Time–Frequency Domain Analysis of the Ground Vibration of an Elevated Railway and Study on the Elliptic Polarization Dispersion Characteristics of Rayleigh Waves

Abstract

Elevated railways are a crucial component of railway lines, characterized by their widespread distribution, simple structure, and low cost, while actively promoting local economic development. However, they also cause significant ground vibrations when trains pass. Similarly, considerable vibration levels are transmitted to the subgrade and surrounding structures when trains operate on viaducts within the Loess Plateau region. However, research on mitigating these vibration effects remains relatively scarce. This study focused on the impacts of such vibrations on surrounding buildings and stratum structures and evaluated the effectiveness of a vibration isolation trench in mitigating these effects. Time frequency domain analysis of ground vibrations during train passage revealed that the characteristic frequency of the train-induced pulse excitation in the track structure had a pronounced peak in the spectrum curve. The introduction of a vibration isolation trench effectively blocked the propagation of vibration waves in the soil, reduced soil vibration, and significantly lowered the peak value in the spectrum. Numerical simulations were employed to analyze the elliptical polarization dispersion characteristics of surface wave propagation with the vibration isolation trench in place, confirming the effective damping performance of the trench. These findings could offer a valuable reference for high-speed railway vibration isolation and significantly advance the application of surface wave theory in high-speed railway technology.

1. Introduction

Since the 1990s, high-speed railways have been widely constructed and developed worldwide owing to their high speed and low energy consumption. However, this development is accompanied by various issues including noise pollution, electrical radiation, and ground vibration [1,2,3]. In particular, the vibration caused by high-speed trains during their operation significantly affects residents’ quality of life and the manufacturing or application of precision equipment, thereby presenting a considerable environmental hazard [4,5]. In response to these challenges, numerous researchers have dedicated efforts to exploring and applying various methods to address these problems.

The essence of ground vibration lies in the propagation of wave energy, which is similar to that observed in earthquakes [6,7]. Time–frequency domain analysis of displacement can be an effective method for studying ground vibration and is widely employed in various applications, including mechanical fault analysis, mechanical shock vibration signal analysis, and the study of small deformations in tire self-excitation [8,9,10]. In civil engineering, this analytical technique is also used to detect foundation deformation and tunnel vault deformation and to perform other structural assessments [11,12,13].

Similarly to the propagation process of sound waves, measures can be taken along the wave propagation path to achieve shock absorption or vibration isolation [14,15]. The most conventional method uses barriers [16], which function based on the principles of wave refraction and reflection. Common forms of barriers include open trenches and row piles [17]. Among these, vibration isolation trenches are most frequently employed in roads, bridges, and other applications. This technique involves excavating a channel near the road to interrupt the propagation path of vibration or to alter the propagation form of surface waves, thereby reducing the energy of wave propagation and minimizing the impact of vibration on buildings behind the trench [18,19]. Celebi and Kumar verified the damping effect of vibration isolation trenches using numerical simulations and field tests [20,21,22]. In recent years, owing to the advancement of large-scale computational software, such as finite element analysis, and the high cost of field tests, many researchers have relied on numerical simulations to investigate and analyze the effectiveness of vibration isolation trenches [15,23]. Zhang Huilai conducted physical model experiments using organic glass simulation slots to investigate this phenomenon [24]. Research has indicated that the empty trenches can provide superior vibration isolation compared with other methods.

The study of the dispersion characteristics of the elliptic polarization rate of Rayleigh surface waves dates back to early research in geophysical exploration. Notably, in 1969, the American geophysicists Boore and Toksöz explored the relationship between the formation media structure and Rayleigh waves [25]. A significant advancement in this field was the introduction of the single-point spectral ratio method (H/V spectral ratio method), which calculated the Fourier spectral amplitude ratio of the horizontal and vertical components of microtremor information collected by a single three-component geophone to estimate the site characteristics [26,27]. Building on this foundation, Zhang Li applied the elliptical characteristics of Rayleigh surface waves to invert the velocity model of formation structures [28] and studied Rayleigh wave behavior in a two-dimensional undulating terrain [29]. In 2015, research on the elliptic polarization dispersion characteristics of Rayleigh surface waves in a two-dimensional transverse medium at the Wenfu site established a basis for three-dimensional spatial numerical simulations [19]. Liu Shijie applied the radial grid method to investigate the elliptic polarization dispersion characteristics of surface wave propagation in three-dimensional space [30,31]. Isolation trenches find predominant application in highway engineering, while their implementation in railway systems-particularly viaduct railways-remains relatively limited. This gap is exemplified by the scarcity of experimental investigations, such as those conducted by [32,33], examining the vibration isolation efficacy of trenches on subgrade and underlying soil strata.

