# Thermal Effects on Vibration and Control of Piezocomposite Kirchhoff Plate Modeled by Finite Elements Method

Authors: M. Sanbi; R. Saadani; K. Sbai; M. Rahmoune
Journal: Smart Materials Research
 (2015)

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## Abstract

Theoretical and numerical results of the modeling of a smart plate are presented for optimal active vibration control. The smart plate consists of a rectangular aluminum piezocomposite plate modeled in cantilever configuration with surface bonded thermopiezoelectric patches. The patches are symmetrically bonded on top and bottom surfaces. A generic thermopiezoelastic theory for piezocomposite plate is derived, using linear thermopiezoelastic theory and Kirchhoff assumptions. Finite element equations for the thermopiezoelastic medium are obtained by using the linear constitutive equations in Hamilton’s principle together with the finite element approximations. The structure is modelled analytically and then numerically and the results of simulations are presented in order to visualize the states of their dynamics and the state of control. The optimal control LQG-Kalman filter is applied. By using this model, the study first gives the influences of the actuator/sensor pair placement and size on the response of the smart plate. Second, the effects of thermoelastic and pyroelectric couplings on the dynamics of the structure and on the control procedure are studied and discussed. It is shown that the effectiveness of the control is not affected by the applied thermal gradient and can be applied with or without this gradient at any time of plate vibrations.


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## Body

## 1. Introduction

In the piezoelectric sensors applications, mechanically or thermally induced deformations can be determined from measurement of the induced electrical potential, whereas in piezoelectric actuator applications deformation of strain can be controlled through the introduction of appropriate electric potential. By integrating distributed piezoelectric sensors/actuators and advanced composites, the potential exists for forming high-strength, high stiffness, lightweight structures capable of self-monitoring and self-controlling. Typical applications of such structures are envisioned in the thermal distortion management of propulsion components and space structures. Before they can be utilized in these applications, the performance of piezoelectric structures in thermal environment must be quantified. Thermal effects become important when the piezoelectric structure has to operate in either extremely hot or cold temperature environments. These extreme conditions may severely affect the response of piezoelectric elements by induction of thermal stresses resulting from thermoelastic and pyroelectric coefficients.Recently, there have been various mathematical models developed to describe the behavior of the piezocomposite or laminated plates actuated and sensed by piezoelectric materials. Görnandt and Gabbert [1] have analysed the finite element method of thermopiezoelectric smart structures. Vel and Batra [2] presented a generalized plane strain thermopiezoelectric analysis of multilayered plates. de Abreu et al. [3] have implemented finite element modeling of a plate with localized piezoelectric sensors and actuators. Deü and Benjeddou [4] have studied free-vibration analysis of laminated plates with embedded shear-mode piezoceramic layers. Tanaka and Sanada [5] have detailed modal control of a rectangular plate using smart sensors and smart actuators. Trindade and Benjeddou [6] have discussed the critical evaluation and optimization of the effective electromechanical coupling coefficients of piezoelectric adaptive structures. Sanbi et al. [7] have analyzed thermoelastic and pyroelectric couplings effects on dynamics and active control of piezolaminated beam. A general solution for piezothermoelasticity of transversely isotropic piezoelectric materials and its applications have been deduced by Ding et al. [8].In this work, fundamental equations governing the behavior of smart structures are at first given, and, based on the work of de Abreu et al. [3], the generalized linear finite element formulation of the problem is described. Thermopiezoelastic characteristics of aluminium rectangular plate and the theory of active structures in sensing and optimal control are established. A generic thermopiezoelastic theory for piezocomposite plate is derived, using linear thermopiezoelasticity andKirchhoff assumptions. Generalized finite element equations for the thermopiezoelasticity are obtained by using the linear constitutive equations in Hamilton’s principle together with the finite element approximations. AKirchhoff four-node rectangular element with one electrical, one temperature, and three mechanical degrees of freedom is considered. The structure consists of a modeling of cantilevered piezocomposite plate with perfectly surface bonded thermopiezoelectric elements. The structure is modelled analytically and then numerically and the results of simulations are presented in order to visualize the states of their dynamics and the state of control. The optimal controlLQG-Kalman accompanied is applied. The effects of thermoelastic and pyroelectric couplings coefficients on the dynamics of the structure and on the control effectiveness are discussed. We show that the control procedure cannot be perturbed by applying a thermal gradient and the control can be applied at any time during the period of vibration of the plate.


