Research Article Volume 3 Issue 5
Department of Intellectual Technical Systems, National Research University of Electronic Technology (MIET), Russia
Correspondence: Sergey M Afonin Department of Intellectual Technical Systems, National Research University of Electronic Technology (MIET), Moscow, 124498 Moscow, Russia
Received: April 21, 2017 | Published: September 4, 2018
Citation: Afonin SM. Structural–parametric model electroelastic actuator nano– and microdisplacement of mechatronics systems for nanotechnology and ecology research. MOJ Eco Environ Sci. 2018;3(5):306-309 DOI: 10.15406/mojes.2018.03.00104
The structural–parametric model, the decision of the wave equation, the parametric structural schematic diagram, the transfer functions of the electroelastic actuator of the mechatronics system for the nanotechnology and the ecology research are obtained. Effects of geometric and physical parameters of the piezoactuator and the external load on its dynamic characteristics are determined. The parametric structural schematic diagram and the transfer functions of the piezoactuator for the transverse, longitudinal, shift piezoelectric effects are obtained from the structural–parametric model of the piezoactuator. For calculation of the mechatronics systems for the nanotechnology with the piezoactuator it’s the parametric structural schematic diagram and the transfer functions are determined. The generalized parametric structural schematic diagram of the electroelastic actuator is constructed.
Keywords: electroelastic actuator, piezoactuator, deformation, structural–parametric model, parametric structural schematic diagram, decision wave equations, transfer functions
For the nanotechnology, the ecology research, the nanobiology, the power engineering, the microelectronics, the astronomy for the large compound telescopes, the antennas satellite telescopes and the adaptive optics equipment is promising for use the mechatronics system with the actuator based on the electroelasticity for the piezoelectric or the electrostriction effects. The piezoactuator is the piezomechanical device intended for the actuation of mechanisms, systems or the management based on the piezoelectric effect, the converts electrical signals into the mechanical movement or the force.1–5 In the present work is solving the problem of building the structural parametric model of the electroelastic actuator in contrast Cady and Mason electrical equivalent circuits for calculation of piezoelectric transmitter and receiver.6–9 The structural–parametric model of the piezoactuator describes the structure and conversion the energy electric field into the mechanical energy and the corresponding displacements and forces at its the faces. The structural–parametric model of the electroelastic actuator of the mechatronics system is determined by using the method of the mathematical physics. The transfer functions and the parametric structural schematic diagrams of the electroelastic actuator are obtained from its structural–parametric model.3–14 The piezoactuator for the nano– and microdisplacement of the mechatronics system operates based on the inverse piezoeffect. The displacement is achieved due to deformation of the piezoactuator when the external electric voltage is applied to it. The piezoactuator for the drives of nano– and micrometric movements provide a movement range from several nanometers to tens of micrometers, a sensitivity of up to 10 nm/V, a loading capacity of up to 1000 N, a transmission band of up to 100 Hz. The piezoactuator provides high speed and force, its return to the initial state when switched off. The use of the piezoactuator solves the problems of the precise alignment and the compensation of the temperature and gravitational deformations. The piezoactuator is used in the majority mechatronic systems for the nanotechnology, the ecology research in the scanning tunneling microscopes and the atomic force microscopes.11–16
The deformation of the electroelastic actuator corresponds to its stressed state. In the piezoactuator there are six stress components T1T1 , T2T2 T3T3 , T4T4 , T5T5 , T6T6 where the components T1−T3T1−T3 are related to extension–compression stresses and the components T4−T6T4−T6 to shear stresses. The matrix state equations8,11 connecting the electric and elastic variables for the polarized piezoceramics have the following form:
D=dT+εTED=dT+εTE , (1)
S=sET+dtES=sET+dtE , (2)
where the first equation describes the direct piezoelectric effect, and the second - the inverse piezoelectric effect; DD is the column matrix of electric induction along the coordinate axes; SS is the column matrix of relative deformations; TT is the column matrix of mechanical stresses; E E is the column matrix of electric field strength along the coordinate axes; sE sE is the elastic compliance matrix for E=constE=const ; εTεT is the matrix of dielectric constants for T=constT=const ; dt dt is the transposed matrix of the piezoelectric modules.
