# Key Mechanism Research of Top Plate Thickening of the Box-Girder Bridge for Noise Reduction Design in High-Speed Railway

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Two Kinds of Top Plate Thickening Low-Acoustic Radiation Bridge Design Scheme

- (1)
- The design of the thickened BGB top plate can enhance key parameters, such as the bending stiffness of the BGB, and thus reduce its vibrational response;
- (2)
- The top plate of the BGB is the most important component contributing to its acoustic radiation, and local vibration within it significantly affects its acoustic radiation. Thickening the top plate can change the overall vibrational distribution of the BGB, especially in cases of the partial thickening of the top plate, which will not have a uniform thickness, and this has a stronger impact on the vibrational distribution. Resonance at certain frequencies can be avoided to a certain extent, and the change in vibrational pattern can affect the modal acoustic radiation efficiency of the BGB, thus weakening its acoustic radiation capacity.

## 3. Numerical Analysis

**M**,

**C**, and

**K**are the mass, stiffness, and damping matrices, respectively; $\ddot{u}$, $\dot{u}$ and $u$ are the displacement, velocity, and acceleration vectors, respectively; and F is the generalized load. The vehicle–track–bridge interaction model is shown in Figure 3. In the overall system, the vehicle submodel uses the classical 35-degree-of-freedom model. Each vehicle section includes a vehicle body, two steering racks, and four wheels sets, all considered to be rigid-body structures, connected by the first and second suspension systems; the suspension system exhibits linear and nonlinear characteristics. Each rigid body comprises lateral, vertical, side-roll, rocker, and nodding degrees of freedom; the vehicle model dynamic equations and related parameters are described in the literature [40]. The finite element model is used for determining the structure of the elevated infrastructure. The theoretical finite element model of the track and the BGB structure is shown in Figure 3. The relevant parameters and finite element type are shown in Table 1.

^{−4}s, and the bridge system can be solved using the implicit Newmark-β method with a time step of 10

^{−3}s. To calculate the vibrations of the BGB, the irregularity of the high-speed tracks is used to represent the excitation; the foregoing is also used as the acoustic boundary condition in the second step of the analysis. For these calculations, we assume that a CRH380B high-speed train under tare conditions is moving at a speed of 200 km/h.

## 4. Comparative Analysis of Acoustic Vibration Characteristics of BGBs for Three Working Conditions

- The stronger frequency band of BGB acoustic radiation can, in general, be divided into two bands, namely, the 0–40 Hz lower-frequency band and the 80–100 Hz severe vibration band. In these two bands, the BGB shows strong acoustic radiation characteristics. The phenomenon of concern is that the acoustic radiation of BGBs in the lower frequency band of different sound fields is very strong—even stronger than that seen in BGBs under the effects of severe external excitation. This phenomenon is contradictory to the general law of strong acoustic radiation caused by severe vibration. Therefore, the reasons for this phenomenon and the key issues to be addressed when designing a thickened top plate to reduce noise are analyzed in detail in the following, providing a basis for the further optimization of the subsequent design;
- The two noise reduction designs with top plate thickening have obvious noise reduction effects, the most significant being in the lower frequency bands, and there are also substantial reductions in sound pressure at certain frequencies. From the vibrational characteristics of the BGB, it can be seen that the vibration of the bottom plate intensifies at some frequencies in the lower frequency band, and so the sound pressure in the sound field directly below the BGB increases at the corresponding frequency (refer to the curve of the sound pressure spectrum of the sound field SF1). However, the noise reduction effect around 6.8 Hz is worse compared with those at other frequency bands, especially in the sound field under the bridge. Although the BGB has an obvious noise reduction effect in the main vibration band, the sound pressure in the sound field below the slope of the two BGBs with thickened top plates is enhanced (refer to the curve of the sound pressure spectrum of sound field SF2), and the reducing effect on the sound field above the bridge is poor compared with other sound fields (refer to the curve of the sound pressure spectrum of sound field SF4).

## 5. Key Issues of BGB Top Plate Thickening Design in Acoustic Vibration Control

#### 5.1. Analysis of Self-Vibration Characteristics of BGB under Three Working Conditions

#### 5.2. Vibration Acceleration Distribution Characteristics of BGB Subjected to External Excitation under Three Working Conditions

^{2}and 0.72 m/s

^{2}, respectively, while the maximum amplitude of the improved BGB will be seen in the rest of the plate. That is, the vibration of the rest of the plate is obviously increased, and the vibration of the top plate is effectively suppressed. When thickening the top plate of the BGB, in addition to the overall vibration of the BGB, its vibration distribution will be changed, while in other cases, such as when applying 15.8 Hz, 24.8 Hz and 25.9 Hz, not only will the local vibration distribution of the top plate be changed, but that of the rest of the plate will be changed too.

