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Archive of Applied Mechanics

, Volume 81, Issue 3, pp 361–383 | Cite as

Nonlinear vibration analysis of piezo-thermo-electrically actuated functionally graded circular plates

  • F. EbrahimiEmail author
  • A. Rastgoo
Original

Abstract

A theoretical model for geometrically nonlinear vibration analysis of thermo-piezoelectrically actuated circular plates made of functionally grade material (FGM) is presented based on Kirchhoff’s–Love hypothesis with von-Karman type geometrical large nonlinear deformations. The material properties of the FG core plate are assumed to be graded in the thickness direction according to the power-law distribution in terms of the volume fractions of the constituents. Dynamic equations and boundary conditions including thermal, elastic and piezoelectric couplings are formulated and solutions are derived. An exact series expansion method combined with perturbation approach is used to model the nonlinear thermo-electro-mechanical vibration behavior of the structure. Control of the FG plate’s nonlinear deflections and natural frequencies using high control voltages is studied and their nonlinear effects are evaluated. Numerical results for FG plates with various mixtures of ceramic and metal are presented in dimensionless forms. A parametric study is also undertaken to highlight the effects of the thermal environment, applied actuator voltage and material composition of the FG core plate on the nonlinear vibration characteristics of the composite structure.

Keywords

Functionally graded plate Piezo-thermo-electrically actuated plate Circular plate vibration 

List of symbols

a

Plate radius

\({\hat {a}}\)

Dimensionless vibration amplitude

e

Permeability constant of piezoelectric material

E

Young’s modulus

f(t)

Temporal function

hf, hp

FG plate and Piezoelectric layers thickness

n

FGM volume fraction index

\({\overline{N}}\)

Non-dimensional force

Qij

Transverse shear component

Rd, Sd

Eigen-functions (mode shape function)

r, θ, z

Radial, circumferential and transverse direction

T

Temperature

T*

Non-dimensional temperature load

ur, uθ, w

Radial, circumferential, transverse displacements

V

Non-dimensional voltage

\({V_z^t, V_z^b }\)

Applied control voltages to the top and bottom piezoelectric layers

\({\overline{w}}\)

Non-dimensional transverse deflection

x, y

Non-dimensional radical distances

\({{X}_{\rm s},Y_{\rm s}^m}\)

Non-dimensional slope and force

α

Coefficient of thermal expansion

\({\overline{\varepsilon }_{ij}, \kappa _{ij}}\)

Membrane and bending strains

κ

Thermal conductivity

ρ

Density

λ

Non-dimensional eigenvalue

μ1μ2

Nonlinear coefficient functions

ν

Poisson’s ratio

τ

Non-dimensional time

ω

Nonlinear frequency

ωn

Natural frequency

Superscripts

e,m

Electric, mechanical and temperature

L,n

Linear and nonlinear force

f,p

The variable in FGM and Piezo layer

Subscripts

m,c

Metal and ceramic

s, d

Static and dynamic condition

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Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Mechanical Engineering DepartmentTehran UniversityTehranIran

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