The compound longitudinal piezodrive is used in scanning microscopy, adaptive optics of compound telescope, nanotechnology. For nanorobotics research the compound longitudinal piezodrive is applied. The characteristics of the compound longitudinal piezodrive are determined. Its time constant are founded. The parameters of the compound longitudinal piezodrive for control system with negative feedback on the displacement inductive sensor or on the piezoaccelerometer are determined.

Keywords: compound longitudinal piezodrive, characteristic, time constant correction

The compound longitudinal piezodrive is used for nanorobotics research in the nano displacement.¹⁻¹² This compound longitudinal piezodrive based on the piezoelectric effect.⁶⁻²¹ The compound longitudinal piezodrive is applied in nanotechnology, scanning microscopy, adaptive optics, interferometry, microsurgery, nanopump.²²⁻⁵⁴ The deformation this compound longitudinal piezodrive is described with the structural parametric model. By using of mathematical physics method for the compound longitudinal piezodrive with one fixed end its characteristics are determined at elastic inertial load. In the work consider the correction the time constant of the compound longitudinal piezodrive in control system.

Characteristics compound piezodrive

The mathematic model⁶⁻⁵⁴ of the compound longitudinal piezodrive with the control of voltage is determined by using the inverse piezoeffect equation

$S_3=d_{33}{E}_3+s_{33}^E{T}_3$

here $S_3$ , $d_{33}$ , $E_3$ , $s_{33}^E$ , $T_3$  are the relative deformation on 3 axis, the piezomodule, the electric field strength, the elastic compliances at $E=\text{const}$ , the mechanic field strength.

By using of mathematical physics method we have the structural parametric model of the compound longitudinal piezodrive with one fixed end^21,22 and its function at elastic inertial load in the form the second order oscillatory link.

Therefore, for the control of voltage the transfer function of the compound longitudinal piezodrive with one fixed end is obtained in the form

$W(s)=\frac{{\Xi }_e(s)}{U(s)}=\frac{d_{33}n}{\left(1+C_e+C_{33}^E\right)\left(T_t^2{s}^2+2T_t{\xi }_{t}s+1\right)}$

after transformation this function is determined in the form

$W(s)=\frac{{\Xi }_e(s)}{U(s)}=\frac{k_t}{T_t^2{s}^2+2T_t{\xi }_{t}s+1}$

$k_t=\frac{d_{33}n}{1+C_e/C_{33}^E},$ $T_t=\sqrt{M/\left(C_e+C_{33}^E\right)}$

here ${\Xi }_e(s)$ , $U(s)$ , $k_t$   $T_t$ , ${\xi }_t$ , $M$   $C_e$ , $C_{33}^E$  are the Laplace transformation of the displacement second end, the Laplace transformation of the voltage, the transfer coefficient of the compound longitudinal piezodrive, the time constant, the attenuation coefficient, the load mass, the load stiffness, the longitudinal piezodrive stiffness.

Then the transient characteristic of the compound longitudinal piezodrive is determined in the form

${\xi }_e(t)=k_t{U}_m\left(1-\frac{e^{-\frac{{\xi }_{t}t}{T_t}}}{\sqrt{1-{\xi }_t^2}}\sin ({\omega }_{t}t+{\phi }_t)\right)$

${\omega }_t=\sqrt{1-{\xi }_t^2}/T_t,$ ${\phi }_t=arctg\left(\sqrt{1-{\xi }_t^2}/{\xi }_t\right)$ ,${\xi }_m=k_t{U}_m$

here ${\xi }_e$ , $U_m$ , ${\omega }_t$ , ${\phi }_t$   ${\xi }_m$  are the displacement second end, the voltage amplitude, the circular frequency, the phase of oscillations, the steady state displacement.

Microscope MIN-8 was used for experimental observation of resonance frequency. Electronic measuring system model 214 of Caliber plant was used for experimental data of the steady state displacement. For the compound longitudinal PZT piezodrive at $U_m$  = 110 V, $d_{33}$  = 4∙10^-10 m/V, n = 8, M = 2 kg, $C_{33}^E$  = 6∙10⁷ N/m, $C_e$  = 0.6∙10⁷ N/m its parameters $k_t$  = 2.9 nm/V, ${\xi }_m$  = 320 nm and $T_t$  = 1.74∙10^-4 s are obtained with error 20%.

