# Dynamic Response of Transmission Tower-Line Systems Due to Ground Vibration Caused by High-Speed Trains

^{*}

## Abstract

**:**

## 1. Introduction

^{2}. With increasing train speed, the ground vibration gradually increases. Hesami et al. [14] used a two-dimensional finite element method to analyze the influence of train vibration on residential buildings near the Qaemshahr railway. The train–ground dynamic model is preliminarily verified by the measured data. The results show that the vibration level decreases significantly with increasing distance from the track centerline to the building. Zhou et al. [15] sampled vibration data from the proposed site near the railway, and the measured ground acceleration was used as the excitation for the proposed building. The law of some trains’ impact on the vibration of nearby buildings was obtained. Erkan et al. [16] studied the ground vibration caused by high-speed trains and its impact on surrounding residential areas through a large amount of field work and many field measurements in Türkiye. The above studies mainly focus on the impact of ground vibration caused by high-speed trains on adjacent high-rise buildings, bridge structures, and residential areas.

## 2. Establishment and Verification of the Finite Element Model

#### 2.1. Establishment and Verification of the Train-Track-Foundation-Soil Coupling Model

#### 2.1.1. Model of the Train

#### 2.1.2. Train-Track Model

#### 2.1.3. Track Irregularities

- (1)
- German low-interference track irregularity spectrum:

_{c}is the spatial cutoff frequency, and ${\Omega}_{\mathrm{c}}$ is the corresponding time truncation frequency (${f}_{\mathrm{c}}=v{\Omega}_{\mathrm{c}}/2\pi $).

- (2)
- Simulated track irregularity spectrum for 350 km/h [32]:$${S}_{v}(f)=\frac{a({f}^{2}{v}^{-3})+b(f{v}^{-2})}{(1+bf{v}^{-1}+c{f}^{3}{v}^{-3})}$$

#### 2.1.4. Subgrade and Foundation Soil Model

- (1)
- Subgrade model parameters

- (2)
- Parameters of the foundation soil model

^{−5}. At this time, the soil is almost completely in the elastic stage. Therefore, the following assumptions [2] are adopted for the soil model in this study. The foundation soil is assumed to be a layered elastic body, and the material of each layer of soil is consistent and simplified as isotropic. The atomic and molecular motions and internal pores of soil particles in the soil are not considered, and continuous functions can be used to describe the changing laws of physical quantities such as soil stress, deformation, and displacement. The initial stress of the soil is neglected.

#### 2.1.5. Calculation of Damping

#### 2.1.6. Wheel–Rail Contact and Track-Subgrade Connection

^{−8}.

#### 2.1.7. Verification of the Finite Element Model

#### 2.2. Analysis of Three-Dimensional Vibration Characteristics and Attenuation Law

#### 2.3. Finite Element Model of the Transmission Tower-Line System

#### 2.3.1. Parameters of the Transmission Tower-Line System

#### 2.3.2. Modal Analysis of the Transmission Tower-Line System

## 3. Results and Discussion

#### 3.1. Working Conditions

#### 3.2. Monitoring Points

#### 3.3. Influence of Different Soil Qualities and Different Track Distances on the Vibration Response of the Transmission Tower-Line System

