Research Article Volume 7 Issue 1

National Research University of Electronic Technology MIET, Russia

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

Received: October 10, 2023 | Published: November 14, 2023

**Citation: **Afonin SM. Absolute stability of system with nano piezoengine for biomechanics. *MOJ App Bio Biomech*. 2023;7(1):211-213. DOI: 10.15406/mojabb.2023.07.00197

The nano piezoengine is used for biomechanics and nano sciences in dosing device, scanning microscopy, nano manipulator, nano pump. For the nano piezoengine with hysteresis in control system its set of equilibrium positions is the segment of line. The frequency method for studying the stability of system is used. By applying Yakubovich criterion for system with the nano piezoengine the absolute stability of system is calculated for biomechanics. The ratio of the piezomodules of the nano piezoengine with transverse, longitudinal, shear piezoelectric effects is proportional the ratio of its tangents of the angle of inclination to the hysteresis.

**Keywords:** absolute stability system, nano piezoengine, hysteresis, set equilibrium positions, biomechanics

Many equilibrium positions are found in system with nano piezoengine for biomechanics and nano science. For calculation absolute stability of system with the nano piezoengine is using Yakubovich criterion, which is the development of the Lyapunov and Popov criterions.^{1–20} The nano piezoengine is used for biomechanics and nano sciences in dosing device, scanning microscopy, nano manipulator, nano pump.^{3–30}

The frequency method for studying the stability of system is used to study the absolute stability of control system with the nano piezoengine.

In this work for discussions stability of system with the nano piezoengine are used three main problems: the set of equilibrium positions, the Yakubovich criterion for the absolute stability of system, the maximum of the tangent the angle of inclination to the hysteresis loop of the nano piezoengine.

For written the hysteresis of the nano piezoengine for biomechanics and nano science the Preisach model is used for its hysteresis deformation. The hysteresis Preisach function of the relative deformation the nano piezoengine on Figure 1 is determined.^{3–28}

${S}_{i}=F\left[{{E}_{m}|}_{0}^{t},t,{S}_{i}\left(0\right),sign{\dot{E}}_{m}\right]$

here ${S}_{i}$
- the hysteresis deformation, *t* - time, ${S}_{i}\left(0\right)$
- the initial condition, ${E}_{m}$
- the strength of electric field, $\text{sign}\text{\hspace{0.17em}}{\dot{E}}_{m}$
- the sign for velocity of change strength of electric field.

In control system the set of equilibrium positions is the set of points *M* of intersection of the line *L* with the hysteresis characteristic of nano piezo engine on Figure 1 in the form of the selected line segment.^{3} Respectively, the equation of the line *L* is evaluated

${E}_{m}+k{S}_{i}=0$

here $k$ - the transfer coefficient for the linear part of control system.

The expression for the symmetric main hysteresis loop^{28 }of the characteristic of nano piezoengine on Figure 1 is determined in the form

${S}_{i}={d}_{mi}{E}_{m}-{\gamma}_{mi}{E}_{m\text{\hspace{0.17em}}\text{max}}{\left(1-\frac{{E}_{m}^{2}}{{E}_{m\text{\hspace{0.17em}}\text{max}}^{2}}\right)}^{{n}_{mi}}\text{sign}{\dot{E}}_{m}$

here ${d}_{mi}$ - the piezo module, ${\gamma}_{mi}={S}_{i}^{0}/{E}_{m\text{\hspace{0.17em}}\text{max}}$ - the coefficient of hysteresis, ${S}_{i}^{0}$ - the relative deformation at ${E}_{m}=0$ , ${n}_{mi}$ - the coefficient for the nano piezoengine from PZT ${n}_{mi}$ = 1.

The width of the resting zone at $\Delta {E}_{m\text{\hspace{0.17em}}\text{max}}$ is obtained

$\Delta {E}_{m\text{\hspace{0.17em}}\text{max}}+k{S}_{i}^{+}\left(\Delta {E}_{m\text{\hspace{0.17em}}\text{max}}\right)=0$

here $\Delta $ - the relative value of electric field strength; ${S}_{i}^{+}\left(\Delta {E}_{m\text{\hspace{0.17em}}\text{max}}\right)$ - the value of the relative deformation on the ascending branch for ${\dot{E}}_{m}>0$ , ${S}_{i}^{-}\left(-\Delta {E}_{m\text{\hspace{0.17em}}\text{max}}\right)$ - the value of the relative deformation on the descending branch for ${\dot{E}}_{m}<0$ on Figure 1.