## 2. Basic Equations and FEM Implementation

The physical and dimensional characteristics of the plate material are given in Table1. Properties for the two piezoelectric materials used in the simulation can be found in the literature as in [9, 10]. The piezoelectric and elastic properties of the sensor are chosen smaller compared to those of the actuator. This is justified by the fact that the sensor must have high sensitivity to capture the deformation of the structure, while the actuator is required to act as a secondary source on the plate to deaden its vibrations. In order to model the structure by finite element method, we consider an element with four-node quadrilateral flexible element type “plate.” Each node has three mechanical degrees of freedom (dofs), the displacement in z-direction and two rotations in x- and y-direction, one electric dof Φ, and one temperature dof Θ (Figure 1).Table 1
Properties of the plate, sensor, and actuator.
Properties/labels/units
Plate
Sensor
Actuator
Al
PVDF
PZT-5H32
LengthL (m)
0.6
0.25
0.25
Widthl (m)
0.4
0.25
0.25
Thicknesst (m)  ×10-3
1
0.2
0.2
Densityρ(g/cm3)
8.03
1.78
7.7
Young modulusE (GPa)
68
5.04
62
Reduced elastic constants (GPa)
c¯11E
—
5.04
66.2
c¯12E
—
3.25
19.2
c¯66E
—
—
23.5
Piezoelectric stress constant
—
0.15
−9.11
g
31 (Vm/N)
Piezoelectric strain constant
—
4.34
−274
d
31  ×10−12 (m/V)
Reduced piezoelectric constant
—
0.018
−23.4
e
¯
31 (C/m2)
Pyroelectric constant
—
−1.25
—
p  ×10−5 Cm−2K−1
Dielectric constantϵ33S/ϵ0
—
12
3300
(
ϵ
0
=
8.8541
0
-
12
F/m
)
Thermal conductivity
—
1.5
1.5
K
d (W/m·K)
Thermal expansion
—
140
3
α  ×10−6 (K−1)
Specific heatc0 J·kg−1·K−1
—
350
1.5
Dampingα  ×10−3
1
—
—
Constantsβ  ×10−3
0.1
—
—Figure 1
The finite element type “plate” used in the formulation and different degrees of freedom.In the present formulation, we assume that the piezoelectric layers are perfectly bonded and the formulation is restricted to linear elastic material behavior with small displacement and strains. This formulation uses the Kirchhoff assumption for thin plate in which the transverse normal remains straight and normal to the midsurface after deformation. The thickness of the plate does not change during a deformation. Depending on these assumptions and as shown in Figure1, displacements u, v, and w can be expressed by(1)u=-z∂w∂x,v=-z∂w∂y,w=wx,y,where x and y are the longitudinal and transverse axes located on the medial surface of the plate and z is the thickness direction (Figure 1). The displacements u, v, and w are taken along the axes x, y, and z, respectively. Assuming that the transverse deformation is negligible, the deformation in terms of displacements can be written as(2)ɛ=ɛxɛyɛxyT=-z∂2w∂x2∂2w∂y2∂2w∂x∂yT.For an isotropic material, the constraints are related to deformation by(3)σ=Cɛ,where σ=[σxσyτxy]T and the matrix C is given by(4)C=EP1-ν21ν0ν10001-ν2,with ν being the Poisson coefficient and EP the plateYoung modulus.In the finite element modeling, a four-node bending plate rectangular element based on the classical theory of plates is considered [11]. Each node element has three mechanical degrees of freedom: the displacement w¯ in z-direction, x-rotation θ¯y, and y-rotation θ¯x (Figure 1). The function of the displacement w can be written as(5)wxi,yi=c1+c2xi+c3yi+c4xi2+c5xiyi+c6yi2+c7xi3+c8xi2yi+c9xiyi2+c10yi3+c11xi3yi+c12xiyi3,with i=1⋯4;x1=-a; y1=-b; x2=a; y2=b;x3=a; y3=b; x4=-a; y4=-b.The transverse displacement can be expressed by(6)w=PTc,where(7)c=c1c2c3c4c5⋯c12,P=1xyx2xyy2x3x2yxy2y3x3yxy3T.We define the vector q(i) that contains the nodal displacements of the rectangular element by(8)qi=w¯1θ¯x1θ¯y1w¯2θ¯x2θ¯y2w¯3θ¯x3θ¯y3w¯4θ¯x4θ¯y4T,where(9)w¯i=wxi,yi,θ¯xi=∂w∂yxi,yi,θ¯yi=∂w∂yxi,yi.By combining (5) and (8) with (9) for the four nodes, we obtain the following matrix expression:(10)qi=Xc,where [X] is a (12×12) matrix given by (11)X=1x1y1x12x1y1y12x13x12y1x1y12y13x13y1x1y130010x12y10x122x1y13y13x133x1y120-10-2x1-y10-3x122x1y1-y120-3x12y1-y131x1y1x12x1y1y12x13x12y1x1y12y13x13y1x1y130010x12y10x122x1y13y13x133x1y120-10-2x1-y10-3x122x1y1-y120-3x12y1-y131x1y1x12x1y1y12x13x12y1x1y12y13x13y1x1y130010x12y10x122x1y13y13x133x1y120-10-2x1-y10-3x122x1y1-y120-3x12y1-y13.Therefore, the coefficient vector [c] can be calculated from (10) as follows:(12)c=X-1qi.Substituting (12) into (6) leads to(13)w=Nwqi,where [Nw] is the z-direction shape function matrix given by(14)Nw=PTX-1.Substituting (3) into (2) yields (15)ɛ=-z∂2PT∂x2∂2PT∂y22∂2PT∂x∂yTX-1qi.By manipulating (15), we can obtain(16)ɛ=-zLkX-1qi,with(17)Lk=0002006x2y006xy0000002002x6y06xy00002004x4y06x26y2.The displacements u, v, and w can be expressed by the vector {q} as(18)q=wuvT.The substitution of (1) into (18) yields (19)q=w-z∂w∂x-z∂w∂yTand substituting (13) into (19), we get(20)q=HLMTX-1qi,where [LM]T and [H] are given by(21)LMT=PT∂PT∂x∂PT∂yT,H=1000-z000-z.Considering thermopiezoelastic material, the constitutive equations of linear thermopiezoelasticity and temperature are written as [12](22)σ=CEɛ-eE-λΘ,(23)D=eTɛ-ϵSE+pΘ,(24)s=λTɛ+pTE+α~Θ,where the exponent S means that the values are measured at constant strain and the exponent E means they are measured at a constant electric field, {σ} is the stress tensor, {D} is the vector of electric displacement, {s} is the entropy, {Θ} is the temperature vector, {ɛ} is the tensor deformation, {E} is the electric field, [C]E is the matrix of elastic coefficients at constant electric field, [e] is the matrix of piezoelectric coefficients, [ϵ]S is the tensor dielectric constant mechanical strain, [p] is the pyroelectric tensor, and [λ] is the thermoelastic tensor. (α~) is the expansion coefficient given by α~=ρpc0/Θ0, where c0 and Θ0 are the specific heat and the initial temperature, respectively.On the other hand, for the relation between[e] and [d], the coefficients of the piezoelectric strain are written as(25)e=CEd.If the polarization direction is held in, the electrical potential applied or captured through the elementssensor oractuator is given by the following equation [13]:(26)Φz=z-hp/2hΦ,where h and Φ are the thickness and the maximum electric potential at the outer surface of the corresponding thermopiezoelectric element (actuator or sensor) and z (za or zc) is defined in the following two intervals:(27)hp2≤za≤hp2+ha,hp2≤zc≤hp2+hc.Assuming that the electric field E is constant along the thickness of the actuator and sensor, we can therefore write(28)E=-dΦzdz=-BzΦ=-Φh.Hamilton’s principle is used here to derive finite element equations and is given by(29)∫t1t2δT-U+We-Wth+δWdt=0,where t1 and t2 are two arbitrary instants, T is the kinematic energy, U is the energy potential, We is the work due to electrical forces, and Wth is the work due to thermal stresses. The total kinematic energy T and energy potential U are given by the following relations:(30)T=12∫Vρq˙Tq˙dV,(31)U=12∫VɛTσdV-∫VɛTλΘdV,with {q˙} being the derivative with respect to time t and a volume element dV given by(32)dV=dVp+dVa+dVc,where the indexes p, a, and c refer to the plate, the actuator, and the sensor, respectively, and dVp, dVa, and dVc are given by(33)dVp=∫-hp/2hp/2∫-bb∫-aadxdydz,dVa=∫-hp/2hp/2+ha∫-bb∫-aadxdydz,dVc=∫-hp/2-hchp/2∫-bb∫-aadxdydz.The energy of electric forces and elementary virtual work produced by the volume forces {fv}, surface forces {fA}, and the applied electrical loads are written as(34)We=12∫VETD+pΘdV,(35)δW=∫VδqTfvdV+∫AδqTfAdA-∫AδΦσqdA,with D being the electric displacement vector, fv the volume force, fA the surface force, and σq the electric surface stress.Substituting (3) and (22) into (31) and (23) into (34), respectively, leads to(36)U=12∫VɛTCEɛ-12∫VɛTeEdV-12∫VɛTλΘdV,We=12∫VETeTɛ+12∫VETϵTEdV+12∫VETpTΘdV.The substitution of (30), (35), and (36) into (29) gives(37)∫t1t2∫VρδqTq¨dV-∫VδɛTCEɛ+∫VδɛTeTEdV+∫VδɛTλTΘdV+∫VδETeɛdV+∫VδETϵTEdV+∫VδETpΘdV+∫VδqTfvdV+∫VδqTfAdV-∫VδΦσqdAdt=0.Substituting (20), (16), and (28) into (37) leads to(38)∫t1t2δqkTMqqeq¨k+Kqqeqk+KqΦeΦ-KqΘeΘ-f¯+δΦKΦqeqk+KΦΦeΦ+KΦΘeΘ+Qadt=0,where(39)Mqqe=ρ∫VX-TLMHTHLMTX-1dV,Kqqe=X-T∫Vz2LKTDLKX-1dV,KqΦe=KΦqeT=-X-T∫VzLKTeTBzdV,KqΘe=KΘqeT=-X-T∫VzLKTλTdV,KΦΦe=-∫VBzTϵTBzdV,KΦΘe=-∫VBzTpTdV,f¯=∫VfvdV+∫AfAdV,Qa=∫AσqdA.Considering that the variation of {qk} and {Φ} is arbitrary, we obtain two equilibrium equations rewritten in generalized coordinates for the kth element:(40)Mqqeq¨k+Kqqeqk+KqΦeΦ-KΘΦeΘ-f¯=0,KΦqeqk+KΦΘeΘ+KΦΦeΦ+Qa=0,with Kqqe being the extended stiffness matrix and Mqqe the elementary mass matrix. By integrating Kqqe in the