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Research Papers

Nonlinear Dynamics of Carbon Nanotubes Under Large Electrostatic Force

[+] Author and Article Information
Tiantian Xu

Department of Mechanical Engineering,
State University of New York at Binghamton,
Binghamton, NY 13902
e-mail: txu2@binghamton.edu

Mohammad I. Younis

Department of Mechanical Engineering,
State University of New York at Binghamton,
Binghamton, NY 13902;
Department of Mechanical Engineering,
King Abdullah University of Science
and Technology KAUST,
Thuwal 23955-6900, Saudi Arabia
e-mails: myounis@binghamton.edu;
mohammad.younis@kaust.edu.sa

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received February 27, 2015; final manuscript received May 29, 2015; published online August 26, 2015. Assoc. Editor: Stefano Lenci.

J. Comput. Nonlinear Dynam 11(2), 021009 (Aug 26, 2015) (12 pages) Paper No: CND-15-1055; doi: 10.1115/1.4030830 History: Received February 27, 2015

Because of the inherent nonlinearities involving the behavior of carbon nanotubes (CNTs) when excited by electrostatic forces, modeling and simulating their behavior are challenging. The complicated form of the electrostatic force describing the interaction of their cylindrical shape, forming upper electrodes, to lower electrodes poises serious computational challenges. This presents an obstacle against applying and using several nonlinear dynamics tools typically used to analyze the behavior of complicated nonlinear systems undergoing large motion, such as shooting, continuation, and integrity analysis techniques. This work presents an attempt to resolve this issue. We present an investigation of the nonlinear dynamics of CNTs when actuated by large electrostatic forces. We study by expanding the complicated form of the electrostatic force into enough number of terms of the Taylor series. Then, we utilize this form along with an Euler–Bernoulli beam model to study for the first time the dynamic behavior of CNTs when excited by large electrostatic force. The geometric nonlinearity and the nonlinear electrostatic force are considered. An efficient reduced-order model (ROM) based on the Galerkin method is developed and utilized to simulate the static and dynamic responses of the CNTs. Several results are generated demonstrating softening and hardening behavior of the CNTs near their primary and secondary resonances. The effects of the DC and AC voltage loads on the behavior have been studied. The impacts of the initial slack level and CNT diameter are also demonstrated.

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Figures

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Fig. 1

Schematic of an initially curved CNT

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Fig. 2

Comparison of the Taylor series expansion and the exact form of the nondimensional electrostatic force

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Fig. 3

Percentage error of the Taylor series expansion of the nondimensional electrostatic force

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Fig. 4

Frequency-response curves of (a) 2 nm and (b) 5 nm slacked CNT showing hardening and softening behaviors, respectively. Results are shown for VDC = VAC = 1 V and a quality factor of 100.

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Fig. 5

Frequency-response curve of 2 nm slacked CNT. Shown for VDC = VAC = 1 V and a quality factor of 100.

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Fig. 6

Frequency-response curves of 5 nm slacked CNT. Shown for VDC = VAC = 1 V and a quality factor of 100.

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Fig. 7

Frequency-response curves for (a) VDC = 2 V, VAC = 1 V; (b) VDC = 3 V, VAC = 1 V; (c) VDC = 10 V, VAC = 1 V; (d) VDC = 20 V, VAC = 1 V; and (e) VDC = 30 V, VAC = 1 V. Results are shown for 5 nm slacked CNT and a quality factor of 100.

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Fig. 8

Frequency-response curves for (a) VDC = 1 V, VAC = 2 V; (b) VDC = 1 V, VAC = 3 V. Results are shown for 5 nm slacked CNT and a quality factor of 100.

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Fig. 9

Frequency-response curves for (a) VDC = 3 V, VAC = 1 V; (b) VDC = 3 V, VAC = 3 V. Results are shown for 5 nm slacked CNT and a quality factor of 100.

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Fig. 10

Frequency-response curves for (a) VDC = 10 V, VAC = 1 V; (b) VDC = 10 V, VAC = 5 V; (c) VDC = 10 V, VAC = 10 V. Results are shown for 5 nm slacked CNT and a quality factor of 100.

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Fig. 11

Frequency-response curves for (a) VDC = 30 V, VAC = 1 V; (b) VDC = 30 V, VAC = 10 V. Results are shown for 5 nm slacked CNT and a quality factor of 100.

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Fig. 12

Frequency-response curves for slack level (a) 5 nm, (b) 10 nm, and (c) 20 nm. Results are shown for VDC = 10 V, VAC = 1 V and a quality factor of 100.

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Fig. 13

Frequency-response curves for slack level 100 nm. Results are shown for VDC = 10 V, VAC = 10 V and a quality factor of 100.

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Fig. 14

Frequency-response curves for VDC = 1 V, VAC = 10 V. Results are shown for 100 nm slacked CNT and a quality factor of 100.

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Fig. 15

Frequency-response curves for (a) VDC = 10 V, VAC = 1 V; (b) VDC = 10 V, VAC = 10 V. Results are shown for 100 nm slacked CNT and a quality factor of 100.

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Fig. 16

Frequency-response curves for VDC = 30 V, VAC = 10 V. Results are shown for 100 nm slacked CNT and a quality factor of 100.

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