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Vibration analysis of multiple-layer microbeams based on the modified couple stress theory: analytical approach

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Abstract

The modified couple stress theory (MCST) is used to capture size effect on dynamic response in multiple-layer microbeams in the present article. Governing equations of the system are obtained based on the MCST and using Hamilton’s principle. The natural frequencies of the multiple-layer microbeam are calculated using the analytical method. Then, the results of the natural frequencies are presented with respect to different values of the system parameters such as the geometric layers and also the dimensionless material length-scale parameter. The results show that the material length-scale parameter values and also the length, width, and thickness of each layer are extremely effective on the vibration characteristic of the multiple-layer microbeams.

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References

  1. Rezazadeh, G., Tahmasebi, A., Zubstov, M.: Application of piezoelectric layers in electrostatic MEM actuators: controlling of pull-in voltage. Microsyst. Technol. 12(12), 1163–1170 (2006)

    Article  Google Scholar 

  2. Hu, Y.C., Chang, C.M., Huang, S.C.: Some design considerations on the electrostatically actuated microstructures. Sens. Actuat. A 112(1), 155–161 (2004)

    Article  Google Scholar 

  3. Mahdavi, M.H., Farshidianfar, A., Tahani, M., Mahdavi, S., Dalir, H.: A more comprehensive modeling of atomic force microscope cantilever. Ultramicroscopy 109(1), 54–60 (2008)

    Article  Google Scholar 

  4. Lun, F.Y., Zhang, P., Gao, F.B., Jia, H.G.: Design and fabrication of micro-optomechanical vibration sensor. Microfabr. Technol. 120(1), 61–64 (2006)

    Google Scholar 

  5. McMahan, L.E., Castleman, B.W.: Characterization of vibrating beam sensors during shock and vibration. In: Position Location and Navigation Symposium. PLANS 2004. IEEE, pp. 102–110 (2004)

  6. Coutu, R.A., Kladitis, P.E., Starman, L.A., Reid, J.R.: A comparison of micro-switch analytic, finite element, and experimental results. Sens. Actuat. A 115(2), 252–258 (2004)

    Article  Google Scholar 

  7. Duc, T.C., Creemer, J.F., Sarro, P.M.: Piezoresistive cantilever beam for force sensing in two dimensions. Sens. J. IEEE 7, 96–104 (2007)

    Article  Google Scholar 

  8. Abdel-Rahman, E.M., Younis, M.I., Nayfeh, A.H.: Characterization of the mechanical behavior of an electrically actuated microbeam. J. Micromech. Microeng. 12(6), 759–766 (2002)

    Article  Google Scholar 

  9. Zand, M.M., Ahmadian, M.T.: Vibrational analysis of electrostatically actuated microstructures considering nonlinear effects. Commun. Nonlinear Sci. Numer. Simul. 14(4), 1664–1678 (2009)

    Article  Google Scholar 

  10. Orhan, S.: Analysis of free and forced vibration of a cracked cantilever beam. NDT & E Int. 40, 443–450 (2007)

    Article  Google Scholar 

  11. Barad, K.H., Sharma, D.S., Vyas, V.: Crack detection in cantilever beam by frequency based method. Proc. Eng. 51, 770–775 (2013)

    Article  Google Scholar 

  12. Lam, D.C., Yang, F., Chong, A.C.M., Wang, J., Tong, P.: Experiments and theory in strain gradient elasticity. J. Mech. Phys. Solids 51(8), 1477–1508 (2003)

    Article  MATH  Google Scholar 

  13. Fleck, N.A., Muller, G.M., Ashby, M.F., Hutchinson, J.W.: Strain gradient plasticity: theory and experiment. Acta Metall. Mater. 42(2), 475–487 (1994)

    Article  Google Scholar 

  14. Stölken, J.S., Evans, A.G.: A microbend test method for measuring the plasticity length scale. Acta Mater. 46(14), 5109–5115 (1998)

    Article  Google Scholar 

  15. Lam, D.C., Chong, A.C.: Indentation model and strain gradient plasticity law for glassy polymers. J. Mater. Res. 14(09), 3784–3788 (1999)

    Article  Google Scholar 

  16. Chong, A.C., Lam, D.C.: Strain gradient plasticity effect in indentation hardness of polymers. J. Mater. Res. 14(10), 4103–4110 (1999)

    Article  Google Scholar 

  17. Eringen, A.C.: Nonlocal polar elastic continua. Int. J. Eng. Sci. 10(1), 1–16 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  18. Mindlin, R.D., Tiersten, H.F.: Effects of couple-stresses in linear elasticity. Arch. Ration. Mech. Anal. 11(1), 415–448 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  19. Yang, F.A.C.M., Chong, A.C.M., Lam, D.C., Tong, P.: Couple stress based strain gradient theory for elasticity. Int. J. Solids Struct. 39(10), 2731–2743 (2002)

