberger A et al (2005) A study of creep in polysilicon MEMS devices. J Eng Mater Technol 127:90–96CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Mechanical EngineeringUniversity of AdelaideAdelaideAustralia

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