direction of z, we obtain(41)Kqqe=∑i=13hiX-T∫ALKTDiLKdAX-1,where hi are given by(42)h1=hahp2+ha22+ha312,h2=hp312,h3=hchp2+hc22+hc312,and Da, Dp, and Dc, for i=1,2,3, are calculated from (4) for the thermopiezoelectric material and the plate, respectively, and dA=dxdy.The integration inz of the elementary mass matrix gives(43)Mqqe=∑i=13ρX-T∫ALMTHiLMdAX-1,with ρ1=ρa, ρ2=ρp, and ρ3=ρc and [Hi] (for i=1,2,3) are(44)H1=Ha=ha000h1000h1,H2=Hp=hp000h2000h2,H3=Hc=hc000h3000h3.By integrating the direction z, the thermoelastic coupling matrix [KqΘe] for the three materials is given by(45)KqΘe=∑i=13hiX-T∫ALKTλiLKdAX-1,with λi being the matrix of each thermoelastic layer given in Table 1. The elementary stiffness matrix is due to electromechanical coupling; [KqΦe] and pyroelectric [KΘΦe] are integrated along the direction z with respect to the thickness of each thermopiezoelectric layer (dV=dVa or dVc). Thus,(46)KqΦe=-12hpha+ha2X-T∫ALKTeaTBzdA,(47)KΦΦea=-4abhaϵaS,(48)KqΦe=12hphc+hc2X-T∫ALKTecTBzdA,(49)KΦΦec=-4abhcϵcS,KΘΦea=-12hpha+ha2X-T∫ALKTpaTBzdA,KΘΦec=12hphc+hc2X-T∫ALKTpcTBzdA.Equations (41), (43), (46), and (48) are integrated numerically using the quadratic integration method [11] as follows: (50)Kqqe=∑i=13hiX-T∑η∑ξLKTDiLKWηWξX-1,Mqqe=∑i=13ρiX-T∑η∑ξLMTHiLMWηWξX-1,KqΘe=∑i=13ρiX-T∑η∑ξLMTλiLMWηWξX-1,KqΦea=haX-T∑η∑ξLKTeaTLKWηWξBz,KqΦec=hcX-T∑η∑ξLKTecTLKWηWξBz,KΘΦe=haX-T∑η∑ξLKTpaTLKWηWξBz,KΘΦe=hcX-T∑η∑ξLKTpcTLKWηWξBz,where (ξ,η) are the coordinates of Gaussian point integration and Wξ, Wη are the associated weight factors with ha=-1/2hpha+ha2 and hc=1/2hphc+hc2.All elementary matrices constructed may be assembled into global matrices as follows:(51)M=∑k=1NTkTMqqeTk,Kqq=∑k=1NTkTKqqeTk,where N is the number of finite elements and Tk is a distribution matrix defined by(52)Tki,j=0ifj≠mki1ifj=mki,for i=1,2,…,12 and j=1,2,…,ndof with (dof) being the number of degrees of freedom of the entire structure. mk is the indicator vector containing the degrees of freedom 3(dof) of nth node (1, 2, 3, or 4; Figure 1) in kth finite element given by (53)mk=3nk-23nk-13nk.Considering that there are na actuators and nc sensors distributed on the surface of the plate, (40) including damping matrix [Cqq] can be written in the following global shape:(54)Mg¨+Cqqg˙+Kqqg+∑k=1neiTkiT·KqΦiΦ-KqΘΘ-F=0,(55)∑k=1neiKΦqiTkig+KΦΘiTkiΘ+KΦΦiΦ+Qa=0,where [Tk]i is the distribution matrix indicating the position of the kth element on the plate and is composed of numbers 1 or 0. The figure 0 means that no thermopiezoelectric element (actuator or sensor) is present while the figure 1 indicates the presence of the element. nei is the number of finite elements of ith thermopiezoelectric actuator/sensor and {g} is the vector of nodal displacement of the entire structure.We have no voltage applied to the sensor (Qa=0), so the electrical potential generated is calculated through (55) for i=1,2,…,ns as follows:(56)Φc=-∑k=1neiKΦΦci-1·KΦqciTkig+KΦΘciTkiΘ=0.The total voltage Φ is composed of tension Φs sensed by the sensor, tension Φsa sensed by the actuator, and finally the applied voltage Φa. So, the total voltage Φ is given by(57)Φ=Φs+Φsa+Φa.The global dynamic equation can be formed by substitution of (56) into (57) and then into (54). The introduction of electrical forces due to the actuator forces, pyroelectric thermoelastic forces, and external mechanical forces leads to (j=1,2,…,na)(58)Mg¨+Kqq∗g=F+Felj+Feth+Fpthj,where {Feth} and {Fpth}j are the thermoelastic and pyroelectric forces, respectively, given by(59)Feth=KqΘΘ,Fpthj=-KqΦKΦΦ-1KΦΘΘ,and [Kqq∗] and {Fel} (electrical force) are given, respectively, by(60)Kqq∗=Kqq-Kel,Felj=-∑i=1nejTkjTKqΦaj,for j=1,2,…,na. The electric stiffness matrix [Kel] can be written as(61)Kel=∑i=1nc∑ks=1neiTkiTKqΦciKΦΦci-1KΦqciTki+∑j=1na∑k=1neiTkjTKqΦajKΦΦaj-1KΦqajTkj.The static equation can be written using (33) as follows:(62)ζ=Kqq∗-1F+Fel+Feth+Fpth.For the transformation into a state space model, equations of motion can be written as (63)x˙=Ax+BΦa+Bpertupert+Bth,y=Cx,where [A] is the matrix structure, [B] is the control matrix, [Bpert] is the matrix of disturbances, and [Bth] is the thermal disturbance, given, respectively, by(64)A=-M¯-1Cqq¯-M¯-1Kqq¯I0,B=-M¯-1K¯qϕ0,Bpert=M¯-1F¯0,Bth=M¯-1F¯eth+F¯pth0.


