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Optimal control of laminated plate integrated with piezoelectric sensor and actuator considering TSDT and meshfree method
Accepted Manuscript Optimal control of laminated plate integrated with piezoelectric sensor and actuator considering TSDT and meshfree method R. Talebitooti, Asst. Prof, K. Daneshjoo, Prof, S.A.M. Jafari, MSc., Stu PII:
S0997-7538(15)00125-4
DOI:
10.1016/j.euromechsol.2015.09.004
Reference:
EJMSOL 3234
To appear in:
European Journal of Mechanics / A Solids
Received Date: 21 April 2015 Revised Date:
28 July 2015
Accepted Date: 10 September 2015
Please cite this article as: Talebitooti, R., Daneshjoo, K., Jafari, S.A.M, Optimal control of laminated plate integrated with piezoelectric sensor and actuator considering TSDT and meshfree method, European Journal of Mechanics / A Solids (2015), doi: 10.1016/j.euromechsol.2015.09.004. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT Optimal control of laminated plate integrated with piezoelectric sensor and actuator considering TSDT and meshfree method R. Talebitootia, K. Daneshjoob, S.A.M.Jafaric Prof, Sch. of Mech. Eng., Iran Univ. of Sci. and Tech., Teheran 16844, Iran;
bProf,
Sch. of Mech. Eng., Iran Univ. of Sci. and Tech., Teheran 16844, Iran;
cMSc.
Stu, Sch. of New Technologies, Iran Univ. of Sci. and Tech., Teheran 16844, Iran;
Abstract
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aAsst.
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In this paper, the element free galerkin (EFG) method based on third-order shear deformation theory (TSDT) is used to investigate shape and vibration control of piezoelectric laminated plate bonded with
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piezoelectric actuator and sensor layers. The electric potential distributions through the thickness for each piezoelectric layer are assumed to vary linearly. In addition, a closed-loop velocity feedback control and optimal steady-state regulator with output feedback algorithm is used for the active control of the static deflection as well as the dynamic response of the plates with bonded distributed piezoelectric sensors and actuators. Furthermore, the effects of the size of support and nodal density on the numerical
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accuracy are also investigated. The results indicate that, the accuracy and reliability of presented work have an excellent agreement with those of other available numerical approaches such as finite element and FSDT meshfree method.
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Keywords: Optimal control, Meshfree method, TSDT theory, Sensor and actuator
Introduction
The study of embedded or surface-mounted piezoelectric materials in structures has received considerable attention in recent two decades [1]. It is due to possibility of creating certain types of structures and systems capable of adapting to or correcting for changing operating conditions. The advantage of incorporating these special types of material into the structure is due to the fact that the sensing and actuating mechanism becomes part of the structure by sensing and actuating strains directly.
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ACCEPTED MANUSCRIPT Classical mathematical models have already been well established for the phenomena in the areas of mechanics of solids and structures. Meanwhile, different types of differential or partial differential equations (PDEs) that govern these phenomena have also been derived. There are largely two categories of numerical methods for solving these PDEs [2]: direct approach and indirect approach. The direct approach known as strong form methods (such as the finite difference method (FDM) and collocation
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method) discretizes and solves the PDEs directly, and the indirect approach known as weak form methods (such as finite element method (FEM)) establishes first an alternative weak form system equation that governs the same physical phenomena and then solves it. The weak form equations are
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usually in an integral form, implying that they need to be satisfied only in an integral (averaged) sense. The (EFG), one of known category of meshfree, is a standard weak formulation that is variationally
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consistent due to the use of compatible moving least squares (MLS) shape functions and the Galerkin approach with constraints to impose the essential boundary conditions. In the EFG method, in order to derive the stiffness matrix, some complex integrals should be solved via numerical procedures. Therefore, a need for background mesh to perform the numerical integration is unavoidable. Subsequently, in the EFG method the integration cells do not require to be attuned with the scattered nodes distributed on the
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problem domain. Hence, the generation of background mesh is more easily than the FEM methods. Liew et. al. [3], [4] represented a formulation by employing the EFG method based on the first-order shear deformation theory to study the shape control and vibration suppression of piezo-laminated composite beams and plates. Liu et. al. [5] developed the previous work with employing the point interpolation
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method using radial basis functions (RPIM) based on the first-order shear deformation theory.
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In EFG method, the shape functions constructed by MLS approximation do not consider the property of delta functions. Hence, the essential boundary conditions cannot be imposed as conveniently as the standard FEM method. Liu and Chen [6] consider the orthogonal transformation techniques to impose the essential boundary conditions for free vibration analysis of a thin plate. Dai et. al. [7] used the penalty method to impose the essential boundary conditions for the static deflection and free vibration analysis of composite plates. This proposed method presented by [7] is more efficiently than orthogonal transform method [6].
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ACCEPTED MANUSCRIPT Samanta et. al. [8] represented a generalized finite element formulation with an eight noded twodimensional quadratic quadrilateral isoparametric element for active vibration control of a laminated plate integrated with piezoelectric polymer layers acting as distributed sensors and actuators based on third-order shear deformation theory. Similar works developed finite element formulation for active
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vibration control of smart structure [9–12]. The modern control techniques are to be widespread in designing the stability augmentation systems. This is accomplished by regulating certain states of the system to zero while obtaining desirable closedloop response characteristics. Output feedback will allow engineers to design plant controllers of any
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desired structure. In the constant gain velocity feedback control strategy (CGVF), the stability of the system is guaranteed only when the actuators and sensors are truly collocated. Moreover, the linear
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quadratic regulator (LQR) algorithm does not require collocated actuator–sensor pairs for stability, but it requires the measurement of all state variables, which is a difficult proposition. This is another reason for preferring output feedback over full-state feedback [13]. Ray [14] presented a simple method of closedform solution for optimal control of thin symmetric laminated plates with output feedback using distributed piezoelectric sensors and actuators. Bhattacharya et. al. [15]proposed a new control strategy
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based on Independent Modal Space Control (IMSC) technique and used for the vibration suppression of spherical shells made of laminated composites. Zabihollah et.al. [16] investigated the vibration control of the new generation of smart structures using the LQR strategy with finite element model based on the
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layerwise theory of Reddy [17]. Kusculuoglu and Royston [18], Kapuria and Yasin [19] used a reducedorder state space model for active vibration suppression of piezoelectric laminate plates using both
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classical (CGVF) and optimal control strategies (LQR). In the literature reviewed above in active vibration control of piezoelectric composite plates with meshfree method, the first order shear deformation theory are used, whereas this theory for thick plate and high frequency response is less accurate. Thus, in present job the higher order theory is accomplished. Controllers used, in above literature are not appropriate. In some works, the CVGF controller is employed which is a simple controller, but the stability of the system is not guaranteed in general conditions. Moreover, in some works the LQR controllers are used. In spite of efficient stability LQR controller, it requires the measurement of all state variables, which is essentially troublesome. In the 3
ACCEPTED MANUSCRIPT present work, in order to cope with this recent drawback, the LQR with output feedback is recommended. In this controller both stability of the system as well as the swiftness of the time response expenditure are satisfied. Therefor the objective of this paper is to develop the EFG method based on the third-order shear deformation theory for static, free vibration and dynamic control of piezoelectric composite plates integrated with sensors and actuators. The active vibration control capability is studied using a simple,
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CGVF and LQR with output feedback. The accuracy and reliability of the proposed method is verified by
2.1
Theory and formulation
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comparing its numerical predictions with those of other available numerical approaches.
Linear piezoelectric constitutive equations
The quasi-static linear piezoelectric constitutive equations can be defined as: = +
= − .
(1) (2)
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where D, E, σ and ε represent the vectors of electric displacement, electric field, stress, and strain,
respectively; and c, k, e denote the matrices of plane-stress reduced elastic for a constant electric field, dielectric constant at constant mechanical strain and piezoelectric stress constant, respectively.
(3)
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= −
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The electric field potential relation is given by:
The plane-stress elastic constantsc are given as:
"" ! #" = 0 0 0
"#
## 0 0 0
0 0
%% 0 0
0 0 0
&& 0
0 0* ) 0) 0)
'' (
(4)
where the material constants are given by:
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"" =
" ,"# # # , "# = , ## = , 1 − ,"# ,#" 1 − ,"# ,#" 1 − ,"# ,#"
(5)
%% = ."# , && = ."/ , '' = .#/
2.2
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in which is Young moduli, . is the shear moduli and ν is the Poisson’s ratios.
Displacements and strains based on TSDT
For the TSDT, we assume the following displacement field [17]:
92 , 3, 45 = 96 2 , 35
96 : 3
(6)
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;2 , 3, 45 = ;6 2 , 35 + 47< 2 , 35 − 4 / " 87< +
96 :
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12 , 3, 45 = 16 2 , 35 + 47 2 , 35 − 4 / " 87 +
where c" = 4/3h# and 2u6 , v6 , w6 5 denote the displacements of a point on the midplane in 2x, y, z5 direction andGφI , φJ K are rotation about 2y, x5 respectively.
The in-plane strains are thus expressed by the following equation: <<
L< = 265 + 4 2"5 + 4 / 2/5
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= where:
16, 7, 7, + 96, 2"5 2/5 ; 7 7<,
(10)
For an orthotropic lamina, the strain–stress relations in local coordinates can be denoted in the form of
5
2.3
a"# a## a%# 0 0
a"% a#% a%% 0 0
0 0 0 a&& a'&
0 ` ` _ 0 * [ ) YL<< Y 0 ) < LT ^ a&' ) Z Y Y a'' ( X L
(11)
Approximation of displacement by MLS method
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` a"" !a [ _ << #" Y Y \< = a%" Z \T ^ 0 Y\ Y X
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The function of a field variable u2x5 in the domain Ω is approximated with an approximation function
ub 2x5. The MLS approximates the field function in the following form [20]: =e
f 2g5h2g5
l
= i j 2d5k 2d5
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1
c 2d5
mno
(12)
e2g5 = j" 2d5 h2g5 = k" 2d5
j# 2d5
k# 2d5
… …
jl 2d5 ,
kl 2d5 .
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where the basis p2d5 and the related coefficients a2d5 are define as
(13)
where p2x5 is a vector of basis functions that consists of monomials of the lowest order of m. The basis function p2x5 can be represented as the following form:
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ef 2d5 = 1, , 3, # , 3, 3 # , s = 6
(14)
Noting that a2x5 is an arbitrary functions of x. in order to determine this value, a function of weighted x
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residual is constructed with using nodal and approximated value as: u = i 92g − g v 5Sub 2d5 − 1w U , wn"
#
(15)
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where n is the number of nodes in the influence domain of x and w2g − g y 5 is a weight function.
In MLS Method, the value of a(x) should be chosen in a form to satisfy the following relationship as: u = 0; z
(16)
|2g5z2g5 = }2g5~
(17)
which results in the following linear equation system:
from which 6
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z2g5 = |o 2g5}2g5~
(18)
where
|2d5 = i 92d − dw 5 e2dw 5e 2dw 5, no
(19)
}2d5 = 92d − d" 5e2d" 5, … , 92d − dx 5e2dx 5
functions as: 1
c 2g5
x
= i w 2g5 1w wn"
2g5 = ef 2g5o 2g5w 2g5
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where the nodal shape function N 2g5 is determined by:
(20)
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Substituting Eq.(17) into Eq.(12), the approximated value of ub 2x5 can be expressed in terms of the shape
(21)
In this paper circular influence domains are used to construct MLS shape function. Meanwhile, the weight functions should be continuous and positive in their support domain. Therefore, the quartic spline weight function is used as:
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1 − 6 # + 8 / − 3 & , < 1 925 = 0 > 1 ‖dw − d‖ w
=
(23)
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with
(22)
2.4
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where dv is the support size of node I.
Variational form of system equations
The equations of motion of a piezoelectric composite plate can be derived by Hamilton Variational principle:
− + = 0
(24)
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potential is denoted by W. as the MLS shape functions 2d5 do not satisfy the Kronecker delta condition
generally, i.e. Gd K ≠ δ a penalty method is used to satisfy the essential boundary conditions.
¢
£
¢
¡
ª 5 «2©~ − ~ ª 5¨ = 0. − 2©~ − ~ ¢
¡
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G~ ~ + − K − 2~ ¤¥ + ¦ ¤§ 5¨
(25)
where {u} is displacement vector, ¦ is the electric potential vector and is the density, respectively..
