A multi-layer electro elastic drive of robotics is used in adaptive optics of compound telescope, scanning microscopy, interferometry and nanotechnology. For micro and nano robotics a multi-layer electro elastic drive is applied. The parametric model of a multi-layer electro elastic drive is determined. Its functions and matrix deformations are founded. The parameters of the multi-layer longitudinal PZT drive are determined.

Keywords: multi-layer electro elastic drive, multi-layer piezo drive, parametric model, micro and nano robotics.

A multi-layer electro elastic drive is used promising for micro and nano robotics in the micro and nano displacement.¹⁻⁸ This drive based on the piezoelectric or electrostriction effects.^9s−19 A multi-layer electro elastic drive of robotics is applied in adaptive optics of compound telescope, scanning microscopy, micro surgery, interferometry, nano pump, nano stabilization and nanotechnology.²⁰⁻⁵³ The deformations of a multi-layer drive are described with the matrix equation. Its parametric model, scheme and functions are obtained by using of mathematical physics method.

Parametric model

The parametric model⁶⁻⁵³ of multi-layer piezo actuator with the voltage or current controlled are determined using the equation of inverse piezo effect in the form at the control of voltage

$S_i=d_{mi}{E}_m+s_{ij}^E{T}_j$

at the control of current

$S_i=g_{mi}{D}_m+s_{ij}^D{T}_j$

here $S_i$ , $E_m$ , $D_m$ , $T_j$ , $d_{mi}$ , $g_{mi}$ , $s_{ij}^E$ , $s_{ij}^D$  are the relative deformation, the strength of electric field, the electric induction, the strength of mechanic field, the piezo module, the piezo constant, the elastic compliances at $E=\text{const}$  and at $D=\text{const}$ , and $i,j,m$  are the indexes.

The equation of the direct piezo effect has the form⁶⁻⁵³

$D_m=d_{mi}{T}_i+{\epsilon }_{mk}^T{E}_k$

here ${\epsilon }_{mk}^T$  is the dielectric constants at $T=\text{const}$ , k is the index

Then the electroelasticity equation of a multi-layer drive⁶⁻⁵³ has the form

$S_i=v_{mi}{\psi }_m+s_{ij}^{\psi }{T}_j$

here $\Psi =E,D$  is control parameter at the control of voltage and the control of current.

A multi-layer drive consist from the piezo layers connected the series mechanically and the parallel electrically [6 − 44]. We have the system of equations for T-form quadripole of k piezo layer

$F_{k\text{inp}}\left(s\right)=-\left(Z_1+Z_2\right){\Xi }_k\left(s\right)+Z_2{\Xi }_{k+1}\left(s\right)$

$-F_{k\text{out}}\left(s\right)=-Z_2{\Xi }_k\left(s\right)+\left(Z_1+Z_2\right){\Xi }_{k+1}\left(s\right)$

$Z_1=\frac{S_o\gamma th(\delta \gamma )}{s_{ij}^{\psi }},Z_2=\frac{S_o\gamma }{s_{ij}^{\psi }sh(\delta \gamma )}$

where $Z_1$ , $Z_2$ , $s$ , $\delta$ , $\gamma$ , $F_{k\text{inp}}\left(s\right)$ , $F_{k\text{out}}\left(s\right)$ , ${\Xi }_k\left(s\right)$ , ${\Xi }_{k+1}\left(s\right)$  are the resistances of quadripole k piezo layer, the transform parameter, the thickness, the coefficient wave propagation, the Laplace transform of the forces at the input and output ends of k piezo layer, the transforms of the displacements at input and output ends of k piezo layer.

The system of the equations for k piezo layer has the form

$-F_{k\text{inp}}\left(s\right)=\left(1+\frac{Z_1}{Z_2}\right)F_{k_{out}}\left(s\right)+Z_1\left(2+\frac{Z_1}{Z_2}\right){\Xi }_{k+1}\left(s\right)$

${\Xi }_k\left(s\right)=\frac{1}{Z_1}{F}_{k\text{out}}\left(s\right)+\left(1+\frac{Z_1}{Z_2}\right){\Xi }_{k+1}\left(s\right)$

This system is founded in the matrix form

$\left[\begin{array}{c}-F_{k\text{inp}}\left(s\right)\\ {\Xi }_k\left(s\right)\end{array}\right]=\left[M\right]\left[\begin{array}{c}{F}_{k\text{out}}\left(s\right)\\ {\Xi }_{k+1}\left(s\right)\end{array}\right]$

