National Research University of Electronic Technology, MIET, Moscow, Russia

*Corresponding Author: Afonin SM, National Research University of Electronic Technology, MIET, Moscow, Russia, Tel: 4997102233, Email: [email protected]

Received Date: October 17, 2022

Published Date: December 05, 2022

Citation: Afonin SM. (2022). A Piezo Drive for Nano Chemistry Research. Catalysis Research. 2(1):03.

Copyright: Afonin SM © 2022

ABSTRACT

The mathematical model of a piezo drive is determined for nano chemistry research. The structural schemes of a piezo drive are obtained for nano chemistry research. The matrix equation is constructed for a piezo drive.

Keywords: Piezo drive, Structural scheme, Nano chemistry research

INTRODUCTION

The piezo drive works on basis of the reverse piezoelectric effect [8-52]

where , , , , , are the relative deformation, piezo module, strength electric field, elastic compliance, strength mechanical field, i, j, m are the indexes.

The differential equation is written [8-52]

here, , s, x, y are the transform of the deformation, the parameter Laplace transform, the coordinate, the propagation factor. For the longitudinal piezo drive we have at  x = 0 the deformation

  and at     .

Its decision is written

The system for the longitudinal piezo drive is obtained [14 − 26] for   and 

The mathematical model is written

where  ,   are the transforms of the deformations,   is cross sectional area.

The system for the transverse piezo drive is determined for   and 

The mathematical model of this drive has the form

The system for the shift piezo drive is written for   and 

The mathematical model is written

At   and   for   the system in general is obtained

Therefore, the mathematical model in general of a piezo drive is determined on Figure 1

where,

 

Figure 1: In general, structural scheme of piezo drive.

The mathematical model of drive on Figure 1 is used for nano chemistry research. The matrix of deformations is written

where the functions are

The settled longitudinal deformations are determined

For   = 4×10^-10 m/V, = 25 V,   = 1 kg,   = 4 kg we have the settled deformations   = 8 nm,   = 2 nm and   = 10 nm at error 10%.

To calculate the back electromotive force of the piezo drive, we use the equation of the direct piezoelectric effect [8-16]

where  ,   are the electric induction and the permittivity, i, m, k are the indexes, The direct coefficient   for the piezo drive is written

The transform of the voltage for the back electromotive force of the piezo drive on Figure 2 has the form

 ,   

where   is the number of the face.

Figure 2: Structural scheme of piezo drive with back electromotive force.

Consider the influence of the back electromotive force of the piezo drive on its static deformation.

At voltage control the maximum mechanical stress and the maximum force are written

At current control the maximum force has the form

where  ,   are the sectional area of the capacitor and the capacitor capacitance.

Therefore,

and

where   is the electromechanical coupling coefficient.

For current control of the piezo drive we get the expressions

The elastic compliance   takes the form  ,

where  . Therefore,   is the stiffness of drive at voltage control,   is the stiffness of drive at current control,  ,   is the stiffness of drive. The stiffness of a piezo drive at open electrodes increases then the stiffness at closed electrodes.

From the equation of electroelasticity the mechanical characteristic   [11-26] is determined

and the adjustment characteristic   [11-26] is obtained

The mechanical characteristic is written

where   is the maximum of the deformation and   is the maximum of the force. The mechanical characteristic of the transverse piezo drive is determined

At   = 2∙10^-10 m/V,   = 0.5∙10⁵ V/m,   = 2.5∙10^-2 m,   = 1.5∙10^-5 m²,   = 15∙10^-12 m²/N the parameters are found   = 250 nm and   = 10 N at error 10%

The deformation of a piezo drive at elastic load has the form

The adjustment characteristic of a piezo drive is written

We get in general the elastic compliance  and the coefficient   of the change of elastic compliance

The direct and reverse coefficients of a piezo drive in the form

From Figure 2 we get the structural scheme Figure 3 of a piezo drive at one fixed face and elastic-inertial load.

Figure 3: Structural scheme of drive.

The expression on voltage for Figure 3 has form

 , 

 , 

here   is the damping coefficient.

For the transverse piezo drive at   the expression on voltage is obtained

 ,   

For   = 1 kg,   = 0.1×10⁷ N/m,   = 1.5×10⁷ N/m we have   = 0.25×10^-3 s,   = 4×10³ s^-1 at error 10%.

The settled transverse deformation has the form

For   = 2∙10^-10 m/V,   = 25,   = 0.1 the coefficient is determined   = 4.5 nm/V at error 10%

CONCLUSION

The mathematical model and the structural schemes of a piezo drive are obtained for nano chemistry research. The matrix of the deformations of a piezo drive is constructed. The parameters of a piezo drive are determined.

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