Study on the Application of Determinant-MTMD(TMDD) Vibration Reduction in Cable-Supported Pedestrian Suspension Bridge

Abstract

In this study, multiple tuned mass dampers (MTMDs) were studied to understand their impact on the human-induced vibration response and comfort level of a pedestrian cable-supported suspension bridge. A spatial finite element model based on a specific engineering case was established. The dynamic characteristics of the bridge under human-induced loads were investigated, and its comfort level under human-induced vibrations was analyzed using the time-history method. Then, this study adjusted the design parameters of the dampers based on various optimal damper parameter expressions. Furthermore, the damping effectiveness of MTMD under different mass ratios ($\mu$) was evaluated, and it was found that increasing the mass ratio significantly impacts damping performance. Finally, determinant-TMD (TMDD) was introduced, and a comparison between the damping effect, robustness, and performance of TMDD and MTMD was conducted. The results indicate that while increasing the mass ratio does not linearly affect maximum vibration acceleration, the damping effect increases initially and then stabilizes, with a damping rate converging at approximately 55%. However, with the TMDD approach, the maximum damping rate can reach approximately 70%, enhancing comfort levels from the “minimum CL3” achieved with MTMD to the “medium CL2” level. Additionally, while TMDD’s robustness is slightly inferior to MTMD at lower mass ratios, it demonstrates superior robustness at higher mass ratios.

1. Introduction

With the widespread use of high-strength lightweight materials and continuous advancements in structural analysis techniques, bridge design is evolving toward longer spans, slender, and lighter weights. However, large-span bridges, characterized by their lightness, flexibility, and low-damping properties, are prone to vibration issues. When conducting dynamic analyses of bridge structures, researchers often prioritize seismic and wind loads, while crowd loads are typically simplified as static loads, underestimating their potential dynamic effects. In bridges for railways and highways, where live loads caused by trains or vehicles constitute a significant portion and crowd loads are minimal, simplifying crowd loads as static loads is appropriate. Nevertheless, for pedestrian bridges, especially those with larger spans, the dynamic response generated by crowd loads has a notable impact. Ignoring the human-induced vibrations that may arise during operation can lead to potential bridge safety issues, as exemplified by the Millennium Bridge in London and the T Bridge in Japan [1].

When the natural frequency of a bridge is similar to the frequency of human walking, there are two main strategies to deal with vibrations: frequency adjustment and suppression [2,3,4]. The frequency adjustment method reduces vibration by increasing stiffness and optimizing the frequency range sensitive to the structure’s dynamics. However, it is only suitable when a single mode falls within the range of human walking frequency and can be expensive. On the other hand, the vibration suppression method reduces the resonance response by improving the damping performance of the pedestrian bridge structure. This can be achieved by adding viscous dampers, tuned mass dampers (TMDs), multiple tuned mass dampers (MTMDs), tuned liquid cylindrical dampers (LTCDs and ACTCDs) [5,6], and other types of semi-active and active TMDs (SATMDs and ATMDs) to the bridge structure [7,8].

Den Hartog [9] examined the optimal parameter design of TMDs under external harmonic excitation. Chen Zhengqing [10,11] summarized the design theory of pedestrian bridges, proposed an improved random time-domain model of pedestrian walking force based on instantaneous walking parameters, studied the distribution of pedestrian walking parameters through experiments, and conducted vibration reduction analysis on a cable-stayed bridge. Poovarodom [12] explored the mechanical properties of a steel pedestrian bridge with and without TMDs using experimental and numerical simulation methods. The results showed that TMDs can reduce structural responses caused by pedestrians, and the type of human excitation influences the degree of reduction. Song [13] evaluated the comfort of a bridge treated with TMDs for vibration reduction using time history analysis based on the German guide for the design of pedestrian bridges. Werkle [14] verified the accuracy of the optimization criteria proposed by Den Hartog using a 45 m span pedestrian bridge as an example. Fan [15] studied the placement of TMDs, the optimal frequency ratio, the optimal damping ratio of the system, and the vibration reduction effect and provided relevant design suggestions.

While there are a few studies on human-induced vibration reduction in cable-supported pedestrian suspension bridges, the dynamic characteristics differ from ordinary pedestrian suspension bridges. Cable-supported pedestrian suspension bridges usually have lower mass, damping, and stiffness. As a result, multiple natural frequencies within the crowd load’s step frequency range may lead to multiple resonances and affect the vibration response. Liu Feng [16] studied the laws of the vibration response of pedestrian suspension bridges by combining field measurements with finite element simulations. Zhang Yanling [17] and Chen Sitian [18] et al. studied the impact of pedestrian vibration response and pedestrian comfort on pedestrian glass suspension bridges. Zou Zhuo [19] studied a double-tower, three-span self-anchored steel box girder pedestrian suspension bridge and conducted a dynamic characteristic analysis. Jiang Wang [20] investigated the influence of wind-resistant cables with different inclination angles on the dynamic characteristics of pedestrian suspension bridges and considered that the comfort improvement of the pedestrian suspension bridge by the wind-resistant cable is limited. Hu Xibing [21] proposed a simplified calculation method for the vertical load of a pedestrian suspension bridge and verified it on a 117 m pedestrian suspension bridge. Li [22] discussed the human-induced vibration effects of pedestrian suspension bridges with different deck heights and considered this kind of suspension bridge to significantly improve lateral pedestrian comfort.

