Structural model of a piezo engine for composite telescope

doi:10.15406/paij.2022.06.00243

eISSN: 2576-4543

Physics & Astronomy International Journal

Review Article Volume 6 Issue 1

Structural model of a piezo engine for composite telescope

Afonin SM

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National Research University of Electronic Technology, Russia

Correspondence: Afonin Sergey Mikhailovich, National Research University of Electronic Technology, MIET, 124498, Moscow, Russia

Received: February 01, 2022 | Published: February 22, 2022

Citation: Afonin SM. Structural model of a piezo engine for composite telescope. Phys Astron Int J. 2022;6(1):12-15. DOI: 10.15406/paij.2022.06.00243

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Abstract

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

Introduction

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

Structural scheme of a piezo engine

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

$(D) = (d) (T) + (ε^{T}) (E)$ $(D) = (d) (T) + (ε^{T}) (E)$

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

where $(D)$ $(D)$ , $(S)$ $(S)$ , $(T)$ $(T)$ , $(E)$ $(E)$ 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

$(d) = ⎛ ⎜ ⎝ \begin{matrix} \begin{matrix} 0 & 0 & 0 & 0 & d_{15} & 0 \end{matrix} \begin{matrix} 0 & 0 & 0 & d_{15} & 0 & 0 \end{matrix} \begin{matrix} d_{31} & d_{31} & d_{33} & 0 & 0 & 0 \end{matrix} \end{matrix} ⎞ ⎟ ⎠$ $(d) = (\begin{matrix} \begin{matrix} 0_{} & 0_{} & 0_{} & 0_{} & d_{15} & 0 \end{matrix} \\ \begin{matrix} 0_{} & 0_{} & 0_{} & d_{15} & 0_{} & 0 \end{matrix} \\ \begin{matrix} d_{31} & d_{31} & d_{33} & 0_{} & 0_{} & 0_{} \end{matrix} \end{matrix})$

${(d)}^{t} = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ \begin{matrix} 0 & 0 & d_{31} 0 & 0 & d_{31} 0 & 0 & d_{33} 0 & d_{15} & 0 d_{15} & 0 & 0 0 & 0 & 0 \end{matrix} ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠$ ${(d)}^{t} = (\begin{matrix} 0 & 0 & d_{31} \\ 0 & 0 & d_{31} \\ 0 & 0 & d_{33} \\ 0 & d_{15} & 0 \\ d_{15} & 0 & 0 \\ 0 & 0 & 0 \end{matrix})$

$(ε^{T}) = ⎛ ⎜ ⎜ ⎝ \begin{matrix} ε_{11}^{T} & 0 & 0 0 & ε_{22}^{T} & 0 0 & 0 & ε_{33}^{T} \end{matrix} ⎞ ⎟ ⎟ ⎠$ $(ε^{T}) = (\begin{matrix} ε_{11}^{T} & 0 & 0 \\ 0 & ε_{22}^{T} & 0 \\ 0 & 0 & ε_{33}^{T} \end{matrix})$

$(s^{E}) = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ \begin{matrix} s_{11}^{E} & s_{12}^{E} & s_{13}^{E} & 0 & 0 & 0 s_{12}^{E} & s_{11}^{E} & s_{13}^{E} & 0 & 0 & 0 s_{13}^{E} & s_{13}^{E} & s_{33}^{E} & 0 & 0 & 0 0 & 0 & 0 & s_{55}^{E} & 0 & 0 0 & 0 & 0 & 0 & s_{55}^{E} & 0 0 & 0 & 0 & 0 & 0 & 2 (s_{11}^{E} - s_{12}^{E}) \end{matrix} ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠$ $(s^{E}) = (\begin{matrix} s_{11}^{E} & s_{12}^{E} & s_{13}^{E} & 0 & 0 & 0 \\ s_{12}^{E} & s_{11}^{E} & s_{13}^{E} & 0 & 0 & 0 \\ s_{13}^{E} & s_{13}^{E} & s_{33}^{E} & 0 & 0 & 0 \\ 0 & 0 & 0 & s_{55}^{E} & 0 & 0 \\ 0 & 0 & 0 & 0 & s_{55}^{E} & 0 \\ 0 & 0 & 0 & 0 & 0 & 2 (s_{11}^{E} - s_{12}^{E}) \end{matrix})$

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

$S_{i} = d_{m i} E_{m} + s_{i j}^{E} T_{j}$ $S_{i} = d_{m i} E_{m} + s_{i j}^{E} T_{j}$

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.