The widespread distribution of elevated railways significantly differs from that of urban rail transit systems. This study was conducted on the Loess Plateau, an area characterized by a thick loess cover, and addressed the relatively underexplored topic of vibration isolation for high-speed railways in loess regions, particularly for elevated lines. The study began with an analysis of the nature of ground vibration in the time–frequency domain. This study first analyzed the properties of ground vibrations in the time–frequency domain, utilizing three-component geophones to examine displacement variations in each direction. This included analyses of velocity dispersion characteristics and ellipticity dispersion characteristics. Numerical simulations were subsequently conducted using a large-scale nonlinear finite element analysis software package. The vibration displacement patterns were compared both before and after the incorporation of a 10-m-deep vibration isolation trench to evaluate the effectiveness of the open trench as a vibration mitigation measure. These findings were validated using field data. This study could offer a valuable theoretical foundation for selecting high-speed railway vibration isolation technologies and significantly advancing the application of surface wave theory in high-speed railway technology.

2. Basic Method

This chapter mainly introduces the basic theories and methods used in the article, as well as their relationship with the research content.

2.1. Time Domain Analysis

Time domain analyses involve examining the stability, transient response, and steady-state performance of a system based on the time domain expression of the output for a given input. In this study, time domain analysis was applied to the test results of the displacement time history of ground vibration. Using Equation (1), the maximum peak displacement at each measurement point in the X, Y, and Z directions was calculated [34]. Higher train speeds induce greater vibrations, as evidenced by the variations in peak displacement values across different directions at various monitoring locations with distance. The attenuation patterns in all three directions at these locations are similar. Specifically, for a train speed of $v$ within 0–75 m from the embankment, vibrations decay rapidly, with peak displacements generally exhibiting Y > Z > X magnitudes; beyond 75 m to 180 m, attenuation becomes more gradual, and peak displacement values in all three directions become comparable.

(1)$D_{max}=MAX|D\left(t\right)|$

2.2. Frequency Domain Analysis

The vibration spectrum effectively reflects the frequency components of vibration and the distribution of vibrational energy, allowing for intuitive observation of frequency and energy concentration regions. This allows for the extraction of patterns that are not easily discernible in the time domain, summarizing the dominant frequency distribution at ground measurement points and the frequency characteristics of vibrations induced by periodic loading from high-speed trains. By applying the Fast Fourier Transform (FFT) to the time domain data sequence obtained from the tests, the vibration spectrum for ground measurement points at various distances and in various directions caused by trains traveling at different speeds was obtained according to the following equation [35].

(2)$F_{\mathrm{A}}(\omega )={\displaystyle {\int }_{-\infty }^{+\infty }A(t)}\cdot e^{-i\omega t}dt$

where A(t) is a time domain data sequence; $F_A\left(\omega \right)$ is a sequence of spectral functions in the frequency domain; and ω = 2πf.

2.3. Elliptic Polarization Rate

The elliptic polarization rate exhibits constant behavior within homogeneous elastic half-space media, whereas in layered media, it exhibits frequency-dependent behavior. The authors of [36] proposed that this parameter can be derived by calculating the spectral ratio between the horizontal and vertical component spectra, denoted as H(f) and V(f), respectively, following Fourier transformation. This approach yields the defining equation:

(3)$Er\left(f\right)=H\left(f\right)/V\left(f\right)$

Equation (3) reveals the frequency dependence of the elliptic polarization rate. This dependence constitutes a fundamental characteristic common to all modes of Rayleigh surface waves and is designated as elliptic polarization dispersion.

Time domain analysis reveals localized concentration characteristics in the vibration response across measurement points, alongside discernible trends in ground vibration displacement amplitude and vibration duration with increasing distance from the source. Frequency domain (spectral) analysis extracts patterns not statistically discernible in the time domain, enabling the identification of dominant frequency distributions at ground measurement points. This analysis also elucidates the frequency characteristics of vibrations induced by the periodic loading of high-speed trains and reveals the vibration spectra across different measurement points, directions, and distances resulting from trains operating at varying speeds. Consequently, time domain analysis and frequency domain analysis serve as the primary diagnostic tools in this study.

