The Principal Parametric Resonance of Coupled van der Pol Oscillators under Feedback Control
The principal parametric resonance of two van der Pol oscillators under coupled position and velocity feedback control with time delay is investigated analytically and numerically on the assumption that only one of the two oscillators is parametrically excited and the feedback control is linear. The...
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| Format: | Article |
| Language: | English |
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Wiley
2012-01-01
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| Series: | Shock and Vibration |
| Online Access: | http://dx.doi.org/10.3233/SAV-2010-0636 |
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| _version_ | 1832562746144587776 |
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| author | Xinye Li Huabiao Zhang Lijuan He |
| author_facet | Xinye Li Huabiao Zhang Lijuan He |
| author_sort | Xinye Li |
| collection | DOAJ |
| description | The principal parametric resonance of two van der Pol oscillators under coupled position and velocity feedback control with time delay is investigated analytically and numerically on the assumption that only one of the two oscillators is parametrically excited and the feedback control is linear. The slow-flow equations are obtained by the averaging method and simplified by truncating the first term of Taylor expansions for those terms with time delay. It is found that nontrivial solutions corresponding to periodic motions exist only for one oscillator if no feedback control is applied although the two oscillators are nonlinearly coupled. Based on Levenberg-Marquardt method, the effects of excitation and control parameters on the amplitude of periodic solutions of the system are graphically given. It can be seen that both of the two oscillators can be excited in periodic vibration with proper feedback. However, the amplitudes of the periodic vibrations are independent of the sign of feedback gains. In addition, the influence of time delay on the response of the system is periodic. In terms of numerical simulations, it is shown that both of the two oscillators can also have quasi-periodic motions, periodic motions about a new equilibrium position and other complex motions such as relaxation oscillation when feedback control is considered. |
| format | Article |
| id | doaj-art-7fffc4d5108c4ede8074d7853dde8a3e |
| institution | Kabale University |
| issn | 1070-9622 1875-9203 |
| language | English |
| publishDate | 2012-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Shock and Vibration |
| spelling | doaj-art-7fffc4d5108c4ede8074d7853dde8a3e2025-02-03T01:21:56ZengWileyShock and Vibration1070-96221875-92032012-01-0119336537710.3233/SAV-2010-0636The Principal Parametric Resonance of Coupled van der Pol Oscillators under Feedback ControlXinye Li0Huabiao Zhang1Lijuan He2School of Mechanical Engineering, Hebei University of Technology, Tianjin, ChinaSchool of Astronautics, Harbin Institute of Technology, Harbin, ChinaSchool of Mechanical Engineering, Tianjin University of Science and Technology, Tianjin, ChinaThe principal parametric resonance of two van der Pol oscillators under coupled position and velocity feedback control with time delay is investigated analytically and numerically on the assumption that only one of the two oscillators is parametrically excited and the feedback control is linear. The slow-flow equations are obtained by the averaging method and simplified by truncating the first term of Taylor expansions for those terms with time delay. It is found that nontrivial solutions corresponding to periodic motions exist only for one oscillator if no feedback control is applied although the two oscillators are nonlinearly coupled. Based on Levenberg-Marquardt method, the effects of excitation and control parameters on the amplitude of periodic solutions of the system are graphically given. It can be seen that both of the two oscillators can be excited in periodic vibration with proper feedback. However, the amplitudes of the periodic vibrations are independent of the sign of feedback gains. In addition, the influence of time delay on the response of the system is periodic. In terms of numerical simulations, it is shown that both of the two oscillators can also have quasi-periodic motions, periodic motions about a new equilibrium position and other complex motions such as relaxation oscillation when feedback control is considered.http://dx.doi.org/10.3233/SAV-2010-0636 |
| spellingShingle | Xinye Li Huabiao Zhang Lijuan He The Principal Parametric Resonance of Coupled van der Pol Oscillators under Feedback Control Shock and Vibration |
| title | The Principal Parametric Resonance of Coupled van der Pol Oscillators under Feedback Control |
| title_full | The Principal Parametric Resonance of Coupled van der Pol Oscillators under Feedback Control |
| title_fullStr | The Principal Parametric Resonance of Coupled van der Pol Oscillators under Feedback Control |
| title_full_unstemmed | The Principal Parametric Resonance of Coupled van der Pol Oscillators under Feedback Control |
| title_short | The Principal Parametric Resonance of Coupled van der Pol Oscillators under Feedback Control |
| title_sort | principal parametric resonance of coupled van der pol oscillators under feedback control |
| url | http://dx.doi.org/10.3233/SAV-2010-0636 |
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