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Resonance frequencies of size dependent perforated nonlocal nanobeam

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Abstract

Perforation is a common procedure in fabrication process of micro/nano-electromechanical systems (M/NEMS). Therefore, this paper presents an effort to study the resonance frequencies of size dependent regular square perforated nonlocal nanobeam, which not be discussed before. Equivalent characteristic parameters of perforated beam such as, bending stiffness, shear stiffness, mass, and rotary inertia are presented. The size-scale effect of long-range atomic interaction of nanobeam is described by using nonlocal differential form of Eringen model. Constitutive and governing equations of local and nonlocal perforated Timoshenko and Euler–Bernoulli nanobeam are derived. Analytical solution are exploited to solve the proposed model and derived closed form frequency equations as function of nanoscale and perforation parameters. The verification of current model is presented and compared with published works. Numerical results are illustrated to present the influences of length scale parameter, number of perforated holes, perforation size, and shear effects on the natural frequencies of both nanobeams theories. The obtained results are supportive in mechanical design of high-precision measurement nanobeams sensor and actuators.

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References

  • Aissani K, Bouiadjra MB, Ahouel M, Tounsi A (2015) A new nonlocal hyperbolic shear deformation theory for nanobeams embedded in an elastic medium. Struct Eng Mech 55(4):743–763

    Article  Google Scholar 

  • Apuzzo A et al (2017) Free vibrations of Bernoulli–Euler nano-beams by the stress-driven nonlocal integral model. Compos Part B Eng 123:105–111

    Article  Google Scholar 

  • Bourouina H, Yahiaoui R, Sahar A, Benamar MEA (2016) Analytical modeling for the determination of nonlocal resonance frequencies of perforated nanobeams subjected to temperature-induced loads. Phys E 75:163–168

    Article  Google Scholar 

  • Chen Y, Lee JD, Eskandarian A (2004) Atomistic viewpoint of the applicability of microcontinuum theories. Int J Solids Struct 41(8):2085–2097

    Article  MATH  Google Scholar 

  • Demir Ç, Civalek Ö (2017) A new nonlocal FEM via Hermitian cubic shape functions for thermal vibration of nano beams surrounded by an elastic matrix. Compos Struct 168:872–884

    Article  Google Scholar 

  • Deng CS, Peng HG, Gao YS, Zhong JX (2014) Ultrahigh-Q photonic crystal nanobeam cavities with H-shaped holes. Phys E 63:8–13

    Article  Google Scholar 

  • Deotare P (2012) Nanobeam cavities for reconfigurable photonics. Doctoral dissertation, Cambridge, USA

  • Deotare PB, Bulu I, Frank IW, Quan Q, Zhang Y, Ilic R, Loncar M (2012) All optical reconfiguration of optomechanical filters. Nat Commun 3:846. https://doi.org/10.1038/ncomms1830

  • Ebrahimi F, Barati MR (2018) Axial magnetic field effects on dynamic characteristics of embedded multiphase nanocrystalline nanobeams. Microsyst Technol 1–16. https://doi.org/10.1007/s00542-018-3771-z

  • Ebrahimi F, Daman M (2017) Dynamic characteristics of curved inhomogeneous nonlocal porous beams in thermal environment. Struct Eng Mech 64(1):121–133

    Google Scholar 

  • El-Sinawi AH, Bakri-Kassem M, Landolsi T, Awad O (2015) A novel comprehensive approach to feedback control of membrane displacement in radio frequency micro-electromechanical switches. Sens Actuators A 221:123–130

    Article  Google Scholar 

  • Eltaher MA, Emam SA, Mahmoud FF (2012) Free vibration analysis of functionally graded size-dependent nanobeams. Appl Math Comput 218(14):7406–7420

    MathSciNet  MATH  Google Scholar 

  • Eltaher MA, Alshorbagy AE, Mahmoud FF (2013a) Vibration analysis of Euler–Bernoulli nanobeams by using finite element method. Appl Math Model 37(7):4787–4797

