Skip to main content
SpringerLink
Log in

Computer Analysis of a Model of a Synchronous No-Current Electric Motor

  • MATHEMATICS
  • Published:
Vestnik St. Petersburg University, Mathematics Aims and scope Submit manuscript

Abstract

We consider a simplified model of a synchronous electric motor, which is described by a second-order differential equation, not containing electrical currents. F. Tricomi found that the phase portrait of this equation refers to one of three types, depending on whether the damping coefficient it contains is greater than, less than or equal to some critical value. Since an explicit expression for the critical value is not available, the efforts of many mathematicians were focused on deriving explicit upper and lower analytical estimates for this value. We use a computer to obtain phase portraits of this equation; the features of its phase trajectories are revealed, which are difficult to notice in known phase portraits obtained by analytical methods. Computer calculations are used to plot the critical value of the damping factor in this equation as a function of the main steady-state value of the angular variable. Linear and sinusoidal approximations of this curve are proposed, and the absolute and relative errors of such approximations are computed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1.
Fig. 2.
Fig. 3.
Fig. 4.
Fig. 5.
Fig. 6.
Fig. 7.

Similar content being viewed by others

REFERENCES

  1. G. A. Leonov, “Phase synchronization. Theory and applications,” Avtom. Telemekh. 10, 47–85 (2006).

    MathSciNet  Google Scholar 

  2. G. A. Leonov, “Lyapunov’s second method in the theory of phase synchronization,” Prikl. Mat. Mekh. 40, 238–244 (1976).

    Google Scholar 

  3. A. Kh. Gelig, G. A. Leonov, and V. A. Yakubovich, Stability of Nonlinear Systems with a Nonunique Equilibrium State (Nauka, Moscow, 1978) [in Russian].

    MATH  Google Scholar 

  4. B. I. Konosevich and Yu. B. Konosevich, “Sufficient global stability condition for a model of the synchronous electric motor under nonlinear load moment,” Vestn. St. Petersburg Univ.: Math. 51, 57–65 (2018). https://doi.org/10.3103/S1063454118010053

    Article  MathSciNet  MATH  Google Scholar 

  5. F. Tricomi, “Integrazione di unequazione differenziale presentasi in electrotechnica,” Ann. Della Roma Schuola Norm. Super. Pisa 2 (2), 1–20 (1933).

    Google Scholar 

  6. E. A. Barbashin and V. A. Tabueva, Dynamical Systems with Cylindrical Phase Space (Nauka, Moscow, 1969) [in Russian].

    MATH  Google Scholar 

  7. D. M. Klimov and S. A. Kharlamov, Dynamics of a Gyroscope with a Gimbal Suspension (Nauka, Moscow, 1978) [in Russian].

    Google Scholar 

  8. B. I. Konosevich and Yu. B. Konosevich, “On stability of steady-state motions of a gimbals mounted gyroscope supplied with the electric motor,” Mech. Solids 48, 285–297 (2013). https://doi.org/10.3103/S0025654413030059

    Article  Google Scholar 

  9. B. I. Konosevich and Yu. B. Konosevich, “Stability criterion for stationary solutions of multi-current model equations for a synchronous gimbal-mounted gyroscope. Part 1,” Mech. Solids 55, 258–272 (2020). https://doi.org/10.3103/S0025654420020119

    Article  Google Scholar 

  10. B. I. Konosevich and Yu. B. Konosevich, “Stability criterion for stationary solutions of the equations of a multi-current model of a synchronous gyroscope in a gimbal. 2,” Mech. Solids 56, 40–54 (2021). https://doi.org/10.3103/S0025654421010088

    Article  Google Scholar 

  11. G. A. Leonov and A. M. Zaretskiy, “Global stability and oscillations of dynamical systems describing synchronous electrical machines,” Vestn. St. Petersburg Univ.: Math. 45, 157–163 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  12. B. I. Konosevich and Yu. B. Konosevich, “Approximation of the critical value of the damping parameter for the synchronous electric motor,” Tr. Inst. Prikl. Mat. Mekh. 29, 121–126 (2014).

    MATH  Google Scholar 

  13. V. G. Karmanov, Mathematical Programming, 2nd ed. (Nauka, Moscow, 1980; Mir, Moscow, 1989).

  14. L. Amerio, “Determinazione delle condizioni di stabilit`a per gli integrali di un’equazione interessante l’electrotecnica,” Ann. Mat. Pura Appl. 2, 75–90 (1949).

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Applied Mathematics and Mechanics, 283050, Donetsk, Russia

    B. I. Konosevich & Yu. B. Konosevich

Authors
  1. B. I. Konosevich
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yu. B. Konosevich
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to B. I. Konosevich or Yu. B. Konosevich.

Ethics declarations

The authors declare that they have no conflicts of interest.

Additional information

Translated by M. Shmatikov

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Konosevich, B.I., Konosevich, Y.B. Computer Analysis of a Model of a Synchronous No-Current Electric Motor. Vestnik St.Petersb. Univ.Math. 56, 350–361 (2023). https://doi.org/10.1134/S1063454123030056

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1063454123030056

Keywords:

Search

Navigation