For SS1 simply supported:
© = ±k²1
1
0
1
1
0
1
k 1 k = ± , 2 ´ 0 k 3 = ± , 2
For SS2 simply supported:
© = ±k²0
0 1
1 1
0 1
k 1 k = ± , 2 ´ 0 k 3 = ± , 2
(26)
(27)
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© = ±k²1
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© = ±k²0
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is the matrix depends on boundary conditions for example[21]:
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¤¬ , ¤® ¯, 1°, are prescribed surface force, surface charge and displacement on boundary, respectively. R
2.5
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α is a diagonal matrix of penalty factors whose their elements are 1 × 10"/ .
Discrete model construction
In order to construct the discrete model of the problem using the MLS method, the displacements and electric potential fields should be expressed in terms of nodal variables. Since the thickness of the piezoelectric layer is negligible compared to the other layers, thus, the electric potential functions,
ϕ¸ 2x, y, z, t5, have linear variations across the thickness of the piezoelectric layer. Moreover, the values of 8
ACCEPTED MANUSCRIPT potential function on the surfaces of the piezoelectric layer and substrate are assumed to be zero. Therefore, the potential function is followed as: 2 , 3, 4, 5 =
4 − ℎ/2 6 2 , 3, 5 ℎ»
(28)
where
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6 = S¼ U ½
(29)
where ℎ, ℎ» , 6 are substrate thickness, piezoelectric thickness and electric potential at any point on the
surface of the actuator and sensor layers respectively, while S¼ U is the EFG shape function matrix.
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Moreover, the electric field potential is given by Substituting Eq.(28) into Eq.(3):
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¾ = S¿¼ US}¼ U ½
where 4 − ℎ/2 ℎ» 0 0
−
0
4 − ℎ/2 ℎ» 0
0 * ) ¼, ) 0 ) , S}¼ U = O¼,< R ¼ ) 1) − ℎ» (
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S¿¼ U =
!−
(30)
(31)
The displacements can be defined in terms of nodal variables
where 0
0
0
/
" 4 3 1
" 4 /
4 − " 4 /
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¿ =
!1 0
1
(32)
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~ = ¿À ~½
0 0
* ) 4 − " 4 / ) ) 0 ( 0
(33)
Using infinitesimal engineering strain, the strain vector at any point can be defined as: = } ~½
(34)
Substituting Eqs.(30-34) in Eq. (25), results in final form of equations of motion written as:
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 Á 0
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ª Å 0 ~ à V W + Ä Å¼ 0
ż ~ Ç¥ Æ Á à = VÇ W −ż¼ ¦ §
(35,36)
where V
Šż
} É } ż W = ∬Ì Ä Å¼¼ }¼ ¿¼ Ê}~
Î = Ï Ï } Î 4 Ò, Î «} Å Ð
} Ê ¿¼ }¼ Æ 4Ω, }¼ ¿¼ Ë¿¼ }¼
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Ñ
ª = Å + Å Î , Å ÇÓ±
V Ç W = ÏÒ Ä Ô
¿ À~ ¤Ó± ¿ À ¤Ô
Æ Ò,
(37)
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 = ÏÌ Ï ¿À ¿À 4Ω,
Â,Å ,SÅ~¼ U Õ= Sż U Ö,Sż¼ U are global mass, global mechanical stiffness, global electrical-
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mechanical coupling stiffness and global piezoelectric permittivity matrix, respectively. Ç¥ , ǧ ¯ are
Î U attained from external mechanical force and electrical force vector. The additional stiffness matrix SÅ
penalty technique is used to satisfy the essential boundary conditions. According to the Eq.(26) for SS1 boundary condition:
Î w = ±k²w }
0
×w
×w
0
ØÙ w ¯ k = ±k/2,
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Î w = ±k²0 }
ØÚ w
0 k 3 = ±´/2,
(38)
Substituting Eq.(36) in Eq.(35):
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ª + Sż USż¼ Uo Sż UÖ ~ = Ç¥ + Sż USż¼ Uo ǧ ¯ Â~ + ÕÅ
(39)
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This model is valid for any number and location of piezoelectric patch on thick plates. In addition the applied voltage on the actuators can have any general distribution.
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Active vibration control
The piezoelectric laminate composite plate consists of layers as shown in Fig.1. In this paper, in order to control the active vibration of the plate, the velocity feedback and LQR with output feedback controller
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3.1
Displacement and velocity feedback
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In distributed sensor layer, the electric charges are induced when the structure is oscillating. These charges are amplified, using the appropriate electronic circuit. In addition, the output voltage of the amplifier is fed back into the distributed actuator through closed loop control algorithm. In actuator layer the generated force due to the converse piezoelectric effect can damp the vibration of the plate. With
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assuming that the applied charge F® ¯ is zero and also the converse piezoelectric effect is negligible, the ¬
¦¥ = Sż¼ U Sż U ~ o ¥
¥
The sensor charge due to the deformation:
ܧ ¯¥ = Sݼ U ~ ¥
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generated potential on the sensor layer can be derived from Eq.(36) as:
(40)
(41)
The actuating voltage ¦Þ with closed loop control is implemented as:
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Þ = .ß ¥ + .à á¥
(42)
where Gã and Gä are the displacement and velocity feedback control gains. The input voltage for the
distributed actuator is expressed by substituting the Eqs.(40,42) into Eq.(36) : ǧ ¯Þ = Sż U ~ − .ß Sż¼ U Sż¼ U Sż U ~ − .à Sż¼ U Sż¼ U Sż U ~á o
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Þ
Þ
¥
¥
Þ
o ¥
¥
(43)
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With substituting Eqs.(40,43) into Eq.(39), the dynamic equation become: Â~ + 2åÞ + åæ 5~á + Å∗ ~ = Ç¥
(44)
åÞ = .à Sż U Sż¼ U Sż U
(45)
where åÞ , åæ are active damping, Rayleigh damping matrix, respectively. These matrixes are given by: Þ
ª åæ = è + éÅ
o ¥
¥
(46)
ª + .ß Sż U Sż¼ U Sż U Å∗ = Å Þ
o ¥
¥
(47)
è, é are Rayleigh coefficients that can be determined from experiments. 11
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3.2 Optimal control with output feedback In continues structure, the size of a discretized model may become very large [22]. Then, before solving the eigenvalue problem it may be advisable to reduce the size of the model by condensing the degrees of
freedom. Hence, a truncated modal matrix ê for the first m modes can be assumed to transform the
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element free displacement vector ~ to the so called modal displacement vector ë as shown below:
~25 = êë25
(48)
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Since the potential distribution is known on the actuators layers [8], the equation of motion can be defined in term of mechanical displacement and voltage distribution on the actuator layer, followed as: ª + Sż U Sż¼ Uo Sż U Ö ~ = Ç¥ − Sż U ¦Þ Â~ + ÕÅ ¥
¥
Þ
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¥
(49)
As the converse piezoelectric effect is negligible, the voltages induced into the sensor are not adequate to have any considerable effect on the dynamic response. Therefore, the electrical stiffness matrix in Eq.(49) can be omitted. Consequently, the charge generated in the sensor layer can be obtained as:
ïð
where
ïð
(51)
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Å¥ = } Tnc î
(50)
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Q ¥ = T |Tnc î = Å¥ ~
The sensor output voltage V¬ can be expressed in term of mechanical displacement rate of the sensor
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layer. Hence:
ò¥ 25 = .ó Å¥ ~á
(52)
.ó is the coefficient of the charge amplifier.
With considering the Rayleigh damping, substituting Eq.(48) in Eq.(49) then pre-multiplying with ê : ª æ½ß ~ = ê Ç¥ − ê Sż U ¦Þ Âæ½ß ë + åæ½ß ëá + Å Þ
where
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(53)
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ª æ½ß = ê Å ª ê Âæ½ß = ê Âê, åæ½ß = ê åê, Å
(54)
In the lack of an applied mechanical load, this equation can be expressed into state-space form as: ôá = |õ + }òÞ
(55)
ö = åô
(56)
0 |=V −Âo æ½ß Åæ½ß
.ó Å¥ ê,
å = 0
0 W, } = V−Âo ê SÅ U W, õ = ë −Âo å ¼ Þ æ½ß æ½ß æ½ß
ëá
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where
(57)
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Through closed loop feedback, the output voltage sensor is negatively fed back into the distributed
òÞ = −Ýò¥ = −Ý0
.ó Å¥ êô
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actuator. Thus, the amplitude of actuator voltage can be derived in following form:
(58)
where K is the optimal gain and achieved by selecting the control input voltage to minimize a quadratic performance index:
1 ø 1 ø ô ùô + úÞ ©úÞ ûü = 2ô 2ù + å Ý ©Ýå5õ5 2 6 2 ý
(59)
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÷=
where Q and R are symmetric positive semi definite weighting matrices. This dynamical optimization problem may be converted into an equivalent static one that is easier to solve as follows. Suppose P is a
(60)
1 õ 205 þõ205 2
(61)
1 2þ5 2
(62)
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2õ þõ5 = −õ 2ù + å Ý ©Ýå5õ
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constant, symmetric, positive semi-definite matrix so that:
Assuming that the closed-loop system is asymptotically stable. Then, ÷ may be written as:
÷=
In order to relieve index ÷ of its dependence on the initial vector, the value ÷ replaced by ÷.
÷ =
where = õ205õ 205 is the autocorrelation of the initial state vector. The necessary conditions for the solution of the LQR problem with output feedback are given by:
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|ó þ + þ|ó + å6 Ý ©Ýå6 + É = 0
(63)
|ó + |ó + = 0
(64)
Ý = ©o } þå6 2å6 å6 5o
(65)
where |ó = | − }Ýå6
(66)
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In order to obtain the output feedback gain Ý minimizing the index ÷, need to solve the three coupled equations. Therefore an iterative solution algorithm is employed to solve Eqs.(63,64)[23] : 1.
set = 0.
2.
set |ó` = | − }Ý` å6
4. 5.
4
evaluate ∆Ý` = ©" } þ` ` åý Gåý åý K
o
− Ý` .
set Ý P" = Ý` + è∆Ý` , where is chosen at each iteration such that ÷`P" < ÷` .
set = + 1, go to step 2, and repeat the process until the convergence is achieved.
Numerical results
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3.
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solve for P¸ and S¸ in Eq.(63) and Eq.(64)
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Determine a gain Ý6 so that | − }Ý6 å6 is asymptotically stable.
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In the EFG method, only in performing the integration for computing the stiffness and mass matrix, a background cell structure is needed. The accuracy of the integrals evaluated depends on the background
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mesh and nodal density of the domain. Therefore, the number of gauss quadrature points in the cells depends on total number of unconstrained field nodes in the problem domain. In this study quadrilateral cells and 4 × 4 Gaussian points per cell are used as a result of its accuracy and the appropriate time
expenditure in numerical procedure. Other parameters which influence on accuracy of computations are the dimensions and the shape of the support domain. In this simulation, the support domain that specifies
the number of nodes for integration point is a circle with radius r. The radius r is defined as r = dI h ,
where dI is a dimensionless coefficient known as support size and h is distance between two adjacent
nodes. In order to demonstrate the accuracy and stability of the EFG method based on the third-order
14
ACCEPTED MANUSCRIPT shear deformation theory, various numerical examples are performed; then the results are compared with previous works done in this field.
4.1.1
Static analysis
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4.1
Static verification of the results
To accomplish static analysis for a piezoelectric composite plate, a plate with dimensions 220 cm ×
20 cm5 and simply supported boundary condition in all four edges is considered. Figure1, shows a
T300/976 graphite–epoxy laminated plate consists of four composite layer that stacking sequence of
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these layers are −θ/θ¬ and −θ/θ¬ in which subscripts ‘‘s’’ and ‘‘as’’ indicate symmetric and anti-
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symmetric ply sequence, respectively. The total thickness of this laminated plate specified as h = 1. In order to actively control the static deflection of the plate that subjected to a uniformly distributed
transverse load q = 100N/m# , two PZTG1195N layers with thickness t = 0.2 mm bonded on the top and bottom surfaces of the substrate; where these layers are firstly used as actuators. The mechanical properties of these materials are listed in Table 1. In order to control the static deflection of the plate
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subjected to an external uniform load, the voltages (0, 5, 8 and 10) are applied to the actuator layers. To verify the results, the maximum deflections of the piezoelectric laminated plate are estimated. The results are compared with those of finite element [10] and first order RPIM [5]. As listed in Table 2, an excellent
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agreement is observed due to this comparison.
