Structural-Parametric Model Electromagnetoelastic Actuator Nano- and Microdisplacement for Precision Engineering

Sergey Mikhailovich Afonin

Department of Intellectual Technical Systems, National Research University of Electronic Technology (MIET), Moscow, Russia

Email address

Citation

Sergey Mikhailovich Afonin. Structural-Parametric Model Electromagnetoelastic Actuator Nano- and Microdisplacement for Precision Engineering. Engineering and Technology. Vol. 3, No. 6, 2016, pp. 110-119.

Abstract

Structural-parametric model, parametric structural schematic diagram, transfer functions of electromagnetoelastic actuator nano- and microdisplacement for precision engineering are obtained. Effects of geometric and physical parameters of electromagnetoelastic actuators and external load on its dynamic characteristics are determined. For calculation of precision systems with piezoactuators the parametric structural schematic diagram and the transfer functions of piezoactuators are obtained. A generalized parametric structural schematic diagram and transfer functions of the piezoactuators are constructed.

Keywords

Electromagnetoelastic Actuators, Structural-Parametric Model, Piezoactuator, Deformation, Parametric Structural Schematic Diagrams, Nano- and Microdisplacement, Transfer Functions

1. Introduction

For precision engineering, nanotechnology, electronic engineering, microelectronics, nanobiology, power engineering, astronomy, antennas satellite telescopes and adaptive optics are promising the electromechanical actuators based on electromagnetoelasticity (piezoelectric, piezomagnetic, electrostriction, and magnetostriction effects). Piezoactuator - piezomechanical device intended for actuation of mechanisms, systems or management based on the piezoelectric effect, converts electrical signals into mechanical movement or force. Piezoactuators are used in the majority of scanning tunneling microscopes (STMs), atomic force microscopes (AFMs) [1-16].

In the present paper is solving the problem of building the structural parametric model of the electromagnetoelastic actuator nano- and microdisplacement in contrast solving its electrical equivalent circuit [5-7]. Equivalent circuits of the piezoelectric transducers are designed for calculation of piezoelectric transmitters and receivers. By solving the wave equation with allowance methods of mathematical physics for the corresponding equation of electromagnetoelasticity, the boundary conditions on loaded working surfaces of actuators, and the strains along the coordinate axes, it is possible to construct a structural-parametric model of the actuator.

The transfer functions and the parametric structural schematic diagrams of the actuators are obtained from a set of equations describing the corresponding structural-parametric model [17-30].

2. The Structural-Parametric Model of the Electromagnetoelastic Actuator

For the piezoactuator its deformation corresponds to stressed state. If the mechanical stress is created in the piezoelectric element, the deformation  is formed in it. There are six stress components , , , , , , the components  -  are related to extension-compression stresses,  -  to shear stresses.

The matrix state equations [7] connecting the electric and elastic variables for polarized ceramics have the form

,                 (1)

,                 (2)

where the first equation describes the direct piezoelectric effect, and the second - the inverse piezoelectric effect;  is the column matrix of electric induction along the coordinate axes;  is the column matrix of relative deformations;  is the column matrix of mechanical stresses;  is the column matrix of electric field strength along the coordinate axes;  is the dielectric permeability matrix for T = const;  is the elastic compliance matrix for ;  is the transposed matrix of the piezoelectric modules. In polarized ceramics PZT there are five independent components , , , ,  in the elastic compliance matrix for polarized piezoelectric ceramics, three independent components of the piezoelectric modules , ,  in the transposed matrix of the piezoelectric modules and three independent components of the dielectric constants , ,  in the matrix of dielectric constants.

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 for the longitudinal and transverse piezoelectric effects.

The generalized electromagnetoelasticity equation of the actuator [7] has the form

,     (3)

where  is the relative deformation along the axis i, E is the electric field strength, H is the magnetic field strength,  is the temperature,  is the elastic compliance for , , ,  is the mechanical stress along the axis j,  is the piezomodule, i.e., the partial derivative of the relative deformation with respect to the electric field strength for constant magnetic field strength and temperature, i.e., for , ,  is the electric field strength along the axis m,  is the magnetostriction coefficient,  is the magnetic field strength along the axis m,  is the coefficient of thermal expansion,  is deviation of the temperature  from the value , i = 1, 2, …, 6, j = 1, 2, …, 6, m = 1, 2, 3.