3. Data Acquisition and Processing

This chapter details the data acquisition methodology, including the experimental scheme, site selection, and instrumentation. Furthermore, it presents the analysis of ground vibration experimental results obtained from a representative viaduct track segment without vibration isolation trenches.

3.1. Experimental Background and Plan

The test site was located at the elevated section near K988 + 848 of Weinan North Station on the Zhengzhou-Xi’an High-Speed Railway. The strata at the test site are predominantly quaternary, with a total thickness of approximately 200 m. The lithology from top to bottom consists of approximately 19 m of self-weight collapsible loess from the Upper Pleistocene, 28 m of non-collapsible loess from the Upper Pleistocene, and a stable sand layer from the Middle Pleistocene. The groundwater in the test area is quaternary pore phreatic water, with a depth of approximately 30 m. The selected test area features a wide, flat terrain surrounded by extensive farmland, with no buildings or other traffic vibrations affecting the site. This setting thus provided favorable conditions for arranging the measurement points and analyzing the ground vibration effects caused by trains (Figure 1). The main beam of the elevated bridge in the experimental area is made of C50 concrete, with a beam top width of 11.4 m, a beam height of 2 m, and a beam bottom width of 5.4 m. The pier columns are simply supported beams with equal heights and are made of C40 concrete. The length of the pier bottom in the longitudinal direction of the bridge is 2.2 m, and the length in the transverse direction of the bridge is 2.8 m. The high-speed train operated by the Zhengxi High speed Railway is the “Harmony” CRH2C high-speed train, and the characteristic length of the test train is shown in (Figure 2).

Numerical simulations adopt proportional modeling. The test points were located within the straight section of the interval. There was no obvious waveform wear to the rail and no damage to the track within 200 m of the test section in either direction. On the ground, perpendicular to the direction of the railway line, a measurement point was placed every 15 m, and the total length of the survey line was 180 m. In total, 12 points were selected. The arrangement of the measurement points is illustrated in Figure 3. The test instrument was a three-component EPS portable digital seismograph (Figure 4), The EPS portable digital seismometer incorporates an internal data logger and a high-sensitivity three-component sensor. With a frequency bandwidth of 0.2–200 Hz, it encompasses the 1–80 Hz range typically required for ambient vibration measurements. Configurable sampling rates include 1000, 500, 250, 100, and 50 sps. and it was ensured that the bottom of the sensor was in close contact with the ground during the setting up of the instrument. To ensure consistency between experimental testing and numerical simulations, the following coordinate system is defined: the X-direction aligns with the railway track’s forward travel direction, the Y-direction is perpendicular to the survey line, and the Z-direction runs parallel to the survey line.

3.2. Analysis of Ground Vibration Test Results for the Elevated Track Section

In this study, data collected as eight EMU trains passed through the test section on the elevated track at a speed of 240 km/h were selected for analysis. The running direction of these trains was Xi’an-Zhengzhou (near-track passing).

3.2.1. No Isolation Ditch Time Domain Analysis

Figure 5 illustrates the ground vibration–displacement time history curves for the measurement points 3, 6, and 9 in the X, Y, and Z directions, respectively, during the passage of trains at a speed of 240 km/h.

Figure 5 demonstrates that the vibration response of each measurement point exhibited local concentration characteristics in the time domain and was unstable. The time history curve for the measurement point nearest the pier clearly demonstrates a periodic peak caused by the wheelset as a train passes, which was less noticeable at the measurement point further away from the pier, where the vibration amplitude was small and background vibration was more prominent. The amplitude of the ground vibration displacement decreased with increasing distance, with a notable vibration rebound at 90 m. Significant differences were observed in the time history curves for the X, Y, and Z directions, with the overall displacement being the smallest in the X direction (along the track), followed by the Y direction (along the survey line, perpendicular to the track), and the largest in the Z direction (vertically, perpendicular to the ground). The vibration caused by trains increased at higher speeds. The attenuation patterns in the three directions were similar, with the peak displacements following the order Z > Y > X. The attenuation was more pronounced between 0 and 75 m, becoming more gradual from 105 to 180 m, with a noticeable rebound increase at 90 m in all three directions.

