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Dynamic transverse vibration characteristics and vibro-buckling analyses of axially moving and rotating nanobeams based on nonlocal strain gradient theory

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Abstract

Dynamic transverse vibration characteristics and vibro-buckling analyses of axially moving nanobeam and rotating nanobeam based on nonlocal strain gradient theory are investigated. The axially moving model involves axial velocity in kinetic energy, and the effect of rotating is added as Coriolis acceleration to energy relation for rotating model. Governing equations of two dynamical models are derived by Hamilton’s principle. Meanwhile, corresponding classical and high order non-classical boundaries are also given. Numerical results are obtained by employing Galerkin approach for double simply supported boundary. Results for influence of axial velocity on transverse vibration characteristics illustrate that natural frequencies decrease with increasing axial velocity. However, frequencies of rotating nanobeam model are splitted into forward and backward frequencies with increasing rotating velocity. It is found that forward frequencies increase and backward frequencies decrease with elevated rotating velocity. It is revealed that the nonlocal effect weakens the rigidity and the strain gradient effect strengthens the rigidity. Finally, vibro-buckling analyses of two models, referring to fundamental frequencies vanishing for axially moving model and frequencies corresponding to rotating velocity for rotating model, are shown.

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References

  • Abbas BAH (1986) Dynamic stability of a rotating Timoshenko beam with a flexible root. J Sound Vib 108:25–32

    Article  Google Scholar 

  • 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 

  • Aydogdu M (2009) A general nonlocal beam theory: its application to nanobeam bending, buckling and vibration. Phys E 41:1651–1655

    Article  Google Scholar 

  • Bourlon B, Glattli DC, Miko C, Forró L, Bachtold A (2004) Carbon nanotube based bearing for rotational motions. Nano Lett 4:709–712

    Article  Google Scholar 

  • Chen L-Q (2005) Analysis and control of transverse vibrations of axially moving strings. Appl Mech Rev 58:91–116

    Article  Google Scholar 

  • Chen J, Guo J, Pan E (2017) Reflection and transmission of plane wave in multilayered nonlocal magneto-electro-elastic plates immersed in liquid. Compos Struct 162:401–410

    Article  Google Scholar 

  • Dehrouyeh-Semnani AM, Dehrouyeh M, Zafari-Koloukhi H, Ghamami M (2015) Size-dependent frequency and stability characteristics of axially moving microbeams based on modified couple stress theory. Int J Eng Sci 97:98–112

    Article  MathSciNet  Google Scholar 

  • Ding H, Chen L-Q (2010) Galerkin methods for natural frequencies of high-speed axially moving beams. J Sound Vib 329:3484–3494

    Article  Google Scholar 

  • Drexler KE (1992) Nanosystems: molecular machinery, manufacturing, and computation. Wiley, Hoboken

    Google Scholar 

  • Ebrahimi F, Barati MR (2016) Wave propagation analysis of quasi-3D FG nanobeams in thermal environment based on nonlocal strain gradient theory. Appl Phys A 122:843

    Article  Google Scholar 

  • Ebrahimi F, Barati MR, Dabbagh A (2016) A nonlocal strain gradient theory for wave propagation analysis in temperature-dependent inhomogeneous nanoplates. Int J Eng Sci 107:169–182

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Eringen AC, Edelen D (1972) On nonlocal elasticity. Int J Eng Sci 10:233–248

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  Google Scholar 

  • Fung R-F, Lu P-Y, Tseng C-C (1998) Non-linearly dynamic modelling of an axially moving beam with a tip mass. J Sound Vib 218:559–571

    Article  Google Scholar 

  • 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 

  • Ghayesh MH, Balar S (2010) Non-linear parametric vibration and stability analysis for two dynamic models of axially moving Timoshenko beams. Appl Math Model 34:2850–2859

    Article  MathSciNet  MATH  Google Scholar 

  • Guo X, Wang Z (2010) Thermoelastic coupling vibration characteristics of the axially moving beam with frictional contact. J Vib Acoust 132:051010