For the electric and magnetic fields acting on the electromagnetoelastic actuator separately, we have [2,7,9]

the equation of inverse piezoelectric effect:

for the longitudinal deformation when the electric field along axis 3 causes deformation along axis 3

,                (4)

for the transverse deformation when the electric field along axis 3 causes deformation along axis 1

,                (5)

for the shift deformation when the electric field along axis 1 causes deformation in the plane perpendicular to this axis

,                (6)

the equation of magnetostriction:

for the longitudinal deformation when the magnetic field along axis 3 causes deformation along axis 3

,                (7)

for the transverse deformation when the magnetic field along axis 3 causes deformation along axis 1

,                (8)

for the shift deformation when the magnetic field along axis 1 causes deformation in the plane perpendicular to this axis

.                (9)

Let us consider the longitudinal piezoelectric effect in a piezoactuator shown in Figure 1, where  is the thickness. The electrodes deposited on its faces perpendicular to axis 3, the area of face is equal to . In the equation (4) of the inverse longitudinal piezoelectric effect are the following parameters:  is the relative displacement of the cross section of the piezoactuator,  is the piezomodule for the longitudinal piezoelectric effect,  is the electric field strength,  is the voltage between the electrodes of actuator,  is the thickness,  is the elastic compliance along axis 3, and  is the mechanical stress along axis 3.

Figure 1. Piezoactuator for the longitudinal piezoelectric effect.

For constructing a structural parametric model of the voltage-controlled piezoactuator, let us solve simultaneously the wave equation, the equation of the inverse longitudinal piezoeffect, and the equation of forces acting on the faces of the piezoactuator. Calculations of the piezoactuators are performed using a wave equation [2,7,9] describing the wave propagation in a long line with damping but without distortions, which can be written as

,     (10)

where  is the displacement of the section, x is the coordinate, t is time,  is the sound speed for ,  is the damping coefficient. Using the Laplace transform, we can reduce the original problem for the partial differential hyperbolic equation of type (10) to a simpler problem for the linear ordinary differential equation [8, 9]. Applying the Laplace transform to the wave equation (10) and setting the zero initial conditions, we obtain the linear ordinary second-order differential equation with the parameter p

,   (11)

with its solution being the function

,            (12)

where  is the Laplace transform of the displacement of the section of the piezoelectric actuator,  is the propagation coefficient. Determining coefficients C and B from the boundary conditions as

 for ,           (13)

 for .

Then, the constant coefficients

, .                   (14)

The solution (11) can be written as

.                    (15)

The equations for the forces on the faces of the piezoactuator

 for ,     (16)

 for ,

where  and  are determined from the equation of the inverse piezoelectric effect.

For  and , we obtain the following set of equations for determining stresses in the piezoactuator [24-26]:

,      (17)

.

The set of equations (17) yield the set of equations for the structural-parametric model of the piezoactuator Figure 2:

,                    (18)

,

where .

Figure 2. Parametric structural schematic diagram of a voltage-controlled piezoactuator for longitudinal piezoelectric effect.

In the equation (5) of the inverse transverse piezoeffect [7, 9] are the following parameters:  is the relative displacement of the cross section along axis 1 Figure 3,  is the piezomodule for the transverse piezoeffect,  is the elastic compliance along axis 1,  is the stress along axis 1.

Figure 3. Piezoactuator for the transverse piezoelectric effect.

The solution of the linear ordinary differential equation (11) can be written as (12), where the constants C and B in the form

 for ,          (19)

 for ,

 (20)

Then, the solution (11) can be written as

 (21)

The equations of forces acting on the faces of the piezoactuator

 for x = 0,    (22)

 for x = h,

where

    (23)

The set of equations describing the structural-parametric model and parametric structural schematic diagram Figure 4

,                     (24)

,

where .

Figure 4. Parametric structural schematic diagram of a voltage-controlled piezoactuator for transverse piezoelectric effect.