Correction time constant

Let us consider the correction the time constant of the compound longitudinal piezodrive at elastic inertial load in control system. The time constant of the compound longitudinal piezodrive at elastic inertial load is determined for control system with negative feedback in control system Figure 1 on the displacement inductive sensor in the form $T_{tdis}=\sqrt{\frac{M/\left(C_e+C_{33}^E\right)}{1+k_t{k}_{dis}}}$ and after transformation $T_{tdis}=T_t\sqrt{\frac{1}{1+k_t{k}_{dis}}}$

Figure 1 Сontrol system with negative feedback on displacement inductive sensor.

here $T_{tdis}$ , $k_{dis}$  are the time constant compound drive in control systems with negative feedback on the displacement inductive sensor and the transfer coefficient of the displacement inductive sensor.

For the compound longitudinal PZT drive at $k_{dis}$  = 0.1 V/nm, M = 2 kg, $C_{33}^E$  = 6∙10⁷ N/m, $C_e$  = 0.6∙10⁷ N/m the time constant $T_{tdis}$  = 1.53∙10^-4 s is obtained. The time constant decreases $T_{tdis}<T_t$ .

Respectively, the time constant compound longitudinal piezodrive in control system Figure 2 with negative feedback on the piezoaccelerometer in the form piezoplate on end of the compound piezodrive has the form

$T_{tacc}=T_t\sqrt{1+\frac{k_t{k}_{acc}}{T_t^2}}$

Figure 2 Control system with negative feedback on piezoaccelerometer.

here $T_{tacc}$ , $k_{acc}$  are the time constant compound drive in control systems with negative feedback on the piezoaccelerometer and the transfer coefficient of the piezoaccelerometer.

For the compound longitudinal PZT drive at $k_{acc}$  = 1 Vs²/m, M = 2 kg, $C_{33}^E$  = 6∙10⁷ N/m, $C_e$  = 0.6∙10⁷ N/m the time constant $T_{tacc}$  = 1.82∙10^-4 s is obtained. The time constant increases $T_{tacc}>T_t$ .

We have transfer function of the compound longitudinal piezodrive with one fixed end and its characteristics by using of mathematical physics method. The deformation the compound longitudinal piezodrive is described with using the structural parametric model. By using mathematical physics method the time constants of the compound longitudinal piezodrive at elastic inertial load are determined in control systems.

The compound longitudinal piezodrive is used for nanorobotics research in the nanodisplacement for nanotechnology, scanning microscopy, adaptive optics, interferometry, microsurgery, nanopump. The parameters of the compound longitudinal PZT drive are obtained.

The characteristics of the compound longitudinal piezodrive are obtained by mathematical physics method. The time constants of the compound longitudinal piezodrive at elastic inertial load are determined for control systems with negative feedbacks on the displacement inductive sensor or the piezoaccelerometer.

None.

Author declares that there is no conflict of interest.