Figure 2 illustrates the center line deflection of piezoelectric composite plate with various applied
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voltages. It can be found that with enhancing actuator voltage, the plate deflections intend to be completely flat. This condition can be achieved when the actuator voltage is approximately 8 volts. In addition, the effects of fiber orientation of the laminate on deflection of the plate are observed. As it is anticipated, with increasing the ply angles the maximum deflections of the plate at the similar actuator voltage have been decreased as a result of enhancing the laminate stiffness matrix.
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Active control of laminate plate
In this section, in order to actively control the static response of a simply supported piezoelectric laminated plate with stacking sequence j/−45/45¥ and similar geometry, material noted in previous
section is considered. This plate is sandwiched between two outer piezoelectric layers where the upper
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and lower layers act as actuator and sensor, respectively. Figure 3 shows the static deflection of the simply supported plate with different displacement feedback gains. As illustrated, the maximum deflection of the plate becomes smaller when the displacement feedback control gain increases. It is also observe that while Gã = 30 the plate is well flattened. Here, two
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methods are introduced for static control of the plate. The first method applies an open loop to control the actuator voltage according to laminate deflection. However, the second one employs the vibration of
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the plate to generate the electrical power. Therefore the later one is more suitable; thus it is confirmed in this work.
4.2.1
Dynamic analysis
Dynamic verification of the results
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4.2
The performance of the EFG method to evaluate the natural frequencies of the structure is considered here. The composite plate is including three layers of Graphite/Epoxy with the stacking sequence 0/90/0 and two piezoelectric (PZT-4) layers. The similar material properties listed in Table 1 are held
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in this problem. The laminate aspect and thickness ratios are considered as ℎ = 0.20 and uℎ = 50 ,
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respectively. Moreover, the density is assumed to be = 1 kg/m/ for all layers. The electric boundary
condition in the piezo layer is closed circuit where the electric potential is kept zero. The dimensionless first natural frequency of the piezoelectric composite plate ( ) are given in Table 3 for various nodal density. The obtained results are in agreements with the other methods particularly are close with the finite element method; nevertheless, there is an inevitable gap between numerical results and analytical solution. In the EFG method, with increasing nodal density in the domain of the problem, distribution shape function become smoother. Therefore, the accuracy of the approximated values of the field function at these nodes improved. The fact should be noted here is that, with enhancing the nodal density, the 16
ACCEPTED MANUSCRIPT results are going to be converged. The size of support domain as well as the nodal density are both investigated in this work according to the flowchart of Fig.4. Figure 5 shows the convergence of the dimensionless natural frequency of simply supported piezoelectric composite. In this case the support size of influence domain is constant and the number of discrete nodes
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in the domain is increased. As illustrated, is converged where the nodal density equals to 17 × 17.
In order to investigate the effect of influence domain on accuracy and stability of the results, dimensionless natural frequencies against various support size of influence domain are depicted in Fig.6.
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In this case the convergence is occurred in influence domain equals to lÞ = 3.90. Comparing the Figs.
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(5-6) indicates that enhancing the size of support domain will results in more accurate results rather than using the nodal density improvement. The first four vibration modes of the simply supported piezoelectric laminate plate are illustrated in Fig.7. As depicted, the amplitude of 4th mode could be ignored comparing with the first three modes. Therefore, the modal matrices are assembled considering
4.2.2
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these first three modes.
Active control with using velocity feedback controller
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The presented mesh-less model is extended to the investigation of dynamic active vibration control of simply supported piezoelectric laminated plate. Therefore, the effect of choosing various velocity gains on
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active damping and vibration suppression is investigated. Here, the velocity feedback controller is used for active vibration control of the laminated plate. As previously noted, in this method the stability of the system is guaranteed only when the actuators and sensors are truly collocated. The piezo-laminated composite plate consist of three layers 218cm × 18cm5 of graphite epoxy composite 20,90,05 and the substrate covered with two layers of PVDF that the upper layer served as actuator and the lower ones served as sensor. The material properties of the laminates are given in Table 1. Each layer of the substrate is 2mm thick and each PVDF layer is 0.10mm thick. In this simulation in order to model the vibration
control, firstly the plate is subjected to a uniform load of q6 = 100 N⁄m# ; then within 1 × 10& seconds after, the plate is experienced the free vibration. Moreover, the proportional damping with the values of 17
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è = 1 × 10% and é = 0.965 × 10/ is considered. In order to analyze the dynamic response of the plate the modal superposition is used to reduce the size of the problem. Figure 8 illustrates the transient response of the plate using various velocity feedback gains. It can be seen that enhancing the control gain cause the dynamic response to be faster and the magnitude of the
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response to be decreased. Moreover, in a situation where the velocity feedback gain is zero, although there is not active vibration control, the response will be decreased in a time domain due to structural damping. In addition, the presence of variety of velocity feedback gain, ranging from 0 to 1.0, result into damping ratio listed in Table 4. It should be noted that the input voltage in actuator layer is restricted as a
4.2.3
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the velocity feedback gain should be bounded properly.
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result of the fact that enhancing the voltage causes the response to be steady and saturated. Therefore,
Optimal control with LQR output feedback
In order to simulate the LQR output controller for active vibration control, a piezoelectric laminated plate with characteristic given in previous section is considered. The weighting matrices Q and R should be
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selected in an appropriate way. The elements of these matrices can be adjusted based on the priority of minimizing the vibration energy as well as the requirements on minimizing the control voltage. In the present work, the output matrix [C] is obtained with considering the velocity feedback. Whereas, the time
constant is in the range of microsecond then the value of charge amplifier gain is Gó = 1.6 × 10 . In this
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simulation, the plate is initially subjected to a uniform load of intensify Ô = 100 ⁄s# . Then, the load is
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removed which leads into free vibration of the plate as well as switching on the LQR controller. Here, the effects of weighting matrices on the dynamic response and control voltage are investigated. The dynamic response of the system in presence of the active vibration control and without control showed in Fig. 9. In this case, the response is obtained based on the weighting matrices in the following form: ù = Ô Á ã ý
ý Ã , = y y
(67)
where q and r denote the design parameters that can influence on response and control voltage and I is
identity matrix. In order to achieve the proper weighting parameters q and r, firstly, for a fixed design
18
ACCEPTED MANUSCRIPT parameter of r the value of q is increased. Then, the parameter q is designated and the value of r will be increased. Figure 10-11 demonstrate the influence of various weighting matrix Q on LQR output feedback controller. It can be seen that a greater [Q] puts higher demand on control voltage and the regulator takes a shorter
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time to achieve the zero state variables. It can be also deduced that with enhancing this weighting matrix the response will be faster. Moreover, as results of choosing the weighting matrices in a high level, the optimal gain will be placed in a high level. Therefore, the close loop poles of the system are going to be far from the open loop poles. In this situation, the system will be sensitive in respect to the background
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noises. Consequently, the peak value of the actuator voltage will exceed the breakdown voltage.
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Figure 12 demonstrates the control voltage with respect to various weighting matrix R while the weighting matrix Q is constant. It can be deduced that, with increasing the weighting matrix R the control voltage will be reduced. However, as illustrated in Fig. 13, the amplitude of response for the simply supported piezo-laminated plate is increased. Considering both response and input voltage, it is found
ù = Á ã ý
ý Ã , = 0.0001y y
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that the appropriate controller can be acquired by selecting the weighting matrices in following form:
(68)
For these values of the weighting matrices the convergence of cost function (J) against the iteration numbers demonstrates in Fig. 14. As illustrated, the cost function is converged into the minimum point
5
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with increasing the successive iteration numbers.
Conclusion
In this paper an efficient method, known as the element free galerkin (EFG), was presented to control the vibration of piezoelectric laminated plate bonded with piezoelectric actuator and sensor layers. The equation of motion for piezo-laminated composite plate was obtained from third order shear deformation theory. The derived shape function could not satisfy the delta function property. Therefore, the penalty technique was used because it has higher efficiency than Lagrange multipliers and orthogonal 19
ACCEPTED MANUSCRIPT transformation method. A constant displacement feedback control algorithm was introduced to regulate static deflection of simply supported plate. The influence of stacking sequence and various input actuator voltage on static deflection was also investigated. In addition, the convergence of fundamental natural frequency with various nodal density and support size was demonstrated. In order to control the dynamic behavior of piezolaminated composite plate, a velocity feedback and optimal steady-state regulator with
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output feedback algorithm were designed. Moreover, the presented results are compared with those of other researches which indicated an excellent agreement. Furthermore, several numerical examples were presented for static control, convergence of natural frequency, vibration mode and dynamic control of
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piezoelectric laminated plates with different stacking sequences. Particularly, the remarkable conclusions from the present formulation and numerical results are followed as:
In the EFG method the integration cells do not require to be attuned with the scattered nodes
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1.
distributed on the problem domain. Therefore, the generation of background mesh is more easily than the FEM methods. 2.
The results of piezo-laminated composite plate indicate that the HSDT can give more accurate results rather than classical or FSDT. Moreover, it does not require the shear correction factor. In dynamic control using CVGF controller, the actuators and sensors have to truly be collocated.
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3.
However, this issue is not important in LQR controller. In order to design the efficient LQR controller, it requires the measurement of all state variables. This
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is a difficult suggestion to do. Therefore, the LQR with output feedback is designed in the present work. Consequently, the need to measure all state variables is avoided.
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4.
20
ACCEPTED MANUSCRIPT References
J. A. Mitchell and J. N. Reddy, “A refined hybrid plate theory for composite laminates with piezoelectric laminae,” Int. J. Solids Struct., vol. 32, no. 16, pp. 2345–2367, 1995.
[2]
G.-R. Liu and Y.-T. Gu, An introduction to meshfree methods and their programming. Springer, 2005.
[3]
K. M. Liew, X. Q. He, M. J. Tan, and H. K. Lim, “Dynamic analysis of laminated composite plates with piezoelectric sensor/actuator patches using the FSDT mesh-free method,” Int. J. Mech. Sci., vol. 46, no. 3, pp. 411–431, Mar. 2004.
[4]
K. M. Liew, H. K. Lim, M. J. Tan, and X. Q. He, “Analysis of laminated composite beams and plates with piezoelectric patches using the element-free Galerkin method,” Comput. Mech., vol. 29, no. 6, pp. 486–497, 2002.
[5]
G. R. Liu, K. Y. Dai, and K. M. Lim, “Static and vibration control of composite laminates integrated with piezoelectric sensors and actuators using the radial point interpolation method,” Smart Mater. Struct., vol. 13, no. 6, p. 1438, 2004.
[6]
G. R. Liu and X. L. Chen, “A mesh-free method for static and free vibration analyses of thin plates of complicated shape,” J. Sound Vib., vol. 241, no. 5, pp. 839–855, 2001.
[7]
K. Y. Dai, G. R. Liu, K. M. Lim, and X. L. Chen, “A mesh-free method for static and free vibration analysis of shear deformable laminated composite plates,” J. Sound Vib., vol. 269, no. 3, pp. 633– 652, 2004.
[8]
B. Samanta, M. C. Ray, and R. Bhattacharyya, “Finite element model for active control of intelligent structures,” AIAA J., vol. 34, no. 9, pp. 1885–1893, 1996.
[9]
J. M. S. Moita, C. M. M. Soares, and C. A. M. Soares, “Active control of forced vibrations in adaptive structures using a higher order model,” Compos. Struct., vol. 71, no. 3, pp. 349–355, 2005.
[10]
P. Phung-Van, T. Nguyen-Thoi, T. Le-Dinh, and H. Nguyen-Xuan, “Static and free vibration analyses and dynamic control of composite plates integrated with piezoelectric sensors and actuators by the cell-based smoothed discrete shear gap method (CS-FEM-DSG3),” Smart Mater. Struct., vol. 22, no. 9, p. 095026, Sep. 2013.
[11]
P. Phung-Van, L. De Lorenzis, C. H. Thai, M. Abdel-Wahab, and H. Nguyen-Xuan, “Analysis of laminated composite plates integrated with piezoelectric sensors and actuators using higherorder shear deformation theory and isogeometric finite elements,” Comput. Mater. Sci., 2014.
