Free vibrations analysis of stepped nanobeams using nonlocal elasticity theory

Document Type : Research Note

Authors

1 Department of Electronic and Automation, Soma Vocational School, Manisa Celal Bayar University, 45500 Soma, Manisa, Turkiye

2 Department of Mechanical Engineering, Manisa Celal Bayar University Yunusemre, 45140 Manisa, Turkiye

3 Department of Machinery and Metal Technologies, Soma Vocational School, Manisa Celal Bayar University, 45500 Soma, Manisa, Turkiye

Abstract

Free vibration of stepped nanobeams was investigated using Eringen's nonlocal elasticity theory. Beam analysis is based on Bernoulli-Euler theory and nanoscale analysis is based on Eringen's nonlocal elasticity theory. The system boundary conditions were determined as simple-simple. The equations of motion of the system were obtained using Hamilton's principle. For the solution of the obtained state equations, a multi-time scale, which is one of the perturbation methods, was used. The results part of the study, it is aimed to observe the nano-size effect and the effects of the step state. For this purpose, the natural frequency values of the first three modes of the system were obtained for different non-local parameter values, step rates, and step positions. When the results were examined, it was determined that the non-local parameter value, step ratio, and natural frequency were inversely proportional to each other. In addition, to strengthen the accuracy of the results, the results obtained were compared with the results of other studies in the literature conducted under the specified conditions, and a perfect agreement was observed. The current beam model, on the other hand, could help design and manufacture ICs such as nano-sensors and nano-actuators.