In polarized piezoceramics from lead zirconate titanate PZT for the piezoactuator on Figure 1 there are five independent components sE11sE11 sE12sE12 , sE13sE13 ,sE33sE33 , sE11sE11 in the elastic compliance matrix, three independent components d33d33 ,d31d31 , d15d15 in the transposed matrix of the piezoelectric modules and three independent components, εT11εT11 ,εT33εT33 εT22εT22 in the matrix of dielectric constants.
Let us consider the piezoactuator for the longitudinal piezoelectric effect, where δδ is thickness and the electrodes deposited on its faces perpendicular to axis 3, the area of which is equalS0S0 . The direction of the polarization axis Р, i.e., the direction along which polarization was performed, is usually taken as the direction of axis 3. The equation of the inverse longitudinal piezoelectric effect8,11 has the form:
S3=d33E3(t)+sE33T3(x,t)S3=d33E3(t)+sE33T3(x,t) , (3)
where S3=∂ξ(x,t)/∂xS3=∂ξ(x,t)/∂x is the relative displacement of the cross section of the piezoactuator, d33d33 is the piezomodule for the longitudinal piezoeffect, E3(t)=U(t)/δE3(t)=U(t)/δ is the electric field strength, U(t)U(t) c is the voltage between the electrodes of actuator, δδ is the thickness, sE33sE33 is the elastic compliance along axis 3, and T3T3 is the mechanical stress along axis 3.
The equation of equilibrium for the force acting on the piezoactuator on Figure 1 can be written as
T3S0=F+M∂2ξ(x,t)∂t2T3S0=F+M∂2ξ(x,t)∂t2 , (4)
Where F is the external force applied to the piezoactuator, S0S0 is the cross section area and M is the displaced mass.
the equation of the inverse longitudinal piezoeffect, the wave equation using Laplace transform, the equations of the forces acting on the faces of the piezoactuator. The calculations of the piezoactuators are performed using the wave equation8,11,12 describing the wave propagation in the long line with damping but without distortions in the following form:
1(сE)2∂2ξ(x,t)∂t2+2αcE∂ξ(x,t)∂t+α2ξ(x,t)=∂2ξ(x,t)∂x2, (5)
where ξ(x,t) is the displacement of the section, x is the coordinate, t is the time, cE is the sound speed for E=const, α is the damping coefficient. We can reduce the original problem for the partial differential hyperbolic equation of type (5) using Laplace transform to a simpler problem for the linear ordinary differential equation[10,12]. Applying the Laplace transform to the wave equation (5)
Ξ(x,p)=L{ξ(x,t)}=∞∫0ξ(x,t)e−ptdt, (6)
Setting the zero initial conditions we obtain the linear ordinary second–order differential equation with the parameter p in the form
d2Ξ(x,p)dx2−γ2Ξ(x,p)=0, (7)
With its solution being the function
Ξ(x,p)=Ce−xγ+Bexγ, (8)
Where Ξ(x,p) is the Laplace transform of the displacement of the section of the piezoelectric actuator, γ=p/cE+α is the propagation coefficient.
We denote for the faces of the piezoactuator
Ξ(0,p)=Ξ1(p) for x=0,(9)
Ξ(δ,p)=Ξ2(p) for x=δ.
Then we get the coefficients C and B
C=(Ξ1eδγ−Ξ2)/[2sh(δγ)] ,B=(Ξ2−Ξ1e−δγ)/[2sh(δγ)] , (10)
The solution (7) can be written as
Ξ(x,p)={Ξ1(p)sh[(δ−x)γ]+Ξ2(p)sh(xγ)}/sh(δγ) , (11)
The equations for the forces on the faces of the piezoactuator
T3(0,p)S0=F1(p)+M1p2Ξ1(p) for x=0 , (12)
T3(δ,p)S0=−F2(p)−M2p2Ξ1(p) for x=δ ,
Where T3(0,p) and T3(δ,p) are determined from the equation of the inverse piezoelectric effect. For x=0 and T3(δ,p) , we obtain the set of equations for determining stresses in the piezoactuator:11−14
T3(0,p)=1sE33dΞ(x,p)dx|x=0−d33sE33E3(p) , (13)
T3(δ,p)=1sE33dΞ(x,p)dx|x=δ−d33sE33E3(p) .