$\left[M\right]=\left[\begin{array}{cc}{m}_{11}& m_{12}\\ m_{21}& m_{22}\end{array}\right]=\left[\begin{array}{cc}1+\frac{Z_1}{Z_2}& Z_1\left(2+\frac{Z_1}{Z_1}\right)\\ \frac{1}{Z_2}& 1+\frac{Z_1}{Z_2}\end{array}\right]$

$m_{11}=m_{22}=1+\frac{Z_1}{Z_2}=ch(\delta \gamma ),$ $m_{12}=Z_1\left(2+\frac{Z_1}{Z_1}\right)=Z_{0}sh(\delta \gamma )$

$m_{21}=\frac{1}{Z_2}=\frac{sh(\delta \gamma )}{Z_0},$ $Z_0=\frac{S_0\gamma }{s_{ij}^{\psi }}$

The equation of forces at the boundary between two layers is obtained in the form

$F_{k\text{out}}\left(s\right)=-F_{k+1\text{inp}}\left(s\right)$

For a multi-layer electro elastic drive with n layers and l length its system has the matrix form

$\left[\begin{array}{c}-F_{1\text{inp}}\left(s\right)\\ {\Xi }_1\left(s\right)\end{array}\right]={\left[M\right]}^n\left[\begin{array}{c}{F}_{n\text{out}}\left(s\right)\\ {\Xi }_{n+1}\left(s\right)\end{array}\right]$

${\left[M\right]}^n=\left[\begin{array}{cc}ch(n\delta \gamma )& Z_{0}sh(n\delta \gamma )\\ \frac{sh(n\delta \gamma )}{Z_0}& ch(n\delta \gamma )\end{array}\right]$

then

${\left[M\right]}^n=\left[\begin{array}{cc}ch(l\gamma )& Z_{0}sh(l\gamma )\\ \frac{sh(l\gamma )}{Z_0}& ch(l\gamma )\end{array}\right]$

The equations of forces a multi-layer drive we have in the form

at$x=0,T_j\left(0,s\right)S_0=F_1\left(s\right)+M_1{s}^2{\Xi }_1\left(s\right)$

at$x=l,T_j\left(l,s\right)S_0=-F_2\left(s\right)-M_2{s}^2{\Xi }_2\left(s\right)$

The transform of its force causes deformation has the equation in the form

$F\left(s\right)=\frac{v_{mi}{S}_0{\psi }_m\left(s\right)}{s_{ij}^{\psi }}$

Then the parametric model and scheme on Figure 1 of a multi-layer electro elastic drive are obtained in the form

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

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

Figure 1 Parametric scheme multi layer electro elastic drive.

here $v_{mi}=\{\begin{array}{c}{d}_{33},d_{31},d_{15}\\ g_{33,}{g}_{31},g_{15}\end{array},$ , ${\Psi }_m=\{\begin{array}{c}{E}_3,E_1\\ D_3,D_1\end{array}$ , $s_{ij}^{\psi }=\{\begin{array}{ccc}{s}_{33}^E& s_{11}^E& s_{55}^E\\ s_{33}^D& s_{11}^D& s_{55}^D\end{array}$ , $c^{\Psi }=\{\begin{array}{c}{c}^E\\ c^D\end{array}$ , $\gamma =\{\begin{array}{c}{\gamma }^E\\ {\gamma }^E\end{array}$ , ${\chi }_{ij}^{\psi }=s_{ij}^{\psi }/S_0$ .

The matrix function of a multi-layer drive has the form

$\left[\begin{array}{c}{\Xi }_1(s)\\ {\Xi }_2(s)\end{array}\right]=\left[\begin{array}{ccc}{W}_{11}\left(s\right)& W_{12}\left(s\right)& W_{13}\left(s\right)\\ W_{21}\left(s\right)& W_{22}\left(s\right)& W_{23}\left(s\right)\end{array}\right]\left[\begin{array}{c}{\psi }_m\left(s\right)\\ F_1(s)\\ F_2(s)\end{array}\right]$

then

$\left[\Xi \left(s\right)\right]=\left[W\left(s\right)\right]\left[P\left(s\right)\right]$

$\left[\Xi \left(s\right)\right]=\left[\begin{array}{c}{\Xi }_1\left(s\right)\\ {\Xi }_2\left(s\right)\end{array}\right]$ ,$\left[P\left(s\right)\right]=\left[\begin{array}{c}{\Psi }_m\left(s\right)\\ F_1\left(s\right)\\ F_2\left(s\right)\end{array}\right]$

