Energy-Preserving/Group-Preserving Schemes for Depicting Nonlinear Vibrations of Multi-Coupled Duffing Oscillators
Abstract
:1. Introduction
- For the single, two-coupled and three-coupled undamped and unforced Duffing equations, novel methods to automatically preserve energy were developed.
- Detailed formulations of energy invariants, variable transformations, Lie algebras and Lie groups used in long-term computations of nonlinear free vibrations were derived.
- For the damped and unforced Duffing equations, group-preserving schemes were developed at the first time.
- Highly accurate solutions of responses were obtained.
2. An Automatically Energy-Preserving Scheme
2.1. Lie-Group for
- (i)
- We give , h, and .
- (ii)
- For ,
- (iii)
- We compute
2.2. Lie-Group for
2.3. Testing the Efficiency of AEPS
2.4. General Setting
- (i)
- We give , , , step size h, and a final time .
- (ii)
- For , , we predict, by an Euler step,
- (iii)
- We compute
3. The GPS for Equation (1)
4. The GPS for a Duffing–van der Pol Oscillator
5. Two Coupled Duffing Equations
5.1. Hamiltonian Form
5.2. Lie-Type Forms
5.3. Automatically Energy-Preserving Scheme
- (i)
- We give , h, and initial values, and compute by Equations (67)–(70).
- (ii)
- We perform for
- (iii)
- We iteratively solve the new by
5.4. Group-Preserving Scheme for Damped and Forced System
- (i)
- We give , initial values at initial time and time stepsize h, and compute the initial values of by Equations (67)–(70).
- (ii)
- For , we repeat
- (iii)
- The new is iterated by
6. Three Coupled Duffing Equations
7. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Appendix A
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t | Exact | DTM | GPS |
---|---|---|---|
0.1 | −0.2769871994 | −0.27743 | −0.2769872174 |
0.2 | −0.2659399328 | −0.26759 | −0.2659399639 |
0.3 | −0.2554793822 | −0.259 | −0.2554794220 |
0.4 | −0.2455581425 | −0.2515 | −0.2455581874 |
0.5 | −0.2361343462 | −0.24495 | −0.2361343933 |
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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
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 StyleLiu, 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