1 Introduction

The high proportion of bridges and fast operation speed are the typical characteristics of high-speed railways (HSRs), resulting into more serious low-frequency structural noise in elevated railways. Low-frequency noise seriously affects the human body health and living environment. Meanwhile, it will have a great impact on the survival environment and mutual communication of the wild animals [1,2,3,4,5]. At present, the bridge structure noise even becomes a key problem limiting the development of the new generation high-speed railways HSRs of operation speed of 400 km/h in China. For better understanding and thus addressing this problem, many theoretical prediction methods [6,7,8,9,10] have been developed, and many vibration and noise reductions have been adopted [11,12,13,14].

The noise research on HSR bridges starts from the operation of Shinkansen in Japan, because the passing trains brought great noise impact to the surrounding residents’ living environment. Due to the difficulty in noise control, the cost of controlling bridge noise of Shinkansen even exceeds the construction cost. Therefore, the bridge structure noise problem always attracts the great attention of scholars in the world. In the early stage, due to the lack of theoretical research methods, the bridge structural noise is mainly studied by experiment [15, 16]. With the development of research, the acoustic boundary element method (BEM) and statistical energy analysis (SEA) method have been adopted to predict the noises radiated by railway bridges in the low- and middle–high-frequency ranges, respectively [17]. In order to improve the computational efficiency of BEM, the mode acoustic transfer vector method [18] and 2.5 dimensional (2.5D) technology [19] are combined with the traditional BEM. Subsequently, the hybrid numerical calculation methods have been proposed to study bridge structural noises, such as finite element method (FEM) and SEA [20, 21], BEM and SEA [22], power flow method and finite or infinite element method [23], waveguide FE and 2.5 BEM [24], and FEM and equivalent infinitesimal sound source summation method [25]. In these methods, indirect BEM method has been widely used because of its applicability and versatility. These researches reveal that the noise radiation of the concrete box-girder bridges (BGBs) is mainly in low frequency, and due to its own structural characteristics, even slight vibrations can cause strong sound radiation.

The theoretical research also reveals that the train-induced bridge noise radiation is a system problem involving the train, track, and bridge subsystems. However, at current, the vibration and noise reduction is over-reliant on measures adopted in the track system, such as the vibration absorber of rail, reducing vibration fastener, reducing vibration sleeper, and reducing vibration track bed [26,27,28,29]. These vibration reduction measures adopted in the track system have good vibration reduction effects on the bridge, but their effect on reducing the bridge structural noises is very limited and even bring about other adverse problems [30]. Meanwhile, for train running safety and track structure stability, the HSR track system uses very few vibration reduction measures. Therefore, it is difficult to control the structural noise radiated by concrete BGB in HSR by adopting vibration reduction measures in the track system, especially for the low-frequency noises caused by slight vibrations.

For the vibration and noise reduction of HSR BGB, it is a more feasible way to directly adopt measures on the bridge structure. Thickening the bridge plates have effective noise reduction effect [31,32,33]. In addition, laying damping materials on the bridge can obtain a good noise reduction effect [34, 35]. Since concrete bridges mainly radiate low-frequency structural noise, installing the tuned mass damper (TMD), composed of a mass block and the spring-damping element, has an effective vibration reduction effect [36]. On this basis, the multiple tuned mass dampers (MTMDs) with higher robustness are applied to reduce the low-frequency vibrations of bridges caused by running train [37,38,39,40,41]. However, there is little research on the bridge noise control due to MTMDs [42, 43]. In the previous studies, there are two typical manners to reduce the low-frequency vibrations of concrete BGB using MTMDs, i.e., installing MTMDs on the top or bottom plate of the bridge in the cavity. However, the frequency for the controlled vibrations is relatively single. Furthermore, the acoustic principle poses a greater complexity compared to the vibration principle for the BGB. Therefore, the current installation manners of MTMDs cannot result in effective noise reduction for the concrete BGB. It is imperative to improve the noise reduction effect MTMDs on BGB.