## 3. Results and Discussions

Thermopiezoelastic behavior and the active control of cantileveraluminium plate are presented. The model considered is a flexible piezocomposite rectangular plate size (0.6 m × 0.4 m × 0.001 m) whose geometric and physical properties are shown in Table 1 and in which two piezoelectric layers (size 0.25 m × 0.25 m × 0.0002 m) are perfectly bonded (Figure 2). In the finite element modeling, the plate is divided into 60 × 40 finite elements of equal size and each element has three mechanical, one electrical, and one temperature dofs. The study consists of three phases; the first is to verify the actuation in the static case by exciting the actuator by constant and harmonic inputs; the second phase corresponds to analyse the responses of the structure dynamics under a center plate pulse and study the effect of the location of piezoelectric elements on the effectiveness of control; and the third phase was dedicated to analyzing the effect of thermoelastic and pyroelectric coupling, by applying thermal gradient, on the dynamics of the structure and the control procedure.Figure 2
Actuation by application of constants or harmonic tensions.


### 3.1. Case of Absence of Thermal Effect

To validate the mathematical formulation and the finite element model in the static case, the configuration shown in Figure2 gives an illustration of the mode of actuation of the actuator(s) bonded on the surface of the plate. This allows us to check the actuation states of the actuators and their acting on the plate. To this end, we apply a voltage Φ¯a (constant or harmonic) at the surfaces of the actuators causing a vibration plate.We have given in Figure3 the spectra of the three first modes 11, 12, and 21 and in Figure 4, for illustration, some examples of the first modes shape of vibration of the plate, that is, modes 11, 22, 33, 44, 12, 21, 13, and 14. Figure 5 shows the response to the actuator excitations by tensions with constant or different amplitudes. Figure 6 illustrates the state of the plate after a harmonic input voltage whose amplitudes and frequencies are chosen arbitrarily.Figure 3
Spectra of first modes 11, 12, and 21 of plate dynamics in open and closed loop. Mode 11 at 3.31 Hz, mode 12 at 34.7 Hz, and mode 21 at 56.4 Hz.Figure 4
First modes shape of a plate drawn in 3D.Figure 5
Static test of the used finite element and control methods by giving to the actuator in different sizes pulses with different amplitudes.Figure 6
Plate dynamics under actuator harmonic input with change in position and size of actuator patches.In order to verify the characteristics of the actuators, and by varying their size and position, and applying the two types of voltage excitations, it was shown that the plate can generate dynamics which can be controllable. In Figure7, the symmetrical distributed pairs of patches (actuator/sensor) in several positions and with different sizes on both surfaces of the plate are placed. The center plate displacement in mode 11 with the corresponding actuator voltage and sensor output is shown in Figures 8 and 9. TheLQR-Kalman method parameters (weight matrices Q and R with the gain control Gc) are taken equal to 106, 1, and 200, respectively. Figure 10 gives the first modes of plate vibration with and without control. Moreover, we conclude that the control can be applied during the vibration of the plate without diminishing its quality or effectiveness as shown in Figure 11. The control is applied 0.5 seconds after the start of the oscillation. We note that the plate is quickly damped and consequently the actuator input tension begins to decrease after the application of control. In another side, the position and size of piezoelectric patches may increase or decrease the quality control. Indeed, we deduce from Figure 12 that the wider and nearer to the center of plate the piezoelectric patches are (point undergoing large deformations), the better the structure is controlled. This is justified by the fact that the sensor (mounted in front with the actuator) captures the highest deformations in the plate vibration so that the patches near to the sides of the plate capture the low vibration amplitudes, which affects the output voltage sensor, and therefore the responses of the actuator would be less effective. This is explained by the fact that the patches must be placed as possible near to the the pulse application point. However, if external perturbations are unpredictable, trials must be established to obtain the best positions to the control.Figure 7
Closed loop of the control (sensor-controller-actuator) used in simulation.Figure 8
Response of a pulse (3 N) of displacement of center of plate and corresponding actuator input voltage (Q=106, R=1, and Gc=200).Figure 9
Response of sensor output voltage in open and closed loop (Q=106, R=1, and Gc=200).Figure 10
