Electroelastic Actuator Nano- and Microdisplacement for Precision Mechanics

In the present work the structural-parametric model of the piezoactuator is determined in contrast electrical equivalent circuit types Cady or Mason for the calculation of the piezoelectric transmitter and receiver, the vibration piezoactuator and the vibration piezomotor with the mechanical parameters in form the velosity and the pressure. The aim of this work is to obtain the structural-parametric model of the electroelastic actuator with the mechanical parameters the displacement and the force. The method of mathematical physics is used. Structural scheme of electroelastic actuator for nanotechnology is obtained. The transfer functions of the actuators are determined. For calculations control systems for nanotechnology with piezoactuator the structural scheme and the transfer functions of piezoactuator are obtained. The generalized structural-parametric model, the generalized structural scheme, the generalized matrix equation for the electroelastic actuator nanoand microdisplacement are obtained in the matrix form. The deformations of the electroelastic actuator for the precision mechanics are described by the matrix equation.


Introduction
The electroelastic actuator based on the electroelasticity in the form the piezoelectric, piezomagnetic, electrostriction effects is used for the precision mechanics in the nanotechnology, the nanobiology, the microelectronics, the astronomy and the adaptive optics. This actuator are solved problems of the compensation of the temperature and gravity deformations, the correction of the wave front and the precision alignment [1 − 10]. Piezoactuator is the piezomechanical device intended for the actuation of the mechanisms, the systems or the management based on the piezoelectric effect, converts the electrical signals into the mechanical movement and the force [1,6,9].
Piezoactuator nano-and microdisplacement for the precision mechanics provide the movement range from several nanometers to tens of microns, the sensitivity of up to 1 nm/V, the loading capacity of up to 1000 N. Piezoactuator give high stress and speed of operation, return to the initial state when switched off and have very low relative displacement less than 1%. Piezoactuator nano-and microdisplacement is used in the majority of the scanning tunneling microscopes, the scanning force microscopes, the atomic force microscopes [1 − 20].
The structural-parametric model of the piezoactuator is determined in contrast electrical equivalent circuit types Cady or Mason for the calculation of the piezoelectric transmitter and receiver, the vibration piezoactuator and the vibration piezomotor with the mechanical parameters in form the velosity and the pressure [2 − 5, 11, 12]. By using the method of mathematical physics and solving the wave equation with the Laplace transform for the corresponding equations of the piezoeffect [6,9,10,20], the boundary conditions on loaded faces of the piezoactuator, the strains along the coordinate axes, it is possible to construct the structural parametric model of the piezoactuator [14,15]. Its transfer functions and structural scheme are determined.
The generalized structural-parametric model and structural scheme, the generalized matrix equation for the electroelastic actuator nano-and microdisplacement are obtained in the matrix form in general from the wave equation of the actuator and the equation of the electroelasticity.

Structural Model and Scheme
For clarity, let us consider the problems of the piezoelasticity. As the result of the joint solution of the wave equation of the piezoactuator nano-and microdisplacement equation with the Laplace transform, the piezoeffect equation and the boundary conditions on the two loaded working surfaces of the piezoactuator, we obtain the corresponding structural-parametric model of the piezoactuator [15,16]. The aim of this work is to obtain the structural-parametric model of the electroelastic actuator with the mechanical parameters the displacement and the force.
For piezoactuator the deformation corresponds to stressed state. If stress T is created in piezoactuator, the deformation S is formed in it. There are six stress components 1 T , 2 T , s , 55 E s in the elastic compliance matrix.
Let us consider the electroelastic actuator.
In general the equation of electroelasticity [10,12,15] has following form  For calculation of the electroelastic actuator nano-and microdisplacement is used the wave equation [10,12,16,19] for the wave propagation in a long line with damping but without distortions. After Laplace transform is obtained the linear ordinary second-order differential equation with the parameter p, where the original problem for the partial differential equation of hyperbolic type using the Laplace transform is reduced to the simpler problem [10,13,14] for the linear ordinary differential equation is the Laplace transform of the displacement of the section of the electroelastic actuator, p c The constants C and B of the solution the linear ordinary second-order differential equation [7] are determined from the boundary conditions for the electroelastic actuator Therefore, the solution the linear ordinary second-order differential equation (5) can be written in the form The system of the equations for the forces on the faces of the electroelastic actuator are determined in the following form From equations of forces acting on the faces of the electroelastic actuator nano-and microdisplacement we obtain the boundary conditions on loaded surfaces where 0 S is the cross section area and 1 M , 2 M are the displaced mass on the faces of the electroelastic actuator.
From (4), the boundary conditions on loaded surfaces (5), the strains along the axes the system of equations for the generalized structural-parametric model and the generalized parametric structural scheme are determined for Figure 1 of the electroelastic actuator with the output parameters the Laplace transform for the displacements ( )