The vibration reduction performance of a tuned mass damper (TMD) depends on accurately capturing the natural frequency of the main structure. Only when the TMD frequency is tuned to the controlled frequency of the structure and the external excitation covers this frequency component can it have the best control effect. The difference between the expected and actual values of design parameters can greatly impact the vibration reduction effect. Studies by Rana [23], Werkle [14], Lievens [24], and Miguel [25] have examined this inconsistency. Van Nimmen [26] found that even detailed finite element model analysis can lead to prediction errors in frequency of up to 10%. Traditional TMD design methods did not account for variations in load or structural natural vibration characteristics. Studies have been conducted on optimizing TMD performance under uncertain conditions. Zhang [27] compared the traditional TMD design using the embedded polynomial chaotic expansion (PCE) method and evaluated the effectiveness of TMD in controlling human-induced vibrations when the modal parameters of the footbridge were uncertain. Li [28] developed a TMD robust optimization method based on the perturbation method, which can significantly reduce computational costs. However, this method is not suitable for systems with strong uncertainty or nonlinearity. In these robust optimization problems, typically only the load is considered an uncertain factor [29]. In most structural dynamics problems, the uncertainty of system parameters often has a greater impact on the response than the uncertainty of the load [30], which is particularly important for pedestrian bridges.

In this study, firstly, numerical simulation methods were used to study the dynamic characteristics of a large-span cable-supported pedestrian suspension bridge. The bridge is subjected to a simple harmonic load based on the German EN03 specification, and the time-history analysis method is used to evaluate the dynamic response. Then, this study proposes the determinant tuned mass damper (TMDD), and it is developed from the MTMD. It decomposes a single TMD in the MTMD into a matrix distribution TMD array, which is equivalent to a single TMD in the original MTMD. It determines the design parameters of the damper based on existing research results of the damper parameter optimization formula. Lastly, the differences in robustness between the TMDD and MTMD were compared with the natural frequency of the bridge changing.

2. Basic Equation of Pedestrian Bridge TMDD

The excitation generated by pedestrians mainly causes the vibration phenomenon in pedestrian bridges. When a crowd moves, pedestrian bridges experience a dynamic force that varies over time due to changes in the position and acceleration of pedestrians. This force has components in three spatial directions: first, the vertical force acting along the direction of gravity; the second is the lateral force orthogonal to the direction of travel; and the third is the vertical force along the direction of travel.

To simplify the problem, a simply supported beam with constant bending stiffness is used for derivation, and the pedestrian bridge TMDD system is shown in Figure 1. When the pedestrian load $P\left(t\right)$ was loaded vertically on the pedestrian bridge, the dynamic equation of the system is as follows.

$$\left\{\begin{array}{c}EI{\displaystyle \frac{{\partial }^{4}z}{\partial x^4}}+C{\displaystyle \frac{\partial z}{\partial t}}+m{\displaystyle \frac{{\partial }^{2}z}{\partial t^2}}=P\left(t\right)+{\displaystyle \sum _{i=1}^n}\delta \left(x-x_i\right)p_i\left(t\right)\\ p_i\left(t\right)=k_i\left[y_i\left(t\right)-z\left(x_i-t\right)\right]+c_i\left[\stackrel{·}{y_i}\left(t\right)-{\displaystyle \frac{z\left(x_i,t\right)}{\partial t}}\right], i = 1, 2, \dots , n\\ m_i\stackrel{··}{y_i}\left(t\right)+k_i\left[y_i\left(t\right)-z\left(x_i-t\right)\right]+c_i\left[\stackrel{·}{y_i}\left(t\right)-{\displaystyle \frac{z\left(x_i,t\right)}{\partial t}}\right]= 0, i = 1, 2, \dots , n\end{array}\right.$$

(1)

where z is the vertical displacement of the beam, $y_i$ is the vertical displacement of TMDs, C is the damping of the bridge, m is the bridge mass, $EI$ is the bending stiffness of the bridge, $t$ is the time, and $P\left(t\right)$ is the force of TMDs on the bridge. The definitions of other parameters are shown in Figure 1.

