The multi-layer longitudinal piezo engine with parallel and coded control is used for nano biomedical research. The characteristics of the multi-layer longitudinal piezo engine with parallel and coded control are determined for nano biomedical research. The characteristics of the multi-layer longitudinal piezo engine are obtained by applied method of mathematical physics.

Keywords: Multi-layer longitudinal piezo engine, parallel and coded control, characteristics, biomedical research

The use of the multi-layer longitudinal piezo engine for nano- and micro displacements is promising in nano biomedical research for the compensation of gravitational and temperature deformations, precise alignment,¹⁻¹⁰ Nano pumps, microsurgery, scanning microscopy, adaptive optics, interferometers.⁸⁻³⁸

Increasing the range of displacement to tens of micrometers is achieved by using the multi-layer longitudinal piezo engine in the form the composite, stack or block piezo engine.¹⁻⁸

At present the use of the multi-layer longitudinal piezo engine with parallel and coded control is relevant, which requires determining the characteristics of this piezo engine. The application of the multi-layer longitudinal piezo engine at coded control makes it possible to effectively use electromechanical digital-to-analog conversion proportional to the control code for nano- and microdisplacements.¹¹⁻³⁴

In contrast to the simple piezo engine the multi-layer longitudinal piezo engine in static without load has the range of the movement increased in n times, where n – the number of the piezo layers. The characteristics of the multi-layer longitudinal piezo engine for parallel and coded control are calculated by applied method of mathematical physics.

Characteristics multi-layer longitudinal piezo engine at parallel control

Structurally the multi-layer longitudinal piezo engine, depending on the manufacturing technology, can be made in the form: the composite piezo engine made of individual elastically pressed piezo plates; packaged or block piezo engine made of piezo plates sintered using silver paste; the composite piezo engine made of the piezo packages with elastic reinforcement; the glued multi-layer piezo engine made of the piezo plates; the multi-layer piezo engine with the layers by using thick-film or thin-film.^1–18

The equation³⁻⁶ of the inverse longitudinal piezo effect has the form

$S_{\text{3}}=d_{\text{33}}{E}_{\text{3}}+s_{\text{33}}^E{T}_{\text{3}}$

here $S_{\text{3}},E_{\text{3}},T_{\text{3}},d_{\text{33}},s_{\text{33}}^E$ − the relative displacement, the electric field stress, the mechanical stress, the piezo module, the elastic compliance with $E=\text{const}$ , index 3 for 3 axis.

We have the equation of the mechanical characteristic at parallel control in the form $\Delta l=d_{\text{33}}nU-s_{\text{33}}^{E}Fl/S_{\text{0}}=d_{\text{33}}nU-F/C_{\text{33}}^E$ and after the transformation we have the equation of the mechanical characteristic

$\Delta l=\Delta l{}_{\text{3max}}\left(\text{1}-F/F_{\text{3max}}\right)$

$\Delta l{}_{\text{3max}}=d_{\text{33}}nU,F_{\text{3max}}=d_{\text{33}}U{S}_{\text{0}}/\left(s_{\text{33}}^E\delta \right)$

here $l=n\delta$  − the length, $C_{\text{33}}^E=S_{\text{0}}/\left(s_{\text{33}}^{E}l\right)$  − the rigidity of the of the multi-layer longitudinal piezo engine, $\Delta l$  − the displacement, $F$  − the force. Let us consider the mechanical characteristic on Figure 1 of the multi-layer longitudinal piezo engine at parallel control from ceramic PZT.

Figure 1 Mechanical characteristic of multi-layer longitudinal piezo engine at parallel control.

The measurements of the mechanical characteristic were made on the Universal testing machine UMM-5 Russia in the range of working loads under mechanical stresses in the multi layered longitudinal piezo engine up to 100 MPa. At $d_{\text{33}}$  = 0.4 nm/V, n = 50, $C_{\text{33}}^E$  = 2×10⁸ N/m for 1) U = 50 V; 2) U = 100 V; 3) U = 150 V the parameters of the multi-layer longitudinal piezo engine from ceramic PZT are determined on Figure 1 in the form 1) $\Delta l_{\text{3max}}$  = 1000 nm, $F_{\text{3max}}$  = 200 N; 2) $\Delta l_{\text{3max}}$  = 2000 nm, $F_{\text{3max}}$  = 400 N; 3) $\Delta l_{\text{3max}}$  = 3000 nm, $F_{\text{max}}$  = 600 N. The discrepancy between the experimental data and the calculation results is 10%.

