DSpace at VNU: Analysis and control of FGM plates integrated with piezoelectric sensors and actuators using cell-based smoothed discrete shear gap method (CS-DSG3)

Accepted ManuscriptAnalysis and control of FGM plates integrated with piezoelectric sensors and actuators using cell-based smoothed discrete shear gap method CS-DSG3 K.. In this paper, t

Accepted Manuscript

Analysis and control of FGM plates integrated with piezoelectric sensors and

actuators using cell-based smoothed discrete shear gap method (CS-DSG3)

K Quang, H Dang-Trung, V Ho-Huu, H Luong-Van, T

Nguyen-Thoi

DOI: http://dx.doi.org/10.1016/j.compstruct.2017.01.006

To appear in: Composite Structures

Received Date: 9 November 2016

Revised Date: 29 December 2016

Accepted Date: 4 January 2017

Please cite this article as: Nguyen-Quang, K., Dang-Trung, H., Ho-Huu, V., Luong-Van, H., Nguyen-Thoi, T.,Analysis and control of FGM plates integrated with piezoelectric sensors and actuators using cell-based smoothed

discrete shear gap method (CS-DSG3), Composite Structures (2017), doi: http://dx.doi.org/10.1016/j.compstruct.

2017.01.006

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Division of Computational Mathematics and Engineering, Institute for Computational Science, Ton Duc Thang

University, Ho Chi Minh City, Vietnam 2

Faculty of Civil Engineering, Ton Duc Thang University, Ho Chi Minh City, Vietnam

3

Faculty of Aerospace Engineering, Delft University of Technology, Delft, The Netherlands

4 Faculty of Civil Engineering, Ho Chi Minh City University of Technology, Ho Chi Minh City, Vietnam

E-mail addresses: nguyenquangkha.ce@gmail.com (K Nguyen-Quang); dangtrunghau@tdt.edu.vn (H

Dang-Trung); vinh.ho.h@gmail.com (V Ho-Huu); lvhai@hcmut.edu.vn (H Luong-Van);

nguyenthoitrung@tdt.edu.vn (T Nguyen-Thoi)

Abstract

A cell-based smoothed discrete shear gap method (CS-DSG3) based on the first-order shear deformation theory was recently proposed for static and dynamics analyses of Mindlin plates In this paper, the CS-DSG3 is extended for analysis and active vibration control of the functionally graded material (FGM) plates integrated with piezoelectric sensors and actuators In the piezoelectric FGM plates, the properties of core material are assumed to be graded through the thickness by the power law distribution while the electric potential is assumed to be a linear function through the thickness of each piezoelectric sub-layer A closed-loop control algorithm based on the displacement and velocity feedbacks is used to control static deflection and active vibration of piezoelectric FGM plates Several numerical examples are conducted to demonstrate the reliability and accuracy of the proposed method compared to other available numerical results

Keywords: Cell-based smoothed discrete shear gap method (CS-DSG3); FGM plates;

sensor/ actuator layers; displacement/ velocity feedback control algorithm; active deflection/ vibration control.

1 Introduction

Being first discovered in Sendai by a group of Japanese scientists in 1984, functionally graded materials (FGMs) have been then developed rapidly around the world [1–7] The FGMs are usually manufactured by combining metals and ceramics following a certain distribution rule of the material fraction This enables FGMs to inherit the best properties of these two materials including: (1) low thermal conductivity and high thermal resistance from

*

Corresponding author Tel.: +84 933 666 226

E-mail addresses: nguyenthoitrung@tdt.edu.vn (T Nguyen-Thoi)

2

ceramics and (2) durability and high loading resistance from metals Subsequently, various progress has also been developed over the last decade for the active vibration suppression and shape control of FGM plates by using bonded or embedded piezoelectric materials One of the essential features of piezoelectric materials is the ability of transformation between mechanical energy and electric energy When a structure integrated with the piezoelectric material is deformed, the piezoelectric material generates an electric charge, which is termed the direct piezoelectric effect In contrast, when an electric field is applied, it produces mechanical deformation in the structure due to the converse piezoelectric effect [8] As a result, if the force that causes deformation of the structure is controlled appropriately, the vibration of the structure may be suppressed adequately Therefore, the integration of FGM plates with piezoelectric materials to give active lightweight smart structures has attracted the considerable interest of researchers in various industries such as automotive sensors, actuators, transducers and active damping devices, etc