Using the mode decomposition method [31], the vertical displacement $z$ of the pedestrian bridge is represented as a set of the nth orders of vibration mode. Multiplying both sides of the equation by the mode function ${\varphi }_k\left(x\right)$ and integrating the entire bridge yields, we can obtain

$$\left\{\begin{array}{c}\begin{array}{c}{M}_k\stackrel{··}{q_k}\left(t\right)+2M_k{\omega }_k{\xi }_k\stackrel{·}{q_k}\left(t\right)+K_k{q}_k\left(t\right)-\hfill \\ {\sum }_{i=1}^n{\varphi }_k\left(x_i\right)\left\{\begin{array}{c}{k}_i\left[y_i\left(t\right)-{\sum }_{i=1}^n{\varphi }_i\left(x_i\right)q_i\left(t\right)\right]+\\ c_i\left[\stackrel{·}{y_i}\left(t\right)-{\sum }_{i=1}^n{\varphi }_i\left(x_i\right)\stackrel{·}{q_i}\left(t\right)\right]\end{array}\right\}=P_k\left(t\right), k = 1, \dots , n\end{array}\\ \begin{array}{c}{m}_i\stackrel{··}{y_i}\left(t\right)+k_i\left[y_i\left(t\right)-{\sum }_{i=1}^n{\varphi }_i\left(x_i\right)q_i\left(t\right)\right]+\\ c_i\left[\stackrel{·}{y_i}\left(t\right)-{\sum }_{i=1}^n{\varphi }_i\left(x_i\right)\stackrel{·}{q_i}\left(t\right)\right]= 0, i = 1, 2, \dots , m\end{array}\end{array}\right.$$

(2)

where $M_k$, ${\omega }_k, {\xi }_k, K_k, {\varphi }_k$ represent the k-order modal mass, frequency, damping ratio, modal stiffness, and mode function of the pedestrian bridge, while $P_k\left(t\right)$ represents the k-order generalized modal load. The motion state of the system is represented as column vectors $s$,

$$\left\{s\left(t\right)\right\}=\left\{s_1\left(t\right), \dots , s_n\left(t\right), y_1\left(t\right), \dots , y_m\left(t\right)\right\}$$

(3)

The entire pedestrian bridge system can be expressed in a unified matrix form as

$$\left[M\right]\left\{\ddot{s}\left(t\right)\right\}+\left[C\right]\left\{\dot{s}\left(t\right)\right\}+\left[K\right]\left\{s\left(t\right)\right\}=\left\{P\left(t\right)\right\}$$

(4)

where $C_i=2M_i{\omega }_i{\xi }_i$ is the ith order’s modal damping of pedestrian bridges, $K_i=M_i{\omega }_i^2$ is the ith order’s modal stiffness of pedestrian bridges, and the mass matrix [M], damping matrix [C], and stiffness matrix [K] are as follows:

$$\begin{array}{c}\left[M\right]=\left[\begin{array}{cccccc}{M}_1& & & & & \\ & \ddots & & & 0& \\ & & M_n& & & \\ & & & m_1& & \\ & 0& & & & \\ & & & & & m_m\end{array}\right]\\ \left[C\right]=\left[\begin{array}{cccccc}{C}_1+{\displaystyle \sum _{j=1}^m}{c}_j{\displaystyle {\varphi }_1^2}\left(x_j\right)& \cdots & C_1+{\displaystyle \sum _{j=1}^m}{c}_j{\displaystyle {\varphi }_1^2}\left(x_j\right)& -{\varphi }_1\left(x_1\right)c_1& \cdots & -{\varphi }_1\left(x_m\right)c_m\\ & \ddots & ⋮& ⋮& ⋮& ⋮\\ & & C_n+{\displaystyle \sum _{j=1}^m}{c}_j{\displaystyle {\varphi }_n^2}\left(x_j\right)& -{\varphi }_n\left(x_1\right)c_1& \cdots & -{\varphi }_n\left(x_m\right)c_m\\ & & & c_1& \cdots & 0\\ & symmetric& & & \ddots & ⋮\\ & & & & & c_m\end{array}\right]\\ \left[K\right]=\left[\begin{array}{cccccc}{K}_1+{\displaystyle \sum _{j=1}^m}{k}_j{\displaystyle {\varphi }_1^2}\left(x_j\right)& \cdots & K_1+{\displaystyle \sum _{j=1}^m}{k}_j{\displaystyle {\varphi }_1^2}\left(x_j\right)& -{\varphi }_1\left(x_1\right)k_1& \cdots & -{\varphi }_1\left(x_m\right)k_m\\ & \ddots & ⋮& ⋮& ⋮& ⋮\\ & & K_n+{\displaystyle \sum _{j=1}^m}{k}_j{\displaystyle {\varphi }_n^2}\left(x_j\right)& -{\varphi }_n\left(x_1\right)k_1& \cdots & -{\varphi }_n\left(x_m\right)k_m\\ & & & k_1& \cdots & 0\\ & symmetric& & & \ddots & ⋮\\ & & & & & k_m\end{array}\right]\end{array}$$

The stiffness matrix [K] and damping matrix [C] have non-zero non-diagonal elements, making decoupling and finding a theoretical solution challenging. Using numerical methods such as the central difference method or finite element method simplifies the process. This article employs the pedestrian load model recommended by the German EN03 standard [32] to create a finite element model for simulation calculations. With a time step of t = 0.01 s, it meets the accuracy requirements for analyzing pedestrian bridge fundamental frequencies (f ≤ 10 Hz), as stated in reference [33].