Shape of first modes 11 and 33 with and without control in 3D (Q=106, R=1, and Gc=200).Figure 11
Application of control after 0.5 seconds from the vibration beginning (Q=106, R=1, and Gc=200).Figure 12
Influence of the position distribution of actuators and sensors on the quality of control (Q=106, R=1, and Gc=200).


### 3.2. Case of Presence of the Thermal Effect

We assume at first that the plate and the layers thermopiezoelectric actuator/sensor are placed in a thermal field and the temperature of the plate reached quickly the studied case. From (23), the change in temperature can generate a voltage in the thermopiezoelectric sensor and the mechanical deformation created under the effect of temperature also produces an electrical signal in the sensor. In order to verify these changes in the deformation, we apply successively, in static case, a thermal gradient in the surface of the plate so that the top surface is hotter than the bottom surface. Because of this gradient, the centerline of the plate is deformed upward. Figure 13 gives the deformation of the plate for different values of the gradient of temperature and illustrates the thermoelastic effect. We recall that the sensor must be more sensitive than the actuator (the coefficient of rigidity of the actuator must be higher compared to the sensor; see Table 1).Figure 13
Plate neutral plan deformation for different values of temperature.We apply, now, at the center of the flexible plate a pulse with amplitude 2 N. After a period of 0.5 seconds of applying the impulse, a control based on the methodLQG-Kalman is applied. The thermal effect due to a temperature gradient of 4∘C is considered in this case (Figure 14). The weight matrices Q and R and gain control Gc are taken equal to 1010, 1, and 200, respectively. We assume that the material with which the piezoelectric patches are bonded has a good resistance to temperature change (the patches remain perfectly bonded to the surfaces of the plate). Figure 15 illustrates the compensation due to the application of the gradient without control. The amplitudes of vibration of the neutral plane centre stabilize at the amplitude −2.5 mm. This is logical because the temperature above the plate is high compared to that below. The plate continues to oscillate in the same amplitude. Consequently, the tension in the actuator increases at the application of the gradient due to the increased mechanical deformations by thermal expansion in the sensor and also the tension due to pyroelectric effect. Figure 15 gives also the corresponding actuator voltage that increases when the gradient is applied.Figure 14
Application of a thermal gradient between surfaces of the plate.Figure 15
Displacement and the corresponding actuator input voltage of plate center and the actuator tension after the application of a 4∘C thermal gradient.We take the same position as before, and we apply the control (Figure16). We note that the plate is checked just to the deviation amplitude (−2.5 mm). The corresponding tension in the actuator is shown in Figure 16. The only precaution is to not let the gradient be applied for a long time; otherwise, we lose the controllability monitoring. It should be noted here that the tension needed to be applied to piezoelectric layers is very important to offset a small gradient of 4∘C, for example. In addition, excessively high voltages applied to the actuator must be avoided in order to not depolarize the thermopiezoelectric material (PZT). This means that thermal gradients applied must be limited by the used material of the actuator. We also note that the application of a constant gradient for a long period cannot be monitored or controlled by theLQG method, but a sudden applied gradient is controllable. This is one of the advantages of this method, since all the thermal disturbances are unpredictable. Figure 17 shows this effect. After a period of 0.5 seconds of applying the pulse (2 N), the plate is subjected suddenly to a temperature gradient of 4°C. The control is applied a second later. The structure is controlled during and after the application of the gradient. This shows that the used control method is effective for sudden changes in temperature. The tension with which the actuator acts is due to the pyroelectric effect (by increasing the temperature of the actuator itself) and the amplified voltage from the controller. It receives a voltage sensor that is the result of the thermal effect (pyroelectric effect) and the effect of increasing mechanical and thermal deformations. The voltage then increases when a thermal gradient is applied and the intensity of control increases accordingly (Figure 17).Figure 16
The noninfluence of the application of thermal gradient on control effectiveness (Q=1010, R=1, and Gc=200).Figure 17
Application of sudden thermal gradient during the application of control (Q=1010, R=1, and Gc=200).