$Δ δ_{max} = d_{33} E_{3} δ = d_{33} U$ $Δ δ_{max} = d_{33} E_{3} δ = d_{33} U$ , $F_{max} = d_{33} E_{3} S_{0} / s_{33}^{E}$ $F_{max} = d_{33} E_{3} S_{0} / s_{33}^{E}$

At $d_{33}$ $d_{33}$ = 4∙10^‒10 m/V, $E_{3}$ $E_{3}$ = 0.8∙10⁵ V/m, $δ$ $δ$ = 2.5∙10^‒3 m, $S_{0}$ $S_{0}$ = 1.5∙10^‒4 m², $s_{33}^{E}$ $s_{33}^{E}$ = 15∙10^‒12 m²/N its maximum values of deformation and force are received in the form $Δ δ_{max}$ $Δ δ_{max}$ = 80 nm, $F_{max}$ $F_{max}$ = 320 N with error 10%.

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

$\frac{d^{2} Ξ (x, s)}{d x^{2}} - γ^{2} Ξ (x, s) = 0$ $\frac{d^{2} Ξ (x, s)}{d x^{2}} - γ^{2} Ξ (x, s) = 0$

where $x$ $x$ , $s$ $s$ , $γ$ $γ$ are coordinate, operator and coefficient.

Its solution has form

$Ξ (x, s) = {Ξ_{1} (s) sh [(l - x) γ] + Ξ_{2} (s) sh (x γ)} / sh (l γ)$ $Ξ (x, s) = {Ξ_{1} (s) sh [(l - x) γ] + Ξ_{2} (s) sh (x γ)} / sh (l γ)$

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

$T_{j} (0, s) = \frac{1}{s_{i j}^{Ψ}} {\frac{d Ξ (x, s)}{d x} ∣ ∣}_{x = 0} - \frac{d_{m i}}{s_{i j}^{Ψ}} Ψ_{m} (s)$ $T_{j} (0, s) = \frac{1}{s_{i j}^{Ψ}} {\frac{d Ξ (x, s)}{d x} |}_{x = 0} - \frac{d_{m i}}{s_{i j}^{Ψ}} Ψ_{m} (s)$

$T_{j} (l, s) = \frac{1}{s_{i j}^{Ψ}} {\frac{d Ξ (x, s)}{d x} ∣ ∣}_{x = l} - \frac{d_{m i}}{s_{i j}^{Ψ}} Ψ_{m} (s)$ $T_{j} (l, s) = \frac{1}{s_{i j}^{Ψ}} {\frac{d Ξ (x, s)}{d x} |}_{x = l} - \frac{d_{m i}}{s_{i j}^{Ψ}} Ψ_{m} (s)$

where $Ψ = E$ $Ψ = E$ or $Ψ = D$ $Ψ = D$ .

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.

$Ξ_{1} (s) = [1 / (M_{1} s^{2})] {- F_{1} (s) + (1 / χ_{i j}^{Ψ}) [\begin{matrix} d_{m i} Ψ_{m} (s) - [γ / sh (l γ)] \times [ch (l γ) Ξ_{1} (s) - Ξ_{2} (s)] \end{matrix}]}$ $Ξ_{1} (s) = [1 / (M_{1} s^{2})] {- F_{1} (s) + (1 / χ_{i j}^{Ψ}) [\begin{array}{l} d_{m i} Ψ_{m} (s) - [γ / sh (l γ)] \\ \times [ch (l γ) Ξ_{1} (s) - Ξ_{2} (s)] \end{array}]}$

$Ξ_{2} (s) = [1 / (M_{2} s^{2})] {- F_{2} (s) + (1 / χ_{i j}^{Ψ}) [\begin{matrix} d_{m i} Ψ_{m} (s) - [γ / sh (l γ)] \times [ch (l γ) Ξ_{2} (s) - Ξ_{1} (s)] \end{matrix}]}$ $Ξ_{2} (s) = [1 / (M_{2} s^{2})] {- F_{2} (s) + (1 / χ_{i j}^{Ψ}) [\begin{array}{l} d_{m i} Ψ_{m} (s) - [γ / sh (l γ)] \\ \times [ch (l γ) Ξ_{2} (s) - Ξ_{1} (s)] \end{array}]}$

