Bridge Vibration Research Articles

Modal Frequency Shifts Measured on Xinjiang Bridge Excited by Traveling Vehicles

To excite sufficient vibrations of bridges for more effective modal experiments to take place, traveling vehicles were utilized in many bridge field inspection campaigns in history. However, the dynamics of a bridge subjected to the traveling vehicles are of changed resonant properties compared with those of the same structure with the ambient excitation in nature. Therefore, the field modal experiments employing vehicle excitations might unfavorably lead to shifted measurement results of low-order natural frequencies for long-span bridges of large mass. Using Xinjiang Bridge (a single-tower pre-stressed concrete cable-stayed bridge with spans [Formula: see text]) as the engineering background, this paper validates this technical notion based on reliable physical experiments and numerical simulations and seeks a practical approach to deal with the issue. It is found that the maximum deviation between the low-order modal frequencies measured on Xinjiang Bridge for the vehicle excitation case and those measured for the ambient excitation case (the accurate values) reaches 0.65[Formula: see text]Hz, and the deviation between the lower limit of the usual vehicle excitation’s energy band and the modal frequency measured (factor 1) and the traveling vehicles’ speed (factor 2) are the two key factors influencing the modal frequency shifts. An empirical model to compensate for modal frequencies’ deviations is formulated considering the two key factors accordingly, which is of high-practical significance in future applications of the modal experiments using vehicle excitations to other long-span bridges.

Deep learning-based automated identification on vortex-induced vibration of long suspenders for the suspension bridge

Large amplitude vortex-induced vibration (VIV) of long suspenders, caused by vortex-shedding and wake flow, will decrease the service life of high-strength wires and its anchorage systems, hence listed as an essential monitoring indicator of the structural health monitoring system in the suspension bridges. Traditionally, a static vibration amplitude limitation is usually predefined and used to identify VIV from continuously monitored acceleration data of bridge suspenders. Due to the complexity of the starting vibration of the suspenders, it may not be able to flexibly adapt to the suspenders VIV under different conditions. In addition, transient but significant vibration events may not be captured since it relies only on static vibration amplitude limits without considering the temporal information of vibration. To address the above-mentioned issues, a novel framework to autonomously identify the suspenders VIV is proposed using the multimodal fusion techniques with deep neural networks. The main contributions of the methodology are: i) by calculating the wind field and vibration characteristic, two vibration response parameters are identified as indexes for the identification of suspenders VIV based on the significant difference method of big data analysis; ii) Two different modal information of time history images and power spectral density (PSD) sequences are used as inputs to the convolutional neural network (CNN) and bidirectional long short-term memory (BiLSTM) to extract the suspenders vibration response features from the time and frequency domain perspectives, respectively; and iii) The multimodal information is concentrated and enhanced by the multi-head attention (MHA) mechanism to achieve automated identification of the suspenders VIV. The proposed framework is validated with data collected from a full-scale long-span suspension bridge in comparison with the base model. The results show that the proposed framework can efficiently identify the full-cycle development process of the suspenders VIV. This study provides a convenient strategy for suspenders VIV identification, which can be used for bridge management and vibration mitigation device design.

Analytical model analysis of suspension bridges with tower-girder-connected dampers

Recently the tower-girder-connected damper has been utilized to suppress multi-mode vertical vortex-induced vibration (VIV) of long-span suspension bridges. In this paper, an analytical solution is proposed for the damping ratio of vertical bending modes of a single-span suspension bridge, with a vertical or/and a rotational damper mounted near the end of the stiffening girder, which significantly simplifies the design procedure for tower-girder-connected dampers on suspension bridges. The damping effect and the accuracy of the analytical solution are validated for Humen Bridge. It is found that both the vertical and rotational damper with an appropriate parameter selected can supply significantly additional damping ratio to multiple bending modes of the suspension bridge. The rotational and vertical dampers work in a similar way and provide nearly the same maximum additional damping ratio for each mode. In particular, for the first two symmetric modes, regions of reduced motion near the ends of the stiffening girder are developed due to the effects of cable geometry and elasticity, and this reduces the effect of the tower-girder-connected damper. Excluding these modes, the damper effect on a suspension bridge is similar to that on a tension beam with the same beam stiffness and horizontal tension forces.