At the symmetric main hysteresis loop characteristic of the nano piezo engine the equation is evaluated

${S}_{i}^{+}\left(\Delta {E}_{m\text{\hspace{0.17em}}\text{max}}\right)={d}_{mi}\Delta {E}_{m\text{\hspace{0.17em}}\text{max}}-{\gamma}_{mi}{E}_{m\text{\hspace{0.17em}}\text{max}}\left(1-\frac{{\left(\Delta {E}_{m\text{\hspace{0.17em}}\text{max}}\right)}^{2}}{{E}_{m\text{\hspace{0.17em}}\text{max}}^{2}}\right)$

After transformation this expression is determined

${S}_{i}^{+}\left(\Delta {E}_{m\text{\hspace{0.17em}}\text{max}}\right)={d}_{mi}\Delta {E}_{m\text{\hspace{0.17em}}\text{max}}-{\gamma}_{mi}{E}_{m\text{\hspace{0.17em}}\text{max}}\left(1-{\Delta}^{\text{2}}\right)$

From equation for the width of the resting zone the expression is calculated

$\Delta {E}_{m\text{\hspace{0.17em}}\text{max}}+k{E}_{m\text{\hspace{0.17em}}\text{max}}{\left[{d}_{mi}\Delta -{\gamma}_{mi}\left(1-{\Delta}^{2}\right)\right]}_{}^{}=0$

Then the equation is determined

$\Delta +k{\left[{d}_{mi}\Delta -{\gamma}_{mi}\left(1-{\Delta}^{2}\right)\right]}_{}^{}=0$

The quadratic equation is calculated

${\Delta}^{\text{2}}+\frac{\left(1+k{d}_{mi}\right)}{k{\gamma}_{mi}}\Delta -1=0$

The relative width of the rest zone of system with the nano piezoengine for biomechanics and nano sciences is obtained from this quadratic equation for the symmetric loop characteristic in the form

$2\Delta =-\frac{\left(1+k{d}_{mi}\right)}{k{\gamma}_{mi}}+\sqrt{\frac{{\left(1+k{d}_{mi}\right)}^{2}}{{k}^{2}{\gamma}_{mi}^{2}}+4}$

and for the asymmetric loop characteristic its relative width of the rest zone of system is evaluated in the form

$\begin{array}{l}{\Delta}^{+}+{\Delta}^{-}=-\frac{\left(1+k{d}_{mi}\right)}{2k}\left(\frac{1}{{\gamma}_{mi}^{+}}+\frac{1}{{\gamma}_{mi}^{-}}\right)\\ +\frac{1}{2}\sqrt{\frac{{\left(1+k{d}_{mi}\right)}^{2}}{{k}^{2}{\left({\gamma}_{mi}^{+}\right)}^{2}}+4}+\frac{1}{2}\sqrt{\frac{{\left(1+k{d}_{mi}\right)}^{2}}{{k}^{2}{\left({\gamma}_{mi}^{-}\right)}^{2}}+4}\end{array}$

From the Yakubovich criterion,^{1–4} which is the development of the Lyapunov and Popov criterions, the absolute stability of system with the nano piezoengine for biomechanics is obtained. The condition for the absolute stability of system with nano piezoactuator from PZT for biomechanics on Figure 2 is evaluated in the form

$\text{Re}{\nu}_{mi}W\left(j\omega \right)\ge -1$

here ω - the frequency, *j* - the imaginary unit, ${\nu}_{mi}$
- maximum of the tangent the angle of inclination to the hysteresis loop. The amplitude-phase frequency characteristic on Figure 2 shows the frequency transfer function $W\left(j\omega \right)$
with boundary vertical line *B*, passing point -1 on real axis.

At the maximum strength of electric field in the nano piezoengine the minimum for the tangent the angle of inclination has the form $\text{min}\left[d{S}_{i}/d{E}_{m}\right]=\text{0}$ and maximum has the form $\text{max}\left[d{S}_{i}/d{E}_{m}\right]={\nu}_{mi}.$ For the nano piezoengine from PZT for biomechanics we have its maximum tangents ${\nu}_{31}$ = 0.55 nm/V for transverse piezoeffect, ${\nu}_{33}$ = 1 nm/V for longitudinal piezoeffect, and ${\nu}_{15}$ = 1.25 nm/V for shear piezoeffect at error 10%.

Therefore, the ratio of the piezomodules of the nano piezoengine from PZT with transverse, longitudinal, shear piezoelectric effects is proportional the ratio of its tangents of the angle of inclination to the hysteresis in the form: ${d}_{31}\text{:}{d}_{33}\text{:}{d}_{15}={\nu}_{31}\text{:}{\nu}_{33}\text{:}{\nu}_{15}$

For the nano piezoengine with hysteresis in the control system for biomechanics its set of equilibrium positions of the control system is the segment of line. The frequency method for studying the stability of system is used. By using Yakubovich criterion for system with the nano piezoengine the absolute stability of control system is obtained for biomechanics. The ratio of the piezomodules of the nano piezoengine with transverse, longitudinal, shear piezoelectric effects is proportional the ratio of its tangents of the angle of inclination to its hysteresis deformation.

None.

None.

The authors declare that there are no conflicts of interest.

