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Research Papers

Nonlinear Modes of Vibration and Internal Resonances in Nonlocal Beams

[+] Author and Article Information
Pedro Ribeiro

DEMec/INEGI,
Faculdade de Engenharia,
Universidade do Porto,
R. Dr. Roberto Frias, s/n,
Porto 4200-465, Portugal
e-mail: pmleal@fe.up.pt

Olivier Thomas

Arts et Métiers Paris Tech,
LSIS UMR CNRS 7296,
8 Boulevard Louis XIV,
Lille 59046, France
e-mail: Olivier.THOMAS@ensam.eu

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received March 4, 2016; final manuscript received October 14, 2016; published online January 11, 2017. Assoc. Editor: Sotirios Natsiavas.

J. Comput. Nonlinear Dynam 12(3), 031017 (Jan 11, 2017) (11 pages) Paper No: CND-16-1116; doi: 10.1115/1.4035060 History: Received March 04, 2016; Revised October 14, 2016

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.

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Figures

Grahic Jump Location
Fig. 6

Time histories (a) and phase plane plots (b) of transverse oscillations of point ξ = 0.5, local and nonlocal (e0a = 2 nm and ζ=0.04) CNTs, when ω/ωl2 = 1.163, and the nondimensional first harmonic amplitude is close to 0.33 (point P1 in Fig. 5)

Grahic Jump Location
Fig. 7

Shapes assumed by local (—) and nonlocal (◇, e0a = 2 nm, ζ=0.04) CNTs when ω/ωl2 = 1.163 and V1/d ≅ 0.33, point P1 in Fig. 5, along half a vibration cycle

Grahic Jump Location
Fig. 1

Transverse vibration displacement in the beginning of a vibration cycle of the CNT with L = 25 nm. The nonlocal parameter takes values: + e0a = 0 (ζ=0), ○ e0a = 1.0 nm (ζ=0.04), and × e0a = 2.0 nm (ζ=0.08):(a) first mode, (b) second mode, and (c) third mode.

Grahic Jump Location
Fig. 2

Transverse vibration displacement in the beginning of a vibration cycle, of the CNT with L = 50 nm. The nonlocal parameter takes values: + e0a = 0 (ζ=0), ○ e0a = 1.0 nm (ζ=0.02), and × e0a = 2.0 nm (ζ=0.04):(a) first mode, (b) second mode, and (c) third mode.

Grahic Jump Location
Fig. 3

Transverse vibration displacement in the beginning of a vibration cycle, of the CNT with L = 100 nm. The nonlocal parameter takes values: + e0a = 0 (ζ=0), ○ e0a = 1.0 nm (ζ=0.01), and × e0a = 2.0 nm (ζ=0.02).

Grahic Jump Location
Fig. 4

First nonlinear mode and bifurcations; the figures show amplitudes of harmonics of the CNT with L = 50 nm, when e0a = 0 nm □ and when e0a = 2.0 nm (ζ=0.04) : (a) first harmonic, (b) third harmonic, (c) detail of figure (b), and (d) fifth harmonic

Grahic Jump Location
Fig. 5

Amplitudes of harmonics of the CNT with L = 50 nm, second nonlinear mode and interactions, when e0a = 0 nm □ and when e0a = 2.0 nm (ζ=0.04) . The subfigures represent: (a)—the first harmonic, (b)—detail of figure (a), (c)—the third harmonic, and (d)—the fifth harmonic.

Grahic Jump Location
Fig. 8

Shapes assumed along half a vibration cycle by local (—) and nonlocal (◇, e0a = 2 nm, ζ=0.04) CNTs, solutions corresponding, respectively, to points P2 and P3 of Fig. 5

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