3. Establishment of Finite Element Model

3.1. Engineering Background

A single-span cable-supported suspension bridge with a main cable span of 376 m was set as the engineering background. The deck width was 1.8 m, the sag-to-span ratio was 1/20, and the sag was 18.8 m. The main cable is comprised of 32 mm (6 × 37 + IWR) steel-wire rope. There are two parts to the wind-resistant cable. The part spanning the canyon is 230 m with an in-plane rise-to-span ratio of 1/20 and uses a 32 mm (6 × 37 + IWR) steel wire rope. In addition, six anti-wind cables are installed on the right side of the mountain, spaced 18.0 m apart and using 16 mm (6 × 37 + IWR) steel-wire ropes. The intersections of the wind-resistant cable and the main cable are set to (2 × 14 + 12 + 8 × 10 + 8 × 8 + 8 × 10 + 2 × 12 + 30 + 2 × 18 + 22) m, and the cables are arranged in triangle. The wind-resistant cables are made of 12 mm (6 × 37 + IWR) steel wires and set at a 45° angle to the horizontal plane of the main beam. The bridge uses a tower-free structure, with the tower saddle on the hills on both sides of the canyon. The bridge deck consists of stainless-steel plates, wooden bridge decks, and angle steel, with each standard section measuring 2.0 m in length and 1.5 m in width. The bridge deck is connected to the limit plate using M10 bolts. The bridge layout is shown in Figure 2.

3.2. Finite Element Model

3.2.1. Unit and Boundary Conditions

The finite element model was created using Midas/Civil 2022. In this model, the main cable, wind-resistant cable, and stay cable were simulated using cable elements with an elastic modulus E of 110 MPa, and the beam consisted of stainless-steel plates and angle steel, which were simulated using beam elements with an elastic modulus E of 210 GPa. The wooden bridge deck was treated as an equivalent static load and the average mass per linear meter of cableway bridge span direction m = 125.0 kg. The boundary conditions were as follows: the anchorage of the main cable, the anchorage of the wind-resistant cable, and the bottom of the cable saddle were consolidation connections; the main cable had displacement constraints released in the x-direction (along the bridge) at the saddle; and rigid connections were used between the main cable and stainless-steel plate, the stainless-steel plate and angle steel, the wind-resistant cable and stainless-steel plate, and the wind-resistant and wind-resistant cables. Considering the small weight of the pedestrian suspension bridge and the large aspect ratio of the bridge beam, the pedestrian mass was approximately equivalent to the self-weight load of the bridge for dynamic analysis of the bridge’s structural system. The finite element analysis model is shown in Figure 3.

The analysis focused on the first 200 vibration modes. It was found that the vertical vibration frequency of the first 10 modes ranged from 0.251 to 0.584 Hz, indicating high flexibility and prominent bending–torsion coupling characteristics of the bridge. According to the literature [34], the coupling characteristics of transverse bending and torsion of the cable-supported footbridge are obvious, and vertical bending did not occur with bending torsion. This study mainly discusses the working condition of vertical vibration, which is directly related to the vertical vibration mode, so only the vertical vibration mode is listed to simplify the discussion.

Pedestrian loads on bridges typically have narrow-band frequency excitations. When the frequency of pedestrian loads aligns closely with the vibration frequency of the bridge, resonance can occur, affecting pedestrian comfort. Therefore, when studying pedestrian-induced vibrations, it is important to consider vibration modes within the pedestrian step frequency range for comfort research. According to the German EN03 standard, the vertical resonance frequency range for pedestrians in the vertical direction is 1.25 to 2.3 Hz. The analysis of the bridge model is shown in

Table 1. The natural frequency and vibration mode of bridges within the vertical frequency range.

Vibration Order	Vibration Frequency/Hz	Vibration Mode Diagram
8	1.261
9	1.404
10	1.578
11	1.688
12	1.818
13	1.968
14	2.130
15	2.240

, and the eight main beam vertical vibration modes are in this range.

3.2.2. Comfort Index

The experience of vibration for pedestrians is a complex problem, as different people may have different experiences of the same vibration and even the same person may have different experiences of the same vibration at different times [8]. The Chinese specification for pedestrian bridges does not define a comfort index. Meanwhile, in the British BS5400 [35] specification and the international ISO 10137 standard [36], the acceleration limit is related to the natural frequency; in the Swedish Bro2004 code and the European EN1990 code [37], the acceleration is limited as a constant; German EN03 (2007) [32] divides the different ranges of peak acceleration values into four comfort levels and conducts a more comprehensive and detailed evaluation of the vibration comfort of pedestrian bridges, as shown in

Table 2. Evaluation standard of pedestrian bridge comfort index in German EN03 code.