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[1]
T. I. Thinh and L. K. Ngoc, “Static behavior and vibration control of piezoelectric cantilever composite plates and comparison with experiments,” Comput. Mater. Sci., vol. 49, no. 4, pp. S276– S280, 2010.
[13]
F. L. Lewis, D. Vrabie, and V. L. Syrmos, Optimal control. John Wiley & Sons, 2012.
[14]
M. C. Ray, “Optimal control of laminated plate with piezoelectric sensor and actuator layers,” AIAA J., vol. 36, no. 12, pp. 2204–2208, 1998.
[15]
P. Bhattacharya, H. Suhail, and P. K. Sinha, “Finite element analysis and distributed control of laminated composite shells using LQR/IMSC approach,” Aerosp. Sci. Technol., vol. 6, no. 4, pp. 273– 281, 2002. 21
ACCEPTED MANUSCRIPT A. Zabihollah, R. Sedagahti, and R. Ganesan, “Active vibration suppression of smart laminated beams using layerwise theory and an optimal control strategy,” Smart Mater. Struct., vol. 16, no. 6, p. 2190, 2007.
[17]
J. N. Reddy, Mechanics of laminated composite plates and shells: theory and analysis. CRC press, 2004.
[18]
Z. K. Kusculuoglu and T. J. Royston, “Finite element formulation for composite plates with piezoceramic layers for optimal vibration control applications,” Smart Mater. Struct., vol. 14, no. 6, p. 1139, 2005.
[19]
S. Kapuria and M. Y. Yasin, “Active vibration suppression of multilayered plates integrated with piezoelectric fiber reinforced composites using an efficient finite element model,” J. Sound Vib., vol. 329, no. 16, pp. 3247–3265, 2010.
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K. M. Liew, Y. Q. Huang, and J. N. Reddy, “Analysis of general shaped thin plates by the moving least-squares differential quadrature method,” Finite Elem. Anal. Des., vol. 40, no. 11, pp. 1453– 1474, 2004.
[21]
S. Kapuria and P. Kumari, “Boundary layer effects in Levy-type rectangular piezoelectric composite plates using a coupled efficient layerwise theory,” Eur. J. Mech., vol. 36, pp. 122–140, 2012.
[22]
A. Preumont, Vibration control of active structures: an introduction, vol. 179. Springer, 2011.
[23]
D. D. Moerder and A. J. Calise, “Convergence of a numerical algorithm for calculating optimal output feedback gains,” IEEE Trans. Automat. Contr., vol. 30, no. 9, pp. 900–903, 1985.
[24]
A. Alibeigloo, “Free vibration analysis of functionally graded carbon nanotube-reinforced composite cylindrical panel embedded in piezoelectric layers by using theory of elasticity,” Eur. J. Mech., vol. 44, pp. 104–115, 2014.
[25]
V. M. Franco Correia, M. A. Aguiar Gomes, A. Suleman, C. M. Mota Soares, and C. A. Mota Soares, “Modelling and design of adaptive composite structures,” Comput. Methods Appl. Mech. Eng., vol. 185, no. 2, pp. 325–346, 2000.
[26]
D. A. Saravanos, P. R. Heyliger, and D. A. Hopkins, “Layerwise mechanics and finite element for the dynamic analysis of piezoelectric composite plates,” Int. J. Solids Struct., vol. 34, no. 3, pp. 359–378, 1997.
[27]
P. Heyliger and D. A. Saravanos, “Exact free-vibration analysis of laminated plates with embedded piezoelectric layers,” J. Acoust. Soc. Am., vol. 98, no. 3, pp. 1547–1557, 1995.
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ACCEPTED MANUSCRIPT Table 1 Relevant mechanical properties of respective materials (Electrical permittivity of air, = 8.85 × 10 /) Piezoelectric layer PZT-G1195N PVDF 63.0 2 63.0 2 63.0 2 24.2 1 24.2 1 24.2 1 0.3 0.29 0.3 0.29 0.3 0.29 7600 1800 2.54e-10 0.046 1728.8 12 1728.8 12 1694.9 12
TE D EP AC C
PZT-4[24] 81.3 81.3 64.5 30.6 25.6 25.6 0.329 0.432 0.432 7600 -1.23e-10 1475 1475 1300
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Gr/Ep 132.38 10.76 10.76 3.61 5.65 5.65 0.24 0.24 0.49 1578 -
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) ) ) ) ) ) kg/m ) = m/V) = C/m ) / F/m) / F/m) / F/m)
Core plate T300/979 150 9.0 9.0 7.1 7.1 2.5 0.3 0.3 0.3 1600 -
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property
ACCEPTED MANUSCRIPT Table 2 Central node deflection of the simply supported piezoelectric laminate plate under different actuator voltage× 10 m Actuator voltage
method
⁄−15⁄15
⁄−30⁄30
⁄−45⁄45
⁄−45⁄45
0V
EFG CS-DSG3[10] RPIM [5]
−0.7549 −0.7442 −0.7222
−0.6657 −0.6688 −0.6542
−0.6338 −0.6323 −0.6217
−0.6340 −0.6326 −0.6038
5V
EFG CS-DSG3[10] RPIM [5]
−0.3329 −0.3259 −0.3134
−0.2984 −0.2957 −0.2862
−0.2857 −0.2801 −0.2717
−0.2892 −0.2863 −0.2717
10V
EFG CS-DSG3[10] RPIM [5]
0.0931 0.0924 0.0954
0.0782 0.0774 0.0819
0.0615 0.0601 0.0604
0.0739 0.0721 0.0757
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scheme
ACCEPTED MANUSCRIPT plate [piezo/0/90/0/piezo] Method Present method Present method Present method IGA [11] Q9–HSDT [25] HSDT-layerwise [26] CS-FEM-DSG3 [10] Analytical solution [27]
234.174 234.125 234.221 235.100 230.461 234.533 234.500 245.941
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nodes 11 × 11 13 × 13 15 × 15 -
of simply supported piezoelectric composite
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Table 3 non dimensional natural frequency =
ACCEPTED MANUSCRIPT Table 4 damping ratio of piezolaminated composite plate for various velocity feedback gains G୴ ܩ௩ Mode 1
0.0268 0.0395 0.0687
0.0512 0.1730 0.3511
0.0853 0.3136 0.6310
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0.5
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1 2 3
0
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AC C
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Fig. 1 Schematic illustration of a laminated plate with integrated piezoelectric and regularly distributed nodes
ACCEPTED MANUSCRIPT -5
2
x 10
1
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0
-2 -3
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Deflection(m)
-1
-5 -6 -7 0
0.05
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-4
0.1
0.15
0.2
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Distance x(m)
Fig. 2(a) Centerline deflections of the simply supported laminates ⁄−45⁄45 under uniform loading
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and different actuator input voltages (
0V
5V
8V
10V)
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1
x 10
0
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Deflection(m)
-1
0.1
0.15
0.2
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Distance x(m)
Fig. 2(b) Centerline deflections of the simply supported laminates ⁄−45⁄45 under uniform loading
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and different actuator input voltages (
0V
5V
8V
10V)
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1
x 10
0
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-3
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-5
-6
-7 0
0.05
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Deflection(m)
-1
0.1
0.15
0.2
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Distance x(m)
Fig. 2(c) Centerline deflections of the simply supported laminates ⁄−30⁄30 under uniform loading
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and different actuator input voltages (
0V
5V
8V
10V)
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1
x 10
0
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-1
-3 -4
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Deflection(m)
-2
-6 -7 -8 0
0.05
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-5
0.1
0.15
0.2
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Distance x(m)
Fig. 2(d) Centerline deflections of the simply supported laminates ⁄−15⁄15 under uniform loading
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and different actuator input voltages (
0V
5V
8V
10V)
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1
x 10
0 Gd=30
-3
Gd=5
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Gd=10
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maximum Deflection(m)
-1
Gd=0
-7 0
0.05
0.1
0.15
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Distance x(m)
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Fig. 3 static deflection of simply supported plate with different displacement control gain Gୢ .
0.2
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Geometry generation
For all the cells in the background mesh
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For all the quadrature points in the cell
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Node and background mesh generation
Search in the current cell and its surrounding cells of integration for all the nodes
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that fall in the support domain of quadrature point
MLS shape function creation-based selected nodes
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Calculate nodal matrices
Assemble the nodal matrix into the global matrix
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Solve eigenvalue problem
If ߱ ሺ݊ + 1ሻ − ߱ ሺ݊ሻ < 0.01
Yes
Plot convergence of Eigen value
Fig. 4 Flowchart of the converging algorithm
No
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234.2
234.1 12
14
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Non dimensional natural frequency
234.3
16
18
20
Nodal density
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Fig.5 convergence of the dimensionless natural frequency with respect to various nodal density
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234
233.5 2.5
3
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Non dimention natural frequency
234.5
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dmax
4
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Fig. 6convergence of the dimensionless natural frequency with respect to various sizes of the influence domains
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0.035 0.03 0.025
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z
0.02 0.015 0.01 0.005
1
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0
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
y
0
x
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0
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Fig. 7a First mode natural frequency of simply supported plate piezo⁄0 /90/ 0⁄piezo
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0.015
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0.01
z
0.005 0 -0.005 -0.01
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-0.015
1 0.8
1
0.8
0.6
0.4
0.2
y
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0.6
0.4
0.2
0
0
x
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Fig. 7b Second mode natural frequency of simply supported plate piezo⁄0 /90/ 0⁄piezo
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0.02
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z
0.01
0
-0.01
-0.02 0.8
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1
1
0.6
0.8
0.6
0.4
0.4
0.2 0
0.2 0
x
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y
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Fig. 7c Third mode natural frequency of simply supported plate piezo⁄0 /90/ 0⁄piezo
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-3
x 10
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z
5
-5
1
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0.8 0.6
1
0.4
0.8
y
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0.6
0.2
0.4
0
0.2
0
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Fig. 7d Fourth mode natural frequency of simply supported plate piezo⁄0 /90/ 0⁄piezo
ACCEPTED MANUSCRIPT -7
3
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Gv=0 Gv=0.5 Gv=1
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0.01
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maximum deflection(m)
2
0.02
0.025
0.03
0.035
0.04
0.045
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time(sec)
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Fig. 8maximum deflection of the simply supported laminated plate covered with piezoelectric layers with respect to various velocity feedback gains.
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5
x 10
Without Control With active Control
4
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2 1 0 -1
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maximum deflection(m)
3
-2
-4 -5 0
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0.01
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0.02
0.025
0.03
0.035
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time(sec)
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Fig. 9center deflection of simply supported piezoelectric laminated plate without control and with active control (LQR controller).
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x 10
-7
q=1 q=10 q=100
4
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0 -1 -2 -3 -4
0.01
0.02 0.03 time(sec)
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maximum deflection(m)
3
0.04
0.05
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Fig.10 center deflection of simply supported plate using LQR output feedback controller and various design parameter q
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300 q=1 q=10 q=100
200
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0
-100
-200
-300
0.01
0.02 0.03 time(sec)
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control voltage(Volt)
100
0.04
0.05
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Fig.11 input voltage of simply supported plate using LQR output feedback controller and various design parameter q
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80 r=0.0001 r=0.0003
60
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20
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0 -20
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control voltage(Volt)
40
-40 -60
-100 0
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-80
0.01
0.02 0.03 time(sec)
0.04
0.05
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Fig.12 control voltage of simply supported plate using LQR output feedback controller and various design parameter r
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x 10
-7
r=0.0001 r=0.0003
4
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2 1
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0 -1 -2 -3 -4 0.01
0.02 0.03 time(sec)
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3
0.04
0.05
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Fig.13 center deflection of simply supported plate using LQR output feedback controller and various design parameter r
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5.5
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5 4.5
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4
3
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J
3.5
2.5
1.5 1 0.5 0
5
10
15
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20
25
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Itration Number
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Fig.14 convergent procedure of cost function
30
35
40
45
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highlights
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1. We model a piezoelectric plate with TSDT and meshfree galerkin method. 2. We design optimal (LQR with output feedback) controller with using meshfree model. 3. The dynamic response of the system is more accurate than FEM methods.
