Calculation of structural scheme nano piezoengine with lumped parameters at voltage and current control for aerospace

doi:10.15406/aaoaj.2025.09.00227

eISSN: 2576-4500

Aeronautics and Aerospace Open Access Journal

Research Article Volume 9 Issue 2

Calculation of structural scheme nano piezoengine with lumped parameters at voltage and current control for aerospace

SM Afonin

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National Research University of Electronic Technology, MIET, Moscow, Russia

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

Received: May 12, 2025 | Published: May 28, 2025

Citation: Afonin SM. Calculation of structural scheme nano piezoengine with lumped parameters at voltage and current control for aerospace. Aeron Aero Open Access J. 2025;9(2):113-116. DOI: 10.15406/aaoaj.2025.09.00227

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Abstract

In the work the structural scheme of nano piezoengine with lumped parameters is determined for aerospace. The PZT nano engines have small dimensions, high accuracy, simple design. Calculation of structural scheme nano piezoengine with lumped parameters was made at voltage and current control for aerospace. Method mathematical physics is used for calculation of structural scheme nano piezoengine. Schemes nano piezoengines with lumped parameters at voltage and current control for aerospace are determined.

Keywords nano piezoengine, structural scheme, aerospace

Introduction

The nano piezoengine is used for aerospace.^1–20 This nano piezoengine is used in control system aerospace for satellite and compound telescope, laser interferometer, deformable mirror, damping vibration, scanning microscopy.^14–64 The schemes of nano piezoengines are constructed at voltage and current control.

Scheme piezoengine at voltage control

The scheme of the piezoengine with lumped parameters has big interes in calculation nano mechatronics systems for aerospace. The scheme of the piezoengine is determined from the equation of the reverse piezoeffect and the equations forces of engine in dynamics by using method mathematical physics.^11–64

The equation reverse piezoeffect has the form^1–64

$S_{i} = d_{m i} E_{m} + s_{i j}^{E} T_{j}$ (1)

Here the indexes i, j, m and $S_{i}$ , $E_{m}$ , $T_{j}$ , $d_{m i}$ , $s_{i j}^{E}$ are the relative displacement, the strength electric field, the strength mechanical field, the piezo module, and the elastic compliance at $E = const$ .

Therefore, we have the Laplace transform general force^46–62 at voltage control $F (s)$ of the nano piezoengine in the form

$F (s) = d_{m i} S_{0} E_{m} (s) / s_{i j}^{E}$ (2)

Also we receive the Laplace transform general force at voltage control from the voltage $U_{C} (s) = U (s) / (R C_{0} s + 1)$ on its piezo capacitor

$F (s) = \frac{d_{m i} S_{0} U_{C} (s)}{δ s_{i j}^{E}} = \frac{d_{m i} S_{0} U (s)}{δ s_{i j}^{E} (R C_{0} s + 1)}$ (3)

Than this Laplace transform general force of the piezoengine is obtained

$F (s) = k_{r} U_{C} (s)$ (4)

Here $k_{r}$ is the reverse coefficient in the form

$k_{r} = \frac{d_{m i} S_{0}}{δ s_{i j}^{E}}$ (5)

The equation direct piezo effect ^1-64 has the form

$D_{m} = d_{m i} T_{i} + ε_{m k}^{E} E_{k}$ (6)

Here $ε_{m k}^{E}$ , $k$ are the permittivity, index

Than the direct coefficient $k_{d}$ at voltage control is founded in the form

$k_{d} = k_{r} = \frac{d_{m i} S_{0}}{δ s_{i j}^{E}}$ (7)

Therefore, we have the Laplace transform voltage $U_{d} (s)$ for the feedback on the velocity of the second end at voltage control on Figure 1 in the form

$U_{d} (s) = \frac{d_{m i} S_{0}}{δ s_{i j}^{E}} {\dot{Ξ}}_{2} (s) R = k_{d} {\dot{Ξ}}_{2} (s) R = k_{d} R s Ξ_{2} (s)$ (8)

Here ${\dot{Ξ}}_{2} (s) = s Ξ_{2} (s)$ is the Laplace transform velocity the second end of the nano piezoengine.

The Laplace transform force at the second end $F_{2} (s)$ of the nano piezoengine and the load force are used (Figure 1). The transform force for the feedback on the load force at voltage control is obtained in the form

$F_{l} (s) = (C_{e} + C_{i j}^{E} + k_{ν} s) Ξ_{2} (s)$ (9)

Here $C_{e}$ , $C_{i j}^{E}$ , $k_{ν}$ are the elastic load stiffness, the stiffness piezoengine at $E = const$ , the coefficient of viscous friction.

This Laplace transform the load force is founded in the form the sum of the elastic force and the viscous friction force.

The scheme for lumped parameters of the nano piezoengine (Figure 1) at voltage control is constructed for the first fixed end, the elastic inertial load with the mass load $M_{2}$ . The inertia force $F_{i} (s)$ has form

$F_{i} (s) = M_{2} s^{2} Ξ_{2} (s)$ (10)

Here $Ξ_{2} (s)$ is the Laplace transform displacement the second end of the nano piezoengine.