#### 5.3. Exploration of the Key Influence of Vibration Distribution of BGB on Its Sound Radiation

- (1)
- The BGBs under the three conditions have similar inherent frequencies at 6.8 Hz. The vibration pattern and external excitation vibration patterns take the form of first-order overall vertical bending, forming overall resonance in the BGBs. This vibration pattern does not correlate with strong acoustic radiation efficiency, and the BGBs under the three conditions show a high acoustic radiation capacity when subjected to very small vibration instead. This shows that, at this time, in the BGB subjected to resonance, a small vibration can give rise to a strong acoustic radiation capacity. Although the vibration of the BGB at 6.8 Hz is small, the thickened top plate also attenuates the vibration amplitude of the BGB, which can serve to reduce the acoustic radiation capacity of the BGB to a certain extent;
- (2)
- The 10th order vibration patterns of the BGBs under the three operating conditions change significantly, with the BGB with a thickened top plate showing bottom plate local self-oscillation characteristics. The acoustic radiation efficiency of the BGB with a thickened top plate is substantially reduced, which in turn reduces the acoustic radiation capacity of the BGB—for example, the maximum amplitudes of the initial BGB and that with a thickened top plate are 0.60 m/s
^{2}and 0.54 m/s^{2}, respectively, which is a smaller difference, but with a 9.6 dB sound power attenuation. Further, the sound radiation efficiency and the maximum vibration amplitude of the locally thickened BGB under the top rail plate have been reduced to a greater extent compared with the initial BGB, so the noise reduction effect is better. The comparison of the vibroacoustic characteristics between the two types of top plate-thickened BGBs and the initial BGB shows that the sound radiation efficiency of the attenuated BGBs in the low frequency band is better in terms of noise reduction as opposed to vibration reduction; - (3)
- The vibroacoustic characteristics of the BGB in the 12th order mode under three operating conditions further affirm the results of the second analysis, and the significant reduction in sound radiation efficiency directly reduces the sound radiation capacity of the BGB. The web of locally thickened BGBs under the top rail plate also shows a certain resonance under these conditions, so the noise reduction effect of the design with the thickening of the whole top plate is better;
- (4)
- The natural vibration characteristics of the BGBs in the 18th order mode under the three operating conditions are quite different. As discussed above, the small-amplitude vibration of BGBs in this frequency band under external excitation will be shifted, and the maximum vibration amplitude of the two BGBs with thickened top plates will be increased substantially. Still, the problem is that the acoustic radiation capacity of the BGB with a thickened top plate under the rail will be slightly enhanced. In contrast, the sound radiation capacity of the BGB with a whole thickened top plate will be weakened. At this time, the natural vibration characteristics of the two top plate-thickened BGBs can be shown as the local natural vibration characteristics of the bottom plate and the top wing plate, respectively, and these affect the sound radiation efficiency of the BGB. This shows that the top plate of the BGB plays a particularly important role in its sound radiation and is a key area to control when aspiring to noise reduction. It also shows that, in the low-frequency band, the vibration of the BGB is not the most critical factor contributing to sound radiation;
- (5)
- The vibration patterns of the BGBs subjected to severe external excitation are similar under each of the three working conditions, and their acoustic radiation capacities decrease as the vibration is reduced. Therefore, the acoustic radiation of BGBs in the severe vibration band can be controlled mainly by reducing vibration.

## 6. Conclusions

- (1)
- When designing a low-acoustic-radiation BGB, it is recommended to first focus on changing its vibration distribution. Thickening the top plate of the BGB can significantly change its vibration distribution—not only do the self-oscillation characteristics change significantly, but the vibrational frequency of similar vibration characteristics caused by external excitation will be shifted to the weaker frequency band of the initial BGB’s acoustic radiation. This evades the resonance effect caused by the small vibration resulting from the peak of acoustic radiation, and weakens the acoustic radiation efficiency of the BGB, thus reducing its acoustic radiation capacity. This will effectively reduce the acoustic radiation performance of a BGB in the lower frequency band;
- (2)
- Regarding the sound radiation, compared with the initial BGB, the two noise reduction schemes can limit the sound radiation to the weaker frequency band of the initial BGB, although the designs enhance the sound radiation characteristics and expand its range in this frequency band. Regarding the sound radiation in BGBs induced by severe vibration, compared to small vibration, it is entirely possible to achieve noise reduction by reducing vibration, and the implementation of noise reduction control measures is also easier under this approach. Therefore, thickening the top plate is a good solution for achieving low-sound-radiating BGBs, and the two design solutions involving top plate thickening described in this paper can contribute effectively to vibration and noise reduction;
- (3)
- Regarding the acoustic radiation characteristics of BGBs related to small-amplitude vibration in the lower frequency band, the design of vibration and noise reduction schemes is more difficult, and it is necessary to combine sound radiation mechanisms to formulate a more effective noise reduction scheme. If only a single damping measure is used, aiming at reducing the vibration, it is difficult to reduce the BGB radiation acoustic waves. The mechanisms involved in the two types of BGB noise reduction via top plate thickening and the key issues show that, by changing the vibration distribution of the BGB, the peak sound radiation, as determined by the characteristics of the BGB itself, can be reduced, so damping the vibration and changing the vibration distribution can be combined to achieve a better noise reduction effect.

