A nonlocal Bernoulli–Euler p-version finite-element (p-FE) is developed to investigate nonlinear modes of vibration and to analyze internal resonances of beams with dimensions of a few nanometers. The time domain equations of motion are transformed to the frequency domain via the harmonic balance method (HBM), and then, the equations of motion are solved by an arc-length continuation method. After comparisons with published data on beams with rectangular cross section and on carbon nanotubes (CNTs), the study focuses on the nonlinear modes of vibration of CNTs. It is verified that the p-FE proposed, which keeps the advantageous flexibility of the FEM, leads to accurate discretizations with a small number of degrees-of-freedom. The first three nonlinear modes of vibration are studied and it is found that higher order modes are more influenced by nonlocal effects than the first mode. Several harmonics are considered in the harmonic balance procedure, allowing us to discover modal interactions due to internal resonances. It is shown that the nonlocal effects alter the characteristics of the internal resonances. Furthermore, it is demonstrated that, due to the internal resonances, the nonlocal effects are still noticeable at lengths that are longer than what has been previously found.
Issue Section:
Research Papers
References
1.
Eom
, K.
, Park
, H. S.
, Yoon
, D. S.
, and Kwon
, T.
, 2011
, “Nanomechanical Resonators and Their Applications in Biological/Chemical Detection: Nanomechanics Principles
,” Phys. Rep.
, 503
(4–5
), pp. 115
–163
.2.
Li
, C.
, Thostenson
, E. T.
, and Chou
, T. W.
, 2008
, “Sensors and Actuators Based on Carbon Nanotubes and Their Composites: A Review
,” Compos. Sci. Technol.
, 68
(6
), pp. 1227
–1249
.3.
Karabalin
, R. B.
, Masmanidis
, S. C.
, and Roukes
, M. L.
, 2010
, “Efficient Parametric Amplification in High and Very High Frequency Piezoelectric Nanoelectromechanical Systems
,” Appl. Phys. Lett.
, 97
(18
), p. 183101
.4.
Lazarus
, A.
, Thomas
, O.
, and Deü
, J. F.
, 2012
, “Finite Element Reduced Order Models for Nonlinear Vibrations of Piezoelectric Layered Beams With Applications to NEMS
,” Finite Elem. Anal. Des.
, 49
(1
), pp. 35
–51
.5.
Nguyen
, V. N.
, Baguet
, S.
, Lamarque
, C. H.
, and Dufour
, R.
, 2015
, “Bifurcation-Based Micro-/Nanoelectromechanical Mass Detection
,” Nonlinear Dyn.
, 79
(1
), pp. 647
–662
.6.
Gibson
, R. F.
, Ayorinde
, E. O.
, and Wen
, Y.-F.
, 2007
, “Vibrations of Carbon Nanotubes and Their Composites: A Review
,” Compos. Sci. Technol.
, 67
(1
), pp. 1
–28
.7.
Ansari
, R.
, and Sahmani
, S.
, 2012
, “Small Scale Effect on Vibrational Response of Single-Walled Carbon Nanotubes With Different Boundary Conditions Based on Nonlocal Beam Models
,” Commun. Nonlinear Sci. Numer. Simul.
, 17
(4
), pp. 1965
–1979
.8.
Yang
, F.
, Chong
, A. C. M.
, Lam
, D. C. C.
, and Tong
, P.
, 2002
, “Couple Stress Based Strain Gradient Theory for Elasticity
,” Int. J. Solids Struct.
, 39
(10
), pp. 2731
–2743
.9.
Eringen
, A. C.
, 1972
, “Linear Theory of Nonlocal Elasticity and Dispersion of Plane Waves
,” Int. J. Eng. Sci.
, 10
(5
), pp. 425
–435
.10.
Eringen
, A.
, 1983
, “On Differential Equations of Nonlocal Elasticity and Solutions of Screw Dislocation and Surface Waves
,” J. Appl. Phys.
, 54
(9
), p. 4703
.11.
Wang
, L. F.
, and Hu
, H. Y.
