Research PaperNo Access

Nonlinear Vibration Movements of the Mid-Supported Micro-Beam

Şevki Akkoca

Department of Mechanical Engineering, Manisa Celal Bayar University, 45140 Yunusemre, Manisa, Turkey

E-mail Address: sevkiakkoca@gmail.com

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Süleyman Murat Bağdatli

Department of Mechanical Engineering, Manisa Celal Bayar University, 45140 Yunusemre, Manisa, Turkey

E-mail Address: murat.bagdatli@cbu.edu.tr

Corresponding author.

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, and

Necla Kara Toğun

Department of Mechanical Engineering, Gaziantep University, 27310 Şehitkamil, Gaziantep, Turkey

E-mail Address: nkara@gantep.edu.tr

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https://doi.org/10.1142/S0219455422501747Cited by:3 (Source: Crossref)

Abstract

This study analyzes the vibration movements of multi-support micro beams placed in an electrically smooth area using the modified couple stress theory. It has been assumed that the potential voltage that creates the electrical field strength varies harmonically. Large number of experiments in recent years have indicated that classical continuum theory is unable to predict the mechanical behavior of microstructure with small size. However, nonclassical continuum theory should be used to accurately design and analyze the microstructures. Modified couple stress theory models the micro and nanomechanical systems with higher accuracy because they employ additional material parameters to the equation considering size dependent behavior. The most general nonlinear motion equations for multi-support microbeams have been obtained by considering the material size parameter, the number of support and support positions, damping effect, axial stresses, electrical field strength, and nonlinear effects resulting from elongations. The nonlinear equations of motion are obtained according to the Hamilton method using the modified couple stress theory (MCST). The resulting equations of motion are nondimensionalized. In this way, the mathematical model has been made independent of the type and geometric structure of the material. Approximate solutions of the obtained dimensionless motion equation are obtained by the multi-scale method, which is one of the perturbation methods. As a result, an increase occurs in the first mode frequencies ( $ω_{1}$ $ω_{1}$ ) and nonlinear correction effect parameters ( $λ (ω_{1})$ $λ (ω_{1})$ ) with the progress of the center support position gradually towards $η = 0.5$ $η = 0.5$ and the increase of the microbeam elasticity coefficient ( $α_{2}$ $α_{2}$ ).