    Article  Google Scholar 

  • Guo S, He Y, Liu D, Lei J, Shen L, Li Z (2016) Torsional vibration of carbon nanotube with axial velocity and velocity gradient effect. Int J Mech Sci 119:88–96

    Article  Google Scholar 

  • Han J, Globus A, Jaffe R, Deardorff G (1997) Molecular dynamics simulations of carbon nanotube-based gears. Nanotechnology 8:95–102

    Article  Google Scholar 

  • Hoa SV (1979) Vibration of a rotating beam with tip mass. J Sound Vib 67:369–381

    Article  MATH  Google Scholar 

  • Hosseini-Hashemi S, Ilkhani M (2016a) Nonlocal modeling for dynamic stability of spinning nanotube under axial load. Meccanica 52:1107–1121

    Article  MathSciNet  MATH  Google Scholar 

  • Hosseini-Hashemi S, Ilkhani M (2016b) Exact solution for free vibrations of spinning nanotube based on nonlocal first order shear deformation shell theory. Compos Struct 157:1–11

    Article  Google Scholar 

  • Hosseini-Hashemi S, Khaniki HB (2016) Dynamic behavior of multi-layered viscoelastic nanobeam system embedded in a viscoelastic medium with a moving nanoparticle. J Mech 91:1–17

    Google Scholar 

  • Ilkhani MR, Hosseini-Hashemi S (2016) Size dependent vibro-buckling of rotating beam based on modified couple stress theory. Compos Struct 143:75–83

    Article  Google Scholar 

  • Kazemirad S, Ghayesh MH, Amabili M (2013) Thermo-mechanical nonlinear dynamics of a buckled axially moving beam. Arch Appl Mech 83:25–42

    Article  MATH  Google Scholar 

  • Khaniki HB, Hosseini-Hashemi Sh (2017a) The size-dependent analysis of multilayered microbridge systems under a moving load/mass based on the modified couple stress theory. Eur Phys J Plus 132(5):1–18

    Google Scholar 

  • Khaniki HB, Hosseini-Hashemi Sh (201b7) Dynamic response of biaxially loaded double-layer viscoelastic orthotropic nanoplate system under a moving nanoparticle. Int J Eng Sci 115:51–72

    Article  MathSciNet  Google Scholar 

  • Kiani K (2013) Longitudinal, transverse, and torsional vibrations and stabilities of axially moving single-walled carbon nanotubes. Curr Appl Phys 13:1651–1660

    Article  Google Scholar 

  • Kiani K (2014) Longitudinal and transverse instabilities of moving nanoscale beam-like structures made of functionally graded materials. Compos Struct 107:610–619

    Article  Google Scholar 

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

    Article  MATH  Google Scholar 

  • Lee SY, Kuo YH (1992) Bending vibrations of a rotating non-uniform beam with an elastically restrained root. J Sound Vib 154:441–451

    Article  MATH  Google Scholar 

  • Lei J, He Y, Zhang B, Liu D, Shen L, Guo S (2015) A size-dependent FG micro-plate model incorporating higher-order shear and normal deformation effects based on a modified couple stress theory. Int J Mech Sci 104:8–23

    Article  Google Scholar 

  • Lei J, He Y, Guo S, Li Z, Liu D (2016) Size-dependent vibration of nickel cantilever microbeams: experiment and gradient elasticity. AIP Adv 6:105202

    Article  Google Scholar 

  • Li L, Hu Y (2015) Buckling analysis of size-dependent nonlinear beams based on a nonlocal strain gradient theory. Int J Eng Sci 97:84–94

    Article  MathSciNet  Google Scholar 

  • Li L, Hu Y, Ling L (2015) Flexural wave propagation in small-scaled functionally graded beams via a nonlocal strain gradient theory. Compos Struct 133:1079–1092

    Article  Google Scholar 

  • Li X, Li L, Hu Y, Ding Z, Deng W (2017) Bending, buckling and vibration of axially functionally graded beams based on nonlocal strain gradient theory. Compos Struct 165:250–265