, 2005
, “Flexural Wave Propagation in Single-Walled Carbon Nanotubes
,” Phys. Rev. B
, 71
(19
), p. 195412
.12.
Reddy
, J. N.
, 2007
, “Nonlocal Theories for Bending, Buckling and Vibration of Beams
,” Int. J. Eng. Sci.
, 45
(2–8
), pp. 288
–307
.13.
Hu
, Y.-G.
, Liew
, K. M.
, and Wang
, Q.
, 2009
, “Nonlocal Elastic Beam Models for Flexural Wave Propagation in Double-Walled Carbon Nanotubes
,” J. Appl. Phys.
, 106
(4
), p. 044301
.14.
Yang
, J.
, Ke
, L. L.
, and Kitipornchai
, S.
, 2010
, “Nonlinear Free Vibration of Single-Walled Carbon Nanotubes Using Nonlocal Timoshenko Beam Theory
,” Physica E
, 42
(5
), pp. 1727
–1735
.15.
Arash
, B.
, and Wang
, Q.
, 2012
, “A Review on the Application of Nonlocal Elastic Models in Modeling of Carbon Nanotubes and Graphenes
,” Comput. Mater. Sci.
, 51
(1
), pp. 303
–313
.16.
Thai
, H.-T.
, 2012
, “A Nonlocal Beam Theory for Bending, Buckling, and Vibration of Nanobeams
,” Int. J. Eng. Sci.
, 52
, pp. 56
–64
.17.
Eltaher
, M. A.
, Alshorbagy
, A. E.
, and Mahmoud
, F. F.
, 2013
, “Vibration Analysis of Euler–Bernoulli Nanobeams by Using Finite Element Method
,” Appl. Math. Modell.
, 37
(7
), pp. 4787
–4797
.18.
Oliveira
, M. A. M.
, 2014
, “Vibrações de Nanoplacas
,” M.Sc. thesis, Universidade do Porto, Porto, Portugal.19.
Şimşek
, M.
, 2014
, “Large Amplitude Free Vibration of Nanobeams With Various Boundary Conditions Based on the Nonlocal Elasticity Theory
,” Composites, Part B
, 56
, pp. 621
–628
.20.
Ghayesh
, M. H.
, and Farokhi
, H.
, 2016
, “Coupled Nonlinear Dynamics of Geometrically Imperfect Shear Deformable Extensible Microbeams
,” ASME J. Comput. Nonlinear Dyn.
, 11
(4
), p. 041001
.21.
Houmat
, A.
, 2016
, “Nonlinear Free Vibration of Non-Prismatic Single-Walled Carbon Nanotubes by a Non-Local Shear Deformable Beam p-Element
,” Acta Mech.
, 227
(4
), pp. 1051
–1065
.22.
Ghoniem
, N. M.
, Busso
, E. P.
, Kioussis
, N.
, and Huang
, H.
, 2003
, “Multiscale Modelling of Nanomechanics and Micromechanics: An Overview
,” Philos. Mag.
, 83
(31–34
), pp. 3475
–3528
.23.
Lewandowski
, R.
, 1994
, “Nonlinear Free-Vibrations of Beams by the Finite-Element and Continuation Methods
,” J. Sound Vib.
, 170
(5
), pp. 577
–593
.24.
Ribeiro
, P.
, and Petyt
, M.
, 1999
, “Non-Linear Vibration of Beams With Internal Resonance by the Hierarchical Finite-Element Method
,” J. Sound Vib.
, 224
(4
), pp. 591
–624
.25.
Touzé
, C.
, Thomas
, O.
, and Huberdeau
, A.
, 2004
, “Asymptotic Non-Linear Normal Modes for Large-Amplitude Vibrations of Continuous Structures
,” Comput. Struct.
, 82
(31–32
), pp. 2671
–2682
.26.
Meirovitch
, L.
, and Baruh
, H.
, 1983
, “On the Inclusion Principle for the Hierarchical Finite Element Method
,” Int. J. Numer. Methods Eng. Struct.
, 19
(2
), pp. 281
–291
.27.
Stoykov
, S.