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References

1. M. Kandaz and H. Dal, A comparative study of modified strain gradient theory and modified couple stress theory for gold microbeams. Arch. Appl. Mech. 88 (2018) 2051–2070. Crossref, Web of Science, Google Scholar
2. A. M. Dehrouyeh-Semnani and M. Nikkhah-Bahrami, A discussion on incorporating the Poisson effect in microbeam models based on modified couple stress theory, Int. J. Eng. Sci. 86 (2014) 20–25, http://dx.doi.org/10.1016/j.ijengsci.2014.10.003. Crossref, Web of Science, Google Scholar
3. R. Damghanian, K. Asemi and M. Babaei, A new beam element for static, free and forced vibration responses of microbeams resting on viscoelastic foundation based on modified couple stress and third-order beam theories, Iran. J. Sci. Technol. Trans. Mech. Eng. 46 (2022) 131–147, https://doi.org/10.1007/s40997-020-00407-z. Crossref, Web of Science, Google Scholar
4. F. Ebrahimi and M. R. Barati, Axial magnetic field effects on dynamic characteristics of embedded multiphase nanocrystalline nanobeams, Microsyst. Technol. 24 (2018) 3521–3536, https://doi.org/10.1007/s00542-018-3771-z. Crossref, Web of Science, Google Scholar
5. Z. T. Beni and Y. T. Beni, Dynamic stability analysis of size-dependent viscoelastic/piezoelectric nano-beam, Int. J. Struct. Stab. Dyn. 22 (2022) 2250050. Link, Web of Science, Google Scholar
6. E. R. Estabragh and G. H. Baradaran, Large amplitude free vibration analysis of nanobeams based on modified couple stress theory, Int. J. Struct. Stab. Dyn. 21(9) (2021) 2150129. Link, Web of Science, Google Scholar
7. M. Simsek, Dynamic analysis of an embedded microbeam carrying a moving microparticle based on the modified couple stress theory, Int. J. Eng. Sci. 48 (2010) 1721–1732. Crossref, Web of Science, Google Scholar
8. X. Wang, C. Li, Y. Zhang, Z. Said, S. Debnath, S. Sharma, M. Yang and T. Gao, Influence of texture shape and arrangement on nanofluid minimum quantity lubrication turning, The Int. J. Adv. Manuf. Technol. 119 (2022) 631–646, https://doi.org/10.1007/s00170-021-08235-4. Crossref, Web of Science, Google Scholar
9. X. Xiao, G. Bu, Z. Ou and Z. Li, Nonlinear in-plane instability of the confined FGP arches with nanocomposites reinforcement under radially-directed uniform pressure, Eng. Struct. 252 (2022) 113670, https://doi.org/10.1016/j.engstruct.2021.113670. Crossref, Web of Science, Google Scholar
10. M. Arefi, E. M. R. Bidgoli and T. Rabczuk, Effect of various characteristics of graphene nanoplatelets on thermal buckling behavior of FGRC micro plate based on MCST, Eur. J. Mech. A Solids 77 (2019) 103802, https://doi.org/10.1016/j.euromechsol.2019.103802. Crossref, Web of Science, Google Scholar
11. M. Arefi, M. Kiani and A. M. Zenkour, Size-dependent free vibration analysis of a three-layered exponentially graded nano-/micro-plate with piezomagnetic face sheets resting on Pasternak’s foundation via MCST, J. Sandwich Struct. Meter. 22(1) (2017) 55–86, https://doi.org/10.1177/1099636217734279. Crossref, Web of Science, Google Scholar
12. M. Arefi, Size-dependent bending behavior of three-layered doubly curved shells: Modified couple stress formulation, J. Sandwich Struct. Meter. 22(7) (2020) 2210–2249, https://doi.org/10.1177/1099636218793993. Crossref, Web of Science, Google Scholar
13. M. L. Dehsaraji, M. Arefi and A. Loghman, Size dependent free vibration analysis of functionally graded piezoelectric micro/nano shell based on modified couple stress theory with considering thickness stretching effect, Def. Technol. 17(1) (2021) 119–134, https://doi.org/10.1016/j.dt.2020.01.001. Crossref, Web of Science, Google Scholar
14. M. Arefi and M. Kiani, Magneto-electro-mechanical bending analysis of three-layered exponentially graded microplate with piezomagnetic face-sheets resting on Pasternak’s foundation via MCST, Mech. Adv. Mater. Struct. 27(5) (2020) 383–395, https://doi.org/10.1080/15376494.2018.1473538. Crossref, Web of Science, Google Scholar
15. L. Wang, W. Liu and H. Dai, Dynamics and instability of current-carrying microbeams in a longitudinal magnetic field, Physica E Low Dimens. Syst. Nanostruct. 66 (2014) 87–92. Crossref, Web of Science, Google Scholar
16. M. A. Attia and S. A. Emam, Electrostatic nonlinear bending, buckling and free vibrations of viscoelastic microbeams based on the modified couple stress theory, Acta Mech. 229 (2018) 3235–3255. Crossref, Web of Science, Google Scholar
17. M. Vaezi, M. M. Shirbani and A. Hajnayeb, Free vibration analysis of magneto-electro-elastic microbeams subjected to magneto-electric loads, Physica E Low Dimens. Syst. Nanostruct. 75 (2016) 280–286. Crossref, Web of Science, Google Scholar
18. A. Ghanbari, A. Babaei and F. Vakili-Tahamic, Free vibration analysis of micro beams based on the modified couple stress theory, using approximate methods, Int. J. Eng. & Technol. Sci. 3(2) (2015) 136–143. Google Scholar
19. H. B. Khaniki and S. H. Hashemi, Free vibration analysis of nonuniform microbeams based on modified couple stress theory: An analytical solution, Int. J. Eng. 30 (2017) 311–320. Google Scholar
20. M. K. Khabaz, S. A. Eftekhari and M. Hashemian, Free vibration analysis of sandwich micro beam with piezoelectric based on modified couple stress theory and surface effects, J. Simul. Anal. Nov. Technol. Mech. Eng. 10(4) (2018) 0033–0048. Google Scholar
