Smart structures with integrated sensors, actuators, and control electronics are of importance to the next generation high-performance structural systems. In this study, thermopiezoelastic characteristics of piezoelectric beam continua are studied and applications of the theory to active structures in sensing and optimal control are discussed. Using linear thermopiezoelastic theory and Timoshenko assumptions, a generic thermopiezoelastic theory for piezolaminated composite beam is derived. Finite element equations for the thermopiezoelastic media are obtained by using the linear constitutive equations in Hamilton's principle together with the finite element approximations. The structure consists of a modeling of cantilevered piezolaminated Timoshenko beam with integrated thermopiezoelectric elements between two aluminium layers. 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 accompanied by the Kalman filter is applied. The effects of thermoelastic and pyroelectric couplings on the dynamics of the structure and on the control procedure are studied and 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 beam.
1. Introduction
In the development of distributed sensors, actuators, and thin-film devices, thin layer piezoelectrics (either laminated, deposited, or embedded) are of importance in many applications, for example, dynamic measurement, control, actuation, and so forth. In case studies of electromechanical coupling and implementation of the finite element method, several studies have been the subject of research on the topic covered in this paper. Aldraihem and Khdeir [1] have studied smart beams with extension and thickness-shear piezoelectric actuators. Trindade et al. [2] have investigated the piezoelectric active vibration control of damped sandwich beams. Gabbert et al. [3] have implemented the modeling, control, and simulation of piezoelectric smart structures using finite element method and optimal LQ control. Raja et al. [4] have analysed the active vibration control of composite sandwich beams with piezoelectric extension-bending and shear actuators. Moita et al. [5] have studied the active control of adaptative laminated structures with bonded piezoelectric sensors and actuators. Manjunath and Bandyopadhyay have used the technique of fast output sampling feedback in the control of vibrations in SISO based Timoshenko structures [6, 7]. Trindade and Benjeddou [8] have evaluated and optimized the effective electromechanical coupling coefficients of piezoelectric adaptive structures.
Besides mechanical and electric couplings and interactions, temperature can also influence the performance of piezoelectric devices and its variation can introduce voltage/charge generation in piezoelectric sensors. In addition, control voltage can cause temperature rise in piezoelectric actuators. Temperature can introduce the pyroelectric effect and the thermal strain effect to the distributed sensors and also thermal deflection in dynamic oscillations. Aouadi [9] has discussed the generalized thermopiezoelectric problems with temperature-dependent properties. Ganesan and Sethuraman [10] have studied the thermally induced vibrations of piezothermoviscoelastic composite beam with relaxation times and system response. Sadek and Abukhaled [11] have implemented the optimal control of thermoelastic beam vibrations by piezoelectric actuation.
In this study, distributed sensing and control of a piezolaminated composite beam under sudden and intense thermal gradient have been studied and sensing/control demonstrated. The present work investigates the influence of the thermal and pyroelectric coupling on the dynamic behavior of the flexible composite piezolaminated beam and on the control procedure by the application of a thermal gradient on the faces of the structure. For this purpose, we consider a piezolaminated Timoshenko’s beam with sandwiched thermopiezoelectric sensors and actuators placed at different positions. The structure is modelled by finite element method where the linear constitutive equations in Hamilton’s principle together with the finite element approximations are used. We look first for the effects of changes in temperature and position sensor on quality control by varying the sensor position along the beam. We are secondly looking if there is a decrease in the control quality when it is applied after the start of vibration of the beam. We demonstrate that the active control of the beam is influenced by the variation of temperature.
2. Basic Equations and FE Method Implementation
In this study, we consider a structure consisting of a cantilever composite piezolaminated beam (L × l × t = 0.2 × 0.03 × 0.001) with shear thermopiezoelectric layers (actuator or sensor) and a rigid foam both placed at the core of the structure and sandwiched between two relatively thick aluminum layers (Figure 1). The foam is introduced to fill the space between aluminium layers in order to obtain a compact beam. To obtain a better coupling between the main structure and piezoelectric layers, the center layers (thermopiezoelectric + foam) are considered perfectly bonded. Thickness, mass, and stiffness of the adhesive are considered relatively negligible. The beam is devised in 5 two-node finite beam elements (FE) where the actuator is fixed at the FE1 while the sensor occupies successively the four other finite elements (Figure 1). The physical, piezoelectric, pyroelectric, and thermal characteristics of the used beam and thermopiezoelectric elements (sensor and actuator) are given in the Table 1.
