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Axial control for nonlinear resonances of electrostatically actuated nanobeam with graphene sensor

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Abstract

The nonlinear resonance response of an electrostatically actuated nanobeam is studied over the near-half natural frequency with an axial capacitor controller. A graphene sensor deformed by the vibrations of the nanobeam is used to produce the voltage signal. The voltage of the vibration graphene sensor is used as a control signal input to a closed-loop circuit to mitigate the nonlinear vibration of the nanobeam. An axial control force produced by the axial capacitor controller can transform the frequency-amplitude curves from nonlinear to linear. The necessary and sufficient conditions for guaranteeing the system stability and a saddle-node bifurcation are studied. The numerical simulations are conducted for uniform nanobeams. The nonlinear terms of the vibration system can be transformed into linear ones by applying the critical control voltage to the system. The nonlinear vibration phenomena can be avoided, and the vibration amplitude is mitigated evidently with the axial capacitor controller.

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Abbreviations

w :

nanobeam deflection

h :

a half of the nanobeam thickness

t′:

time

t :

nondimensional time

s :

width of the nanobeam

g :

distance between the two poles of the capacitor excitatory

l :

beam length

u :

nondimensional displacement

x :

longitudinal coordinate

z :

nondimensional longitudinal coordinate

d 0 :

distance between the two poles of the capacitor controller

A d :

axial cross section area of the graphene film

U :

control voltage

V :

applied voltage

g d,g f :

linear and nonlinear feedback gains of the control voltages

K :

potential energy

W :

external work

Π:

kinetic energy

m :

nanobeam mass per unit length

b*:

viscous damping coefficient per unit length

A :

cross-sectional area

ρ :

material density

σ :

detuning parameter

r :

graphene resistance

R :

divider resistance

ρ d :

equivalent resistance coefficient of graphene

ε 0 :

permittivity of vacuum

T 0/T 1 :

fast/slow time scale

Ω*:

nondimensional frequency of excitation

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Authors and Affiliations

  1. School of Transportation and Vehicle Engineering, Shandong University of Technology, Zibo, 255049, Shandong Province, China

    Canchang Liu, Qingmei Gong, Chicheng Ma & Shuchang Yue

  2. School of Mechanical Engineering, Tianjin University, Tianjin, 300072, China

    Qian Ding

Authors
  1. Canchang Liu
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  2. Qian Ding
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  3. Qingmei Gong
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  4. Chicheng Ma
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  5. Shuchang Yue
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Corresponding author

Correspondence to Canchang Liu.

Additional information

Project supported by the National Natural Science Foundation of China (Nos. 51275280 and 51575325)

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Liu, C., Ding, Q., Gong, Q. et al. Axial control for nonlinear resonances of electrostatically actuated nanobeam with graphene sensor. Appl. Math. Mech.-Engl. Ed. 38, 527–542 (2017). https://doi.org/10.1007/s10483-017-2184-6

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  • DOI: https://doi.org/10.1007/s10483-017-2184-6

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