1. Uchino K. Piezoelectric actuator and ultrasonic motors. Boston, MA: Kluwer Academic Publisher. 1997: 350 p.
2. Zhao C. Ultrasonic Motors. Technologies and applications. Springer: Berlin, Germany. 2011: 494 p.
3. Afonin SM. Absolute stability conditions for a system controlling the deformation of an elecromagnetoelastic transduser. Doklady Mathematics. 2006;74(3):943–948.
4. Bhushan B. Springer handbook of nanotechnology. New York: Springer. 2004: 1222 p.
5. Shevtsov SN, Soloviev AN, Parinov IA, et al. Piezoelectric actuators and generators for energy harvesting. Research anddevelopment. Springer: Switzerland, Cham. 2018: 182 p.
6. Afonin SM. Generalized parametric structural model of a compound elecromagnetoelastic transduser. Doklady Physics. 2005;50(2):77–82.
7. Afonin SM. Structural parametric model of a piezoelectric nanodisplacement transducer. Doklady Physics. 2008;53(3):137–143.
8. Afonin SM.Solution of the wave equation for the control of an elecromagnetoelastic transduser. Doklady Mathematics. 2006;73(2):307–313.
9. Afonin SM. Optimal control of a multilayer electroelastic engine with a longitudinal piezoeffect for nanomechatronics systems. Appl Syst Innov. 2020;3(4):53.
10. Afonin SM. Coded сontrol of a sectional electroelastic engine for nanomechatronics systems. Appl Syst Innov. 2021;4(3):47.
11. Afonin SM. Structural-parametric model of electromagnetoelastic actuator for nanomechanics. Actuators. 2018;7(1):6.
12. Afonin SM. Structural-parametric model and diagram of a multilayer electromagnetoelastic actuator for nanomechanics. Actuators. 2019;8(3):52.
13. Mason W. Physical acoustics. Principles and methods. methods and devices. Academic Press, New York. 1964;8(pt A):515.
14. Liu Y, Zhang S, Yan P, et al. Finite element modeling and test of piezo disk with local ring electrodes for micro displacement. Micromachines. 2022;13(6):951.
15. Jang S, Yang Su Chul. Highly piezoelectric BaTiO3 nanorod bundle arrays using epitaxially grown TiO2 nanomaterials. Nanotechnology. 2018;29(23):235602.
16. Afonin SM. Structural-parametric model and transfer functions of electroelastic actuator for nano- and microdisplacement. Parinov IA, editors. Chapter 9 in Piezoelectrics and Nanomaterials: Fundamentals, Developments and Applications. Nova Science: New York; 2015:225–242.
17. Afonin SM. Structural-parametric model electromagnetoelastic actuator nanodisplacement for mechatronics. International Journal of Physics. 2016;5(1):9–15.
18. Afonin SM. Structural-parametric model multilayer electromagnetoelastic actuator for nanomechatronics. International Journal of Physics. 2019;7(2):50–57.
19. Afonin SM. Rigidity of a multilayer piezoelectric actuator for the nano and micro range. Russian Engineering Research. 2021;41(4):285–288.
20. Afonin SM. Structural scheme of an electromagnetoelastic actuator for nanotechnology research. Springer Proceedings in Materials Physics and Mechanics of New Materials and Their Applications. 2024;41:486–501.
21. Afonin SM. Deformation of electromagnetoelastic actuator for nano robotics system. Int Rob Auto J. 2020;6(2):84–86.
22. Afonin SM. A multi-layer electro elastic drive for micro and nano robotics. Int Rob Auto J. 2024;10(2):73–76.
23. Afonin SM. Electromagnetoelastic actuator for large telescopes. Aeron Aero Open Access J. 2018;2(5):270–272.
24. Afonin SM. Piezoactuator of nanodisplacement for astrophysics. Aeron Aero Open Access J. 2022;6(4):155–158.
25. Afonin SM. Condition absolute stability of system with nano piezoactuator for astrophysics research. Aeron Aero Open Access J. 2023;7(3):99–102.
26. Afonin SM. Piezoengine for nanomedicine and applied bionics. MOJ App Bio Biomech. 2022;6(1):30–33.
27. Afonin SM. System with nano piezoengine under randomly influences for biomechanics. MOJ App Bio Biomech. 2024;8(1):1–3.
28. Afonin SM. DAC electro elastic engine for nanomedicine. MOJ App Bio Biomech. 2024;8(1):38–40.
29. Afonin SM. Characteristics of an electroelastic actuator nano- and microdisplacement for nanotechnology. Bartul Z, Trenor J, editors. Nova Science. Chapter 8 in Advances in Nanotechnology. Volume 25. New York; 2021:251-266.
30. Afonin SM. Structural model of nano piezoengine for applied biomechanics and biosciencess. MOJ Applied Bionics and Biomechanics. 2023;7(1):21–25.
31. Afonin SM. Multilayer piezo engine for nanomedicine research. MOJ App Bio Biomech. 2020;4(2):30–31.