S0997-7538(15)00125-4
DOI:
10.1016/j.euromechsol.2015.09.004
Reference:
EJMSOL 3234
To appear in:
European Journal of Mechanics / A Solids
Received Date: 21 April 2015 Revised Date:
28 July 2015
Accepted Date: 10 September 2015
Please cite this article as: Talebitooti, R., Daneshjoo, K., Jafari, S.A.M, Optimal control of laminated plate integrated with piezoelectric sensor and actuator considering TSDT and meshfree method, European Journal of Mechanics / A Solids (2015), doi: 10.1016/j.euromechsol.2015.09.004. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT Optimal control of laminated plate integrated with piezoelectric sensor and actuator considering TSDT and meshfree method R. Talebitootia, K. Daneshjoob, S.A.M.Jafaric Prof, Sch. of Mech. Eng., Iran Univ. of Sci. and Tech., Teheran 16844, Iran;
bProf,
Sch. of Mech. Eng., Iran Univ. of Sci. and Tech., Teheran 16844, Iran;
cMSc.
Stu, Sch. of New Technologies, Iran Univ. of Sci. and Tech., Teheran 16844, Iran;
Abstract
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aAsst.
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In this paper, the element free galerkin (EFG) method based on third-order shear deformation theory (TSDT) is used to investigate shape and vibration control of piezoelectric laminated plate bonded with
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piezoelectric actuator and sensor layers. The electric potential distributions through the thickness for each piezoelectric layer are assumed to vary linearly. In addition, a closed-loop velocity feedback control and optimal steady-state regulator with output feedback algorithm is used for the active control of the static deflection as well as the dynamic response of the plates with bonded distributed piezoelectric sensors and actuators. Furthermore, the effects of the size of support and nodal density on the numerical
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accuracy are also investigated. The results indicate that, the accuracy and reliability of presented work have an excellent agreement with those of other available numerical approaches such as finite element and FSDT meshfree method.
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Keywords: Optimal control, Meshfree method, TSDT theory, Sensor and actuator
Introduction
The study of embedded or surface-mounted piezoelectric materials in structures has received considerable attention in recent two decades [1]. It is due to possibility of creating certain types of structures and systems capable of adapting to or correcting for changing operating conditions. The advantage of incorporating these special types of material into the structure is due to the fact that the sensing and actuating mechanism becomes part of the structure by sensing and actuating strains directly.
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ACCEPTED MANUSCRIPT Classical mathematical models have already been well established for the phenomena in the areas of mechanics of solids and structures. Meanwhile, different types of differential or partial differential equations (PDEs) that govern these phenomena have also been derived. There are largely two categories of numerical methods for solving these PDEs [2]: direct approach and indirect approach. The direct approach known as strong form methods (such as the finite difference method (FDM) and collocation
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method) discretizes and solves the PDEs directly, and the indirect approach known as weak form methods (such as finite element method (FEM)) establishes first an alternative weak form system equation that governs the same physical phenomena and then solves it. The weak form equations are
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usually in an integral form, implying that they need to be satisfied only in an integral (averaged) sense. The (EFG), one of known category of meshfree, is a standard weak formulation that is variationally
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consistent due to the use of compatible moving least squares (MLS) shape functions and the Galerkin approach with constraints to impose the essential boundary conditions. In the EFG method, in order to derive the stiffness matrix, some complex integrals should be solved via numerical procedures. Therefore, a need for background mesh to perform the numerical integration is unavoidable. Subsequently, in the EFG method the integration cells do not require to be attuned with the scattered nodes distributed on the
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problem domain. Hence, the generation of background mesh is more easily than the FEM methods. Liew et. al. [3], [4] represented a formulation by employing the EFG method based on the first-order shear deformation theory to study the shape control and vibration suppression of piezo-laminated composite beams and plates. Liu et. al. [5] developed the previous work with employing the point interpolation
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method using radial basis functions (RPIM) based on the first-order shear deformation theory.
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In EFG method, the shape functions constructed by MLS approximation do not consider the property of delta functions. Hence, the essential boundary conditions cannot be imposed as conveniently as the standard FEM method. Liu and Chen [6] consider the orthogonal transformation techniques to impose the essential boundary conditions for free vibration analysis of a thin plate. Dai et. al. [7] used the penalty method to impose the essential boundary conditions for the static deflection and free vibration analysis of composite plates. This proposed method presented by [7] is more efficiently than orthogonal transform method [6].
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ACCEPTED MANUSCRIPT Samanta et. al. [8] represented a generalized finite element formulation with an eight noded twodimensional quadratic quadrilateral isoparametric element for active vibration control of a laminated plate integrated with piezoelectric polymer layers acting as distributed sensors and actuators based on third-order shear deformation theory. Similar works developed finite element formulation for active
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vibration control of smart structure [9–12]. The modern control techniques are to be widespread in designing the stability augmentation systems. This is accomplished by regulating certain states of the system to zero while obtaining desirable closedloop response characteristics. Output feedback will allow engineers to design plant controllers of any
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desired structure. In the constant gain velocity feedback control strategy (CGVF), the stability of the system is guaranteed only when the actuators and sensors are truly collocated. Moreover, the linear
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quadratic regulator (LQR) algorithm does not require collocated actuator–sensor pairs for stability, but it requires the measurement of all state variables, which is a difficult proposition. This is another reason for preferring output feedback over full-state feedback [13]. Ray [14] presented a simple method of closedform solution for optimal control of thin symmetric laminated plates with output feedback using distributed piezoelectric sensors and actuators. Bhattacharya et. al. [15]proposed a new control strategy
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based on Independent Modal Space Control (IMSC) technique and used for the vibration suppression of spherical shells made of laminated composites. Zabihollah et.al. [16] investigated the vibration control of the new generation of smart structures using the LQR strategy with finite element model based on the
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layerwise theory of Reddy [17]. Kusculuoglu and Royston [18], Kapuria and Yasin [19] used a reducedorder state space model for active vibration suppression of piezoelectric laminate plates using both
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classical (CGVF) and optimal control strategies (LQR). In the literature reviewed above in active vibration control of piezoelectric composite plates with meshfree method, the first order shear deformation theory are used, whereas this theory for thick plate and high frequency response is less accurate. Thus, in present job the higher order theory is accomplished. Controllers used, in above literature are not appropriate. In some works, the CVGF controller is employed which is a simple controller, but the stability of the system is not guaranteed in general conditions. Moreover, in some works the LQR controllers are used. In spite of efficient stability LQR controller, it requires the measurement of all state variables, which is essentially troublesome. In the 3
ACCEPTED MANUSCRIPT present work, in order to cope with this recent drawback, the LQR with output feedback is recommended. In this controller both stability of the system as well as the swiftness of the time response expenditure are satisfied. Therefor the objective of this paper is to develop the EFG method based on the third-order shear deformation theory for static, free vibration and dynamic control of piezoelectric composite plates integrated with sensors and actuators. The active vibration control capability is studied using a simple,
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CGVF and LQR with output feedback. The accuracy and reliability of the proposed method is verified by
2.1
Theory and formulation
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2
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comparing its numerical predictions with those of other available numerical approaches.
Linear piezoelectric constitutive equations
The quasi-static linear piezoelectric constitutive equations can be defined as: = +
= − .
(1) (2)
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where D, E, σ and ε represent the vectors of electric displacement, electric field, stress, and strain,
respectively; and c, k, e denote the matrices of plane-stress reduced elastic for a constant electric field, dielectric constant at constant mechanical strain and piezoelectric stress constant, respectively.
(3)
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= −
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The electric field potential relation is given by:
The plane-stress elastic constantsc are given as:
"" ! #" = 0 0 0
"#
## 0 0 0
0 0
%% 0 0
0 0 0
&& 0
0 0* ) 0) 0)
'' (
(4)
where the material constants are given by:
4
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"" =
" ,"# # # , "# = , ## = , 1 − ,"# ,#" 1 − ,"# ,#" 1 − ,"# ,#"
(5)
%% = ."# , && = ."/ , '' = .#/
2.2
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in which is Young moduli, . is the shear moduli and ν is the Poisson’s ratios.
Displacements and strains based on TSDT
For the TSDT, we assume the following displacement field [17]:
92 , 3, 45 = 96 2 , 35
96 : 3
(6)
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;2 , 3, 45 = ;6 2 , 35 + 47< 2 , 35 − 4 / " 87< +
96 :
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12 , 3, 45 = 16 2 , 35 + 47 2 , 35 − 4 / " 87 +
where c" = 4/3h# and 2u6 , v6 , w6 5 denote the displacements of a point on the midplane in 2x, y, z5 direction andGφI , φJ K are rotation about 2y, x5 respectively.
The in-plane strains are thus expressed by the following equation: <<
L< = 265 + 4 2"5 + 4 / 2/5
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= where:
16, 7, 7, + 96, 2"5 2/5 ; 7 7<,
(10)
For an orthotropic lamina, the strain–stress relations in local coordinates can be denoted in the form of
5
2.3
a"# a## a%# 0 0
a"% a#% a%% 0 0
0 0 0 a&& a'&
0 ` ` _ 0 * [ ) YL<< Y 0 ) < LT ^ a&' ) Z Y Y a'' ( X L
(11)
Approximation of displacement by MLS method
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` a"" !a [ _ << #" Y Y \< = a%" Z \T ^ 0 Y\ Y X
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The function of a field variable u2x5 in the domain Ω is approximated with an approximation function
ub 2x5. The MLS approximates the field function in the following form [20]: =e
f 2g5h2g5
l
= i j 2d5k 2d5
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1
c 2d5
mno
(12)
e2g5 = j" 2d5 h2g5 = k" 2d5
j# 2d5
k# 2d5
… …
jl 2d5 ,
kl 2d5 .
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where the basis p2d5 and the related coefficients a2d5 are define as
(13)
where p2x5 is a vector of basis functions that consists of monomials of the lowest order of m. The basis function p2x5 can be represented as the following form:
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ef 2d5 = 1, , 3, # , 3, 3 # , s = 6
(14)
Noting that a2x5 is an arbitrary functions of x. in order to determine this value, a function of weighted x
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residual is constructed with using nodal and approximated value as: u = i 92g − g v 5Sub 2d5 − 1w U , wn"
#
(15)
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where n is the number of nodes in the influence domain of x and w2g − g y 5 is a weight function.
In MLS Method, the value of a(x) should be chosen in a form to satisfy the following relationship as: u = 0; z
(16)
|2g5z2g5 = }2g5~
(17)
which results in the following linear equation system:
from which 6
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z2g5 = |o 2g5}2g5~
(18)
where
|2d5 = i 92d − dw 5 e2dw 5e 2dw 5, no
(19)
}2d5 = 92d − d" 5e2d" 5, … , 92d − dx 5e2dx 5
functions as: 1
c 2g5
x
= i w 2g5 1w wn"
2g5 = ef 2g5o 2g5w 2g5
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where the nodal shape function N 2g5 is determined by:
(20)
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Substituting Eq.(17) into Eq.(12), the approximated value of ub 2x5 can be expressed in terms of the shape
(21)
In this paper circular influence domains are used to construct MLS shape function. Meanwhile, the weight functions should be continuous and positive in their support domain. Therefore, the quartic spline weight function is used as:
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1 − 6 # + 8 / − 3 & , < 1 925 = 0 > 1 ‖dw − d‖ w
=
(23)
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with
(22)
2.4
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where dv is the support size of node I.
Variational form of system equations
The equations of motion of a piezoelectric composite plate can be derived by Hamilton Variational principle:
− + = 0
(24)
7
ACCEPTED MANUSCRIPT where T is kinetic energy and U is potential energy and the work done by external applied force and
potential is denoted by W. as the MLS shape functions 2d5 do not satisfy the Kronecker delta condition
generally, i.e. Gd K ≠ δ a penalty method is used to satisfy the essential boundary conditions.
¢
£
¢
¡
ª 5 «2©~ − ~ ª 5¨ = 0. − 2©~ − ~ ¢
¡
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G~ ~ + − K − 2~ ¤¥ + ¦ ¤§ 5¨
(25)
where {u} is displacement vector, ¦ is the electric potential vector and is the density, respectively..