#### 3.4. Influence of Different Train Speeds on the Vibration Response of the Transmission Tower-Line System

## 4. Conclusions

- (1)
- The ground vibration characteristics of trains are mainly influenced by factors such as track irregularity, soil quality, and train speed. The irregularity of the track has a significant impact on the vibration response of structures near the track, and considering the irregularity of the track, the high-frequency components in the roadbed response are significantly higher than those in the smooth state. The roadbed structure also has a great inhibitory effect on high-frequency vibration at the vibration source, and the attenuation of vibration waves through the roadbed structure to the ground surface vibration beyond 4.5 m of the track can be ignored due to the influence of track irregularity. The soil quality of a free field has a significant impact on vehicle-induced surface vibration: the amplitude of the vehicle-induced vibration response on the surface corresponding to a soft soil foundation is significantly greater than that of hard soil, while the frequency distribution of ground vibration on a hard soil foundation is wider than that on soft soil. The vibration response amplitude of the ground surface increases significantly with increasing vehicle speed, but with increasing vehicle speed, the impact effect of wheel sets on the ground surface near the source gradually weakens;
- (2)
- The predominant frequency of the acceleration responses of the transmission tower under soft and hard soil foundations is mainly within 10 Hz, and the main frequency of both is close to the first-order natural frequency (2.99 Hz) of a single transmission tower. The tower-line structure vibrates mainly in the low-frequency range, and the vibration of trains is distributed in a wide frequency range. The amplitude of the high-level displacement response transfer function of the X-direction and Y-direction tower tops is concentrated in the range of 2~4 Hz. This indicates that the vibration of the tower is sensitive in this frequency range. In addition, the effective displacement along the top of the tower (X direction) is greater than the dynamic response in the vertical direction (Y direction);
- (3)
- The effective value of ground vibration acceleration caused by trains will increase with increasing train speed and decrease with increasing distance to the track. Due to the influence of various factors, such as the frequency distribution, acceleration amplitude, and vibration duration of ground vibration, the dynamic response of the transmission tower-line system is not the same. From the point of view of the effective displacement value of the tower top, when the speed is 350 km/h, the effective displacement value of the tower top under the two kinds of soil is the largest. At a close distance (4.5 m), the acceleration amplitude of hard soil is greater than that of soft soil, so the effective displacement of the top of the tower under hard soil is also greater than that of soft soil. However, due to the different Rayleigh wave velocities of the two soils, the Rayleigh wave velocity corresponding to the soft soil foundation soil is obviously smaller than that of the hard soil foundation soil; therefore, at a relatively long distance (40.5 m), the effective displacement value of the tower top under soft soil is significantly greater than that of hard soil. Overall, for the crossing areas of the soft soil foundation soil in this article, the vibration response of the transmission tower is the highest when the train speed is 250~400 km/h and the distance to the track is within 40 m. The transmission tower in the crossing section of hard soil foundation soil has the highest vibration response when the train speed is 250~350 km/h and the distance to the track is within 30 m.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 4.**Details of the track section. (

**a**) Simplified track cross-section size diagram (mm). (

**b**) Finite element model of ballastless track.

**Figure 5.**Vertical irregularity amplitude simulation time series. (

**a**) Simulated time series of low-interference high–low irregularity amplitude in Germany. (

**b**) The simulated 350 km/h track irregularity amplitude time series.

**Figure 6.**Comparison of the simulated and analytical values of the irregularity power spectrum. (

**a**) The spectrum of German low interference. (

**b**) The spectrum of 350 km/h.

**Figure 7.**The relationship between the amplitude vertical irregularity and the forward distance of the train. (

**a**) The relationship between the amplitude of German low-interference vertical irregularity and the forward distance of the train. (

**b**) The relationship between the amplitude of 350 km/h irregularity amplitude and the forward distance of the train.

**Figure 13.**Comparison of vertical acceleration between the present results and the results in ref. [30]. (

**a**) Vertical acceleration time history diagram. (

**b**) Vertical acceleration amplitude frequency diagram.

**Figure 14.**Time history of ground acceleration at different distances from the center of the track. (

**a**) Time history of ground X-direction acceleration at different distances from the center of the track. (

**b**) Time history of ground Z-direction acceleration at different distances from the center of the track. (

**c**) Time history of ground Y-direction acceleration at different distances from the center of the track.

**Figure 15.**Three-dimensional dynamic response diagram of the roadbed surface. (

**a**) The acceleration time history. (

**b**) Amplitude frequency of the acceleration.

**Figure 16.**Time history and amplitude frequency of ground three-dimensional acceleration at distances of 4.5 m, 19.5 m, and 49.5 m from the track center. (

**a**) The acceleration time history. (

**b**) Amplitude frequency of the acceleration.

**Figure 18.**Partial vibration mode diagram of the transmission tower-line structure and single transmission tower. (

**a**) First-order vibration mode of single tower (2.99 Hz). (

**b**) Second-order vibration mode of single tower (3.73 Hz). (

**c**) The first mode of vibration (0.200 Hz).

**Figure 21.**Transmission tower monitoring point diagram. (

**a**) Transmission tower main material monitoring points. (

**b**) Transmission tower top monitoring points.

**Figure 22.**The X- and Y-direction acceleration time history diagram from 250 km/h to different distances from the track center. (

**a**) Soft soil foundation. (

**b**) Hard soil foundation. (

**c**) Soft soil foundation. (

**d**) Hard soil foundation.

**Figure 23.**The X- and Y-direction acceleration amplitude-frequency diagram at different distances from the track center at 250 km/h. (

