Vibration Suppression of Two Adjacent Cables Using an Interconnected Tuned Mass Damper/Nonlinear Energy Sink
Abstract
:1. Introduction
2. Geometric Shape of the Stay Cable–Damper System under Self-Weight
2.1. System Description
2.2. Derivation of the Cable Shape
2.3. Numerical Example One
3. Dynamic Characteristics of the Cable–Damper System
Equations of Motion for Two Adjacent Cables Interconnected with an NES or a TMD
4. Optimal Design of the Damper
4.1. Optimal Problem Description
4.2. External Excitation Description
4.3. Estimation of Fundamental Frequency of the Two Cables
4.4. Initial Guess of Damping and Stiffness Coefficients
4.5. Optimization Result Analysis
5. Performance Analysis of the Optimal Damper
5.1. Robustness Analysis
5.1.1. Amplitude Robustness
Amplitude Robustness under Harmonic Excitation at the First-Mode Frequency
Amplitude Robustness under Swept-Sine Excitation (Excitation of the First Mode)
Amplitude Robustness under Harmonic Excitation at the Third-Mode Frequency
5.1.2. Frequency Robustness
5.2. Dynamic Response Characteristics
5.2.1. Time–History Analysis
Time–History Analysis under Harmonic Excitation at the First-Mode Frequency
Time–History Analysis under Swept-Sine Excitation (Excitation of the First Mode)
Time–History Analysis under Harmonic Excitation at the Third-Mode Frequency
5.2.2. Phase Diagram Analysis
Phase Diagram Analysis under Harmonic Excitation at the First-Mode Frequency
Phase Diagram Analysis under Swept-Sine Excitation (Excitation of the First Mode)
Phase Diagram Analysis under Harmonic Excitation at the Third-Mode Frequency
5.2.3. Spectral Characteristics
Spectral Analysis under Harmonic Excitation at the First-Mode Frequency
Spectral Analysis under Swept-Sine Excitation (Excitation of the First Mode)
Spectral Analysis under Harmonic Excitation at the Third-Mode Frequency
6. Conclusions
- (1)
- The damper mass has a non-negligible impact on the static cable shape.
- (2)
- It is shown that TMDs and NESs with appropriate damping and stiffness parameters can achieve satisfactory control performance by substantially reducing the vibration of two adjacent cables under a harmonic or swept-sine excitation targeting at the first or third mod. In particular, the NES performs slightly better under harmonic excitation of the first-mode frequency, and the TMD performs slightly better under swept-sine excitation targeting at the first mode, and under harmonic excitation of the third-mode frequency.
- (3)
- An NES exhibits better robustness to the variations in the amplitude of the external harmonic excitation of the first-mode frequency, while a TMD exhibits better robustness to the variations in the amplitude of swept-sine excitation targeting at the first mode. Both dampers exhibit good robustness to variations in the amplitude of the external harmonic excitation of the third-mode frequency. An NES shows better frequency robustness.
- (4)
- The steady-state time histories of the two adjacent cables with an optimal TMD are typically periodic responses of constant amplitude under a harmonic excitation of the first mode frequency. The phase diagram of the cables interconnected with an NES is more complicated than the cases of TMD and no damper. Additionally, the spectra show that the NES has an obvious frequency coupling capability and can shift the vibration frequency from one cable to another. Under swept-sine excitations, both the TMD and NES can effectively suppress energy input, and the TMD can form a stable phase difference region. It can be seen from the frequency domain results that the area enclosed by the spectral curve is substantially reduced, indicating a decrease in total vibration energy. Under harmonic excitations of the third-mode frequency, the two dampers have similar effects in that both can effectively suppress vibration, produce almost the same drag effect on the phase difference, and generate a vibration frequency shift between the two cables.
- (5)
- From dynamic characteristic analyses, some nonlinear dynamic behaviors of the system, such as multiple-frequency resonance, multiple-branch response, and different frequency components, can be identified. Coupling effects and drag motion phenomena, such as energy transfer (especially when the phase is opposite), phase diagram boundary rotation, frequency shift, and phase difference, can also be observed.