Due to these attractive properties, various numerical methods have been proposed to simulate the behavior of piezoelectric FGM plates He et al [9] investigated the shape and vibration control of FGM plates integrated with sensors and actuators In this work, a finite element formulation based on classical plate theory (CPT) was presented Liew et al [10] developed finite element formulations to study the behaviors of FGM plates containing sensors/actuators patches under environments subjected to a temperature gradient, using linear piezoelectric theory and first-order shear deformation theory (FSDT) Balamurugan and Narayanan [11] used the nine-node piezo-laminated plate finite element incorporating the FGM material model and the electromechanical coupling constitutive relations of the piezoelectric sensors/actuators to investigate the active control of piezoelectric FGM plates Ebrahimi and Rastgoo [12] presented a nonlinear free vibration of a thin annular FGM plate integrated with two uniformly distributed actuator layers on the top and bottom surfaces based on Kirchhoff plate theory by analytical solutions Ahmad Gharib [13] also developed

an analytical solution for analyzing deflection control of FGM beams with embedded piezoelectric sensors and actuators by using FSDT Saidi et al [14] presented another analytical approach for free vibration analysis of moderately thick functionally graded rectangular plates coupled with piezoelectric layers Liu et al [15,16] presented active vibration control of laminated composite beams and plates containing distributed sensors and actuators based on CPT and the radial point interpolation method (RPIM) Shakeri and Mirzaeifar [17] proposed a general finite element formulation based on the layerwise theory for static and dynamic analysis of thick FGM plates with piezoelectric layers Aryana et al

[18] used a finite element formulation based on CPT and an efficient method based on and second-order approximations in Taylor expansion to modify dynamic characteristics of piezoelectric FGM plates Hosseini-Hashemi et al [19] studied a 3-D Ritz solution for free vibration of circular/annular functionally graded plates integrated with piezoelectric layers Recently, Selim et al [20] proposed a Reddy’s higher-order shear deformation theory with the element-free IMLS-Ritz method for active vibration control of piezoelectric FGM plates Phung-Van et al [21] presented a generalized shear deformation theory in combination with isogeometric approach for nonlinear transient analysis of FGM plates under thermo-electro-mechanical loads In general, it can be seen from the above mentioned literature review that there has been only a few published papers carried out to study behaviors of piezoelectric FGM plates in terms of deformable characteristic, stress distribution and vibration characteristics In addition, the use of simple linear plate elements such as three-node triangular Mindlin plate elements for analysis of piezoelectric FGM plates is somewhat still limited

first-In the other front of the development of numerical methods, Liu et al [22] have integrated the strain smoothing technique [23] into the FEM to create a series of smoothed FEM (S-FEM) such as a cell/element-based smoothed FEM (CS-FEM) [24], a node-based smoothed FEM (NS-FEM) [25], an edge-based smoothed FEM (ES-FEM) [26] and a face-based smoothed FEM (FS-FEM) [27] Each of these S-FEM has different properties and has been used to produce desired solutions for a wide class of benchmark and practical mechanics problems Several earlier and related works of the S-FEM models have been provided in Refs [28–32] Among these S-FEM models, the CS-FEM [24,32–34] shows some interesting properties in the solid mechanics problems Extending the idea of the CS-FEM to plate structures, Nguyen-Thoi et al [35] have recently formulated a cell-based smoothed discrete shear gap method (CS-DSG3) which belongs to the group of simple three-node triangular Mindlin plate elements In the CS-DSG3, each triangular element is divided into three sub-triangles, and in each sub-triangle, the stabilized DSG3 is used to compute the strains and avoid the transverse shear locking Then the cell-based strain smoothing technique on whole the triangular element is used to smooth the strains on these three sub-triangles The numerical results showed that the CS-DSG3 is free of shear locking and achieves the high accuracy compared to others existing elements It has been successfully extended to analyze various plate and shell problems such as flat shells [36], stiffened plates [37], composite and sandwich plates [38], piezoelectric composite plates [39], plates resting on viscoelastic

by a simple power rule of the volume fractions of the constituents The electric potential is assumed to be a linear function through the thickness of each piezoelectric sub-layer A closed-loop control algorithm based on the displacement and velocity feedbacks is used to control static deflection and active vibration of piezoelectric FGM plates Equilibrium equation is derived from the principle of virtual displacements based on FSDT The accuracy and reliability of the proposed method are verified by comparing its numerical results with those of other available numerical approaches The static analysis of piezoelectric FGM plate

is investigated with different voltages and boundary conditions The numerical results are presented in both tabular and graphical forms For dynamic vibration control, the effect of various types of load, and the influence of feedback control gain on static and dynamic response are also studied

