Step piezoengine for nano and micro bionics

doi:10.15406/mojabb.2026.10.00238

MOJ

eISSN: 2576-4519

Applied Bionics and Biomechanics

Research Article Volume 10 Issue 1

Step piezoengine for nano and micro bionics

Afonin SM

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learner01@mail.ru

National Research University of Electronic Technology, Russia

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

Received: February 27, 2026 | Published: March 13, 2026

Citation: Afonin SM. Step piezoengine for nano and micro bionics. MOJ App Bio Biomech. 2026;10(1):22-24. DOI: 10.15406/mojabb.2026.10.00238

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Abstract

The parameters of the step piezoengine for nano and micro bionics, adaptive optics, laser systems, nanorobotics are obtained. The method mathematical physics are used. The transfer functions of the central piezoengine and piezolock are determined. The transfer functions for the step piezoengine for nano and micro bionics are investigated.

Keywords: step piezoengine, parameter, central piezoengine, piezolock, nano and micro bionics

Introduction

The step piezoengine is used for alignment and positioning in nano and micro bionics, adaptive optics, laser systems, nanorobotics. The step piezoengine is a piezomechanical device for converting electrical energy into mechanical energy and for actuating mechanisms and systems based on the piezoeffect^1–15 in the nanometer-to-millimeter range. Increasing the travel range to millimeters with the positioning error on the order of the nanometer is achieved by using the step piezoengine with the multilayer central piezoengine and piezolocks made of PZT ceramics.^16–27

Method

The parameters of the step piezoengine are obtained by method mathematical physics with using the equation the inverse piezoeffect and the ordinary second-order differential equation.

The transverse inverse piezoeffect^1–12 is written as

$S_{1} = d_{31} E_{3} + s_{11}^{E} T_{1}$ ,

and longitudinal inverse piezoeffect - as

$S_{3} = d_{33} E_{3} + s_{33}^{E} T_{3}$ ,

where $S_{1}, S_{3}, d_{31}, d_{33}, E_{3}, s_{11}^{E}, s_{33}^{E}, T_{1}, T_{3}$ are the relative deformation along axes 1 and 3, the transverse and longitudinal piezomodules, the electric field strength along axes 3, the elastic compliances at the transverse and longitudinal piezoeffect and the mechanical stress along axes 1 and 3, then the equation for the inverse piezoeffect has the general form^1–12

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

where $S_{i}$ , $d_{m i}$ , $E_{m}$ , $s_{i j}^{E}$ , $T_{j}$ are the relative deformation, the piezomodule, the electric field strength, the elastic compliance at $E = const$ , the mechanical stress, and the indexes i, j = 1, 2, … , 6; m = 1, 2, 3, … ,.6.

The ordinary second-order differential equation for central piezoengine and the piezolock has the form¹²⁻¹⁵

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

where $Ξ (x, s)$ , $x$ , $s$ , $γ$ are the Laplace transform of the displacement, the coordinate, the transformation operator, the propagation coefficient.

Parameters of central piezoengine and piezolock

The expression for the Laplace transform of the relative deformation under the elastic inertial load has the form at $x = 0$ , $Ξ_{1} (s) = Ξ (0, s) = 0$ and at $x = l$ , $Ξ_{2} (s) = Ξ (l, s)$ , we have at the one fixed end the solution of the ordinary differential equation of the piezoengine in the form

$Ξ (x, s) = Ξ_{2} (s) sh (x γ) / sh (l γ)$ ,

where $l$ is the length of the piezoengine.

The expression for the Laplace transform of the relative deformation under the elastic inertial load at $x = l$ has the form

{d Ξ (x, s) / d x |}_{x = l} = d_{m i} E_{m} (s) - s_{i j}^{E} M ​ p^{2} Ξ_{2} (s) / S_{0} - s_{i j}^{E} C_{e} Ξ_{2} (s) / S_{0},

where $M, C_{e}, S_{0}$ are the mass and the stiffness of the load, the cross-sectional area of the piezoengine.

Therefore, we have the expression in the form

Ξ_{2} (s) γ cth (l γ) + Ξ_{2} (s) s_{i j}^{E} M ​ s^{2} / S_{0} + Ξ_{2} (s) s_{i j}^{E} C_{e} / S_{0} = d_{m i} E_{m} (s) .

Then the transfer function of the piezoengine with distributed parameters has the form

$W_{U} (s) = \frac{Ξ_{2} (s)}{U (s)} = \frac{d_{m i} l / δ}{M s^{2} / C_{p} + l γ cth (l γ) + C_{e} / C_{p}}$ ,

where $C_{p} = \frac{S_{0}}{s_{i j}^{E} l}$ , $δ$ , $l = n δ$ , $n$ are the stiffness of the piezoengine, the thickness of the piezolayer, the length of the piezoengine and the number of the piezolayers in the piezoengine.

The transfer function on the electric field strength for the multilayer longitudinal piezoengne for the central piezoengne or the piezolock of the step piezoengne in Figure 1 has the form

W_{E} (s) = \frac{Ξ_{2} (s)}{E_{m} (s)} = \frac{d_{33} n δ}{M s^{2} / C_{p} + l γ cth (l γ) + C_{e} / C_{p}} .