$\mathbf{Vertical} \mathbf{Acceleration} (\mathbf{m}/{\mathbf{s}}^2)$	$\mathbf{Lateral} \mathbf{Acceleration} (\mathbf{m}/{\mathbf{s}}^2)$	Notes
<0.5	<0.1	CL1, best
0.5–1.0	0.1–0.3	CL2, medium
1.0–2.5	0.3–0.8	CL3, minimum
>2.5	>0.8	CL4, unacceptable

.

3.2.3. Pedestrian Loads

Currently, Chinese pedestrian bridge specifications need further improvement, and the existing Chinese standards do not define the mathematical model of pedestrian loads. This article analyzes how pedestrians respond to first-order harmonic wave loads based on the calculation method of pedestrian loads in the German EN03 standard. The German EN03 standard equates a random number of pedestrian loads to a synchronously moving pedestrian load model and provides a calculation formula for this equivalent load model, as follows:

$$P\left(t\right)=P_0\mathit{cos}\left(2\pi f\cdot t\right)n^{\prime }\psi$$

(5)

where $P\left(t\right)$ is the harmonic wave load measured in N/m² and P₀ is the amplitude value; P₀ is taken as 280 N for the vertical vibration mode and P₀ is taken as 35 N for the lateral vibration mode. f is the vibration mode frequency, t is time, and n′ is the equivalent pedestrian density, which in this study is less than 1 pedestrian per unit area and valued as $n^{\prime }=10.8\sqrt{\xi n}/S$. ξ is the structural damping coefficient, set as 0.02 for this steel structure bridge [15]. S is the actual loading area of the crowd load, valued at 1.5 m × 376 m, n is the converted number of pedestrians with the loading area S, and Ψ is the reduction coefficient of amplitude change caused by the probability of pedestrian frequency approaching the modal frequency range, with a value of 1.0 in this study.

The pedestrian harmonic wave load values for each frequency are obtained through the harmonic wave attenuation coefficient calculation method [31], using eigenvalue analysis to select pedestrian step frequency resonance modes that meet the requirements. The load is evenly distributed to each main cable, and the design load of the bridge is 0.25 person/m². The pedestrian harmonic load is shown in Figure 4, and the calculation results are detailed in

Table 3. Pedestrian harmonic load $P\left(t\right)$.

Vibration Order	Include Pedestrian Quality		Maximum Acceleration
	Step Frequency (Hz)	$\mathit{P}\left(\mathit{t}\right)$ (N/m2)	Amax (m/s2)
8	1.261	0.660 cos(7.923t)	0.06
9	1.404	9.243 cos(8.822t)	0.89
10	1.578	19.688 cos(9.915t)	1.55
11	1.688	26.291 cos(10.606t)	2.34
12	1.818	27.011 cos(11.423t)	2.35
13	1.968	27.011 cos(12.365t)	2.38
14	2.130	22.960 cos(13.383t)	1.95
15	2.240	8.103 cos(14.074t)	0.94

. The maximum acceleration occurs at the 13th order, reaching 2.38 m/s², which, according to the comfort standard recommended by German EN03, falls under the “minimum CL3” level and requires further vibration reduction design.

4. MTMD Vibration Reduction Analysis

The Multi-Tuned Mass Damper (MTMD) system is widely used for dissipating energy and reducing vibration in engineering structures. It consists of multiple TMDs arranged in a specific manner. When the main structure vibrates, the TMD system is designed to align the subsystem’s natural frequency with the main structure’s fundamental frequency. This alignment causes the substructure to produce an inertial force opposite to the direction of the main structure’s vibration, effectively reducing the vibration response of the main structure.

In reference [38], it was found that the method for calculating the best tuning mass damper (TMD) parameters varies based on different excitation sources and optimization goals. When dealing with a sine function load, the primary optimization methods are those shown in

Table 4. Optimal parameters of TMD with different optimization objectives under sine function.

Working Condition	Optimization Objectives	fopt	ξ2opt
1	displacement	${\displaystyle \frac{1}{1+\mu }}$	${\displaystyle \frac{3\mu }{8\left(1+\mu \right)}}$
2	speed	${\displaystyle \frac{{\left(1+\mu /2\right)}^{0.5}}{1+\mu }}$	${\displaystyle \frac{3\mu \left(1+\mu +5/24{\mu }^2\right)}{8\left(1+\mu \right){\left(1+0.5\mu \right)}^2}}$
3	acceleration	${\displaystyle \frac{1}{{\left(1+\mu \right)}^{0.5}}}$	${\displaystyle \frac{3\mu }{8\left(1+\mu /2\right)}}$

.

In the above table, $\mu$ is the mass ratio, which is defined as the ratio of TMD mass to total structure mass. The main control factor for pedestrian bridges is the magnitude of acceleration felt by pedestrians, so acceleration needs to be the priority for control. However, cable-supported pedestrian suspension bridges have great flexibility, and displacement and velocity must also be controlled. Assuming that the importance of displacement, velocity, and acceleration control are equal, this article chooses f_opt from the table above for calculation and selects the average value for research. For research purposes, the article selects $\mu$ = 0.02, 0.04, 0.06, 0.08, 0.10, 0.15, 0.20, 0.25, and 0.30 for research, and the calculated results of individual TMD physical parameters are shown in

Table 5. TMD physical parameters.