    Article  MathSciNet  Google Scholar 

  • Eltaher MA, Alshorbagy AE, Mahmoud FF (2013b) Determination of neutral axis position and its effect on natural frequencies of functionally graded macro/nanobeams. Compos Struct 99:193–201

    Article  Google Scholar 

  • Eltaher MA, Hamed MA, Sadoun AM, Mansour A (2014) Mechanical analysis of higher order gradient nanobeams. Appl Math Comput 229:260–272

    MathSciNet  MATH  Google Scholar 

  • Eltaher MA, Khater ME, Emam SA (2016a) A review on nonlocal elastic models for bending, buckling, vibrations, and wave propagation of nanoscale beams. Appl Math Model 40(5):4109–4128

    Article  MathSciNet  Google Scholar 

  • Eltaher MA, El-Borgi S, Reddy JN (2016b) Nonlinear analysis of size-dependent and material-dependent nonlocal CNTs. Compos Struct 153:902–913

    Article  Google Scholar 

  • Eltaher MA, Kabeel AM, Almitani KH, Abdraboh AM (2017) Static bending and buckling of perforated nonlocal size-dependent nanobeams. Microsyst Technol. https://doi.org/10.1007/s00542-018-3905-3

  • Eringen AC (1972) Nonlocal polar elastic continua. Int J Eng Sci 10(1):1–16

    Article  MathSciNet  MATH  Google Scholar 

  • Eringen AC (1983) On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J Appl Phys 54(9):4703–4710

    Article  Google Scholar 

  • Eringen AC (2002) Nonlocal continuum field theories. Library of congress cataloging-in-publication data. Springer, New York

  • Fang DM, Li XH, Yuan Q, Zhang HX (2010) Effect of etch holes on the capacitance and pull-in voltage in MEMS tunable capacitors. Int J Electron 97(12):1439–1448

    Article  Google Scholar 

  • Ghadiri M, Shafiei N (2016) Nonlinear bending vibration of a rotating nanobeam based on nonlocal Eringen’s theory using differential quadrature method. Microsyst Technol 22(12):2853–2867

    Article  Google Scholar 

  • Huang Z (2012) Nonlocal effects of longitudinal vibration in nanorod with internal long-range interactions. Int J Solids Struct 49(15):2150–2154

    Article  Google Scholar 

  • Jeong KH, Amabili M (2006) Bending vibration of perforated beams in contact with a liquid. J Sound Vib 298(1):404–419

    Article  Google Scholar 

  • Joshi AY, Sharma SC, Harsha SP (2011) Zeptogram scale mass sensing using single walled carbon nanotube based biosensors. Sens Actuators A 168(2):275–280

    Article  Google Scholar 

  • Kaghazian A, Hajnayeb A, Foruzande H (2017) Free vibration analysis of a piezoelectric nanobeam using nonlocal elasticity theory. Struct Eng Mech 61(5):617–624

    Article  Google Scholar 

  • Khadem SE, Rasekh M, Toghraee A (2012) Design and simulation of a carbon nanotube-based adjustable nano-electromechanical shock switch. Appl Math Model 36(6):2329–2339

    Article  Google Scholar 

  • Luschi L, Pieri F (2012) A simple analytical model for the resonance frequency of perforated beams. Proc Eng 47:1093–1096

    Article  Google Scholar 

  • Luschi L, Pieri F (2014) An analytical model for the determination of resonance frequencies of perforated beams. J Micromech Microeng 24(5):055004

    Article  Google Scholar 

  • Luschi L, Pieri F (2016) An analytical model for the resonance frequency of square perforated Lamé-mode resonators. Sens Actuators B Chem 222:1233–1239

    Article  Google Scholar 

  • Miura R, Imamura S, Ohta R, Ishii A, Liu X, Shimada T, Iwamoto S, Arakawa Y, Kato YK (2014) Ultralow mode-volume photonic crystal nanobeam cavities for high-efficiency coupling to individual carbon nanotube emitters. Nat Commun 5:5580. https://doi.org/10.1038/ncomms6580