Let us consider transformation transfer function for scheme (Figure 1) at $R$ about several Om. Than for no the feedback on the velocity the Laplace transform equation for forces on the second end of the piezoengine has the form

$\frac{d_{m i} S_{0} U (s)}{δ s_{i j}^{E} (R C_{0} s + 1)} = M_{2} s^{2} Ξ_{2} (s) + (C_{e} + C_{i j}^{E} + k_{ν} s) Ξ_{2} (s)$ (11)

The transfer function of the nano piezoengine has the form

$W_{1} (s) = \frac{Ξ_{2} (s)}{U (s)} = \frac{d_{m i} S_{0}}{δ s_{i j}^{E} (R C_{0} s + 1) (M_{2} s^{2} + k_{ν} s + C_{e} + C_{i j}^{E})}$ (12)

And

$W_{1} (s) = \frac{Ξ_{2} (s)}{U (s)} = \frac{k_{r}}{(R C_{0} s + 1) (M_{2} s^{2} + k_{ν} s + C_{e} + C_{i j}^{E})}$ (13)

By using the feedback on the velocity $U_{d} (s) = k_{d} R s Ξ_{2} (s)$ after conversion transfer function of the nano piezoengine) at more 100 Om we have the transfer function in general for lumped parameters of the nano piezoengine for voltage control (Figure 1) in the form

$W (s) = Ξ_{2} (s) / U (s) = k_{r} / (a_{3} s^{3} + a_{2} s^{2} + a_{1} s + a_{0})$ (14)

$a_{3} = R C_{0} M_{2}$ , $a_{2} = M_{2} + R C_{0} k_{v}$ (15)

$a_{1} = k_{v} + R C_{0} C_{i j}^{E} + R C_{0} C {}_{e}+ k_{r} k_{d} R$ , $a_{0} = C {}_{e}+ C_{i j}^{E}$ (16)

Figure 1 Scheme piezoengine for lumped parameters at voltage control.

Than the transfer function of the nano piezoengine at voltage control for $R = 0$ is founded in the form

$W (s) = Ξ_{2} (s) / U (s) = k_{31}^{U} / (T_{t}^{2} s^{2} + 2 T_{t} ξ_{t} s + 1)$ (17)

$k_{31}^{U} = d_{31} (h / δ) / (1 + C_{e} / C_{11}^{E})$ (18)

$T_{t} = \sqrt{M / (C_{e} + C_{11}^{E})}$ , $ω_{t} = 1 / T_{t}$ (19)

For PZT engine $d_{31}$ = 0.2∙nm/V, $h / δ$ = 20, $C_{e}$ = 0.5×10⁷ N/m, $C_{11}^{E}$ = 1.1×10⁷ N/m its transfer coefficient at voltage control is determined $k_{31}^{U}$ = 2.75 nm/V with error of 10%. For PZT engine $M_{2}$ = 1 kg, $C_{e}$ = 0.5×10⁷ N/m, $C_{11}^{E}$ = 1.1×10⁷ N/m its time const at voltage control is obtained $T_{t}$ = 0.25×10^-3 s with error of 10%.

Scheme piezoengine at current control

For direct feed of the nano piezoengine at current control we have the transform general force by using the voltage $U_{C} (s) = I (s) / (C_{0} s)$ on the piezo capacitor in the form

$F (s) = \frac{d_{m i} S_{0} I (s)}{δ s_{i j}^{D} C_{0} s}$ (20)

Here $I (p)$ , $s_{i j}^{D}$ re the current, the the elastic compliance at $D = const$

The Laplace transform current $I_{d} (s)$ for the feedback on the velocity the second end of the piezoengine is founded at current control on Figure 2 in the form

$I_{d} (s) = \frac{d_{m i} S_{0}}{δ s_{i j}^{D}} {\dot{Ξ}}_{2} (s) = k_{d} {\dot{Ξ}}_{2} (s)$ (21)

The direct coefficient at current control $k_{d}$ is founded

$k_{d} = k_{r} = \frac{d_{m i} S_{0}}{δ s_{i j}^{D}}$ (22)

The Laplace transform load force $F_{l} (s)$ for the feedback at current control has the form

$F_{l} (s) = (C_{e} + C_{i j}^{D} + k_{ν} s) Ξ_{2} (s)$ (23)

The scheme for lumped parameters of the piezoengine at the current control is constructed (Figure 2) at the first fixed end, the elastic inertial load.

Figure 2 Scheme piezoengine for lumped parameters at current control.

The schemes of the piezoengine for lumped parameters at the voltage or current control are used in calculation nano mechatronics systems for aerospace. Let us consider parameters of the piezoengine for lumped parameters at the current control in static regime.

$F_{j max} = U \frac{1}{δ} d_{m i} \frac{S_{0}}{s_{i j}^{E}} + F_{j max} \frac{1}{S_{0}} d_{m i} n S_{p} \frac{1}{ε_{m k}^{T} n S_{p} / δ} \frac{1}{δ} d_{m i} \frac{S_{0}}{s_{i j}^{E}}$ (24)

After conversion we have equation in the form

$\frac{F_{j max}}{S_{0}} s_{i j}^{E} (1 - \frac{d_{m i}^{2}}{ε_{m k}^{T} s_{i j}^{E}}) = \frac{F_{j max}}{S_{0}} s_{i j}^{E} (1 - k_{m i}^{2}) = \frac{F_{j max}}{S_{0}} s_{i j}^{D} = E_{m} d_{m i}$ (25)

$k_{m i}^{2} = \frac{d_{m i}^{2}}{ε_{m k}^{T} s_{i j}^{E}}$ , $s_{i j}^{D} = (1 - k_{m i}^{2}) s_{i j}^{E}$ (26)

Here $k_{m i}$ is the electromechanical coupling coefficient. The elastic compliance of PZT engines was measured using the UMM-5 press $s_{i j}^{E} / s_{i j}^{D} \leq 1, 2$ with error of 10%.

Discussion

The structural scheme of nano piezoengine with lumped parameters is determined for aerospace. The PZT nano engines is used for aerospace. Method mathematical physics is used for calculation scheme nano piezoengine.

Conclusion

The schemes nano piezoengines with lumped parameters at voltage and current control for aerospace are determined. The transfer function and its lumped parameters of the nano piezoengine are obtained. The parameters PZT nano engines are found.