We received the structural circuit of the multilayer piezo engine for nanomedicine research. The characteristics of the multilayer piezo engine are obtained.

Keywords: multilayer piezo engine; structural circuit, transfer function, characteristic

The multilayer piezo engine with the piezoelectric effect is used for nanomedicine research in scanning microscopy, microdosing devices, nanomanipulates, nanopumps.^1‒5 The structural circuit of the multilayer piezo engine is obtained in contrast to Cady's and Mason's equivalent circuits.^6‒27

The structural circuit of the multilayer piezo engine is received from the equation of the electro elasticity and its matrix of the quadripole. The equation of the electro elasticity,^{6 ,10,11} has the form

$S_i={\nu }_{mi}{\Psi }_m+s_{ij}^{\Psi }{T}_j$

where $S_i$ , ${\nu }_{mi}$ , ${\Psi }_m$ , $s_{ij}^{\Psi }$ , $T_j$  are the relative displacement, the coefficient of electro elasticity, the control parameter, the elastic compliance and the mechanical stress.

For the multilayer piezo engine the causes force is obtained in the form

$F={\nu }_{mi}{S}_0{\Psi }_m/s_{ij}^{\Psi }$

where S_o is the area of the multilayer piezo engine.

The matrix of the quadripole for the multilayer piezo engine,^{6, 24‒26} is received in the form

${\left[M\right]}^n=\left[\begin{array}{cc}\text{ch}\left(l\gamma \right)& Z_0\text{sh}\left(l\gamma \right)\\ \frac{\text{sh}\left(l\gamma \right)}{Z_0}& \text{ch}\left(l\gamma \right)\end{array}\right]$

where l is the length of the multilayer piezo engine and $\gamma$  is the coefficient propagation.

The structural circuit of the multilayer piezo engine for nanomedicine research on Figure 1 is obtained in the form

$\begin{array}{l}{\Xi }_1\left(p\right)={\left[1/\left(M_1{p}^2\right)\right]}^{}\times \\ \times \left\{-F_1\left(p\right)+{\left(1/{\chi }_{ij}^{\Psi }\right)}_{}^{}\left[{\nu }_{mi}{\Psi }_m\left(p\right)-{\left[\gamma /\text{sh}\left(l\gamma \right)\right]}_{}\left[\text{ch}\left(l\gamma \right){\Xi }_1\left(p\right)-{\Xi }_2\left(p\right)\right]\right]\right\}\end{array}$

$\begin{array}{l}{\Xi }_2\left(p\right)={\left[1/\left(M_2{p}^2\right)\right]}^{}\times \\ \times \left\{-F_2\left(p\right)+{\left(1/{\chi }_{ij}^{\Psi }\right)}_{}^{}\left[{\nu }_{mi}{\Psi }_m\left(p\right)-{\left[\gamma /\text{sh}\left(l\gamma \right)\right]}_{}\left[\text{ch}\left(l\gamma \right){\Xi }_2\left(p\right)-{\Xi }_1\left(p\right)\right]\right]\right\}\end{array}$

Figure 1 Structural circuit of multilayer piezo engine for nanomedicine research.

where $v_{mi}=\{\begin{array}{c}{d}_{33},d_{31},d_{15}\\ g_{33},g_{31},g_{15}\end{array}$ ; ${\Psi }_m=\{\begin{array}{c}{E}_3,E_1\\ D_3,D_1\end{array}$ ; $s_{ij}^{\Psi }==\{\begin{array}{c}{s}_{33}^E,s_{11}^E,s_{55}^E\\ s_{33}^D,s_{11}^D,s_{55}^D\end{array}$ ;

$c^{\Psi }=\{\begin{array}{c}{c}^E\\ c^D\end{array}$ ; $\gamma =p/c^{\Psi }+\alpha$ ; ${\chi }_{ij}^{\Psi }=s_{ij}^{\Psi }/S_0$ ; ${\Xi }_1\left(p\right)$ , ${\Xi }_2\left(p\right)$ , $F_1\left(p\right)$ , $F_2\left(p\right)$ are Laplace transforms of the displacements and the forces on the faces 1, 2 of the multilayer piezo engine; $M_1$ , $M_1$  are the masses of the loads.

Accordingly, in the practical application of the multilayer piezo engine, we have its parameters: displacement 1 nm -10μm, fast response 1-10ms, force 100-1000 N.

For the system with the lumped parameters the transfer function of the multilayer engine with longitudinal piezoeffect and one fixed face is obtained in the form

$W\left(p\right)=\frac{{\Xi }_2\left(p\right)}{U\left(p\right)}=\frac{d_{33}n}{{\left(1+C_e/C_{33}^E\right)}_{}^{}\left(T_t^2{p}^2+2T_t{\xi }_{t}p+1\right)}$

$T_t=\sqrt{M_2/\left(C_e+C_{33}^E\right)}$

${\xi }_t=\alpha {\left(n\delta \right)}^2{C}_{33}^E/\left(3c^E\sqrt{M_2\left(C_e+C_{33}^E\right)}\right)$

where $T_t$ , ${\xi }_t$  are the time constant, the damping coefficient of the multilayer engine.

At $d_{33}$ =4∙10^-10 m/V, =16,  = 60 V, $M_2$  = 4 kg, $C_{33}^E$  = 1.5∙10⁷ N/m,  = 0.3∙10⁷ N/m for the multilayer piezo engine from PZT are received the maximum static displacement of its second face ${\text{ξ}}_2$ =320 nm and the time constant  = 0.47∙10^-3 s. The discrepancy between the experimental data and the calculation results is 5%.

We received the structural circuit of the multilayer piezo engine for nanomedicine research with used the equation of the electro elasticity and the matrix of the quadripole. We obtained the static displacement and the time constant of the multilayer piezo engine for nanomedicine research. The static and dynamic characteristics of the multilayer piezo engine are used in the calculation of the mechatronic control system.

None.

The authors declare there are no conflicts of interest.

None.

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