$\mathit{\mu }$	mTMD/kg	kTMD/(N/m)	cTMD/(Ns/m)
0.02	101.7	7083.6	196.6
0.04	203.3	13,762.4	543.5
0.06	305.0	20,066.8	976.6
0.08	406.7	26,024.2	1471.3
0.10	508.3	31,659.6	2013.0
0.15	762.5	44,484.2	3512.6
0.20	1016.7	55,739.3	5148.9
0.25	1270.8	65,671.1	6865.6
0.30	1525.0	74,479.0	8627.8

. To improve the computational efficiency, the MTMD system in this article is equipped with 13 sub-TMDs (TMD1~TMD13), uniformly arranged along the bridge direction at a spacing of 28 m. TMD7 is located at the mid-span, as shown in Figure 5.

The results of the calculations show that the configuration of multiple tuned mass dampers (MTMDs) is vital in enhancing the structure’s seismic performance. Increasing the mass of the TMD can improve its damping efficiency; however, there is no direct linear correlation between the TMD mass and damping efficiency. Instead, the inflection point is approximately at $\mu$ = 0.1, as illustrated in Figure 6. Within the typical mass ratio range (0.01 <$\mu$ ≤ 0.05), the damping rate significantly increases with the rise in mass ratio. Still, at this point, the damping rate is below 30%, and the damping effect is limited. As the mass ratio reaches 0.05 < $\mu$ ≤ 0.2, the damping rate continues to increase with the mass ratio, but at a slower pace. At this stage, the damping rate ranges from around 42% to 52%. When the mass ratio is 0.2 ≤ $\mu$ ≤ 0.3, it can be observed that the damping rate hardly increases with the rise in mass ratio and remains stable at approximately 55%.

The mass ratio plays a crucial role in controlling the vibration reduction effect of the MTMD system and its maximum acceleration. However, considering comfort, as the mass ratio of the MTMD system increases from 0 to 0.3, the maximum acceleration decreases from 2.384 m/s² to 1.094 m/s². According to the German EN03 specification, the comfort level before and after vibration reduction remains at level CL3. This indicates that while the MTMD vibration reduction effect reduces the peak acceleration, it does not improve the comfort level. Therefore, a new TMD layout is needed to reduce vibration further.

5. TMDD Vibration Reduction Analysis

Based on the results in Section 4, it is evident that increasing the mass of a single tuned mass damper (TMD) in a multiple tuned mass damper (MTMD) system is advantageous for reducing vibrations in cable-supported pedestrian suspension bridges. The optimal mass ratio falls within the range of 0.05–0.2. Beyond this range, further increasing the mass ratio constrains the effectiveness of vibration reduction. However, when the mass ratio reaches 0.2, the mass of a single TMD becomes 1016.7 kg, which poses challenges for installation. Consequently, this paper proposes a method of dispersing a single large TMD into several TMD arrays arranged in a matrix form. In this arrangement, the mass, damping, and stiffness parameters of each sub-TMD in the array are taken as 1/n of the original single TMD in the MTMD. The equivalent mass, stiffness, and damping of the array sub-TMDs remain consistent with those of the original MTMD single TMD. This approach aims to reduce the mass of a single TMD and simplify installation and maintenance processes.

We selected 1 × 1 (n = 1, i.e., MTMD), 2 × 2 (n = 4), 3 × 3 (n = 9), 4 × 4 (n = 16), and 5 × 5 (n = 25) for the study. As shown in Section 4, nine mass ratios of $\mu$= 0.02–0.30 were selected for research and compared with the MTMD. The TMD layout in the TMDD is shown in Figure 7, where the spacing between each crossbeam is 2 m and the spacing between the longitudinal main cables is 0.375 m. The longitudinal direction of the bridge is from left to right.

The results indicate that the TMDD is similar to the MTMD, and increasing the mass of the TMD can improve its vibration reduction effect. The maximum acceleration gradually decreases with the increase in u and eventually stabilizes. However, when using the MTMD, as the mass ratio $\mu$ increases, the lower limit of its maximum acceleration is around 1.1 m/s². When using the TMDD, the lower limit of its maximum acceleration is lower than 1.0 m/s², indicating that the TMDD has a better vibration-reduction effect.

In Figure 8, the variation curve of the TMDD damping rate with the mass ratio follows a similar trend to the variation curve of the MTMD damping rate. As the mass ratio increases, the damping rate gradually increases, and when $\mu$ ≥ 0.2, the variation of the damping rate tends to become steady. For a 2 × 2 (n = 4) TMDD, when $\mu$ = 0.1, the damping rate is 0.481, which is 14.2% higher than the 0.421 damping rate of MTMD. When $\mu$= 0.2, the damping rate of TMDD is 0.634, 22.4% higher than the 0.518 damping rate of MTMD. As $\mu$ = 0.3, the damping rate of TMDD is 0.699, which is 29.2% higher than the 0.541 damping rate of MTMD. This shows that as the mass ratio increases, the advantage of using the TMDD over the MTMD regarding the damping effect increases.