, and Ribeiro
, P.
, 2010
, “Nonlinear Forced Vibrations and Static Deformations of 3D Beams With Rectangular Cross Section: The Influence of Warping, Shear Deformation and Longitudinal Displacements
,” Int. J. Mech. Sci.
, 52
(11
), pp. 1505
–1521
.28.
Stoykov
, S.
, and Ribeiro
, P.
, 2013
, “Vibration Analysis of Rotating 3D Beams by the p-Version Finite Element Method
,” Finite Elem. Anal. Des.
, 65
, pp. 76
–88
.29.
Ribeiro
, P.
, 2009
, “Asymmetric Solutions in Large Amplitude Free Periodic Vibrations of Plates
,” J. Sound Vib.
, 322
(1–2
), pp. 8
–14
.30.
Stoykov
, S.
, and Ribeiro
, P.
, 2011
, “Nonlinear Free Vibrations of Beams in Space Due to Internal Resonance
,” J. Sound Vib.
, 330
(18–19
), pp. 4574
–4595
.31.
dos Santos
, J. V. A.
, and Reddy
, J. N.
, 2012
, “Vibration of Timoshenko Beams Using Non-Classical Elasticity Theories
,” Shock Vib.
, 19
(3
), pp. 251
–256
.32.
Fu
, Y. M.
, Hong
, J. W.
, and Wang
, X. Q.
, 2006
, “Analysis of Nonlinear Vibration for Embedded Carbon Nanotubes
,” J. Sound Vib.
, 296
, pp. 746
–756
.33.
Woinowsky-Krieger
, S.
, 1950
, “The Effect of an Axial Force on the Vibration of Hinged Bars
,” ASME J. Appl. Mech.
, 17
(1
), pp. 35
–36
.34.
Cheung
, Y. K.
, and Lau
, S. L.
, 1982
, “Incremental Time-Space Finite Strip Method for Non-Linear Structural Vibrations
,” Earthquake Eng. Struct. Dyn.
, 10
(2
), pp. 239
–253
.35.
Treacy
, M. M. J.
, Ebbesen
, T. W.
, and Gibson
, J. M.
, 1996
, “Exceptionally High Young's Modulus Observed for Individual Carbon Nanotubes
,” Nature
, 381
, pp. 678
–680
.36.
Wang
, Z. L.
, Gao
, R. P.
, Poncharal
, P.
, de Heer
, W. A.
, Dai
, Z. R.
, and Pan
, Z. W.
, 2001
, “Mechanical and Electrostatic Properties of Carbon Nanotubes and Nanowires
,” Mater. Sci. Eng. C
, 16
(1–2
), pp. 3
–10
.37.
Wang
, C. M.
, Tan
, V. B. C.
, and Zhang
, Y. Y.
, 2006
, “Timoshenko Beam Model for Vibration Analysis of Multi-Walled Carbon Nanotubes
,” J. Sound Vib.
, 294
(4–5
), pp. 1060
–1072
.38.
Li
, C. Y.
, and Chou
, T. W.
, 2003
, “A Structural Mechanics Approach for the Analysis of Carbon Nanotubes
,” Int. J. Solids Struct.
, 40
(10
), pp. 2487
–2499
.39.
Xiao
, J. R.
, Gama
, B. A.
, and Gillespie
, J. W.
, 2005
, “An Analytical Molecular Structural Mechanics Model for the Mechanical Properties of Carbon Nanotubes
,” Int. J. Solids Struct.
, 42
(11–12
), pp. 3075
–3092
.40.
Yoon
, J.
, Ru
, C. Q.
, and Mioduchowski
, A.
, 2003
, “Vibration of an Embedded Multiwall Carbon Nanotube
,” Compos. Sci. Technol.
, 63
(11
), pp. 1533
–1542
.41.
Wang
, Q.
, 2005
, “Wave Propagation in Carbon Nanotubes Via Nonlocal Continuum Mechanics
,” J. Appl. Phys.
, 98
(12
), p. 124301
.Copyright © 2017 by ASME
You do not currently have access to this content.