3.2.2. No Isolation Ditch Frequency Domain Analysis

Figure 6 presents the Fourier vibration spectra for three measurement points at a train speed of 240 km/h, illustrating how the spectrum varied with the distance in different directions. This Figure 6 reveals that the high-frequency vibrations attenuated more rapidly than the low-frequency vibrations, with the spectrum at longer distances predominantly controlled by the low frequencies centered around 10 Hz. At a speed of 240 km/h, the ground vibration frequencies remained below 50 Hz, with the main frequencies in the X, Y, and Z directions being generally similar. Specifically, the main frequency at 15 m ranged from 35 to 43 Hz, while at 30 m it was approximately 30 Hz. At 90 m, there was a noticeable vibration rebound phenomenon with a rebound frequency of approximately 26 Hz, indicating a trend towards a richer frequency component and increased displacement amplitude. Beyond 90 m, the frequency spectrum attenuated to less than 20 Hz, and the corresponding ground vibration displacement amplitude decreased. This shift was due to the increasing dominance of low-frequency components and the attenuation of high-frequency components, resulting in a main frequency for the far-field measurement points closer to the low frequencies.

When the trains were in operation, their characteristic frequency generated pulse excitation in the track structure, causing the spectrum curve to exhibit peak values near the characteristic frequency of the train. Near the track, the ground vibration was significantly influenced by the fixed wheelbase excitation frequency ($f_1$), the vehicle fixed distance excitation frequency ($f_2$), the center distance excitation frequency ($f_3$), and the length excitation frequency ($f_4$) of the adjacent bogies of the front and rear cars. As the distance from the track increased, the influence of the vehicle fixed distance excitation frequency ($f_2$) and the vehicle length excitation frequency ($f_4$) on the frequency amplitude decreased, primarily due to the cut-off effect of the soil’s natural frequency. At greater distances, the fixed wheelbase excitation frequency ($f_1$), the center distance excitation frequency ($f_3$), and their frequency doubling effects from the adjacent bogies of the front and rear vehicles became more dominant in determining the vibration response amplitude at the ground measurement points.

4. Empty Trench Vibration Isolation Effect Test

This chapter presents the experimental time–frequency domain results for ground vibrations measured at a viaduct track segment equipped with vibration isolation trenches.

4.1. Time Domain Analysis with the Implementation of the Vibration Isolation Trench

Figure 7 presents the ground vibration–displacement time history curves for measurement points 3, 6, and 9 in the X, Y, and Z directions, respectively, when trains passed at a speed of 240 km/h with the implementation of the vibration isolation trench.

From Figure 7, in the near-field source area close to the pier at a distance of 45 m, the vibration characteristics resembled those of an unexcavated area, with a notable local concentration. At greater distances, the periodic peak caused by the wheelset of each passing train was less pronounced, with reduced vibration amplitude and increased background vibration effects. Notably, at 90 m, the presence of a vibration isolation trench significantly reduced the peak value compared with the condition without a trench, demonstrating that the trench effectively mitigated the vibrations. Specifically, the displacement of the soil behind the trench was reduced by at least 10% in the X, Y, and Z directions compared with the area before the trench. However, at 135 m, the effect of the vibration isolation trench was comparable to that without the trench, indicating that this area represented the far-field zone where ground vibration became stable.

4.2. Frequency Domain Analysis with the Implementation of the Vibration Isolation Trench

Figure 8 illustrates the vibration spectra recorded at 12 measurement points in different directions when trains passed at 240 km/h with the implementation of the vibration isolation trench. The Fourier transformation of the displacement time history curves for each component revealed that with the implementation of the vibration isolation trench, the amplitude of the vibration displacement at frequencies above 20 Hz was significantly reduced. This indicated that the vibration isolation trench was more effective in isolating high-frequency vibrations, whereas its effectiveness in isolating low-frequency vibrations, particularly at approximately 10 Hz and below, was limited. Comparison with Figure 6 reveals that the incorporation of the isolation trench significantly enhanced high-frequency attenuation, with particularly pronounced effects observed within the 40–50 Hz frequency band.

5. Analysis of Elliptical Polarization Characteristics of Empty Trench Vibration Isolation

This chapter primarily investigates the synthetic seismic records of elevated railway tracks both with and without vibration isolation trenches, focusing on the dispersion characteristics of elliptical polarization ratios.

The elliptic characteristics of the surface wave propagation can be determined by analyzing the vector direction of a single measurement point. Therefore, to obtain better results while adhering to the sampling theorem, it could be essential to collect additional sample data. To explore the polarization characteristics of the far-field region, the position of the measurement point was appropriately adjusted based on the numerical model and field acquisition conditions while maintaining a suitable sampling interval length. The length of the survey line was increased to 195 m, the number of measurement points was increased to 19, and the spacing of measurement points was adjusted to 10 m. However, the position of the vibration isolation trench remained unchanged at 75 m from the pier.