Then the transfer functionon the voltage for the multilayer longitudinal piezoengine with distributed parameters has the form

W_{U} (s) = \frac{Ξ_{2} (s)}{U (s)} = \frac{d_{33} n}{M s^{2} / C_{p} + l γ cth (l γ) + C_{e} / C_{p}} .

The transfer function on the electric field strength for the multilayer longitudinal piezoengne for the central piezoengne or the piezolock of the step piezoengne in Figure 1 has the form.

Figure 1 Step piezoengine: 1 – central piezoengine; 2 – piezolock.

Therefore, we obtain the steady state displacement of the free end of the multilayer central piezoengne in the form

$ξ_{2} (\infty) = \frac{d_{33} n U_{0}}{1 + C_{e} / C_{p}} = k_{U} U_{0}$ ,

where $k_{U} = \frac{d_{33} n}{1 + C_{e} / C_{p}}$ is the transmission coefficient on the voltage, $t \to \infty$ is the time, and $U_{0}$ is the voltage amplitude.

For the piezoelectric actuator made of PZT ceramics with the elastic inertial load at $d {}_{33}$ ;= 0.4 nm/V, $n$ = 8, $C_{p}$ = 2×10⁷ N/m, $C_{e}$ = 0.5×10⁷ N/m, $U_{0}$ = 100 V we obtain the transfer coefficient $k_{U}$ = 2.56 nm/V and the steady state value $ξ_{2} (\infty)$ = 256 nm with the error 10 %.

Then, the transfer function $W_{U} (s)$ of the multilayer central piezoengne with the lumped parameters at the low matching circuit resistance $R \to 0$ and the elastic inertial load has the form

$W_{U} (s) = \frac{Ξ_{2} (s)}{U (s)} = \frac{d_{33} n}{(1 + C_{e} / C_{p}) (T_{t}^{2} s^{2} + 2 T_{t} ξ_{t} s + 1)}$ ,

$W_{U} (s) = \frac{k_{U}}{T_{t}^{2} s^{2} + 2 T_{t} ξ_{t} s + 1}$ ,

where $T_{t} = \sqrt{\frac{M}{C_{e} + C_{p}}}$ is the time constant for the oscillatory link.

For the PZT ceramic multilayer central piezoengne with the lumped parameters at the elastic inertial load $C_{p}$ = 2×10⁷ N/m, $C_{e}$ = 0.5×10⁷ N/m and $M$ = 1 kg we have the const time $T_{t}$ = 0.2×10^-3 s with the error 10 %.

Taking into account the capacitance of the multilayer central piezoengne and the high resistance $R > > 0$ of the matching circuit we have the time constant for the aperiodic link $T_{a} = R C_{n}$ ,

where

W_{U} (s) = \frac{Ξ_{2} (s)}{U (s)} = \frac{k_{U}}{T_{a} s + 1}

at $T_{a} > > T_{t}$ .

For the PZT ceramic multilayer longitudinal central piezoengne at $R$ = 20 kΩ, $C_{n}$ = 1 μF we have the const time for the aperiodic link $T_{a}$ = 20×10^-3 s with the error 10 % and $T_{a} / T_{t} = 10$ .

Displacement for step piezoengine

The amplitude step $β$ of the step piezoengine is written as

$β = \frac{n d_{33} U_{0}}{1 + C_{e} / C_{p}} = k_{U} U_{0}$ .

Let us consider the time diagrams of the operation of the step piezoengne on Figure 2 and Figure 3, where $T$ , $T_{1}$ , $t$ , $τ$ are the period of the clocking pulses, the duration of the pulse, the time from the start of the movement and the time in the current step.

Figure 2 Time diagram step piezoengine in oscillatory mode.

Figure 3 Time diagram step piezoengine in aperiodic mode.

For the central piezoengine in the oscillatory link mode at $R \to 0$ the displacement of the step piezoengine on k step Figure 2 is given as

ξ (τ) = β {k - 1 + (1 - \frac{e^{- \frac{ξ_{t} τ}{T_{t}}}}{\sqrt{1 - ξ_{t}^{2}}} sin (ω_{t} τ + ϕ_{t}))} .

For the central piezoengine in the oscillatory link mode at $R > > 0$ the displacement of the step piezoengine on k step Figure 3 is written as

$ξ (τ) = β {k - 1 + (1 - e^{- τ / T_{a}})}$ .

Therefore, we have the parameters of the step piezoengine for nano and micro bionics.

Results

We begin by making the assumption that the weights (wi; i=1,…,N) of individual fish are exchangeable for all values of N. This assumption means in particular that the joint predictive distribution of the weights, describing beliefs about the weights of the fish in the sample, is always the same regardless which particular N fish would have been sampled, and how they would be ordered within the sample. According to the celebrated representation theorem of de Finetti, the assumption of exchangeability allows us to write the joint predictive distribution (density) in the form

Conclusion

The step piezoengine is used for alignment and positioning in nano and micro bionics, for adaptive optics and laser systems. Increasing the displasement range to millimeters with the positioning error on the order of the nanometer is achieved. The step piezoengine with the multilayer central piezoengine and piezolocks made of PZT ceramics and used due to its high rigidity and nano precision. The parameters of the step piezoengine for nano and micro bionics are determined. The transfer functions of the central piezoengine and piezolock are derived.