For SS1 simply supported:
© = ±k²1
1
0
1
1
0
1
k 1 k = ± , 2 ´ 0 k 3 = ± , 2
For SS2 simply supported:
© = ±k²0
0 1
1 1
0 1
k 1 k = ± , 2 ´ 0 k 3 = ± , 2
(26)
(27)
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© = ±k²1
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© = ±k²0
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is the matrix depends on boundary conditions for example[21]:
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¤¬ , ¤® ¯, 1°, are prescribed surface force, surface charge and displacement on boundary, respectively. R
2.5
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α is a diagonal matrix of penalty factors whose their elements are 1 × 10"/ .
Discrete model construction
In order to construct the discrete model of the problem using the MLS method, the displacements and electric potential fields should be expressed in terms of nodal variables. Since the thickness of the piezoelectric layer is negligible compared to the other layers, thus, the electric potential functions,
ϕ¸ 2x, y, z, t5, have linear variations across the thickness of the piezoelectric layer. Moreover, the values of 8
ACCEPTED MANUSCRIPT potential function on the surfaces of the piezoelectric layer and substrate are assumed to be zero. Therefore, the potential function is followed as: 2 , 3, 4, 5 =
4 − ℎ/2 6 2 , 3, 5 ℎ»
(28)
where
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6 = S¼ U ½
(29)
where ℎ, ℎ» , 6 are substrate thickness, piezoelectric thickness and electric potential at any point on the
surface of the actuator and sensor layers respectively, while S¼ U is the EFG shape function matrix.
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Moreover, the electric field potential is given by Substituting Eq.(28) into Eq.(3):
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¾ = S¿¼ US}¼ U ½
where 4 − ℎ/2 ℎ» 0 0
−
0
4 − ℎ/2 ℎ» 0
0 * ) ¼, ) 0 ) , S}¼ U = O¼,< R ¼ ) 1) − ℎ» (
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S¿¼ U =
!−
(30)
(31)
The displacements can be defined in terms of nodal variables
where 0
0
0
/
" 4 3 1
" 4 /
4 − " 4 /
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¿ =
!1 0
1
(32)
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~ = ¿À ~½
0 0
* ) 4 − " 4 / ) ) 0 ( 0
(33)
Using infinitesimal engineering strain, the strain vector at any point can be defined as: = } ~½
(34)
Substituting Eqs.(30-34) in Eq. (25), results in final form of equations of motion written as:
9
 Á 0
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ª Å 0 ~ à V W + Ä Å¼ 0
ż ~ Ç¥ Æ Á à = VÇ W −ż¼ ¦ §
(35,36)
where V
Šż
} É } ż W = ∬Ì Ä Å¼¼ }¼ ¿¼ Ê}~
Î = Ï Ï } Î 4 Ò, Î «} Å Ð
} Ê ¿¼ }¼ Æ 4Ω, }¼ ¿¼ Ë¿¼ }¼
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Ñ
ª = Å + Å Î , Å ÇÓ±
V Ç W = ÏÒ Ä Ô
¿ À~ ¤Ó± ¿ À ¤Ô
Æ Ò,
(37)
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 = ÏÌ Ï ¿À ¿À 4Ω,
Â,Å ,SÅ~¼ U Õ= Sż U Ö,Sż¼ U are global mass, global mechanical stiffness, global electrical-
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mechanical coupling stiffness and global piezoelectric permittivity matrix, respectively. Ç¥ , ǧ ¯ are
Î U attained from external mechanical force and electrical force vector. The additional stiffness matrix SÅ
penalty technique is used to satisfy the essential boundary conditions. According to the Eq.(26) for SS1 boundary condition:
Î w = ±k²w }
0
×w
×w
0
ØÙ w ¯ k = ±k/2,
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Î w = ±k²0 }
ØÚ w
0 k 3 = ±´/2,
(38)
Substituting Eq.(36) in Eq.(35):
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ª + Sż USż¼ Uo Sż UÖ ~ = Ç¥ + Sż USż¼ Uo ǧ ¯ Â~ + ÕÅ
(39)
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This model is valid for any number and location of piezoelectric patch on thick plates. In addition the applied voltage on the actuators can have any general distribution.
3
Active vibration control
The piezoelectric laminate composite plate consists of layers as shown in Fig.1. In this paper, in order to control the active vibration of the plate, the velocity feedback and LQR with output feedback controller
10
ACCEPTED MANUSCRIPT is used. In these methods, the actuator layer, denoted with the subscript a, is located on the top layer of the substrate and the sensor layer, denoted with the subscript s, is located on the other surface.
3.1
Displacement and velocity feedback
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In distributed sensor layer, the electric charges are induced when the structure is oscillating. These charges are amplified, using the appropriate electronic circuit. In addition, the output voltage of the amplifier is fed back into the distributed actuator through closed loop control algorithm. In actuator layer the generated force due to the converse piezoelectric effect can damp the vibration of the plate. With
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assuming that the applied charge F® ¯ is zero and also the converse piezoelectric effect is negligible, the ¬
¦¥ = Sż¼ U Sż U ~ o ¥
¥
The sensor charge due to the deformation:
ܧ ¯¥ = Sݼ U ~ ¥
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generated potential on the sensor layer can be derived from Eq.(36) as:
(40)
(41)
The actuating voltage ¦Þ with closed loop control is implemented as:
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Þ = .ß ¥ + .à á¥
(42)
where Gã and Gä are the displacement and velocity feedback control gains. The input voltage for the
distributed actuator is expressed by substituting the Eqs.(40,42) into Eq.(36) : ǧ ¯Þ = Sż U ~ − .ß Sż¼ U Sż¼ U Sż U ~ − .à Sż¼ U Sż¼ U Sż U ~á o
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Þ
Þ
¥
¥
Þ
o ¥
¥
(43)
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With substituting Eqs.(40,43) into Eq.(39), the dynamic equation become: Â~ + 2åÞ + åæ 5~á + Å∗ ~ = Ç¥
(44)
åÞ = .à Sż U Sż¼ U Sż U
(45)
where åÞ , åæ are active damping, Rayleigh damping matrix, respectively. These matrixes are given by: Þ
ª åæ = è + éÅ
o ¥
¥
(46)
ª + .ß Sż U Sż¼ U Sż U Å∗ = Å Þ
o ¥
¥
(47)
è, é are Rayleigh coefficients that can be determined from experiments. 11
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3.2 Optimal control with output feedback In continues structure, the size of a discretized model may become very large [22]. Then, before solving the eigenvalue problem it may be advisable to reduce the size of the model by condensing the degrees of
freedom. Hence, a truncated modal matrix ê for the first m modes can be assumed to transform the
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element free displacement vector ~ to the so called modal displacement vector ë as shown below:
~25 = êë25
(48)
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Since the potential distribution is known on the actuators layers [8], the equation of motion can be defined in term of mechanical displacement and voltage distribution on the actuator layer, followed as: ª + Sż U Sż¼ Uo Sż U Ö ~ = Ç¥ − Sż U ¦Þ Â~ + ÕÅ ¥
¥
Þ
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¥
(49)
As the converse piezoelectric effect is negligible, the voltages induced into the sensor are not adequate to have any considerable effect on the dynamic response. Therefore, the electrical stiffness matrix in Eq.(49) can be omitted. Consequently, the charge generated in the sensor layer can be obtained as:
ïð
where
ïð
(51)
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Å¥ = } Tnc î
(50)
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Q ¥ = T |Tnc î = Å¥ ~
The sensor output voltage V¬ can be expressed in term of mechanical displacement rate of the sensor
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layer. Hence:
ò¥ 25 = .ó Å¥ ~á
(52)
.ó is the coefficient of the charge amplifier.
With considering the Rayleigh damping, substituting Eq.(48) in Eq.(49) then pre-multiplying with ê : ª æ½ß ~ = ê Ç¥ − ê Sż U ¦Þ Âæ½ß ë + åæ½ß ëá + Å Þ
where
12
(53)
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ª æ½ß = ê Å ª ê Âæ½ß = ê Âê, åæ½ß = ê åê, Å
(54)
In the lack of an applied mechanical load, this equation can be expressed into state-space form as: ôá = |õ + }òÞ
(55)
ö = åô
(56)
0 |=V −Âo æ½ß Åæ½ß
.ó Å¥ ê,
å = 0
0 W, } = V−Âo ê SÅ U W, õ = ë −Âo å ¼ Þ æ½ß æ½ß æ½ß
ëá
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where
(57)
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Through closed loop feedback, the output voltage sensor is negatively fed back into the distributed
òÞ = −Ýò¥ = −Ý0
.ó Å¥ êô
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actuator. Thus, the amplitude of actuator voltage can be derived in following form:
(58)
where K is the optimal gain and achieved by selecting the control input voltage to minimize a quadratic performance index:
1 ø 1 ø ô ùô + úÞ ©úÞ ûü = 2ô 2ù + å Ý ©Ýå5õ5 2 6 2 ý
(59)
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÷=
where Q and R are symmetric positive semi definite weighting matrices. This dynamical optimization problem may be converted into an equivalent static one that is easier to solve as follows. Suppose P is a
(60)
1 õ 205 þõ205 2
(61)
1 2þ5 2
(62)
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2õ þõ5 = −õ 2ù + å Ý ©Ýå5õ
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constant, symmetric, positive semi-definite matrix so that:
Assuming that the closed-loop system is asymptotically stable. Then, ÷ may be written as:
÷=
In order to relieve index ÷ of its dependence on the initial vector, the value ÷ replaced by ÷.
÷ =
where = õ205õ 205 is the autocorrelation of the initial state vector. The necessary conditions for the solution of the LQR problem with output feedback are given by:
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|ó þ + þ|ó + å6 Ý ©Ýå6 + É = 0
(63)
|ó + |ó + = 0
(64)
Ý = ©o } þå6 2å6 å6 5o
(65)
where |ó = | − }Ýå6
(66)
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In order to obtain the output feedback gain Ý minimizing the index ÷, need to solve the three coupled equations. Therefore an iterative solution algorithm is employed to solve Eqs.(63,64)[23] : 1.
set = 0.
2.
set |ó` = | − }Ý` å6
4. 5.
4
evaluate ∆Ý` = ©" } þ` ` åý Gåý åý K
o
− Ý` .
set Ý P" = Ý` + è∆Ý` , where is chosen at each iteration such that ÷`P" < ÷` .
set = + 1, go to step 2, and repeat the process until the convergence is achieved.
Numerical results
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3.
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solve for P¸ and S¸ in Eq.(63) and Eq.(64)
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Determine a gain Ý6 so that | − }Ý6 å6 is asymptotically stable.
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In the EFG method, only in performing the integration for computing the stiffness and mass matrix, a background cell structure is needed. The accuracy of the integrals evaluated depends on the background
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mesh and nodal density of the domain. Therefore, the number of gauss quadrature points in the cells depends on total number of unconstrained field nodes in the problem domain. In this study quadrilateral cells and 4 × 4 Gaussian points per cell are used as a result of its accuracy and the appropriate time
expenditure in numerical procedure. Other parameters which influence on accuracy of computations are the dimensions and the shape of the support domain. In this simulation, the support domain that specifies
the number of nodes for integration point is a circle with radius r. The radius r is defined as r = dI h ,
where dI is a dimensionless coefficient known as support size and h is distance between two adjacent
nodes. In order to demonstrate the accuracy and stability of the EFG method based on the third-order
14
ACCEPTED MANUSCRIPT shear deformation theory, various numerical examples are performed; then the results are compared with previous works done in this field.
4.1.1
Static analysis
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4.1
Static verification of the results
To accomplish static analysis for a piezoelectric composite plate, a plate with dimensions 220 cm ×
20 cm5 and simply supported boundary condition in all four edges is considered. Figure1, shows a
T300/976 graphite–epoxy laminated plate consists of four composite layer that stacking sequence of
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these layers are −θ/θ¬ and −θ/θ¬ in which subscripts ‘‘s’’ and ‘‘as’’ indicate symmetric and anti-
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symmetric ply sequence, respectively. The total thickness of this laminated plate specified as h = 1. In order to actively control the static deflection of the plate that subjected to a uniformly distributed
transverse load q = 100N/m# , two PZTG1195N layers with thickness t = 0.2 mm bonded on the top and bottom surfaces of the substrate; where these layers are firstly used as actuators. The mechanical properties of these materials are listed in Table 1. In order to control the static deflection of the plate
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subjected to an external uniform load, the voltages (0, 5, 8 and 10) are applied to the actuator layers. To verify the results, the maximum deflections of the piezoelectric laminated plate are estimated. The results are compared with those of finite element [10] and first order RPIM [5]. As listed in Table 2, an excellent
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agreement is observed due to this comparison.