Indirect measurement of bridge surface roughness using vibration responses of a two-axle moving vehicle based on physics-constrained generative adversarial network

This study addresses the challenge of indirectly measuring bridge surface roughness through the vibration responses of a moving vehicle, which is crucial for pavement maintenance and bridge safety assessment. A physics-constrained generative adversarial network (PC-GAN) was proposed for the probabilistic estimation of surface roughness. The method consists of two steps: initially, a GAN informed by physics-based knowledge extracts combined information of bridge vibration deflection and surface roughness from vehicle accelerations. Subsequently, a feed-forward network isolates the bridge surface roughness from the combined data. Numerical examples validate the PC-GAN method, demonstrating sustained high accuracy under challenging conditions, including ISO 8608 level C road roughness, vehicle speeds up to 8 m s-1, 10 % deviation in vehicle parameters, 10 % environmental noise, and 10 % vehicle damping ratio. Laboratory tests further confirmed the method's efficacy, with the successful detection of artificial barriers on the bridge surface and a mean relative error of 3.33 % in height estimation. The PC-GAN method is demonstrated to be a robust tool for estimating bridge surface roughness under various numerical and laboratory conditions. These findings provide valuable insights for the rapid inspection of bridge pavement conditions using vibration responses from moving test vehicles.

Field observation and cause investigation of low-frequency cable vibrations in a cable-stayed bridge

Considerable levels of stay cable vibration in cable-supported bridges pose significant challenges for safety, operation, and maintenance. An accurate understanding of the specific causes and types of vibration is essential for effectively addressing this problem with an optimal design for mitigation measures. This study aims to investigate the cause of an extraordinary low-frequency vibration that occurred at a stay cable by employing comprehensive methodologies. The modal properties and vibrational characteristics of cable vibrations were first obtained through finite element modeling, operational modal analysis, and vision-based displacement measurements. Subsequently, the primary cause of low-frequency vibrations was investigated using field observations and wind tunnel tests. Field-monitoring data revealed that stay cables were excited by small deck vibrations. The critical wind speeds at which deck vibrations occurred in a wind tunnel experiment were aligned with the ranges of wind speeds that produce low-frequency cable vibrations, which confirmed that these two phenomena are manifested under similar wind conditions. A calculated support excitation amplitude, determined via the use of an empirical equation, exhibited consistency with vision-based measured displacement obtained in field experiments. These findings underscore the potential risk of support excitation between minor deck vibrations and cables, which provides valuable insights that could mitigate cable vibrations.

Estimation of bridge noise during the passage of trains with worn wheel treads from rail vibration in a different section

Bridge noise may increase when a train with worn wheel treads passes. This study attempted to estimate bridge noise when a train with worn wheels passed based on rail vibration in a different section, i.e., using accelerometers on the rail near a station and a vehicle maintenance depot. The rail vibration in a different section was converted into that in a section where bridge noise was required to be estimated. In the estimation, the vibration properties of the wheel, track, and contact spring in both sections were considered, and it was assumed that wheel roughness exceeded rail roughness at spatial frequencies where the influence of the wear was observed. The bridge noise generated by one wheel was estimated from the bridge vibration and converted from the rail vibration with transfer functions by considering the radiation area of the viaduct. The overall bridge noise was obtained by summing the noise generated by all the wheels in a train. The estimated results were consistent with the measured values at frequencies where the wear had a strong influence.

Bridge modal identification method based on adaptive VMD-HT algorithm

ABSTRACT Bridge engineering structures play a crucial role in transportation. Traditional methods of road and bridge closures for inspection and evaluation, while effective, increase bridge inspection costs and disrupt normal traffic flow. Therefore, a rapid detection and evaluation method for bridge structural modal parameters in the operational state is more suitable for practical engineering applications. However, under real operating conditions, bridge vibration signals typically contain a large amount of noise and interference, and modal identification must overcome issues of non-stationarity and non-linearity in the vibration signals caused by vehicle loads and environmental changes to ensure the accuracy of the identification. This study replaces the traditional Empirical Mode Decomposition algorithm in the Hilbert-Huang Transform with the Variational Mode Decomposition algorithm. It proposes a new method that combines the VMD algorithm with the Hilbert Transform, referred to as the Variational Mode Decomposition-Hilbert Transform method. Compared to the EMD algorithm, it offers significant improvements in suppressing mode mixing and endpoint effects. A vehicle-bridge coupling model was constructed, and the bridge’s natural modal information was successfully extracted from the vibration response signals. The research results indicate that the improved algorithm offers higher accuracy and faster computational speed compared to traditional identification algorithms.