5.1. Numerical Model

Abaqus is a large-scale finite element analysis software package with advanced computational capabilities and an extensive library of elements. Given the requirements of the research object in this study, Abaqus was selected as the numerical simulation software.

In this study, the Rake wavelet with a central frequency of 20 Hz was employed as the source. The mathematical expression is as follows (3):

(4)$u\left(t\right)=[1-2{\left(\pi f_0\right)}^2-{(t-t_0)}^2]\ast exp[-{\left(\pi f_0\right)}^2-{(t-t_0)}^2]$

where $t$ represents the propagation time of the Rake wavelet, $f_0$ represents the dominant frequency of the Racker wavelet, and $t_0$ denotes the time when the dominant frequency of the Racker wavelet can be observed.

In real life, the stratum structure exists in a three-dimensional space. Thus, a three-dimensional space model was established for numerical simulation. The model had a length of 132.8 m along the track, a horizontal width of 410 m perpendicular to the track, and a soil layer thickness of 50 m (Figure 9). The model used the infinite element boundaries, with the soil bottom fixed and the four sides defined as viscoelastic boundaries (ground damping springs), while the ground surface remained free. To ensure high computational efficiency, the grid was finely discretized near the track and coarser further from it. The bridge section was a simply supported beam bridge with a single 32.5 m span and a total of four spans, supported by 10 m tall piers and a 17 m deep pile foundation with a cap. The components of the pile foundation and cap were modeled using C3D8 grid elements. A vibration isolation trench (20 m long, 1 m wide, and 6 m deep) was placed 74–75 m from the pier, parallel to the train’s direction of travel. The measurement points were arranged along the vertical track starting 15 m from the pier and spaced 10 m apart, resulting in 19 points over a survey line length of 180 m, as shown in Figure 10. The model parameters are listed in Table 1.

5.2. Analysis and Processing of Synthetic Seismogram

Figure 11 illustrates the synthetic seismogram without trenching at a train speed of 240 km/h. In Figure 11a, the direct P-wave was prominent, which was consistent with the characteristics of the wave-field snapshot. Before 1000 ms, the propagation characteristics exhibited clear surface wave propagation similar to that of a single-point source. However, after 1000 ms, as the train passed, each pier acted as an approximate point source, resulting in significant wave interference and the loss of distinct surface wave characteristics. Figure 11b displays the horizontal X component along the survey line direction, where similar properties were observed. The analysis revealed that under the same amplification factor, the Y component perpendicular to the survey line exhibited the lowest energy intensity compared to the other two components, exhibiting an order of magnitude difference. This observation aligned with the shear direction propagation characteristics observed with a single-point source, highlighting the rationale for using horizontal and vertical components in two-dimensional space modeling. Despite the influence of multiple seismic sources, the energy intensity of the Y component in the direction of the train remained low. Analysis of Figure 9 indicates that in the absence of an isolation trench, surface wave energy is relatively concentrated, as evidenced by the propagation characteristics of the DP wave. This concentration is particularly apparent within the high-frequency region (depicted by the red-shaded area in Figure 6), which corresponds to the waveform following the direct P-wave arrival in Figure 11. These observations collectively validate the numerical model. Furthermore, Figure 3 and Figure 9 illustrate greater waveform instability with increasing propagation distance.

Figure 12 illustrates the synthetic seismogram at a velocity of 240 km/h. The diagram demonstrates that under the same amplification coefficient, the energy intensity of the three-direction synthetic seismogram with trenching decreased significantly compared to that without trenching. This reduction in energy intensity was observed not only in the horizontal Y component in the direction of the train, but also in the Z and X components owing to the presence of the trench. Figure 12a indicates that although the direct P-wave (DP) was still distinguishable, it was much weaker than under the trenchless condition, indicating that trenching effectively mitigated the wave energy and achieved vibration isolation. Furthermore, Figure 12a reveals that, while the energy intensity was reduced, the vertical Z component remained the strongest among the three components, and the energy intensity in the horizontal direction was much lower, which was consistent with single-point source observations, demonstrating significant scientific research value.