Keywords

Main Subjects


References:
1.Wei, L., Kuai, X., Bao, Y., et al. “The recent progress ofmems/nems resonators”, Micromachines, 12(6), 724 (2021). https://doi.org/10.3390/mi12060724.
2.Forouzanfar, S., Pala, N., Madou, M., et al. “Perspectiveson C-MEMS and C-NEMS biotech applications”, Biosens Bioelectron, 180, 113119 (2021). https://doi.org/10.1016/j.bios.2021.113119.
3.Farag, N., Mattossovich, R., Merlo, R., et al. “Folding-upon-repair DNA nanoswitches for monitoring the activity of DNA repair enzymes”, Angew. Chemie - Int. Ed, 60(13), pp. 7359-7365 (2021). https://doi.org/10.1002/ange.202016223.
4.Afonin, S.M. “Rigidity of a multilayer piezoelectricactuator for the nano and micro range”, Russ. Eng. Res., 41, pp. 285-288 (2021). https://doi.org/10.3103/S1068798X21040031.
5. Arefi, M. “Analysis of a doubly curved piezoelectric nanoshell: Nonlocal electro-elastic bending solution”, Eur. J. Mech. A/Solids, 70, pp. 226-237 (2018). https://doi.org/10.1016/j.euromechsol.2018.02.012.
6.Shaat, M. and Abdelkefi, A. “New insights on theapplicability of Eringen’s nonlocal theory”, Int. J. Mech. Sci., (2017). https://doi.org/10.1016/j.ijmecsci.2016.12.013.
7.Liu, Y.P. and Reddy, J.N. “A nonlocal curved beam modelbased on a modified couple stress theory”, Int. J. Struct. Stab. Dyn., 11(03), pp. 495-512 (2011). https://doi.org/10.1142/S0219455411004233.
8.Li, Y.S. and Xiao, T. “Free vibration of the one-dimensional piezoelectric quasicrystal microbeams based on modified couple stress theory”, Appl. Math. Model, 96, pp. 733-750 (2021). https://doi.org/10.1016/j.apm.2021.03.028.
9.Faraji-Oskouie, M., Norouzzadeh, A., Ansari, R., et al.“Bending of small-scale Timoshenko beams based on the integral/differential nonlocal-micropolar elasticity theory: a finite element approach”, Appl. Math. Mech. English Ed., 40, pp. 767-782 (2019). https://doi.org/10.1007/s10483-019-2491-9.
10.Thai, C.H., Ferreira, A.J.M., Nguyen-Xuan, H., et al. “Asize dependent meshfree model for functionally graded plates based on the nonlocal strain gradient theory”, Compos. Struct., 272, 114169 (2021). https://doi.org/10.1016/j.compstruct.2021.114169.
11.Tocci Monaco, G., Fantuzzi, N., Fabbrocino, F., et al.“Hygro-thermal vibrations and buckling of laminated nanoplates via nonlocal strain gradient theory”, Compos. Struct., 262, 113337 (2021). https://doi.org/10.1016/j.compstruct.2020.113337.
12.Abdelrahman, A.A., Esen, I., Özarpa, C., et al.“Dynamics of perforated higher order nanobeams subject to moving load using the nonlocal strain gradient theory”, Smart Struct. Syst, 28(4), pp. 515-533 (2021). https://doi.org/10.12989/sss.2021.28.4.515.
13.Lu, L., Guo, X., and Zhao, J. “On the mechanics ofKirchhoff and Mindlin plates incorporating surface energy”, Int. J. Eng. Sci., 124, pp. 24-40 (2018). https://doi.org/10.1016/j.ijengsci.2017.11.020.
14.Eltaher, M.A., Abdelrahman, A.A., and Esen, I.,“Dynamic analysis of nanoscale Timoshenko CNTs based on doublet mechanics under moving load”, Eur. Phys. J. Plus, 136, 705 (2021). https://doi.org/10.1140/epjp/s13360-021-01682-8.
15. Cemal Eringen, A. “On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves”, J. Appl. Phys, 54, pp. 4703-4710 (1983). https://doi.org/10.1063/1.332803.
16. Dresselhaus, M.S., Dresselhaus, G., and Saito, R. “Physics of carbon nanotubes”, Carbon N. Y., 33(7), pp. 883-891 (1995). https://doi.org/10.1016/0008-6223(95)00017-8.
17. Zenkour, A.M. “Torsional dynamic response of a carbon nanotube embedded in Visco-Pasternak’s medium”, Math. Model. Anal, 21(6), pp. 852-868 (2016). https://doi.org/10.3846/13926292.2016.1248510.
18. Judy, J.W. “Microelectromechanical systems (MEMS): Fabrication, design and applications”, Smart Mater. Struct., 10(06), 1115 (2001). https://doi.org/10.1088/0964-1726/10/6/301.
19. Xie, W.C., Lee, H.P., and Lim, S.P. “Nonlinear dynamic analysis of MEMS switches by nonlinear modal analysis”, Nonlinear Dyn, 31, pp. 243-256 (2003). https://doi.org/10.1023/A:1022914020076.
20. Alkharabsheh, S.A. and Younis, M.I. “The dynamics of MEMS arches of non-ideal boundary conditions”, Proc. ASME Des. Eng. Tech. Conf., 54846, pp. 197-207 (2011). https://doi.org/10.1115/DETC2011-48501.
21. Fu, Y., Zhang, J., and Wan, L. “Application of the energy balance method to a nonlinear oscillator arising in the microelectromechanical system (MEMS)”, Curr. Appl. Phys, 11(3), pp. 482-485 (2011). https://doi.org/10.1016/j.cap.2010.08.037.