The displacement of the multi-layer longitudinal piezo engine at parallel control and elastic load on Figure 2 has the form

$\Delta l=d_{\text{33}}nU-F/C_{\text{33}}^E$

$F=F_{\text{0}}+C_a\Delta l+C_e\Delta l,F_{\text{0}}={\sigma }_a{S}_{\text{0}}$

Figure 2 Multi-layer longitudinal piezo engines at parallel control and elastic load.

Here $F_{\text{0}}$  − the force of initial compression by the elastic element; ${\sigma }_a$  − the mechanical stress of the initial reinforcement in the piezo engine; $C_a$  − the rigidity of the reinforcing element; $C_e$  − the load rigidity.

Consequently, the equation for the adjustment characteristic of the multi-layer longitudinal piezo engine at parallel control and elastic load has the form

$\Delta l=\frac{d_{\text{33}}nU-{\sigma }_{a}l{s}_{\text{33}}^E}{\text{1}+\left(C_a+C_e\right)/C_{\text{33}}^E}=\frac{l\left(d_{\text{33}}{E}_{\text{3}}-{\sigma }_a{s}_{\text{33}}^E\right)}{\text{1}+\left(C_a+C_e\right)/C_{\text{33}}^E}$

For ${\sigma }_a=\text{0}$  and $C_a=\text{0}$  the equation the adjustment characteristic on Figure 3 of the multi-layer longitudinal piezo engine at parallel control and elastic load has the form

$\Delta l=\frac{d_{\text{33}}nU}{\text{1}+C_e/C_{\text{33}}^E}=\frac{\Delta l_{\text{3max}}}{\text{1}+C_e/C_{\text{33}}^E}$

Figure 3 Adjustment characteristic at parallel control and elastic load.

The adjustment characteristics on Figure 3 are determined by using electronic measuring system of displacement Model 214 Russia for the multi-layer longitudinal piezo engine from PZT for parallel control and elastic load at $d_{\text{33}}$ = 0.4 nm/V, n = 25, $C_{\text{33}}^E$  = 4×10⁸ N/m, $C_a=\text{0}$  for 1) $C_e=\text{0}$ ; 2) $C_e$  = 0.4×10⁸ N/m with error 10%.

Characteristics multi-layer longitudinal piezo engine at coded control

The length of the multi-layer longitudinal piezo engine at coded control has the form

$l={\displaystyle \sum _{k=\text{1}}^N{l}_k=}\left({\text{2}}^N-\text{1}\right)\delta$

The maximum displacement of the multi-layer longitudinal piezo engine at coded control has the form

$\Delta l_{\max }=d_{\text{33}}\left({\text{2}}^N-\text{1}\right)U=d_{\text{33}}nU$

here $n={\text{2}}^N-\text{1}$  is the number of the piezo layers.

In static conditions at the force $F=\text{0}$ and the binary code $a_k=\text{0,}\text{1}$  we have displacement of the multi-layer longitudinal piezo engine at coded control in the form

$\Delta l={\displaystyle \sum _{k=\text{1}}^N{a}_k\Delta l_k}$

Therefore, its displacement has the form

$\Delta l={\displaystyle \sum _{k=\text{1}}^N{a}_k{d}_{\text{33}}{\text{2}}^{k-\text{1}}U=}{d}_{\text{33}}\left({\displaystyle \sum _{k=\text{1}}^N{a}_k{\text{2}}^{k-\text{1}}}\right)U$

We have the mechanical characteristic at coded control¹¹⁻³⁴ in the form $\Delta l=d_{\text{33}}\left({\displaystyle \sum _{k=\text{1}}^N{a}_k{\text{2}}^{k-\text{1}}}\right)U-s_{\text{33}}^{E}Fl/S_{\text{0}}=d_{\text{33}}\left({\displaystyle \sum _{k=\text{1}}^N{a}_k{\text{2}}^{k-\text{1}}}\right)U-F/C_{\text{33}}^E$ after transformation, the normalized mechanical characteristic has the form