- It is difficult in the current study to accurately determine the optimal thickness of the top plate of a low-noise-radiating BGB that meets all other requirements. An optimization algorithm can be adopted to determine the optimal top plate thickness for the low-frequency noise control of a BGB, the reference [46] maybe give us a great inspiration. Meanwhile, it is necessary to address all other engineering requirements.
- It will be difficult to use an optimization algorithm to determine the optimal thickness without mastering the key mechanisms of the noise reduction schemes.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Schematics of a BGB with a thickened top plate in a high-speed railway: (

**a**) initial; (

**b**) whole top plate thickened; (

**c**) top plate locally thickened under the track.

**Figure 2.**Schematic representation of the vibration-acoustic radiation characteristics analysis of box-girder bridge.

**Figure 4.**In situ measurement of the noise radiation of high-speed railway BGB: (

**a**) photograph of measurement site; (

**b**) calculated and measured sound pressure levels.

**Figure 6.**The vibration acceleration comparison of the top plat-thickening design schemes: (

**a**) top plate center point; (

**b**) bottom plate center point.

**Figure 7.**Comparison of the sound pressure level at different sound field points in three kinds of BGBs: (

**a**) SF1; (

**b**) SF2; (

**c**) SF3; (

**d**) SF4.

**Figure 9.**The distribution of sound field patterns of BGBSs under the same reference values for three case conditions (Unit: dB).

**Figure 11.**Vibration acceleration characteristics and distribution under external excitation of three BGBs (Unit: m/s

^{2}).

**Figure 12.**Comparison of vibration acceleration clouds of the remaining plate members at 93.6 Hz (Unit: m/s

^{2}).

**Figure 14.**Acoustic vibration characteristics of BGBs at 27.1 Hz under the three case conditions (unit of vibration: m/s

^{2}).

**Figure 15.**Correspondence diagram of vibrational acoustic radiation of the BGB at 85.7 Hz under the three working conditions.

Track Components | Finite Element Type | Related Parameters | Value |
---|---|---|---|

Rails | Beam188 | Modulus of elasticity/(Pa) | $2.1\times {10}^{11}$ |

Sectional moment of inertia/(m^{4}) | $3.215\times {10}^{-5}$ | ||

Poisson ratio | 0.3 | ||

Line Density(kg·m^{−1}) | 60.64 | ||

Fasteners | Combin14 | Fastener stiffness/(N·m^{−1}) | $4\times {10}^{7}$ |

Fastener damping/(N·s·m^{−1}) | $2.2656\times {10}^{4}$ | ||

Vertical spacing/(m) | 0.625 | ||

Track | Solid45 | Length/(m) | 4.93 |

Width/(m) | 2.4 | ||

Thickness/(m) | 0.2 | ||

Modulus of elasticity/(Pa) | $3.6\times {10}^{10}$ | ||

Poisson’s ratio | 0.25 | ||

Density/(kg·m^{−3}) | 2500 | ||

CA mortar layer | Combin14 | Stiffness/(N·m^{−1}) | $9.375\times {10}^{9}$ |

Damping/(N·s·m^{−1}) | $7.5\times {10}^{5}$ | ||

Base | Solid45 | Modulus of elasticity/(Pa) | $3.3\times {10}^{10}$ |

Poisson’s ratio | 0.2 | ||

Density/(kg·m^{−3}) | 2500 | ||

Bridge | Shell63 | Length/(m) | 32.5 |

Thickness of top plate/(m) | 0.315 | ||

Thickness of web/(m) | 0.480 | ||

Thickness of base plate/(m) | 0.300 | ||

Modulus of elasticity/(Pa) | $3.8\times {10}^{10}$ | ||

Poisson’s ratio | 0.25 | ||

Density/(kg·m^{−3}) | 2500 |

**Table 2.**Comparison of self-vibration characteristics of three types of BGBs used in high-speed railways.

Number of Steps | Characteristics | Initial BGB | Whole Thickened Top Plate | Local Thickened Top Plate |
---|---|---|---|---|

