Energy-Preserving/Group-Preserving Schemes for Depicting Nonlinear Vibrations of Multi-Coupled Duffing Oscillators

Liu, Chein-Shan; Kuo, Chung-Lun; Chang, Chih-Wen

doi:10.3390/vibration7010006

Open AccessArticle

Energy-Preserving/Group-Preserving Schemes for Depicting Nonlinear Vibrations of Multi-Coupled Duffing Oscillators

by

Chein-Shan Liu

¹

,

Chung-Lun Kuo

¹ and

Chih-Wen Chang

^2,*

¹

Center of Excellence for Ocean Engineering, National Taiwan Ocean University, Keelung 202301, Taiwan

²

Department of Mechanical Engineering, National United University, Miaoli 360302, Taiwan

^*

Author to whom correspondence should be addressed.

Vibration 2024, 7(1), 98-128; https://doi.org/10.3390/vibration7010006

Submission received: 21 November 2023 / Revised: 18 December 2023 / Accepted: 12 January 2024 / Published: 18 January 2024

(This article belongs to the Special Issue Feature Papers in Vibration)

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Abstract

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

O (h^{4})

for a small time stepsize h.

Keywords: multi-coupled Duffing equations; Duffing–van der Pol equation; automatically energy-preserving scheme; group-preserving scheme; nonlinear vibration

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MDPI and ACS Style

Liu, C.-S.; Kuo, C.-L.; Chang, C.-W. Energy-Preserving/Group-Preserving Schemes for Depicting Nonlinear Vibrations of Multi-Coupled Duffing Oscillators. Vibration 2024, 7, 98-128. https://doi.org/10.3390/vibration7010006

AMA Style

Liu C-S, Kuo C-L, Chang C-W. Energy-Preserving/Group-Preserving Schemes for Depicting Nonlinear Vibrations of Multi-Coupled Duffing Oscillators. Vibration. 2024; 7(1):98-128. https://doi.org/10.3390/vibration7010006

Chicago/Turabian Style

Liu, Chein-Shan, Chung-Lun Kuo, and Chih-Wen Chang. 2024. "Energy-Preserving/Group-Preserving Schemes for Depicting Nonlinear Vibrations of Multi-Coupled Duffing Oscillators" Vibration 7, no. 1: 98-128. https://doi.org/10.3390/vibration7010006

APA Style

Liu, C.-S., Kuo, C.-L., & Chang, C.-W. (2024). Energy-Preserving/Group-Preserving Schemes for Depicting Nonlinear Vibrations of Multi-Coupled Duffing Oscillators. Vibration, 7(1), 98-128. https://doi.org/10.3390/vibration7010006

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Energy-Preserving/Group-Preserving Schemes for Depicting Nonlinear Vibrations of Multi-Coupled Duffing Oscillators

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