5.3. Elliptic Polarization

By performing the one-dimensional Fourier transform on the horizontal and vertical components of the seismic wave from the homogeneous model, the spectral ratio was calculated to obtain the corresponding elliptic polarization rate Er. The exploration depth was then determined through the time–depth conversion, which allowed for the generation of an elliptical polarization dispersion profile (the y-axis labels is depth). In the near-field region of the source, the interference from body waves and other factors caused the elliptic polarization rate Er to exhibit radial variations. As the distance from the source increased, this radial variation decreased, and Er approached a stable, constant value. In a homogeneous medium, Er remained constant, with no dispersion phenomena. The elliptic polarization rate collected from a single point proved to be more sensitive than that of the synthetic seismogram. Figure 13 presents the dispersion analysis profile of Er at a train speed of 240 km/h without a trench. Based on the characteristics of the elliptic polarization rate in Figure 13, in the near-field region, particularly within 75 m from the source, Er exhibited high extreme values due to the interference of body waves, preventing the formation of a stable surface wave region. Conversely, beyond 135 m in the far-field region, Er stabilized and a stable surface wave region was established. This analysis helped to determine the horizontal polarization characteristics.

Figure 14 illustrates the elliptic polarization frequency dispersion profile with a trench located at 75 m at a train speed of 240 km/h. Comparing with Figure 13, it is found that This figure shows that when the surface wave reached the trench, there was a sharp decrease in the elliptic polarization rate. With the same color scale, this rate approached the stable area observed in the absence of a trench, indicating that the vibration isolation trench effectively mitigated the surface wave effects, achieving a reduction in the elliptic polarization rate to a level lower than that observed in the uniform stratum without a trench. Nevertheless, the high-speed trains’ operation introduced multi-source interference, causing abnormal elliptic polarization rate values in the regions between 75 and 135 m and between 135 and 195 m at depths of −4 to −6 m. This interference resulted in an increase in the elliptic polarization rate in certain areas, which aligned with the results of synthetic seismic records. However, by comparing Figure 11, Figure 12, Figure 13 and Figure 14, it is found that the elliptic polarization rate exhibited greater sensitivity than synthetic seismic records, allowing for more immediate detection of anomalies.

5.4. Field Test of the Vibration Isolation Trench

Combining the results of the numerical simulation with the research background, the same location was selected as the research object and used in the numerical simulation. The first measurement point was set at 15 m from the pier, with additional points arranged at 10-m intervals, resulting in a total of 19 points along a 180-m survey line. The offset distance between the points was 15 m and the channel spacing was 10 m.

5.4.1. Earthquake Record

Figure 15 shows the measured seismic record as the train passed through, and the field data analysis was highly consistent with the numerical simulation. As seen in Figure 15a,b, the direct P wave was prominent, and the energy intensity decreased sharply at 75 m, indicating the presence of the trench. This decrease in the energy intensity confirmed that the trenching effectively weakened the wave energy and achieved vibration isolation. Additionally, under the same amplification factor, the Y component, which was perpendicular to the survey line, exhibited the lowest energy intensity of the three components, demonstrating that despite multiple seismic sources, the energy intensity of the Y component along the train’s running direction remained low.

5.4.2. Elliptical Polarization

Figure 16 shows the elliptical polarization frequency dispersion profile obtained from the measured data when the train passed by with the implementation of the vibration isolation trench 75 m away from the railway line. The diagram indicates that the elliptic polarization rate was relatively high in the near-field source area from 15 m to 75 m, reflecting the significant impact of train vibration on the ground. However, when the surface wave reached the trench, there was a significant reduction in the elliptical polarization rate. Under the same color scale, the rate approached a stable value beyond the trench. Despite this, the high-speed trains’ operation introduced multi-source interference, causing abnormal elliptic polarization rate values in the 75–195 m range at depths of −4 to −6 m (the y-axis labels is depth). In some areas, such as the yellow area from 135 to 195 m in Figure 16, the elliptic polarization rate increased, whereas in other deep areas, indicated by a green color, the rate decreased. These variations may be influenced by other on-site factors and may reflect the sensitivity of the elliptic polarization rate, allowing for the quick detection of anomalies.

6. Conclusions

The vibration caused by the studied elevated railway on the Loess Plateau differed significantly from that observed in other areas owing to the thick loess cover. By analyzing the time–frequency domain with and without the implementation of a vibration isolation trench and examining the characteristics of the elliptical polarization dispersion, the main conclusions were obtained, as follows:

(1). In the near-field source area, that is, the region close to the bridge pier, the time–frequency domain analysis revealed significant amplitude changes based on the presence of a vibration isolation trench, with a higher periodic peak value being reported without a vibration isolation trench. However, the amplitude tended to stabilize at a certain distance from the bridge pier.