The set of equations (13) yield the set of the equations for the structural–parametric model of the piezoactuator and the parametric structural schematic diagram of the voltage–controlled piezoactuator for the longitudinal piezoelectric effect on Figure 2.
Ξ1(p)=[1/(M1p2)]⋅{−F1(p)+(1/χE33) [d33E3(p)−[γ/sh(δγ)] [ch(δγ)Ξ1(p)−Ξ2(p)]]}, (14)
Ξ2(p)=[1/(M2p2)]⋅{−F2(p)+(1/χE33) [d33E3(p)−[γ/sh(δγ)] [ch(δγ)Ξ2(p)−Ξ1(p)]]},
WhereχE33=sE33/S0 .
From (2), (3), (14) we obtain the system of the equations describing the generalized structural–parametric model of the electroelastic actuator
Ξ1(p)=[1/(M1p2)]⋅{−F1(p)+(1/χΨij)[dmiΨm(p)−[γ/sh(lγ)][ch(lγ)Ξ1(p)−Ξ2(p)]]}, (15)
Ξ2(p)=[1/(M2p2)]⋅{−F2(p)+(1/χΨij)[dmiΨm(p)−[γ/sh(lγ)][ch(lγ)Ξ2(p)−Ξ1(p)]]},
where dmi={d33,d31,d15g33,g31,g15 , Ψm={E3,E3,E1D3,D3,D1 , sΨij={sE33,sE11,sE55sD33,sD11,sD55 , l={ δ,h,b , cΨ={ cE,cD , γΨ={ γE,γD ,
χΨij=sΨij/S0 , i = 1, 2…, 6, j = 1, 2, … , 6, m = 1, 2, 3,
Then the parameter Ψ of the control parameter for the electroelastic actuator: E for the voltage control, D for the current control. On Figure 3 is shown the generalized parametric structural schematic diagram of the electroelastic actuator corresponding to the set (15) of the equations.
From the generalized structural–parametric model (15) of the electroelastic actuator after the algebraic transformations we obtain the transfer functions in matrix form.11−14 The transfer functions are the ratio of the Laplace transform of the displacement of the face for the electroelastic actuator and the Laplace transform of the corresponding control parameter or force at zero initial conditions.
Ξ1(p)=W11(p)Ψm(p)+W12(p)F1(p)+W13(p)F2(p) , (16)
Ξ2(p)=W21(p)Ψm(p)+W22(p)F1(p)+W23(p)F2(p) ,
where the generalized transfer functions
W11(p)=Ξ1(p)/Ψm(p)=dmi [M2χΨijp2+γth(lγ/2)]/Aij ,
Aij=M1M2(χΨij)2p4+{ (M1+M2)χΨij/[cΨth(lγ)] }p3+[(M1+M2)χΨijα/th(lγ)+1/(cΨ)2]p2+2αp/cΨ+α2,
W21(p)=Ξ2(p)/Ψm(p)=dmi [M1χΨijp2+γth(lγ/2)]/Aij ,
W12(p)=Ξ1(p)/F1(p)=−χΨij[M2χΨijp2+γ/th(lγ)]/Aij ,
W13(p)=Ξ1(p)/F2(p)=W22(p)=Ξ2(p)/F1(p)=[χΨijγ/sh(lγ)]/Aij,
W23(p)=Ξ2(p)/F2(p)=−χΨij[M1χΨijp2+γ/th(lγ)]/Aij .
From the set (15) of the equations we obtain the generalized matrix equation for the electroelastic actuator
(Ξ1(p)Ξ2(p))=(W11(p)W12(p)W13(p)W21(p)W22(p)W23(p)) (Ψm(p)F1(p)F2(p)) . (17)
Let us find the displacement of the faces for the electroelastic actuator in the stationary regime for the inertial load at Ψm(t)=Ψm0×1(t) , F1(t)=F2(t)=0 .