Therefore, in order to suppress the low-frequency noise radiation of the concrete BGB caused by the small vibrations, this paper proposes a design scheme for installing MTMDs locally on the top plate of the BGB in its cavity. For this purpose, an acoustic analysis model is developed for the concrete BGB using the train–track–bridge interaction theory and acoustic boundary element theory. Then, combined with the sound radiation mechanism of BGB [44], the installation positions and parameters of MTMDs for the BGB are redesigned. Finally, the noise reduction effect and relevant mechanism of MTMDs installed on the top plate of the BGB in the cavity are investigated. The research results can provide the theoretical guidance for the MTMDs application in the low-frequency structure noise control of the HSR BGB.

2 Acoustic analysis model of BGB with MTMDs

2.1 Outline of acoustic analysis model

As the most used in China’s HSR, the 32-m concrete simply supported BGB is selected as the research object, of which geometric dimensions are listed in Table 1. In order to systematically analyze the influences of MTMDs on the acoustic-vibration characteristics of BGBs, a train–track–bridge–MTMDs coupled dynamics model and acoustic boundary element model of a BGB are developed, based on the train–track–bridge interaction theory [45, 46] and the acoustic boundary element theory.

Table 1 Finite element types and dynamic parameters of track and bridge
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The developed train–track–bridge–MTMDs coupled dynamics model is illustrated in Fig. 1. In the model, each vehicle of the train is simplified as multi-rigid-body system of 35 degrees of freedom. The CRTS-I ballastless track, the 32 m simply supported BGB, and the MTMDs are modeled by FEM. Their motion equations can be written in the matrix form as follows:

Fig. 1
figure 1

Train–track–bridge interaction schematic diagram and railway structure finite element model on HSR

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Train subsystem

$${\varvec{M}}_{{\text v}} {\varvec{\ddot{u}}}_{{\text v}} + {\varvec{C}}_{{\text v}} {\dot{\varvec{u}}}_{{\text v}} + {\varvec{K}}_{{\text v}} {\varvec{u}}_{{\text v}} = {\varvec{F}}_{\text {wr}} ({\varvec{u}}_{{\text v}} ,{\dot{\varvec{u}}}_{{\text v}} ,{\varvec{u}}_{{\text t}} ,{\dot{\varvec{u}}}_{{\text t}} ).$$
(1)

Track subsystem

$$\begin{aligned} {\varvec{M}}_{{\text t}} {\varvec{\ddot{u}}}_{{\text t}} + {\varvec{C}}_{{\text t}} {\dot{\varvec{u}}}_{{\text t}} + {\varvec{K}}_{{\text t}} {\varvec{u}}_{{\text t}} = & {\varvec{F}}_{\text {wr}} ({\varvec{u}}_{{\text v}} ,{\dot{\varvec{u}}}_{{\text v}} ,{\varvec{u}}_{\text {t}} ,{\dot{\varvec{u}}}_{\text {t}} ) \\ & - {\varvec{F}}_{\text {tb}} ({\varvec{u}}_{\text {t}} ,{\dot{\varvec{u}}}_\text{t} ,{\varvec{u}}_{\text {b}} ,{\dot{\varvec{u}}}_{\text {b}} ) .\\ \end{aligned}$$
(2)

Bridge subsystem with MTMDs

$$\begin{aligned} {\varvec{M}}_{\text b} {\varvec{\ddot{u}}}_{\text b} + {\varvec{C}}_{\text b} {\dot{\varvec{u}}}_{\text b} + {\varvec{K}}_{\text b} {\varvec{u}}_{\text b} & = {\varvec{F}}_\text {tb} ({\varvec{u}}_{\text t} ,{\dot{\varvec{u}}}_{\text t} ,{\varvec{u}}_{\text b} ,{\dot{\varvec{u}}}_{\text b} ) \\ & - \sum\limits_{i = 1}^{N_m} F_{\text{bm},\text i} \delta ({\varvec{x}}_{\text{bm},{\text i}} ) .\end{aligned}$$
(3)