Properties of the aluminium and piezoelectric layers.
Properties
Labels
Units
Beam aluminium
Sensor PVDFbiaxially oriented
ActuatorPZT-5H32
Foam
Length
Lb,s,a
(m)
0.2
0.04
0.04
0.04
Width
lb,s,a
(m)
0.03
0.03
0.03
0.03
Thickness
tb,s,a
(m)
0.001
0.001
0.001
0.001
Density
ρ
(g/cm^{3})
8.03
1.78
7.7
0.7689
Young modulus
E
(GPa)
68
5.04
62
0.69
Piezoelectric stress constant
g31
(Vm/N)
—
0.15
−9.11
—
Piezoelectric strain constant
d31
×10^{−12} (m/V)
—
4.34
−274
—
Pyroelectric constant
p
×10^{−5} Cm^{−2} K^{−1}
—
−1.25
—
—
Dielectric constant
ϵ33S/ϵ0
(ϵ0=8.85410-12 F/m)
—
12
3300
—
Damping
α
×10^{−3}
1
—
—
—
Constants
β
×10^{−3}
0.1
—
—
—
Thermal conductivity
Kd
(W/m·K)
—
1.5
1.5
—
Thermal expansion
α
×10^{−6} (m/m·K)
—
140
3
—
Specific heat
c0
J·kg^{−1}·K^{−1}
—
350
1.5
—
Cantilever beam with embedded thermopiezoelectric elements and the different models.
We start by giving equations of displacement, strain, and stress; then we give the constitutive equations and Hamilton’s principle to the structure. In this work the following linear constitutive relations for thermopiezoelectric materials are employed [12]:
(1)σ=CEɛ-eE-λΘ,D=eTɛ-ϵE+pΘ,{s}=λTɛ+pTE+α~Θ,
where the superscript S means that the values are measured at constant strain and the superscript E means that the values are measured at constant electric field, {σ} is the stress tensor, {D} is the electric displacement vector, {Θ} is temperature, {s} is the entropy, and {E} is the electric field. {ɛ} is the strain tensor, [CE] is the elastic constants at constant electric field, [e] denotes the piezoelectric stress coefficients, [ϵ] is the dielectric tensor at constant mechanical strain, [λ] is the thermoelastic tensor, [p] is the pyroelectric tensor, and α~ is the expansion coefficient with α~=ρpc0/Θ0 where c0 and Θ0 are the specific heat and initial temperature, respectively.
Hamilton’s principle is employed here to derive the finite element equations:
(2)∫t1t2δT-U+We-Wth+δWdt=0,
where t1 and t2 are two arbitrary instants, T is the kinetic energy, U is the potential energy, We denotes the work done by electrical forces, and Wth is the work done by thermal forces. The total kinetic energy T and the potential energy U of the composite beam are described by the following relations:
(3)δT=I1u˙+I2θ˙∂u˙+I1w˙∂w˙+I2u˙+I3θ˙∂θ˙,δU=Nx∂δU∂x+Mx∂δθ∂x+Qxzθ+∂δw∂x.
The work done by electrical forces and thermal forces and the element virtual works done by appliqued surface forces {fA} are given by
(4)δWe=ETD+pΘ,δWth=ɛTλΘ,δW=δqTfA-δΦσq.
The mass moments characteristics of the cross-section of the beam are defined as
(5)I1,I2,I3=c∫-h1h2ρb1,z,z2dz,
with ρb and h1,2,3 being the mass density and the height of the beam + piezoelectric patches (the thickness of the total structure), respectively.
The dynamic equation can be found in [13], and the displacements u(x) and w(x) are written as
(6)ux,z=u0x+zθx,t,wx,z=w0x,
where u0, w0(x), and θ(x) are the axial, transverse midplane displacements and y-rotation, respectively [14]. Assuming that there is no compressibility in the z direction, the normal and transverse components of strain are:
(7)εx=∂u0∂x+z∂θ∂x,εz=0,γxz=∂u∂x+∂w0∂x=θx+∂w0∂x.