$\begin{array}{l}\Delta l/\Delta l{}_{\text{3max}}=\text{1}-F/F_{\text{3max}}\\ {\Delta }_{\text{3max}}=d_{\text{33}}\left({\displaystyle \sum _{k=\text{1}}^N{a}_k{\text{2}}^{k-\text{1}}}\right)U,F_{\text{3max}}=d_{\text{33}}\left({\displaystyle \sum _{k=\text{1}}^N{a}_k{\text{2}}^{k-\text{1}}}\right)U{S}_{\text{0}}/\left(s_{\text{33}}^{E}l\right)\end{array}$

here $C_{\text{33}}^E=S_{\text{0}}/\left(s_{\text{33}}^{E}l\right)$ .

Consequently, the equation for the adjustment characteristic of the multi-layer longitudinal piezo engine at coded control and elastic load on Figure 4 has the form

$\Delta l=\frac{d_{\text{33}}\left({\displaystyle \sum _{k=\text{1}}^N{a}_k{\text{2}}^{k-\text{1}}}\right)U}{\text{1}+\left(C_a+C_e\right)/C_{\text{33}}^E}$

Figure 4 Multi-layer longitudinal piezo engines at coded control and elastic load.

Therefore, the displacement of the multi-layer longitudinal piezo engine elastic load has the form

$\Delta l=k_{c}U$

here $k_c$ is the coefficient

$k_c=\{\begin{array}{c}\frac{d_{\text{33}}n}{\text{1}+\left(C_a+C_e\right)/C_{\text{33}}^E}-\text{with parallel control, }\\ \frac{d_{\text{33}}\left({\displaystyle \sum _{k=\text{1}}^N{a}_k{\text{2}}^{k-\text{1}}}\right)}{\text{1}+\left(C_a+C_e\right)/C_{\text{33}}^E}-\text{with codedl control}.\end{array}$

The measurements of the parameters mechanical characteristic were made on the Universal testing machine UMM-5 Russia for the multi-layer longitudinal piezo engine from PZT for coded control at $d_{\text{33}}$ = 0.4 nm/V, n = 7, $C_{\text{33}}^E$  = 8×10⁸ N/m, and U = 200 V for 1) $a_{\text{1}}$  = 1, $a_{\text{2}}$  = 0, $a_{\text{3}}$  = 0; 2) $a_{\text{1}}$ = 1, $a_{\text{2}}$ = 1, $a_{\text{3}}$ = 0; 3) $a_{\text{1}}$ = 1, $a_{\text{2}}$ = 1, $a_{\text{3}}$ = 1. The maximum displacements and the maximum forces on Figure 5 are obtained 1) $\Delta l_{\text{max}}$  = 80 nm, $F_{\text{max}}$  = 64 N; 2) $\Delta l_{\text{max}}$  = 240 nm, $F_{\max }$ = 192 N; 3) $\Delta l_{\max }$  = 560 nm, $F_{\max }$  = 448 N with error 10%.

Figure 5 Mechanical characteristic at coded control.

Thus, the adjustment and mechanical characteristics of the multi-layer longitudinal piezo engine at parallel and coded control are found.

Through the use of mathematical physics we have obtained the adjustment and mechanical characteristics of the multi-layer longitudinal piezo engine at parallel and coded control for nano biomedical research. The generalized adjustment and mechanical characteristics of the multi-layer longitudinal piezo engine at parallel and coded control are determined by using the equations of the inverse longitudinal piezo effect and the mechanical force.

The multi-layer longitudinal piezo engine is used in nano biomedical research for the compensation of gravitational and temperature deformations, scanning microscopy, adaptive optics. The characteristics of the multi-layer longitudinal piezo engine at parallel and coded control are obtained by using method of mathematical physics. The parameters and the characteristics of this multi-layer longitudinal piezo engines are determined.

The adjustment and mechanical characteristics in general of the multi-layer longitudinal piezo engine at parallel and coded control are found for nano biomedical research. Future works are planned to investigate the characteristics of multi-layer piezo engines in various applications.

None.

None.

The author declares that there is no conflict of interest.

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