On the construction of stable periodic solutions for the dynamical motion of AC machines

Mohamed El-Borhamy; Essam Eddin M. Rashad; Arafa A. Nasef; Ismail Sobhy; Samah M. Elkholy; Mohamed El-Borhamy; Essam Eddin M. Rashad; Arafa A. Nasef; Ismail Sobhy; Samah M. Elkholy

doi:10.3934/math.2023446

AIMS Mathematics

2023, Volume 8, Issue 4: 8902-8927. doi: 10.3934/math.2023446

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On the construction of stable periodic solutions for the dynamical motion of AC machines

1.
Department of Engineering Mathematics and Physics, Faculty of Engineering, University of Tanta, Tanta 31527, Egypt
2.
Department of Basic Sciences, Faculty of Engineering, Horus University, New Damietta, Egypt
3.
Department of Electrical Power and Machines Engineering, Faculty of Engineering, University of Tanta, Tanta 31527, Egypt
4.
Department of Engineering Mathematics and Physics, Faculty of Engineering, University of Kafrelsheikh, Kafer Elsheikh 33516, Egypt

Received: 19 September 2022 Revised: 14 January 2023 Accepted: 19 January 2023 Published: 09 February 2023
MSC : 34A30, 34C25, 34D20, 70K60

This article discusses the stability of periodic responses for the dynamical motion of AC machines from the perspective of Lyapunov function approach. The dynamical motion of AC machines is prototypically modeled as an equivalent linear RLC series circuit with time-variant inductance represented by a linear differential equation with periodic coefficients. Based on the deduced stability conditions, some special identities among the equivalent circuit parameters to ensure the stability of responses and their periodic structures are concluded. Through these conditions, the periodic structure of responses is obtained by using the method of strained parameters. Through a comparison with the experimental results from the specialized practical literatures, a strong agreement with the obtained analytical results is achieved. In addition, from a practical point of views, some future points within the discussion are raised to improve the mathematical modeling of AC machines to obtain a better model and simulation.
- linear differential equations,
- stability theory,
- periodic solutions,
- perturbation techniques
Citation: Mohamed El-Borhamy, Essam Eddin M. Rashad, Arafa A. Nasef, Ismail Sobhy, Samah M. Elkholy. On the construction of stable periodic solutions for the dynamical motion of AC machines[J]. AIMS Mathematics, 2023, 8(4): 8902-8927. doi: 10.3934/math.2023446

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Abstract

This article discusses the stability of periodic responses for the dynamical motion of AC machines from the perspective of Lyapunov function approach. The dynamical motion of AC machines is prototypically modeled as an equivalent linear RLC series circuit with time-variant inductance represented by a linear differential equation with periodic coefficients. Based on the deduced stability conditions, some special identities among the equivalent circuit parameters to ensure the stability of responses and their periodic structures are concluded. Through these conditions, the periodic structure of responses is obtained by using the method of strained parameters. Through a comparison with the experimental results from the specialized practical literatures, a strong agreement with the obtained analytical results is achieved. In addition, from a practical point of views, some future points within the discussion are raised to improve the mathematical modeling of AC machines to obtain a better model and simulation.

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