The ith TMD

$$ M_{\text{m},{i}} \ddot{u}_{\text{m},{i}} + C_{\text{m},{i}} \dot{u}_{\text{m},{i}} + K_{\text{m},{i}} u_{\text{m},{i}} = F_{\text{bm},{i}}, $$
(4)

where M, C, and K represent the mass, damping, and stiffness matrices, respectively; u is the displacement vector; and subscripts v, t, b, and m denote the vehicle, track, bridge, and MTMDs subsystems, respectively; \({\varvec{F}}_{\text{wr}} ({\varvec{u}}_{\text{v}} ,{\dot{\varvec{u}}}_{\text{v}} ,{\varvec{u}}_{\text{t}} ,{\dot{\varvec{u}}}_{\text{t}} )\) is the vector of dynamic wheel–rail forces, which are calculated by the dynamic wheel/rail interaction model; \({\varvec{F}}_{\text{tb}} ({\varvec{u}}_{\text{t}} ,{\dot{\varvec{u}}}_{\text{t}} ,{\varvec{u}}_{\text{b}} ,{\dot{\varvec{u}}}_{\text{b}} )\) is the load vector composed of the dynamic track–bridge interaction forces, which are determined by the relative motions between the track and bridge subsystems; Nm is the total number of MTMDs, xbm,i is the coordinate of the ith TMD on the bridge, and \(F_{\text{bm},{i}}\) is the interaction force between the bridge and the ith TMD, which can be calculated by

$$ F_{\text{bm},{i}} = K_{i} (u_{\text{b},{i}}- u_{\text{m},{i}} ) + C_{i} (\dot{u}_{\text{b},{i}} - \dot{u}_{\text{m},{i}} ), $$
(2)

where Ki and Ci are the stiffness and damping of the ith TMD, respectively; and ub,i is the displacement of bridge at the location of ith TMD.

By taking the track irregularities as the excitation, the dynamic responses of the whole system can be obtained using the hybrid explicit–implicit numerical integration method. Then, taking the bridge vibration accelerations as the acoustic boundary condition, the noise radiation of the bridge can be calculated by the developed acoustic BE model. The details of the modeling process can be found in our previous works [44, 47] and are not detailed explained here.

2.2 Model verification

To verify the developed model, the calculated noise radiation is compared with the measured one. Because the MTMDs have not yet been applied in the China’s HSR bridge, the measurement and calculation are implemented in the case that the BGB is not installed with MTMDs. The main parameters of the track and BGB are shown in Table 1. The measurements are performed on the viaduct line of a high-speed railway in China, as shown in Fig. 2a. In the measurements and calculations, the CRH380B high-speed train is moving at a speed of 200 km/h.

Fig. 2
figure 2

Calculated and measured sound pressure level radiated by HSR BGB: a field test photo; b comparison of results

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Figure 2 demonstrates the calculated and measured sound pressure level at the 3 m and 5 m directly below the bridge. It can be found that the calculated and measured results are in well agreement in the studied frequency range. Their change trend with time is basically consistent, and their peak values appear at the similar frequencies. Therefore, the developed acoustic prediction model of the BGB is reliable.

3 Parameter design of MTMDs

MTMDs are primarily designed to reduce vibration and noise of the controlled structure by suppressing its vibration and noise radiation at the predominant frequencies. Therefore, in this section, the vibration and acoustic characteristics of BGBs are systematically analyzed first. On this basis, the key design parameters of MTMDs are identified according to the predominant frequencies and vibration-sensitive locations of BGB.

3.1 Acoustic-vibration characteristics of BGBs

In order to reveling the acoustic-vibration characteristics of BGB, the bridge vibration accelerations at center points of top, web, and bottom plates are selected as the analysis object. The selected sound field points are illustrated in Fig. 3. The vibration accelerations, the sound pressures of different sound field points, the sound power, and the sound radiation efficiency of the bridge are shown in Fig. 4. Since the frequency range of BGB acoustic radiation is mainly concentrated in the low-frequency range below 200 Hz, which has been proved by many existing research. Therefore, for more effective investigation of acoustic-vibration characteristics of BGB, the upper limit of the analysis frequency is 260 Hz in this paper. In addition, the train running speed is set at 200 km/h in the analysis.