**a**) Soft soil foundation. (

**b**) Hard soil foundation. (

**c**) Soft soil foundation. (

**d**) Hard soil foundation.

**Figure 25.**The maximum compressive stress distribution of the main material at different distances from the track center at 250 km/h. (

**a**) Soft soil. (

**b**) Hard soil.

**Figure 26.**The maximum tensile stress distribution of the main material at different distances from the track center at 250 km/h. (

**a**) Soft soil. (

**b**) Hard soil.

**Figure 27.**X-direction acceleration time history response at 13.5 m under soft and hard soil at different speeds. (

**a**) Soft soil foundation. (

**b**) Hard soil foundation.

**Figure 28.**X-direction acceleration amplitude-frequency diagram at 13.5 m under soft and hard soil at different speeds. (

**a**) Soft soil foundation. (

**b**) Hard soil foundation.

**Figure 30.**The maximum compressive stress of the main material is distributed along the height at different speeds. (

**a**) Soft soil foundation. (

**b**) Hard soil foundation.

**Figure 31.**The maximum tensile stress of the main material at different speeds was distributed along the height. (

**a**) Soft soil foundation. (

**b**) Hard soil foundation.

**Figure 32.**Transmission tower top displacement effective value fitting diagram. (

**a**) Data fitting of effective displacement of tower top in X direction. (

**b**) Data fitting of effective displacement of tower top in Y direction.

**Figure 33.**Transfer function of the displacement at the top of the tower response under a soft soil foundation. (

**a**) Transfer function of X-direction displacement response at the top of the tower. (

**b**) Transfer function of Y-direction displacement response at the top of the tower.

Vehicle Parameters | Size |
---|---|

${M}_{\mathrm{c}}\left(\mathrm{kg}\right)$ | 47,900 |

${J}_{\mathrm{c}}\left({\mathrm{kg}/\mathrm{m}}^{2}\right)$ | Lateral is 8.224 × 10^{6} |

Vertical is 8.232 × 10^{6} | |

Longitudinal is 2.751 × 10^{5} | |

${M}_{\mathrm{t}}\left(\mathrm{kg}\right)$ | 1381 |

${J}_{\mathrm{t}}\left({\mathrm{kg}/\mathrm{m}}^{2}\right)$ | Lateral is 1695 |

Vertical is 2844 | |

Longitudinal is 1378 | |

${M}_{\mathsf{\omega}}\left(\mathrm{kg}\right)$ | 1400 |

${K}_{\mathrm{s}1}\left(\mathrm{N}/\mathrm{m}\right)$ | 1.87 × 10^{6} |

${C}_{\mathrm{s}1}\left(\mathrm{N}\cdot \mathrm{s}/\mathrm{m}\right)$ | 5 × 10^{5} |

${K}_{\mathrm{s}2}\left(\mathrm{N}/\mathrm{m}\right)$ | 1.72 × 10^{5} |

${C}_{\mathrm{s}2}\left(\mathrm{N}\cdot \mathrm{s}/\mathrm{m}\right)$ | 1.92 × 10^{5} |

Tire size (m) | 0.46 |

Distance from coupler to coupler (m) | 2.50 |

Wheel base 2d (m) | 2.50 |

Structure Layer Name | Width (m) | Depth (m) |
---|---|---|

Track board | 2.4 | 0.20 |

CA mortar bed | 2.4 | 0.05 |

Concrete base | 3 | 0.30 |

Rail bearing | 0.25 | 0.16 |

Structure Layer Name | Density (kg/m ^{3}) | Elastic Modulus (Pa) | Poisson’s Ratio |
---|---|---|---|

Steel rail | 7800 | 2.06 × 10^{11} | 0.25 |

Rail bearing | 2500 | 3.60 × 10^{10} | 0.20 |

Track board | 2600 | 3.50 × 10^{10} | 0.17 |

CA mortar bed | 1800 | 9.20 × 10^{6} | 0.40 |

Concrete base | 2500 | 2.40 × 10^{10} | 0.20 |

Names of Each Foundation Bed | Thickness (m) | Dynamic Elastic Modulus (MPa) | Poisson’s Ratio | Density (kg/m^{3}) | Cohesion (Pa) | Internal Friction Angle (°) | Damping Ratio |
---|---|---|---|---|---|---|---|