22. Ebrahimy, F. and Hosseini, S.H.S., “Nonlinear electroelastic vibration analysis of NEMS consisting of double-viscoelastic nanoplates”, Appl. Phys. A Mater. Sci. Process, 122, 922 (2016).https://doi.org/10.1007/s00339-016-0452-6.
23. Eltaher, M.A., Almalki, T.A., Almitani, K.H., et al., “Participation factor and vibration of carbon nanotube with vacancies”, J. Nano Res., 57, pp. 158-174 (2019). https://doi.org/10.4028/www.scientific.net/JNanoR.57.158.
24. Bornassi, S. and Haddadpour, H. “Nonlocal vibration and pull-in instability analysis of electrostatic carbon-nanotube based NEMS devices”, Sensors Actuators, A Phys, 266, pp. 185-196 (2017). https://doi.org/10.1016/j.sna.2017.08.020.
25. Sharma, J.N. and Grover, D. “Thermoelastic vibration analysis of Mems/Nems plate resonators with voids”, Acta Mech, 223, pp. 167-187 (2012). https://doi.org/10.1007/s00707-011-0557-0.
26. Ebrahimi, F. and Hosseini, S.H.S. “Double nanoplate-based NEMS under hydrostatic and electrostatic actuations”, Eur. Phys. J. Plus, 131, p. 160 (2016). https://doi.org/10.1140/epjp/i2016-16160-1.
27. Wong, E.W., Sheehan, P.E., and Lieber, C.M. “Nanobeam mechanics: Elasticity, strength, and toughness of nanorods and nanotubes”, Science, 80(5334), pp. 1971-1975 (1997). https://doi.org/10.1126/science.277.5334.1971.
28. Aydogdu, M. “A general nonlocal beam theory: Its application to nanobeam bending, buckling and vibration”, Phys. E Low-Dimensional Syst. Nanostructures, 41(9), pp. 1651-1655 (2009). https://doi.org/10.1016/j.physe.2009.05.014.
29. Akkoca, Ş., Bağdatli, S.M., And Toğun, N. “Ortadan mesnetli mikro kirişin doğrusal titreşim hareketleri”, Gazi Üniversitesi Mühendislik-Mimarlık Fakültesi Derg, 36(2), pp. 1089-1104 (2020). https://doi.org/10.17341/gazimmfd.734809.
30. Khaniki, H.B. “On vibrations of nanobeam systems”, Int. J. Eng. Sci, 124, pp. 85-103 (2018). https://doi.org/10.1016/j.ijengsci.2017.12.010.
31. Salari, E., Ashoori, A.R., Sadough Vanini, S.A., et al. “Nonlinear dynamic buckling and vibration of thermally post-buckled temperature-dependent FG porous nanobeams based on the nonlocal theory”, Phys. Scr, 97(8), 085216 (2022). https://doi.org/10.1088/14024896/ac81.
32. Ebrahimi, F. and Salari, E. “Effect of non-uniform temperature distributions on nonlocal vibration and buckling of inhomogeneous size-dependent beams”, Adv. Nano Res., 6(4), pp. 377-397 (2018). https://doi.org/10.12989/anr.2018.6.4.377.
33. Salari, E., Ashoori, A., and Vanini, S.A.S. “Porosity-dependent asymmetric thermal buckling of inhomogeneous annular nanoplates resting on elastic substrate”, Adv. Nano Res., 7(1), pp. 25-38 (2019). https://doi.org/10.12989/anr.2019.7.1.025.
34. Ebrahimi, F. and Salari, E. “Semi-Analytical vibration analysis of functionally graded size-dependent nanobeams with various boundary conditions”, Smart Struct. Syst, 19(3), pp. 243-257 (2017). https://doi.org/10.12989/sss.2017.19.3.243.
35. Ashoori, A.R., Vanini, S.A.S., and Salari, E. “Size-dependent axisymmetric vibration of functionally graded circular plates in bifurcation/limit point instability”, Appl. Phys. A Mater. Sci. Process, 123, p. 226 (2017). https://doi.org/10.1007/s00339-017-0825-5.
36. Salari, E., Vanini, S. A. S., and Ashoori, A. “Nonlinear thermal stability and snap-through buckling of temperature-dependent geometrically imperfect graded nanobeams on nonlinear elastic foundation”, Mater. Res. Express, 6(12), 1250j6 (2019). https://doi.org/10.1088/2053-1591/ab5e50.
37. Assadi, A. and Nazemizadeh, M. “Size-dependent vibration analysis of stepped nanobeams based on surface elasticity theory”, Int. J. Eng. Trans. C Asp, 34(3), pp. 744-749 (2021). https://doi.org/10.5829/ije.2021.34.03c.20.
38. Tekin, A. and Özkaya, E. “Ankastre mesnetli çok kademeli kirişlerin lineer enine titreşimleri”, Celal Bayar Univ. J. Sci, 3(2), pp. 143-152 (2007). https://dergipark.org.tr/en/pub/cbayarfbe/issue/4045/53327.
39.Lu, Z.R., Huang, M., Liu, J.K., et al. “Vibration analysisof multiple-stepped beams with the composite element model”, J. Sound Vib, 322(4-5), pp.1070-1080 (2009). https://doi.org/10.1016/j.jsv.2008.11.041.
40.Dong, X.J., Meng, G., Li, H.G., et al. “Vibration analysisof a stepped laminated composite Timoshenko beam”, Mech. Res. Commun, 32(5), pp. 572-581 (2005). https://doi.org/10.1016/j.mechrescom.2005.02.014.
41.Mao, Q. and Pietrzko, S. “Free vibration analysis ofstepped beams by using Adomian decomposition method”, Appl. Math. Comput, 217(7), pp. 3429-3441 (2010). https://doi.org/10.1016/j.amc.2010.09.010.