Fig. 3
figure 3

Sound field analysis points of initial BGB at mid-span

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Fig. 4
figure 4

Acoustic-vibration radiation characteristics of initial BGB: a vibration accelerations; b sound pressures; c sound power of BGB; d acoustic radiation efficiency

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As shown in Fig. 4a, the prominent vibrations of the BGB are concentrated in the frequency range below 100 Hz, among which the most evident vibrations are concentrated in 80–100 Hz, and some local peaks appear in 0–80 Hz. According to Fig. 4b, the prominent amplitudes of sound pressure of the BGB are mainly concentrated in the low-frequency range below 250 Hz. The sound pressure amplitudes are more distinct in the frequency range below 100 Hz. In the frequency range of 80–100 Hz, the distinct amplitudes of sound pressure are caused by the severe vibrations (see Fig. 4a). Meanwhile, in the frequency range below 80 Hz, the quite large sound pressure at frequencies around the local peaks of vibrations are caused by small-amplitude vibrations. The reason for this phenomenon is that the sound radiation of the BGB in this frequency range is dominant by the BGB structural characteristics [44].

The BGB noise caused by large vibrations can be effectively reduced by vibration reduction measures which do not change the BGB structural characteristics. However, due to its structural characteristics, the small vibrations of BGB can even result in great noise radiation that is difficult to mitigate through vibration reduction measures. Focusing on this issue, the MTMDs are adopted in this paper to change the vibration distribution and structural characteristics of the BGB, thereby reducing its ability to radiate the noises of this type rather than simply reducing the vibrations. The parameter design of MTMDs for noise reduction can be performed according to this phenomenon, which will be explained in the following sections.

As shown in Fig. 4c, the sound power has two prominent frequency ranges of 0–40 and 80–100 Hz, illustrating that the BGB has great acoustic radiation capacity in these two frequency ranges. Combined with Fig. 4a, it can be found that these two frequency ranges are corresponding to small and severe vibrations, respectively. The acoustic radiation efficiency represents the ability of BGB to radiate sound. Figure 4d shows that the BGB has very strong acoustic radiation efficiency in low-frequency range of 10–30 Hz. The acoustic radiation efficiency reaches the peak value at 15.8 Hz and thereafter approaches 1. Our previous works [44, 47] demonstrate that the vibration modes of BGB (as shown in Fig. 5) play a crucial role on these strong acoustic radiation abilities caused by small vibrations at 10–30 Hz.

Fig. 5
figure 5

Vibration modes and distribution of BGB: a–c displacements in mm at 7.0, 15.8, and 23.7 Hz; d–f accelerations in m/s2 at 6.8, 15.8, and 27.1 Hz

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In summary, the main acoustic-vibration frequency range of BGB can be divided into two notable frequency ranges of 0–80 and 80–100 Hz. In the former frequency range, the sound radiations are caused by the small vibrations and its own structural characteristics; while, in the latter frequency range, the sound radiation of the bridge is mainly caused by severe vibrations. Therefore, the parameters of MTMDs can be designed based on the acoustic and vibration characteristics of BGB, to suppress the resonance phenomenon, the acoustic radiation efficiency, and the severe vibration of BGB, thereby lowering the sound radiation ability of BGB.

3.2 Design scheme of low-noise BGB by installing MTMDs on top plate

Based on the above analysis in Sect. 3.1, the acoustic radiation efficiency of BGB is larger at 7.0, 15.8, and 23.7 Hz. Accordingly, the vibration modes nearby these frequencies are selected as the main control modes. For this purpose, the natural vibration frequencies of the MTMDs are tuned to these control frequencies. Moreover, the MTMDs are installed in the middle span and both ends of top plate, in which the TMDs in middle span are designed to mainly control the vibration mode at frequency of 7.0 Hz, and the TMDs at plate ends are designed to mainly suppress the vibration modes at frequencies of 15.8 and 23.7 Hz. The vibrations of bridge end at 15.8 and 23.7 Hz are very significantly. Hence, the parameters of TMDs at plate ends are designed to be different for a more comprehensive suppression effect on the sound radiation at these frequencies.

