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Research Papers

Nonlinear Dynamics of Electrically Actuated Carbon Nanotube Resonators

[+] Author and Article Information
Hassen M. Ouakad

Department of Mechanical Engineering, State University of New York at Binghamton, Binghamton, NY 13902

Mohammad I. Younis

Department of Mechanical Engineering, State University of New York at Binghamton, Binghamton, NY 13902myounis@binghamton.edu

J. Comput. Nonlinear Dynam 5(1), 011009 (Dec 08, 2009) (13 pages) doi:10.1115/1.4000319 History: Received October 27, 2008; Revised June 17, 2009; Published December 08, 2009; Online December 08, 2009

This work presents an investigation of the nonlinear dynamics of carbon nanotubes (CNTs) when actuated by a dc load superimposed to an ac harmonic load. Cantilevered and clamped-clamped CNTs are studied. The carbon nanotube is described by an Euler–Bernoulli beam model that accounts for the geometric nonlinearity and the nonlinear electrostatic force. A reduced-order model based on the Galerkin method is developed and utilized to simulate the static and dynamic responses of the carbon nanotube. The free-vibration problem is solved using both the reduced-order model and by solving directly the coupled in-plane and out-of-plane boundary-value problems governing the motion of the nanotube. Comparison of the results generated by these two methods to published data of a more complicated molecular dynamics model shows good agreement. Dynamic analysis is conducted to explore the nonlinear oscillation of the carbon nanotube near its fundamental natural frequency (primary-resonance) and near one-half, twice, and three times its natural frequency (secondary-resonances). The nonlinear analysis is carried out using a shooting technique to capture periodic orbits combined with the Floquet theory to analyze their stability. The nonlinear resonance frequency of the CNTs is calculated as a function of the ac load. Subharmonic-resonances are found to be activated over a wide range of frequencies, which is a unique property of CNTs. The results show that these resonances can lead to complex nonlinear dynamics phenomena, such as hysteresis, dynamic pull-in, hardening and softening behaviors, and frequency bands with an inevitable escape from a potential well.

Copyright © 2010 by American Society of Mechanical Engineers
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Figures

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Figure 1

Schematic of an electrically actuated (a) clamped-clamped and (b) cantilever CNT resonators

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Figure 2

Variation of the normalized static deflection of the carbon nanotube with the dc voltage for case 2 of Table 1. In the figure: (○) one-mode ROM, (●) two-mode ROM, and ( ⋆) three-mode ROM.

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Figure 3

Variation of the normalized static deflection of clamped-clamped carbon nanotubes with the dc voltage, (a) without including midplane stretching and (b) with including midplane stretching. In the figure: (—) stable branch, (– –) unstable branch, and (○) results of Pugno (22).

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Figure 4

Variation of the normalized static deflection of cantilever carbon nanotubes with the dc voltage for (a) case 4 of Table 1 and (b) case 5 of Table 1. In the figure: ( ⋆) ROM (one mode, linear case), (◻) ROM (two modes, linear case), (○) ROM (two modes, nonlinear case), and (●) experimental data of (a) Ref. 42 and (b) Ref. 25.

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Figure 5

Variation of the fundamental natural frequency ω1 normalized with that at zero voltage ω0 for various values of dc voltage. The results are shown for the carbon nanotube of case 2 of Table 1. In the figure: (○) are the results obtained by solving the boundary-value problem directly and (—) are those obtained using a one-mode ROM.

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Figure 6

Variation of the fundamental natural frequency with the dc voltage for two clamped-clamped carbon nanotubes. (a) is for a CNT of length 20.7 nm and gap width 3 nm and (b) is for a CNT of length 20.7 nm and gap width 1 nm. In the figure: (○) ROM without including van der Waals forces, (◇) ROM with including van der Waals forces, and (●) results of Desquesnes (10).

