⚡ electrical engineering

Vibration Control in Cable-Stayed Bridge Towers Using an EL-FRF Model for MTLD-Structure Interaction

This study introduces a novel Euler-Lagrange Frequency Response Function (EL-FRF) framework that optimizes multi-tuned liquid damper (MTLD) systems for superior broadband vibration control in cable-stayed bridge towers, demonstrating a 20–30% improvement in amplitude reduction over traditional single-tuned dampers through application to Vietnam's Bai Chay bridge.

Original authors: Duc-Thi-Thu-Dinh Nguyen, Dang Viet Duc, Truong Viet Hung

Published 2026-11-02

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CC BY 4.0

Original authors: Duc-Thi-Thu-Dinh Nguyen, Dang Viet Duc, Truong Viet Hung

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a tall, slender bridge tower swaying in the wind. To the naked eye, the movement might look like a gentle, rhythmic breathing, but inside the steel and concrete, the structure is fighting a constant battle against invisible forces. When strong gusts hit these towers, they can begin to vibrate in a way that feels like a ringing bell. If this shaking continues unchecked, it can lead to metal fatigue, excessive movement that makes the bridge feel unsafe, or even structural failure over time. Engineers have long known that adding a "tuned liquid damper" can help. This is essentially a large tank of water placed on the structure. As the tower sways, the water inside sloshes back and forth. This movement creates a counter-force that pushes against the tower's motion, absorbing the energy and calming the shake. It is a simple, passive solution that requires no electricity, just the physics of water moving in a container.

However, there is a catch. A single tank of water is tuned to a very specific frequency, like a radio station set to one precise number. If the wind changes speed or the bridge's natural rhythm shifts slightly due to temperature or construction changes, the water might stop matching the tower's movement perfectly. When that happens, the damper becomes ineffective, and the tower begins to shake again. This is the problem researchers at the University of Transport and Communications and Thuyloi University in Vietnam set out to solve. They looked at the Bai Chay Bridge, a major cable-stayed bridge in Vietnam known for its tall towers, which currently uses hundreds of small water tanks to dampen vibrations. While these tanks work, they are all tuned to the exact same frequency, making the system vulnerable if the wind conditions change.

The team developed a new way of thinking about how these water tanks interact with a building, creating a mathematical model they call the Euler-Lagrange Frequency Response Function. Instead of treating the water tanks as a single, uniform block, they proposed using an array of many tanks, each filled with slightly different amounts of water. This creates a "multi-tuned" system where the tanks are not all singing the same note, but rather a chord of slightly different notes. By spreading the tuning across a small range of frequencies, the system can catch the tower's vibrations even if the wind speed changes or the structure shifts slightly. The researchers used their new model to simulate how this multi-tank system would behave compared to the traditional single-frequency setup. They found that by arranging the tanks so that the central group holds the most water—about 60% of the total liquid mass—and the surrounding tanks hold the rest, the system becomes much more robust.

In their simulations, this new arrangement proved significantly more effective than the current setup on the Bai Chay Bridge. The multi-tuned system reduced the amplitude of the vibrations by 20% to 30% more than the single-frequency tanks could achieve. More importantly, the new design required fewer layers of tanks to achieve this superior performance. The researchers discovered that there is a sweet spot for the number of tanks: having between 5 and 15 different tanks in the array works best. If there are too few, the system loses its ability to catch a wide range of wind speeds. If there are too many, the tanks end up so close in frequency that they start acting like a single, less effective unit again. The study also showed that the spacing between the frequencies of the tanks needs to be precise, with a specific bandwidth that allows the system to absorb energy across a broad spectrum of wind forces without losing its damping power.

The implications of this work extend beyond just one bridge in Vietnam. The researchers demonstrated that their method can be applied to any tall, slender structure, such as wind turbine towers or high-rise buildings, where wind-induced shaking is a concern. By using a mathematical framework that accounts for the complex, non-linear movement of water waves, they provided engineers with a tool to design vibration control systems that are less likely to fail when conditions change. The study suggests that the future of protecting these structures lies not in adding more of the same, but in diversifying the way we tune our dampers. Instead of relying on a single, rigid solution, the most effective approach is a flexible, multi-layered system that can adapt to the shifting rhythms of the wind, keeping the bridge steady and safe for the people who cross it.

Technical Summary: Vibration Control in Cable-Stayed Bridge Towers Using an EL-FRF Model for MTLD-Structure Interaction

Problem Statement
Cable-stayed bridges, particularly those with single-plane cable arrangements, are inherently susceptible to lateral wind-induced vibrations in their towers. These dynamic responses can lead to structural fatigue, excessive displacements, and serviceability failures, especially in regions prone to typhoons and monsoons. While Tuned Liquid Dampers (TLDs) have been widely adopted for vibration control due to their simplicity and efficiency, existing applications often rely on Single-Tuned Liquid Dampers (STLDs). STLDs are typically modeled as two-degree-of-freedom (2-DOF) systems. A critical limitation of STLDs is their sensitivity to frequency detuning; slight deviations between the damper's natural frequency and the structure's natural frequency (caused by environmental factors or structural modifications) can lead to catastrophic reductions in damping performance. Although Multi-Tuned Liquid Dampers (MTLDs)—arrays of tanks with detuned frequencies—have been proposed to broaden the suppression bandwidth, there is a lack of a unified, analytically rigorous frequency response function (FRF) specifically tailored for MTLD-structure interactions. Current approaches often rely on empirical data or resource-intensive finite element simulations, lacking a closed-form, non-dimensional transfer function that enables rapid parametric optimization.