Step 1 | Inherent frequency (Hz) | 3.4 | 3.1 | 3.2 |

Description of vibration model | Bridge overall lateral tilt | Bridge overall lateral tilt | Bridge overall lateral tilt | |

Step 2 | Inherent frequency (Hz) | 7.0 | 6.5 | 6.7 |

Description of vibration model | Bridge first-step vertical bend | Bridge first-step vertical bend | Bridge first-step vertical bend | |

Step 3 | Inherent frequency (Hz) | 12.4 | 15.5 | 13.8 |

Description of vibration model | Top plate local vibration | Bridge overall torsion | Top plate local vibration | |

Step 4 | Inherent frequency (Hz) | 13.2 | 15.6 | 15.3 |

Description of vibration model | Top plate local vibration | All plates local vibration | Top plate local vibration | |

Step 5 | Inherent frequency (Hz) | 14.8 | 17.2 | 16.6 |

Description of vibration model | Top plate local vibration | Overall bridge vibration | Overall bridge torsion | |

Step 6 | Inherent frequency (Hz) | 15.8 | 17.6 | 17.3 |

Description of vibration model | Top plate local vibration | Top plate local vibration | Overall bridge vibration | |

Step 7 | Inherent frequency (Hz) | 16.2 | 19.1 | 18.1 |

Description of vibration model | Overall bridge vibration | Top plate local vibration | Top plate local vibration | |

Step 8 | Inherent frequency (Hz) | 17.1 | 22.3 | 20.1 |

Description of vibration model | Overall bridge vibration | All plates local vibration | All plates local vibration | |

Step 9 | Inherent frequency (Hz) | 19.2 | 24.1 | 22.5 |

Description of vibration model | Top plate local vibration | Top plate local vibration | Top plate local vibration | |

Step 10 | Inherent frequency (Hz) | 21.0 | 24.3 | 23.8 |

Description of vibration model | Overall bridge vibration | Bottom plate local vibration | Bottom plate local vibration | |

Step 11 | Inherent frequency (Hz) | 21.9 | 28.1 | 27.4 |

Description of vibration model | All plates local vibration | All plates local vibration | All plates local vibration | |

Step 12 | Inherent frequency (Hz) | 23.7 | 29.0 | 27.8 |

Description of vibration model | Top plate local vibration | All plates local vibration | All plates local vibration | |

Step 13 | Inherent frequency (Hz) | 26.1 | 32.6 | 29.5 |

Description of vibration model | All plates local vibration | Top plate local vibration | Top plate local vibration | |

Step 14 | Inherent frequency (Hz) | 27.2 | 35.8 | 34.7 |

Description of vibration model | Overall bridge vibration | All plates local vibration | All plates local vibration | |

Step 15 | Inherent frequency (Hz) | 29.6 | 38.8 | 37.8 |

Description of vibration model | Top plate local vibration | Overall local vibration of bridge | Overall local vibration of bridge | |

Step 16 | Inherent frequency (Hz) | 31.4 | 42.6 | 38.1 |

Description of vibration model | All plates local vibration | Top plate local vibration | Top plate local vibration | |

Step 17 | Inherent frequency (Hz) | 34.8 | 44.4 | 40.5 |

Description of vibration model | Top plate local vibration | All plates local vibration | All plates local vibration | |

Step 18 | Inherent frequency (Hz) | 36.7 | 46.6 | 42.1 |

Description of vibration model | Top plate local vibration | Bottom plate local vibration | Wing plate local vibration | |

Step 19 | Inherent frequency (Hz) | 36.8 | 46.7 | 42.7 |

Description of vibration model | Top plate local vibration | All plates local vibration | All plates local vibration | |

Step 20 | Inherent frequency (Hz) | 37.6 | 47.1 | 43.1 |

Description of vibration model | Top plate local vibration | Bottom plate local vibration | Top plate local vibration |

**Table 3.**Comparison of acoustic vibration characteristics of BGBs used in a high-speed railway under three working conditions.