where $v_{m i} = {\begin{matrix} d_{33}, d_{31}, d_{15} g_{33}, g_{31}, g_{15} \end{matrix}$ $v_{m i} = {\begin{matrix} d_{33}, d_{31}, d_{15} \\ g_{33}, g_{31}, g_{15} \end{matrix}$ , $Ψ_{m} = {\begin{matrix} E_{3}, E_{1} D_{3}, D_{1} \end{matrix}$ $Ψ_{m} = {\begin{matrix} E_{3}, E_{1} \\ D_{3}, D_{1} \end{matrix}$ , $s_{i j}^{Ψ} = {\begin{matrix} s_{33}^{E}, s_{11}^{E}, s_{55}^{E} s_{33}^{D}, s_{11}^{D}, s_{55}^{D} \end{matrix}$ $s_{i j}^{Ψ} = {\begin{matrix} s_{33}^{E}, s_{11}^{E}, s_{55}^{E} \\ s_{33}^{D}, s_{11}^{D}, s_{55}^{D} \end{matrix}$ , $l = {δ, h, b$ $l = {δ, h, b$ , $γ = {γ^{E}, γ^{D}$ $γ = {γ^{E}, γ^{D}$ , $c^{Ψ} = {c^{E}, c^{D}$ $c^{Ψ} = {c^{E}, c^{D}$ , $χ_{i j}^{Ψ} = s_{i j}^{Ψ} / S_{0}$ $χ_{i j}^{Ψ} = s_{i j}^{Ψ} / S_{0}$ , $v_{m i}$ $v_{m i}$ is the piezo coefficient.

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

$(\begin{matrix} Ξ_{1} (s) Ξ_{2} (s) \end{matrix}) = (\begin{matrix} \begin{matrix} W_{11} (s) & W_{12} (s) & W_{13} (s) \end{matrix} \begin{matrix} W_{21} (s) & W_{22} (s) & W_{23} (s) \end{matrix} \end{matrix}) ⎛ ⎜ ⎝ \begin{matrix} Ψ_{m} (s) F_{1} (s) F_{2} (s) \end{matrix} ⎞ ⎟ ⎠$ $(\begin{matrix} Ξ_{1} (s) \\ Ξ_{2} (s) \end{matrix}) = (\begin{matrix} \begin{matrix} W_{11} (s) & W_{12} (s) & W_{13} (s) \end{matrix} \\ \begin{matrix} W_{21} (s) & W_{22} (s) & W_{23} (s) \end{matrix} \end{matrix}) (\begin{matrix} Ψ_{m} (s) \\ F_{1} (s) \\ F_{2} (s) \end{matrix})$

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

$ξ_{1} (\infty) = d_{33} U M_{2} / (M_{1} + M_{2})$ $ξ_{1} (\infty) = d_{33} U M_{2} / (M_{1} + M_{2})$

$ξ_{2} (\infty) = d_{33} U M_{1} / (M_{1} + M_{2})$ $ξ_{2} (\infty) = d_{33} U M_{1} / (M_{1} + M_{2})$

At $d_{33}$ $d_{33}$ = 4×10^‒10 m/V, U= 250 V, $M_{1}$ $M_{1}$ = 1 kg and $M_{2}$ $M_{2}$ = 4 kg its displacements are obtained $ξ_{1} (\infty)$ $ξ_{1} (\infty)$ = 80 nm, $ξ_{2} (\infty)$ $ξ_{2} (\infty)$ = 20 nm, $ξ_{1} (\infty) + ξ_{2} (\infty)$ $ξ_{1} (\infty) + ξ_{2} (\infty)$ = 100 nm with error 10%.

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

$W (s) = \frac{Ξ (s)}{U (s)} = \frac{d_{31} h / δ}{(1 + C_{l} / C_{11}^{E}) (T_{t}^{2} p^{2} + 2 T_{t} ξ_{t} p + 1)}$ $W (s) = \frac{Ξ (s)}{U (s)} = \frac{d_{31} h / δ}{(1 + C_{l} / C_{11}^{E}) (T_{t}^{2} p^{2} + 2 T_{t} ξ_{t} p + 1)}$

$T_{t} = \sqrt{M / (C_{l} + C_{11}^{E})}$ $T_{t} = \sqrt{M / (C_{l} + C_{11}^{E})}$ , $ω_{t} = 1 / T_{t}$ $ω_{t} = 1 / T_{t}$

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

Conclusion

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.

Acknowledgments

None.

Conflicts of interest

None.

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