    Article  MATH  Google Scholar 

  20. Liang, L.N., Ke, L.L., Wang, Y.S., Yang, J., Kitipornchai, S.: Flexural vibration of an atomic force microscope cantilever based on modified couple stress theory. Int. J. Struct. Stab. Dyn. 15(07), 1540025 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  21. Park, S.K., Gao, X.L.: Bernoulli–Euler beam model based on a modified couple stress theory. J. Micromech. Microeng. 16(11), 2355–2359 (2006)

    Article  Google Scholar 

  22. Dai, H.L., Wang, Y.K., Wang, L.: Nonlinear dynamics of cantilevered microbeams based on modified couple stress theory. Int. J. Eng. Sci. 94, 103–112 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  23. Ma, H.M., Gao, X.L., Reddy, J.N.: A microstructure-dependent Timoshenko beam model based on a modified couple stress theory. J. Mech. Phys. Solids 56(12), 3379–3391 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  24. Ghiasi, E.K.: Application of modified couple stress theory to study dynamic characteristics of electrostatically actuated micro-beams resting upon squeeze-film damping under mechanical shock. Int. J. Adv. Mech. Eng. 6(1), 1–15 (2016)

    MathSciNet  Google Scholar 

  25. Asghari, M., Kahrobaiyan, M.H., Rahaeifard, M., Ahmadian, M.T.: Investigation of the size effects in Timoshenko beams based on the couple stress theory. Arch. Appl. Mech. 81(7), 863–874 (2011)

    Article  MATH  Google Scholar 

  26. Simsek, M., Aydın, M.: Size-dependent forced vibration of an imperfect functionally graded (FG) microplate with porosities subjected to a moving load using the modified couple stress theory. Compos. Struct. 160, 408–421 (2017)

    Article  Google Scholar 

  27. He, D., Yang, W., Chen, W.: A size-dependent composite laminated skew plate model based on a new modified couple stress theory. Acta Mech. Solida Sin. 30(1), 75–86 (2017)

    Article  Google Scholar 

  28. Guo, J., Chen, J., Pan, E.: Free vibration of three-dimensional anisotropic layered composite nanoplates based on modified couple-stress theory. Physica E 87, 98–106 (2017). https://doi.org/10.1016/j.physe.2016.11.025

    Article  Google Scholar 

  29. Akbas, S.D.: Free vibration of edge cracked functionally graded microscale beams based on the modified couple stress theory. Int. J. Struct. Stab. Dyn. 17(03), 1750033 (2017)

    Article  MathSciNet  Google Scholar 

  30. Alinaghizadeh, F., Shariati, M., Fish, J.: Bending analysis of size-dependent functionally graded annular sector microplates based on the modified couple stress theory. Appl. Math. Model. (2017). https://doi.org/10.1016/j.apm.2017.02.018

    Article  MathSciNet  MATH  Google Scholar 

  31. Askari, A.R., Tahani, M.: Size-dependent dynamic pull-in analysis of geometric non-linear micro-plates based on the modified couple stress theory. Physica E 86, 262–274 (2017). https://doi.org/10.1016/j.physe.2016.10.035

    Article  Google Scholar 

  32. Ghayesh, M.H., Farokhi, H.: Nonlinear dynamics of microplates. Int. J. Eng. Sci. 86, 60–73 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  33. Shoaib, M., Hisham, N., Basheer, N., Tariq, M.: Frequency analysis of electrostatic cantilever-based MEMS sensor. In: Symposium on Design, Test, Integration and Packaging of MEMS/MOEMS (DTIP), pp. 1–6 (2015)

  34. Shoaib, M., Hisham, N., Basheer, N., Tariq, M.: Frequency and displacement analysis of electrostatic cantilever-based MEMS sensor. In: Analog Integrated Circuits and Signal Processing, pp. 1–11 (2016)

  35. Shoaib, M., Hamid, N.H., Jan, M.T., Ali, N.B.Z.: Effects of crack faults on the dynamics of piezoelectric cantilever-based MEMS sensor. IEEE Sens. J. 17(19), 6279–6294 (2017)

    Article  Google Scholar 

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Authors and Affiliations

  1. Faculty of Mechanical and Energy Engineering, Shahid Beheshti University, Tehran, Iran

    Abbas Rahi

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Rahi, A. Vibration analysis of multiple-layer microbeams based on the modified couple stress theory: analytical approach. Arch Appl Mech 91, 23–32 (2021). https://doi.org/10.1007/s00419-020-01795-z

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