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Nonlinear primary resonance analysis for a coupled thermo-piezoelectric-mechanical model of piezoelectric rectangular thin plates

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Abstract

A model of piezoelectric rectangular thin plates with the consideration of the coupled thermo-piezoelectric-mechanical effect is established. Based on the von Karman large deflection theory, the nonlinear vibration governing equation is obtained by using Hamilton’s principle and the Rayleigh-Ritz method. The harmonic balance method (HBM) is used to analyze the first-order approximate response and obtain the frequency response function. The system shows non-linear phenomena such as hardening nonlinearity, multiple coexistence solutions, and jumps. The effects of the temperature difference, the damping coefficient, the plate thickness, the excited charge, and the mode on the primary resonance response are theoretically analyzed. With the increase in the temperature difference, the corresponding frequency jumping increases, while the resonant amplitude decreases gradually. Finally, numerical verifications are carried out by the Runge-Kutta method, and the results agree very well with the theoretical results.

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Abbreviations

L x :

length of the plate

L y :

width of the plate

h m :

thickness of the substrate material

hp:

thickness of the piezoelectric material

\(c_{ij}^{\rm{m}}\) :

stiffness coefficient of the substrate material

\(c_{ij}^{\rm{P}}\) :

stiffness coefficient of the piezoelectric material

E m :

Young’s modulus

υ m :

Poisson’s ratio

σ i :

normal stress

T ij :

shear stress

ε i :

normal strain

γ ij :

shear strain

e 31 :

piezoelectric constant

ε 33 :

dielectric permittivity

λ :

thermo-mechanical coupling constant

α:

coefficient of linear thermal expansion

d 3 :

thermal-piezoelectric coupling constant

E z :

electric field component

D z :

electric displacement component

V (t):

voltage

q(t):

surface charge of the piezoelectric layer

η :

entropy

a T :

material constant aT =cE/To

c E :

heat capacity

θ :

temperature difference

T o :

initial temperature

T c :

Curie temperature.

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Authors and Affiliations

  1. Department of Mechanics, School of Science, North University of China, Taiyuan, 030051, China

    Xun Wang, Chunxia Xue & Haitao Li

  2. Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai, 200072, China

    Haitao Li

Authors
  1. Xun Wang
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  2. Chunxia Xue
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  3. Haitao Li
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Corresponding author

Correspondence to Chunxia Xue.

Additional information

Citation: WANG, X., XUE, C. X., and LI, H. T. Nonlinear primary resonance analysis for a coupled thermo-piezoelectric-mechanical model of piezoelectric rectangular thin plates. Applied Mathematics and Mechanics (English Edition) 40(8), 1155–1168 (2019) https://doi.org/10.1007/s10483-019-2510-6

Project supported by the National Natural Science Foundation of China (No. 11202190), the Natural Science Foundation for Young Scientists of Shanxi Province of China (No. 201801D221037), and the China Postdoctoral Science Foundation (No. 2018M640373)

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Wang, X., Xue, C. & Li, H. Nonlinear primary resonance analysis for a coupled thermo-piezoelectric-mechanical model of piezoelectric rectangular thin plates. Appl. Math. Mech.-Engl. Ed. 40, 1155–1168 (2019). https://doi.org/10.1007/s10483-019-2510-6

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  • DOI: https://doi.org/10.1007/s10483-019-2510-6

Key words

Chinese Library Classification

2010 Mathematics Subject Classification

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