Surface layer of foundation bed | 0.4 | 120 | 0.3 | 2184 | 7 × 10^{4} | 27 | 0.045 |

Bottom layer of subgrade bed | 2.3 | 70 | 0.3 | 1939 | 5 × 10^{4} | 23 | 0.039 |

Embankment | 3.6 | 50 | 0.35 | 1837 | 4 × 10^{4} | 20 | 0.035 |

Name of the Subgrade Structure of Each Layer | Angle of Friction (°) | Flow Stress Ratio $\mathit{K}$) | Expansion Angle (°) | Compression Yield Stress (Pa) | Absolute Plastic Strain |
---|---|---|---|---|---|

Surface layer of foundation bed | 27 | 0.855 | 0 | 177847.90 | 0 |

Bottom foundation bed | 23 | 0.876 | 0 | 122247.00 | 0 |

Embankment | 20 | 0.892 | 0 | 95136.97 | 0 |

Soil Type | Name of Each Layer of Soil | Thickness (m) | Dynamic Elastic Modulus (MPa) | Poisson’s Ratio | Shear Wave Velocity (m/s) | Density (kg/m^{3}) | Damping Ratio |
---|---|---|---|---|---|---|---|

Soft soil | Silty clay | 6 | 30 | 0.290 | 78.27 | 1898 | 0.050 |

Silt clay | 9 | 14 | 0.300 | 56.21 | 1704 | 0.050 | |

Sandy silt | 24 | 74 | 0.310 | 123.66 | 1847 | 0.050 | |

Uniform elastic half-space soil layer | 21 | 141 | 0.330 | 167.03 | 1900 | 0.023 | |

Hard soil | Silty clay | 6 | 124 | 0.302 | 158.40 | 1898 | 0.020 |

Silt clay | 9 | 111 | 0.310 | 157.82 | 1704 | 0.020 | |

Sandy silt | 24 | 159 | 0.318 | 180.83 | 1847 | 0.020 | |

Uniform elastic half-space soil layer | 21 | 141 | 0.330 | 167.03 | 1900 | 0.020 |

Soil Quality | Name | $\mathit{\xi}$ | ${\mathit{\omega}}_{\mathit{i}}$ | ${\mathit{\omega}}_{\mathit{j}}$ | $\mathit{\alpha}$ | $\mathit{\beta}$ |
---|---|---|---|---|---|---|

Soft soil | Surface layer of foundation bed | 0.045 | 3.0536 | 3.9663 | 0.1553 | 0.0128 |

Bottom layer of subgrade bed | 0.039 | 3.0536 | 3.9663 | 0.1346 | 0.0111 | |

Embankment | 0.035 | 3.0536 | 3.9663 | 0.1208 | 0.0100 | |

Silty clay | 0.050 | 3.0536 | 3.9663 | 0.1725 | 0.0142 | |

Silt clay | 0.050 | 3.0536 | 3.9663 | 0.1725 | 0.0142 | |

Sandy silt | 0.050 | 3.0536 | 3.9663 | 0.1725 | 0.0142 | |

Uniform elastic half-space soil layer | 0.023 | 3.0536 | 3.9663 | 0.0794 | 0.0066 | |

Hard soil | Surface layer of foundation bed | 0.045 | 3.9628 | 6.3627 | 0.2198 | 0.0087 |

Bottom layer of subgrade bed | 0.039 | 3.9628 | 6.3627 | 0.1905 | 0.0076 | |

Embankment | 0.035 | 3.9628 | 6.3627 | 0.1709 | 0.0068 | |

Silty clay | 0.020 | 3.9628 | 6.3627 | 0.0977 | 0.0039 | |

Silt clay | 0.020 | 3.9628 | 6.3627 | 0.0977 | 0.0039 | |

Sandy silt | 0.020 | 3.9628 | 6.3627 | 0.0977 | 0.0039 | |

Uniform elastic half-space soil layer | 0.020 | 3.9628 | 6.3627 | 0.0977 | 0.0039 |

**Table 8.**Comparison of acceleration and displacement amplitude between the present results and the results in ref. [38].

