The structural model of a piezo engine for composite telescope is constructed. This structural model clearly shows the conversion of electrical energy by a piezo engine into mechanical energy of the control element of a composite telescope. The structural scheme of a piezo engine is determined. For the control systems with a piezo engine its deformations are obtained in the matrix form. This structural model, structural scheme and matrix equation of a piezo engine are applied in calculation the parameters of the control systems for composite telescope.

Keywords: Piezo engine, Structural model, Structural scheme, Matrix equation, Deformation, Composite telescope

A piezo engine based on the piezoelectric effect is used in the control systems for composite telescope and adaptive optics.^1‒14 A piezo engine is applied for precise adjustment, compensation the deformations of composite telescope and scanning microscope.^15‒21 For decisions the displacements and the forces of a piezo engine in the control systems for composite telescope is used the structural model of a piezo engine. The structural model clearly shows the conversion of electrical energy by a piezo engine into mechanical energy of the control element of a composite telescope with using the physical parameters of a engine and its load.^16‒28 The structural model and the structural scheme of a piezo engine for composite telescope are determined in difference from Cady’s and Mason’s electrical equivalent circuits of a piezo transducer.^7‒28

The matrix state equations [8, 11‒17] of a piezo engine have the form

$\left(D\right)=\left(d\right)\left(T\right)+\left({\epsilon }^T\right)\left(E\right)$

$\left(S\right)=\left(s^E\right)\left(T\right)+{\left(d\right)}^t\left(E\right)$

where $\left(D\right)$ , $\left(S\right)$ , $\left(T\right)$ , $\left(E\right)$  are the matrices of electric induction, relative deformation, mechanical field and electric field stresses, and t is transpose operator. For PZT engine the matrices have the form

$\left(d\right)=\left(\begin{array}{c}\begin{array}{cccccc}{0}_{}& 0_{}& 0_{}& 0_{}& d_{15}& 0\end{array}\\ \begin{array}{cccccc}{0}_{}& 0_{}& 0_{}& d_{15}& 0_{}& 0\end{array}\\ \begin{array}{cccccc}{d}_{31}& d_{31}& d_{33}& 0_{}& 0_{}& 0_{}\end{array}\end{array}\right)$

${\left(d\right)}^t=\left(\begin{array}{ccc}0& 0& d_{31}\\ 0& 0& d_{31}\\ 0& 0& d_{33}\\ 0& d_{15}& 0\\ d_{15}& 0& 0\\ 0& 0& 0\end{array}\right)$

$\left({\epsilon }^T\right)=\left(\begin{array}{ccc}{\epsilon }_{11}^T& 0& 0\\ 0& {\epsilon }_{22}^T& 0\\ 0& 0& {\epsilon }_{33}^T\end{array}\right)$

$\left(s^E\right)=\left(\begin{array}{cccccc}{s}_{11}^E& s_{12}^E& s_{13}^E& 0& 0& 0\\ s_{12}^E& s_{11}^E& s_{13}^E& 0& 0& 0\\ s_{13}^E& s_{13}^E& s_{33}^E& 0& 0& 0\\ 0& 0& 0& s_{55}^E& 0& 0\\ 0& 0& 0& 0& s_{55}^E& 0\\ 0& 0& 0& 0& 0& 2\left(s_{11}^E-s_{12}^E\right)\end{array}\right)$

The equation of the reverse piezo effect [8‒51] has the form

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

where  m, i, j are axises.

For the longitudinal piezo engine on Figure 1 its parameters are determined in the form

Figure 1 A piezo engine for composite telescope.

$\Delta {\delta }_{\text{max}}=d_{33}{E}_3\delta =d_{33}U$ ,  $F_{\text{max}}=d_{33}{E}_3{S}_0/s_{33}^E$

At $d_{33}$  = 4∙10^‒10 m/V,  $E_3$ = 0.8∙10⁵ V/m,  $\delta$ = 2.5∙10^‒3 m,  $S_0$ = 1.5∙10^‒4 m²,  $s_{33}^E$ = 15∙10^‒12 m²/N its maximum values of deformation and force are received in the form  $\Delta {\delta }_{\text{max}}$ = 80 nm,  $F_{\text{max}}$ = 320 N with error 10%.

The differential equation for a piezo engine has the form^11‒51

$\frac{d^2\Xi \left(x,s\right)}{d{x}^2}-{\gamma }^2\Xi \left(x,s\right)=0$

where $x$ , $s$ , $\gamma$  are coordinate, operator and coefficient.