Let us consid er the piezoactuator for the shift piezoelectric effect (6) on Figure 5.

Figure 5. Piezoactuator for the shift piezoelectric effect.

We obtain the following set of equations describing the structural-parametric model and schematic diagram Figure 6

,                    (25)

,

where .

Figure 6. Parametric structural schematic diagram of a voltage-controlled piezoactuator for shift piezoelectric effect.

From (3), (18), (24), (25) we obtain the system of equations describing the generalized structural-parametric model of the electromagnetoelastic actuator

,                   (26)

,

, , ,

, , , ,

where parameters  of the control for the electromagnetoelastic actuator: E for voltage control, D for current control, H for magnetic field strength control. On Figure 7 shows the generalized parametric structural schematic diagram of the electromagnetoelastic actuator corresponding to the set of equations (26) of the actuator for precision engineering.

Figure 7. Generalized parametric structural schematic diagram of the electromagnetoelastic actuator.

3. Transfer Functions of Electromagnetoelastic Actuator

Let us consider the generalized structural-parametric model (26) of the actuator after algebraic transformations provides the transfer functions of the electromagnetoelastic actuator in matrix form [24-26], where the transfer functions are the ratio of the Laplace transform of the displacement of the face actuator and the Laplace transform of the corresponding parameter or force at zero initial conditions.

,(27)

,

where the generalized transfer functions of the electromagnetoelastic actuator are the following form:

,                 (28)

,                        (29)

,                        (30)

,                             (31)

.                        (32)

Therefore, we obtain from equations (27) the generalized matrix equaion for the electromagnetoelastic actuator

.  (33)

Let us find the displacement of the faces the electromagnetoelastic actuator in a stationary regime for ,  and inertial load. The static displacement of the faces the electromagnetoelastic actuator  and  can be written in the form

,       (34)

,      (35)

,     (36)

where m is the mass of the electromagnetoelastic actuator,  are the load masses.

Let us consider a numerical example of the calculation of static characteristics of the piezoactuator from piezoceramics PZT under the longitudinal piezoelectric effect at  and . For m/V, V, kg and kg we obtain the static displacement of the faces of the piezoactuator nm, nm, nm.

The static displacement the faces of the piezoactuator for the transverse piezoelectric effect and inertial load at ,  and  can be written in the following form:

,   (37)

,   (38)

.          (39)

Consider a numerical example for the calculation of static characteristics of the piezoactuator from piezoceramics PZT under the transverse piezoelectric effect at  and . For m/V, m, m, V, kg and kg we obtain the static displacement of the faces of the piezoactuator μm, μm,  μm.

For the description of the piezoactuator for the longitudinal piezoelectric effect for one rigidly fixed face of the transducer at  we obtain from equation (29) and (32) the transfer functions  and of the piezoactuator for the longitudinal piezoelectric effect in the following form:

,(40)

. (41)

Accordingly, the static displacement  of the piezoactuator under the longitudinal piezoeffect in the form

,      (42)

.        (43)

Consider a numerical example for the calculation of static characteristics of the piezoactuator under the longitudinal piezoeffects. For m/V, V we obtain nm. For m, m²/N, N, m² we obtain nm. The experimental and calculated values for the piezoactuator are in agreement to an accuracy of 5%.

Let us consider the operation at low frequencies for the piezoactuator with one face rigidly fixed so that  and . Using the approximation of the hyperbolic cotangent by two terms of the power series in transfer functions (40) and (41), at  we obtain the expressions in the frequency range of

,           (44)

,          (45)

, ,

.

where  is the time constant and  is the damping coefficient,  - is the is rigidity of the piezoactuator under the longitudinal piezoeffect.

In the static mode of operation the piezoactuator for elastic load we obtain the equation the following form

,             (46)

where  is the displacement of the piezoactuator in the case of the elastic load,  is the maximum displacement of the piezoactuator,  is the load rigidity.

From (44), (46) we obtain the transfer functions of the piezoactuator with a fixed end and elastic inertial load

,                 (47)

where the time constant  and the damping coefficient  are determined by the formulas

, .