When $\mu$ = 0.1, the damping rates of 2 × 2 (n = 4), 3 × 3 (n = 9), 4 × 4 (n = 16), and 5 × 5 (n = 25) are 0.481, 0.477, 0.478, and 0.473, respectively, which are 14.2%, 13.3%, 13.5%, and 12.6% higher than the damping rates of the MTMD. When $\mu$ = 0.2, the damping rates are 0.634, 0.612, 0.609, and 0.597, respectively, which are 22.4%, 18.1%, 17.6%, and 15.2% higher than the damping rates of the MTMD. When $\mu$ = 0.3, the damping rates are 0.699, 0.667, 0.662, and 0.644, respectively, which are 29.2%, 23.3%, 22.4%, and 19.0% higher than the damping rates of the MTMD. This indicates that for this bridge, a 2 × 2 (n = 4) arrangement is optimal. When the mass ratio $\mu$ is small, the difference between the four TMDDs is insignificant, and the damping effect of TMDD gradually decreases with the increase in n.

According to Figure 8, the maximum acceleration decreases as the mass ratio increases. However, when $\mu$ ≤ 0.15, whether using tuned mass damper devices (TMDDs) or multiple tuned mass dampers (MTMDs), the maximum acceleration of the TMDD is smaller than that of the MTMD, but its maximum acceleration is still greater than 1.0 m/s², resulting in a CL3 comfort level. However, when $\mu$ ≥ 0.2, using the MTMD leads to a CL3 comfort level, while using the TMDD reduces the maximum acceleration to below 1.0 m/s², resulting in a CL2 comfort level, indicating better comfort. This difference might be because a single TMD has a specific range of vibration reduction effects, with less overlap in the impact range when arranged in a 2 × 2 configuration. On the other hand, TMDs arranged in 3 × 3, 4 × 4, and 5 × 5 configurations have poorer vibration-reduction effects due to their proximity and greater impact range overlap.

6. TMDD Robustness Analysis

The actual parameters of the structure, such as stiffness and mass parameters, are difficult to describe accurately using mathematical models. The uncertainty of the structure can lead to instability in system control and the deterioration of control performance. Considering the gap between the finite element model and the actual structure, the control robustness of structural stiffness parameter perturbations is studied. The quality of bridge structures generally does not change much [39], and this paper studies the situation where the parameters of the TMD remain unchanged when the natural frequency $\omega$ changes by +10%, +5%, −5%, and −10%, while four different mass ratios of u = 0.02, 0.1, 0.2, and 0.3 are taken according to reference [26]. The pedestrian loading method is the same as when the natural frequency $\omega$ remains unchanged. Comparing the robustness differences between the TMDD and the MTMD under crowd load excitation, the maximum acceleration changes under the four stiffness variations are shown in

Table 6. Comparison of maximum acceleration between TMDD and MTMD with different mass ratios under stiffness changes (m/s²).

Bridge Stiffness Variation	$\mathit{\mu }$	MTMD	2 × 2	3 × 3	4 × 4	5 × 5
Δw = −10%	0.02	1.832	1.785	1.736	1.748	1.755
	0.10	1.514	1.218	1.248	1.237	1.265
	0.20	1.222	0.880	0.930	0.943	0.978
	0.30	1.153	0.720	0.804	0.830	0.863
Δw = −5%	0.02	1.959	1.928	1.919	1.938	1.935
	0.10	1.408	1.209	1.231	1.233	1.235
	0.20	1.163	0.867	0.927	0.925	0.936
	0.30	1.111	0.708	0.799	0.811	0.833
Δw = +5%	0.02	2.101	2.122	2.112	2.106	2.101
	0.10	1.295	1.205	1.208	1.199	1.256
	0.20	1.104	0.864	0.910	0.914	0.941
	0.30	1.060	0.713	0.786	0.795	0.835
Δw = +10%	0.02	1.999	2.032	2.199	2.010	1.998
	0.10	1.213	1.177	1.189	1.185	1.266
	0.20	1.055	0.864	0.889	0.897	0.931
	0.30	1.033	0.733	0.788	0.799	0.827

and Figure 9.

Figure 9 shows that when the stiffness remains constant, the maximum acceleration decreases with an increase in mass ratio. However, the rate of decrease becomes smaller and eventually stabilizes at a specific value. This trend is unaffected by changes in bridge stiffness. Table 6 shows that when the natural frequency varies from −10% to +10%, regardless of whether using the MTMD or TMDD, the maximum acceleration decreases with an increase in the mass ratio. This aligns with the findings in Section 5 and is a common characteristic of cable-supported suspension bridge structures. For the MTMD at the same mass ratio, when the stiffness changes from −10% to +10%, the maximum acceleration first increases and then decreases. This phenomenon is likely due to the change in bridge vibration mode when the stiffness changes and the loading method of the vibration mode is inconsistent with the actual vibration mode, rather than the MTMD having a better vibration reduction effect when the natural frequency of the bridge changes.

