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Dynamic stability in parametric resonance of axially excited Timoshenko microbeams

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Abstract

The dynamic stability in parametric resonance of a Timoshenko microbeam subject to a time-dependent axial excitation load (comprised of a mean value along with time-dependent variations) is analysed in the subcritical regime. Based on the modified couple stress theory, continuous expressions for the elastic potential and kinetic energies are developed using kinematic and kinetic relations. The continuous model of the system is obtained via use of Hamilton’s principal. A model reduction procedure is carried out by applying the Galerkin scheme, in conjunction with an assumed-mode technique, yielding a high-dimensional second-order reduced-order model. A liner analysis is carried out upon the linear part of this model in order to obtain the linear natural frequencies and critical buckling loads. For the system in the subcritical regime, the parametric nonlinear responses are analysed by exciting the system at the principal parametric resonance in the first mode of transverse motion; this analysis is performed via use of a continuation technique, the Floquet theory, and a direct time integration method. Results are shown in the form of parametric frequency–responses, parametric force–responses, time traces, phase-plane diagrams, and fast Fourier transforms. The validity of the numerical simulations is tested via comparing our results, for simpler models for buckling response, with those given in the literature.

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References

  1. Li H, Piekarski B, DeVoe DL, Balachandran B (2008) Nonlinear oscillations of piezoelectric microresonators with curved cross-sections. Sens Actuators, A 144:194–200

    Article  Google Scholar 

  2. Yu Y, Wu B, Lim CW (2012) Numerical and analytical approximations to large post-buckling deformation of MEMS. Int J Mech Sci 55:95–103

    Article  Google Scholar 

  3. Farokhi H, Ghayesh MH (2015) Nonlinear dynamical behaviour of geometrically imperfect microplates based on modified couple stress theory. Int J Mech Sci 90:133–144

    Article  Google Scholar 

  4. Lam DCC, Yang F, Chong ACM, Wang J, Tong P (2003) Experiments and theory in strain gradient elasticity. J Mech Phys Solids 51:1477–1508

    Article  ADS  MATH  Google Scholar 

  5. Fleck NA, Muller GM, Ashby MF, Hutchinson JW (1994) Strain gradient plasticity: theory and experiment. Acta Metall Mater 42:475–487

    Article  Google Scholar 

  6. Ghayesh MH, Farokhi H, Amabili M (2014) In-plane and out-of-plane motion characteristics of microbeams with modal interactions. Compos B Eng 60:423–439

    Article  Google Scholar 

  7. Ghayesh M, Farokhi H, Amabili M (2013) Coupled nonlinear size-dependent behaviour of microbeams. Appl Phys A 112:329–338

    Article  ADS  MATH  Google Scholar 

  8. Kong S, Zhou S, Nie Z, Wang K (2008) The size-dependent natural frequency of Bernoulli–Euler micro-beams. Int J Eng Sci 46:427–437

    Article  MATH  Google Scholar 

  9. Asghari M, Ahmadian MT, Kahrobaiyan MH, Rahaeifard M (2010) On the size-dependent behavior of functionally graded micro-beams. Mater Des 31:2324–2329

    Article  Google Scholar 

  10. Akgöz B, Civalek Ö (2011) Strain gradient elasticity and modified couple stress models for buckling analysis of axially loaded micro-scaled beams. Int J Eng Sci 49:1268–1280

    Article  MathSciNet  Google Scholar 

  11. Akgöz B, Civalek Ö (2013) Free vibration analysis of axially functionally graded tapered Bernoulli–Euler microbeams based on the modified couple stress theory. Compos Struct 98:314–322

    Article  Google Scholar 

  12. Ghayesh MH, Farokhi H, Amabili M (2013) Nonlinear dynamics of a microscale beam based on the modified couple stress theory. Compos B Eng 50:318–324

    Article  MATH  Google Scholar 

  13. Ma HM, Gao XL, Reddy JN (2008) A microstructure-dependent Timoshenko beam model based on a modified couple stress theory. J Mech Phys Solids 56:3379–3391

    Article  ADS  MathSciNet  MATH  Google Scholar 

  14. Nateghi A, Salamat-talab M (2013) Thermal effect on size dependent behavior of functionally graded microbeams based on modified couple stress theory. Compos Struct 96:97–110

    Article  Google Scholar 

  15. Ansari R, Gholami R, Sahmani S (2011) Free vibration analysis of size-dependent functionally graded microbeams based on the strain gradient Timoshenko beam theory. Compos Struct 94:221–228

    Article  MATH  Google Scholar 

  16. Ansari R, Faghih Shojaei M, Gholami R, Mohammadi V, Darabi MA (2013) Thermal postbuckling behavior of size-dependent functionally graded Timoshenko microbeams. Int J Non-Linear Mech 50:127–135