Data Sources | Monitoring Point | Vertical Acceleration Amplitude (m/s ^{2}) | Vertical Displacement Amplitude (mm) |
---|---|---|---|

Results of ref. [38] | Ground surface at a distance of 5 m from the track | 0.15 | 1.3 |

Present results | Ground surface at a distance of 4.5 m from the track | 0.18 | 1.4 |

Numbering | Tower Parts | Rod Specifications | Numbering | Tower Parts | Rod Specification |
---|---|---|---|---|---|

1 | Tower leg main material | L80 × 7 | 8 | Inner main material of upper crank arm | L45 × 4 |

2 | Tower leg inclined material | L56 × 5 | 9 | Outer main material of lower crank arm | L63 × 5 |

3 | Tower leg diagonal brace | L40 × 4 | 10 | Inner main material of lower crank arm | L56 × 5 |

4 | Main material of tower body | L80 × 7 | 11 | Tower leg top surface cross- section main material | L56 × 4 |

5 | Tower body inclined material | L45 × 4, L40 × 4 | 12 | Tower body top surface cross- section main material | L100 × 8 |

14 | Cross-arm inclined material | L40 × 4 | 13 | Outer main material of upper crank arm | L63 × 5 |

13 | Cross-arm main material | L50 × 4 | 14 | Tower body | L40 × 4 |

Item | Cross- Sectional Area (mm ^{2}) | Diameter (mm) | Line Density (kg/m) | Elastic Modulus (MPa) | Average Operating Tension (N) | Rupture Force ×0.95 (N) |
---|---|---|---|---|---|---|

LGJ—400/35 | 425.24 | 26.82 | 1.349 | 65,000 | 21,870 | 98,705 |

JLB40-150 | 148.07 | 15.75 | 0.6967 | 103,600 | 23,847 | 90,620 |

Distance to Track (m) | Soft Soil $\mathbf{X}-\mathbf{Direction}\text{}{\mathit{u}}_{\mathbf{rms}}$ (mm) | Soft Soil $\mathbf{Y}-\mathbf{Direction}\text{}{\mathit{u}}_{\mathbf{rms}}$ (mm) | Hard Soil $\mathbf{X}-\mathbf{Direction}\text{}{\mathit{u}}_{\mathbf{rms}}$ (mm) | Hard Soil $\mathbf{Y}-\mathbf{Direction}\text{}{\mathit{u}}_{\mathbf{rms}}$ (mm) |
---|---|---|---|---|

4.5 m | 4.07 | 0.56 | 3.49 | 2.46 |

13.5 m | 4.25 | 0.66 | 4.06 | 2.16 |

22.5 m | 4.25 | 0.84 | 4.17 | 1.75 |

31.5 m | 4.23 | 0.98 | 3.51 | 1.57 |

40.5 m | 3.03 | 0.90 | 3.44 | 1.48 |

Train Speed (km/h) | Soft Soil X-Direction ${\mathit{u}}_{\mathbf{rms}}$ (mm) | Soft Soil Y-Direction ${\mathit{u}}_{\mathbf{rms}}$ (mm) | Hard Soil X-Direction ${\mathit{u}}_{\mathbf{rms}}$ (mm) | Hard Soil Y-Direction ${\mathit{u}}_{\mathbf{rms}}$ |
---|---|---|---|---|

250 | 4.25 | 0.70 | 3.37 | 0.92 |

300 | 3.08 | 0.72 | 3.21 | 1.00 |

350 | 10.10 | 2.22 | 10.00 | 3.47 |

400 | 3.16 | 0.70 | 3.02 | 1.15 |

450 | 3.37 | 0.66 | 4.06 | 2.16 |

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**MDPI and ACS Style**

Zhao, G.; Wang, M.; Liu, Y.; Zhang, M.
Dynamic Response of Transmission Tower-Line Systems Due to Ground Vibration Caused by High-Speed Trains. *Buildings* **2023**, *13*, 2884.
https://doi.org/10.3390/buildings13112884

**AMA Style**

Zhao G, Wang M, Liu Y, Zhang M.
Dynamic Response of Transmission Tower-Line Systems Due to Ground Vibration Caused by High-Speed Trains. *Buildings*. 2023; 13(11):2884.
https://doi.org/10.3390/buildings13112884

**Chicago/Turabian Style**

Zhao, Guifeng, Meng Wang, Ying Liu, and Meng Zhang.
2023. "Dynamic Response of Transmission Tower-Line Systems Due to Ground Vibration Caused by High-Speed Trains" *Buildings* 13, no. 11: 2884.
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