According to the analysis in Table 6, the TMDD appears to be more effective in reducing vibration when the natural frequency changes from −10% to +10%. For instance, when Δw = −10% and $\mu$ = 0.2, the maximum acceleration of the TMDD in the 2 × 2 (n = 4) configuration is only 72.0% of the MTMD. Similarly, when Δw = +10% and $\mu$ = 0.2, under the same conditions, the maximum acceleration is only 81.9% of the MTMD. Compared to the scenario where the natural frequency increases, the TMDD shows a better vibration reduction when the natural frequency decreases. This improvement may be associated with the bridge’s vibration mode change. Specifically, when the natural frequency decreases, the vibration mode change is more substantial than when the natural frequency increases.

Moreover, to evaluate the robustness of MTMD and TMDD, the vibration control efficiency is defined as

$$\eta ={\displaystyle \frac{A_f-A_0}{A_0}}$$

(6)

where $A_f$ and $A_0$ are the maximum vibration acceleration values of the bridge when the frequency changes and when the frequency remains constant, respectively.

Table 7. Comparison of maximum $\eta$ values between TMDD and MTMD with different mass ratios under stiffness changes.

Mass Ratio	MTMD	2 × 2	3 × 3	4 × 4	5 × 5
0.02	0.173	0.192	0.251	0.207	0.203
0.1	0.218	0.049	0.047	0.048	0.025
0.2	0.145	0.018	0.044	0.049	0.049
0.3	0.110	0.035	0.023	0.043	0.042

shows the variation of the maximum value of $\eta$ when the natural frequency $\omega$ changes from −10% to +10% under the same mass ratio conditions.

When the natural frequency of the bridge changes, there is a significant relationship between the magnitude of $\eta$ and the mass ratio. The overall robustness improves as the mass ratio $\mu$ increases. When the mass ratio is relatively small (e.g., $\mu$ = 0.02), the MTMD demonstrates a certain advantage in robustness compared to the TMDD. However, when the mass ratio is relatively large (e.g., $\mu$ ≥ 0.1), the TMDD has an advantage in robustness compared to the MTMD. This is reflected in the fact that when the mass ratios are the same, the $\eta$ values of the MTMD are all greater than 0.1, while the $\eta$ values of the TMDD are all greater than 0.05. The final $\eta$ value of the TMDD hardly changes with the number of n, and its robustness is not affected by mass ratio $\mu$.

7. Conclusions

In this study, a spatial finite element model of a cable-supported pedestrian bridge is established to evaluate the vibration reduction effect of the MTMD and TMDD under different mass ratios. Through time domain dynamic analysis and comparison of the vibration reduction effect of the MTMD and TMDD in cable-supported footbridges, the following conclusions are obtained:

• Due to pedestrian loads, the maximum vertical acceleration of the bridge reached 2.35 m/s², exceeding the EN03 standard’s allowable value and negatively affecting pedestrian comfort. Using the MTMD and selecting suitable parameters reduced the vertical acceleration to about 1.10 m/s², achieving a limited vibration reduction effect of approximately 55%. The comfort level reached CL3, hence the MTMD has a limited damping effect.
• The vibration-reduction effect of cable-supported suspension bridge is limited in the common mass ratio range and increasing the mass ratio $\mu$ can effectively improve its vibration reduction efficiency. When using the MTMD layout, the mass of a single TMD is huge and difficult to install. So, the layout of a TMDD is proposed based on an MTMD in this study, and the dynamic response of the structure is analyzed. When the mass ratio $\mu$ ≥ 0.2, compared with the traditional MTMD, the maximum acceleration with the TMDD method was further reduced to a level lower than 0.90 m/s ², and the comfort level reached the “medium CL2” level.
• With a mass ratio $\mu$ ≥ 0.2, the damping effect of the TMDD was more potent than that of the MTMD due to the influence range of each sub-TMD in the TMDD. Different layout forms such as 2 × 2, 3 × 3, 4 × 4, and 5 × 5 were analyzed. The 2 × 2 layout demonstrated the minimum acceleration, while the comfort index difference between the 2 × 2 layout and the other layouts was small.
• The study examined the robustness of the MTMD and TMDD to natural frequency changes by establishing bridge models with Δw = ±10% and Δw = ±5%. The results showed that when the mass ratio is small ($\mu$ = 0.02), the MTMD is more robust than the TMDD. However, when the mass is relatively large ($\mu$ ≥ 0.1), the TMDD demonstrated significant robustness advantages compared to the MTMD, with the $\eta$ value of TMDD hardly changing, indicating independence from the number of the single TMD.