    Article  Google Scholar 

  • Lim CW, Li C, Yu J-L (2010) Dynamic behaviour of axially moving nanobeams based on nonlocal elasticity approach. Acta Mech Sin 26:755–765

    Article  MathSciNet  MATH  Google Scholar 

  • Lim CW, Li C, Yu J (2012) Free torsional vibration of nanotubes based on nonlocal stress theory. J Sound Vib 331:2798–2808

    Article  Google Scholar 

  • Lim C, Zhang G, Reddy J (2015) A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation. J Mech Phys Solids 78:298–313

    Article  MathSciNet  MATH  Google Scholar 

  • Lin SC, Hsiao KM (2001) Vibration analysis of a rotating Timoshenko beam. J Sound Vib 240:303–322

    Article  MATH  Google Scholar 

  • Liu D, He Y, Tang X, Ding H, Hu P, Cao P (2012) Size effects in the torsion of microscale copper wires: experiment and analysis. Scripta Mater 66:406–409

    Article  Google Scholar 

  • Liu J, Li C, Fan X, Tong L (2017) Transverse free vibration and stability of axially moving nanoplates based on nonlocal elasticity theory. Appl Math Model 45:65–84

    Article  MathSciNet  Google Scholar 

  • Marynowski K (2014) Axially moving microscale panel model based on modified couple stress theory. J Nanomechanics Micromechanics 5:A4014002

    Article  Google Scholar 

  • Mote C Jr (1972) Dynamic stability of axially moving materials. Shock Vib Dig 4:2–11

    Article  Google Scholar 

  • Ni Q, Li M, Tang M, Wang L (2014) Free vibration and stability of a cantilever beam attached to an axially moving base immersed in fluid. J Sound Vib 333:2543–2555

    Article  Google Scholar 

  • Öz H, Pakdemirli M (1999) Vibrations of an axially moving beam with time-dependent velocity. J Sound Vib 227:239–257

    Article  Google Scholar 

  • Pellicano F, Vestroni F (2002) Complex dynamics of high-speed axially moving systems. J Sound Vib 258:31–44

    Article  Google Scholar 

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

    Article  MATH  Google Scholar 

  • Rezaee M, Lotfan S (2015) Non-linear nonlocal vibration and stability analysis of axially moving nanoscale beams with time-dependent velocity. Int J Mech Sci 96:36–46

    Article  Google Scholar 

  • Schilhansl MJ (1958) Bending frequency of a rotating cantilever beam. ASME J Appl Mech 25:28–30

    MathSciNet  MATH  Google Scholar 

  • Şimşek M (2016) Nonlinear free vibration of a functionally graded nanobeam using nonlocal strain gradient theory and a novel Hamiltonian approach. Int J Eng Sci 105:12–27

    Article  MathSciNet  Google Scholar 

  • Sinha SK (2007) Combined torsional-bending-axial dynamics of a twisted rotating Cantilever Timoshenko beam with contact-impact loads at the free end. J Appl Mech 74:505–522

    Article  MATH  Google Scholar 

  • Stylianou M, Tabarrok B (1994) Finite element analysis of an axially moving beam, Part II: stability analysis. J Sound Vib 178:455–481

    Article  Google Scholar 

  • Thai H-T (2012) A nonlocal beam theory for bending, buckling, and vibration of nanobeams. Int J Eng Sci 52:56–64

    Article  MathSciNet  Google Scholar 

  • Thai H-T, Vo TP (2012) A nonlocal sinusoidal shear deformation beam theory with application to bending, buckling, and vibration of nanobeams. Int J Eng Sci 54:58–66

    Article  MathSciNet  Google Scholar 

  • Torkaman-Asadi M, Rahmanian M, Firouz-Abadi R (2015) Free vibrations and stability of high-speed rotating carbon nanotubes partially resting on Winkler foundations. Compos Struct 126:52–61

    Article  Google Scholar 

  • Wang Q (2005) Wave propagation in carbon nanotubes via nonlocal continuum mechanics. J Appl Phys 98:124301