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Figure 7

Variation of the normalized fundamental natural frequency ω1 with the dc voltage for clamped-clamped carbon nanotubes for case 1 (○), case 2 ( ⋆), and case 3 (●) of Table 1

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Figure 8

Variation of the fundamental natural frequency with the dc voltage for a cantilever carbon nanotube (case 4 of Table 1)

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Figure 9

Frequency-response curve of the CNT for case 2 of Table 1 showing the shift in the nonlinear resonance frequency in the hardening-behavior case. The results are shown for Vdc=5 V, Vac=2 V, and Q=100. In the figure, wmax=w(0.5,t) is the midpoint/maximum deflection of a clamped-clamped CNT, (—) shooting method (stable branch), (– –) shooting method (unstable branch), ( ⋆) long-time integration (one-mode ROM), and (○) long-time integration (two-mode ROM).

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Figure 10

Frequency-response curve of the CNT for case 4 of Table 1 showing the shift in the nonlinear resonance frequency in the softening-behavior case. The results are shown for Vdc=0.5 V, Vac=0.13 V, and Q=100. In the figure, wmax=w(1,t) is the tip/maximum deflection of a cantilever CNT, (—) stable branch, and (– –) unstable branch.

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Figure 11

The normalized nonlinear resonance frequency for two cases of CNTs calculated using the shooting technique for Q=100. (a) Case 2 of Table 1 for (●) Vdc=2 V and (○) Vdc=5 V; and (b) case 4 of Table 1 for (●) Vdc=2 V and (○) Vdc=5 V.

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Figure 12

Frequency-response curve of the clamped-clamped carbon nanotube of case 2 of Table 1 and for Vdc=25 V, Vac=5 V, and Q=100. In the figure: (—) stable branch and (– –) unstable branch.

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Figure 13

Frequency-response curve of the clamped-clamped carbon nanotube of case 3 of Table 1 and for Vdc=100 V, Vac=5 V, and Q=100. In the figure: (—) stable branch and (– –) unstable branch.

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Figure 14

The calculated instability tongues (inevitable escape bands) in the case of primary-resonance of a clamped-clamped carbon nanotube (a) hardening; for case 2 of Table 1 with Vdc=25 V, and (b) softening; for case 3 of Table 1 with Vdc=100 V and Q=100

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Figure 15

Frequency-response curve of a cantilever carbon nanotube for case 4 of Table 1 and for Vdc=1 V, Vac=0.1 V, and Q=100. In the figure, wmax=w(1,t) is the tip/maximum deflection of the carbon nanotube, (—) stable branch, and (– –) unstable branch.

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Figure 16

Frequency-response curve of the clamped-clamped carbon nanotube for the superharmonic-resonance of order 2 for case 2 of Table 1 for Vdc=20 V, Vac=16 V, and Q=150. In the figure: (—) stable branch and (– –) unstable branch.

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Figure 17

Frequency-response curve for the subharmonic-resonance of order 1/2 of the carbon nanotube of case 2 of Table 1 for Vdc=20 V, Vac=1.2 V, and Q=150. In the figure: (—) stable branch and (– –) unstable branch.

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Figure 18

Frequency-response curve for the subharmonic-resonance of order 1/2 of the carbon nanotube of case 2 of Table 1 for Vdc=20 V, Vac=16 V, and Q=150. In the figure: (—) stable branch and (– –) unstable branch.

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Figure 19

Frequency-response curve for the subharmonic-resonance of order 1/3 of the carbon nanotube of case 2 of Table 1 for Vdc=20 V, Vac=16 V, and Q=150. In the figure: (—) stable branch and (– –) unstable branch.

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Figure 20

The calculated instability tongues in the case of subharmonic-resonance of order 1/2 (a) hardening; for case 2 of Table 1 with Vdc=20 V, and (b) softening; for case 3 of Table 1 with Vdc=100 V and Q=150

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Figure 21

Frequency-response curves for the subharmonic-resonance of order 1/2 of the carbon nanotube for case 2 of Table 1 when Vdc=20 V and Q=150, and for (a) hardening; Vac=37 V and (b) softening; Vac=40 V. In the figure: (—) stable branch and (– –) unstable branch.

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Figure 22

Frequency-response curve for the subharmonic-resonance of order 1/2 of the cantilever carbon nanotube for case 4 of Table 1 when Vdc=1 V, Vac=0.2 V, and Q=150. In the figure: (—) stable branch and (– –) unstable branch.

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