Figure 2 illustrates the center line deflection of piezoelectric composite plate with various applied
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voltages. It can be found that with enhancing actuator voltage, the plate deflections intend to be completely flat. This condition can be achieved when the actuator voltage is approximately 8 volts. In addition, the effects of fiber orientation of the laminate on deflection of the plate are observed. As it is anticipated, with increasing the ply angles the maximum deflections of the plate at the similar actuator voltage have been decreased as a result of enhancing the laminate stiffness matrix.
15
ACCEPTED MANUSCRIPT 4.1.2
Active control of laminate plate
In this section, in order to actively control the static response of a simply supported piezoelectric laminated plate with stacking sequence j/−45/45¥ and similar geometry, material noted in previous
section is considered. This plate is sandwiched between two outer piezoelectric layers where the upper
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and lower layers act as actuator and sensor, respectively. Figure 3 shows the static deflection of the simply supported plate with different displacement feedback gains. As illustrated, the maximum deflection of the plate becomes smaller when the displacement feedback control gain increases. It is also observe that while Gã = 30 the plate is well flattened. Here, two
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methods are introduced for static control of the plate. The first method applies an open loop to control the actuator voltage according to laminate deflection. However, the second one employs the vibration of
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the plate to generate the electrical power. Therefore the later one is more suitable; thus it is confirmed in this work.
4.2.1
Dynamic analysis
Dynamic verification of the results
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4.2
The performance of the EFG method to evaluate the natural frequencies of the structure is considered here. The composite plate is including three layers of Graphite/Epoxy with the stacking sequence 0/90/0 and two piezoelectric (PZT-4) layers. The similar material properties listed in Table 1 are held
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in this problem. The laminate aspect and thickness ratios are considered as ℎ = 0.20 and uℎ = 50 ,
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respectively. Moreover, the density is assumed to be = 1 kg/m/ for all layers. The electric boundary
condition in the piezo layer is closed circuit where the electric potential is kept zero. The dimensionless first natural frequency of the piezoelectric composite plate ( ) are given in Table 3 for various nodal density. The obtained results are in agreements with the other methods particularly are close with the finite element method; nevertheless, there is an inevitable gap between numerical results and analytical solution. In the EFG method, with increasing nodal density in the domain of the problem, distribution shape function become smoother. Therefore, the accuracy of the approximated values of the field function at these nodes improved. The fact should be noted here is that, with enhancing the nodal density, the 16
ACCEPTED MANUSCRIPT results are going to be converged. The size of support domain as well as the nodal density are both investigated in this work according to the flowchart of Fig.4. Figure 5 shows the convergence of the dimensionless natural frequency of simply supported piezoelectric composite. In this case the support size of influence domain is constant and the number of discrete nodes
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in the domain is increased. As illustrated, is converged where the nodal density equals to 17 × 17.
In order to investigate the effect of influence domain on accuracy and stability of the results, dimensionless natural frequencies against various support size of influence domain are depicted in Fig.6.
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In this case the convergence is occurred in influence domain equals to lÞ = 3.90. Comparing the Figs.
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(5-6) indicates that enhancing the size of support domain will results in more accurate results rather than using the nodal density improvement. The first four vibration modes of the simply supported piezoelectric laminate plate are illustrated in Fig.7. As depicted, the amplitude of 4th mode could be ignored comparing with the first three modes. Therefore, the modal matrices are assembled considering
4.2.2
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these first three modes.
Active control with using velocity feedback controller
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The presented mesh-less model is extended to the investigation of dynamic active vibration control of simply supported piezoelectric laminated plate. Therefore, the effect of choosing various velocity gains on
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active damping and vibration suppression is investigated. Here, the velocity feedback controller is used for active vibration control of the laminated plate. As previously noted, in this method the stability of the system is guaranteed only when the actuators and sensors are truly collocated. The piezo-laminated composite plate consist of three layers 218cm × 18cm5 of graphite epoxy composite 20,90,05 and the substrate covered with two layers of PVDF that the upper layer served as actuator and the lower ones served as sensor. The material properties of the laminates are given in Table 1. Each layer of the substrate is 2mm thick and each PVDF layer is 0.10mm thick. In this simulation in order to model the vibration
control, firstly the plate is subjected to a uniform load of q6 = 100 N⁄m# ; then within 1 × 10& seconds after, the plate is experienced the free vibration. Moreover, the proportional damping with the values of 17
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è = 1 × 10% and é = 0.965 × 10/ is considered. In order to analyze the dynamic response of the plate the modal superposition is used to reduce the size of the problem. Figure 8 illustrates the transient response of the plate using various velocity feedback gains. It can be seen that enhancing the control gain cause the dynamic response to be faster and the magnitude of the
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response to be decreased. Moreover, in a situation where the velocity feedback gain is zero, although there is not active vibration control, the response will be decreased in a time domain due to structural damping. In addition, the presence of variety of velocity feedback gain, ranging from 0 to 1.0, result into damping ratio listed in Table 4. It should be noted that the input voltage in actuator layer is restricted as a
4.2.3
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the velocity feedback gain should be bounded properly.
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result of the fact that enhancing the voltage causes the response to be steady and saturated. Therefore,
Optimal control with LQR output feedback
In order to simulate the LQR output controller for active vibration control, a piezoelectric laminated plate with characteristic given in previous section is considered. The weighting matrices Q and R should be
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selected in an appropriate way. The elements of these matrices can be adjusted based on the priority of minimizing the vibration energy as well as the requirements on minimizing the control voltage. In the present work, the output matrix [C] is obtained with considering the velocity feedback. Whereas, the time
constant is in the range of microsecond then the value of charge amplifier gain is Gó = 1.6 × 10 . In this
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simulation, the plate is initially subjected to a uniform load of intensify Ô = 100 ⁄s# . Then, the load is
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removed which leads into free vibration of the plate as well as switching on the LQR controller. Here, the effects of weighting matrices on the dynamic response and control voltage are investigated. The dynamic response of the system in presence of the active vibration control and without control showed in Fig. 9. In this case, the response is obtained based on the weighting matrices in the following form: ù = Ô Á ã ý
ý Ã , = y y
(67)
where q and r denote the design parameters that can influence on response and control voltage and I is
identity matrix. In order to achieve the proper weighting parameters q and r, firstly, for a fixed design
18
ACCEPTED MANUSCRIPT parameter of r the value of q is increased. Then, the parameter q is designated and the value of r will be increased. Figure 10-11 demonstrate the influence of various weighting matrix Q on LQR output feedback controller. It can be seen that a greater [Q] puts higher demand on control voltage and the regulator takes a shorter
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time to achieve the zero state variables. It can be also deduced that with enhancing this weighting matrix the response will be faster. Moreover, as results of choosing the weighting matrices in a high level, the optimal gain will be placed in a high level. Therefore, the close loop poles of the system are going to be far from the open loop poles. In this situation, the system will be sensitive in respect to the background
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noises. Consequently, the peak value of the actuator voltage will exceed the breakdown voltage.
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Figure 12 demonstrates the control voltage with respect to various weighting matrix R while the weighting matrix Q is constant. It can be deduced that, with increasing the weighting matrix R the control voltage will be reduced. However, as illustrated in Fig. 13, the amplitude of response for the simply supported piezo-laminated plate is increased. Considering both response and input voltage, it is found
ù = Á ã ý
ý Ã , = 0.0001y y
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that the appropriate controller can be acquired by selecting the weighting matrices in following form:
(68)
For these values of the weighting matrices the convergence of cost function (J) against the iteration numbers demonstrates in Fig. 14. As illustrated, the cost function is converged into the minimum point
5
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with increasing the successive iteration numbers.
Conclusion
In this paper an efficient method, known as the element free galerkin (EFG), was presented to control the vibration of piezoelectric laminated plate bonded with piezoelectric actuator and sensor layers. The equation of motion for piezo-laminated composite plate was obtained from third order shear deformation theory. The derived shape function could not satisfy the delta function property. Therefore, the penalty technique was used because it has higher efficiency than Lagrange multipliers and orthogonal 19
ACCEPTED MANUSCRIPT transformation method. A constant displacement feedback control algorithm was introduced to regulate static deflection of simply supported plate. The influence of stacking sequence and various input actuator voltage on static deflection was also investigated. In addition, the convergence of fundamental natural frequency with various nodal density and support size was demonstrated. In order to control the dynamic behavior of piezolaminated composite plate, a velocity feedback and optimal steady-state regulator with
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output feedback algorithm were designed. Moreover, the presented results are compared with those of other researches which indicated an excellent agreement. Furthermore, several numerical examples were presented for static control, convergence of natural frequency, vibration mode and dynamic control of
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piezoelectric laminated plates with different stacking sequences. Particularly, the remarkable conclusions from the present formulation and numerical results are followed as:
In the EFG method the integration cells do not require to be attuned with the scattered nodes
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1.
distributed on the problem domain. Therefore, the generation of background mesh is more easily than the FEM methods. 2.
The results of piezo-laminated composite plate indicate that the HSDT can give more accurate results rather than classical or FSDT. Moreover, it does not require the shear correction factor. In dynamic control using CVGF controller, the actuators and sensors have to truly be collocated.
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3.
However, this issue is not important in LQR controller. In order to design the efficient LQR controller, it requires the measurement of all state variables. This
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is a difficult suggestion to do. Therefore, the LQR with output feedback is designed in the present work. Consequently, the need to measure all state variables is avoided.
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4.
20
ACCEPTED MANUSCRIPT References
J. A. Mitchell and J. N. Reddy, “A refined hybrid plate theory for composite laminates with piezoelectric laminae,” Int. J. Solids Struct., vol. 32, no. 16, pp. 2345–2367, 1995.
[2]
G.-R. Liu and Y.-T. Gu, An introduction to meshfree methods and their programming. Springer, 2005.
[3]
K. M. Liew, X. Q. He, M. J. Tan, and H. K. Lim, “Dynamic analysis of laminated composite plates with piezoelectric sensor/actuator patches using the FSDT mesh-free method,” Int. J. Mech. Sci., vol. 46, no. 3, pp. 411–431, Mar. 2004.
[4]
K. M. Liew, H. K. Lim, M. J. Tan, and X. Q. He, “Analysis of laminated composite beams and plates with piezoelectric patches using the element-free Galerkin method,” Comput. Mech., vol. 29, no. 6, pp. 486–497, 2002.
[5]
G. R. Liu, K. Y. Dai, and K. M. Lim, “Static and vibration control of composite laminates integrated with piezoelectric sensors and actuators using the radial point interpolation method,” Smart Mater. Struct., vol. 13, no. 6, p. 1438, 2004.
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G. R. Liu and X. L. Chen, “A mesh-free method for static and free vibration analyses of thin plates of complicated shape,” J. Sound Vib., vol. 241, no. 5, pp. 839–855, 2001.
[7]
K. Y. Dai, G. R. Liu, K. M. Lim, and X. L. Chen, “A mesh-free method for static and free vibration analysis of shear deformable laminated composite plates,” J. Sound Vib., vol. 269, no. 3, pp. 633– 652, 2004.
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B. Samanta, M. C. Ray, and R. Bhattacharyya, “Finite element model for active control of intelligent structures,” AIAA J., vol. 34, no. 9, pp. 1885–1893, 1996.
[9]
J. M. S. Moita, C. M. M. Soares, and C. A. M. Soares, “Active control of forced vibrations in adaptive structures using a higher order model,” Compos. Struct., vol. 71, no. 3, pp. 349–355, 2005.
[10]
P. Phung-Van, T. Nguyen-Thoi, T. Le-Dinh, and H. Nguyen-Xuan, “Static and free vibration analyses and dynamic control of composite plates integrated with piezoelectric sensors and actuators by the cell-based smoothed discrete shear gap method (CS-FEM-DSG3),” Smart Mater. Struct., vol. 22, no. 9, p. 095026, Sep. 2013.
[11]
P. Phung-Van, L. De Lorenzis, C. H. Thai, M. Abdel-Wahab, and H. Nguyen-Xuan, “Analysis of laminated composite plates integrated with piezoelectric sensors and actuators using higherorder shear deformation theory and isogeometric finite elements,” Comput. Mater. Sci., 2014.