The beam constitutive equations can be written as
(8)NxMxQxz=A11B110B11D11000A55∂u0∂x∂θ∂xθ+∂w0∂xT-E11F11G55T,
where
(9)Nx=∫-h/2h/2cσxdz,Mx=∫-h/2h/2cσxzdz,-Qxz=∫-h/2h/2cτxzdz,
where σx=Q¯11 is the extension stress, τxz=Q¯55, γxz is the shear stress, c is the beam width, z is the distance measured between the plane of the structure and that of the kth layer laminate, h is the total thickness of the structure (beam + actuator/sensor + beam), Nx is the axial force, Mx is the bending moment, Qxz is the shear force, A11, B11, D11, A55 are the extension, extension-bending, bending, and transverse shear stiffness coefficients given by [13, 15]
(10)A11=c∑k=1nQ¯11kzk-zk-1,B11=c2∑k=1nQ¯11kzk2-zk-12,D11=c3∑k=1nQ¯11kzk3-zk-13,A55=cK∑k=1nQ¯55kzk-zk-1,
where zk is the distance of the kth layer relative to the x-axis, n is the number of layers, K is the correction shear deformation factor generally taken to be 5/6, and Q¯11, Q¯55 are calculated based on the physical properties of piezoelectric material [13, 15, 16]:
(11)Q¯11=Q11cos4λ+Q22sin4λ+2Q12+Q66sin2λcos2λ,Q¯55=G13cos2λ+G23sin2λ.
The angle λ is the angle between the direction of the fibers and the longitudinal axis of the beam. The physical constants Q11, Q22, Q12, Q66, Q13, Q23 relative to the foam, aluminum, and the piezoelectric material are
(12)Q11=E111-ν12ν12,Q22=E221-ν12ν12,Q12=ν12E111-ν12ν12,Q66=G12,ν12E11=ν12E11,
where ν is the poisson coefficient and G is the rigidity transverse modulus. Respectively, E11, F11, and G55 are the induced piezoelectric axial force, bending moment due to deformation of the actuator, and the shear strength given by [13, 15]
(13)E11=c∑k=1naQ¯11kaVkx,td31k,F11=c2∑k=1naQ¯11kaVkx,td31kzk+a-zk-a,G55=cK∑k=1naQ¯55kaVkx,td15k.
Here, E11=F11=0 since the piezoelectric layers are polarized longitudinally. Vk(x,t)= the voltage applied to the kth actuator with thickness (zk+a-zk-a) and piezoelectric constants d31k and d15k. Na= the number of actuators. Consider
(14)u,w,θ=Nu,Nw,Nθq,
where q=u1w1θ1u2w2θ2T is the vector of nodal displacements and [Nu], [Nw], [Nθ] are the mode shape functions due to the axial displacement, transverse displacement, and the slop, which are defined as [17]
(15)Nu=N1N2N3N4N5N6,Nw=N7N8N9N10,Nθ=N11N12N13N14.
The elements of the shape functions are given in [17]. The inertial forces vector N can be written as
(16)N=N1N2N3N4N5N60N7N80N9N100N11N120N13N14q.
The mass matrix of the regular beam element is given by
(17)Mb=∫0lbNTINdx,
where
(18)I=I10I20I10I20I3
is the inertia matrix. Similarly, the stiffness matrix of the regular beam element can be written as
(19)Kb=∫0lbBTDBAbdx,
where Ab is the area of cross-section of the beam element and
(20)B=dNdx,D=A11B110B11B11000A55.
The mass and stiffness element matrices of the piezoelectric element are obtained in the same manner with respect to the physical characteristics of the material. The mass and stiffness of the finite elements which together contain the element of the beam and the piezoelectric element are given by M=Mp+2Mb and K=Kp+2Kb.
The sensor output voltage, due to thermal and mechanical deformations and temperature (pyroelectric effect), can be written as
(21)Vst=STq˙+pΘ˙,
where
(22)ST=6cη-12η+lb2Gce1502-lp0-2lp,
and lb is the beam element length which is equal to piezoelectric element length lp, Gc is the controller gain, p is the pyroelectric constant, and η is a constant given by
(23)η=D11A55γB11D11-1,γ=B11A11;
the control force developed by the thermopiezoelectric actuator is written
(24)fctrl=Gd15h¯∫0lpNθdxVat=hVat,
where G is the transverse module, h¯=(ha+hb)/2 is the distance between neutral axes of the beam and the thermopiezoelectric layer, Nθ is the shape function of rotations, and Va(t) is the actuator input voltage.
Similarly, the forces due to thermoelastic and pyroelectric couplings feth and fpth are given by
(25)feth=∫NΘTλNΘΘdA,fpth=∫NΘTpNΘΘdA,
where NΘ is thermal shape function, [λ] is the thermoelastic tensor, and [p] is pyroelectric tensor. If an external force fext is applied, the total force acting on the beam is
(26)ft=fext+fctrl+feth+fpth.