21. R. Sourki and S. H. Hoseini, Free vibration analysis of size-dependent cracked microbeam based on the modified couple stress theory, Appl. Phys. A 122 (2016) 413. Crossref, Web of Science, Google Scholar
22. B. Habibi, Y. T. Beni and F. Mehralian, Free vibration of magneto-electro-elastic nanobeams based on modified couple stress theory in thermal environment, Mech. Adv. Mater. Struct. 26(7) (2019) 601–613. Crossref, Web of Science, Google Scholar
23. Z. Saadatnia, H. Askari and E. Esmailzadeh, Multi-frequency excitation of microbeams supported by Winkler and Pasternak foundations, J. Vib. Control 24 (2018) 2894–2911. Crossref, Web of Science, Google Scholar
24. M. H. Ghayesh, H. Farokhi and M. Amabili, Nonlinear dynamics of a micro scale beam based on the modified couple stress theory, Compos. B 50 (2013) 318–324. Crossref, Web of Science, Google Scholar
25. H. L. Dai, Y. K. Wang and L. Wang, Nonlinear dynamics of cantilevered microbeams based on modified couple stress theory, Int. J. Eng. Sci. 94 (2015) 103–112. Crossref, Web of Science, Google Scholar
26. Y. G. Wang, W. H. Lin and N. Liu, Nonlinear free vibration of a microscale beam based on modified couple stress theory, Physica E 47 (2012) 80–85. Crossref, Google Scholar
27. M. Simsek, Nonlinear static and free vibration analysis of microbeams based on the nonlinear elastic foundation using modified couple stress theory and He’s variational method, Compos. Struct. 112 (2014) 264–272. Crossref, Web of Science, Google Scholar
28. M. Hadian, K. Torabi and S. H. Jazi, Nonlinear vibration analysis of an elastically connected double-non-classical Timoshenko microbeam subject to moving particle based on the modified couple stress theory, J. Braz. Soc. Mech. Sci. Eng. 42 (2020) 246. Crossref, Web of Science, Google Scholar
29. V. H. Dang, Q. D. Le and K. T. Nguyen, Nonlinear vibration of microbeams under magnetic field using the modified couple stress theory, Asian Res. J. Math. 12(1) (2019) 1–14. Crossref, Google Scholar
30. A. M. D. Semnani, M. Dehrouyeh, H. Z. Koloukhi and M. Ghamami, Size-dependent frequency and stability characteristics of axially moving microbeams based on modified couple stress theory, Int. J. Eng. Sci. 97 (2015) 98–112. Crossref, Web of Science, Google Scholar
31. N. Ding, X. Xu, Z. Zheng and E. Li, Size-dependent nonlinear dynamics of a microbeam based on the modified couple stress theory, Acta Mech. 228 (2017) 3561–3579. Crossref, Web of Science, Google Scholar
32. L. Wang, Y. Y. Xu and Q. Ni, Size-dependent vibration analysis of three-dimensional cylindrical microbeams based on modified couple stress theory: A unified treatment, Int. J. Eng. Sci. 68 (2013) 1–10. Crossref, Web of Science, Google Scholar
33. M. Mohammadimehr, M. Mehrabi, H. Hadizadeh and H. Hadizadeh, Surface and size dependent effects on static, buckling, and vibration of micro composite beam under thermo-magnetic fields based on strain gradient theory, Steel Compos. Struct. 26(4) (2018) 513–531, https://doi.org/10.12989/scs.2018.26.4.513. Web of Science, Google Scholar
34. Z. Heidary and A. Mojra, Numerical study of thermal effect on the stability of carbon nanotubes resting on a viscoelastic foundation subjected to magnetic field, Int. J. Struct. Stab. Dyn. 22(2) (2022) 2250024. Link, Web of Science, Google Scholar
35. S. Kural and E. Özkaya, Size-dependent vibrations of a micro beam conveying fluid and resting on an elastic foundation, J. Vib. Control 23(7) (2015) 1106–1114. Crossref, Web of Science, Google Scholar
36. F. Yang, A. C. M. Chong, D. C. C. Lam and P. Tong, Couple stress based strain gradient theory for elasticity, Int. J. Solids Struct. 39(10) (2002) 2731–2743. Crossref, Web of Science, Google Scholar
37. S. K. Park and X.-L. Gao, Bernoulli–Euler beam model based on a modified couple stress theory, J. Micromech. Microeng. 16(11) (2006) 2355–2359. Crossref, Web of Science, Google Scholar
38. H. M. Ma, X.-L. Gao and J. N. Reddy, A microstructure-dependent Timoshenko beam model based on a modified couple stress theory, J. Mech. Phys. Solids 56 (2008) 3379–3391. Crossref, Web of Science, Google Scholar
39. S. Ahangar, G. Rezazadeh, R. Shabani, G. Ahmadi and A. Toloei, On the stability of microbeam conveying fluid considering modified couple stress theory, Int. J. Mech. Mater. Des. 7(4) (2011) 327–342. Crossref, Web of Science, Google Scholar
40. G. Y. Zhang, X. L. Gao and S. R. Ding, Band gaps for wave propagation in 2-D periodic composite structures incorporating microstructure effects, Acta Mech. 229 (2018), 4199–4214. Crossref, Web of Science, Google Scholar
41. A. H. Nayfeh and D. T. Mook, Nonlinear Oscillations (John Wiley, New York, 1979). Google Scholar
42. S. M. Bagdatli, H. R. Öz and E. Özkaya, Non-linear transverse vibrations and 3:1 internal resonances of a tensioned beam on multiple supports, Math. Comput. Appl. 16(1) (2011) 203–215. Google Scholar
43. N. Togun and S. M. Bağdatli, Size dependent nonlinear vibration of the tensioned nanobeam based on the modified couple stress theory, Compos. B 97 (2016) 255–262. Crossref, Web of Science, Google Scholar
44. M. H. Kahrobaiyan, M. Asghari, M. Hoore and M. T. Ahmadian, Nonlinear size-dependent forced vibrational behavior of microbeams based on a non-classical continuum theory. J. Vib. Control 18(5) (2012) 696–711, http://doi.org/10.1177/1077546311414600. Crossref, Web of Science, Google Scholar