$\left[W\left(s\right)\right]=\left[\begin{array}{ccc}{W}_{11}\left(s\right)& W_{12}\left(s\right)& W_{13}\left(s\right)\\ W_{21}\left(s\right)& W_{22}\left(s\right)& W_{23}\left(s\right)\end{array}\right]$

The functions of a multi-layer drive are determined

$W_{11}\left(s\right)={\Xi }_1(s)/{\psi }_m\left(s\right)=v_{mi}\left[M_2{\chi }_{ij}^{\psi }{s}^2+\gamma th(l\gamma /2)\right]/A_{ij}$

${\chi }_{ij}^{\psi }=s_{ij}^{\psi }/S_0$

$A_{ij}=\begin{array}{c}{M}_1{M}_2{\left({\chi }_{ij}^{\psi }\right)}^2{s}^4+\left\{\left(M_1+M_2\right)\left({\chi }_{ij}^{\psi }\right)/\left[c^{\psi }th(l\gamma )\right]\right\}{s}^3+\\ +\left[\left(M_1+M_2\right){\chi }_{ij}^{\psi }\alpha /th(l\gamma )+1/{\left(c^{\psi }\right)}^2\right]s^2+2\alpha s/c^{\psi }+{\alpha }^2\end{array}$

$W_{21}\left(s\right)={\Xi }_2(s)/{\psi }_m\left(s\right)=-v_{ij}^{\psi }{M}_1{\chi }_{ij}^{\psi }{s}^2+\gamma th(l\gamma /2)/A_{ij}$

$W_{12}\left(s\right)={\Xi }_1(s)/F_1\left(s\right)=-{\chi }_{ij}^{\psi }\left[M_2{\chi }_{ij}^{\psi }{s}^2+\gamma th(l\gamma )\right]/A_{ij}$

$\begin{array}{l}{W}_{13}\left(s\right)={\Xi }_1(s)/F_2\left(s\right)=\\ =W_{22}\left(s\right)={\Xi }_2(s)/F_1\left(s\right)=\left[{\chi }_{ij}^{\psi }\gamma /sh\left(l\gamma \right)\right]/A_{ij}\end{array}$

$W_{23}\left(s\right)={\Xi }_2(s)/F_2\left(s\right)=-{\chi }_{ij}^{\psi }\left[M_1{\chi }_{ij}^{\psi }{s}^2+\gamma /th(l\gamma )\right]/A_{ij}$

The function of the multi-layer longitudinal piezo drive with one fixed face is determined at the elastic-inertial load and the control of voltage in the form

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

$T_t=\sqrt{M_2/\left(C_e+C_{33}^E\right)},{\xi }_t=\alpha l^2{C}_{33}^E/\left[3c^E\sqrt{M_2\left(C_e+C_{33}^E\right)}\right]$

Its transient response has the form

$\xi \left(t\right)={\xi }_m{\left(1-\frac{e^{-\frac{{\xi }_{t}t}{T_t}}}{\sqrt{1-{\xi }_t^2}}\text{sin}\left({\omega }_{t}t+{\varphi }_t\right)\right)}_{}$

${\xi }_m=\frac{d_{33}n{U}_m}{1+C_e/C_{33}^E}$ , ${\omega }_t=\sqrt{1-{\xi }_t^2}/T_t$ , ${\varphi }_t=\text{arctg}\left(\sqrt{1-{\xi }_t^2}/{\xi }_t\right)$

For the multi-layer longitudinal PZT drive at $U_m$  =60 V, $d_{33}$ = 4∙10^-10 m/V, n= 8, M= 1 kg, $C_{33}^E$ = 5.8∙10⁷ N/m, $C_e$ = 0.6∙10⁷ N/m its parameters ${\xi }_m$  = 174 nm and $T_t$  = 1.25∙10^-4 s are obtained with error 10%.

We have the parametric model and scheme of a multi-layer electro elastic drive of robotics by using of mathematical physics method. The system of equations for T-form quadripole of piezo layer is used. For a multi-layer electro elastic drive its system has the matrix form. By using the equations of forces on ends of a multi-layer drive and the equation of force causes deformation its parametric model and scheme, functions are determined.

A multi-layer electro elastic drive of robotics is used in adaptive optics, scanning microscopy, micro surgery, interferometry, nanotechnology, nano pump, nano stabilization, compensation of deformations. The parameters of the multi-layer longitudinal PZT drive at the control of voltage are obtained.

The characteristics in of the multi-layer piezo drive are obtained by applied of mathematical physics method. Future works are planned to explore the multi-layer piezo drive for adaptive optics of compound telescope.

None.

Author declares that there is no conflict of interest.

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