    Article  Google Scholar 

  17. Ke L-L, Wang Y-S, Wang Z-D (2011) Thermal effect on free vibration and buckling of size-dependent microbeams. Physica E 43:1387–1393

    Article  ADS  Google Scholar 

  18. Ramezani S (2012) A micro scale geometrically non-linear Timoshenko beam model based on strain gradient elasticity theory. Int J Non-Linear Mech 47:863–873

    Article  Google Scholar 

  19. Asghari M, Kahrobaiyan MH, Ahmadian MT (2010) A nonlinear Timoshenko beam formulation based on the modified couple stress theory. Int J Eng Sci 48:1749–1761

    Article  MathSciNet  MATH  Google Scholar 

  20. Mohammad-Abadi M, Daneshmehr A (2014) Size dependent buckling analysis of microbeams based on modified couple stress theory with high order theories and general boundary conditions. Int J Eng Sci 74:1–14

    Article  MathSciNet  Google Scholar 

  21. Şimşek M, Reddy JN (2013) A unified higher order beam theory for buckling of a functionally graded microbeam embedded in elastic medium using modified couple stress theory. Compos Struct 101:47–58

    Article  Google Scholar 

  22. Ghayesh MH, Kazemirad S, Amabili M (2012) Coupled longitudinal-transverse dynamics of an axially moving beam with an internal resonance. Mech Mach Theory 52:18–34

    Article  Google Scholar 

  23. Reddy JN (2003) Mechanics of laminated composite plates and shells: theory and analysis, 2nd edn. Taylor & Francis, Abingdon

    Google Scholar 

  24. Yang F, Chong ACM, Lam DCC, Tong P (2002) Couple stress based strain gradient theory for elasticity. Int J Solids Struct 39:2731–2743

    Article  MATH  Google Scholar 

  25. Ghayesh MH, Amabili M, Farokhi H (2013) Three-dimensional nonlinear size-dependent behaviour of Timoshenko microbeams. Int J Eng Sci 71:1–14

    Article  MathSciNet  MATH  Google Scholar 

  26. Emam SA, Nayfeh AH (2009) Postbuckling and free vibrations of composite beams. Compos Struct 88:636–642

    Article  Google Scholar 

  27. Ghayesh MH, Farokhi H (2015) Internal energy transfer in dynamical behaviour of Timoshenko microarches. Math Comput Simul 112:28–39

    Article  ADS  MathSciNet  Google Scholar 

  28. Ghayesh MH (2012) Coupled longitudinal–transverse dynamics of an axially accelerating beam. J Sound Vib 331:5107–5124

    Article  ADS  Google Scholar 

  29. 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 36:3299–3311

    Article  MathSciNet  MATH  Google Scholar 

  30. Ghayesh MH (2010) Parametric vibrations and stability of an axially accelerating string guided by a non-linear elastic foundation. Int J Non-Linear Mech 45:382–394

    Article  Google Scholar 

  31. Ghayesh MH, Amabili M (2013) Steady-state transverse response of an axially moving beam with time-dependent axial speed. Int J Non-Linear Mech 49:40–49

    Article  MATH  Google Scholar 

  32. Park SK, Gao XL (2006) Bernoulli–Euler beam model based on a modified couple stress theory. J Micromech Microeng 16:2355

    Article  Google Scholar 

  33. Reddy JN (2007) Nonlocal theories for bending, buckling and vibration of beams. Int J Eng Sci 45:288–307

    Article  MATH  Google Scholar 

  34. Xia W, Wang L, Yin L (2010) Nonlinear non-classical microscale beams: static bending, postbuckling and free vibration. Int J Eng Sci 48:2044–2053

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The financial support to this research by the start-up grant of the University of Wollongong is gratefully acknowledged.

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Authors and Affiliations

  1. Department of Mechanical Engineering, McGill University, Montreal, QC, H3A 0C3, Canada

    Hamed Farokhi

  2. School of Mechanical, Materials and Mechatronic Engineering, University of Wollongong, Wollongong, NSW, 2522, Australia

    Mergen H. Ghayesh & Shahid Hussain

Authors
  1. Hamed Farokhi
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  2. Mergen H. Ghayesh
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  3. Shahid Hussain
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Correspondence to Mergen H. Ghayesh.

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Farokhi, H., Ghayesh, M.H. & Hussain, S. Dynamic stability in parametric resonance of axially excited Timoshenko microbeams. Meccanica 51, 2459–2472 (2016). https://doi.org/10.1007/s11012-016-0380-8

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  • DOI: https://doi.org/10.1007/s11012-016-0380-8

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