Investigation of Resonance Modes in Iced Transmission Lines Using Two Discrete Methods
Abstract
1. Introduction
2. Galloping Equation
2.1. Mathematical Model
2.2. Discrete Method I
2.3. Discrete Method II
3. The Finite Element Method
4. Multiscale Method
5. Different Resonance Analysis
5.1. The Primary Resonance
5.2. The Harmonic Resonance
5.2.1. The 1/2 Subharmonic Resonance
5.2.2. The 1/3 Subharmonic Resonance
5.2.3. The Two Superharmonic Resonance
5.2.4. The Three Superharmonic Resonance
5.3. The Internal Resonance
5.3.1. The 2:1:2 Internal Resonance
5.3.2. The 1:1:1 Internal Resonance
6. Conclusions
- (1)
- For primary resonance, the galloping characteristics are significantly influenced by the two discrete methods. As the excitation amplitude increases, the response amplitude also increases, and the amplitude-frequency response exhibits pronounced geometric nonlinearity. As the detuning parameter increases, the response amplitude transitions from single-valued to multi-valued, leading to unstable nonlinear vibrations.
- (2)
- For harmonic resonance, the galloping characteristics across the four types of harmonic resonance are similar. In the amplitude-frequency response, when the excitation amplitude is too small, superharmonic resonance exhibits no geometric nonlinearity, and the system is more prone to 1/2 subharmonic and two superharmonic resonances. The excitation amplitude corresponding to primary resonance is significantly smaller than that of harmonic resonances. Regarding the influence of the detuning parameter, except for the 1/3 subharmonic resonance response, where the response amplitude remains multi-valued, the remaining curves are consistent with primary resonance.
- (3)
- During the 2:1:2 internal resonance, the differences between methods have a greater impact on amplitude in the z-direction. For the 1:1:1 internal resonance, the differences between methods show a relatively minor influence on amplitudes in both the y- and z-directions, although the overall trends in the displacement response differ from those observed during primary resonance. As the excitation amplitude and detuning parameter increase, the overall characteristics of the curves resemble those of primary resonance, yet the amplitude of the internal resonance decreases.
- (4)
- When comparing the two discrete methods, the results from DMII are closer to the FEM due to its equivalent treatment of dynamic tension, which is more consistent with FEM principles. DMII yields larger galloping amplitudes and trajectories compared to DMI. Furthermore, the frequency and geometric nonlinearity of DMII are smaller than those of DMI. From the larger galloping response observed with DMII, it can be inferred that, for practical engineering applications, DMII is more conservative.
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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Parameter | Symbol | Value |
---|---|---|
Axial rigidity (N) | EA | 13.30 × 106 |
Horizontal component of tension (kN) | H | 21.73 |
Transmission lines diameter (mm) | d | 18.8 |
Span (m) | L | 125.9 |
Damping ratio in the y direction | ξy | 0.08 × 10−2 |
Damping ratio in the z direction | ξθ | 3.79 × 10−2 |
Mass per unit length (kg∙m−1) | m | 1.53 |
Order | DMI | DMII | FEM | |
---|---|---|---|---|
Natural frequency | 1st order | 0.6022 | 0.5559 | 0.5554 |
2nd order | 1.1373 | 0.9466 | 0.9464 | |
3rd order | 1.6866 | 1.4232 | 1.4232 | |
4th order | 2.2398 | 1.8933 | 1.8926 | |
5th order | 2.7945 | 2.3674 | 2.3658 | |
6th order | 3.0619 | 2.8402 | 2.8369 | |
7th order | 3.9062 | 3.3140 | 3.3085 | |
8th order | 4.4626 | 3.7873 | 3.7787 | |
9th order | 5.0193 | 4.2612 | 4.2486 |
Order | DMI | DMII | |
---|---|---|---|
Relative Error (%) | 1st order | 0.0842 | 0.0003 |
2nd order | 0.2017 | 0.0002 | |
3rd order | 0.1850 | 0 | |
4th order | 0.1834 | 0.0003 | |
5th order | 0.1812 | 0.0006 | |
6th order | 0.0793 | 0.0011 | |
7th order | 0.1806 | 0.0016 | |
8th order | 0.1809 | 0.0022 | |
9th order | 0.1814 | 0.0029 | |
Mean Relative Error (%) | 0.1457 | 0.0009 |
Coefficient | Value | Coefficient | Value |
---|---|---|---|
α1 | −0.2992 | β1 | −0.3290 |
α2 | 0.4362 | β2 | −0.9235 |
α3 | 0.2766 | β3 | 0.1930 |
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Chen, R.; Bao, W.; Cai, M. Investigation of Resonance Modes in Iced Transmission Lines Using Two Discrete Methods. Mathematics 2025, 13, 2376. https://doi.org/10.3390/math13152376
Chen R, Bao W, Cai M. Investigation of Resonance Modes in Iced Transmission Lines Using Two Discrete Methods. Mathematics. 2025; 13(15):2376. https://doi.org/10.3390/math13152376
Chicago/Turabian StyleChen, Rui, Wanyu Bao, and Mengqi Cai. 2025. "Investigation of Resonance Modes in Iced Transmission Lines Using Two Discrete Methods" Mathematics 13, no. 15: 2376. https://doi.org/10.3390/math13152376
APA StyleChen, R., Bao, W., & Cai, M. (2025). Investigation of Resonance Modes in Iced Transmission Lines Using Two Discrete Methods. Mathematics, 13(15), 2376. https://doi.org/10.3390/math13152376