Then we get the static displacement of the faces for the electroelastic actuator
ξ1(∞)=limt→∞ξ1(t)=limp→0α→0pW11(p)Ψm0/p=dmilΨm0(M2+m/2)/(M1+M2+m), (18)
ξ2(∞)=limt→∞ξ2(t)=limp→0α→0pW21(p)Ψm0/p=dmilΨm0(M1+m/2)/(M1+M2+m), (19)
ξ1(∞)+ξ2(∞)=limt→∞(ξ1(t)+ξ2(t))=dmilΨm0 , (20)
where m is the mass of the electroelastic actuator, M1, M2 are the load masses.
Let us consider the static characteristics of the piezoactuator from the piezoceramics PZT under the longitudinal piezoelectric effect at m<<M1 andm<<M2 . For d33=4⋅10−10 m/V, U=50 V, M1=2 kg and M2=8 kg we obtain the static displacement of the faces of the piezoactuatorξ1(∞)=16 nm,ξ2(∞)=4 nm, ξ1(∞)+ξ2(∞)=20 nm. The displacements in the stationary regime of the faces for the piezoactuator under the transverse piezoelectric effect and the inertial load at U(t)=U0⋅1(t) ,E3(t)=E30⋅1(t)=(U0/δ)⋅1(t) , F1(t)=F2(t)=0 can be written in the following form
ξ1(∞)=limt→∞ξ1(t)=limp→0α→0pW11(p)(U0/δ)/p=d(h/δ)31U0(M2+m/2)/(M1+M2+m), (21)
ξ2(∞)=limt→∞ξ2(t)=limp→0α→0pW21(p)(U0/δ)/p=d31(h/δ)U0(M1+m/2)/(M1+M2+m) (22)
ξ1(∞)+ξ2(∞)=limt→∞(ξ1(t)+ξ2(t))=d31(h/δ)U0 . (23)
From (21), (22) we obtain the static displacements of the faces of the piezoactuator under the transverse piezoeffect at m<<M1 , m<<M2 in the form
ξ1(∞)=limt→∞ξ1(t)=limp→0α→0pW11(p)(U0/δ)/p=d31(h/δ)U0M2/(M1+M2), (24)
ξ2(∞)=limt→∞ξ2(t)=limp→0α→0pW21(p)(U0/δ)/p=d31(h/δ)U0M1/(M1+M2). (25)
Let us consider the static characteristics of the piezoactuator from piezoceramics PZT under the transverse piezoelectric effect atm<<M1 and m<<M2 . For d31=2⋅10−10 m/V, h=4⋅10−2 m, δ=2⋅10−3 m, U=50 V, M1=2 kg and M2=8 M2=8 kg we obtain the static displacement of the faces of the piezoelectric actuator ξ1(∞)=160 nm, ξ2(∞)=40 nm, ξ1(∞)+ξ2(∞)=200 nm.
From (16) we obtain the transfer functions of the piezoactuator with the fixed end and the elastic inertial load so thatM1→∞ and m<<M2 in the following form
W2(p)=Ξ2(p)U(p)=d33(1+Ce/CE33) (T2tp2+2Ttξtp+1) , (26)
where the time constant Tt and the damping coefficient ξt are determined by the formulas
Tt=√M2/(C+eCE33) , ξt=αδ2CE33/(3cE√M(Ce+CE33)) .
Let us consider the operation of the piezoactuator from piezoceramics PZT with one face rigidly fixed and the elastic inertial load so that M1→∞ and m<<M2 for M2=10 kg,C33=2.1⋅106 N/m, Ce=0.4⋅106 N/m we obtain Tt=2⋅10−3 c. The experimental and calculated values for the piezoactuator are in agreement to an accuracy of 5%.
The structural–parametric models, the decision of the wave equation, the parametric structural schematic diagram, the transfer functions of the electroelastic actuator are obtained using Laplace transform. The parametric structural schematic diagram and the transfer functions of the piezoactuator for the transverse, longitudinal, shift piezoelectric effects are determined from the structural–parametric model of the electroelastic actuator. The transfer functions in matrix form are describes deformations of the piezoactuator during its operation as part of the mechatronics system for the nanotechnology and the ecology research. From the decision of the electroelasticity equation, the wave equation and the features of the deformations along the coordinate axes we obtain the generalized structural–parametric model and the parametric structural schematic diagram of the electroelastic actuator for the mechatronics system and its dynamic and static properties.
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The author declares there is no conflict of interest.
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