Transfer Functions
The transfer functions of the electroelastic actuator nanoand microdisplacement are determined from its generalized structural-parametric model, taking into account the generalized equation of electroelasticity, its wave equation and the equation of the forces on its faces. Therefore, the Laplace transforms of displacements for two faces of the actuator are dependent from the Laplace transforms of the general parameter of control and forces on two faces and are written in the matrix form. From (10) for the Laplace transforms of the displacements of two faces of the actuator yields the matrix equation in the following form where the transfer functions ( ) ( ) ( ) Let us find the displacement of the faces the electroelastic actuator in the stationary regime for where m is the mass of the electroelastic actuator, 1 2 , M M are the load masses.
Let us consider a numerical example for the time constant and t ξ is the damping coefficient of the piezoactuator. Therefore, we obtain on Figure 3 for (17) the structural scheme of the voltage-controlled piezoactuator at zero source resistance with one fixed face under the transverse piezoeffect for the elastic-inertial load.
where m ξ is the steady-state value of displacement for the voltage-controlled piezoactuator, m U is the amplitude of the voltage in the steady-state.
Let us consider a numerical example for the voltagecontrolled piezoactuator from the piezoceramics PZT under the transverse piezoelectric effect with one fixed face for the elastic-inertial load 1 M → ∞ ,

Results and Discussions
We obtain the structural scheme of the electroelastic actuator nano-and microdisplacement for the precision mechanics. From generalized structural-parametric model of the electroelastic actuator after algebraic transformations we obtain the transfer functions of the electroelastic actuator.
It is possible to construct the generalized structuralparametric model using the solutions of the wave equation of the actuator and taking into account the features of its deformations along the coordinate axes.
For calculations control systems in the nanotechnology, the nanobiology, the microelectronics, the astronomy and the adaptive optics with the electroelastic actuator nano-and microdisplacement for the precision mechanics its transfer functions are obtained.

Conclusions
Taking into account the features of the deformations along the axes and using the solutions of the wave equation, it is possible to construct the structural-parametric model and structural scheme of the electroelastic actuator nano-and microdisplacement for the precision mechanics and to describe its dynamic and static properties.
The structural scheme and the transfer functions of the piezoactuator are obtained from structural parametric model of the piezoactuator for the precision mechanics.
The generalized structural-parametric model, the generalized structural scheme, the generalized matrix equaion for the electroelastic actuator nano-and microdisplacement in the matrix form with the output parameters displacements are obtained.
The structural-parametric models, the structural schemes of the piezoactuator for the transverse, longitudinal, shift piezoelectric effects are determined from the generalized structural-parametric model of the electroelastic actuator nano-and microdisplacement for the precision mechanics.
From the solution of the wave equation, the equations of the electroelasticity and the deformations along the axes with using the Laplace transform, the generalized structuralparametric model and the generalized structural scheme of the electroelastic actuator nano-and microdisplacement with the mechanical parameters the displacement and the force are constructed for the precision mechanics.
The deformations of the electroelastic actuator for the precision mechanics are described by the matrix equation for the transfer functions of the actuator.