Since the BGBs are the cavity structure and the BGB top plate is the main sound radiation contributor [43, 44, 47], it is more appropriate to install the MTMDs on the top plate of the BGB inside the cavity for reducing the cost and installing difficulty. In general, the MTMDs are installed with equal spacing for vibration reduction [37,38,39]. However, the vibration modes of BGB top plate have both whole and local vertical vibrations. The amplitudes of modal shape and train-induced vibrations of top plate often occur in the middle span and both ends of the top plate. Therefore, the arrangement of MTMDs is as follows: (1) In the transverse direction, the MTMDs are installed with equal spacing; and (2) the middle span and both ends of top plate are taken as the main installation positions of the MTMD.

In general, it is favored to set a larger mass ratio of TMD to the controlled object for better vibration reduction effects. However, too large mass will increase the bridge bearing weight, installation difficulty, economic costs, and other negative effects. Here, the mass ratio of TMD to the top plate is set as 0.01. The TMD stiffness can be decided by the TMD mass and the control frequency. The design parameters of TMDs are shown in Table 2.

Table 2 Design parameters of MTMDs
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In summary, the parameters and installation locations of MTMDs are designed as those given in Table 2 and Fig. 6.

Fig. 6
figure 6

Schematic diagram of MTMDs local installation on the top plate of a BGB

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4 Noise reduction mechanism of BGB installed with MTMDs

4.1 Vibration and noise reduction caused by MTMDs

In order to analyze the effects of MTMDs on acoustic-vibration characteristics of BGB, the natural vibration characteristics and train-induced vibrations of BGBs are investigated, as shown in Table 3 and Fig. 7. According to Table 3, installing MTMDs on top plate can obviously change the natural vibration characteristics of BGB, in which the most obvious effect is that the local vibrations of BGB are enhanced. From Fig. 7, it can be observed that installing MTMDs on top plate has a very significant vibration reduction effect, especially in the frequency range of severe vibration.

Table 3 Natural vibration characteristics of BGBs with and without MTMDs
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Fig. 7
figure 7

Comparison of vibration reduction effect for the BGB

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Figure 8 shows the effects of MTMDs on the sound pressure radiated by BGB. It can be found that installing MTMDs on top plate has a certain reduction effect on the noise radiation of BGB. However, the reduction effect is limited in the frequency range of 80–100 Hz which is corresponding to the severe vibration. In the frequency range of below 80 Hz, the noise reduction effect of MTMDs on top plate is relatively better, and except at the frequencies of global modes of BGB, the sound radiation is significantly reduced.

Fig. 8
figure 8

Comparison of sound pressure at different sound points

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The effects of MTMDs on the sound pressure radiated by BGB at typical frequencies are extracted from Fig. 8 into Table 4. It can be seen that installing MTMDs on top plate has a good noise reduction effect at most frequencies. Especially for those in Table 4 italics, the noise reduction effect is very significant. For example, the sound pressures of all field points are greatly reduced at frequencies of 16.9, 18.0, 25.9, 27.1, 29.3, and 30.4 Hz. Especially at 27.1 Hz, the sound pressures of SF1–4 are reduced by 15.5, 8.2, 15.1, and 12.3 dB, respectively. The sound pressures of sound field are enhanced at the main controlled frequencies, but the noise reduction effect is better at the nearby frequencies, which are related to the tuning frequencies of TMDs.

Table 4 Comparison sound pressure of sound field points at different frequencies
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According to above analysis, the noise reduction mechanism of BGBs due to installing MTMDs on top plate can be summarized as the following two points:

  • (1) Installing MTMDs on top plate can reduce the acoustic radiation efficiency of BGB by changing the natural vibration mode. As shown in Fig. 9, installing MTMDs on top plate has a certain suppression effect on the acoustic radiation efficiency of the BGB at 16–70 Hz, and thus can inhibit the sound radiation ability of BGB in this frequency range. However, the noise control effect is poor when the BGB vibration is dominated by the global vertical bending mode.