Let us consider the operation at low frequencies for the piezoactuator with one face rigidly fixed and elastic inertial load so that  and  for kg,N/m, N/m we obtain c.

4. Results and Discussions

The structural-parametric model and the generalized parametric structural schematic diagram of electromagnetoelastic actuator for precision engineering are obtained taking into account equation of generalized electromagnetoelasticity (piezoelectric, piezomagnetic, electrostriction, and magnetostriction effects) and decision wave equation. The results of constructing the generalized structural-parametric model and the generalized parametric structural schematic diagram of electromagnetoelastic actuator for the longitudinal, transverse and shift deformations are shown in Figure 7. The parametric structural schematic diagrams piezoelectric actuator for longitudinal, transverse, shift piezoelectric effects Figure 2, Figure 4, Figure 6 converts to the generalized parametric structural schematic diagram of the electromagnetoelastic actuator for precision engineering Figure 7 with the replacement of the following parameters:

, , , .

The generalized structural-parametric model and the generalized parametric structural schematic diagram of the electromagnetoelastic actuator after algebraic transformations provides the transfer functions of the electromagnetoelastic actuator actuator for precision engineering. The piezoelectric actuator with the transverse piezoelectric effect compared to the piezoelectric actuator for the longitudinal piezoelectric effect provides a greater range of static displacement and a less working force. The magnetostriction actuators provides a greater range of static working forces.

It is possible to construct the generalized structural-parametric model, the generalized parametric structural schematic diagram and the transfer functions in matrix form of the electromagnetoelastic actuator for precision engineering using the solutions of the wave equation of the electromagnetoelastic actuator and taking into account the features of the deformations actuator along the coordinate axes.

5. Conclusions

The systems of equations are determined for the structural-parametric models of the piezoactuators. The transfer functions and the parametric structural schematic diagrams of the piezoactuators for the transverse, longitudinal, shift piezoelectric effects are obtained from structural-parametric models of the piezoactuators. Using the obtained solutions of the wave equation and taking into account the features of the deformations along the coordinate axes, it is possible to construct the generalized structural-parametric model and parametric structural schematic diagram of the electromagnetoelastic actuator for precision engineering and to describe its dynamic and static properties.

The transfer functions in matrix form of the electromagnetoelastic are describe deformations actuator during its operation as a part of the control systems for precision engineering.