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[12]
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[1]
T. I. Thinh and L. K. Ngoc, “Static behavior and vibration control of piezoelectric cantilever composite plates and comparison with experiments,” Comput. Mater. Sci., vol. 49, no. 4, pp. S276– S280, 2010.
[13]
F. L. Lewis, D. Vrabie, and V. L. Syrmos, Optimal control. John Wiley & Sons, 2012.
[14]
M. C. Ray, “Optimal control of laminated plate with piezoelectric sensor and actuator layers,” AIAA J., vol. 36, no. 12, pp. 2204–2208, 1998.
[15]
P. Bhattacharya, H. Suhail, and P. K. Sinha, “Finite element analysis and distributed control of laminated composite shells using LQR/IMSC approach,” Aerosp. Sci. Technol., vol. 6, no. 4, pp. 273– 281, 2002. 21
ACCEPTED MANUSCRIPT A. Zabihollah, R. Sedagahti, and R. Ganesan, “Active vibration suppression of smart laminated beams using layerwise theory and an optimal control strategy,” Smart Mater. Struct., vol. 16, no. 6, p. 2190, 2007.
[17]
J. N. Reddy, Mechanics of laminated composite plates and shells: theory and analysis. CRC press, 2004.
[18]
Z. K. Kusculuoglu and T. J. Royston, “Finite element formulation for composite plates with piezoceramic layers for optimal vibration control applications,” Smart Mater. Struct., vol. 14, no. 6, p. 1139, 2005.
[19]
S. Kapuria and M. Y. Yasin, “Active vibration suppression of multilayered plates integrated with piezoelectric fiber reinforced composites using an efficient finite element model,” J. Sound Vib., vol. 329, no. 16, pp. 3247–3265, 2010.
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K. M. Liew, Y. Q. Huang, and J. N. Reddy, “Analysis of general shaped thin plates by the moving least-squares differential quadrature method,” Finite Elem. Anal. Des., vol. 40, no. 11, pp. 1453– 1474, 2004.
[21]
S. Kapuria and P. Kumari, “Boundary layer effects in Levy-type rectangular piezoelectric composite plates using a coupled efficient layerwise theory,” Eur. J. Mech., vol. 36, pp. 122–140, 2012.
[22]
A. Preumont, Vibration control of active structures: an introduction, vol. 179. Springer, 2011.
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D. D. Moerder and A. J. Calise, “Convergence of a numerical algorithm for calculating optimal output feedback gains,” IEEE Trans. Automat. Contr., vol. 30, no. 9, pp. 900–903, 1985.
[24]
A. Alibeigloo, “Free vibration analysis of functionally graded carbon nanotube-reinforced composite cylindrical panel embedded in piezoelectric layers by using theory of elasticity,” Eur. J. Mech., vol. 44, pp. 104–115, 2014.
[25]
V. M. Franco Correia, M. A. Aguiar Gomes, A. Suleman, C. M. Mota Soares, and C. A. Mota Soares, “Modelling and design of adaptive composite structures,” Comput. Methods Appl. Mech. Eng., vol. 185, no. 2, pp. 325–346, 2000.
[26]
D. A. Saravanos, P. R. Heyliger, and D. A. Hopkins, “Layerwise mechanics and finite element for the dynamic analysis of piezoelectric composite plates,” Int. J. Solids Struct., vol. 34, no. 3, pp. 359–378, 1997.
[27]
P. Heyliger and D. A. Saravanos, “Exact free-vibration analysis of laminated plates with embedded piezoelectric layers,” J. Acoust. Soc. Am., vol. 98, no. 3, pp. 1547–1557, 1995.
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[16]
22
ACCEPTED MANUSCRIPT Table 1 Relevant mechanical properties of respective materials (Electrical permittivity of air, = 8.85 × 10 /) Piezoelectric layer PZT-G1195N PVDF 63.0 2 63.0 2 63.0 2 24.2 1 24.2 1 24.2 1 0.3 0.29 0.3 0.29 0.3 0.29 7600 1800 2.54e-10 0.046 1728.8 12 1728.8 12 1694.9 12
TE D EP AC C
PZT-4[24] 81.3 81.3 64.5 30.6 25.6 25.6 0.329 0.432 0.432 7600 -1.23e-10 1475 1475 1300
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Gr/Ep 132.38 10.76 10.76 3.61 5.65 5.65 0.24 0.24 0.49 1578 -
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) ) ) ) ) ) kg/m ) = m/V) = C/m ) / F/m) / F/m) / F/m)
Core plate T300/979 150 9.0 9.0 7.1 7.1 2.5 0.3 0.3 0.3 1600 -
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property
ACCEPTED MANUSCRIPT Table 2 Central node deflection of the simply supported piezoelectric laminate plate under different actuator voltage× 10 m Actuator voltage
method
⁄−15⁄15
⁄−30⁄30
⁄−45⁄45
⁄−45⁄45
0V
EFG CS-DSG3[10] RPIM [5]
−0.7549 −0.7442 −0.7222
−0.6657 −0.6688 −0.6542
−0.6338 −0.6323 −0.6217
−0.6340 −0.6326 −0.6038
5V
EFG CS-DSG3[10] RPIM [5]
−0.3329 −0.3259 −0.3134
−0.2984 −0.2957 −0.2862
−0.2857 −0.2801 −0.2717
−0.2892 −0.2863 −0.2717
10V
EFG CS-DSG3[10] RPIM [5]
0.0931 0.0924 0.0954
0.0782 0.0774 0.0819
0.0615 0.0601 0.0604
0.0739 0.0721 0.0757
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scheme
ACCEPTED MANUSCRIPT plate [piezo/0/90/0/piezo] Method Present method Present method Present method IGA [11] Q9–HSDT [25] HSDT-layerwise [26] CS-FEM-DSG3 [10] Analytical solution [27]
234.174 234.125 234.221 235.100 230.461 234.533 234.500 245.941
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nodes 11 × 11 13 × 13 15 × 15 -
of simply supported piezoelectric composite
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Table 3 non dimensional natural frequency =
ACCEPTED MANUSCRIPT Table 4 damping ratio of piezolaminated composite plate for various velocity feedback gains G୴ ܩ௩ Mode 1
0.0268 0.0395 0.0687
0.0512 0.1730 0.3511
0.0853 0.3136 0.6310
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0.5
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1 2 3
0
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Fig. 1 Schematic illustration of a laminated plate with integrated piezoelectric and regularly distributed nodes
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2
x 10
1
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0
-2 -3
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Deflection(m)
-1
-5 -6 -7 0
0.05
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-4
0.1
0.15
0.2
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Distance x(m)
Fig. 2(a) Centerline deflections of the simply supported laminates ⁄−45⁄45 under uniform loading
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and different actuator input voltages (
0V
5V
8V
10V)
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1
x 10
0
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-2
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-4
-5
-6
-7 0
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Deflection(m)
-1
0.1
0.15
0.2
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Distance x(m)
Fig. 2(b) Centerline deflections of the simply supported laminates ⁄−45⁄45 under uniform loading
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and different actuator input voltages (
0V
5V
8V
10V)
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1
x 10
0
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-2
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-3
-4
-5
-6
-7 0
0.05
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Deflection(m)
-1
0.1
0.15
0.2
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Distance x(m)
Fig. 2(c) Centerline deflections of the simply supported laminates ⁄−30⁄30 under uniform loading
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and different actuator input voltages (
0V
5V
8V
10V)
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1
x 10
0
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-1
-3 -4
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Deflection(m)
-2
-6 -7 -8 0
0.05
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-5
0.1
0.15
0.2
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Distance x(m)
Fig. 2(d) Centerline deflections of the simply supported laminates ⁄−15⁄15 under uniform loading
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and different actuator input voltages (
0V
5V
8V
10V)
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1
x 10
0 Gd=30
-3
Gd=5
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Gd=10
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-4
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maximum Deflection(m)
-1
Gd=0
-7 0
0.05
0.1
0.15
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Distance x(m)
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Fig. 3 static deflection of simply supported plate with different displacement control gain Gୢ .
0.2
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Geometry generation
For all the cells in the background mesh
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For all the quadrature points in the cell
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Node and background mesh generation
Search in the current cell and its surrounding cells of integration for all the nodes
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that fall in the support domain of quadrature point
MLS shape function creation-based selected nodes
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Calculate nodal matrices
Assemble the nodal matrix into the global matrix
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Solve eigenvalue problem
If ߱ ሺ݊ + 1ሻ − ߱ ሺ݊ሻ < 0.01
Yes
Plot convergence of Eigen value
Fig. 4 Flowchart of the converging algorithm
No
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234.2
234.1 12
14
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Non dimensional natural frequency
234.3
16
18
20
Nodal density
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Fig.5 convergence of the dimensionless natural frequency with respect to various nodal density
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234
233.5 2.5
3
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Non dimention natural frequency
234.5
3.5
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dmax
4
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Fig. 6convergence of the dimensionless natural frequency with respect to various sizes of the influence domains
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0.035 0.03 0.025
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z
0.02 0.015 0.01 0.005
1
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0
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
y
0
x
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0
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Fig. 7a First mode natural frequency of simply supported plate piezo⁄0 /90/ 0⁄piezo
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0.015
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0.01
z
0.005 0 -0.005 -0.01
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-0.015
1 0.8
1
0.8
0.6
0.4
0.2
y
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0.6
0.4
0.2
0
0
x
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Fig. 7b Second mode natural frequency of simply supported plate piezo⁄0 /90/ 0⁄piezo
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0.02
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z
0.01
0
-0.01
-0.02 0.8
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1
1
0.6
0.8
0.6
0.4
0.4
0.2 0
0.2 0
x
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y
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Fig. 7c Third mode natural frequency of simply supported plate piezo⁄0 /90/ 0⁄piezo
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-3
x 10
0
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z
5
-5
1
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0.8 0.6
1
0.4
0.8
y
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0.6
0.2
0.4
0
0.2
0
x
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Fig. 7d Fourth mode natural frequency of simply supported plate piezo⁄0 /90/ 0⁄piezo
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3
x 10
Gv=0 Gv=0.5 Gv=1
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1
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0
-1
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0.01
0.015
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maximum deflection(m)
2
0.02
0.025
0.03
0.035
0.04
0.045
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time(sec)
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Fig. 8maximum deflection of the simply supported laminated plate covered with piezoelectric layers with respect to various velocity feedback gains.
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5
x 10
Without Control With active Control
4
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2 1 0 -1
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maximum deflection(m)
3
-2
-4 -5 0
0.005
0.01
0.015
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0.02
0.025
0.03
0.035
0.04
0.045
0.05
time(sec)
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Fig. 9center deflection of simply supported piezoelectric laminated plate without control and with active control (LQR controller).
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x 10
-7
q=1 q=10 q=100
4
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2 1
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0 -1 -2 -3 -4
0.01
0.02 0.03 time(sec)
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maximum deflection(m)
3
0.04
0.05
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Fig.10 center deflection of simply supported plate using LQR output feedback controller and various design parameter q
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300 q=1 q=10 q=100
200
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0
-100
-200
-300
0.01
0.02 0.03 time(sec)
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-400 0
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control voltage(Volt)
100
0.04
0.05
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Fig.11 input voltage of simply supported plate using LQR output feedback controller and various design parameter q
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80 r=0.0001 r=0.0003
60
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20
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0 -20
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control voltage(Volt)
40
-40 -60
-100 0
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-80
0.01
0.02 0.03 time(sec)
0.04
0.05
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Fig.12 control voltage of simply supported plate using LQR output feedback controller and various design parameter r
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x 10
-7
r=0.0001 r=0.0003
4
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2 1
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0 -1 -2 -3 -4 0.01
0.02 0.03 time(sec)
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-5 0
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3
0.04
0.05
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Fig.13 center deflection of simply supported plate using LQR output feedback controller and various design parameter r
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5.5
x 10
5 4.5
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4
3
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J
3.5
2.5
1.5 1 0.5 0
5
10
15
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20
25
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Itration Number
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Fig.14 convergent procedure of cost function
30
35
40
45
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highlights
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1. We model a piezoelectric plate with TSDT and meshfree galerkin method. 2. We design optimal (LQR with output feedback) controller with using meshfree model. 3. The dynamic response of the system is more accurate than FEM methods.