  • (2) Installing MTMDs on top plate can reduce the BGB vibration and change the distribution of train-induced vibration in top plate. As shown in Fig. 10, the top plate of initial BGB demonstrates a typical local vibration mode at 27.1 Hz, which has a great influence on the sound radiation. The vibration amplitudes and the vibration distribution in top plate are significantly suppressed by installing MTMDs, and the acoustic radiation efficiency is effectively suppressed.

Fig. 9
figure 9

Comparison of acoustic radiation efficiency

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Fig. 10
figure 10

Vibration acceleration distribution at 27.1 Hz (unit: m/s2): a initial BGB; b BGB with MTMDs

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The noise radiation of BGB is greatly reduced at 27.1 Hz through the together action of the above two points. By analyzing the change of sound power contribution of different plates due to installing MTMDs (as shown in Fig. 11 and Table 5), it can be concluded that MTMDs mainly realize the noise reduction purpose by reducing the acoustic radiation contribution of plates. The sound power contribution of top plate decreases significantly at 20–70 Hz. The sound power contribution of web plate is effectively suppressed at 0–30 Hz. However, the influence of MTMDs on the contribution of bottom plate is very small.

Fig. 11
figure 11

Comparison of sound power contribution of BGB plates

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Table 5 Comparison sound power contribution of BGB plates at different frequencies
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In addition, a comparison between Figs. 7 and 8 reveals that, in the frequency range of 80–100 Hz, the MTMDs have a notable vibration reduction effect, but the noise reduction effect is not such obvious. The main reason is that the BGB radiated noises are not only related to the structure vibration intensity, but also to the structure topological shape, radiation efficiency, vibration distribution, transmission medium, and transmission path, etc. [43]. Therefore, the bridge radiated noises are potentially not lowered simply by reducing the BGB vibrations. As demonstrated by Figs. 9 and 11, in this frequency range, the acoustic radiation efficiency of the BGB is not reduced by MTMDs, and the BGB still has strong acoustic radiation capability. Additionally, in this frequency range, sound power contributions from all plates are also not significantly weakened. These indicate that in this frequency range, the acoustic radiation of the BGB is not directly related to its vibration intensity.

4.2 Influences of MTMDs on acoustic radiation efficiency of BGB

As shown from Fig. 9, the acoustic radiation efficiencies of BGB are significantly suppressed by installing MTMDs on top plate at 21.4, 23.7, and 27.1 Hz. The noise reduction effects caused by MTMDs are good at 27.1 Hz, and the relevant mechanism has been elaborated in the above analysis. Nevertheless, at 21.4 and 23.7 Hz, the sound pressures are increased. The relevant mechanism will be explained below.

Figures 12 and 13 show the vibration characteristics of BGB nearby 23.7 Hz. It can be found that the natural frequency of BGB installed with MTMDs deviates from 23.7 Hz, and the vibration distribution on all plates is greatly increased. Therefore, the sound pressure of SF4 is decreased, while the sound pressures of SF1 – 3 are increased.

Fig. 12
figure 12

Natural vibration characteristics of BGB nearby 23.7 Hz in mm: a the 12th mode of initial BGB, 23.7 Hz; b the 14th mode of BGB with MTMDs, 22.1 Hz; c the 15th mode of BGB with MTMDs, 25.1 Hz

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Fig. 13
figure 13

Vibration acceleration distribution at 23.7 Hz (unit: m/s2): a initial BGB; b BGB with MTMDs

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In addition, the sound radiation efficiency of BGB is obviously decreased at 21.4 Hz, while the sound pressures in sound field are increased. It can be seen that at this frequency, installing MTMDs on top plate changes the natural vibration mode of the BGB from the global mode to the local mode located at ends of top plate, as shown in Fig. 14. According to our previous works [44, 47], the acoustic radiation performance of BGB is related to these modes. At 21.4 Hz, the lateral modal wave number and the acoustic wave number are 0.47 and 0.39, respectively. Therefore, the acoustic wave number is less than the modal wave number on the lateral direction. Moreover, the edge-type vibration modes of BGB top plate result into that the sound wave cancels each other in lateral direction. The acoustic radiation efficiency of BGB is effectively weaken because MTMDs suppress lateral acoustic waves of top plate. However, in the changed vibration mode, the ends of top plate occur local resonance phenomenon, and the vibrations of other plates are intensified. Therefore, the BGB structural noise increases.

