Energy-Preserving/Group-Preserving Schemes for Depicting Nonlinear Vibrations of Multi-Coupled Duffing Oscillators
In the paper, we first develop a novel automatically energy-preserving scheme (AEPS) for the undamped and unforced single and multi-coupled Duffing equations by recasting them to the Lie-type systems of ordinary differential equations. The AEPS can automatically preserve the energy to be a constant...
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| Format: | Article |
| Language: | English |
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MDPI AG
2024-01-01
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| Series: | Vibration |
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| Online Access: | https://www.mdpi.com/2571-631X/7/1/6 |
| _version_ | 1797239128410030080 |
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| author | Chein-Shan Liu Chung-Lun Kuo Chih-Wen Chang |
| author_facet | Chein-Shan Liu Chung-Lun Kuo Chih-Wen Chang |
| author_sort | Chein-Shan Liu |
| collection | DOAJ |
| description | In the paper, we first develop a novel automatically energy-preserving scheme (AEPS) for the undamped and unforced single and multi-coupled Duffing equations by recasting them to the Lie-type systems of ordinary differential equations. The AEPS can automatically preserve the energy to be a constant value in a long-term free vibration behavior. The analytical solution of a special Duffing–van der Pol equation is compared with that computed by the novel group-preserving scheme (GPS) which has fourth-order accuracy. The main novelty is that we constructed the quadratic forms of the energy equations, the Lie-algebras and Lie-groups for the multi-coupled Duffing oscillator system. Then, we extend the GPS to the damped and forced Duffing equations. The corresponding algorithms are developed, which are effective to depict the long term nonlinear vibration behaviors of the multi-coupled Duffing oscillators with an accuracy of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">O</mi><mo>(</mo><msup><mi>h</mi><mn>4</mn></msup><mo>)</mo></mrow></semantics></math></inline-formula> for a small time stepsize <i>h</i>. |
| first_indexed | 2024-04-24T17:46:37Z |
| format | Article |
| id | doaj.art-d5002919d4e842dd86420e5b5c865f9c |
| institution | Directory Open Access Journal |
| issn | 2571-631X |
| language | English |
| last_indexed | 2024-04-24T17:46:37Z |
| publishDate | 2024-01-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Vibration |
| spelling | doaj.art-d5002919d4e842dd86420e5b5c865f9c2024-03-27T14:07:27ZengMDPI AGVibration2571-631X2024-01-01719812810.3390/vibration7010006Energy-Preserving/Group-Preserving Schemes for Depicting Nonlinear Vibrations of Multi-Coupled Duffing OscillatorsChein-Shan Liu0Chung-Lun Kuo1Chih-Wen Chang2Center of Excellence for Ocean Engineering, National Taiwan Ocean University, Keelung 202301, TaiwanCenter of Excellence for Ocean Engineering, National Taiwan Ocean University, Keelung 202301, TaiwanDepartment of Mechanical Engineering, National United University, Miaoli 360302, TaiwanIn the paper, we first develop a novel automatically energy-preserving scheme (AEPS) for the undamped and unforced single and multi-coupled Duffing equations by recasting them to the Lie-type systems of ordinary differential equations. The AEPS can automatically preserve the energy to be a constant value in a long-term free vibration behavior. The analytical solution of a special Duffing–van der Pol equation is compared with that computed by the novel group-preserving scheme (GPS) which has fourth-order accuracy. The main novelty is that we constructed the quadratic forms of the energy equations, the Lie-algebras and Lie-groups for the multi-coupled Duffing oscillator system. Then, we extend the GPS to the damped and forced Duffing equations. The corresponding algorithms are developed, which are effective to depict the long term nonlinear vibration behaviors of the multi-coupled Duffing oscillators with an accuracy of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">O</mi><mo>(</mo><msup><mi>h</mi><mn>4</mn></msup><mo>)</mo></mrow></semantics></math></inline-formula> for a small time stepsize <i>h</i>.https://www.mdpi.com/2571-631X/7/1/6multi-coupled Duffing equationsDuffing–van der Pol equationautomatically energy-preserving schemegroup-preserving schemenonlinear vibration |
| spellingShingle | Chein-Shan Liu Chung-Lun Kuo Chih-Wen Chang Energy-Preserving/Group-Preserving Schemes for Depicting Nonlinear Vibrations of Multi-Coupled Duffing Oscillators Vibration multi-coupled Duffing equations Duffing–van der Pol equation automatically energy-preserving scheme group-preserving scheme nonlinear vibration |
| title | Energy-Preserving/Group-Preserving Schemes for Depicting Nonlinear Vibrations of Multi-Coupled Duffing Oscillators |
| title_full | Energy-Preserving/Group-Preserving Schemes for Depicting Nonlinear Vibrations of Multi-Coupled Duffing Oscillators |
| title_fullStr | Energy-Preserving/Group-Preserving Schemes for Depicting Nonlinear Vibrations of Multi-Coupled Duffing Oscillators |
| title_full_unstemmed | Energy-Preserving/Group-Preserving Schemes for Depicting Nonlinear Vibrations of Multi-Coupled Duffing Oscillators |
| title_short | Energy-Preserving/Group-Preserving Schemes for Depicting Nonlinear Vibrations of Multi-Coupled Duffing Oscillators |
| title_sort | energy preserving group preserving schemes for depicting nonlinear vibrations of multi coupled duffing oscillators |
| topic | multi-coupled Duffing equations Duffing–van der Pol equation automatically energy-preserving scheme group-preserving scheme nonlinear vibration |
| url | https://www.mdpi.com/2571-631X/7/1/6 |
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