The remainder of this paper is outlined as follows Section 2 describes the weak form of governing equations and finite element formulation for FGM plates related to static, free vibration and dynamic control problems In section 3, the active control analysis is presented Section 4 presents numerical examples to verify the reliability and efficiency of the present method Finally, some conclusions are drawn in section 5

2 Galerkin weak form and finite element formulation for piezoelectric FGM plates

In this section, the Galerkin weak form and finite element formulation for piezoelectric FGM plates are established via a variational formulation [47,48] The piezoelectric FGM plate is assumed to be perfectly bonded, elastic and orthotropic in behavior [49], with small strains and displacements [50], and the deformation taken place under isothermal conditions In addition, the piezoelectric sensors/actuators are made of homogeneous and isotropic dielectric materials [51], and high electric fields as well as cyclic fields are not involved [52] Based on these assumptions, a linear constitutive relationship [53] can be employed for the static and dynamic analysis of the piezoelectric FGM plates

2.1 Formulation of functionally graded material

Functionally graded materials are often composed of a mixture of two distinct material phases: ceramic and metal Ceramic can resist high thermal load because its thermal conductivity is low while the metal component can maintain flexibility of structure under the high-temperature gradient

The material properties are assumed to be graded through the thickness by the power law distribution expressed as

fully metal, and when n→ ∞ , the homogeneous ceramic plate is retrieved, respectively

Fig 1 Variation of the volume fraction against the non-dimensional thickness

2.2 Linear piezoelectric constitutive equations

The linear piezoelectric constitutive equations can be expressed as

6

piezoelectric constant matrix and g denotes the dielectric constant matrix displayed in

2.3 Galerkin weak form of the governing equations

The Galerkin weak form of the governing equations of piezoelectric structures can be derived

by using Hamilton’s variational principle [55] which can be written as

0

L

where L is the general energy functional which describes a summation of kinetic energy,

strain energy, dielectric energy and external work, and is written in the form of

(6)

and point charges

necessary to use efficient numerical methods to approximate the mechanical displacement field and electric potential field In the present work, the CS-DSG3 [35] is used to approximate the mechanical displacement field of piezoelectric FGM plates Additionally, a linear constitutive relationship is also employed [53] for the analysis of the piezoelectric FGM plates, and the formulation for each field will be presented separately

2.4 Approximations of the mechanical displacement field

Consider a FGM plate under bending deformation as shown in Fig 2 The neutral surface of

R

field according to the Reissner-Mindlin model based on FSDT [56] can be expressed by

( , , ) ( , ) ( ) ( , )( , , ) ( , ) ( ) ( , )( , , ) ( , )

x y

ββ

h

h h

h

E z z z z

of the neutral plane around the y - and x -axes, respectively, with the positive directions

s s / 2

h ij h

2.4.2 FEM formulation for FGM plates

Now, by discretizing the bounded domain Ω of a FGM plate into N e finite elements such that N e1

e= e

Ω =∪ Ω and Ω ∩ Ω = ∅ , i i j ≠ , the finite element solution of the FGM plate is j

expressed as

n T

h

i

i i

N N

N N

, b

,

, s

N N

N N

2.4.3 Brief description of the CS-DSG3 formulation for FGM plates

Firstly, the CS-DSG3 is developed by incorporating the CS-FEM [22,24] with the original DSG3 element [58] Details of the CS-DSG3 formulation can be found in [35,37–39,59]

In the CS-DSG3, each triangular element Ωe is divided into three sub-triangles ∆ ∆1, 2

and ∆3 by connecting the central point O of the element to three field nodes as shown in Fig

3 Then, in each sub-triangle ∆j(j=1, 2, 3), the strain fields are calculated similarly to that by the discrete shear gap method (DSG3) to give the membrane, bending and shear strains in each sub-triangle m∆j

B , b∆j

B and s∆j

B , respectively Finally, the cell-based strain smoothing operation in the CS-FEM is applied to give the smoothed strain fields of the element Ωe as follows:

(19)where de =u i v i w i βxi βyiT(i=1, 2, 3) is the nodal displacement vector of element

e

Ω ; , and are the smoothed strain gradient matrices, respectively, given by

CS-2.5 Approximations of the electric potential

In this study, the approximations of the electric potential field of each piezoelectric layer are made by discretization of each piezoelectric layer into finite sublayers along the thickness direction In each sublayer, a linear electric potential function is approximated through the thickness by [50]