Fig. 14
figure 14

Vibration characteristics nearby 21.4 Hz: a initial BGB; b BGB with MTMDs

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In summary, the parameter design of MTMDs for noise reduction of BGB should consider the acoustic radiation efficiency besides the vibration reduction. The acoustic radiation efficiency is closely related to vibration distribution of BGB. Installing MTMDs on the top plate can affect the vibration distribution and sound radiation law of BGB; nevertheless, it has little effect on the sound radiation caused by severe vibrations (80–100 Hz) and vibrations dominated by the global modes of the BGB, as shown in Fig. 15.

Fig. 15
figure 15

Corresponding relationship of sound radiation law and external excitation vibration of BGB

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5 Conclusion

Installing MTMDs on the top plate of BGB in the cavity can effectively reduce the BGB vibrations, but the relevant noise reduction effects are not clear. In the previous studies on vibration reduction of BGB installed with MTMDs, only some certain order modes or the vibrations at some certain frequencies are controlled. However, the sound radiation mechanism of BGB is very complicated. Without clear noise reduction mechanism about BGB installed with MTMDs, the reduction effect on noise radiation of BGB caused by installing MTMDs will be very limited. Therefore, a sound radiation prediction model for the BGB installed with MTMDs is developed here, to study the noise reduction mechanism of BGB installed with MTMDs on the top plate. The main conclusions are as follows:

  1. (1)

    Installing MTMDs on the top plate of concrete BGB can greatly reduce its low-frequency vibrations, especially at frequencies of 80–100 Hz. Meanwhile, at the lower frequencies, the installation of MTMDs can change the acoustic radiation efficiency, modal shapes, and vibration distributions of the BGB. Parameter design of MTMDs for noise reduction of the BGB should consider these factors comprehensively.

  2. (2)

    MTMDs achieve the goal of reducing BGB noise radiation mainly through reducing the acoustic radiation contribution of plates. The sound power contribution of top plate is decreased significantly at 20–70 Hz. The sound power contribution of web plate is effectively suppressed at 0–30 Hz.

  3. (3)

    Installing MTMDs on top plate can affect the vibration distribution and sound radiation law of the BGB; nevertheless, it has little effect on the sound radiation caused by severe vibrations and vibrations dominated by the global modes of the BGB.

  4. (4)

    In low-frequency range, the structural characteristics of BGB play a key role in its acoustic radiation ability, particularly for the great acoustic radiation caused by small vibrations. Installing MTMDs can reduce the noises of this type by changing the vibration frequency and vibration distributions in the BGB to avoid the small vibrations and to disperse the contribution of sound radiation of each plate.

Although the design scheme proposed in this paper has not verified by practical engineering application and the designed parameters are not the optimal, the greatest research significance of this work is to reveal the noise control mechanism of BGB by installing MTMDs on top plate. It can lay some theoretical foundation for the parameter design of MTMDs for noise reduction of the concrete BGB. However, the TMD is difficult to achieve noise reduction effects on BGBs in a wide frequency range, besides its noise reduction effect is usually weaker than its vibration reduction effect, especially for the BGB radiated noise dominant by its own structural characteristics. Therefore, designing the TMD parameters needs to consider its influences on the vibration distributions, the natural frequencies, and mode shapes, etc., resulting in the BGB resonance at other frequencies and affecting the modal acoustic radiation efficiency to enhance its acoustic radiation capability. These are the key problems that affect the practical application of the TMD. In the future, based on BGB vibration characteristics in the main frequency range of its acoustic radiation, we think that the research will focus on the installation distance and number of MTMDs, and the optimization algorithm of TMD to further improve the noise reduction effects of MTMDs on HSR BGB.