Modelling of wake-induced vibration of a long-span bridge with separated parallel nonidentical decks

The separated parallel decks design for long-span bridges can meet higher traffic demands, offering advantages in urban bridge design. However, when exposed to wind, the periodic wake shedding from the front bridge deck significantly affects the rear deck, leading to wake-induced vibrations (WIV). To address this issue, this study proposes and compares four models to comprehensively predict the wake-induced vibration of a long-span bridge with separated parallel nonidentical decks. The predictive capabilities of the four models for WIV are meticulously compared, and the reasons for the observed errors are analyzed. Ultimately, a model capable of accounting for the concurrent influences of wake-induced forces, instantaneous structural motion feedback, and non-linear self-excited forces is obtained. A series of wind tunnel tests are conducted to validate and evaluate the proposed models. Besides, this study encompasses an analysis of how structural damping and the deck layouts influence WIV. These findings bear substantial relevance to engineering practices confronted with similar challenges, thereby contributing to a deeper understanding of the intricate dynamics at play in such complex structural configurations.

Analytical Solution for Dynamic Response of a Reinforced Concrete Beam with Viscoelastic Bearings Subjected to Moving Loads.

To provide a theoretical basis for eliminating resonance and optimizing the design of viscoelastically supported bridges, this paper investigates the analytical solutions of train-induced vibrations in railway bridges with low-stiffness and high-damping rubber bearings. First, the shape function of the viscoelastic bearing reinforced concrete (RC) beam is derived for the dynamic response of the viscoelastic bearing RC beam subjected to a single moving load. Furthermore, based on the simplified shape function, the dynamic response of the viscoelastic bearing RC beam under equidistant moving loads is studied. The results show that the stiffness and damping effect on the dynamic response of the supports cannot be neglected. The support stiffness might adversely increase the dynamic response. Further, due to the effect of support damping, the free vibration response of RC beams in resonance may be significantly suppressed. Finally, when the moving loads leave the bridge, the displacement amplitude of the viscoelastic support beam in free vibration is significantly larger than that of the rigid support beam.

Control Mechanism and Noise Reduction Effect of CVDM on Rail Transit Steel Bridge: Experimental and Numerical Study

Steel bridge, as one of the main structural types in bridge design, has been widely used in rail transit. However, the train-induced vibro-acoustic problem of the bridge has become increasingly prominent. The vibration and noise characteristics must be more significant and complex than steel truss bridge because of too many components. Viscoelastic damping materials (VDMs) have the characteristics of both viscous liquid and elastic solid, and have better damping performance than other damping materials. Therefore, VDM plays an important role in vibration and noise reduction technology. In applications, VDM is usually combined with constraining materials to form composite viscoelastic damping materials (CVDM). In this paper, based on the dynamic receptance method and superposition principle, the coupling model of vehicle–track–bridge in the frequency domain is first established, and then the vibration and noise prediction model of a steel truss bridge with CVDM is established based on RUK theory and statistical energy analysis (SEA) method. Based on the prediction model, the natural dynamic characteristics of typical components of steel truss bridges with CVDM were investigated, and the control mechanism and influence rule of CVDM on train-induced vibration and noise of steel bridges were revealed.

High-Speed Train-Induced Vibration of Bridge–Soft Soil Systems: Observation and MTF-Based ANSYS Simulation

In this paper, a multi-transmitting formula (MTF) was integrated into ANSYS software through secondary development, enabling dynamic finite element simulation of wave propagation in infinite domains. The numerical reliability and accuracy of the MTF were verified through a plane wave problem involving a homogeneous elastic half-space, as well as 3D scattering and source problems in a three-layered soil site. Additionally, a comparative analysis of various artificial boundaries was conducted to highlight the advantages of the MTF. Field observations of environmental vibrations caused by high-speed railway operations revealed localized amplification of vibrations along the depth direction at the Kunshan segment of the Beijing–Shanghai high-speed railway. Based on these observations, a series of numerical analyses were conducted using the customized ANSYS integrated with the MTF to investigate the underlying causes and mechanisms of this phenomenon, as well as the spatial variation characteristics of foundation vibrations induced by bridge vibrations during high-speed train operations. This study reveals the mechanism by which the combined effect of bridge piles and soft soil layers influences the depth variation in peak ground accelerations during site vibrations. It also demonstrates that the presence of bridge piers and pile foundations effectively reduces vibration intensity in the vicinity of the railway, playing a crucial role in mitigating vibrations induced by high-speed train operations.