## 3.1. Case of Absence of Thermal Effect

To validate the mathematical formulation and the finite element model in the static case, the configuration shown in Figure2 gives an illustration of the mode of actuation of the actuator(s) bonded on the surface of the plate. This allows us to check the actuation states of the actuators and their acting on the plate. To this end, we apply a voltage Φ¯a (constant or harmonic) at the surfaces of the actuators causing a vibration plate.We have given in Figure3 the spectra of the three first modes 11, 12, and 21 and in Figure 4, for illustration, some examples of the first modes shape of vibration of the plate, that is, modes 11, 22, 33, 44, 12, 21, 13, and 14. Figure 5 shows the response to the actuator excitations by tensions with constant or different amplitudes. Figure 6 illustrates the state of the plate after a harmonic input voltage whose amplitudes and frequencies are chosen arbitrarily.Figure 3
Spectra of first modes 11, 12, and 21 of plate dynamics in open and closed loop. Mode 11 at 3.31 Hz, mode 12 at 34.7 Hz, and mode 21 at 56.4 Hz.Figure 4
First modes shape of a plate drawn in 3D.Figure 5
Static test of the used finite element and control methods by giving to the actuator in different sizes pulses with different amplitudes.Figure 6
Plate dynamics under actuator harmonic input with change in position and size of actuator patches.In order to verify the characteristics of the actuators, and by varying their size and position, and applying the two types of voltage excitations, it was shown that the plate can generate dynamics which can be controllable. In Figure7, the symmetrical distributed pairs of patches (actuator/sensor) in several positions and with different sizes on both surfaces of the plate are placed. The center plate displacement in mode 11 with the corresponding actuator voltage and sensor output is shown in Figures 8 and 9. TheLQR-Kalman method parameters (weight matrices Q and R with the gain control Gc) are taken equal to 106, 1, and 200, respectively. Figure 10 gives the first modes of plate vibration with and without control. Moreover, we conclude that the control can be applied during the vibration of the plate without diminishing its quality or effectiveness as shown in Figure 11. The control is applied 0.5 seconds after the start of the oscillation. We note that the plate is quickly damped and consequently the actuator input tension begins to decrease after the application of control. In another side, the position and size of piezoelectric patches may increase or decrease the quality control. Indeed, we deduce from Figure 12 that the wider and nearer to the center of plate the piezoelectric patches are (point undergoing large deformations), the better the structure is controlled. This is justified by the fact that the sensor (mounted in front with the actuator) captures the highest deformations in the plate vibration so that the patches near to the sides of the plate capture the low vibration amplitudes, which affects the output voltage sensor, and therefore the responses of the actuator would be less effective. This is explained by the fact that the patches must be placed as possible near to the the pulse application point. However, if external perturbations are unpredictable, trials must be established to obtain the best positions to the control.Figure 7
Closed loop of the control (sensor-controller-actuator) used in simulation.Figure 8
Response of a pulse (3 N) of displacement of center of plate and corresponding actuator input voltage (Q=106, R=1, and Gc=200).Figure 9
Response of sensor output voltage in open and closed loop (Q=106, R=1, and Gc=200).Figure 10
Shape of first modes 11 and 33 with and without control in 3D (Q=106, R=1, and Gc=200).Figure 11
Application of control after 0.5 seconds from the vibration beginning (Q=106, R=1, and Gc=200).Figure 12
Influence of the position distribution of actuators and sensors on the quality of control (Q=106, R=1, and Gc=200).


