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Size-dependent dynamic modeling and vibration analysis of MEMS/NEMS-based nanomechanical beam based on the nonlocal elasticity theory

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Abstract

Accurate modeling and analysis of micro-/nanoelectromechanical systems (MEMS/NEMS) has an immense contribution in identification and improvement of the performance of such systems. This article investigates a nonclassical formulation for dynamicmodeling and vibration analysis of a piezo-actuatedmicrocantilever considering the Euler–Bernoulli beam model. Regarding the size effects in micro- to nanoscales, the size-dependent nonlocal continuum theory is employed to derive dynamic equations of the nonclassical microbeam taking into account the beam discontinuities. The nonlocal formulation of the beam and piezoelectric actuator is taken into consideration. Furthermore, the size effects on the resonant vibration characteristics of the beam are studied and some results are obtained. The results illustrate the size-dependent behavior of the beam particularly at higher resonant modes of vibrations. Also, it is indicated that the nonlocality and piezoelectric characteristics have a significant influence on the resonance behavior of the beam. However, this effect is more significant at higher resonant modes of vibrations.

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Abbreviations

\({\bar{A}_{b}}\) :

Beam cross-sectional area

\({\bar{A}_{p}}\) :

Piezoelectric cross-sectional area

\({\bar{A}_{t}}\) :

Tip cross-sectional area

\({\bar{B}}\) :

Coefficient of viscous air damping

\({\bar{c}_{ijkl}}\) :

Elastic stiffness tensor

\({\bar{c}_{b}}\) :

Young’s modulus of the beam

\({\bar{c}_{p}}\) :

Young’s modulus of the piezoelectric layer along the y-direction

D y :

Electric displacement of the piezoelectric layer along the y-direction

\({\bar{e}_{p}}\) :

Piezoelectric constant

E y :

Electric field of the piezo-layers along the y-direction

H(x):

Heavyside function

k p :

Coefficient of applied voltage

\({\bar{m}_{b}}\) :

Density of the beam

\({\bar{m}_{p}}\) :

Density of the piezoelectric layer

\({\bar{t}_{p}}\) :

Thickness of the piezoelectric layer

\({\bar{t}_{b}}\) :

Thickness of the cantilever

V p :

Piezoelectric applied voltage

\({\bar{v}(x,t)}\) :

Deflection of midplane of the beam

y n :

Neutral axis of each section of the cantilever

\({\bar{w}_{b}}\) :

Width of the cantilever

\({\bar{w}_{t}}\) :

Width of the tip section of the cantilever

\({\bar{w}_{p}}\) :

Width of the piezoelectric layer

\({\bar{x}}\) :

The total length of the nanomechanical cantilever

\({\varepsilon_{xx}}\) :

Strain component along the x-direction

\({\alpha \left|{\vec {r}}\right|}\) :

Nonlocal kernel function

\({\bar{\sigma}_{xx,b}}\) :

Nonlocal stress component in the beam along the x-direction

\({\bar{\sigma}_{xx,p}}\) :

Nonlocal stress component in the piezo-layer along the x-direction

\({\mu}\) :

Nonlocal parameter

\({\mu_{\rm air}}\) :

Viscosity of the air

\({\rho_{\rm air}}\) :

Density of the air

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Authors and Affiliations

  1. Department of Mechanical Engineering, Amirkabir University of Technology, Tehran, Iran

    F. Bakhtiari-Nejad & M. Nazemizadeh

  2. Department of Mechanics, Damavand Branch, Islamic Azad University, Damavand, Iran

    M. Nazemizadeh

Authors
  1. F. Bakhtiari-Nejad
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  2. M. Nazemizadeh
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Correspondence to F. Bakhtiari-Nejad.

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Bakhtiari-Nejad, F., Nazemizadeh, M. Size-dependent dynamic modeling and vibration analysis of MEMS/NEMS-based nanomechanical beam based on the nonlocal elasticity theory. Acta Mech 227, 1363–1379 (2016). https://doi.org/10.1007/s00707-015-1556-3

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  • DOI: https://doi.org/10.1007/s00707-015-1556-3

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