Application of Fractional-Order Calculus to Improve the Mathematical Model of a Two-Mass System with a Long Shaft
Based on the general theory of fractional order derivatives and integrals, application of the Caputo–Fabrizio operator is analyzed to improve a mathematical model of a two-mass system with a long shaft and concentrated parameters. Thus, the real transmission of complex electric drives, which consist...
| Main Authors: | , , , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2021-03-01
|
| Series: | Energies |
| Subjects: | |
| Online Access: | https://www.mdpi.com/1996-1073/14/7/1854 |
| _version_ | 1797539920320921600 |
|---|---|
| author | Andriy Lozynskyy Andriy Chaban Tomasz Perzyński Andrzej Szafraniec Lidiia Kasha |
| author_facet | Andriy Lozynskyy Andriy Chaban Tomasz Perzyński Andrzej Szafraniec Lidiia Kasha |
| author_sort | Andriy Lozynskyy |
| collection | DOAJ |
| description | Based on the general theory of fractional order derivatives and integrals, application of the Caputo–Fabrizio operator is analyzed to improve a mathematical model of a two-mass system with a long shaft and concentrated parameters. Thus, the real transmission of complex electric drives, which consist of long shafts with a sufficient degree of adequacy, is presented as a two-mass system. Such a system is described by ordinary fractional order differential equations. In addition, it is well known that an elastic mechanical wave, propagating along a drive transmission with a long stiff shaft, creates a retardation effect on distribution of the time–space angular velocity, the rotation angle of the shaft, and its elastic moment. The approach proposed in the current work helps to take in account the moving elastic wave along the shaft of electric drive mechanism. On this basis, it is demonstrated that the use of the fractional order integrator in the model for the elastic moment enables it to reproduce real transient processes in the joint coordinates of the system. It also provides an accuracy equivalent to the model with distributed parameters. The distance between the traditional model and the model in which the fractional integral is used for the elastic moment modelling in a two-mass system, with a long shaft, is analyzed. |
| first_indexed | 2024-03-10T12:52:40Z |
| format | Article |
| id | doaj.art-dd75d538a6bf4b7e98382e401475cedc |
| institution | Directory Open Access Journal |
| issn | 1996-1073 |
| language | English |
| last_indexed | 2024-03-10T12:52:40Z |
| publishDate | 2021-03-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Energies |
| spelling | doaj.art-dd75d538a6bf4b7e98382e401475cedc2023-11-21T13:00:09ZengMDPI AGEnergies1996-10732021-03-01147185410.3390/en14071854Application of Fractional-Order Calculus to Improve the Mathematical Model of a Two-Mass System with a Long ShaftAndriy Lozynskyy0Andriy Chaban1Tomasz Perzyński2Andrzej Szafraniec3Lidiia Kasha4Institute of Power Engineering and Control Systems, Lviv Polytechnic National University, 79-013 Lviv, UkraineFaculty of Transport, Electrical Engineering and Computer Science, University of Technology and Humanities in Radom, 26-600 Radom, PolandFaculty of Transport, Electrical Engineering and Computer Science, University of Technology and Humanities in Radom, 26-600 Radom, PolandFaculty of Transport, Electrical Engineering and Computer Science, University of Technology and Humanities in Radom, 26-600 Radom, PolandInstitute of Power Engineering and Control Systems, Lviv Polytechnic National University, 79-013 Lviv, UkraineBased on the general theory of fractional order derivatives and integrals, application of the Caputo–Fabrizio operator is analyzed to improve a mathematical model of a two-mass system with a long shaft and concentrated parameters. Thus, the real transmission of complex electric drives, which consist of long shafts with a sufficient degree of adequacy, is presented as a two-mass system. Such a system is described by ordinary fractional order differential equations. In addition, it is well known that an elastic mechanical wave, propagating along a drive transmission with a long stiff shaft, creates a retardation effect on distribution of the time–space angular velocity, the rotation angle of the shaft, and its elastic moment. The approach proposed in the current work helps to take in account the moving elastic wave along the shaft of electric drive mechanism. On this basis, it is demonstrated that the use of the fractional order integrator in the model for the elastic moment enables it to reproduce real transient processes in the joint coordinates of the system. It also provides an accuracy equivalent to the model with distributed parameters. The distance between the traditional model and the model in which the fractional integral is used for the elastic moment modelling in a two-mass system, with a long shaft, is analyzed.https://www.mdpi.com/1996-1073/14/7/1854electric drivefractional calculustwo-mass systemCaputo-Fabrizio operatormathematical modelsynthesis of control system |
| spellingShingle | Andriy Lozynskyy Andriy Chaban Tomasz Perzyński Andrzej Szafraniec Lidiia Kasha Application of Fractional-Order Calculus to Improve the Mathematical Model of a Two-Mass System with a Long Shaft Energies electric drive fractional calculus two-mass system Caputo-Fabrizio operator mathematical model synthesis of control system |
| title | Application of Fractional-Order Calculus to Improve the Mathematical Model of a Two-Mass System with a Long Shaft |
| title_full | Application of Fractional-Order Calculus to Improve the Mathematical Model of a Two-Mass System with a Long Shaft |
| title_fullStr | Application of Fractional-Order Calculus to Improve the Mathematical Model of a Two-Mass System with a Long Shaft |
| title_full_unstemmed | Application of Fractional-Order Calculus to Improve the Mathematical Model of a Two-Mass System with a Long Shaft |
| title_short | Application of Fractional-Order Calculus to Improve the Mathematical Model of a Two-Mass System with a Long Shaft |
| title_sort | application of fractional order calculus to improve the mathematical model of a two mass system with a long shaft |
| topic | electric drive fractional calculus two-mass system Caputo-Fabrizio operator mathematical model synthesis of control system |
| url | https://www.mdpi.com/1996-1073/14/7/1854 |
| work_keys_str_mv | AT andriylozynskyy applicationoffractionalordercalculustoimprovethemathematicalmodelofatwomasssystemwithalongshaft AT andriychaban applicationoffractionalordercalculustoimprovethemathematicalmodelofatwomasssystemwithalongshaft AT tomaszperzynski applicationoffractionalordercalculustoimprovethemathematicalmodelofatwomasssystemwithalongshaft AT andrzejszafraniec applicationoffractionalordercalculustoimprovethemathematicalmodelofatwomasssystemwithalongshaft AT lidiiakasha applicationoffractionalordercalculustoimprovethemathematicalmodelofatwomasssystemwithalongshaft |