## 3.2. Case of Presence of the Thermal Effect

We assume at first that the plate and the layers thermopiezoelectric actuator/sensor are placed in a thermal field and the temperature of the plate reached quickly the studied case. From (23), the change in temperature can generate a voltage in the thermopiezoelectric sensor and the mechanical deformation created under the effect of temperature also produces an electrical signal in the sensor. In order to verify these changes in the deformation, we apply successively, in static case, a thermal gradient in the surface of the plate so that the top surface is hotter than the bottom surface. Because of this gradient, the centerline of the plate is deformed upward. Figure 13 gives the deformation of the plate for different values of the gradient of temperature and illustrates the thermoelastic effect. We recall that the sensor must be more sensitive than the actuator (the coefficient of rigidity of the actuator must be higher compared to the sensor; see Table 1).Figure 13
Plate neutral plan deformation for different values of temperature.We apply, now, at the center of the flexible plate a pulse with amplitude 2 N. After a period of 0.5 seconds of applying the impulse, a control based on the methodLQG-Kalman is applied. The thermal effect due to a temperature gradient of 4∘C is considered in this case (Figure 14). The weight matrices Q and R and gain control Gc are taken equal to 1010, 1, and 200, respectively. We assume that the material with which the piezoelectric patches are bonded has a good resistance to temperature change (the patches remain perfectly bonded to the surfaces of the plate). Figure 15 illustrates the compensation due to the application of the gradient without control. The amplitudes of vibration of the neutral plane centre stabilize at the amplitude −2.5 mm. This is logical because the temperature above the plate is high compared to that below. The plate continues to oscillate in the same amplitude. Consequently, the tension in the actuator increases at the application of the gradient due to the increased mechanical deformations by thermal expansion in the sensor and also the tension due to pyroelectric effect. Figure 15 gives also the corresponding actuator voltage that increases when the gradient is applied.Figure 14
Application of a thermal gradient between surfaces of the plate.Figure 15
Displacement and the corresponding actuator input voltage of plate center and the actuator tension after the application of a 4∘C thermal gradient.We take the same position as before, and we apply the control (Figure16). We note that the plate is checked just to the deviation amplitude (−2.5 mm). The corresponding tension in the actuator is shown in Figure 16. The only precaution is to not let the gradient be applied for a long time; otherwise, we lose the controllability monitoring. It should be noted here that the tension needed to be applied to piezoelectric layers is very important to offset a small gradient of 4∘C, for example. In addition, excessively high voltages applied to the actuator must be avoided in order to not depolarize the thermopiezoelectric material (PZT). This means that thermal gradients applied must be limited by the used material of the actuator. We also note that the application of a constant gradient for a long period cannot be monitored or controlled by theLQG method, but a sudden applied gradient is controllable. This is one of the advantages of this method, since all the thermal disturbances are unpredictable. Figure 17 shows this effect. After a period of 0.5 seconds of applying the pulse (2 N), the plate is subjected suddenly to a temperature gradient of 4°C. The control is applied a second later. The structure is controlled during and after the application of the gradient. This shows that the used control method is effective for sudden changes in temperature. The tension with which the actuator acts is due to the pyroelectric effect (by increasing the temperature of the actuator itself) and the amplified voltage from the controller. It receives a voltage sensor that is the result of the thermal effect (pyroelectric effect) and the effect of increasing mechanical and thermal deformations. The voltage then increases when a thermal gradient is applied and the intensity of control increases accordingly (Figure 17).Figure 16
The noninfluence of the application of thermal gradient on control effectiveness (Q=1010, R=1, and Gc=200).Figure 17
Application of sudden thermal gradient during the application of control (Q=1010, R=1, and Gc=200).


## 4. Conclusion

The modeling by the finite element method according to the theory ofKirchhoff of a rectangular cantilever piezocomposite plate with bonded thermopiezoelectric elements at its surfaces is presented and implemented. The optimal control based on the method ofLQG-Kalman is applied and discussed. The influence of the location of the patches sensor/actuator and of the application of thermal gradient on the effectiveness of control is demonstrated. It is shown that the application of control during the vibration of the structure does not diminish the control quality; that is, the control can be applied at any time during the vibration of the plate. Moreover, the deformations produced by the sudden application of a thermal gradient can be controlled, to point out that the simulation programs diverge when an intense, or of long duration, thermal gradient is applied. Otherwise, this can infect the pyroelectric effect in the sensor or may deteriorate the polarization of the actuator.



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*Source: 748459-2015-06-03.xml*