Acoustic Vibration Characteristics | Characteristic Index | Initial BGB | Whole Thickened Top Plate of BGB | Local Thickened Top Plate of BGB |
---|---|---|---|---|

Low Frequency Band | ||||

Self-oscillation—2nd order mode | Inherent frequency (Hz) | 7.0 | 6.5 | 6.7 |

Description of vibration type | Bridge first-step vertical bend | Bridge first-step vertical bend | Bridge first-step vertical bend | |

Vibration characteristics by external excitation | Vibration frequency (Hz) | 6.8 | 6.8 | 6.8 |

Vibration pattern (maximum amplitude—m/s^{2}) | First step overall vertical bend (0.20) | First step overall vertical bend (0.10) | First step overall vertical bend (0.12) | |

Acoustic radiation characteristics | Acoustic radiation efficiency | 0.11 | 0.09 | 0.09 |

Sound Power | 115.7 | 106.5 | 107.8 | |

Key factors for noise reduction | Resonance phenomenon formed by BGB | |||

Self-oscillation—10th order mode | Inherent frequency (Hz) | 21.0 | 24.3 | 23.8 |

Description of vibration type | Overall bridge vibration | Bottom plate local vibration | Bottom plate local vibration | |

Vibration characteristics by external excitation | Vibration frequency (Hz) | 21.4 | 24.8 | 23.7 |

Vibration pattern (maximum amplitude—m/s^{2}) | Bridge overall bending vibration (0.60) | Top plate local vibration (0.54) | Local vibration at both ends of the top plate (0.24) | |

Acoustic radiation characteristics | Acoustic radiation efficiency | 3.74 | 0.66 | 0.40 |

Sound Power | 120.2 | 110.6 | 98.7 | |

Key influencing factors of acoustic radiation | Acoustic radiation efficiency of BGBs | |||

Self-oscillation—12th order mode | Inherent frequency (Hz) | 23.7 | 2.0 | 27.8 |

Description of vibration type | Top plate local vibration | All plates local vibration | All plates local vibration | |

Vibration characteristics by external excitation | Vibration frequency (Hz) | 23.7 | 29.3 | 27.1 |

Vibration pattern (maximum amplitude—m/s^{2}) | Local vibration at both ends of the top plate (0.37) | Top plate local vibration (0.16) | Top/web plate Local vibration (0.17) | |

Acoustic radiation characteristics | Acoustic radiation efficiency | 2.54 | 0.60 | 0.78 |

Sound Power | 117.8 | 98.6 | 103.3 | |

Key influencing factors of acoustic radiation | Acoustic radiation efficiency of BGBs | |||

Self-oscillation—18th order mode | Inherent frequency (Hz) | 36.7 | 46.6 | 42.1 |

Description of vibration type | Top plate local vibration | Bottom plate local vibration | Wing plate local vibration | |

Vibration characteristics by external excitation | Vibration frequency (Hz) | 37.2 | 46.2 | 41.7 |

Vibration pattern (maximum amplitude—m/s^{2}) | Top plate local vibration (0.09) | All plates local vibration (0.31) | Top plate local vibration (0.72) | |

Acoustic radiation characteristics | Acoustic radiation efficiency | 0.89 | 0.71 | 0.58 |

Sound Power | 104.1 | 102.7 | 107.3 | |

Key influencing factors of acoustic radiation | Resonance phenomenon formed by BGB | |||

Severe vibration frequency band | ||||

Vibration characteristics by external excitation | Vibration frequency (Hz) | 93.6 | 93.6 | 93.6 |

Vibration pattern (maximum amplitude—m/s^{2}) | All plates local vibration (1.50) | All plates local vibration (1.20) | All plates local vibration (1.07) | |

Acoustic radiation characteristics | Acoustic radiation efficiency | 108.5 | 106.2 | 105.0 |

Key influencing factors of acoustic radiation | Severe vibration formed by BGBs (vibration reduction) |

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## Share and Cite

**MDPI and ACS Style**

Zhang, X.; Zhang, X.; Song, G.; Xu, J.; Yang, L.
Key Mechanism Research of Top Plate Thickening of the Box-Girder Bridge for Noise Reduction Design in High-Speed Railway. *Appl. Sci.* **2023**, *13*, 8958.
https://doi.org/10.3390/app13158958

**AMA Style**

Zhang X, Zhang X, Song G, Xu J, Yang L.
Key Mechanism Research of Top Plate Thickening of the Box-Girder Bridge for Noise Reduction Design in High-Speed Railway. *Applied Sciences*. 2023; 13(15):8958.
https://doi.org/10.3390/app13158958

**Chicago/Turabian Style**

Zhang, Xiaoan, Xiaoyun Zhang, Gao Song, Jiangang Xu, and Li Yang.
2023. "Key Mechanism Research of Top Plate Thickening of the Box-Girder Bridge for Noise Reduction Design in High-Speed Railway" *Applied Sciences* 13, no. 15: 8958.
https://doi.org/10.3390/app13158958