Therefore, the equation of motion can be written as
(27)M*g¨+C*g˙+K*g=fext*+fctrl*+feth*+fpth*,
where M*, K*, C*, and g are the generalized mass, stiffness, damping matrices, and generalized displacement. The above equation can be transformed in state space model as
(28)x˙=Axt+But+Ert+Eth,yt=CTxt+Dut,
with
(29)A=0I-M*-1K*-M*-1C*,B=0M*-1TTh,C=0STT,E=0M*-1TTrt,Eth=0M*-1TTfeth+fpth,
where T is the modal matrix, u(t) is the command vector, and r(t) is the external force vector.
3. Results and Discussion
The graphs presented below correspond to the responses of the free end of the beam. We compare noncontrolled and controlled responses for a disturbance in impulse (2N) before, during, and after applying a thermal gradient so that the faces of the structure are under two different temperatures as illustrated in Figure 2. We assume that the material with which the material bonded the thermopiezoelectric elements is resistant to the temperature change (the patches remain perfectly bonded to the surfaces of the beam). A thermal gradient will be applied to visualize the thermal and pyroelectric effect on the vibration behavior of the structure under thermal perturbations.
Application of a thermal gradient.
Figures 3 and 4 show the three first modes. The different responses, step, impulse, and sinusoidal are illustrated in the Figures 5, 6, and 7. Figure 8 shows 3D control visualisation for the three first modes and for (2N) pulse excitation at free end of the beam.
The first three modes of the beam subjected to a pulse of 2N at its free end.
Spectra of the first three modes for the sensor positions EF5 and EF2 in SISO model, respectively.
Influence of the sensor location on the quality of control in the case of a pulse excitation (Q=107, R=1, and controller gain Gc=200) for the two first modes.
Influence of the sensor location on the quality of control in the case of a step excitation (Q=105, R=1, and controller gain Gc=200) for the first mode.
Displacement (without and with noise) of the beam subjected to a sinusoidal excitation 1.5sin(50t) to its free end for (Q=1010, R=0.01, and controller gain Gc=500).
3D control visualisation for the three first modes and for (2N) pulse excitation at free end of the beam, Q=106, R=0.01, and Gc=500).
Figure 10 shows in 3D the deflection of the beam under a 5°C thermal gradient. In addition to the sensor voltage created by the expansion of the beam under the thermal field, another voltage is created by the temperature increase (pyroelectric effect). In fact, respectively, Figures 9 and 10 show the variation of amplitude and voltage at the input of the actuator for the beam before and during the control applied 0.5 sec after the application of the pulse. The control method can be applied during the vibration without diminishing its quality or effectiveness. The same above figures illustrate the thermal compensation of responses in amplitude and voltage of the actuator. The gradient is applied at the beginning of the pulse after 0.5 sec. We noticed that the application of such a gradient did not alter the quality and effectiveness of control. The effectiveness of control will depend on the intensity of the thermal gradient and its duration. Indeed, the intense gradients can infect the pyroelectric effect in the thermopiezoelectric elements and deteriorate their polarization. We finally conclude that the control method used is better adapted to sudden changes in disturbance types of vibration or for small variations in the thermal gradient. To show the influence of thermal effect on the used control method, we maintain a gradient for 3 sec. We note that the control procedure is effective in applying or removing the thermal gradient. However, the application of a constant gradient for a long duration can not be monitored or controlled by the method LQG. Instead, the sudden application of a gradient is controllable. This is one of the advantages of this method, since all the thermal disturbances are unpredictable. We conclude, therefore, that the control method used is effective for sudden changes in temperature.
Application of control during the vibration of the beam.
The responses of the dynamics of the beam before, during, and after the application of control and thermal gradient.
4. Conclusion
Modeling by the finite element method according to the theory of Timoshenko, of a cantilever piezolaminated composite beam with embedded thermopiezoelectric element, is presented. The optimal control based on the method of LQG-Kalman is applied and discussed. The influence of the location of the sensor and of the application of thermal gradient on the effectiveness of control is analyzed. In fact, our analysis shows that the more the sensor is near the free end of the beam, the more the control is effective. This is due to the the increase of the deformation amplitudes in the sensor, which affects the control voltage of the actuator. We have also 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 beam. Moreover, the deformations produced by the sudden application of a thermal gradient can be controlled. We reported that the application of an intense thermal gradient, or of long duration, can infect the pyroelectric effect in the sensor or may deteriorate the polarization of the actuator.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
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