Nonlinear free vibration analysis of piezoelectric laminated plate with random actuation electric potential difference and thermal loading

https://doi.org/10.1016/j.apm.2021.01.052Get rights and content

Highlights

  • Solutions to moderately large amplitude vibrations of piezoelectric thin laminated plates.

  • Mechanical-thermal-electrical coupled system of nonlinear equations.

  • Displacement-stress mixed formulation.

  • Random actuation electric potential and thermal load on nonlinear frequency ratio.

Abstract

Solutions to nonlinear free vibration analysis of an arbitrarily laminated thin rectangular plate subjected to thermo-piezoelectric load is investigated. The governing equation of motion is based on von Kármán’s deflection of thin plates to capture the effects of moderate geometric nonlinearity. Induced strain actuation assumption is confined to a linear model for applied low electric fields and varying temperature distribution along the thickness direction. The solution for lateral displacement field for simply supported and clamped condition edges are determined by displacement-stress; mixed formulation. The applied Galerkin’s projection reduces the governing nonlinear partial differential equation to a time dependent ordinary differential equation. Solutions are obtained using a predictor-corrector numerical integration scheme. The effects of randomly applied actuation potential and thermal loading on nonlinear frequency ratio are examined. The marginal values provided using various probabilistic methods are found to be in good agreement with each other.

Introduction

The development of adaptive, high-performance structures has been motivated by the increasing demand from modern aeronautical, automobile, and space industries. A composite structure with an embedded actuation unit, sensor unit and a control system can change it's shape and dynamic behaviour in response to any change in the external environment. Such structures use special materials called smart materials as actuating and sensing elements. Bonding or embedding these smart materials into structures gives the inherent self-sensing and self-correcting ability without any separate sensing or actuation unit.

Many studies on free and forced vibration of laminated plates subjected to mechanical, thermal and electrical loads are available in the literature [1], [2], [3], [4]. However, relatively few have been made to analyse the impact of variations in material manufacturing defect and exposure to unknown hostile environments. For critical aerospace structures, it is essential to know the potential variations in the structural response due to the uncertainties in design parameters. This has motivated many researchers to study the dynamic characterization of structural components with randomness in material and physical properties [5], [6], [7], [8]. Nonlinear free vibration of laminated plates with random material properties for various geometrical boundary conditions is studied in Ref. [9], [10], [11].

Huang and Shen [12] used shear deformation theory to study the dynamic response of simply supported laminated plates with piezoelectric actuator and mechanical, thermal and electrical loads. Jayakumar et al. [13] studied the moderately large amplitude dynamic response of a thin piezoelectric laminated rectangular plate. The probabilistic analysis is carried out for simply supported plates with variable material properties and actuation electric potential. However, smart structure components used in orbital vehicles undergo severe heating loads due to solar radiation. Ebrahimi et al. [14] investigated the role of thermal environment on nonlinear static deflections and natural frequencies imposed by the piezoelectric actuators. Alijani et al. [15] provided information on the thermal effects on geometrically nonlinear vibration of functionally graded plates.

Lal et al. [16] presented the effect of randomness in material properties on a geometrically nonlinear piezoelectric laminated conical shell panel subjected to thermo-electro-mechanical loading. Dash and Singh [17] used Green-Lagrange strain-displacement relations to study the stochastic vibration characteristics of the piezoelectric laminated composite plate. They also examined the stochastic post-buckling response with random material properties [18]. Xu and Fu [19] studied the nonlinear vibration behaviour of a smart electrostatic harmonic actuator considering van der Waals forces. Swain et al. [20] used the perturbation method to model uncertainty in material properties and shear deformation theory for transverse deflection to analyse the piezoelectric effect.

Present work, investigates the effect of randomness in material properties, actuation electric potential and thermal environments on nonlinear vibration response of a piezoelectric laminated rectangular plate. The temperature field is assumed to be uniformly distributed over the plate surfaces and is layer-wise varying linearly through the thickness direction. The orthotropic material properties are assumed to be independent of the temperature and applied electric field. Extensive parametric analyses are provided to show the variation of moderately large deflection of the plate for simply supported and clamped conditions. Probabilistic methods, like Monte Carlo simulation and Multivariate methods are utilized to show the extent of variability in nonlinear behaviour.

Section snippets

Mathematical formulation

In this paper, a rectangular piezoelectric laminated plate (shown in Fig. 1) of length a in the x-direction, width b in the y-direction, and thickness h in the z-direction is considered. The plate is thin (i.e., ha; hb) and it is made of N anisotropic layers of arbitrary orientation. Two piezoelectric layers of thickness hp are bonded on the top and bottom surfaces and act as strain actuators on application of electric potential difference δa. It is subjected to thermal loading, ΔT along the

Solution method

In the present work, moderately large transverse displacement of a rectangular laminated plate with surface bonded piezoelectric layers subjected to uniform temperature distribution on its outer surface is considered.

Effect on nonlinear free vibration for simply supported and clamped plate under the influence of actuation electric potential and linear temperature variation along thickness is examined.

  • 1.

    The transverse simply supported conditions are:w=0,Mx=0;atx=0,aw=0,My=0;aty=0,b

  • 2.

    The transverse

Comparison studies

Moderately large amplitude free vibration behavior of thin laminated rectangular plate is compared with available literature results. The results presented in Table 1 for isotropic square plates for nonlinear frequency ratio are found to be in good agreement with those provided by Ebrahimi et al. [14], [26], Chandra and Raju [22], Benamar et al. [27], Pillai et al. [28] and Kadiri et al. [29].

The orthotropic plate results are validated with that of Onkar and Yadav [9] and Singh et al. [30] for

Conclusion

Studies are carried out on nonlinear free vibration of piezoelectric laminated rectangular plates with uniform temperature variation and applied electric potential. The variability of frequency ratio for simply supported and clamped transverse boundary conditions for immovable edges are analysed. The variance observed is more for simply supported cases, whereas it is not very significant for the clamped condition. Piezoelectric laminated thin plate’s response upon changes in electric potential

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