( )

the electric potentials at the top and bottom surfaces of the sublayer, and are defined as

3

13 31 33

( ) 11

k k

d d

p p p

2.6 Elementary governing equation of motion

The elementary governing equation of motion can be derived by substituting Eqs (13), (19), (22) and (24) into Eqs (6) and (5), and assembling the electric potentials along the thickness The final form of this equation is then written by

/ 2 2

3 Active control analysis

We now consider a piezoelectric FGM plate as shown in Fig 4 The top layer is a piezoelectric actuator denoted with subscript “a” and the bottom layer is a piezoelectric sensor labeled with subscript “s” In this work, the displacement feedback control [60], which helps the piezoelectric actuator to generate the charge, is combined with velocity feedback control [15,16,54,55,61–64], which can give a velocity component by using an appropriate electronic circuit

Fig 4 A schematic diagram of a FGM plate with integrated piezoelectric sensors and

actuators

In addition, a consistent method [16,65] which can predict the dynamic responses of smart piezoelectric FGM plate is adopted The constant gains Gd and Gv of the displacement feedback control and velocity feedback control [16] are hence used to couple the input actuator voltage vector φa and the output sensor voltage φs as

(30)Without the external charge Q, the generated potential on the sensor layer can be derived from the second equation of (26) as

convert into the voltage signal The converted signal is then sent and applied to the distributed actuator As a result, stresses and strains are generated through the converse piezoelectric effect, and a resultant force is formed to actively control the dynamic response

of the piezoelectric FGM plate

Substituting Eqs (30) and (31) into the second equation in Eq (26) leads to

(33)Substituting Eq (33) into Eq (29) yields

(34)where

In this section, three numerical examples are conducted to illustrate the accuracy and stability

of the CS-DSG3 compared to some other published methods We first demonstrate the accuracy of the CS-DSG3 for static and free vibration problems We then show the performance of the proposed method for dynamic control of FGM plates integrated with piezoelectric sensors and actuators Here, the properties of piezoelectric FGM plates,

4.1 Free vibration analysis of piezoelectric FGM plates

In this section, we investigate the accuracy and efficiency of the CS-DSG3 for analyzing natural frequencies of piezoelectric FGM plates We now consider a square piezoelectric

2 3

Al/Al O plate The plate is simply supported, and has the ratio of thickness of the FGM core

surfaces of the FGM plate Two sets of electric boundary conditions are considered for inner surfaces of the piezoelectric layers including: (1) a closed-circuit condition in which the electric potential is kept at zero (grounded); and (2) an open-circuit condition in which the electric potential remains free (zero electric displacements)

Table 2 shows the results of the first natural frequencies of the piezoelectric FGM plate

uses the first-order shear deformation theory (FSDT) with only 5 degrees of freedom (dofs) per node while the study in reference [20] used high-order shear deformation theory (HSDT) with 7 dofs per node It is seen that the results by the CS-DSG3 match well with those by [20] and also agree well with the analytical solution [14] Fig 5 illustrates the shapes of the first six lowest eigenmodes by the CS-DSG3

Table 2 The first natural frequency of the simply supported square piezoelectric FGM

plate in different conditions

Power law index h/a

Electrical condition

Method CS-DSG3

(5dofs)

IMLS-Ritz-HSDT (7dofs) [20]

Analytical solution [14]

In this example, we study the static response of a cantilevered square piezoelectric FGM plate

combination of FGM materials consists of Ti-6Al-4V and aluminum oxide material constituents, and the piezoceramic is made of PZT-G1195N All material properties are listed

in Table 1

centerline position of the cantilevered piezoelectric FGM plate that is subjected to only a

three-node triangular elements The numerical results by the CS-DSG3 in Fig 6 show that

Ti-6Al-4V volume fraction while the volume fraction of aluminum oxide with higher elastic modulus value is increased In addition, these numerical results also agree well with those obtained by CPT [9]

piezoelectric FGM plate under a uniformly distributed load by the CS-DSG3

Next, we investigate the case in which all the piezoceramics are served as actuators Fig

7 displays the centerline deflection of piezoelectric FGM plate for various values of power

amplitude voltages with opposite signs are applied across the thickness of the two layers, these piezo-layers will contract or expand depending on whether the active voltage is negative or positive Consequently, strains are induced to generate forces which make the FGM plate bending In other words, the piezoelectric effect makes the plate deflect upward Next, we study the effect of actuator on the shape control to the cantilevered FGM plate