- Yakubovich VA. Frequency conditions for absolute stability of control systems with hysteresis-like non-linearities, Dokl. Akad. Nauk SSSR. 1963;149(2):288–291.
- Andrievsky BR, Barabanov AE, Bondarko VA, et al. Nonlinear systems. Frequency and matrix inequalities. Eds. Gelig AKh, Leonov GA, Fradkov AL. Fizmatlit. Moscow. 2008:608.
- Afonin SM. Absolute stability conditions for a system controlling the deformation of an elecromagnetoelastic transduser.
*Doklady Mathematics*. 2006;74(3):943–948. - Liao X, Yu P.
*Absolute stability of nonlinear control systems*. Springer, Dordrecht. 2008:384. - Shevtsov SN, Soloviev AN, Parinov IA, et al.
*Piezoelectric actuators and generators for energy harvesting*. Res Develop. 2018:182. - Afonin SM. Generalized parametric structural model of a compound elecromagnetoelastic transduser.
*Doklady Physics*. 2005;50(2):77–82. - Afonin SM. Structural parametric model of a piezoelectric nanodisplacement transducer.
*Doklady Physics*. 2008;53(3):137–143. - Afonin SM. Solution of the wave equation for the control of an elecromagnetoelastic transduser.
*Doklady Mathematics*. 2006;73(2):307–313. - Cady WG.
*Piezoelectricity: An introduction to the theory and applications of electromechancial phenomena in crystals*. McGraw-Hill Book Company, New York, London. 1946:806. - Mason WP editor.
*Physical Acoustics*: Principles and Methods. Vol. 1. Part A. Methods and Devices. Academic Press, New York. 1964:515. - Yang Y, Tang L. Equivalent circuit modeling of piezoelectric energy harvesters.
*Journal of Intelligent Material Systems and Structures*. 2009;20(18):2223–2235. - Schultz JA, Ueda J, Asada H.
*Cellular Actuators*. Butterworth-Heinemann Publisher, Oxford. 2017:382. - Afonin SM. Structural-parametric model and transfer functions of electroelastic actuator for nano- and microdisplacement. Chapter 9 in Piezoelectrics and Nanomaterials: Fundamentals, Developments and Applications. Ed. Parinov IA. Nova Science, New York. 2015:225–242.
- Afonin SM. Stability of strain control systems of nano-and microdisplacement piezotransducers.
*Mechanics of Solids*. 2014;49(2):196–207. - Afonin SM. Structural-parametric model electromagnetoelastic actuator nanodisplacement for mechatronics.
*Int J Physics*. 2017;5(1):9–15. - Afonin SM. Structural-parametric model multilayer electromagnetoelastic actuator for nanomechatronics.
*Int J Physics*. 2019;7(2):50–57. - Afonin SM. Structural-parametric model of electromagnetoelastic actuator for nanomechanics.
*Actuators*. 2018;7(1):1–9. - Afonin SM. Structural-parametric model and diagram of a multilayer electromagnetoelastic actuator for nanomechanics.
*Actuators*. 2019;8(3):1–14. - Afonin SM. Electromagnetoelastic actuator for nanomechanics.
*GJRE: A Mechanical and Mechanics Engineering*. 2018;18(2):19–23. - Afonin SM. Condition absolute stability control system of electromagnetoelastic actuator for communication equipment.
*Discoveries in Agriculture and Food Sciences*. 2020;8(1):8–15. - Afonin SM. Optimal control of a multilayer electroelastic engine with a longitudinal piezoeffect for nanomechatronics systems.
*Applied System Innovation*. 2020;3(4):1–7. - Afonin SM. Coded сontrol of a sectional electroelastic engine for nanomechatronics systems.
*Applied System Innovation*. 2021;4(3):1–11. - Afonin SM. Structural–parametric model electroelastic actuator nano- and microdisplacement of mechatronics systems for nanotechnology and ecology research.
*MOJ Eco Environ Sci*. 2018;3(5):306‒309. - Afonin SM. Piezo actuators for nanomedicine research.
*MOJ App Bio Biomech*. 2019;3(2):56–57. - Afonin SM. Piezoengine for nanomedicine and applied bionics.
*MOJ App Bio Biomech*. 2022;6(1):30–33. - Afonin SM. Characteristics of an electroelastic actuator nano- and microdisplacement for nanotechnology. Chapter 8 in Advances in Nanotechnology. Volume 25. Eds. Bartul Z, Trenor J, Nova Science, New York. 2021:251–266.
- Afonin SM. Electroelastic actuator of nanomechatronics systems for nanoscience. Chapter 2 in Recent Progress in Chemical Science Research. Volume 6. Ed. Min HS, B P International, India, UK. London. 2023:15–27.
- Afonin SM. Harmonious linearization of hysteresis characteristic of an electroelastic actuator for nanomechatronics systems. Chapter 34 in Physics and Mechanics of New Materials and Their Applications. Proceedings of the International Conference PHENMA 2021-2022, Springer Proceedings in Materials series. Volume 20. Eds. Parinov IA, Chang SH, Soloviev AN. Springer, Cham. 2023:419–428.
- Nalwa HS editor. Encyclopedia of Nanoscience and Nanotechnology. Los Angeles: American Scientific Publishers. 10 Volumes. 2004.
- Bhushan B editor.
*Springer Handbook of Nanotechnology*. New York: Springer. 2006:1222.

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