Modelling Human-Structure Interaction in Pedestrian Bridges Using a Three-Dimensional Biomechanical Approach

Pedestrian bridges, which are essential in urban and rural infrastructures, are vulnerable to vibrations induced by pedestrian traffic owing to their low mass, stiffness, and damping. This paper presents a novel predictive model of Human-Structure Interaction (HSI) that integrates a three-dimensional biomechanical model of the human body, and a pedestrian bridge represented as a simply supported Euler-Bernoulli beam. Using inverse dynamics, the human model accurately captures three-dimensional gait and its interaction with structural vibrations. The results show that this approach provides precise estimates of human gait kinematics and kinetics, as well as the bridge response under pedestrian loads. The incorporation of a three-dimensional human gait model reflects the changes induced by bridge vibrations, providing a robust tool for evaluating and improving the effect of structural vibrations on the properties and gait patterns.

Stochastic optimization of levitation control system for maglev vehicles subjected to random guideway irregularity

The dynamic performance of maglev vehicle-bridge coupled systems subjected to random guideway irregularity, especially the levitation stability of the running vehicle subsystem, is a critical issue affecting the safe operation and the comfort riding of maglev railways. This study is devoted to stochastic optimization of the levitation control system considering the coupled vibration of maglev vehicle and bridge under the random guideway irregularity. To alleviate the computational burden in the stochastic optimization process, the explicit expressions of the critical responses of the maglev vehicle-bridge coupled systems and the sensitivities of critical responses with respect to the feedback gains of the proportional-integral-derivative (PID) levitation controller are respectively constructed in terms of the guideway irregularity by virtue of the dimension-reduced formulation capability of the explicit time-domain method (ETDM), enabling stochastic dynamic analysis and stochastic sensitivity analysis to be executed at high efficiency. The stochastic optimization problem is formulated as the minimization of the standard deviation of the air gap variation with the constraint on the standard deviation of the current variation, and the optimal values of the PID gains are searched by the gradient-based method of moving asymptotes (MMA), in which the statistical moments and moment sensitivities of the critical responses required in the optimization process are obtained by ETDM. A numerical example with a 2-degree-of-freedom maglev vehicle moving on a multi-span simply-supported bridge is used to demonstrate the efficacy of ETDM for the stochastic analysis of maglev vehicle-bridge coupled systems under guideway irregularity. An engineering example involving a 3-car maglev train traversing a multi-span simply-supported bridge is further presented to show the effectiveness of the ETDM-based approach for the stochastic optimization of large-scale engineering problems. The levitation stability of the maglev train under the action of random guideway irregularity is considerably improved with the use of optimal feedback gains.

Study on Vibration and Noise of Railway Steel–Concrete Composite Box Girder Bridge Considering Vehicle–Bridge Coupling Effect

A steel–concrete composite beam bridge fully exploits the mechanical advantages of the concrete structure and steel structure, and has the advantages of a fast construction speed and large stiffness. It is of certain research value to explore the application of this bridge type in the field of railway bridges. However, with the rapid development of domestic high-speed railway construction, the problem of vibration and noise radiation of high-speed railway bridges caused by train loads is becoming more and more serious. A steel–concrete composite beam bridge combines the tensile characteristics of steel and the compressive characteristics of concrete perfectly. At the same time, it also has the characteristics of a steel bridge and concrete bridge in terms of vibration and noise radiation. This feature makes the study of the vibration and noise of the bridge type more complicated. Therefore, in this paper, the characteristics of vibration and noise radiation of a high-speed railway steel–concrete composite box girder bridge are studied in detail from two aspects: the theoretical basis and a numerical simulation. The main results obtained are as follows: Relying on the idea of vehicle–rail–bridge coupling dynamics, a structural dynamics analysis model of a steel–concrete combined girder bridge for a high-speed railroad was established, and numerical program simulation of the vibration of the vehicle–rail–bridge coupling system was carried out based on the parametric design language of ANSYS 18.0 and the language of MATLAB R2021a, and the structural vibration results were analyzed in both the time domain and frequency domain. By using different time-step loading for the vehicle–rail–bridge coupling vibration analysis, the computational efficiency can be effectively improved under the condition of guaranteeing the accuracy of the result analysis within 100 Hz. Based on the power flow equilibrium equation, a statistical energy method of calculating the high-frequency noise radiation is theoretically derived. Based on the theoretical basis of the statistical energy method, the high-frequency noise in the structure is numerically simulated in the VAONE 2021 software, and the average contribution of the concrete roof plate to the three acoustic field points constructed in this paper is as high as 50%, which is of great significance in the study of noise reduction in steel–concrete composite girders.

