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Research Papers

Scale Effect on Tension-Induced Intermodal Coupling in Nanomechanical Resonators

[+] Author and Article Information
Kai-Ming Hu

State Key Laboratory of Mechanical System
and Vibration,
School of Mechanical Engineering,
Shanghai Jiao Tong University,
800 Dongchuan Road,
Shanghai 200240, China
e-mail hukaiming@sjtu.edu.cn

Wen-Ming Zhang

State Key Laboratory of Mechanical System
and Vibration,
School of Mechanical Engineering,
Shanghai Jiao Tong University,
800 Dongchuan Road,
Shanghai 200240, China
e-mail: wenmingz@sjtu.edu.cn

Xing-Jian Dong

State Key Laboratory of Mechanical System
and Vibration,
School of Mechanical Engineering,
Shanghai Jiao Tong University,
800 Dongchuan Road,
Shanghai 200240, China
e-mail: donxij@sjtu.edu.cn

Zhi-Ke Peng

State Key Laboratory of Mechanical System
and Vibration,
School of Mechanical Engineering,
Shanghai Jiao Tong University,
800 Dongchuan Road,
Shanghai 200240, China
e-mail: z.peng@sjtu.edu.cn

Guang Meng

State Key Laboratory of Mechanical System
and Vibration,
School of Mechanical Engineering,
Shanghai Jiao Tong University,
800 Dongchuan Road,
Shanghai 200240, China
e-mail: gmeng@sjtu.edu.cn

1Corresponding author.

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received June 4, 2014; final manuscript received October 21, 2014; published online December 11, 2014. Assoc. Editor: Jeffrey F. Rhoads.

J. Vib. Acoust 137(2), 021008 (Apr 01, 2015) (14 pages) Paper No: VIB-14-1204; doi: 10.1115/1.4029004 History: Received June 04, 2014; Revised October 21, 2014; Online December 11, 2014

Scale effect on the tension-induced intermodal coupling between the flexural modes in nanomechanical resonators is investigated. Based on the nonlocal theory of elasticity, a theoretical model is developed to depict the scale effect on the intermodal coupling in nanomechanical resonators. The experimental and theoretical validations suggest that the results of the present work are in agreement with the experimental data. The tuning effects of mode coupling on the pull-in voltage and resonant frequency of the doubly clamped beam with the scale effect are analyzed in detail. The results show that the coupling between in-plane and out-of-plane modes increases as the scale reduces since the scale effect could make the energy between mechanical modes transfer more easily. The mode coupling with scale effect can increase the tuning range of the pull-in voltages and positions. The contributions of each term included by the scale effect to the coupling strength, pull-in voltages and frequencies of nanoresonators are discussed. Furthermore, approximate critical formulae are obtained to predict the scale effect on the resonant frequency of nanoresonators. The work demonstrates that the scale effect should be taken into account for the further understanding of the coupling mechanism of nanoresonators.

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References

Figures

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Fig. 1

The schematic of a doubly clamped beam resonator and the semi-infinite electrodes: (a) 3D perspective and (b) side perspective

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Fig. 2

(a) A differential element of the beam with bending in two directions, (b) free body diagram in y direction, and (c) free body diagram in z direction

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Fig. 3

Variations of in-plane and out-of-plane frequencies fI and fO with respect to the dc voltage Vdc1, and the corresponding mode-mixing regions are shown in the insets

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Fig. 4

Scale effect on the coupling strength between the in-plane and out-of-plane modes with respect to (a) the normalized nonlocal parameter μ for different static deflections and (b) the length of the beams for four representative nonlocal parameters

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Fig. 5

Variations of pull-in voltages of the doubly clamped beam versus the normalized nonlocal parameter with different terms included by scale effect under A2 = 0.1 for Vp1 and A1 = 0.1 for Vp2 and N0 = 35.6 nN

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Fig. 6

Variations of pull-in voltages of the doubly clamped beam versus the normalized nonlocal parameter with different preloads N0 under A2 = 0.1 for Vp1 and A1 = 0.1 for Vp2

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Fig. 7

(a) Dispersion curves of the pull-in voltages Vp1 in y direction as a function of the nondimensional in-plane static deflection A1 with different nonlocal parameters, note that μ = 0.00 denotes scale effect is not taken into account (c) shows the 3D dispersion curve of Vdc1 with A1 and A2 under μ = 0.10; (b) dispersion curves of the pull-in voltages Vp2 in the z direction with respect to A2 with different nonlocal parameters and (d) shows the 3D dispersion curve of Vdc2 under μ = 0.10

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Fig. 8

Pull-in performance tuning by altering the gap ratio δ under μ = 0.0: (a) in-plane pull-in voltages and (b) positions of the coupled modes for different out-of-plane static deflections A2; (c) out-of-plane pull-in voltages and (d) positions of the coupled modes for different in-plane static deflections A1

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Fig. 9

The normalized pull-in positions of in-plane and out-of-plane motions with respect to δ for four representative normalized nonlocal parameters

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Fig. 10

Variations of the in-plane (a) and out-of-plane (b) frequencies of the resonators as a function of A1 and A2 when μ = 0.1

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Fig. 11

3D dispersion curves of the fundamental in-plane and out-of-plane frequencies of the resonators as a function of (a) μ and A1 under A2 = 0.0, (b) μ and A2 under A1 = 0.0; scale effect on (c) fO for different A1 under A2 = 0.0, note that A1 = 0.1666 which is calculated by Eq. (52), (d) fI for different A2 under A1 = 0.0, note that A2 = 0.2083 which is calculated by Eq. (51)

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Fig. 12

Variations of (a) the fundamental in-plane frequency fI and (b) out-of-plane frequency fO as a function of μ under A1 = 0.05,A2 = 0.05 and different terms included by the nonlocal effect

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