(2). After the addition of the vibration isolation trench, the time–frequency domain analysis demonstrated a significant reduction in the amplitude of vibration in the soil beyond the trench and an obvious vibration isolation effect, particularly close to the trench. However, the vibration isolation effect was less effective for low-frequency vibrations below 10 Hz.

(3). Both the measured verification and numerical simulation analyses indicated that the energy intensities of the component in the direction of the railway line and the component perpendicular to it were significantly higher than those of the vertical component. The wave-field propagation characteristics of high-speed railway trains passing through the viaduct resembled those of a point source at the initial stage.

(4). The vibration isolation trench clearly demonstrated an effective vibration isolation capability. Although different trains passing over the viaduct at various speeds were affected by multiple factors, such as reflected and refracted waves, the analysis of the elliptic polarization characteristics revealed a significant vibration isolation effect. Both the numerical simulations and field tests confirmed the feasibility and rationality of this vibration isolation method.

Due to the challenging conditions along the railway, constructing a vibration isolation trench could be difficult, and it may be prone to collapse caused by local instability after long-term train vibrations. In practical engineering, it is common to fill empty trenches with various materials to improve their practicality and stability. Vibration isolation trenches should be filled with materials that propagate waves at low speed to ensure that the vibration isolation effect is maintained and to enhance the stability of the trench wall.

Author Contributions

Concepts, Methods, Software, Z.L. (Zhengping Liu), Investigation Writing—original draft preparation, Writing—review and editing, S.L.; Validation, Y.S. and Q.D.; Formal analysis, Z.L. (Zhe Liu); funding acquisition, S.L. All authors have read and agreed to the published version of the manuscript.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

The data presented in this study are available on request from the corresponding author.

Acknowledgments

Special thanks to the reviewers for their helpful comments and suggestions.

Conflicts of Interest

The authors declare no conflicts of interest.

Footnotes

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Figures and Table

Figure 1 Elevated section near K988 + 848 of Weinan North Railway Station.

Figure 2 Testing the characteristic length of the train.

Figure 3 Measurement point layout.

Figure 4 EPS portable digital seismometer.

Figure 5 Ground vibration–displacement time history curve caused by the operation of eight EMUs at 240 km/h in the viaduct section.

Figure 6 Vibration frequency spectra of 12 measurement points in different directions when the train speed was 240 km/h.

Figure 7 Group vibration displacement time history curve caused by the operation of eight EMUs at 240 km/h in the viaduct section with the implementation of the vibration isolation trench.

Figure 8 Ground vibration frequency spectra of 12 measurement points in different directions when the train speed was 240 km/h with the implementation of the vibration isolation trench.

Figure 9 Numerical model.

Figure 10 Schematic diagram of measurement point distribution.

Figure 11 Synthetic seismic record at 240 km/h without trenching (DP is a direct P wave). (a) Vertical Z component; (b) X component along the survey line; (c) horizontal Y component perpendicular to the survey line.

Figure 12 Synthetic seismic record during trenching at 240 km/h (DP is a direct P wave). (a) Vertical Z component; (b) X component along the survey line; (c) horizontal Y component perpendicular to the survey line.

Figure 13 Numerical simulation of the elliptical polarization dispersion profile at a train speed of 240 km/h without a vibration isolation trench.

Figure 14 Numerical simulation of the elliptical polarization dispersion profile at a train speed of 240 km/h with a vibration isolation trench.

Figure 15 Seismic records collected during high-speed train passage (DP is a direct P wave). (a) Vertical Z component; (b) X component along the survey line; (c) horizontal Y component perpendicular to the survey line.

Figure 16 Elliptical polarization frequency dispersion profile as trains pass with the implementation of a vibration isolation trench.

Table 1

Elastic parameters of materials in the model.

Depth	Soil	Density/(kg/m3)	Elastic Modulus/MPa	Poisson Ratio	Force of Cohesion/kPa	Angle of Internal Friction/°
1–10 m	Slightly dense sandy loess	1600	100	0.29	20	23
10–30 m	Medium-compressive clayey loess	1700	120	0.33	26	24
30–50 m	Low-compressive clay loess	1750	140	0.37	30	25
/	concrete	2500	30,000	0.25	/	/

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