An enhanced stochastic subspace identification incorporating time variables and its application to large-span sea-crossing suspension bridges

Structural health monitoring and assessment are essential to large-span sea-crossing bridges, and modal identification is the most effective method for achieving them, with the stochastic subspace identification (SSI) method dominating. However, when analyzed in open traffic scenarios with SSI for a large-span sea-crossing bridge, the contour of the vibration mode is distorted. Related affecting factors such as the measuring conditions, user-defined parameters, and traffic are explored, and traffic is verified as the essential component. To address this issue, an enhanced SSI algorithm incorporating time variables is created to automatically identify and rectify vibration mode shapes. Feature heterotopic signals are first matched with a common reference signal to perform modal identification. Advanced particle swarm optimization is applied to estimate critical parameters. The SSI algorithm with the density-based spatial clustering of applications with noise (DBSCAN) algorithm is then utilized to extract the modal characteristics. Both simulated misalignment and real records are applied to verify the suggested approach's performance. Comparative analysis with state-of-the-art techniques, such as the error norm, cross correlation, phase transform-β, mean phase deviation, and genetic algorithm, shows that the recommended approach is dependable and successful at identifying the large-span sea-crossing bridge's vibration mode shapes and computationally more accurate and efficient.

Bridging the Gap: commodifying infrastructure spatial dynamics with crowdsourced smartphone data

Structural information deficits about aging bridges have led to several avoidable catastrophes in recent years. Data-driven methods for bridge vibration monitoring enable frequent, accurate structural assessments; however, the high costs of widespread deployments of these systems make important condition information a luxury for bridge owners. Smartphone-based monitoring is inexpensive and has produced structural information, i.e., modal frequencies, in crowdsensing applications. Even so, current methods cannot extract spatial vibration characteristics with uncontrolled datasets that are needed for damage identification. Here we present an extensive real-world study with crowdsourced smartphone-vehicle trips within motor vehicles in which we estimate absolute value mode shapes and simulate damage detection capabilities. Our method analyzes over 800 trips across four road bridges with main spans ranging from 30 to 1300 m in length, representing about one-quarter of bridges in the United States. We demonstrate a bridge health monitoring platform compatible with ride-sourcing data streams that check conditions daily. The result has the potential to commodify data-driven structural assessments globally.

Toward Indirect Real-Time Prediction of Bridge Vibration Responses Under Traffic Flow Through a Population of Connected Sensing Vehicles

Condition monitoring of bridge structures as the lifelines of smart cities is high of importance. While indirect vehicle scanning techniques have shown promising results and have less cost compared to mounting fixed sensors on the bridge, they have limitations in predicting response under traffic flow since the crossing time of one vehicle is very short. This paper presents a novel crowsensing-based framework for predicting bridge acceleration responses and identifying the modal characteristics. This method utilizes smartphone data from a diverse population of sensing vehicles as they traverse the bridge to predict bridge acceleration response at various virtual sensing locations. Subsequently, the predicted acceleration is used to identify the mode shapes and natural frequencies of the bridge. The principal innovation in this practical and cost-effective monitoring solution is the utilization of a randomly selected set of sensing vehicles at each timestamp. These selections may differ from one timestamp to the next, reflecting the real-world conditions where certain vehicles may intermittently lose or disconnect their internet connectivity. By consistently updating the set of sensing agents while other vehicles cross the bridge, the proposed framework overcomes the data length limitations of conventional vehicle-based methods by leveraging multiple vehicles and continuous data collection. Comprehensive numerical studies are conducted to evaluate the performance of the method. In the numerical investigations, a three-span bridge is subjected to the continuous passing of a large number of half-car vehicle models with random speeds and initial locations. Vehicle-bridge interaction is considered in the analysis. Utilizing a single randomly selected sensing agent at each timestamp, the results demonstrate the effectiveness of the framework in predicting bridge acceleration response with a relative error of less than 5%. Additionally, the method achieves an accuracy level of 95% in identifying the bridge's initial three mode shapes.