References

1. Uchino, K. (1997) Piezoelectric actuator and ultrasonic motors. Boston, MA: Kluwer Academic Publisher, 347 p.
2. Afonin, S. M. (2006) Solution of the wave equation for the control of an elecromagnetoelastic transduser. Doklady mathematics, 73, 2, 307-313, doi: 10.1134/S1064562406020402.
3. Afonin, S. M. (2008) Structural parametric model of a piezoelectric nanodisplacement transduser. Doklady physics, 53, 3, 137-143, doi: 10.1134/S1028335808030063.
4. Afonin, S. M. (2014) Stability of strain control systems of nano-and microdisplacement piezotransducers. Mechanics of solids, 49, 2, 196-207, doi: 10.3103/S0025654414020095.
5. Yang, Y., Tang, L. (2009) Equivalent circuit modeling of piezoelectric energy harvesters, Journal of intelligent material systems and structures, 20, 18, 2223-2235.
6. Cady, W. G. (1946) Piezoelectricity an introduction to the theory and applications of electromechancial phenomena in crystals. New York, London: McGraw-Hill Book Company, 806 p.
7. Physical Acoustics: Principles and Methods. (1964) Vol. 1. Part A. Methods and Devices. Ed.: W. Mason. New York: Academic Press. 515 p.
8. Zwillinger, D. (1989) Handbook of Differential Equations. Boston: Academic Press. 673 p.
9. Afonin, S. M. (2015) Structural-parametric model and transfer functions of electroelastic actuator for nano- and microdisplacement, Chapter 9 in Piezoelectrics and Nanomaterials: Fundamentals, Developments and Applications. Ed. I. A. Parinov. New York: Nova Science. pp. 225-242.
10. Afonin, S. M. (2005) Generalized parametric structural model of a compound elecromagnetoelastic transduser. Doklady physics, 50, 2, 77-82, doi: 10.1134/1.1881716.
11. Afonin, S. M. (2002) Parametric structural diagram of a piezoelectric converter. Mechanics of solids, 37, 6, 85-91.
12. Afonin, S. M. (2003) Deformation, fracture, and mechanical characteristics of a compound piezoelectric transducer. Mechanics of solids, 38, 6, 78-82.
13. Afonin, S. M. (2004) Parametric block diagram and transfer functions of a composite piezoelectric transducer. Mechanics of solids, 39, 4, 119-127.
14. Afonin, S. M. (2007) Elastic compliances and mechanical and adjusting characteristics of composite piezoelectric transducers. Mechanics of solids, 42, 1, 43-49, doi: 10.3103/S0025654407010062.
15. Afonin, S. M. (2009) Static and dynamic characteristics of a multy-layer electroelastic solid. Mechanics of solids, 44, 6, 935-950, doi: 10.3103/S0025654409060119.
16. Afonin, S. M. (2010) Design static and dynamic characteristics of a piezoelectric nanomicrotransducers. Mechanics of solids, 45, 1, 123-132, doi: 10.3103/S0025654410010152.
17. Afonin, S. M. (2001) Structural-parametric model of nanometer-resolution piezomotor. Russian engineering research, 21, 5, 42-50.
18. Afonin, S. M. (2002) Parametric structure of composite nanometric piezomotor. Russian engineering research, 22, 12, 9-24.
19. Afonin, S. M. (2011) Electromechanical deformation and transformation of the energy of a nano-scale piezomotor. Russian engineering research, 31, 7, 638-642, doi: 10.3103/S1068798X11070033.
20. Afonin, S. M. (2011) Electroelasticity problems for multilayer nano- and micromotors. Russian engineering research, 31, 9, 842-847, doi: 10.3103/S1068798X11090036.
21. Afonin, S. M. (2006) Absolute stability conditions for a system controlling the deformation of an elecromagnetoelastic transduser. Doklady mathematics, 74, 3, 943-948, doi: 10.1134/S1064562406060391.
22. Afonin, S. M. (2015) Dynamic characteristics of multilayer piezoelectric nano- and micromotors. Russian engineering research, 35, 2, 89-93, doi: 10.3103/S1068798X15020045.
23. Afonin, S. M. (2015) Optimal control of a multilayer submicromanipulator with a longitudinal piezo effect. Russian engineering research, 35, 12, 907-910, doi: 10.3103/S1068798X15120035.
24. Afonin, S. M. (2005) Generalized structural parametric model of an elecromagnetoelastic transduser for control system of nano- and microdisplacement: I. Solution of the wave equation for control problem of an elecromagnetoelastic transduser. Journal of computer and systems sciences international, 44, 3, 399-405.
25. Afonin, S. M. (2005) Generalized structural parametric model of an elecromagnetoelastic transduser for control system of nano- and microdisplacements: II. On the generalized structural parametric model of a compound elecromagnetoelastic transduser. Journal of computer and systems sciences international, 44, 4, 606-612.
26. Afonin, S. M. (2006) Generalized structural-parametric model of an elecromagnetoelastic converter for nano- and micrometric movement control systems: III. Transformation parametric structural circuits of an elecromagnetoelastic converter for nano- and micromovement control systems. Journal of computer and systems sciences international, 45, 2, 317-325, doi: 10.1134/S106423070602016X.
27. Afonin, S. M. (2015) Block diagrams of a multilayer piezoelectric motor for nano- and microdisplacements based on the transverse piezoeffect. Journal of computer and systems sciences international, 54, 3, 424-439, doi: 10.1134/S1064230715020021.
28. Afonin, S. M. (2012) Nano- and micro-scale piezomotors. Russian engineering research, 32, 7-8, 519-522, doi: 10.3103/S1068798X12060032.
29. Springer Handbook of Nanotechnology. (2004) Ed. by B. Bhushan. Berlin, New York: Springer, 1222 p
30. Encyclopedia of Nanoscience and Nanotechnology. (2004) Ed. by H. S. Nalwa. Calif.: American Scientific Publishers. 10 Volume Set.