Its solution has form

$\Xi \left(x,s\right)=\left\{{\Xi }_1\left(s\right)\text{sh}\left[\left(l-x\right)\gamma \right]+{\Xi }_2\left(s\right)\text{sh}\left(x\gamma \right)\right\}/\text{sh}\left(l\gamma \right)$

For the stresses acting on two faces a piezo engine its transforms of Laplace have the form

$T_j\left(0,s\right)=\frac{1}{s_{ij}^{\Psi }}{\frac{d\text{Ξ}\left(x,s\right)}{dx}|}_{x=0}-\frac{d_{mi}}{s_{ij}^{\Psi }}{\Psi }_m\left(s\right)$

$T_j\left(l,s\right)=\frac{1}{s_{ij}^{\Psi }}{\frac{d\text{Ξ}\left(x,s\right)}{dx}|}_{x=l}-\frac{d_{mi}}{s_{ij}^{\Psi }}{\Psi }_m\left(s\right)$

where   $\Psi =E$ or $\Psi =D$ .

For the structural model and scheme of a piezo engine for composite telescope on Figure 2 its equations have the form

Figure 2 Structural scheme of a piezo engine for composite telescope.

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

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

where  $v_{mi}=\{\begin{array}{c}{d}_{33},d_{31},d_{15}\\ g_{33},g_{31},g_{15}\end{array}$ , ${\Psi }_m=\{\begin{array}{c}{E}_3,E_1\\ D_3,D_1\end{array}$ , $s_{ij}^{\Psi }=\{\begin{array}{c}{s}_{33}^E,s_{11}^E,s_{55}^E\\ s_{33}^D,s_{11}^D,s_{55}^D\end{array}$ , $l=\{\delta ,h,b$ , $\gamma =\{{\gamma }^E,{\gamma }^D$ , $c^{\Psi }=\{c^E,c^D$ , ${\chi }_{ij}^{\Psi }=s_{ij}^{\Psi }/S_0$ , $v_{mi}$  is the piezo coefficient.

Therefore, the matrix equation of a piezo engine has the form

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

The steady‒state displacements of faces 1 and 2 for the longitudinal piezo engine have the form

${\xi }_1\left(\infty \right)=d_{33}U{M}_2/\left(M_1+M_2\right)$

${\xi }_2\left(\infty \right)=d_{33}U{M}_1/\left(M_1+M_2\right)$

At  $d_{33}$ = 4×10^‒10 m/V,  U= 250 V,  $M_1$ = 1 kg and  $M_2$ = 4 kg its displacements are obtained  ${\xi }_1\left(\infty \right)$ = 80 nm,  ${\xi }_2\left(\infty \right)$ = 20 nm,  ${\xi }_1\left(\infty \right)+{\xi }_2\left(\infty \right)$ = 100 nm with error 10%.

For the transverse piezo engine at elastic‒inertial load the expression has the form

$W\left(s\right)=\frac{\Xi \left(s\right)}{U\left(s\right)}=\frac{d_{31}h/\delta }{\left(1+C_l/C_{11}^E\right)\left(T_t^2{p}^2+2T_t{\xi }_{t}p+1\right)}$

$T_t=\sqrt{M/\left(C_l+C_{11}^E\right)}$ ,  ${\omega }_t=1/T_t$

where $C_l$ , $C_{11}^E$  are the stiffness of load and engine, $T_t$ , ${\xi }_t$ , ${\omega }_t$  are the time constant, the attenuation coefficient and the conjugate frequency of the engine. At  M= 3 kg,  $C_l$ = 0.2×10⁷ N/m,  $C_{11}^E$ = 1×10⁷ N/m its parameters are determined in the form the time constant  $T_t$ = 0.5×10^‒3 s and the conjugate frequency of the engine  ${\omega }_t$ = 2×10³ s^‒1 with error 10%.

The structural model of a piezo engine for composite telescope is obtained. The structural model clearly shows the conversion of electrical energy by a piezo engine into mechanical energy of the control element of a composite telescope using the physical parameters of a piezo engine and its load. The structural scheme of a piezo engine for composite telescope is determined.

The matrix equation of a piezo engine is received for the calculation its displacements and parameters. The structural model, the structural scheme and the matrix equation of a piezo engine are used in decisions of the control systems for composite telescope.

None.

None.

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