    Article  Google Scholar 

  • Wang Q, Wang C (2007) The constitutive relation and small scale parameter of nonlocal continuum mechanics for modelling carbon nanotubes. Nanotechnology 18:075702

    Article  Google Scholar 

  • Wang C, Zhang Y, Ramesh SS, Kitipornchai S (2006) Buckling analysis of micro-and nano-rods/tubes based on nonlocal Timoshenko beam theory. J Phys D Appl Phys 39:3904

    Article  Google Scholar 

  • Wickert J, Mote C (1990) Classical vibration analysis of axially moving continua. J Appl Mech 57:738–744

    Article  MATH  Google Scholar 

  • Xu XJ, Wang XC, Zheng ML, Ma Z (2016) Bending and buckling of nonlocal strain gradient elastic beams. Compos Struct 160:366–377

    Article  Google Scholar 

  • Xu-**a G, Zhong-Min W (2010) Thermoelastic coupling vibration characteristics of the axially moving beam with frictional contact. J Vib Acoust 132:051010

    Article  Google Scholar 

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

    Article  MATH  Google Scholar 

  • Yang JB, Jiang LJ, Chen DC (2004) Dynamic modelling and control of a rotating Euler–Bernoulli beam. J Sound Vib 274:863–875

    Article  MathSciNet  MATH  Google Scholar 

  • Yao G, Zhang Y (2016) Reliability and sensitivity analysis of an axially moving beam. Meccanica 51:491–499

    Article  MathSciNet  MATH  Google Scholar 

  • Yao G, Zhang Y-M, Li C-Y, Yang Z (2016) Stability analysis and vibration characteristics of an axially moving plate in aero-thermal environment. Acta Mech 227:3517–3527

    Article  MathSciNet  Google Scholar 

  • Yigit A, Scott RA, Ulsoy AG (1988) Flexural motion of a radially rotating beam attached to a rigid body. J Sound Vib 121(2):201–210

    Article  MathSciNet  MATH  Google Scholar 

  • Young TH (1991) Dynamic resppnse of a pretwisted, tapered beam with non-constant rotating speed. J Sound Vib 150:435–446

    Article  Google Scholar 

  • Yu M-F, Lourie O, Dyer MJ, Moloni K, Kelly TF, Ruoff RS (2000) Strength and breaking mechanism of multiwalled carbon nanotubes under tensile load. Science 287:637–640

    Article  Google Scholar 

  • Zhang B, He Y, Liu D, Gan Z, Shen L (2013a) A non-classical Mindlin plate finite element based on a modified couple stress theory. Eur J Mech A/Solids 42:63–80

    Article  MathSciNet  Google Scholar 

  • Zhang B, He Y, Liu D, Gan Z, Shen L (2013b) A novel size-dependent functionally graded curved mircobeam model based on the strain gradient elasticity theory. Compos Struct 106:374–392

    Article  Google Scholar 

  • Zhang B, He Y, Liu D, Gan Z, Shen L (2014) Non-classical Timoshenko beam element based on the strain gradient elasticity theory. Finite Elem Anal Des 79:22–39

    Article  MathSciNet  Google Scholar 

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Acknowledgements

This work was financially supported by the National Natural Science Foundation of China (No. 11472114), and the Fundamental Research Funds for the Central Universities, HUST (No. 2015QN138).

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Authors and Affiliations

  1. Department of Mechanics, Huazhong University of Science and Technology, Wuhan, 430074, China

    Song Guo, Yuming He, Dabiao Liu, Jian Lei & Zhenkun Li

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  1. Song Guo
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  2. Yuming He
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Correspondence to Yuming He.

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Guo, S., He, Y., Liu, D. et al. Dynamic transverse vibration characteristics and vibro-buckling analyses of axially moving and rotating nanobeams based on nonlocal strain gradient theory. Microsyst Technol 24, 963–977 (2018). https://doi.org/10.1007/s00542-017-3441-6

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