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A. Line Shape Coefficient of Stay Cables
Appendix B. EOM Coefficients of NES
Appendix C. EOM Coefficients of TMD
References
- Savor, Z.; Radic, J.; Hrelja, G. Cable vibrations at Dubrovnik bridge. Bridge Struct. 2006, 2, 97–106. [Google Scholar] [CrossRef]
- Ni, Y.Q.; Wang, X.Y.; Chen, Z.Q.; Ko, J.M. Field observations of rain-wind-induced cable vibration in cable-stayed Dongting Lake Bridge. J. Wind Eng. Ind. Aerodyn. 2007, 95, 303–328. [Google Scholar] [CrossRef]
- Chang, Y.; Luo, W.; Zhao, L.; Zou, Y.; Liu, Q.; Ge, Y. Influence of reduced drag load on stay cables on the construction cost of cable-stayed bridges: Two study cases. J. Wind Eng. Ind. Aerodyn. 2023, 242, 105584. [Google Scholar] [CrossRef]
- Tang, H.N.; Xu, W.J.; Yi, J. Seismic performance of cable-stayed bridges under large earthquake ground motions considering the loss of stay cables. Case Stud. Constr. Mater. 2024, 20, e02888. [Google Scholar] [CrossRef]
- Nicoletti, V.; Quarchioni, S.; Tentella, L.; Martini, R.; Gara, F. Experimental Tests and Numerical Analyses for the Dynamic Characterization of a Steel and Wooden Cable-Stayed Footbridge. Infrastructures 2023, 8, 100. [Google Scholar] [CrossRef]
- Wen, S.; Fang, G.; Chang, Y.; Zhao, L.; Ge, Y. Probability analysis of rain-wind induced vibration of stay cables due to tropical cyclone. J. Wind Eng. Ind. Aerodyn. 2024, 250, 105754. [Google Scholar] [CrossRef]
- Pacheco, B.M.; Fujino, Y.; Sulekh, A. Estimation Curve for Modal Damping in Stay Cables with Viscous Damper. J. Struct. Eng. 1993, 119, 1961–1979. [Google Scholar] [CrossRef]
- Krenk, S. Vibrations of a Taut Cable With an External Damper. J. Appl. Mech. 2000, 67, 772. [Google Scholar] [CrossRef]
- Tabatabai, H.; Mehrabi, A.B. Design of Mechanical Viscous Dampers for Stay Cables. J. Bridge Eng. 2000, 5, 114–123. [Google Scholar] [CrossRef]
- Shi, X.; Zhu, S. Dynamic characteristics of stay cables with inerter dampers. J. Sound Vib. 2018, 423, 287–305. [Google Scholar] [CrossRef]
- Javanbakht, M.; Cheng, S.; Ghrib, F. Multimode vibration control of stay cables using optimized negative stiffness damper. Struct. Control Health Monit. 2020, 27, e2503. [Google Scholar] [CrossRef]
- Fujino, Y.; Susumpow, T. Active control of cables by axial support motion. Smart Mater. Struct. 1995, 4, A41–A51. [Google Scholar] [CrossRef]
- Johnson, E.A.; Baker, G.A.; Spencer, B.F., Jr.; Fujino, Y. Semiactive damping of stay cables. J. Eng. Mech. 2003, 128, 1–11. [Google Scholar] [CrossRef]
- Weber, F.; Distl, H. Amplitude and frequency independent cable damping of Sutong bridge and Russky bridge by magnetorheological dampers. Struct. Control Health Monit. 2015, 22, 237–254. [Google Scholar] [CrossRef]
- Zhao, Y.L.; Xu, Z.D.; Wang, C. Wind vibration control of stay cables using magnetorheological dampers under optimal equivalent control algorithm. J. Sound Vib. 2018, 443, 732–747. [Google Scholar] [CrossRef]
- Sun, L.; Hong, D.; Chen, L. Cables interconnected with tuned inerter damper for vibration mitigation. Eng. Struct. 2017, 151, 57–67. [Google Scholar] [CrossRef]
- Irvine, H.M. Cable Structures. Can. J. Civ. Eng. 1988, 9, 129–142. [Google Scholar]
- Pei, M.; Zhang, X.; Zhu, B.; Hou, B.; Liu, C. Study on longitudinal wind load calculation method of cables for cable-stayed bridge. Strateg. Study CAE 2009, 11, 5. [Google Scholar]
- Caetano, E.S. Cable Vibrations in Cable-Stayed Bridges, 1st ed.; China Architecture & Building Press: Beijing, China, 2012. [Google Scholar]
Cable 1 with a Damper | Cable 1 without a Damper | Cable 2 with a Damper | Cable 2 without a Damper | |
---|---|---|---|---|
Sag | 4.08617 m | 3.89200 m | 3.69704 m | 3.51252 m |
Sag-to-span ratio | 0.722340% | 0.688014% | 0.687948% | 0.653612% |
Distance from the maximum value position to the centre (offset) | 0 m | 2.59474 m | 0 m | 2.34177 m |
The percentage of increase in sag compared to the no damper case | 4.98895% | / | 5.25321% | / |
TMD (Stiffness; Damping) | NES (Stiffness; Damping) | |
---|---|---|
Harmonic excitation at the first-mode frequency | 0; 538.111 Ns/m | 7971.18 N/m3; 0 |
Swept-sine excitation (excitation of the first mode) | 3457.82 N/m; 542.45 Ns/m | 400.379 N/m3; 1254.14 Ns/m |
Harmonic excitation at the third-mode frequency | 52,424.30 N/m; 3074.28 Ns/m | 67,858.80 N/m3; 3426.30 Ns/m |
Uncontrolled Group (No Control Device) | Best TMD Controller | Best NES Controller | |
---|---|---|---|
Harmonic excitation at the first-mode frequency | 10.9992 m | 3.57421 m | 2.55049 m |
Swept-sine excitation (excitation of the first mode) | 44.8049 m | 7.5476 m | 8.81287 m |
Harmonic excitation at the third-mode frequency | 0.925372 m | 0.309824 m | 0.315037 m |
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |
© 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
Share and Cite
Yao, C.; Li, D.; Wei, X. Vibration Suppression of Two Adjacent Cables Using an Interconnected Tuned Mass Damper/Nonlinear Energy Sink. J. Mar. Sci. Eng. 2024, 12, 1096. https://doi.org/10.3390/jmse12071096
Yao C, Li D, Wei X. Vibration Suppression of Two Adjacent Cables Using an Interconnected Tuned Mass Damper/Nonlinear Energy Sink. Journal of Marine Science and Engineering. 2024; 12(7):1096. https://doi.org/10.3390/jmse12071096
Chicago/Turabian StyleYao, Che, Dejian Li, and Xiaojun Wei. 2024. "Vibration Suppression of Two Adjacent Cables Using an Interconnected Tuned Mass Damper/Nonlinear Energy Sink" Journal of Marine Science and Engineering 12, no. 7: 1096. https://doi.org/10.3390/jmse12071096