Methodology
To address these gaps, the authors propose a novel analytical framework termed the Euler-Lagrange Frequency Response Function (EL-FRF). The methodology proceeds through the following steps:

Theoretical Formulation: The EL-FRF is derived using Euler-Lagrange equations applied to a multi-degree-of-freedom (multi-DOF) system comprising the primary structure and an array of $N$ liquid tanks. The model incorporates:
- Equivalent Linearization: Nonlinear shallow-wave theory is used to model the sloshing dynamics. The system is linearized within a defined validity regime (shallow-water condition where $h_0/L \leq 0.1$ and small-amplitude sloshing where $\eta_{max}/h_0 \leq 0.15$ ).
- NSD-TMD Analogy: The complex fluid-structure interaction is approximated by an equivalent Tuned Mass Damper (TMD) model, where the sloshing liquid acts as a mass with variable stiffness and damping, calibrated via Nonlinear Shallow-water Damper (NSD) theory.
- Matrix Derivation: The Lagrangian ( $L = T - V$ ) is formulated for the structure and $N$ tanks, leading to a matrix equation of motion. The resulting non-dimensional transfer function, $H(\omega)$ , captures the phase lags, amplitude attenuation, and energy dissipation across the frequency bandwidth.
Parametric Optimization: The study conducts extensive parametric simulations to optimize MTLD performance. Key variables include:
- Tank Count ( $N$ ): Ranging from 3 to 21.
- Frequency Bandwidth ( $R$ ): Defined as $(f_{max} - f_{min}) / f_0$ , tested at 0.1, 0.2, and 0.3.
- Mass Distribution: Investigating the ratio of the center-frequency tank mass to the total liquid mass ( $\mu_{cen}$ ).
- Damping Ratios: Evaluating the ratio between structural damping and MTLD damping.
Case Study Application: The framework is applied to the Bai Chay Bridge in Vietnam, a single-plane cable-stayed bridge with a tower prone to lateral vibrations. The study compares the existing STLD system (344 tanks tuned to a single frequency) against a proposed MTLD retrofit using the EL-FRF model.

Key Contributions
The paper identifies four primary contributions:

Closed-Form Analytical Tool: The development of the EL-FRF provides a closed-form, non-dimensional transfer function for MTLD-structure interactions. This allows for efficient sensitivity studies without the need for computationally expensive simulations for every parameter change.
Pole-Zero Configuration Analysis: The authors demonstrate mathematically that while classical STLD models result in two complex conjugate poles (creating a dual-peak response susceptible to detuning), the EL-FRF formulation distributes $N$ pairs of poles and zeroes across the bandwidth. This distribution prevents dangerous resonance amplification and ensures uniform eigenvalue distribution.
Optimization Guidelines: The study identifies specific optimal parameters for MTLD design:
- Tank Count: An optimal range of $N = 5$ to $15$ (specifically $5 \leq N \leq 11$ for practical applications) balances robustness and complexity. Exceeding this range causes frequency clustering, reverting the system's behavior to that of an STLD.
- Bandwidth: Optimal performance is achieved with a frequency bandwidth $R$ between 0.2 and 0.3.
- Mass Allocation: A critical design rule is the allocation of approximately 60% to 80% of the total liquid mass to the center-frequency tank. This "center mass ratio" ensures balanced peak suppression and provides a buffer against aerodynamic detuning.
Practical Retrofit Proposal: The application to the Bai Chay Bridge demonstrates that an optimized MTLD system can achieve superior vibration control with fewer physical damper layers compared to the existing STLD configuration.

Results
Numerical simulations and the Bai Chay Bridge case study yield the following results:

Amplitude Reduction: The MTLD system, when optimized using the EL-FRF, demonstrates 20–30% greater amplitude reduction in lateral vibrations compared to the existing STLD system.
Response Profiles: While the undamped system shows a single sharp resonance peak and the STLD shows two symmetric peaks, the optimized MTLD exhibits a flattened response curve with multiple peaks of uniform amplitude. This indicates effective broadband vibration suppression.
Sensitivity to Parameters:
- Increasing the tank count ( $N$ ) beyond 15 leads to a reversion to a single-peak response, negating the benefits of multi-tuning.
- Narrow bandwidths ( $R=0.1$ ) with high tank counts cause the system to behave like an STLD.
- Deviations from the optimal center mass ratio (60–80%) result in unequal peak amplitudes and reduced suppression efficacy.
Bai Chay Bridge Implementation: The proposed MTLD retrofit for the Bai Chay Bridge utilizes five discrete locations along the tower height. By concentrating the mass in the center-frequency tanks (e.g., 18 layers at the top stage for the center tank vs. 14 for adjacent tanks), the system achieves the desired mass ratio and significantly reduces the total number of required damper layers compared to the current 344-tank STLD setup.

Significance
The paper claims that the EL-FRF framework advances the state of passive damper design by providing a versatile, analytically rigorous tool for structural engineers. Its significance lies in:

Robustness: Offering a solution that is less sensitive to frequency detuning than traditional STLDs, which is crucial for structures in wind-prone or seismic regions where environmental conditions vary.
Efficiency: Enabling rapid parametric optimization and sensitivity analysis through a closed-form solution, reducing reliance on trial-and-error or heavy computational modeling.
Transferability: The non-dimensional nature of the EL-FRF allows the derived optimization principles (mass ratios, bandwidths, tank counts) to be scaled and applied to other slender civil infrastructures, such as high-rise buildings and wind turbine towers, not just the specific bridge case study.

The authors conclude that while the STLD remains a viable alternative due to fabrication constraints, the MTLD optimized via EL-FRF offers a clear advantage in minimizing damper layers while enhancing lateral stability, representing a significant step forward in the vibration control of cable-stayed bridge towers.

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