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Dynamic modeling and vibration analysis of piezoelectric microcantilever in AFM application

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Abstract

Piezoelectric microcantilevers (MCs) are widely used in common piezoelectric micro-electromechanical systems. These systems are used in ultraprecise scanning and characterization applications. Given the widespread use of this type of MCs in nanotechnology, their vibrating motion problem has recently become a matter of interest. Accurate vibration analysis and studying their vibrating behavior can play a key role in increasing their measurement accuracy and optimal design. For this purposes, first, the differential equation of motion of MCs is solved using modal superposition method by using Runge–Kutta in time domain and finite element methods. The extended Fourier amplitude sensitivity test statistical method is used to conduct sensitivity analysis on the main parameters of the vibrating motion to determine the effects of each interaction coefficient on the motion of the MC. Results showed that the numerical finite element method and modal superposition method based on the non-uniform beam model have acceptable accuracy in calculating resonance amplitude and natural frequency of this kind of MC. The results of the sensitivity analysis indicate the high sensitivity of the first vibrating mode to force coefficient, which means that this mode is suitable for topography of the sample surface and the nanoparticle.

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Abbreviations

\( \overline{A} \) :

Fourier cosine

\( \bar{B} \) :

Fourier sine

C e :

Electromechanical coupling coefficient

C :

Damping coefficient

\( \bar{C} \) :

Global damping matrix in FEM

d :

Equilibrium distance between tip and sample/nanoparticle

d 31 :

Piezoelectric constant

E :

Desired value of quantity

E i :

Elasticity module of each layer

f :

Tip/nanoparticle-sample force

F :

Force vector

\( \bar{f} \) :

Interaction force in FEM

\( \bar{F} \) :

Force vector in FEM

Q :

Global displacement vector in FEM

q n :

Generalized time-dependent coordinates

r t :

Tip radius

r p :

Nanoparticle radius

S i :

Sensitivity index

\( S_{i}^{F} \) :

First order sensitivity index

\( S_{{T_{i} }}^{F} \) :

Total effect of sensitivity

V :

Microcantilever transverse deformation

W i :

Layer width

W t :

Tip width

X i :

Input quantity

H :

Hamaker constant

h i :

Layer thickness

I :

Unit matrix

K :

Stifness matrix

K(x):

Microcantilever stifness

\( \bar{K} \) :

Global stiffness matrix in FEM

L :

Microcantilever length

L e :

Element length

L p :

Probe’s length

m :

Mass of unit length

M :

Mass matrix

\( \bar{M} \) :

Global mass matrix in FEM

N j :

Hermitian shape functions

P(t):

Input voltage

P(X):

Multidimensional probability function

\( \bar{X}_{i} \) :

Actual value of input quantity

Y :

Output quantity

γ n :

Coefficient in ordinary differential equation

δ(x):

Dirac function

ρ :

Density

σ :

Conditional variance

σ Y :

Unconditional variance of output quantity

φ n :

nth MC mode shape

ω n :

Natural frequency

Ω :

Frequency

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Authors and Affiliations

  1. Department of Mechanical Engineering, Faculty of Technical and Engineering, Shahrekord Branch, Islamic Azad University, Shahrekord, Iran

    Reza Ghaderi

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  1. Reza Ghaderi
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Correspondence to Reza Ghaderi.

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Ghaderi, R. Dynamic modeling and vibration analysis of piezoelectric microcantilever in AFM application. Int J Mech Mater Des 12, 413–425 (2016). https://doi.org/10.1007/s10999-015-9309-y

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  • DOI: https://doi.org/10.1007/s10999-015-9309-y

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