42.Mao, Q. “Design of piezoelectric modal sensor formultiple-stepped euler-bernoulli beams by using adomian decomposition method”, 20th Int. Congr. Sound Vib. 2013, ICSV 2013 (2013). 43.Shaat, M., Akbarzadeh Khorshidi, M., Abdelkefi, A., etal. “Modeling and vibration characteristics of cracked nano-beams made of nanocrystalline materials”, Int. J. Mech. Sci, 115–116, pp. 574-585 (2016). https://doi.org/10.1016/j.ijmecsci.2016.07.037.
44.Yoon, H.I., Son, I.S., and Ahn, S.J. “Free vibrationanalysis of Euler-Bernoulli beam with double cracks”, J. Mech. Sci. Technol, 21, pp. 476-485 (2007). https://doi.org/10.1007/BF02916309.
45.Lellep, J. and Lenbaum, A. “Free vibrations of steppednano-beams”, Int. J. Comput. Methods Exp. Meas., 6(4), pp. 716-725 (2018). https://doi.org/10.2495/CMEM-V6-N4-716-725.
46.Lellep, J. and Lenbaum, A. “Free vibrations of steppednano-beams with cracks”, Proc. Est. Acad. Sci., 71(1), pp. 103-116 (2022). https://doi.org/10.3176/proc.2022.1.09.
47.Hossain, M. and Lellep, J. “The effect of rotatory inertiaon natural frequency of cracked and stepped nanobeam”, Eng. Res. Express, 2(3), 035009 (2020). https://doi.org/10.1088/26318695/aba48b.
48.Taima, M.S., El-Sayed, T.A., and Farghaly, S.H. “Freevibration analysis of multistepped nonlocal Bernoulli–Euler beams using dynamic stiffness matrix method”, J. Vib. Control, 27(7–8), pp. 774–789 (2021). https://doi.org/10.1177/1077546320933470.
49.Loghmani, M. and Hairi Yazdi, M.R. “An analyticalmethod for free vibration of multi cracked and stepped nonlocal nanobeams based on wave approach”, Results Phys, 11, pp. 166-181 (2018). https://doi.org/10.1016/j.rinp.2018.08.046.
50.Yapanmiş, B.E., Bağdatli, S.M., and Togun, N.“Investigation of linear vibration behavior of middle supported nanobeam”, El-Cezeri J. Sci. Eng., 7(3), pp. 1450 -1459 (2020). https://doi.org/10.31202/ecjse.741269.
51.Oldac, and Olcay, “Nano teknolojide Yerel OlmayanÇubuk Teorisinin Statik ve Dinamik Problemleri”, İstanbul Teknik Üniversitesi (2016). https://polen.itu.edu.tr/items/e476edda-88c8-497a-a2cc-557b5552f1be.
52. Adhikari, S., Karličić, D., and Liu, X. “Dynamic stiffnessof nonlocal damped nano-beams on elastic foundation”, Eur. J. Mech. A/Solids, 86, pp. 104-144 (2021). https://doi.org/10.1016/j.euromechsol.2020.104144.
53.Milazzo, A., Benedetti, I., and Gulizzi, V. “Advancedmodels for nonlocal magneto-electro-elastic multilayered plates based on Reissner mixed variational theorem”, Mech. Adv. Mater. Struct, 28(11), pp. 1170-1186 (2021). https://doi.org/10.1080/15376494.2019.1647480.
54.Pradhan, S.C. and Phadikar, J.K. “Nonlocal elasticitytheory for vibration of nanoplates”, J. Sound Vib., 325(1-2), pp. 206-223 (2009). https://doi.org/10.1016/j.jsv.2009.03.007.
55.Eringen, A.C. “Edge dislocation in nonlocal elasticity”,Int. J. Eng. Sci., 15(3), pp. 177-183 (1977). https://doi.org/10.1016/0020-7225(77)90003-9.
56.Heireche, H., Tounsi, A., Benzair, A., et al. “Sound wavepropagation in single-walled carbon nanotubes using nonlocal elasticity”, Phys. E Low-Dimensional Syst. Nanostructures, 40(8), pp. 2791-2799 (2008). https://doi.org/10.1016/j.physe.2007.12.021.
57.Yang, J., Jia, X.L., and Kitipornchai, S. “Pull-ininstability of nano-switches using nonlocal elasticity theory”, J. Phys. D. Appl. Phys. (2008). https://doi.org/10.1088/0022-3727/41/3/035103.
58.Eringen, A. and Wegner, J., “Nonlocal continuum fieldtheories”, Appl. Mech. Rev, 56(2), pp. B20-B22 (2003). https://doi.org/10.1115/1.1553434.
59.Ibrahim, R. “Book Reviews : Nonlinear Oscillations:A.H. Nayfeh and D.T. Mook John Wiley & Sons, New York, 1979,” Shock Vib. Dig. (1981). https://books.google.com.tr/books?id=T0MzEQAAQBAJ&lpg=PA1&ots=wLlmgQlErh&dq=Nonlinear%20Oscillations%3A%20A.H.%20Nayfeh%20and%20D.T.%20Mook%20&lr&hl=tr&pg=PA1#v=onepage&q&f=false.
60.Eltaher, M.A., Alshorbagy, A.E., and Mahmoud, F.F.“Vibration analysis of Euler-Bernoulli nanobeams by using finite element method”, Appl. Math. Model, 37(7), pp. 4787-4797 (2013). https://doi.org/10.1016/j.apm.2012.10.016.
61.Bagdatli, S.M. “Non-linear vibration of nanobeams withvarious boundary condition based on nonlocal elasticity theory”, Compos. Part B Eng, 80, pp. 43-52 (2015). https://doi.org/10.1016/j.compositesb.2015.05.030.
62.Fan, Z., Kapadia, R., Leu, P.W., et al. “Ordered arrays ofdual-diameter nanopillars for maximized optical absorption”, Nano Lett, 10(10), pp. 3823–3827 (2010). https://doi.org/10.1021/nl1010788.
Volume 32, Issue 3
Transactions on Nanotechnology
January and February 2025 Article ID:7395
  • Receive Date: 20 December 2022
  • Revise Date: 05 July 2023
  • Accept Date: 17 July 2023