  • Mohite SS, Sonti VR, Pratap R (2008) A compact squeeze-film model including inertia, compressibility, and rarefaction effects for perforated 3-D MEMS structures. J Microelectromech Syst 17(3):709–723

    Article  Google Scholar 

  • Nagase T, Kawamura J, Pahlovy SA, Miyamoto I (2010) Ion beam fabrication of natural single crystal diamond nano-tips for potential use in atomic force microscopy. Microelectron Eng 87(5):1494–1496

    Article  Google Scholar 

  • Nguyen TN, Kim NI, Lee J (2017) Static behavior of nonlocal Euler–Bernoulli beam model embedded in an elastic medium using mixed finite element formulation. Struct Eng Mech 63(2):137–146

    Google Scholar 

  • Niknam H, Aghdam MM (2015) A semi analytical approach for large amplitude free vibration and buckling of nonlocal FG beams resting on elastic foundation. Compos Struct 119:452–462

    Article  Google Scholar 

  • Rasekh M, Khadem SE (2011) Pull-in analysis of an electrostatically actuated nano-cantilever beam with nonlinearity in curvature and inertia. Int J Mech Sci 53(2):108–115

    Article  Google Scholar 

  • Reddy JN (2007) Nonlocal theories for bending, buckling and vibration of beams. Int J Eng Sci 45(2–8):288–307

    Article  MATH  Google Scholar 

  • Shao L, Palaniapan M (2008) Effect of etch holes on quality factor of bulk-mode micromechanical resonators. Electron Lett 44(15):938–939

    Article  Google Scholar 

  • Sharma JN, Grover D (2011) Thermoelastic vibrations in micro-/nano-scale beam resonators with voids. J Sound Vib 330(12):2964–2977

    Article  Google Scholar 

  • Şimşek M (2014) Large amplitude free vibration of nanobeams with various boundary conditions based on the nonlocal elasticity theory. Compos B Eng 56:621–628

    Article  Google Scholar 

  • Taghizadeh M, Ovesy HR, Ghannadpour SAM (2015) Nonlocal integral elasticity analysis of beam bending by using finite element method. Struct Eng Mech 54(4):755–769

    Article  MATH  Google Scholar 

  • Tu C, Lee JEY (2012) Increased dissipation from distributed etch holes in a lateral breathing mode silicon micromechanical resonator. Appl Phys Lett 101(2):023504

    Article  Google Scholar 

  • Yakaiah B, Rao AS (2014) Higher order nonlocal strain gradient approach for wave characteristics of carbon nanorod. Nonlinear Anal 19(4):660–668

    Article  MathSciNet  Google Scholar 

  • Zhang X, Wang X, Kong W, Yi G, Jia J (2011) Tribological behavior of micro/nano-patterned surfaces in contact with AFM colloidal probe. Appl Surf Sci 258(1):113–119

    Article  Google Scholar 

Download references

Acknowledgements

This work was supported by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under Grant no. (D-031-135-1438). The authors, therefore, gratefully acknowledge the DSR technical and financial support.

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Authors and Affiliations

  1. Mechanical Engineering Department, Faculty of Engineering, King Abdulaziz University, P.O. Box 80204, Jeddah, Saudi Arabia

    M. A. Eltaher & K. H. Almitani

  2. Mechanical Design and Production Department, Faculty of Engineering, Zagazig University, P.O. Box 44519, Zagazig, Egypt

    M. A. Eltaher

  3. Physics Department, Faculty of Science, Banha University, Banha, Egypt

    A. M. Abdraboh

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  1. M. A. Eltaher
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  2. A. M. Abdraboh
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  3. K. H. Almitani
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Correspondence to M. A. Eltaher.

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Eltaher, M.A., Abdraboh, A.M. & Almitani, K.H. Resonance frequencies of size dependent perforated nonlocal nanobeam. Microsyst Technol 24, 3925–3937 (2018). https://doi.org/10.1007/s00542-018-3910-6

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