32. Afonin SM. Characteristics electroelastic engine for nanobiomechanics. MOJ App Bio Biomech. 2020;4(3):51–53.
33. Afonin SM. Piezo actuators for nanomedicine research. MOJ App Bio Biomech. 2019;3(2):56–57.
34. Afonin SM. Structural scheme of piezoactuator for astrophysics. Phys Astron Int J. 2024;8(1):32‒36.
35. Afonin SM. Nanopiezoactuator for astrophysics equipment. Phys Astron Int J. 2023;7(2):153–155.
36. Afonin SM. Electroelastic actuator of nanomechatronics systems for nanoscience. Chapter 2 in recent progress in chemical science research. Min HS, editors. B P International: India; UK, London, 2023;6:15–27.
37. Afonin SM. Structural scheme of electroelastic actuator for nanomechatronics. Chapter 40 in Advanced Materials. Proceedings of the International Conference on Physics and Mechanics of New Materials and Their Applications, PHENMA 2019. Ivan A Parinov, Shun-Hsyung C, Banh Tien L, editors. Springer Nature: Switzerland, Cham; 2020:487–502.     
38. Afonin SM. Absolute stability of control system for deformation of electromagnetoelastic actuator under random impacts in nanoresearch. Chapter 43 in Physics and Mechanics of New Materials and Their Applications, PHENMA 2020. Parinov IA, Chang SH, Kim YH, et al., editors. Springer Proceedings in Materials. Volume 10.  Springer: Switzerland, Cham; 2021;519–531.
39. 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. Parinov IA, Chang SH, Soloviev AN, editors. Springer Proceedings in Materials series. Vol. 20: Springer, Cham; 2023:419–428.
40. Afonin SM. Structural parametric model and diagram of electromagnetoelastic actuator for nanodisplacement in chemistry and biochemistry research. Chapter 7 in Current Topics on Chemistry and Biochemistry. Baena OJR, editor. Vol. 9, B P International: India, UK; 2023:77–95.
41. Afonin SM. Structural-parametric models of electromagnetoelastic actuators of nano- and microdisplacement for robotics and mechatronics systems. In Proceedings of the 2017 IEEE Conference of Russian Young Researchers in Electrical and Electronic Engineering (EIConRus): St. Petersburg and Moscow, Russia; 2017:769–773.
42. Afonin SM. Multilayer electromagnetoelastic actuator for robotics systems of nanotechnology. In Proceedings of the 2018 IEEE Conference of Russian Young Researchers in Electrical and Electronic Engineering (EIConRus): Moscow and St. Petersburg, Russia; pp. 2018:1698–1701.
43. Afonin SM. Digital analog electro elastic converter actuator for nanoresearch. In Proceedings of the 2020 IEEE Conference of Russian Young Researchers in Electrical and Electronic Engineering (EIConRus): St. Petersburg and Moscow, Russia; 2020:2332–2335.
44. Afonin SM. Frequency method for determination self-oscillations in control systems with a piezo actuator for astrophysical research. Aeron Aero Open Access J. 2024;8(2):115–117.
45. Afonin SM. Parallel and coded control of multi layered longitudinal piezo engine for nano biomedical research. MOJ App Bio Biomech. 2024;8(1):62–65.
46. Afonin SM. Structural model of a nano piezoelectric actuator for nanotechnology. Russian Engineering Research. 2024;44(1):14–19.
47. Afonin SM. Electroelastic actuators for nano- and microdisplacement. SCIREA Journal of Physics. 2018;3(2):81–91.
48. Afonin SM. Structural-parametric models of electromagnetoelastic actuators for nano- and micromanipulators of mechatronic systems.  SCIREA Journal of Mechanics. 2016;1(2):64–80.
49. Afonin SM. Structural-parametric model of electro elastic actuator for nanotechnology and biotechnology. Journal of Pharmacy and Pharmaceutics. 2018;5(1):8–12.
50. Afonin SM. Electroelastic actuator nano- and microdisplacement for precision mechanics. American Journal of Mechanics and Applications. 2018;6(1):17–22.
51. Afonin SM. Structural-parametric models and transfer functions of electromagnetoelastic actuators nano- and microdisplacement for mechatronic systems. International Journal of Theoretical and Applied Mathematics. 2016;2(2):52–59.
52. Afonin SM. Decision wave equation and block diagram of electro magneto elastic actuator nano - and micro displacement for communications systems. International Journal of Information and Communication Sciences. 2016;1(2):22–29.
53. Schultz J, Ueda J, Asada H. Cellular Actuators. Butterworth-Heinemann Publisher, Oxford; 2017. 382 p.
54. Nalwa HS. Encyclopedia of Nanoscience and Nanotechnology. Los Angeles: American Scientific Publishers. 2004;10.