Jellyfish-inspired bistable piezoelectric-triboelectric hybrid generator for low-frequency vibration energy harvesting

In pursuit of enhancing the output power of energy harvesters, this paper introduces a novel jellyfish-inspired bistable piezoelectric-triboelectric hybrid generator (JIB-PTHG) with low potential energy barrier characteristics and low-frequency, broadband energy harvesting capabilities. The JIB-PTHG comprises two flexible beams, and two rigid links, connected via a spring. Utilizing the harmonic balance method and Runge-Kutta fourth-order algorithm, a comprehensive analysis of the mechanical properties of the low-barrier bistable structure is conducted. Static equilibrium bifurcation and dynamic response analysis are performed. Subsequently, an electromechanical coupling model incorporating piezoelectricity and triboelectricity is developed to theoretically predict the output voltage of the two modules, revealing a positive correlation between displacement and output voltage. Experiment validation demonstrated that adjusting the length of the rigid links can effectively increase the deformation of the piezoelectric beam and the relative displacement of the triboelectric layers, thus improving the generator's output performance. Specifically, the TENG module exhibited exceptional energy harvesting performance within the frequency range of 3.5–6.5 Hz, achieving a voltage of 1050 V, a power of 1.84 mW, and the ability to illuminate 70 LED lights under an excitation amplitude of 17.5 mm and a frequency of 5.5 Hz. The PEG module, displayed high output performance within the frequency range of 4-7 Hz, reaching a voltage of 6.2 Vs and a power of 12.8 μW at an excitation frequency of 7 Hz and an amplitude of 17.5 mm. Both theoretical and experimental findings indicate that the JIB-PTHG has achieved improved output power within the frequency range of 3.5–6.5 Hz. Furthermore, the JIB-PTHG has the potential to harness bridge vibration energy and convert it into useful electrical energy for powering wireless sensor nodes in bridge health monitoring systems.

Improved bridge modal identification from vibration measurements using a hybrid empirical Fourier decomposition

Bridge health monitoring has been a prominent focus within the global engineering community. Bridge owners, stakeholders, and engineers face the formidable tasks of ensuring efficient monitoring, conducting reliable data analysis, interpreting data logically, and making timely decisions. With the increasing global infrastructure deficit, there is an ever-increasing need to develop reliable and economical bridge monitoring solutions. In this paper, a bridge condition assessment technique is proposed that can utilize the vibration data collected from the instrumented sensors and provide reliable system identification results. The proposed method develops a hybrid approach by integrating the Natural Excitation Technique (NExT) and Empirical Fourier Decomposition (EFD) to analyze ambient bridge vibration data and determine the modal parameters of the bridge. First, NExT is formulated to determine the cross-correlation functions of the bridge measurements, and then EFD is explored to decompose the signals into their monocomponents to identify the bridge modal parameters. The proposed methodology can overcome mode mixing and perform modal identification of a system with closely spaced frequencies and low energy modes. The estimated modal parameters such as bridge frequencies, mode shapes, and damping ratio are used for condition assessment of numerical, experimental and full-scale structures, including a short-span steel bridge located in Ontario, Canada. The results demonstrate that the proposed methodology can provide accurate and robust estimates of bridge modal parameters. Future research is reserved for real-time implementation of the proposed methodology for a wide range of civil structures.

Robust damping improvement against the vortex-induced vibration in flexible bridges using multiple tuned mass damper inerters

The tuned mass damper inerter (TMDI) is considered as one of countermeasures to mitigate the very-low-frequency vertical bending vortex-induced vibration (VIV) in flexible bridges. However, the control robustness of the TMDI against the system uncertainty is a serious concern due to the limited inerter span capacity. In this paper, the multiple tuned mass damper inerter (MTMDI) is used instead of identical TMDIs to provide a robust equivalent damping ratio against the uncertain VIV within the entire lock-in wind speed range. A design framework of the MTMDI considering the inerter span length is proposed based on a universal VIV model suitable for the fully closed, centrally slotted, and semi-closed box decks. In the framework, the fluctuation of VIV characteristics with the wind speed, the probability of occurrence of the VIV, and the uncertainties in VIV characteristics and deck dynamics are considered, simultaneously. Two objection functions are defined to focus on the maximizing the equivalent damper ratio and reducing the VIV mitigation failure, respectively. Finally, numerical analysis of a flexible bridge-MTMDI system subjected to the VIV is performed. The results show that the wind speed probability distribution plays a role of weighting coefficient in the MTMDI optimization. The choice of the objective function depends on the mass ratio of the MTMDI and the level of the system uncertainty. The well-optimized MTMDI achieved a more robust control result than the TMDI within the entire lock-in wind speed range.